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Existence results of Brezis-Browder type for systems of Fredholm integral equations

Ravi P Agarwal12*, Donal O'Regan3 and Patricia JY Wong4

Author Affiliations

1 Department of Mathematical Sciences, Florida Institute of Technology Melbourne, FL 32901-6975, USA

2 Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

3 Department of Mathematics, National University of Ireland, Galway, Ireland

4 School of Electrical and Electronic Engineering, Nanyang Technological University 50 Nanyang Avenue, Singapore 639798, Singapore

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Advances in Difference Equations 2011, 2011:43  doi:10.1186/1687-1847-2011-43


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2011/1/43


Received:18 March 2011
Accepted:11 October 2011
Published:11 October 2011

© 2011 Agarwal et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, we consider the following systems of Fredholm integral equations:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M1">View MathML</a>

Using an argument originating from Brezis and Browder [Bull. Am. Math. Soc. 81, 73-78 (1975)] and a fixed point theorem, we establish the existence of solutions of the first system in (C[0, T])n, whereas for the second system, the existence criteria are developed separately in (Cl[0,∞))n as well as in (BC[0,∞))n. For both systems, we further seek the existence of constant-sign solutions, which include positive solutions (the usual consideration) as a special case. Several examples are also included to illustrate the results obtained.

2010 Mathematics Subject Classification: 45B05; 45G15; 45M20.

Keywords:
system of Fredholm integral equations; Brezis-Browder arguments; constant-sign solutions

1 Introduction

In this article, we shall consider the system of Fredholm integral equations:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M2">View MathML</a>

(1.1)

where 0 < T <∞, and also the following system on the half-line

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M3">View MathML</a>

(1.2)

Throughout, let u = (u1, u2,..., un). We are interested in establishing the existence of solutions u of the system (1.1) in (C[0, T])n = C[0, T] × C[0, T] × ℙ × C[0, T] (n times), whereas for the system (1.2), we shall seek a solution in (Cl[0, ∞))n as well as in (BC[0, ∞))n. Here, BC[0, ∞) denotes the space of functions that are bounded and continuous on [0, ∞) and Cl[0, ∞) = {x BC[0, ∞) : limt→∞ x(t) exists}.

We shall also tackle the existence of constant-sign solutions of (1.1) and (1.2). A solution u of (1.1) (or (1.2)) is said to be of constant sign if for each 1 ≤ i n, we have θiui(t) ≥ 0 for all t ∈ [0, T] (or t ∈ [0,∞)), where θi ∈ {-1, 1} is fixed. Note that when θi = 1 for all 1 ≤ i n, a constant-sign solution reduces to a positive solution, which is the usual consideration in the literature.

In the literature, there is a vast amount of research on the existence of positive solutions of the nonlinear Fredholm integral equations:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M4">View MathML</a>

(1.3)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M5">View MathML</a>

(1.4)

Particular cases of (1.3) are also considered in [1-3]. The reader is referred to the monographs [[4,5], and the references cited therein] for the related literature. Recently, a generalization of (1.3) and (1.4) to systems similar to (1.1) and (1.2) have been made, and the existence of single and multiple constant-sign solutions has been established for these systems in [6-10].

The technique used in these articles has relied heavily on various fixed point results such as Krasnosel'skii's fixed point theorem in a cone, Leray-Schauder alternative, Leggett-Williams' fixed point theorem, five-functional fixed point theorem, Schauder fixed point theorem, and Schauder-Tychonoff fixed point theorem. In the current study, we will make use of an argument that originates from Brezis and Browder [11]; therefore, the technique is different from those of [6-10] and the results subsequently obtained are also different. The present article also extends, improves, and complements the studies of [5,12-23]. Indeed, we have generalized the problems to (i) systems; (ii) more general form of nonlinearities fi, 1 ≤ i n,; and (iii) existence of constant-sign solutions.

The outline of the article is as follows. In Section 2, we shall state the necessary fixed point theorem and compactness criterion, which are used later. In Section 3, we tackle the existence of solutions of system (1.1) in (C[0, T])n, while Sections 4 and 5 deal with the existence of solutions of system (1.2) in (Cl[0, ∞))n and (BC[0, ∞))n, respectively. In Section 6, we seek the existence of constant-sign solutions of (1.1) and (1.2) in (C[0, T])n, (Cl[0, ∞))n and (BC[0, ∞))n. Finally, several examples are presented in Section 7 to illustrate the results obtained.

2 Preliminaries

In this section, we shall state the theorems that are used later to develop the existence criteria--Theorem 2.1 [24] is Schauder's nonlinear alternative for continuous and compact maps, whereas Theorem 2.2 is the criterion of compactness on Cl[0, ∞) [[16], p. 62].

Theorem 2.1 [24]Let B be a Banach space with E B closed and convex. Assume U is a relatively open subset of E with 0 ∈ U and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M6">View MathML</a>is a continuous and compact map. Then either

(a) S has a fixed point in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M7">View MathML</a>, or

(b) there exist u ∈ ∂U and λ ∈ (0, 1) such that u = λSu.

Theorem 2.2 [[16], p. 62] Let P Cl[0, ∞). Then P is compact in Cl[0, ∞) if the following hold:

(a) P is bounded in Cl[0, ∞).

(b) Any y P is equicontinuous on any compact interval of [0, ∞).

(c) P is equiconvergent, i.e., given ε > 0, there exists T(ε) > 0 such that |y(t) - y(∞)| < ε for any t T(ε) and y P.

3 Existence results for (1.1) in (C[0, T])n

Let the Banach space B = (C[0, T])n be equipped with the norm:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M8">View MathML</a>

where we let |ui|0 = supt∈[0,T] |ui(t)|, 1 ≤ i n. Throughout, for u B and t ∈ [0, T], we shall denote

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M9">View MathML</a>

Moreover, for each 1 ≤ i n, let 1 ≤ pi be an integer and qi be such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M10">View MathML</a>. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M11">View MathML</a>, we shall define

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M12">View MathML</a>

Our first existence result uses Theorem 2.1.

Theorem 3.1 For each 1 ≤ i n, assume (C1)- (C4) hold where

(C1) hi C[0, T], denote Hi ≡ supt∈ [0, T] |hi(t)|,

(C2) fi : [0, T] × ℝn → ℝ is a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13">View MathML</a>-Carathéodory function:

(i) the map u α fi(t, u) is continuous for almost all t ∈ [0, T],;

(ii) the map t α fi(t, u) is measurable for all u ∈ ℝn;

(iii) for any r > 0, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14">View MathML</a>such that |u| ≤ r implies |fi(t, u)| ≤ μr,i(t) for almost all t ∈ [0, T];

(C3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M15">View MathML</a>for each t ∈ [0, T];

(C4) the map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M16">View MathML</a>is continuous from [0, T] to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M17">View MathML</a>.

In addition, suppose there is a constant M > 0, independent of λ, with ||u|| ≠ M for any solution u ∈ (C[0, T])n to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M18">View MathML</a>

for each λ ∈ (0, 1). Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let the operator S be defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M19">View MathML</a>

(3.2)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M20">View MathML</a>

(3.3)

Clearly, the system (1.1) is equivalent to u = Su, and (3.1)λ is the same as u = λSu.

Note that S maps (C[0, T])n into (C[0, T])n, i.e., Si : (C[0, T])n C[0, T], 1 ≤ i n. To see this, note that for any u ∈ (C[0, T])n, there exits r > 0 such that ||u|| < r. Since fi is a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13">View MathML</a>-Carathéodory function, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14">View MathML</a> such that |fi(s, u)| ≤ μr,i(s) for almost all s ∈ [0, T]. Hence, for any t1, t2 ∈ [0, T], we find for 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M21">View MathML</a>

(3.4)

as t1 t2, where we have used (C1) and (C3). This shows that S : (C[0, T])n → (C[0, T])n.

Next, we shall prove that S : (C[0, T])n → (C[0, T])n is continuous. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M22">View MathML</a> in (C[0, T])n, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M23">View MathML</a> in C[0, T], 1 ≤ i n. We need to show that Sum Su in (C[0, T])n, or equivalently Sium Siu in C[0, T], 1 ≤ i n. There exists r > 0 such that ||um||, ||u|| < r. Since fi is a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13">View MathML</a>-Carathéodory function, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14">View MathML</a> such that |fi(s, um)|, |fi(s, u)| ≤ μr,i(s) for almost all s ∈ [0, T]. Using a similar argument as in (3.4), we get for any t1, t2 ∈ [0, T] and 1 ≤ i n:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M24">View MathML</a>

(3.5)

as t1 t2. Furthermore, Sium(t) → Siu(t) pointwise on [0, T], since, by the Lebesgue-dominated convergence theorem,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M25">View MathML</a>

(3.6)

as m → ∞. Combining (3.5) and (3.6) and using the fact that [0, T] is compact, gives for all t ∈ [0, T],

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M26">View MathML</a>

(3.7)

as m → ∞. Hence, we have proved that S : (C[0, T])n → (C[0, T])n is continuous.

Finally, we shall show that S : (C[0, T])n → (C[0, T])n is completely continuous. Let Ω be a bounded set in (C[0, T])n with ||u|| ≤ r for all u ∈ Ω. We need to show that SiΩ is relatively compact for 1 ≤ i n. Clearly, SiΩ is uniformly bounded, since there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14">View MathML</a> such that |fi(s, u)| ≤ μr,i(s) for all u ∈ Ω and a.e. s ∈ [0, T], and hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M27">View MathML</a>

(3.8)

Further, using a similar argument as in (3.4), we see that SiΩ is equicontinuous. It follows from the Arzéla-Ascoli theorem [[5], Theorem 1.2.4] that SiΩ is relatively compact.

We now apply Theorem 2.1 with U = {u ∈ (C[0, T])n : ||u|| < M} and B = E = (C[0, T])n to obtain the conclusion of the theorem. □

Our subsequent results will apply Theorem 3.1. To do so, we shall show that any solution u of (3.1)λ is bounded above. This is achieved by bounding the integral of |fi(t,u(t))| (or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M28">View MathML</a>) on two complementary subsets of [0, T], namely {t ∈ [0, T] : ||u(t)|| ≤ r} and {t ∈ [0, T] : ||u(t)|| > r}, where ρi and r are some constants--this technique originates from the study of Brezis and Browder [11]. In the next four theorems (Theorems 3.2-3.5), we shall apply Theorem 3.1 to the case pi = ∞ and qi = 1, 1 ≤ i n.

Theorem 3.2. Let the following conditions be satisfied for each 1 ≤ i n : (C1)-(C4) with pi = and qi = 1, (C5) and (C6) where

(C5) there exist Bi > 0 such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M29">View MathML</a>

(C6) there exist r > 0 and αi > 0 with i > Hi such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M30">View MathML</a>

Then, (1.1) has at least one solution in (C[0, T])n.

Proof We shall employ Theorem 3.1, and so let u = (u1, u2, l...., un) ∈ (C[0, T])n be any solution of (3.1)λ where λ ∈ (0, 1).

Define

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M31">View MathML</a>

(3.9)

Clearly, [0, T] = I J, and hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M32">View MathML</a>.

Let 1 ≤ i n. If t I, then by (C2), there exists μr,i L1[0, T] such that |fi(t, u(t))| ≤ μr,i(t). Thus, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M33">View MathML</a>

(3.10)

On the other hand, if t J, then it is clear from (C6) that ui(t)fi(t, u(t)) ≥ 0 for a.e. t ∈ [0, T]. It follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M34">View MathML</a>

(3.11)

We now multiply (3.1)λ by fi(t, u(t)), then integrate from 0 to T to get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M35">View MathML</a>

(3.12)

Using (C5) in (3.12) yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M36">View MathML</a>

(3.13)

Splitting the integrals in (3.13) and applying (3.11), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M37">View MathML</a>

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M38">View MathML</a>

where we have used (3.10) in the last inequality. It follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M39">View MathML</a>

(3.14)

Finally, it is clear from (3.1)λ that for t ∈ [0, T] and 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M40">View MathML</a>

(3.15)

where we have applied (3.10) and (3.14) in the last inequality. Thus, |ui|0 li for 1 ≤ i n and ||u|| ≤ max1≤in li L. It follows from Theorem 3.1 (with M = L + 1) that (1.1) has a solution u* ∈ (C[0, T])n. □

Theorem 3.3 Let the following conditions be satisfied for each 1 ≤ i n : (C1)-(C4) with pi = ∞ and qi = 1, (C7) and (C8) where

(C7) there exist constants ai ≥ 0 and bi such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M41">View MathML</a>

(C8) there exist r > 0 and αi > 0 with rαi > Hi + ai such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M42">View MathML</a>

Then, (1.1) has at least one solution in (C[0, T])n.

Proof The proof follows that of Theorem 3.2 until (3.12). Let 1 ≤ i n. We use (C7) in (3.12) to get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M43">View MathML</a>

(3.16)

Splitting the integrals in (3.16) and applying (3.11) gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M44">View MathML</a>

where we have also used (3.10) in the last inequality. It follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M45">View MathML</a>

(3.17)

The rest of the proof follows that of Theorem 3.2. □

Theorem 3.4 Let the following conditions be satisfied for each 1 ≤ i n : (C1)-(C4) with pi = and qi = 1, (C9) and (C10) where

(C9) there exist constants ai ≥ 0, 0 < τi ≤ 1 and bi such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M46">View MathML</a>

(C10) there exist r > 0 and βi > 0 such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M47">View MathML</a>

Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let u = (u1, u2,..., un) ∈ (C[0, T])n be any solution of (3.1)λ where λ ∈ (0, 1). Define

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M48">View MathML</a>

(3.18)

Clearly, [0, T] = I0 J0 and hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M49">View MathML</a>.

Let 1 ≤ i n. If t I0, then by (C2) there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M50">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M51">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M52">View MathML</a>

(3.19)

Further, if t J0, then by (C10) we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M53">View MathML</a>

(3.20)

Now, using (3.20) and (C9) in (3.12) gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M54">View MathML</a>

(3.21)

where in the last inequality, we have made use of the inequality:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M55">View MathML</a>

Now, noting (3.19) we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M56">View MathML</a>

(3.22)

Substituting (3.22) in (3.21) then yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M57">View MathML</a>

Since τi ≤ 1, there exists a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M58">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M59">View MathML</a>

which leads to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M60">View MathML</a>

(3.23)

Finally, it is clear from (3.1)λ that for t ∈ [0, T] and 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M61">View MathML</a>

(3.24)

where we have applied (3.19) and (3.23) in the last inequality. The conclusion now follows from Theorem 3.1. □

Theorem 3.5 Let the following conditions be satisfied for each 1 ≤ i n : (C1), (C2)-(C4) with pi = ∞ and qi = 1, (C10), (C11) and (C12) where

(C11) there exist r > 0, ηi > 0, γi > 0 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M62">View MathML</a>such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M63">View MathML</a>

(C12) there exist ai ≥ 0, 0 < τi < γi + 1, bi, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M64">View MathML</a>with ψi ≥ 0 almost everywhere on [0, T], such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M65">View MathML</a>

Also, ϕi C[0, T], <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M66">View MathML</a>, ψi C[0, T] and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M67">View MathML</a>.

Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let u = (u1, u2,..., un) ∈ (C[0, T])n be any solution of (3.1)λ where λ ∈ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ i n. Applying (C10) and (C11), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M68">View MathML</a>

(3.25)

Using (3.25) and (C12) in (3.12), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M69">View MathML</a>

(3.26)

Now, in view of (3.10) and (C12), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M70">View MathML</a>

(3.27)

Substituting (3.27) into (3.26) and using Hölder's inequality, we find

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M71">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73">View MathML</a>, there exists a constant ki such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M74">View MathML</a>

(3.28)

Finally, it is clear from (3.1)λ that for t ∈ [0, T] and 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M75">View MathML</a>

(3.29)

where we have used (3.28) and (C12) in the last inequality, and li is some constant. The conclusion is now immediate by Theorem 3.1. □

In the next six results (Theorem 3.6-3.11), we shall apply Theorem 3.1 for general pi and qi.

Theorem 3.6 Let the following conditions be satisfied for each 1 ≤ i n : (C1)-(C4), (C5), (C10) and (C13) where

(C13) there exist r > 0, ηi > 0, γi > 0 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M76">View MathML</a>such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M77">View MathML</a>

Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let u = (u1, u2,..., un) ∈ (C[0, T])n be any solution of (3.1)λ where λ ∈ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ i n. If t I, then by (C2), there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M14">View MathML</a> such that |fi(t, u(t))| ≤ μr,i(t). Consequently, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M78">View MathML</a>

(3.30)

On the other hand, using (C10) and (C13), we derive at (3.25).

Next, applying (C5) in (3.12) leads to (3.13). Splitting the integrals in (3.13) and using (3.25), we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M79">View MathML</a>

(3.31)

where (3.30) has been used in the last inequality and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M80">View MathML</a>.

Now, an application of Hölder's inequality gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M81">View MathML</a>

(3.32)

Another application of Hölder's inequality yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M82">View MathML</a>

(3.33)

Substituting (3.33) into (3.32) then leads to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M83">View MathML</a>

(3.34)

Further, using Hölder's inequality again, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M84">View MathML</a>

(3.35)

Substituting (3.34) and (3.35) into (3.31), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M85">View MathML</a>

(3.36)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M86">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M87">View MathML</a>, from (3.36), there exists a constant ki such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M88">View MathML</a>

(3.37)

Finally, it is clear from (3.1)λ that for t ∈ [0, T] and 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M89">View MathML</a>

(3.38)

where in the second last inequality a similar argument as in (3.34) is used, and in the last inequality we have used (3.37). An application of Theorem 3.1 completes the proof. □

Theorem 3.7 Let the following conditions be satisfied for each 1 ≤ i n : (C1)-(C4), (C7), (C10) and (C13). Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let u = (u1, u2,..., un) ∈ (C[0, T])n be any solution of (3.1)λ where λ ∈ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ i n. As in the proof of Theorems 3.3 and 3.6, respectively, (C7) leads to (3.16), whereas (C10) and (C13) yield (3.25).

Splitting the integrals in (3.16) and applying (3.25), we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M90">View MathML</a>

(3.39)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M91">View MathML</a>. Substituting (3.34) and (3.35) into (3.39) then leads to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M92">View MathML</a>

(3.40)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M93">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72">View MathML</a>, from (3.40), we can obtain (3.37) where ki is some constant. The rest of the proof proceeds as that of Theorem 3.6. □

Theorem 3.8 Let the following conditions be satisfied for each 1 ≤ i n : (C1)-(C4), (C10), (C13), and (C14) where

(C14) there exist constants ai ≥ 0, 0 < τi < γi + 1 and bi such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M94">View MathML</a>

Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let u = (u1, u2,..., un) ∈ (C[0, T])n be any solution of (3.1)λ where λ ∈ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ i n. From the proof of Theorem 3.6, we see that (C10) and (C13) lead to (3.25).

Using (3.25) and (C14) in (3.12), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M95">View MathML</a>

(3.41)

Note that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M96">View MathML</a>

(3.42)

where we have used (3.30) in the last inequality. Substituting (3.42) into (3.41) and using (3.34) and (3.35) then provides

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M97">View MathML</a>

(3.43)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73">View MathML</a>, there exists a constant ki such that (3.37) holds. The rest of the proof is similar to that of Theorem 3.6. □

Theorem 3.9 Let the following conditions be satisfied for each 1 ≤ i n : (C1)-(C4), (C10), (C13), and (C15) where

(C15) there exist constants di ≥ 0, 0 < τi < γi + 1 and ei such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M98">View MathML</a>

Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let u = (u1, u2,..., un) ∈ (C[0, T])n be any solution of (3.1)λ where λ ∈ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ i n. As before, we see that (C10) and (C13) lead to (3.25).

Using (3.25) and (C15) in (3.12), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M99">View MathML</a>

(3.44)

Now, it is clear that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M100">View MathML</a>

(3.45)

Moreover, an application of Hölder's inequality gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M101">View MathML</a>

(3.46)

Substituting (3.45) into (3.44) and using (3.34), (3.35) and (3.46) then leads to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M102">View MathML</a>

(3.47)

Noting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73">View MathML</a>, there exists a constant ki such that (3.37) holds. The rest of the proof follows that of Theorem 3.6. □

Theorem 3.10 Let the following conditions be satisfied for each 1 ≤ i n : (C1)-(C4), (C10), (C13) and (C16) where

(C16) there exist constants ci ≥ 0, di ≥ 0, 0 < τi < γi + 1 and ei with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M103">View MathML</a>such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M104">View MathML</a>

Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let u = (u1, u2,..., un) ∈ (C[0, T])n be any solution of (3.1)λ where λ ∈ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ i n. As before, we see that (C10) and (C13) lead to (3.25).

Using (3.25) and (C16) in (3.12) gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M105">View MathML</a>

(3.48)

Now, it is clear that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M106">View MathML</a>

(3.49)

Substituting (3.49) into (3.48) and then using (3.34), (3.35) and (3.46) leads to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M107">View MathML</a>

(3.50)

Noting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73">View MathML</a> as well as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M108">View MathML</a>, from (3.50) there exists a constant ki such that (3.37) holds. The rest of the proof proceeds as that of Theorem 3.6. □

Theorem 3.11 Let the following conditions be satisfied for each 1 ≤ i n : (C1)-(C4), (C10), (C13) and (C17) where

(C17) there exist ai ≥ 0, 0 i < γi + 1, bi, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M109">View MathML</a>with ψi ≥ 0 almost everywhere on [0, T], such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M110">View MathML</a>

Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let u = (u1, u2,..., un) ∈ (C[0, T])n be any solution of (3.1)λ where λ ∈ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ i n. Once again, conditions (C10) and (C13) give rise to (3.25).

Similar to the proof of Theorem 3.5, we apply (3.25) and (C17) in (3.12) to get (3.26). Next, using (3.30) and Hölder's inequality, we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M111">View MathML</a>

(3.51)

Substituting (3.51) into (3.26) and applying (3.34) and (3.35), we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M112">View MathML</a>

(3.52)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M72">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M73">View MathML</a>, from (3.52), there exists a constant ki such that (3.37) holds. The rest of the proof proceeds as that of Theorem 3.6. □

Remark 3.1 In Theorem 3.5, the conditions (C10) and (C11) can be replaced by the following, which is evident from the proof.

(C10)' There exist r > 0 and βi > 0 such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M113">View MathML</a>

where we denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M114">View MathML</a>.

(C11)' There exist r > 0, ηi > 0, γi > 0 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M62">View MathML</a> such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M115">View MathML</a>

Remark 3.2 In Theorems 3.6-3.11, the conditions (C10) and (C13) can be replaced by (C10)' and (C13)' below, and the proof will be similar.

(C13)' There exist r > 0, ηi > 0, γi > 0, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M76">View MathML</a> such that for any u ∈ (C[0, T])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M116">View MathML</a>

4 Existence results for (1.2) in (Cl[0, ∞))n

Let the Banach space B = (Cl[0, ∞))n be equipped with the norm:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M117">View MathML</a>

where we let |ui|0 = supt∈[0,∞) |ui(t)|, 1 ≤ i n. Throughout, for u B and t ∈ [0, ∞), we shall denote that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M118">View MathML</a>

Moreover, for each 1 ≤ i n, let 1 ≤ pi ≤ ∞ be an integer and qi be such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M10">View MathML</a>. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M119">View MathML</a>, we shall define that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M120">View MathML</a>

We shall apply Theorem 2.1 to obtain the first existence result for (1.2) in (Cl[0, ∞))n.

Theorem 4.1 For each 1 ≤ i ≤ n, assume (D1)-(D5) hold where

(D1) hi Cl[0, ∞), denote Hi ≡ supt∈[0,∞) |hi(t)|,

(D2) fi : [0, ∞) × ℝn → ℝ is a L1-Carathéodory function, i.e.,

(i) the map u α fi(t, u) is continuous for almost all t ∈ [0, ∞),

(ii) the map t α fi(t, u) is measurable for all u ∈ ℝn,

(iii) for any r > 0, there exists μr,i L1[0, ∞) such that |u| r implies |fi(t, u)| ≤ μr,i(t) for almost all t ∈ [0, ∞).

(D3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M121">View MathML</a>for each t ∈ [0, ∞),

(D4) the map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M16">View MathML</a>is continuous from [0, ∞) to L[0, ∞),

(D5) there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M122">View MathML</a>such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M123">View MathML</a>in L[0, ∞) as t → ∞, i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M124">View MathML</a>

In addition, suppose there is a constant M > 0, independent of λ, with ||u|| ≠ M for any solution u ∈ (Cl[0, ∞))n to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M125">View MathML</a>

for each λ ∈ (0, 1). Then, (1.2) has at least one solution in (Cl[0, ∞))n.

Proof To begin, let the operator S be defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M126">View MathML</a>

(4.2)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M127">View MathML</a>

(4.3)

Clearly, the system (1.2) is equivalent to u = Su, and (4.1)λ is the same as u = λSu.

First, we shall show that S : (Cl[0, ∞))n → (Cl[0, ∞))n, or equivalently Si : (Cl[0, ∞))n Cl[0, ∞), 1 ≤ i n. Let u ∈ (Cl[0, ∞))n. Then, there exists r > 0 such that ||u|| ≤ r, and from (D2) there exists μr,i L1[0, ∞) such that |fi(s, u)| ≤ μr,i (s) for almost all s ∈ [0, ∞). Let t1, t2 ∈ [0, ∞). Together with (D1) and (D4), we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M128">View MathML</a>

(4.4)

as t1 t2. Hence, Siu C[0, ∞).

To see that Siu is bounded, we have for t ∈ [0, ∞),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M129">View MathML</a>

(4.5)

By (D5), there exists T1 > 0 such that for t > T1,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M130">View MathML</a>

On the other hand, for t ∈ [0, T1], we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M131">View MathML</a>

Hence,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M132">View MathML</a>

(4.6)

It follows from (4.5) that for t ∈ [0, ∞),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M133">View MathML</a>

(4.7)

Hence, Siu is bounded.

It remains to check the existence of the limit limt→∞ Siu(t). We claim that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M134">View MathML</a>

(4.8)

where hi(∞) ≡ limt→∞ hi(t). In fact, it follows from (D5) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M135">View MathML</a>

as t → ∞. This implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M136">View MathML</a>

and so (4.8) is proved. We have hence shown that S : (Cl[0, ∞))n → (Cl[0, ∞))n.

Next, we shall prove that S : (Cl[0, ∞))n → (Cl[0, ∞))n is continuous. Let {um} be a sequence in (Cl[0, ∞))n and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M22">View MathML</a>. In (Cl[0, ∞))n, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M23">View MathML</a>, in Cl[0, ∞), 1 ≤ i n. We need to show that Sum Su in (Cl[0, ∞))n, or equivalently Sium Siu in Cl[0, ∞), 1 ≤ i n. There exists r > 0 such that ||um||, ||u|| < r, Noting (D2), there exists μr,i L1[0, ∞) such that |fi(s, um)|, |fi(s, u)| ≤ μr,i(s) for almost all s ∈ [0, ∞). Denote Siu(∞) ≡ limt→∞ Siu(t) and Sium(∞) ≡ limt→∞ Sium(t). In view of (4.8), we get that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M137">View MathML</a>

(4.9)

Since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M138">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M139">View MathML</a>

by the Lebesgue-dominated convergence theorem, it is clear from (4.9) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M140">View MathML</a>

(4.10)

Further, using (4.8) again we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M141">View MathML</a>

(4.11)

as t → ∞. Similarly, we also have that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M142">View MathML</a>

(4.12)

Combining (4.10)-(4.12), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M143">View MathML</a>

or equivalently, there exist <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M144">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M145">View MathML</a>

(4.13)

It remains to check the convergence in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M146">View MathML</a>. As in (4.4), we find for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M147">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M148">View MathML</a>

(4.14)

as t1 t2. Furthermore, Sium(t) → Siu(t) pointwise on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M146">View MathML</a>, since, by the Lebesgue-dominated convergence theorem,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M149">View MathML</a>

(4.15)

as m → ∞. Combining (4.14) and (4.15) and the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M146">View MathML</a> is compact yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M150">View MathML</a>

(4.16)

Coupling (4.13) and (4.16), we see that Sium Siu in Cl[0, ∞).

Finally, we shall show that S : (Cl[0, ∞))n → (Cl[0, ∞))n is completely continuous. Let Ω be a bounded set in (Cl[0, ∞))n with ||u|| ≤ r for all u ∈ Ω We need to show that SiΩ is relatively compact for 1 ≤ i n. First, we see that SiΩ is bounded; in fact, this follows from an earlier argument in (4.7). Next, using a similar argument as in (4.4), we see that SiΩ is equicontinuous. Moreover, SiΩ is equiconvergent follows as in (4.11). By Theorem 2.2, we conclude that SiΩ is relatively compact. Hence, S : (Cl[0, ∞))n → (Cl[0, ∞))n is completely continuous.

We now apply Theorem 2.1 with U = {u ∈ (Cl[0, ∞))n : ||u|| < M} and B = E = (Cl[0, ∞))n to obtain the conclusion of the theorem. □

Remark 4.1 In Theorem 4.1, the conditions (D2)-(D5) can be stated in terms of general pi and qi as follows, and the proof will be similar:

(D2)' fi : [0, ∞) × ℝn ℝ is a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13">View MathML</a>-Carathéodory function, i.e.,

(i) the map u α fi(t, u) is continuous for almost all t ∈ [0, ∞),

(ii) the map t α fi(t, u) is measurable for all u ∈ ℝn,

(iii) for any r > 0, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M151">View MathML</a> such that |u| r implies |fi(t, u)| ≤ μr,i(t) for almost all t ∈ [0, ∞),

(D3)' <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M152">View MathML</a>, for each t ∈ [0, ∞),

(D4)' the map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M16">View MathML</a> is continuous from [0, ∞) to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M153">View MathML</a>,

(D5)' there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M154">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M123">View MathML</a>, in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M153">View MathML</a> as t → ∞, i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M155">View MathML</a>

Our subsequent Theorems 4.2-4.5 use an argument originating from Brezis and Browder [11]. These results are parallel to Theorems 3.2-3.5 for system (1.1).

Theorem 4.2 Let the following conditions be satisfied for each 1 ≤ i n : (D1)-(D5), (C5), and (C6)where

(C5)there exist Bi > 0 such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M156">View MathML</a>

(C6)there exist r > 0 and αi > 0 with rαi > Hi such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M157">View MathML</a>

Then, (1.2) has at least one solution in (Cl[0, ∞))n.

Proof We shall employ Theorem 4.1, so let u = (u1, u2,..., un) ∈ (Cl[0, ∞))n be any solution of (4.1)λ where λ ∈ (0, 1). The rest of the proof is similar to that of Theorem 3.2 with the obvious modification that [0, T] be replaced by [0, ∞). Also, noting (4.6) we see that the analog of (3.15) holds. □

In view of the proof of Theorem 4.2, we see that the proof of subsequent Theorems 4.3-4.5 will also be similar to that of Theorems 3.3-3.5 with the appropriate modification. As such, we shall present the results and omit the proof.

Theorem 4.3 Let the following conditions be satisfied for each 1 ≤ i n : (D1)-(D5), (C7)and (C8)where

(C7)there exist constants ai ≥ 0 and bi such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M158">View MathML</a>

(C8)there exist r > 0 and αi > 0 with rαi > Hi + ai such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M159">View MathML</a>

Then, (1.2) has at least one solution in (Cl[0, ∞))n.

Theorem 4.4 Let the following conditions be satisfied for each 1 ≤ i n : (D1)-(D5), (C9)and (C10)where

(C9)there exist constants ai ≥ 0, 0 < τi 1 and bi such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M160">View MathML</a>

(C10)there exist r > 0 and βi > 0 such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M161">View MathML</a>

Then, (1.2) has at least one solution in (Cl[0, ∞))n.

Theorem 4.5 Let the following conditions be satisfied for each 1 ≤ i n : (D1)-(D5), (C10), (C11)and (C12)where

(C11)there exist r > 0, ηi > 0, γi > 0 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M162">View MathML</a>such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M163">View MathML</a>

(C12)there exist ai ≥ 0, 0 < τi < γi + 1, bi, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M164">View MathML</a>with ψi ≥ 0 almost everywhere on [0, ∞), such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M165">View MathML</a>

Also, ϕi BC[0, ∞), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M166">View MathML</a>, ψi BC[0, ∞) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M167">View MathML</a>.

Then, (1.2) has at least one solution in (Cl[0, ∞))n.

We also have a remark similar to Remark 3.1.

Remark 4.2 In Theorem 4.5 the conditions (C10)and (C11)can be replaced by the following; this is evident from the proof.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M168">View MathML</a> There exist r > 0 and βi > 0 such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M169">View MathML</a>

where we denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M170">View MathML</a> .

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M171">View MathML</a> There exist r > 0, ηi > 0, γi > 0 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M162">View MathML</a> such that for any u ∈ (Cl[0, ∞))n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M172">View MathML</a>

5 Existence results for (1.2) in (BC[0, ∞))n

Let the Banach space B = (BC[0, ∞))n be equipped with the norm:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M173">View MathML</a>

where we let |ui|0 = supt∈[0,∞) |ui(t)|, 1 < i < n. Throughout, for u B and t ∈ [0, ∞) we shall denote

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M174">View MathML</a>

Moreover, for each 1 ≤ i n, let 1 ≤ pi ≤ ∞ be an integer and qi be such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M10">View MathML</a>. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M119">View MathML</a>, we shall define <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M175">View MathML</a> as in Section 4.

Our first result is a variation of an existence principle of Lee and O'Regan [25].

Theorem 5.1 For each 1 ≤ i n, assume (D2)'-(D4)' and (D6) hold where

(D6) hi BC[0, ∞), denote Hi ≡ supt∈[0, ∞) |hi(t)|.

For each k = 1, 2,..., suppose there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M176">View MathML</a> that satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M177">View MathML</a>

(5.1)

Further, for 1 ≤ i n and k = 1, 2,..., there is a bounded set B ⊆ ℝ such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M178">View MathML</a> for each t ∈ [0, k]. Then, (1.2) has a solution u* ∈ (BC[0, ∞))n such that for 1 ≤ i n, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M179">View MathML</a> for all t ∈ [0, ∞).

Proof First we shall show that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M180">View MathML</a>

(5.2)

The uniform boundedness of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M181">View MathML</a> follows immediately from the hypotheses; therefore, we only need to prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M181">View MathML</a> is equicontinuous. Let 1 ≤ i n. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M178">View MathML</a> for each t ∈ [0, k], there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M182">View MathML</a> such that |fi(s,uk(s))| ≤ μB(s) for almost every s ∈ [0, k].Fix t, t' ∈ [0, λ]. Then, from (5.1) we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M183">View MathML</a>

as t t'. Therefore, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M181">View MathML</a> is equicontinuous on [0, λ].

Let 1 ≤ i n. Now, (5.2) and the Arzéla-Ascoli theorem yield a subsequence N1 of ℕ = {1, 2,...} and a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M184">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M185">View MathML</a> uniformly on [0,1] as k → ∞ in N1. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M186">View MathML</a> . Then, (5.2) and the Arzéla-Ascoli theorem yield a subsequence N2 of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M187">View MathML</a> and a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M188">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M189">View MathML</a> uniformly on [0,2] as k → ∞ in N2. Note that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M190">View MathML</a> on [0,1] since N2 N1. Continuing this process, we obtain subsequences of integers N1, N2,... with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M191">View MathML</a>

(5.3)

and functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M192">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M193">View MathML</a>

(5.4)

Let 1 ≤ i n. Define a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M194">View MathML</a> by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M195">View MathML</a>

(5.5)

Clearly, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M196">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M179">View MathML</a> for each t ∈ [0, λ]. It remains to prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M197">View MathML</a> solves (1.2). Fix t ∈ [0, ∞). Then, choose and fix λ such that t ∈ [0, λ]. Take k λ. Now, from (5.1) we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M198">View MathML</a>

or equivalently

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M199">View MathML</a>

(5.6)

Since fi is a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M13">View MathML</a>-Carathéodory function and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M178">View MathML</a> for each t ∈ [0, k], there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M182">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M200">View MathML</a>

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M201">View MathML</a> . Let k → ∞ (k N) in (5.6). Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M202">View MathML</a> uniformly on [0, ], an application of Lebesgue-dominated convergence theorem gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M203">View MathML</a>

or equivalently (noting (5.5))

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M204">View MathML</a>

(5.7)

Finally, letting → ∞ in (5.7) and use the fact <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M201">View MathML</a> to get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M205">View MathML</a>

Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M197">View MathML</a> is a solution of (1.2). □

It is noted that one of the conditions in Theorem 5.1, namely, (5.1) has a solution in (C[0, k])n, which has already been discussed in Section 3. As such, our subsequent Theorems 5.2-5.5 will make use of Theorem 5.1 and the technique used in Section 3. These results are parallel to Theorems 3.2-3.5 and 4.2-4.5.

Theorem 5.2 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i n. Moreover, suppose the following conditions hold for each 1 ≤ i n and each w ∈ {1, 2,...}:

(C5)w there exist Bi > 0 such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M206">View MathML</a>

(C6)w there exist r > 0 and αi > 0 with rαi > Hi (Hi as in (D6)) such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M207">View MathML</a>

Then, (1.2) has at least one solution in (BC[0, ∞))n.

Proof We shall apply Theorem 5.1. To do so, for w = 1, 2,..., we shall show that the system

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M208">View MathML</a>

(5.8)

has a solution in (C[0, w])n. Obviously, (5.8) is just (1.1) with T = w. Let w ∈ {1, 2,...} be fixed.

Let u = (u1, u2,..., un) ∈ (C[0,w])n be any solution of (3.1)λ (with T = w) where λ ∈ (0, 1). We shall model after the proof of Theorem 3.2 with T = w and Hi given in (D6). As in (3.9), define

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M209">View MathML</a>

Let 1 ≤ i n. If t I, then by (D2) there exists μr,i L1[0, ∞) such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M210">View MathML</a>

[which is the analog of (3.10)]. Proceeding as in the proof of Theorem 3.2, we then obtain the analog of (3.14) as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M211">View MathML</a>

Further, the analog of (3.15) appears as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M212">View MathML</a>

(5.9)

Hence, ||u|| ≤ max1≤in li = L and we conclude from Theorem 3.1 that (5.8) has a solution u* in (C[0, w])n. Using similar arguments as in getting (5.9), we find <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M213">View MathML</a>for each t ∈ [0, w]. All the conditions of Theorem 5.1 are now satisfied, it follows that (1.2) has at least one solution in (BC[0, ∞))n. □

The proof of subsequent Theorems 5.3-5.5 will model after the proof of Theorem 5.2, and will employ similar arguments as in the proof of Theorems 3.3-3.5. As such, we shall present the results and omit the proof.

Theorem 5.3 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w ∈ {1, 2,...} :

(C7)w there exist constants ai ≥ 0 and bi such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M214">View MathML</a>

(C8)w there exist r > 0 and αi > 0 with rαi > Hi + ai (Hi as in (D6)) such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M215">View MathML</a>

Then, (1.2) has at least one solution in (BC[0, ∞))n.

Theorem 5.4 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w ∈ {1, 2,...} :

(C9)w there exist constants ai ≥ 0, 0 i ≤ 1 and bi such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M216">View MathML</a>

(C10)w there exist r > 0 and βi > 0 such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M217">View MathML</a>

Then, (1.2) has at least one solution in (BC[0, ∞))n.

Theorem 5.5 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w ∈ {1, 2,...} : (C10)w,

(C11)w there exist r > 0, ηi > 0, γi > 0 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M218">View MathML</a>such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M219">View MathML</a>

(C12)w there exist ai ≥ 0, 0 < τi < γi + 1, bi, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M220">View MathML</a>with ψi ≥ 0 almost everywhere on [0, w], such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M221">View MathML</a>

Also, ϕi C[0, w], <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M222">View MathML</a>, ψi C[0, w] and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M223">View MathML</a>C[0, w].

Then, (1.2) has at least one solution in (BC[0, ∞))n.

We also have a remark similar to Remark 3.1.

Remark 5.1 In Theorem 5.5 the conditions (C10)w and (C11)w can be replaced by the following, this is evident from the proof.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M224">View MathML</a> There exist r > 0 and βi > 0 such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M225">View MathML</a>

where we denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M226">View MathML</a>.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M227">View MathML</a>There exist r > 0, ηi > 0, γi > 0 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M228">View MathML</a> such that for any u ∈ (C[0, w])n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M229">View MathML</a>

6 Existence of constant-sign solutions

In this section, we shall establish the existence of constant-sign solutions of the systems (1.1) and (1.2), in (C[0, T])n, (Cl[0, ∞))n and (BC[0, ∞))n. Once again, we shall employ an argument originated from Brezis and Browder [11].

Throughout, let θi ∈ {-1, 1}, 1 ≤ i ≤ n be fixed. For each 1 ≤ j ≤ n, we define

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M230">View MathML</a>

6.1 System (1.1)

Our first result is "parallel" to Theorem 3.2.

Theorem 6.1 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)-(C4) with pi = ∞ and qi = 1, (C5), (C6) and (E1)-(E3) where

(E1) θihi(t) ≥ 0 for t ∈ [0, T],

(E2) gi(t, s) ≥ 0 for s, t ∈ [0, T],

(E3) θi fi(t, u) ≥ 0 for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M231">View MathML</a>.

Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Proof First, we shall show that the system

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M232">View MathML</a>

(6.1)

has a solution in (C[0, T])n, where,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M233">View MathML</a>

(6.2)

where for 1 ≤ j ≤ n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M234">View MathML</a>

Clearly, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M235">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M236">View MathML</a>satisfies (C2).

We shall employ Theorem 3.1, so let u = (u1, u2,..., un) ∈ (C[0, T])n be any solution of

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M237">View MathML</a>

where λ ∈ (0, 1). Using (E1)-(E3), we have for t ∈ [0, T] and 1 ≤ i ≤ n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M238">View MathML</a>

Hence, u is a constant-sign solution of (6.3)λ, and it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M239">View MathML</a>

(6.4)

Noting (6.4), we see that (6.3)λ is the same as (3.1)λ. Therefore, using a similar technique as in the proof of Theorem 3.2, we obtain (3.15) and subsequently ||u|| ≤ max1≤in li L. It now follows from Theorem 3.1 (with M = L + 1) that (6.1) has a solution u* ∈ (C[0, T])n.

Noting (E1)-(E3), we have for t ∈ [0, T] and 1 ≤ i ≤ n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M240">View MathML</a>

Thus, u* is of constant sign. From (6.2), it is then clear that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M241">View MathML</a>

Hence, u* is actually a solution of (1.1). This completes the proof of the theorem. □

Based on the proof of Theorem 6.1, we can develop parallel results to Theorems 3.3-3.11 as follows.

Theorem 6.2 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)-(C4) with pi = ∞ and qi = 1, (C7), (C8) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Theorem 6.3 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)-(C4) with pi = ∞ and qi = 1, (C9), (C10) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Theorem 6.4 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)-(C4) with pi = ∞ and qi = 1, (C10)-(C12) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Theorem 6.5 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C5), (C10), (C13) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Theorem 6.6 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C7), (C10), (C13) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Theorem 6.7 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C10), (C13), (C14) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Theorem 6.8 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C10), (C13), (C15) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Theorem 6.9 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C10), (C13), (C16) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Theorem 6.10 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C10), (C13), (C17) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T])n.

Remark 6.1 Similar to Remarks 3.1 and 3.2, in Theorem 6.4 the conditions (C10) and (C11) can be replaced by (C10)' and (C11)'; whereas in Theorems 6.5-6.10, (C10) and (C13) can be replaced by (C10)' and (C13)'.

6.2 System (1.2)

We shall first obtain the existence of constant-sign solutions of (1.2) in (Cl[0, ∞))n. The first result is "parallel" to Theorem 4.2.

Theorem 6.11 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (D1)-(D5), (C5), (C6)and (E1)-(E3)where

(E1)θihi(t) ≥ 0 for t ∈ [0, ∞),

(E2)gi(t, s) ≥ 0 for s, t ∈ [0, ∞),

(E3)θifi(t,u) ≥ 0 for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M242">View MathML</a>.

Then, (1.2) has at least one constant-sign solution in (Cl[0, ∞))n.

Proof First, we shall show that the system

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M243">View MathML</a>

(6.5)

has a solution in (Cl[0, ∞))n. Here,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M244">View MathML</a>

(6.6)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M245">View MathML</a>

Clearly, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M246">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M236">View MathML</a> satisfies (D2).

We shall employ Theorem 4.1, so let u = (u1, u2,..., un) ∈ (Cl[0, ∞))n be any solution of

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M247">View MathML</a>

where λ ∈ (0, 1). Then, using a similar technique as in the proof of Theorem 6.1 (and also Theorem 4.2), we can show that (1.2) has a constant-sign solution u* ∈ (Cl[0, ∞))n. □

Remark 6.2 Similar to Remark 4.1, in Theorem 6.11 the conditions (D2)-(D5) can be replaced by (D2)'-(D5)'.

Based on the proof of Theorem 6.11, we can develop parallel results to Theorems 4.3-4.5 as follows.

Theorem 6.12 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (D1)-(D5), (C7), (C8)and (E1)-(E3). Then, (1.2) has at least one constant-sign solution in (Cl[0, ∞))n.

Theorem 6.13 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (D1)-(D5), (C9), (C10)and (E1)-(E3). Then, (1.2) has at least one constant-sign solution in (Cl[0, ∞))n.

Theorem 6.14 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (D1)-(D5), (C10)-(C12)and (E1)-(E3). Then, (1.2) has at least one constant-sign solution in (Cl[0, ∞))n.

Remark 6.3 Similar to Remark 4.2, in Theorem 6.14 the conditions (C10)and (C11)can be replaced by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M248">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M249">View MathML</a>.

We shall now obtain the existence of constant-sign solutions of (1.2) in (BC[0, ∞))n. The first result is 'parallel' to Theorem 5.1.

Theorem 6.15 For each 1 ≤ i n, assume (D2)'-(D4)' and (D6). For each k = 1, 2,..., suppose there exists a constant-sign <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M250">View MathML</a>that satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M251">View MathML</a>

(6.8)

Further, for 1 ≤ i ≤ n and k = 1, 2,..., there is a bounded set B ⊆ ℝ such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M178">View MathML</a>for each t ∈ [0, k]. Then, (1.2) has a constant-sign solution u*∈ (BC[0, ∞))n such that for 1 ≤ i ≤ n, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M179">View MathML</a>for all t ∈ [0, ∞).

Proof Using a similar technique as in the proof of Theorem 5.1, we can show that (5.2) holds. Let 1 ≤ i ≤ n. Together with the Arzéla-Ascoli theorem, we obtain subsequences of integers N1, N2,... satisfying (5.3), and functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M252">View MathML</a> such that (5.4) holds. Define a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M253">View MathML</a> by (5.5), i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M254">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M255">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M256">View MathML</a> and so <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M257">View MathML</a> . Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M258">View MathML</a> is of constant sign. The rest of the proof is the same as that of Theorem 5.1. □

The next result is "parallel" to Theorem 5.2.

Theorem 6.16 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w ∈ {1, 2,...,} : (C5)w, (C6)w and (E1)w - (E3)w where

(E1)w θihi(t) ≥ 0 for t ∈ [0, w],

(E2)w gi(t, s) ≥ 0 for s, t ∈ [0, w],

(E3)w θifi(t,u) ≥ 0 for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M259">View MathML</a>.

Then, (1.2) has at least one constant-sign solution in (BC[0, ∞))n.

Proof We shall apply Theorem 6.15. To do so, for w = 1, 2,..., we shall show that the system (5.8) has a constant-sign solution u* in (C[0, w])n. The proof of this is similar to that of Theorem 6.1 (with T = w) and Theorem 5.2. As in (5.9) we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M213">View MathML</a> for each t ∈ [0, w] and 1 ≤ i ≤ n. All the conditions of Theorem 6.15 are now satisfied and the conclusion is immediate. □

Based on the proof of Theorem 6.16, we can develop parallel results to Theorems 5.3-5.5 as follows:

Theorem 6.17 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w ∈ {1, 2,...} : (C7)w, (C8)w and (E1)w-(E3)w. Then, (1.2) has at least one constant-sign solution in (BC[0, ∞))n.

Theorem 6.18 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w ∈ {1, 2,...} : (C9)w, (C10)w and (E1)w-(E3)w. Then, (1.2) has at least one constant-sign solution in (BC[0, ∞))n.

Theorem 6.19 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w ∈ {1, 2,...} : (C11)w, (C12)w and (E1)w-(E3)w. Then, (1.2) has at least one constant-sign solution in (BC[0, ∞))n.

Remark 6.4 Similar to Remark 5.1, in Theorem 6.19 the conditions (C10)w and (C11)w can be replaced by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M224">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M227">View MathML</a>.

7 Examples

We shall now illustrate the results obtained through some examples.

Example 7.1 In system (1.1), consider the following fi, 1 ≤ i ≤ n :

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M260">View MathML</a>

(7.1)

Here,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M261">View MathML</a>

where c > 0 is a given constant, and κi is such that

(a) the map u α fi(t, u) is continuous for almost all t ∈ [0, T];

(b) the map t α fi(t, u) is measurable for all u ∈ ℝn;

(c) for any r > 0, there exists μr,i L1[0, T] such that |u| ≤ r implies |κi (t, u)| ≤ μr,i(t) for almost all t ∈ [0, T];

(d) for any u P, ui(t)κi(t, u(t)) ≥ 0 for all t ∈ [0, T].

Next, suppose for each 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M262">View MathML</a>

(7.2)

Clearly, conditions (C1) and (C2) with qi = 1 are fulfilled. We shall check that condition (C6) is satisfied. Pick r > c and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M263">View MathML</a>, 1 ≤ i n. Then, from (7.2) we have i = c > Hi.

Let u P. Then, from (7.1) we have fi(t, u) = κi(t, u). Consider ||u(t)|| > r where t ∈ [0, T]. If ||u(t)|| = |ui(t)|, then noting (d) we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M264">View MathML</a>

(7.3)

If ||u(t)|| = |uk(t)| for some k i, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M265">View MathML</a>

(7.4)

Therefore, from (7.3) and (7.4) we see that condition (C6) holds for u P.

For u ∈ (C[0, T])n\P, we have fi(t, u) = 0 and (C6) is trivially true. Hence, we have shown that condition (C6) is satisfied.

The next example considers a convolution kernel gi(t, s) which arises in nonlinear diffusion and percolation problems; the particular case when n = 1 has been investigated by Bushell and Okrasiński [26].

Example 7.2 Consider system (1.1) with (7.1), (7.2), and for 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M266">View MathML</a>

(7.5)

where γi > 1.

Clearly, gi satisfies (C3) and (C4) with pi = ∞. Next, we shall check condition (C5). For u P (P is given in Example 7.1), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M267">View MathML</a>

(7.6)

since κi(t, u) satisfies (c) (note (c) is stated in Example 7.1). This shows that condition (C5) holds for u P. For u ∈ (C[0, T])n\P, we have fi(t, u) = 0 and (C5) is trivially true. Therefore, condition (C5) is satisfied.

It now follows from Theorem 3.2 that the system (1.1) with (7.1), (7.2) and (7.5) has at least one solution in (C[0, T])n.

The next example considers an gi(t, s) of which the particular case when n = 1 originates from the well known Emden differential equation.

Example 7.3 Consider system (1.1) with (7.1), (7.2), and for 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M268">View MathML</a>

(7.7)

where γi ≥ 0.

Clearly, gi satisfies (C3) and (C4) with pi = ∞. Next, we see that condition (C5) is satisfied. In fact, for u P, corresponding to (7.6) we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M269">View MathML</a>

(7.8)

Hence, by Theorem 3.2 the system (1.1) with (7.1), (7.2) and (7.7) has at least one solution in (C[0, T])n.

Our next example illustrates the existence of a positive solution in (C[0, T])n, this is the particular case of constant-sign solution usually considered in the literature.

Example 7.4 Let θi = 1, 1 ≤ i n. Consider system (1.1) with (7.1), (7.2), and for 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M270">View MathML</a>

(7.9)

Clearly, condition (E1) is met, and noting (d) in Example 7.1 condition (E3) is also fulfilled. Moreover, both gi(t, s) in (7.5) and (7.7) satisfy condition (E2). From Examples 7.1-7.3, we see that all the conditions of Theorem 6.1 are met. Hence, we conclude that

the system (1.1) with (7.1), (7.2), (7.9) and (7.5).

and

the system (1.1) with (7.1), (7.2), (7.9) and (7.7).

each of which has at least one positive solution in (C[0, T])n.

Example 7.5 In system (1.2), consider the following fi, 1 ≤ i n :

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M271">View MathML</a>

(7.10)

Here,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M272">View MathML</a>

where c > 0 is a given constant, and κi is such that

(a)the map u α fi(t, u) is continuous for almost all t ∈ [0, ∞);

(b)the map t α fi(t, u) is measurable for all u ∈ ℝn;

(c)for any r > 0, there exists μr,i L1[0, ∞) such that |u| ≤ r implies |κi(t, u)| ≤ μr,i(t) for almost all t ∈ [0, ∞);

(d)for any u P, ui(t) κi(t, u(t)) ≥ 0 for all t ∈ [0, ∞).

Next, suppose for each 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M273">View MathML</a>

(7.11)

Clearly, conditions (D1) and (D2) are satisfied. Moreover, using a similar technique as in Example 7.1, we see that condition (C6)is satisfied.

Example 7.6 Consider system (1.2) with (7.10), (7.11), and for 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M274">View MathML</a>

(7.12)

where γi 1.

Clearly, gi satisfies (D3), (D4) and (D5) (take <a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M275">View MathML</a>). Next, we shall check condition (C5). For u P(Pis given in Example 7.5), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M276">View MathML</a>

(7.13)

since κi(t, u) satisfies (c)(note (c)is stated in Example 7.5). This shows that condition (C5)holds for u P. For u ∈ (Cl[0, ∞))n\P, we have fi(t, u) = 0 and (C5)is trivially true. Hence, condition (C5)is satisfied.

We can now conclude from Theorem 4.2 that the system (1.2) with (7.10), (7.11) and (7.12) has at least one solution in (Cl[0, ∞))n.

The next example shows the existence of a positive solution in (Cl[0, ∞))n, this is the special case of constant-sign solution usually considered in the literature.

Example 7.7 Let θi = 1, 1 ≤ i n. Consider system (1.2) with (7.10)-(7.12), and for 1 ≤ i n,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2011/1/43/mathml/M277">View MathML</a>

(7.14)

Clearly, conditions (E1)-(E3)are satisfied. Noting Examples 7.5 and 7.6, we see that all the conditions of Theorem 6.11 are met. Hence, the system (1.2) with (7.11)-(7.12) has at least one positive solution in (Cl[0, ∞))n.

Authors' contributions

All authors contributed equally to the manuscript and read and approved the final draft.

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

The authors would like to thank the referee for the comments which help to improve the paper.

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