Abstract
In this article, we consider the following systems of Fredholm integral equations:
Using an argument originating from Brezis and Browder [Bull. Am. Math. Soc. 81, 7378 (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 (C_{l}[0,∞))^{n }as well as in (BC[0,∞))^{n}. For both systems, we further seek the existence of constantsign 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; BrezisBrowder arguments; constantsign solutions1 Introduction
In this article, we shall consider the system of Fredholm integral equations:
where 0 < T <∞, and also the following system on the halfline
Throughout, let u = (u_{1}, u_{2},..., u_{n}). 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 (C_{l}[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 C_{l}[0, ∞) = {x ∈ BC[0, ∞) : lim_{t→∞ }x(t) exists}.
We shall also tackle the existence of constantsign 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 θ_{i}u_{i}(t) ≥ 0 for all t ∈ [0, T] (or t ∈ [0,∞)), where θ_{i }∈ {1, 1} is fixed. Note that when θ_{i }= 1 for all 1 ≤ i ≤ n, a constantsign 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:
and
Particular cases of (1.3) are also considered in [13]. 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 constantsign solutions has been established for these systems in [610].
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, LeraySchauder alternative, LeggettWilliams' fixed point theorem, fivefunctional fixed point theorem, Schauder fixed point theorem, and SchauderTychonoff 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 [610] and the results subsequently obtained are also different. The present article also extends, improves, and complements the studies of [5,1223]. Indeed, we have generalized the problems to (i) systems; (ii) more general form of nonlinearities f_{i}, 1 ≤ i ≤ n,; and (iii) existence of constantsign 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 (C_{l}[0, ∞))^{n }and (BC[0, ∞))^{n}, respectively. In Section 6, we seek the existence of constantsign solutions of (1.1) and (1.2) in (C[0, T])^{n}, (C_{l}[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 criteriaTheorem 2.1 [24] is Schauder's nonlinear alternative for continuous and compact maps, whereas Theorem 2.2 is the criterion of compactness on C_{l}[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 is a continuous and compact map. Then either
(a) S has a fixed point in , or
(b) there exist u ∈ ∂U and λ ∈ (0, 1) such that u = λSu.
Theorem 2.2 [[16], p. 62] Let P ⊂ C_{l}[0, ∞). Then P is compact in C_{l}[0, ∞) if the following hold:
(a) P is bounded in C_{l}[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:
where we let u_{i}_{0 }= sup_{t∈[0,T] }u_{i}(t), 1 ≤ i ≤ n. Throughout, for u ∈ B and t ∈ [0, T], we shall denote
Moreover, for each 1 ≤ i ≤ n, let 1 ≤ p_{i }≤ ∞ be an integer and q_{i }be such that . For , we shall define
Our first existence result uses Theorem 2.1.
Theorem 3.1 For each 1 ≤ i ≤ n, assume (C1) (C4) hold where
(C1) h_{i }∈ C[0, T], denote H_{i }≡ sup_{t∈ [0, T] }h_{i}(t),
(C2) f_{i }: [0, T] × ℝ^{n }→ ℝ is a Carathéodory function:
(i) the map u α f_{i}(t, u) is continuous for almost all t ∈ [0, T],;
(ii) the map t α f_{i}(t, u) is measurable for all u ∈ ℝ^{n};
(iii) for any r > 0, there exists such that u ≤ r implies f_{i}(t, u) ≤ μ_{r,i}(t) for almost all t ∈ [0, T];
(C4) the map is continuous from [0, T] to .
In addition, suppose there is a constant M > 0, independent of λ, with u ≠ M for any solution u ∈ (C[0, T])^{n }to
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
where
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., S_{i }: (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 f_{i }is a Carathéodory function, there exists such that f_{i}(s, u) ≤ μ_{r,i}(s) for almost all s ∈ [0, T]. Hence, for any t_{1}, t_{2 }∈ [0, T], we find for 1 ≤ i ≤ n,
as t_{1 }→ t_{2}, 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 in (C[0, T])^{n}, i.e., in C[0, T], 1 ≤ i ≤ n. We need to show that Su^{m }→ Su in (C[0, T])^{n}, or equivalently S_{i}u^{m }→ S_{i}u in C[0, T], 1 ≤ i ≤ n. There exists r > 0 such that u^{m}, u < r. Since f_{i }is a Carathéodory function, there exists such that f_{i}(s, u^{m}), f_{i}(s, u) ≤ μ_{r,i}(s) for almost all s ∈ [0, T]. Using a similar argument as in (3.4), we get for any t_{1}, t_{2 }∈ [0, T] and 1 ≤ i ≤ n:
as t_{1 }→ t_{2}. Furthermore, S_{i}u^{m}(t) → S_{i}u(t) pointwise on [0, T], since, by the Lebesguedominated convergence theorem,
as m → ∞. Combining (3.5) and (3.6) and using the fact that [0, T] is compact, gives for all t ∈ [0, T],
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 S_{i}Ω is relatively compact for 1 ≤ i ≤ n. Clearly, S_{i}Ω is uniformly bounded, since there exists such that f_{i}(s, u) ≤ μ_{r,i}(s) for all u ∈ Ω and a.e. s ∈ [0, T], and hence
Further, using a similar argument as in (3.4), we see that S_{i}Ω is equicontinuous. It follows from the ArzélaAscoli theorem [[5], Theorem 1.2.4] that S_{i}Ω 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 f_{i}(t,u(t)) (or ) on two complementary subsets of [0, T], namely {t ∈ [0, T] : u(t) ≤ r} and {t ∈ [0, T] : u(t) > r}, where ρ_{i }and r are some constantsthis technique originates from the study of Brezis and Browder [11]. In the next four theorems (Theorems 3.23.5), we shall apply Theorem 3.1 to the case p_{i }= ∞ and q_{i }= 1, 1 ≤ i ≤ n.
Theorem 3.2. Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)(C4) with p_{i }= ∞ and q_{i }= 1, (C5) and (C6) where
(C5) there exist B_{i }> 0 such that for any u ∈ (C[0, T])^{n},
(C6) there exist r > 0 and α_{i }> 0 with rα_{i }> H_{i }such that for any u ∈ (C[0, T])^{n},
Then, (1.1) has at least one solution in (C[0, T])^{n}.
Proof We shall employ Theorem 3.1, and so let u = (u_{1}, u_{2}, l...., u_{n}) ∈ (C[0, T])^{n }be any solution of (3.1)_{λ }where λ ∈ (0, 1).
Define
Clearly, [0, T] = I ∪ J, and hence .
Let 1 ≤ i ≤ n. If t ∈ I, then by (C2), there exists μ_{r,i }∈ L^{1}[0, T] such that f_{i}(t, u(t)) ≤ μ_{r,i}(t). Thus, we get
On the other hand, if t ∈ J, then it is clear from (C6) that u_{i}(t)f_{i}(t, u(t)) ≥ 0 for a.e. t ∈ [0, T]. It follows that
We now multiply (3.1)_{λ }by f_{i}(t, u(t)), then integrate from 0 to T to get
Using (C5) in (3.12) yields
Splitting the integrals in (3.13) and applying (3.11), we get
or
where we have used (3.10) in the last inequality. It follows that
Finally, it is clear from (3.1)_{λ }that for t ∈ [0, T] and 1 ≤ i ≤ n,
where we have applied (3.10) and (3.14) in the last inequality. Thus, u_{i}_{0 }≤ l_{i }for 1 ≤ i ≤ n and u ≤ max_{1≤i≤n }l_{i }≡ 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 p_{i }= ∞ and q_{i }= 1, (C7) and (C8) where
(C7) there exist constants a_{i }≥ 0 and b_{i }such that for any u ∈ (C[0, T])^{n},
(C8) there exist r > 0 and α_{i }> 0 with rα_{i }> H_{i }+ a_{i }such that for any u ∈ (C[0, T])^{n},
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
Splitting the integrals in (3.16) and applying (3.11) gives
where we have also used (3.10) in the last inequality. It follows that
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 p_{i }= ∞ and q_{i }= 1, (C9) and (C10) where
(C9) there exist constants a_{i }≥ 0, 0 < τ_{i }≤ 1 and b_{i }such that for any u ∈ (C[0, T])^{n},
(C10) there exist r > 0 and β_{i }> 0 such that for any u ∈ (C[0, T])^{n},
Then, (1.1) has at least one solution in (C[0, T])^{n}.
Proof Let u = (u_{1}, u_{2},..., u_{n}) ∈ (C[0, T])^{n }be any solution of (3.1)_{λ }where λ ∈ (0, 1). Define
Clearly, [0, T] = I_{0 }∪ J_{0 }and hence .
Let 1 ≤ i ≤ n. If t ∈ I_{0}, then by (C2) there exists such that and
Further, if t ∈ J_{0}, then by (C10) we have
Now, using (3.20) and (C9) in (3.12) gives
where in the last inequality, we have made use of the inequality:
Now, noting (3.19) we find that
Substituting (3.22) in (3.21) then yields
Since τ_{i }≤ 1, there exists a constant such that
which leads to
Finally, it is clear from (3.1)_{λ }that for t ∈ [0, T] and 1 ≤ i ≤ n,
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 p_{i }= ∞ and q_{i }= 1, (C10), (C11) and (C12) where
(C11) there exist r > 0, η_{i }> 0, γ_{i }> 0 and such that for any u ∈ (C[0, T])^{n},
(C12) there exist a_{i }≥ 0, 0 < τ_{i }< γ_{i }+ 1, b_{i}, and with ψ_{i }≥ 0 almost everywhere on [0, T], such that for any u ∈ (C[0, T])^{n},
Also, ϕ_{i }∈ C[0, T], , ψ_{i }∈ C[0, T] and .
Then, (1.1) has at least one solution in (C[0, T])^{n}.
Proof Let u = (u_{1}, u_{2},..., u_{n}) ∈ (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
Using (3.25) and (C12) in (3.12), we obtain
Now, in view of (3.10) and (C12), we have
Substituting (3.27) into (3.26) and using Hölder's inequality, we find
Since and , there exists a constant k_{i }such that
Finally, it is clear from (3.1)_{λ }that for t ∈ [0, T] and 1 ≤ i ≤ n,
where we have used (3.28) and (C12) in the last inequality, and l_{i }is some constant. The conclusion is now immediate by Theorem 3.1. □
In the next six results (Theorem 3.63.11), we shall apply Theorem 3.1 for general p_{i }and q_{i}.
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, η_{i }> 0, γ_{i }> 0 and such that for any u ∈ (C[0, T])^{n},
Then, (1.1) has at least one solution in (C[0, T])^{n}.
Proof Let u = (u_{1}, u_{2},..., u_{n}) ∈ (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 such that f_{i}(t, u(t)) ≤ μ_{r,i}(t). Consequently, we have
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
where (3.30) has been used in the last inequality and .
Now, an application of Hölder's inequality gives
Another application of Hölder's inequality yields
Substituting (3.33) into (3.32) then leads to
Further, using Hölder's inequality again, we get
Substituting (3.34) and (3.35) into (3.31), we obtain
where . Since , from (3.36), there exists a constant k_{i }such that
Finally, it is clear from (3.1)_{λ }that for t ∈ [0, T] and 1 ≤ i ≤ n,
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 = (u_{1}, u_{2},..., u_{n}) ∈ (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
where . Substituting (3.34) and (3.35) into (3.39) then leads to
where . Since , from (3.40), we can obtain (3.37) where k_{i }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 a_{i }≥ 0, 0 < τ_{i }< γ_{i }+ 1 and b_{i }such that for any u ∈ (C[0, T])^{n},
Then, (1.1) has at least one solution in (C[0, T])^{n}.
Proof Let u = (u_{1}, u_{2},..., u_{n}) ∈ (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
Note that
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
Since and , there exists a constant k_{i }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 d_{i }≥ 0, 0 < τ_{i }< γ_{i }+ 1 and e_{i }such that for any u ∈ (C[0, T])^{n},
Then, (1.1) has at least one solution in (C[0, T])^{n}.
Proof Let u = (u_{1}, u_{2},..., u_{n}) ∈ (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
Now, it is clear that
Moreover, an application of Hölder's inequality gives
Substituting (3.45) into (3.44) and using (3.34), (3.35) and (3.46) then leads to
Noting and , there exists a constant k_{i }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 c_{i }≥ 0, d_{i }≥ 0, 0 < τ_{i }< γ_{i }+ 1 and e_{i }with such that for any u ∈ (C[0, T])^{n},
Then, (1.1) has at least one solution in (C[0, T])^{n}.
Proof Let u = (u_{1}, u_{2},..., u_{n}) ∈ (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
Now, it is clear that
Substituting (3.49) into (3.48) and then using (3.34), (3.35) and (3.46) leads to
Noting , as well as , from (3.50) there exists a constant k_{i }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 a_{i }≥ 0, 0 <τ_{i }< γ_{i }+ 1, b_{i}, and with ψ_{i }≥ 0 almost everywhere on [0, T], such that for any u ∈ (C[0, T])^{n},
Then, (1.1) has at least one solution in (C[0, T])^{n}.
Proof Let u = (u_{1}, u_{2},..., u_{n}) ∈ (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
Substituting (3.51) into (3.26) and applying (3.34) and (3.35), we find that
Since and , from (3.52), there exists a constant k_{i }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 β_{i }> 0 such that for any u ∈ (C[0, T])^{n},
(C11)' There exist r > 0, η_{i }> 0, γ_{i }> 0 and such that for any u ∈ (C[0, T])^{n},
Remark 3.2 In Theorems 3.63.11, the conditions (C10) and (C13) can be replaced by (C10)' and (C13)' below, and the proof will be similar.
(C13)' There exist r > 0, η_{i }> 0, γ_{i }> 0, and such that for any u ∈ (C[0, T])^{n},
4 Existence results for (1.2) in (C_{l}[0, ∞))^{n}
Let the Banach space B = (C_{l}[0, ∞))^{n }be equipped with the norm:
where we let u_{i}_{0 }= sup_{t∈[0,∞) }u_{i}(t), 1 ≤ i ≤ n. Throughout, for u ∈ B and t ∈ [0, ∞), we shall denote that
Moreover, for each 1 ≤ i ≤ n, let 1 ≤ p_{i }≤ ∞ be an integer and q_{i }be such that . For , we shall define that
We shall apply Theorem 2.1 to obtain the first existence result for (1.2) in (C_{l}[0, ∞))^{n}.
Theorem 4.1 For each 1 ≤ i ≤ n, assume (D1)(D5) hold where
(D1) h_{i }∈ C_{l}[0, ∞), denote H_{i }≡ sup_{t∈[0,∞) }h_{i}(t),
(D2) f_{i }: [0, ∞) × ℝ^{n }→ ℝ is a L^{1}Carathéodory function, i.e.,
(i) the map u α f_{i}(t, u) is continuous for almost all t ∈ [0, ∞),
(ii) the map t α f_{i}(t, u) is measurable for all u ∈ ℝ^{n},
(iii) for any r > 0, there exists μ_{r,i }∈ L^{1}[0, ∞) such that u ≤ r implies f_{i}(t, u) ≤ μ_{r,i}(t) for almost all t ∈ [0, ∞).
(D4) the map is continuous from [0, ∞) to L^{∞ }[0, ∞),
(D5) there exists such that in L^{∞}[0, ∞) as t → ∞, i.e.,
In addition, suppose there is a constant M > 0, independent of λ, with u ≠ M for any solution u ∈ (C_{l}[0, ∞))^{n }to
for each λ ∈ (0, 1). Then, (1.2) has at least one solution in (C_{l}[0, ∞))^{n}.
Proof To begin, let the operator S be defined by
where
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 : (C_{l}[0, ∞))^{n }→ (C_{l}[0, ∞))^{n}, or equivalently S_{i }: (C_{l}[0, ∞))^{n }→ C_{l}[0, ∞), 1 ≤ i ≤ n. Let u ∈ (C_{l}[0, ∞))^{n}. Then, there exists r > 0 such that u ≤ r, and from (D2) there exists μ_{r,i }∈ L^{1}[0, ∞) such that f_{i}(s, u) ≤ μ_{r,i }(s) for almost all s ∈ [0, ∞). Let t_{1}, t_{2 }∈ [0, ∞). Together with (D1) and (D4), we find that
as t_{1 }→ t_{2}. Hence, S_{i}u ∈ C[0, ∞).
To see that S_{i}u is bounded, we have for t ∈ [0, ∞),
By (D5), there exists T_{1 }> 0 such that for t > T_{1},
On the other hand, for t ∈ [0, T_{1}], we have
Hence,
It follows from (4.5) that for t ∈ [0, ∞),
Hence, S_{i}u is bounded.
It remains to check the existence of the limit lim_{t→∞ }S_{i}u(t). We claim that
where h_{i}(∞) ≡ lim_{t→∞ }h_{i}(t). In fact, it follows from (D5) that
as t → ∞. This implies
and so (4.8) is proved. We have hence shown that S : (C_{l}[0, ∞))^{n }→ (C_{l}[0, ∞))^{n}.
Next, we shall prove that S : (C_{l}[0, ∞))^{n }→ (C_{l}[0, ∞))^{n }is continuous. Let {u^{m}} be a sequence in (C_{l}[0, ∞))^{n }and . In (C_{l}[0, ∞))^{n}, i.e., , in C_{l}[0, ∞), 1 ≤ i ≤ n. We need to show that Su^{m }→ Su in (C_{l}[0, ∞))^{n}, or equivalently S_{i}u^{m }→ S_{i}u in C_{l}[0, ∞), 1 ≤ i ≤ n. There exists r > 0 such that u^{m}, u < r, Noting (D2), there exists μ_{r,i }∈ L^{1}[0, ∞) such that f_{i}(s, u^{m}), f_{i}(s, u) ≤ μ_{r,i}(s) for almost all s ∈ [0, ∞). Denote S_{i}u(∞) ≡ lim_{t→∞ }S_{i}u(t) and S_{i}u^{m}(∞) ≡ lim_{t→∞ }S_{i}u^{m}(t). In view of (4.8), we get that
Since
and
by the Lebesguedominated convergence theorem, it is clear from (4.9) that
Further, using (4.8) again we find that
as t → ∞. Similarly, we also have that
Combining (4.10)(4.12), we have
or equivalently, there exist such that
It remains to check the convergence in . As in (4.4), we find for any ,
as t_{1 }→ t_{2}. Furthermore, S_{i}u^{m}(t) → S_{i}u(t) pointwise on , since, by the Lebesguedominated convergence theorem,
as m → ∞. Combining (4.14) and (4.15) and the fact that is compact yields
Coupling (4.13) and (4.16), we see that S_{i}u^{m }→ S_{i}u in C_{l}[0, ∞).
Finally, we shall show that S : (C_{l}[0, ∞))^{n }→ (C_{l}[0, ∞))^{n }is completely continuous. Let Ω be a bounded set in (C_{l}[0, ∞))^{n }with u ≤ r for all u ∈ Ω We need to show that S_{i}Ω is relatively compact for 1 ≤ i ≤ n. First, we see that S_{i}Ω 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 S_{i}Ω is equicontinuous. Moreover, S_{i}Ω is equiconvergent follows as in (4.11). By Theorem 2.2, we conclude that S_{i}Ω is relatively compact. Hence, S : (C_{l}[0, ∞))^{n }→ (C_{l}[0, ∞))^{n }is completely continuous.
We now apply Theorem 2.1 with U = {u ∈ (C_{l}[0, ∞))^{n }: u < M} and B = E = (C_{l}[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 p_{i }and q_{i }as follows, and the proof will be similar:
(D2)' f_{i }: [0, ∞) × ℝ^{n }→ ℝ is a Carathéodory function, i.e.,
(i) the map u α f_{i}(t, u) is continuous for almost all t ∈ [0, ∞),
(ii) the map t α f_{i}(t, u) is measurable for all u ∈ ℝ^{n},
(iii) for any r > 0, there exists such that u ≤ r implies f_{i}(t, u) ≤ μ_{r,i}(t) for almost all t ∈ [0, ∞),
(D4)' the map is continuous from [0, ∞) to ,
(D5)' there exists such that , in as t → ∞, i.e.,
Our subsequent Theorems 4.24.5 use an argument originating from Brezis and Browder [11]. These results are parallel to Theorems 3.23.5 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 B_{i }> 0 such that for any u ∈ (C_{l}[0, ∞))^{n},
(C6)_{∞ }there exist r > 0 and α_{i }> 0 with rα_{i }> H_{i }such that for any u ∈ (C_{l}[0, ∞))^{n},
Then, (1.2) has at least one solution in (C_{l}[0, ∞))^{n}.
Proof We shall employ Theorem 4.1, so let u = (u_{1}, u_{2},..., u_{n}) ∈ (C_{l}[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 4.34.5 will also be similar to that of Theorems 3.33.5 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 a_{i }≥ 0 and b_{i }such that for any u ∈ (C_{l}[0, ∞))^{n},
(C8)_{∞ }there exist r > 0 and α_{i }> 0 with rα_{i }> H_{i }+ a_{i }such that for any u ∈ (C_{l}[0, ∞))^{n},
Then, (1.2) has at least one solution in (C_{l}[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 a_{i }≥ 0, 0 < τ_{i }≤ 1 and b_{i }such that for any u ∈ (C_{l}[0, ∞))^{n},
(C10)_{∞ }there exist r > 0 and β_{i }> 0 such that for any u ∈ (C_{l}[0, ∞))^{n},
Then, (1.2) has at least one solution in (C_{l}[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, η_{i }> 0, γ_{i }> 0 and such that for any u ∈ (C_{l}[0, ∞))^{n},
(C12)_{∞ }there exist a_{i }≥ 0, 0 < τ_{i }< γ_{i }+ 1, b_{i}, and with ψ_{i }≥ 0 almost everywhere on [0, ∞), such that for any u ∈ (C_{l}[0, ∞))^{n},
Also, ϕ_{i }∈ BC[0, ∞), , ψ_{i }∈ BC[0, ∞) and .
Then, (1.2) has at least one solution in (C_{l}[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.
There exist r > 0 and β_{i }> 0 such that for any u ∈ (C_{l}[0, ∞))^{n},
There exist r > 0, η_{i }> 0, γ_{i }> 0 and such that for any u ∈ (C_{l}[0, ∞))^{n},
5 Existence results for (1.2) in (BC[0, ∞))^{n}
Let the Banach space B = (BC[0, ∞))^{n }be equipped with the norm:
where we let u_{i}_{0 }= sup_{t∈[0,∞) }u_{i}(t), 1 < i < n. Throughout, for u ∈ B and t ∈ [0, ∞) we shall denote
Moreover, for each 1 ≤ i ≤ n, let 1 ≤ p_{i }≤ ∞ be an integer and q_{i }be such that . For , we shall define 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) h_{i }∈ BC[0, ∞), denote H_{i }≡ sup_{t∈[0, ∞) }h_{i}(t).
For each k = 1, 2,..., suppose there exists that satisfies
Further, for 1 ≤ i ≤ n and k = 1, 2,..., there is a bounded set B ⊆ ℝ such that for each t ∈ [0, k]. Then, (1.2) has a solution u* ∈ (BC[0, ∞))^{n }such that for 1 ≤ i ≤ n, for all t ∈ [0, ∞).
Proof First we shall show that
The uniform boundedness of follows immediately from the hypotheses; therefore, we only need to prove that is equicontinuous. Let 1 ≤ i ≤ n. Since for each t ∈ [0, k], there exists such that f_{i}(s,u^{k}(s)) ≤ μ_{B}(s) for almost every s ∈ [0, k].Fix t, t' ∈ [0, λ]. Then, from (5.1) we find that
as t → t'. Therefore, is equicontinuous on [0, λ].
Let 1 ≤ i ≤ n. Now, (5.2) and the ArzélaAscoli theorem yield a subsequence N_{1 }of ℕ = {1, 2,...} and a function such that uniformly on [0,1] as k → ∞ in N_{1}. Let . Then, (5.2) and the ArzélaAscoli theorem yield a subsequence N_{2 }of and a function such that uniformly on [0,2] as k → ∞ in N_{2}. Note that on [0,1] since N_{2 }⊆ N_{1}. Continuing this process, we obtain subsequences of integers N_{1}, N_{2},... with
Let 1 ≤ i ≤ n. Define a function by
Clearly, and for each t ∈ [0, λ]. It remains to prove that 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
or equivalently
Since f_{i }is a Carathéodory function and for each t ∈ [0, k], there exists such that
and . Let k → ∞ (k ∈ N_{ℓ}) in (5.6). Since uniformly on [0, ℓ], an application of Lebesguedominated convergence theorem gives
or equivalently (noting (5.5))
Finally, letting ℓ → ∞ in (5.7) and use the fact to get
Hence, 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 5.25.5 will make use of Theorem 5.1 and the technique used in Section 3. These results are parallel to Theorems 3.23.5 and 4.24.5.
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 B_{i }> 0 such that for any u ∈ (C[0, w])^{n},
(C6)_{w }there exist r > 0 and α_{i }> 0 with rα_{i }> H_{i }(H_{i }as in (D6)) such that for any u ∈ (C[0, w])^{n},
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
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 = (u_{1}, u_{2},..., u_{n}) ∈ (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 H_{i }given in (D6). As in (3.9), define
Let 1 ≤ i ≤ n. If t ∈ I, then by (D2) there exists μ_{r,i }∈ L^{1}[0, ∞) such that
[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
Further, the analog of (3.15) appears as
Hence, u ≤ max_{1≤i≤n }l_{i }= 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 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 5.35.5 will model after the proof of Theorem 5.2, and will employ similar arguments as in the proof of Theorems 3.33.5. 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 a_{i }≥ 0 and b_{i }such that for any u ∈ (C[0, w])^{n},
(C8)_{w }there exist r > 0 and α_{i }> 0 with rα_{i }> H_{i }+ a_{i }(H_{i }as in (D6)) such that for any u ∈ (C[0, w])^{n},
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 a_{i }≥ 0, 0 <τ_{i }≤ 1 and b_{i }such that for any u ∈ (C[0, w])^{n},
(C10)_{w }there exist r > 0 and β_{i }> 0 such that for any u ∈ (C[0, w])^{n},
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, η_{i }> 0, γ_{i }> 0 and such that for any u ∈ (C[0, w])^{n},
(C12)_{w }there exist a_{i }≥ 0, 0 < τ_{i }< γ_{i }+ 1, b_{i}, and with ψ_{i }≥ 0 almost everywhere on [0, w], such that for any u ∈ (C[0, w])^{n},
Also, ϕ_{i }∈ C[0, w], , ψ_{i }∈ C[0, w] and ∈ 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.
There exist r > 0 and β_{i }> 0 such that for any u ∈ (C[0, w])^{n},
There exist r > 0, η_{i }> 0, γ_{i }> 0 and such that for any u ∈ (C[0, w])^{n},
6 Existence of constantsign solutions
In this section, we shall establish the existence of constantsign solutions of the systems (1.1) and (1.2), in (C[0, T])^{n}, (C_{l}[0, ∞))^{n }and (BC[0, ∞))^{n}. Once again, we shall employ an argument originated from Brezis and Browder [11].
Throughout, let θ_{i }∈ {1, 1}, 1 ≤ i ≤ n be fixed. For each 1 ≤ j ≤ n, we define
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 p_{i }= ∞ and q_{i }= 1, (C5), (C6) and (E1)(E3) where
(E1) θ_{i}h_{i}(t) ≥ 0 for t ∈ [0, T],
(E2) g_{i}(t, s) ≥ 0 for s, t ∈ [0, T],
(E3) θ_{i }f_{i}(t, u) ≥ 0 for .
Then, (1.1) has at least one constantsign solution in (C[0, T])^{n}.
Proof First, we shall show that the system
has a solution in (C[0, T])^{n}, where,
where for 1 ≤ j ≤ n,
We shall employ Theorem 3.1, so let u = (u_{1}, u_{2},..., u_{n}) ∈ (C[0, T])^{n }be any solution of
where λ ∈ (0, 1). Using (E1)(E3), we have for t ∈ [0, T] and 1 ≤ i ≤ n,
Hence, u is a constantsign solution of (6.3)_{λ}, and it follows that
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 ≤ max_{1≤i≤n }l_{i }≡ 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,
Thus, u* is of constant sign. From (6.2), it is then clear that
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 3.33.11 as follows.
Theorem 6.2 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)(C4) with p_{i }= ∞ and q_{i }= 1, (C7), (C8) and (E1)(E3). Then, (1.1) has at least one constantsign solution in (C[0, T])^{n}.
Theorem 6.3 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)(C4) with p_{i }= ∞ and q_{i }= 1, (C9), (C10) and (E1)(E3). Then, (1.1) has at least one constantsign solution in (C[0, T])^{n}.
Theorem 6.4 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)(C4) with p_{i }= ∞ and q_{i }= 1, (C10)(C12) and (E1)(E3). Then, (1.1) has at least one constantsign 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 constantsign 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 constantsign 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 constantsign 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 constantsign 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 constantsign 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 constantsign 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 6.56.10, (C10) and (C13) can be replaced by (C10)' and (C13)'.
6.2 System (1.2)
We shall first obtain the existence of constantsign solutions of (1.2) in (C_{l}[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)_{∞ }θ_{i}h_{i}(t) ≥ 0 for t ∈ [0, ∞),
(E2)_{∞ }g_{i}(t, s) ≥ 0 for s, t ∈ [0, ∞),
Then, (1.2) has at least one constantsign solution in (C_{l}[0, ∞))^{n}.
Proof First, we shall show that the system
has a solution in (C_{l}[0, ∞))^{n}. Here,
where
We shall employ Theorem 4.1, so let u = (u_{1}, u_{2},..., u_{n}) ∈ (C_{l}[0, ∞))^{n }be any solution of
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 constantsign solution u* ∈ (C_{l}[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 4.34.5 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 constantsign solution in (C_{l}[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 constantsign solution in (C_{l}[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 constantsign solution in (C_{l}[0, ∞))^{n}.
Remark 6.3 Similar to Remark 4.2, in Theorem 6.14 the conditions (C10)_{∞ }and (C11)_{∞ }can be replaced by and .
We shall now obtain the existence of constantsign 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 constantsign that satisfies
Further, for 1 ≤ i ≤ n and k = 1, 2,..., there is a bounded set B ⊆ ℝ such that for each t ∈ [0, k]. Then, (1.2) has a constantsign solution u*∈ (BC[0, ∞))^{n }such that for 1 ≤ i ≤ n, 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 ArzélaAscoli theorem, we obtain subsequences of integers N_{1}, N_{2},... satisfying (5.3), and functions such that (5.4) holds. Define a function by (5.5), i.e.,
Since , we have and so . Hence, 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 }θ_{i}h_{i}(t) ≥ 0 for t ∈ [0, w],
(E2)_{w }g_{i}(t, s) ≥ 0 for s, t ∈ [0, w],
(E3)_{w }θ_{i}f_{i}(t,u) ≥ 0 for .
Then, (1.2) has at least one constantsign 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 constantsign 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 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 5.35.5 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 constantsign 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 constantsign 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 constantsign 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 and .
7 Examples
We shall now illustrate the results obtained through some examples.
Example 7.1 In system (1.1), consider the following f_{i}, 1 ≤ i ≤ n :
Here,
where c > 0 is a given constant, and κ_{i }is such that
(a) the map u α f_{i}(t, u) is continuous for almost all t ∈ [0, T];
(b) the map t α f_{i}(t, u) is measurable for all u ∈ ℝ^{n};
(c) for any r > 0, there exists μ_{r,i }∈ L^{1}[0, T] such that u ≤ r implies κ_{i }(t, u) ≤ μ_{r,i}(t) for almost all t ∈ [0, T];
(d) for any u ∈ P, u_{i}(t)κ_{i}(t, u(t)) ≥ 0 for all t ∈ [0, T].
Next, suppose for each 1 ≤ i ≤ n,
Clearly, conditions (C1) and (C2) with q_{i }= 1 are fulfilled. We shall check that condition (C6) is satisfied. Pick r > c and , 1 ≤ i ≤ n. Then, from (7.2) we have rα_{i }= c > H_{i}.
Let u ∈ P. Then, from (7.1) we have f_{i}(t, u) = κ_{i}(t, u). Consider u(t) > r where t ∈ [0, T]. If u(t) = u_{i}(t), then noting (d) we have
If u(t) = u_{k}(t) for some k ≠ i, then
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 f_{i}(t, u) = 0 and (C6) is trivially true. Hence, we have shown that condition (C6) is satisfied.
The next example considers a convolution kernel g_{i}(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,
where γ_{i }> 1.
Clearly, g_{i }satisfies (C3) and (C4) with p_{i }= ∞. Next, we shall check condition (C5). For u ∈ P (P is given in Example 7.1), we have
since κ_{i}(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 f_{i}(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 g_{i}(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,
where γ_{i }≥ 0.
Clearly, g_{i }satisfies (C3) and (C4) with p_{i }= ∞. Next, we see that condition (C5) is satisfied. In fact, for u ∈ P, corresponding to (7.6) we have
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 constantsign solution usually considered in the literature.
Example 7.4 Let θ_{i }= 1, 1 ≤ i ≤ n. Consider system (1.1) with (7.1), (7.2), and for 1 ≤ i ≤ n,
Clearly, condition (E1) is met, and noting (d) in Example 7.1 condition (E3) is also fulfilled. Moreover, both g_{i}(t, s) in (7.5) and (7.7) satisfy condition (E2). From Examples 7.17.3, 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 f_{i}, 1 ≤ i ≤ n :
Here,
where c > 0 is a given constant, and κ_{i }is such that
(a)_{∞ }the map u α f_{i}(t, u) is continuous for almost all t ∈ [0, ∞);
(b)_{∞ }the map t α f_{i}(t, u) is measurable for all u ∈ ℝ^{n};
(c)_{∞ }for any r > 0, there exists μ_{r},_{i }∈ L^{1}[0, ∞) such that u ≤ r implies κ_{i}(t, u) ≤ μ_{r,i}(t) for almost all t ∈ [0, ∞);
(d)_{∞ }for any u ∈ P_{∞}, u_{i}(t) κ_{i}(t, u(t)) ≥ 0 for all t ∈ [0, ∞).
Next, suppose for each 1 ≤ i ≤ n,
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,
where γ_{i }≥ 1.
Clearly, g_{i }satisfies (D3), (D4) and (D5) (take ). Next, we shall check condition (C5)_{∞}. For u ∈ P_{∞ }(P_{∞ }is given in Example 7.5), we have
since κ_{i}(t, u) satisfies (c)_{∞ }(note (c)_{∞ }is stated in Example 7.5). This shows that condition (C5)_{∞ }holds for u ∈ P_{∞}. For u ∈ (C_{l}[0, ∞))^{n}\P_{∞}, we have f_{i}(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 (C_{l}[0, ∞))^{n}.
The next example shows the existence of a positive solution in (C_{l}[0, ∞))^{n}, this is the special case of constantsign solution usually considered in the literature.
Example 7.7 Let θ_{i }= 1, 1 ≤ i ≤ n. Consider system (1.2) with (7.10)(7.12), and for 1 ≤ i ≤ n,
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 (C_{l}[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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