### 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**, 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
(*C _{l}*[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:

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

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*])

*[0,*

^{n }= C*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 *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 *θ _{i}u_{i}*(

*t*) ≥ 0 for all

*t*∈ [0,

*T*] (or

*t*∈ [0,∞)), where

*θ*∈ {-1, 1} is fixed. Note that when

_{i }*θ*= 1 for all 1 ≤

_{i }*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:

and

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 *f _{i}*, 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 (

*C*[0, ∞))

_{l}*and (*

^{n }*BC*[0, ∞))

*, respectively. In Section 6, we seek the existence of*

^{n}*constant-sign*solutions of (1.1) and (1.2) in (

*C*[0,

*T*])

*, (*

^{n}*C*[0, ∞))

_{l}*and (*

^{n }*BC*[0, ∞))

*. Finally, several examples are presented in Section 7 to illustrate the results obtained.*

^{n}### 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 *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*[0, ∞)

_{l}*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}

^{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*be such that

_{i }

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*≡ sup

_{i }_{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*];

(C3)
*for each 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*])

*, there exits*

^{n}*r*> 0 such that ||

*u*|| <

*r*. Since

*f*is a

_{i }*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
*C*[0, *T*])* ^{n}*, i.e.,

*C*[0,

*T*], 1 ≤

*i*≤

*n*. We need to show that

*Su*→

^{m }*Su*in (

*C*[0,

*T*])

*, or equivalently*

^{n}*S*→

_{i}u^{m }*S*in

_{i}u*C*[0,

*T*], 1 ≤

*i*≤

*n*. There exists

*r*> 0 such that ||

*u*||, ||

^{m}*u*|| <

*r*. Since

*f*is a

_{i }*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 Lebesgue-dominated 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*Ω is uniformly bounded, since there exists

_{i}*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 Arzéla-Ascoli theorem [[5], Theorem 1.2.4] that

*S*Ω is relatively compact.

_{i}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

*T*], namely {

*t*∈ [0,

*T*] : ||

*u*(

*t*)|| ≤

*r*} and {

*t*∈ [0,

*T*] : ||

*u*(

*t*)|| >

*r*}, where

*ρ*and

_{i }*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

*p*= ∞ and

_{i }*q*= 1, 1 ≤

_{i }*i*≤

*n*.

**Theorem 3.2**. *Let the following conditions be satisfied for each *1 ≤ *i *≤ *n *: *(C1)-(C4) with p _{i }= *∞

*and q*1,

_{i }=*(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*])

*be any solution of (3.1)*

^{n }*where*

_{λ }*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*for 1 ≤

_{i }*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*= 1,

_{i }*(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*≤ 1

_{i }*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*])

*be any solution of (3.1)*

^{n }*where*

_{λ }*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

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

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*= 1,

_{i }*(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*])

*be any solution of (3.1)*

^{n }*where*

_{λ }*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
*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 3.6-3.11), 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*])

*be any solution of (3.1)*

^{n }*where*

_{λ }*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

*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
*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*])

*be any solution of (3.1)*

^{n }*where*

_{λ }*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

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*])

*be any solution of (3.1)*

^{n }*where*

_{λ }*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
*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*+ 1

_{i }< γ_{i }*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*])

*be any solution of (3.1)*

^{n }*where*

_{λ }*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