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R E S E A R C H Open AccessExistence results of Brezis-Browder type for systems of Fredholm integral equations Ravi P Agarwal1,2*, Donal O ’Regan3 and Patricia JY Wong4 * Correspondence:

R E S E A R C H Open Access

Existence results of Brezis-Browder type for

systems of Fredholm integral equations

Ravi P Agarwal1,2*, Donal O ’Regan3

and Patricia JY Wong4

* Correspondence:

Agarwal@tamuk.edu

1 Department of Mathematics, Texas

A&M University - Kingsville,

Kingsville, TX 78363, USA

Full list of author information is

available at the end of the article

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 thefirst system in (C[0, T])n, whereas for the second system, the existence criteria aredeveloped separately in (Cl[0,∞))nas well as in (BC[0,∞))n

For both systems, wefurther seek the existence of constant-sign solutions, which include positive solutions(the usual consideration) as a special case Several examples are also included toillustrate the results obtained

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

Keywords: system of Fredholm integral equations, Brezis-Browder arguments, stant-sign solutions

as well as in(BC[0,∞))n

Here, BC[0,∞) denotes the space of functions that are bounded and tinuous on [0,∞) and Cl[0,∞) = {x Î BC[0, ∞) : limt® ∞x(t) exists}

con-We shall also tackle the existence of constant-sign solutions of (1.1) and (1.2) Asolution u of (1.1) (or (1.2)) is said to be of constant sign if for each 1≤ i ≤ n, we have

© 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

θiui(t)≥ 0 for all t Î [0, T] (or t Î [0,∞)), where θiÎ {-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 tions of the nonlinear Fredholm integral equations:

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 resultssuch as Krasnosel’skii’s fixed point theorem in a cone, Leray-Schauder alternative, Leg-

gett-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

stu-dies of [5,12-23] Indeed, we have generalized the problems to (i) systems; (ii) more

generalform 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 fixedpoint 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

, tively In Section 6, we seek the existence of constant-sign solutions of (1.1) and (1.2)

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

com-pact maps, whereas Theorem 2.2 is the criterion of comcom-pactness 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 with0 Î U andS : U → Eis a continuous and compact

map Then either

(a) S has a fixed point in U, or(b) there exist u Î ∂U and l Î (0, 1) such that u = lSu

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

fol-(a) P is bounded in Cl[0,∞)

(b) Any y Î 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(∞)| <εfor any t≥ T(ε) and y Î P

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

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

Our first existence result uses Theorem 2.1

Theorem 3.1 For each 1 ≤ i ≤ n, assume (C1)- (C4) hold where(C1) hiÎ C[0, T], denote Hi≡ suptÎ [0, T]|hi(t)|,

(C2) fi: [0, T] ×ℝn® ℝ is aL qi-Carathéodory function:

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

(ii) the map t a fi(t, u) is measurable for all u Îℝn

;(iii) for any r > 0, there existsμ r,i ∈ L qi [0, T]such that|u|≤ r implies |fi(t, u)|≤ μr,i

(t) for almost all t Î [0, T];

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

Note that S maps (C[0, T])ninto (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 Î (C[0, T])n, there exits r > 0 such that ||u|| <r Since

fi is aL qi-Carathéodory function, there existsμ r,i ∈ L qi [0, T]such that |fi(s, u)|≤ μr,i(s)

for almost all s Î [0, T] Hence, for any t1, t2 Î [0, T], we find for 1 ≤ i ≤ n,

, or equivalently Sium ® Siu in C[0, T], 1≤ i ≤ n

There exists r > 0 such that ||um||, ||u|| <r Since fi is aL qi-Carathéodory function,

there exists μ r,i ∈ L qi [0, T]such that |fi(s, um)|, |fi(s, u)|≤ μr,i(s) for almost all s Î [0,

T] Using a similar argument as in (3.4), we get for any t1, t2 Î [0, T] and 1 ≤ i ≤ n:

|S i u m (t1)− S i u m (t2)| → 0 and |Si u(t1)− S i u(t2)| → 0 (3:5)

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

|S i u m (t) − S i u(t) | ≤ |S i u m (t) − S i u m (t1)| + |S i u m (t1)− S i u(t1)| + |S i u(t1)− S i u(t)| → 0 (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 Î Ω We need to show that

SiΩ is relatively compact for 1 ≤ i ≤ n Clearly, SiΩ is uniformly bounded, since there

existsμ r,i ∈ L qi [0, T]such that |fi(s, u)| ≤ μr,i(s) for all u ÎΩ and a.e s Î [0, T], and

Our subsequent results will apply Theorem 3.1 To do so, we shall show that anysolution u of (3.1)lis bounded above This is achieved by bounding the integral of |fi(t,

u(t))| (or|f i (t, u(t))| ρi) on two complementary subsets of [0, T], namely {t Î [0, T] : ||u

(t)||≤ r} and {t Î [0, T] : ||u(t)|| >r}, where riand r are some constants–this

techni-que 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 ≤

u i (t)f i (t, u(t)) ≥ rα i |f i (t, u(t))| for ||u(t)|| > r and a.e t ∈ [0, T].

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) Î (C[0, T])nbeany solution of (3.1)lwhere l Î (0, 1)

[0, T] such that |fi(t, u(t))|≤

μr,i(t) Thus, we get

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

i||∞

(||μr,i||1+ k i)≡ l i

(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 ≤i≤nli≡ L It follows from Theorem 3.1 (with M = L + 1) that

(C8) there exist r >0 and ai>0 with rai> Hi+ aisuch that for any u Î(C[0, T])n,

u i (t)f i (t, u(t)) ≥ rα i |f i (t, u(t))| for ||u(t)|| > r and a.e t ∈ [0, T].

Then, (1.1) has at least one solution in (C[0, T])n.ProofThe proof follows that of Theorem 3.2 until (3.12) Let 1≤ i ≤ n We use (C7)

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

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

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

u i (t)f i (t, u(t)) ≥ β i ||u(t)|| · |f i (t, u(t)) | for ||u(t)|| > r and a.e t ∈ [0, T].

Then, (1.1) has at least one solution in (C[0, T])n.Proof Let u = (u1, u2, , un) Î (C[0, T])nbe any solution of (3.1)lwhere l Î (0, 1)

I0 ={t ∈ [0, T] : ||u(t)|| ≤ r0} and J0 ={t ∈ [0, T] : ||u(t)|| > r0 }.

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

Since τi≤ 1, there exists a constantk i such that(β i r0− H i − a i2τ i)

t ∈[0,T] ||g t

i|| ∞

(||μr0,i|| 1+ k i)≡ l i

||u(t)|| ≥ η i |f i (t, u(t)| γi+φ i (t) for ||u(t)|| > r and a.e t ∈ [0, T],

(C12) there exist ai≥ 0, 0 <τi<gi+ 1, bi, andψ i ∈ L γi γi+1[0, T]withψi≥ 0 almost where on[0, T], such that for any u Î (C[0, T])n,

Also, jiÎ C[0, T],h

i ∈ L γi γi+1[0, T],ψiÎ C[0, T] andT

0

|g i (t, s)|γi γi+1ds ∈ C[0, T].Then, (1.1) has at least one solution in (C[0, T])n

Proof Let u = (u1, u2, , un) Î (C[0, T])nbe any solution of (3.1)lwhere l Î (0, 1)

Define the sets I and J as in (3.9) Let 1≤ i ≤ n Applying (C10) and (C11), we get

Since γi1+1< 1andγi τi+1 < 1, there exists a constant kisuch that

where we have used (3.28) and (C12) in the last inequality, and liis 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

||u(t)|| ≥ η i |f i (t, u(t)|γi+φ i (t) for ||u(t)|| > r and a.e t ∈ [0, T].

Then, (1.1) has at least one solution in (C[0, T])n.Proof Let u = (u1, u2, , un) Î (C[0, T])nbe any solution of (3.1)lwhere l Î (0, 1)

Define the sets I and J as in (3.9) Let 1 ≤ i ≤ n If t Î I, then by (C2), there exists

μ r,i ∈ L qi [0, T]such that |fi(t, u(t))|≤ μr,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

Now, an application of Hölder’s inequality gives

Proof Let u = (u1, u2, , un) Î (C[0, T])nbe any solution of (3.1)lwhere l Î (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

Then, (1.1) has at least one solution in (C[0, T])n.Proof Let u = (u1, u2, , un) Î (C[0, T])nbe any solution of (3.1)lwhere l Î (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

Sinceγi1+1 < 1andγi τi+1 < 1, there exists a constant kisuch 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< gi+ 1 and eisuch that for any u Î(C[0, T])n,

Then, (1.1) has at least one solution in (C[0, T])n.Proof Let u = (u1, u2, , un) Î (C[0, T])nbe any solution of (3.1)lwhere l Î (0, 1)

Define the sets I and J as in (3.9) Let 1 ≤ i ≤ n As before, we see that (C10) and

Now, it is clear that

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

Proof Let u = (u1, u2, , un) Î (C[0, T])nbe any solution of (3.1)lwhere l Î (0, 1).

Define the sets I and J as in (3.9) Let 1 ≤ i ≤ n As before, we see that (C10) and

+|e i | ≡ k

i

(3:49)

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

Then, (1.1) has at least one solution in (C[0, T])n.Proof Let u = (u1, u2, , un) Î (C[0, T])nbe any solution of (3.1)lwhere l Î (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

Since γi1+1< 1andγi τi+1 < 1, 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 3.1In Theorem 3.5, the conditions (C10) and (C11) can be replaced by thefollowing, which is evident from the proof

(C10)’ There exist r >0 and bi>0 such that for any u Î (C[0, T])n,

u i (t)f i (t, u(t)) ≥ β i |u i|0· |f i (t, u(t)) | for ||u(t)|| > r and a.e t ∈ [0, T],

where we denote|u i|0= sup

t ∈[0,T] |u i (t)|.(C11)’ There exist r >0, hi>0, gi>0 andφ i ∈ L γi γi+1[0, T]such that for any u Î (C[0,

T])n,

|u i|0≥ η i |f i (t, u(t)| γi+φ i (t) for ||u(t)|| > r and a.e t ∈ [0, T].

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, hi>0, gi>0, andφ i ∈ L pi [0, T]such that for any u Î (C[0, T])

n

,

|u i|0≥ η i |f i (t, u(t))|γi+φ i (t) for ||u(t)|| > r and a.e t ∈ [0, T].

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

Let the Banach space B = (Cl[0,∞))n

be equipped with the norm:

||u|| = max

where we let |ui|0 = suptÎ[0,∞)|ui(t)|, 1≤ i ≤ n Throughout, for u Î B and t Î [0,

∞), we shall denote that

-Carathéodory function, i.e.,

(i) the map u a fi(t, u) is continuous for almost all t Î [0,∞),(ii) the map t a fi(t, u) is measurable for all u Îℝn

,(iii) for any r >0, there existsμr,iÎ L1

[0, ∞) such that |u| ≤ r implies |fi(t, u)|≤ μr,i

(t) for almost all t Î [0,∞)

(D3) g t (s) = g i (t, s) ∈ L∞[0,∞)for each t Î[0,∞),(D4) the mapt → g t

iis continuous from[0,∞) to L∞[0, ∞),(D5) there exists ˜g i ∈ L∞[0,∞)such thatg t → ˜g iin L∞[0,∞) as t ® ∞, i.e.,lim

Clearly, the system (1.2) is equivalent to u = Su, and (4.1)lis the same as u = lSu

First, we shall show that S : (Cl[0,∞))n® (Cl[0,∞))n

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

is continuous Let {um} be asequence in (Cl[0,∞))n

[0, ∞) such that |fi(s, um)|, |fi(s, u)|≤ μr,i(s) for almost all s Î[0,∞) Denote Siu(∞) ≡ limt®∞Siu(t) and Sium(∞) ≡ limt®∞ Sium(t) In view of (4.8),

as t ®∞ Similarly, we also have that

Combining (4.10)-(4.12), we have

|S i u m (t) − S i u(t) | → 0 as t → ∞ and m → ∞

or equivalently, there exist ˆT > 0such that

It remains to check the convergence in [0, ˆT] As in (4.4), we find for any

|S i u m (t1)− S i u m (t2)| → 0 and |S i u(t1)− S i u(t2)| → 0,

|S i u m (t1)− S i u m (t2)| → 0 and |S i u(t1)− S i u(t2)| → 0 (4:14)

as t1 ® t2 Furthermore, Sium(t) ® Siu(t) pointwise on[0, ˆT], since, by the dominated convergence theorem,

as m ® ∞ Combining (4.14) and (4.15) and the fact that[0, ˆT]is compact yields

Coupling (4.13) and (4.16), we see that Sium® Siuin 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 Î 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,

(D2)’ fi: [0, ∞) × ℝn® ℝ is aL qi-Carathéodory function, i.e.,(i) the map u a fi(t, u) is continuous for almost all t Î [0,∞),(ii) the map t a fi(t, u) is measurable for all u Îℝn

,(iii) for any r >0, there existsμ r,i ∈ L qi[0,∞)such that |u| ≤ r implies |fi(t, u)|≤ μr,i

(t) for almost all t Î [0,∞),

(D3)’g t (s) = g i (t, s) ∈ L pi[0,∞), for each t Î [0,∞),(D4)’ the mapt → g t

iis continuous from [0,∞) toL pi[0,∞),(D5)’ there exists ˜g i ∈ L pi[0,∞)such thatg i t → ˜g i, inL pi[0,∞)as t ®∞, i.e.,