Study on the existence of solutions for a generalized functional integral equation in L1 spaces

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R E S E A R C H

Open Access

Study on the existence of solutions for a

generalized functional integral equation in

L



spaces

Lijuan Yang

1

_{, Jing Wang}

1

_{and Ganshan Yang}

2,3*

*_{Correspondence:} [email protected] 2_{Department of Mathematics,} Yunnan Nationalities University, Kunming, 650092, People’s Republic of China

3_{Institute of Mathematics, Yunnan} Normal University, Kunming, 650092, People’s Republic of China Full list of author information is available at the end of the article

Abstract

Using a nonlinear alternative theorem of Krasnosel’skii type proved recently by Smaïl Djebali and Zahira Sahnoun, we investigate, in this paper, the existence of solutions for a generalized mixed-type functional integral equation inL1_{space. We also present}

some examples of the integral equation to conﬁrm the eﬃciency of our results. MSC: 47H10; 45A05

Keywords: measure of weak noncompactness; ﬁxed-point theorem; superposition operator; nonlinear functional-integral equation

1 Introduction

The functional integral equations describe many physical phenomena in various areas of natural science, mathematical physics, mechanics and population dynamics [–]. The theory of integral equations is developing rapidly with the help of tools in functional analy-sis, topology and fixed-point theory (see, for instance, [–]) and it serves as a useful tool, in turn, for other branches of mathematics, for example, for differential equations (see [, ]). A fixed point theorem, frequently used to solve integral equations, is a theorem proved by Krasnosel’skii in  (see, for instance, [, ]). The Krasnosel’skii theorem asserts thatA+Bhas a fixed point in a closed, convex nonempty subsetMofXifA,B

satisfy the following conditions:

• Ais compact and continuous; • Bis a strict contraction; • AM+BM⊆M.

However, the Krasnosel’skii fixed point theorem sometimes turns out to be restrictive for some equations due to the weak topology of the problem. In order to use this result and its variant, one has to find a self-mapped closed convex setMso thatA+BmapsMinto itself or the weaker one:x=Ax+By(y∈M)⇒x∈M. From the application point of view, this condition is also generally strict and is hard to achieve. To relax these conditions, a new effort is made in [] by establishing a new variant of nonlinear Krasnosel’skii type fixed point theorem for nonself maps.

Let us ﬁrst recall the nonlinear alternative Krasnosel’skii ﬁxed point theorem established in [], which plays a central role in our discussion.

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Theorem  Let Sbe an open subset of a Banach space X and let S be the closure of S.

Let A:S→X and B:X→X be two mappings satisfying:

• Ais continuous,A(S)is relatively weakly compact,andAveriﬁes the conditionH. • Bis a contraction and veriﬁes the conditionH.

Then either the equation Ax+Bx=x admits a solution in S,or there exists an element x∈∂S(∂S denotes the boundary of S)such that x=λAx+λB(x_λ)for someλ∈(, ),where conditionsHandHare given in Section.

The advantage of Theorem  lies in that in applying Theorem , one does not need to verify that the involved operator maps a closed convex subset onto itself.

In this paper, we utilize the alternative Theorem  and employ the concept of measure of weak noncompactness deﬁned in [] to study the solvability of a nonlinear generalized mixed-type operator equation of the form

y(t) =gt,y(t)+Tf

t,

ut,s,fs,Ay(s)ds

, ()

wheret∈,gis a function satisfying a contraction condition with respect to the second variable, y belongs toL₍_,_X_{), the space of Lebesgue integrable functions on a}

measur-able subsetofRn_{with values in a ﬁnite-dimensional Banach space}_X_{, while}_T_and_A_are

bounded linear operators onL(,X). Suppose thatNfis the superposition operator

gen-erated by the functionf given by (Nfx)(t) =f(t,x(t)),t∈, andUis a nonlinear Urysohn

integral operator deﬁned by (Ux)(t) =u(t,s,x(s))ds,s,t∈, then Eq. () may be written in the form

y(t) =gt,y(t)+ (TNfUNfAy)(t). ()

The outline of this paper is as follows. In Section , we introduce some basic facts and use them to obtain our aims in Section . In the last section, we present some examples that verify the application of this kind of nonlinear integral equation.

2 Preliminaries

2.1 The weak MNC

We always use (X, · ) to denote a Banach space with the norm · . Denote byB(X) the collection of all nonempty bounded subsets of X and byW(X) the subset ofB(X) consisting of all weakly compact subsets ofX. LetBrbe the closed ball inXcentered at

origin with radiusr. The measure of weak noncompactness introduced by De Blasi is a mapω:B(X)→[,∞) deﬁned by

ω(M) =infr> |there exists aW∈W(X) withM⊆W+Br

for eachM∈B(X).

The following Lemma  comes from [].

Lemma  Let M,M∈B(X).Then we have:

(i) M⊆Mimpliesω(M)≤ω(M).

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(iii) ω(M_w) =ω(M)(whereMw_ is the weak closure ofM). (iv) ω(M∪M) =max{ω(M),ω(M)}.

(v) ω(λM) =|λ|ω(M)for allλ∈R.

(vi) ω(co(M)) =ω(M)(co(M)refers to the convex hull ofM).

(vii) ω(M+M)≤ω(M) +ω(M).

The mapω(·)is called the De Blasi measure of weak noncompactness.

In [], it is shown that in theL_space,_ω₍_·_{) is of the following simple form:}

ω(M) =lim sup ε→

sup ψ∈M

D

ψ(t)_Xdt|D⊂,meas(D)≤ε

()

for all boundedM⊂L(,X), wheremeas(·) represents the Lebesgue measure,X is a ﬁnite-dimensional Banach space.

LetJbe a nonlinear operator fromXinto itself. In what follows, we need the following two conditions:

H. If(xn)n∈Nis a weakly convergent sequence inX, then(Jxn)n∈Nhas a strongly convergent subsequence inX;

H. If(xn)n∈N is a weakly convergent sequence inX, then(Jxn)n∈N has a weakly convergent subsequence inX.

2.2 The superposition operator

In this subsection, we introduce the superposition (Nemytskii’s) operator. Let be a bounded domain ofRn_{and let}_X_and_Y _{be two separable Banach spaces.}_m₍_,_X₎

de-notes the set of all measurable functionsψ:→X. Consider a functionf :×X→Y. We say thatf satisﬁesCarathéodory conditionsif

(i) for anyx∈X, the mapt→f(t,x)is measurable fromtoY; (ii) for almost allt∈, the mapx→f(t,x)is continuous fromXtoY.

Deﬁnition  (Nemytskii’s operator) Letf :×X →Y be a Carathéodory function, Nemytskii’s operator associated with f,Nf :m(,X)→m(,Y) is deﬁned byNfx(t) = f(t,x(t)),∀t∈.

The superposition operator enjoys several nice properties. Speciﬁcally, we have the fol-lowing results.

Lemma [] Let X and Y be two separable Banach spaces.If f is a Carathéodory func-tion,then Nemytskii’s operator Nf maps continuously L(,X)into L(,Y)if and only if there exist a constant b> and a function a(·)∈L

+()such that

f(t,x)_Y≤a(t) +bxX,

where L₊()stands for the positive cone of the space L().

Lemma [] Let X,Y be two ﬁnite-dimensional Banach spaces and letbe a bounded domain ofRn.If f :×X→Y is a Carathéodory function and Nf maps L(,X)into L₍_,_Y_),_{then N}

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We give a ﬁxed point lemma for bilinear forms.

Lemma  Let X be a Banach space and let B:X×X→X be a bilinear map.Let · X

denote the norm in X.If for all x,x∈X,B(x,x)X≤ηxXxX.Then for all y∈ X satisfyingηyX< ,the equation x=y+B(x,x)has a solution x∈X satisfying and uniquely deﬁned by the conditionxX≤ yX.

Remark  The proof of this lemma also shows thatx=limk→∞xk, where the approximate

solutionsxkare deﬁned byx=yandxk=y+B(xk–,xk–). Moreover,xkX≤yXfor

allk.

3 Main results

In this section, we investigate the solvability of the nonlinear functional integral Eq. () in the spaceL₍_,_X_{) by applying Theorem .}

First notice that Eq. () may be written in the abstract form

y=Ay+By,

where (By)(t) =g(t,y(t)), andA=TNfUNfAis the composition of the linear operatorT andAwith the nonlinear Urysohn integral operatorUand the two superposition opera-torsNf,Nf generated byf,f, respectively, whereNfiy(t) =fi(t,y(t)),i= , . Our aim is

to prove thatA+Bhas a ﬁxed point inL₍_,_X_{). To do so, we assume that the following}

conditions are satisﬁed:

(a) The functiong:×X→Xis a measurable function,g(·, )∈L₍_,_X₎_and_g_{is a}

contraction with respect to the second variable,i.e., there exists anL∈[, )such thatg(t,x) –g(t,y) ≤Lx–yfor almost allt∈and allx,y∈X.

(b) fi:×X→X,i= , satisfyCarathéodory conditionsandNfi,i= , , act from L₍_,_X₎_{into itself continuously.}

(c) The operatorsTandAare linear and bounded onL₍_,_X₎_.

(d) The Urysohn operatorUdeﬁned as before maps continuouslyL₍_,_X₎_into L₍_,_X₎_.

(e) u(t,s,x) ≤κ(t,s){ξ(s) +μx}for(t,s)∈_and_x_∈_X_{, where}_ξ_{belongs to}_L +(),

μis a nonnegative constant andκ:×→R+is a measurable function such that its associated integral operatorKdeﬁned by

(Kρ)(t) =

κ(t,s)ρ(s)ds, ρ∈L(),t∈, ()

is continuous and mapsL()into itself.

(f ) There exists a constantN> independent ofλ∗∈(, )such that any solution of the integral equation

y(t) =λ∗g

t,  λ∗y(t)

+λ∗TNfUNfAy(t), t∈,

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Remark  It is deserved to mention that though the Urysohn operatorUmapsL₍_,_X₎

into itself, it does not have to be continuous. Suﬃcient conditions showing thatUmaps

L₍_,_X_{) into itself and is continuous can be found in [].}

Before going on, we give crucial Lemma .

Lemma  Let X be a finite-dimensional Banach space and letbe a compact subset ofRn_. If the conditions(b)-(e)are satisfied,then the operator NfUNfA satisfies the conditionH.

Proof For any nonempty subsetDof, letεbe an arbitrary positive real number. We have

D

NfUNfAy(t)dt

= D f t,

ut,s,fs,Ay(s)dsdt

≤

D

a(t)+b

ut,s,fs,Ay(s)ds

dt

≤ aL₍_D₎+b

D

κ(t,s)ξ(s) +μfs,Ay(s)ds

dt

≤ aL₍_D₎+bK

D

ξ(s) +μfs,Ay(s)ds

≤ a_L₍_D₎+bK

ξ_L +(D)+μ

D

a(s)+bAy(s)ds

≤ aL₍_D₎+bK

ξ_L +(D)+μ

aL₍_D₎+bA

D

y(s)ds

=aL₍_D₎+bKξ_L

+(D)+μaL(D)

+μbbKA

D

y(s)ds.

Now using reference [, Corollary , p.] together with (), we have

lim sup ε→

D

_a₍_t₎+ξ(t)+μa(t)dtmeas(D)≤ε

= . ()

Accordingly,

ω(NfUNfAS)≤μbbKAω(S) ()

for any bounded subsetSofL(,X).

Next, let (yn)n∈Nbe a weakly convergent sequence ofL(,X). Owing to (), we infer that

ω{NfUNfA(yn) :n∈N}= . This shows that the set{NfUNfA(yn) :n∈N}is relatively

weakly compact inL₍_,_X_{). This completes the proof.}

Remark  Due to the assumption (a), we get

By=

_By(t)dt

≤

gt,y(t)–g(t, )dt+

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≤L

y(t)dt+

l(t)dt

=l_L₍₎+L

y(t)dt,

wherel(t) =g(t, ) ∈L₊().

This shows that the operatorBis continuous and maps a bounded set ofL₍_,_X_{) into a}

bounded set ofL₍_,_X_{). According to Lemma , we obtain}_B_{satisﬁes the condition H.}

Now we are in a position to state our main result.

Theorem  Let X be a ﬁnite-dimensional Banach space and letbe a bounded domain ofRn_._{Assume that the conditions}_(a)_-_(f)_{hold true}_._{Then Eq}_{. ()}_{admits at least one solution} in L₍_,_X_).

Proof We apply Theorem  with

S=y∈L(,X) :yL₍_,_X₎<N .

Claim . Letx,y∈L₍_,_X_{). It follows from the assumption (a) that}

_B₍_x_{) –}_B₍_y₎

L₍_,_X₎ =

_g

t,x(t)–gt,y(t)_Xdt

≤L

x(t) –y(t)_Xdt

=Lx–y_L₍_,_X₎.

So,Bis a strict contraction mapping onL₍_,_X_{), and from Remark ,}_B_{satisﬁes the}

con-dition H.

Claim . Clearly, by the assumptions (b)-(d),A=TNfUNfAis continuous onL(,X). Now we check thatAsatisﬁes the condition H. To do this, let (yn)n∈N be a weakly

con-vergent sequence ofL(,X). By Lemma , (NfUNfA(yn))n∈N has a weakly convergent subsequence, say (NfUNfA(ynk))k∈N. Furthermore, the continuity of the linear

opera-tor T implies its weak continuity on L₍_,_X_{) for almost all}_t_∈_{. Thus, the sequence}

(TNfUNfA(ynk))k∈N,i.e., (A(ynk))k∈N converges pointwisely for almost allt∈. By the

Vitali convergence theorem [, p.], we conclude that (A(ynk))k∈N converges strongly

inL₍_,_X_{). Hence,}_A_{satisﬁes the condition H.}

Claim . We show thatA(S) is relatively weakly compact. For this, we need to prove that

ωA(S)=lim sup ε→

sup

y∈S

D

_Ay(t)_Xdtmeas(D)≤ε

= 

for allD⊆, and∀y∈S. By Lemma , we have

D

_A_y(t)_Xdt

=

D

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≤ T

D

≤ T

a_L₍_D₎+bK

ξ_L

+(D)+μaL(D)

+μbbKA

D

y(s)ds

≤ TaL₍_D₎+bKξ_L

+(D)+μaL(D)

+μbbKAN.

Owing to (), we deduce thatω(A(S)) = , and henceA(S) is relatively weakly compact. Finally, thanks to the assumption (f ), the second situation of Theorem  does not occur. Now, applying Theorem , we get thatA+Bhas a ﬁxed point inS, that is to say, Eq. ()

has a solution inS. This completes the proof.

Remark  The requirement thatXshould be a ﬁnite-dimensional Banach space comes from the usage of the relation () proved in [] for bounded subsets in the space of Lebesgue integrable functions with values in a ﬁnite-dimensional Banach space.

By Theorem , we can get a special existence criterion for Eq. ().

Corollary  Let X be a ﬁnite-dimensional Banach space and letbe a bounded domain ofRn_._{Besides the assumptions}_(a)_-_(e),_{we make the following additional assumptions}_:

(i) There exists a continuous functionh: [,∞)→[,∞)such thath(u) > whenever u> and

_N_f

UNfAy(t)Xdt≤h

yL₍_,_X₎

for everyy∈L(,X).

(ii)

sup θ∈[,∞)

( –L)θ l_L

++Th(θ)

> ,

wherel(t) :=g(t, )andLis the constant in the assumption(a).Then Eq. ()has a solution inL₍_,_X₎_.

Proof Thanks to Theorem , we only need to show that (i) and (ii) imply (g). LetN>  satisfy

( –L)N

l_L

++Th(N)

> . ()

The condition (ii) ensures the existence of such anN. Lety∈L₍_,_X_{) be any solution of}

the operator equation

y=λ∗Ay+λ∗B

y

λ∗

, λ∗∈(, ). ()

Then, fort∈, we have the estimate

y(t)≤λ∗g

t,y(t) λ∗

–g(t, )+λ∗g(t, )+λ∗TNfUNfAy(t)

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and so

y(t)dt≤L

y(t)dt+

l(t)dt+Thy.

Therefore

( –L)yL₍_,_X₎ l_L

++Th(yL(,X))

≤. ()

Assuming thatyL₍_,_X₎=N. Equation () implies (–L)N

l_L ++Th(N)

≤ contradicting (). So, each solution of () satisﬁesy_L

+=N. Accordingly, by Theorem , Eq. () has a solution

y∈L(,X). This completes the proof.

Corollary  Let X be a ﬁnite-dimensional Banach space and letbe a bounded domain ofRn_._{Assume that hypotheses}_(a)_-_(f)_{hold true with the additional assumption that}

(iii) L+μbbKA< ,where the constantsb,b,μare these in Lemmaand the hypothesis(f),Kdenotes the norm of the operatorKdeﬁned in().

Then Eq. ()has a solution in L₍_,_X_).

Proof Lety∈L₍_,_X_{). By the formula of Lemma , we have}

_Ay(t)dt

≤ T

≤ TaL₍_D₎+b_K

ξ_L

+(D)+μaL(D)

+μbbKAyL₍_,_X₎

.

On the other hand, with the same arguments as in the proof of Corollary , we have the following estimate:

yL₍_,_X₎≤Ly_L₍_,_X₎+l_L ++λ

∗_A_y

≤LyL₍_,_X₎+l_L ++λ

∗_T_δ_y

L₍_,_X₎,

whereδ(γ) =aL₍_D₎+bK(ξ_L

+(D)+μaL(D)) +μbbKAγ. For the sake of simplicity, we can setaL₍_D₎+bK(ξ_L

+(D)+μaL(D)) =ν. Hence

 –L–μbbTKA yL₍_,_X₎≤ l_L

++Tν. ()

Let

N> lL++Tν  –L–μbbTKA.

IfyL₍_,_X₎=N, then () implies that

N≤ lL++Tν  –L–μbbTKA

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which is a contradiction. So, the hypothesis (g) is satisﬁed and the result then follows from

Theorem . This completes the proof.

4 Examples

In this section, we provide some examples of a classical integral and functional equation considered in nonlinear analysis which are a particular case of Eq. ().

Example  The existence of solutions of the equation

x(t) =gt,x(t)+λ

k(t,s)fs,x(s)ds

has been investigated in [] by this method under proper assumptions. We denote that it is a special case of Eq. () withT=C, whereCis the Fredholm operator deﬁned as

C:L(,Y)→L(,X),

ψ→Cψ:→X; Cψ(t) =

k(t,s)ψ(s)ds.

Example  The following equation proposed in []

ψ(t) =gt,ψ(t)+ (BNfUAψ)(t)

is also a special case of Eq. () withT=Bandf(t,y) =y.

Example  The solvability of the nonlinear integral equation of a mixed type

x(t) =g(t) +





k(t,s)f

s,

s



k(s,τ)fτ,x(τ)dτ

ds, t∈(, )

is discussed in the spaceL(, ) in []. If letX=R,T=K(deﬁned in []) andu(t,s,x) =

k(t,s)xwith= [, ], then the above equation is a particular case of Eq. () and it is applied to solve fractional order integro-diﬀerential equations

y(t) =g(t) +





k(t,s)f

s,

s



(s–τ)–β ( –β)y(τ)dτ

ds, t∈(, ).

Example  Consider the following integral equation of the form

x(t) =  

texp(–t) +tx(t)+T

ln( +t)

 +t +

t



exp(–s) exp(t) + 

exp(s) +sins+ x(s)ds

with ≤s≤t≤.

Let us takeg: [, ]×R→R,fi: [, ]×R→R,i= ,  andu: [, ]×[, ]×R→R

deﬁned by, respectively,

g(t,x) =  

texp(–t) +tx,

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f(t,y) =sint+ y,

u(t,s,x) = exp(–s) exp(t) + 

exp(s) +x.

We can suppose T to be an arbitrary linear bounded operator on L[, ]. It is easy to see that the functiong satisﬁes the assumption (a) withL= _, the function|u(t,s,x)| ≤

exp(–s)

exp(t) (exp(s) +x) withk(t,s) = exp(–s)

exp(t) andμ= .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

LY carried out the main work of this paper; JW and GY participated in work on the part content and modiﬁed work of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematical Sciences, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China.}

2_{Department of Mathematics, Yunnan Nationalities University, Kunming, 650092, People’s Republic of China.}3_{Institute of} Mathematics, Yunnan Normal University, Kunming, 650092, People’s Republic of China.

Acknowledgements

The work is supported by the National Natural Science Foundation of China (No. 10861014, No. 11161057).

Received: 4 November 2012 Accepted: 24 April 2013 Published: 8 May 2013 References

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doi:10.1186/1029-242X-2013-235

Cite this article as:Yang et al.:Study on the existence of solutions for a generalized functional integral equation inL1