R E S E A R C H
Open Access
Fixed-point theorems for the sum of two
operators under
ω
-condensing
Fuli Wang
**Correspondence: [email protected]
School of Mathematics and Physics, Changzhou University, Changzhou, 213164, China
Abstract
The purpose of this paper is to establish fixed-point theorems for the sum of two operatorsAandB, where the operatorAis assumed to be contractive with respect to the measure of weak noncompactness, whileBis an
ϕ
-nonlinear contraction. In the last section, we apply such results to study the existence of solutions to a nonlinear Hammerstein integral equation inL1space.Keywords: measure of weak noncompactness; weak condensing;
ϕ
-nonlinearcontraction; nonlinear Hammerstein integral equations
1 Introduction
The existence of fixed points for the sum of two operators has been followed with interest for a long time. In , to study the existence of solutions of nonlinear equations of the form
Ax+Bx=x, x∈M,
Krasnosel’skii [] first proved operatorA+Bhas a fixed point wheneverMis a nonempty closed convex subset of Banach spaceXand the operatorsAandBsatisfy:
(i) Ais continuous onM, andA(M)is relatively compact, (ii) Bis ak-contraction withk∈[, ),
(iii) A(M) +B(M)⊂M.
In , Darbo [] extended the Schauder fixed-point theorem to the setting of non-compact operators, introducing the notion ofk-set contraction. It is not hard to see that the Krasnosel’skii theorem is a particular case of the Darbo theorem. Namely, it appears thatA+Bis ak-set contraction with respect to the Kuratowskii measure of noncompact-ness. In , Sadovskii [] gave a fixed-point result more general than the Darbo theorem using the concept of condensing operator.
In , De Blasi [] introduced the concept of measure of weak noncompactness. Emmanuele [] established a Sadovskii-type fixed-point result using the concept of ω-condensing with respect to the measure of weak noncompactness, in which the weak continuity of the operator is required. Recently, Garcia-Falset and Latrach established a new version of Sadovskii-type fixed-point theorem for the weakly sequentially continuous operators (see [, Lemma .]).
On the other hand, since the weak continuity condition is usually not easy to verify, Latrachet al.[, ] established generalizations of the Schauder, Darbo and Krasnosel’skii
fixed-point theorems for the weak topology. Their analysis uses the concept of the Blasi measure of weak noncompactness. Moreover, and in contrast to previous works, to prove the new versions of the fixed-point theorems, they neither assume the weak continuity nor the weakly sequential continuity of the operators.
The purpose of this paper is to establish several fixed-point theorems for the sum of two operators underω-condensing. By relaxing the condition of weak compactness of operators, these results extend and supplement some previous ones in the literatures.
This paper is organized as follows. In Section , we gather some notions and prelimi-nary facts which will be needed in our further considerations. In Section , on the basis of a Sadovskii-type fixed-point theorem for ω-condensing operators and its variant on whole space, we discuss several fixed-point theorems for the sum ofA+B, whereAis a ω-contraction andBis anϕ-nonlinear contraction. In Section , we apply such results to study the existence of solutions to a nonlinear Hammerstein integral equation inLspace.
2 Preliminaries
We first gather together some notations and preliminary facts of some weak topology fea-ture which will be needed in our further considerations. LetB(X) be the collection of all nonempty bounded subsets of a Banach spaceX, and letW(X) be the subset ofB(X) con-sisting of all weakly compact subsets ofX. Also, letBrdenote the closed ball inXcentered
inand with radiusr.
De Blasi [] introduced the mapω:B(X)→R+defined by
ω{M}=infr> :∃W∈W(X) :M⊂W+Br
forM∈B(X).
Before we launch into the details, we recall some important properties needed hereafter for the sake of completeness (for the proofs, we refer the reader to [] and []).
Lemma . Let M,Mand Mbe inB(X);we have:
(a) ω{M} ≤ω{M}wheneverM⊂M.
(b) ω{M}= if and only ifMw∈W(X),whereMwis the weak closure ofM. (c) ω{Mw}=ω{M}.
(d) ω{co(M)}=ω{M}whereco(M)refers to the convex hull ofM. (e) ω{λM}=|λ|ω{M},for allλ∈R.
(f ) ω{M+M} ≤ω{M}+ω{M}.
(g) ω{M∪M}=max{ω{M},ω{M}}.
(h) If(Mn)∞n=is a decreasing sequence of nonempty,bounded and weakly closed subsets
ofXwithlimn→∞ω{Mn}= ,thenlimn→∞
∞
k=Mn=ϕandω{
n
k=Mn}= ,i.e.,
∞
k=Mnis relatively weakly compact.
The mapω{·}is called the De Blasi measure of weak noncompactness. In [], Appell and De Pascale proved that inL-spacesω{·}has the following form:
ω{M}=lim sup ε→
sup ψ∈M
D
ψ(t)Xdt:meas(D)≤ε (.)
Throughout this paper,Xdenotes a Banach space;D(T) andR(T), respectively, denote the domain and range of operator T; andω{S} denotes the De Blasi measure of weak noncompactness of bounded subsetS.
Definition . An operator T :D(T)⊂X→Xis said to beω-contractive (or a ω-α -contraction) if it maps bounded sets into bounded sets, and there exists someα∈[, ) such thatω{T(S)} ≤αω{S}for all bounded setsSinD(T).
An operatorT:D(T)⊂X→Xis said to beω-condensing if it maps bounded sets into bounded sets, andω{T(S)}<ω{S}for all bounded setsSinD(T) withω{S}> .
Remark . Obviously, everyω-α-contraction with ≤α< isω-condensing.
LetT be an operator fromD(T)⊂XintoX. Latrachet al.[] introduce the following conditions:
(A) If(xn)n∈Nis a weakly convergent sequence inD(T), then(Txn)n∈Nhas a strongly
con-vergent subsequence inX.
(A) If(xn)n∈Nis a weakly convergent sequence inD(T), then(Txn)n∈Nhas a weakly
con-vergent subsequence inX.
The conditions (A) and (A) were already considered in [, , –].
Remark .
(a) Operators satisfying either (A) or (A) are not necessarily weakly continuous. (b) An operator satisfies (A) if and only if it maps relatively weakly compact sets into
relatively strongly compact ones.
(c) An operator satisfies (A) if and only if it maps relatively weakly compact sets into relatively weakly compact ones (Eberlein-Šmulian theorem, see,e.g.,
[, pp.-]).
(d) Everyω-contractive operator satisfies (A).
(e) The condition (A) holds true for every bounded linear operator.
Lemma . Let operator T :D(T)⊂X→X satisfy(A),and let operator Q:R(T)⊂
X→X be continuous.Then the compound operator Q◦T:D(T)⊂X→X satisfies(A).
Proof Let (xn)n∈Nbe a weakly convergent sequence inD(T). By the hypothesis ofT
satis-fying (A), (Txn)n∈Nhas a strongly convergent subsequence, say (Txnk)k∈N. The continuity
ofQimplies that (QTxnk)k∈Nis also strongly convergent, and thereforeQ◦Tsatisfies (A).
Definition . An operatorT:D(T)⊂X→Xis said to beϕ-nonlinear contractive (or anϕ-nonlinear contraction), if there exists a continuous and nondecreasing functionϕ: R+→R+such that
Tx–Ty ≤ϕx–y for allx,y∈D(T),
whereϕ(r) <rforr> .
Lemma .(see [, Lemma .]) Let operator T:D(T)⊂X→X beϕ-nonlinear con-tractive on a Banach space X and satisfy(A).Then for each bounded subset M of X one hasω{T(M)} ≤ϕ(ω{M}).
Lemma . If an operator T:X→X isϕ-nonlinear contractive,then F:=I–T is a
home-omorphism of X onto X.
Proof For anyy∈X, we define the operatorτyfromXtoX, byτyx=Tx+y. SinceT is
ϕ-nonlinear contractive, it is easy to see thatτyis alsoϕ-nonlinear contractive. According
to Theorem in [],τyhas a unique fixed pointxsuch thatx=Tx+y,i.e. y= (I–T)x,
and thenF:=I–Tis surjective onX. Ifx,y∈Xandx=y, then
Fx–Fy=(x–Tx) – (y–Ty)≥ x–y–Tx–Ty ≥ x–y–ϕx–y> ,
which implies thatFis injective andF–exists onX.
For proving the continuity of F–, suppose that there exists a pointxand a sequence (xn)n∈NinXsuch thatFxn→Fx, andlimn→∞supk≥nxk–x=a. Consequently, from the
inequality
Fxn–Fx ≥ xn–x–ϕ
xn–x
,
we obtain that ≥a–ϕ(a), which implies thata= and, therefore,F–is continuous.
3 Fixed-point theorems for the sum of two operators
The following theorem was proved by Ben Amar and Garcia-Falset [], and its more gen-eral form was presented by Agarwalet al.[] is a variant of the Sadovskii fixed-point theorem for the classes of operators which satisfy (A).
Theorem .(see [, Theorem .] or [, Theorem . and Corollary .]) Let M be a nonempty,bounded,closed and convex subset of a Banach space X.Assume that T:M→ M is continuous and satisfies(A).If T isω-condensing,then it has a fixed point in M.
Our purpose here is to establish a fixed-point theorem for the sum of aω-contractive operator and anϕ-nonlinear contractive operator.
Theorem . Let M be a nonempty, bounded, closed and convex subset of a Banach space X.Suppose that A:M→X and B:X→X are two operators such that
(i) Ais a continuousω-α-contraction withα∈[, ),andAsatisfies(A),
(ii) Bis anϕ-nonlinear contraction withϕ(r) < ( –α)rforr> ,andBsatisfies(A), (iii) (x=Bx+Ay,y∈M)⇒x∈M.
Then there is a point x∈M such that Ax+Bx=x.
For anyy∈M, according to Lemma . there exists a uniquex∈Xsuch thatAy=x–Bx. The hypothesis (iii) shows thatx∈M, which implies thatA(M)⊂(I–B)(M) and, therefore,
J(M)⊂M.
Obviously, the compound operatorJis continuous sinceAand (I–B)–is continuous and by Lemma .,Jsatisfies (A). Now by referring to the formula
(I–B)–A=A+B(I–B)–A (i.e.,J=A+BJ),
for every subsetSofMwithω{S}> , we have
J(S)⊂A(S) +BJ(S).
SinceAisω-α-contractive andBsatisfiesω{BJ(S)} ≤ϕ(ω{J(S)}) by Lemma ., we have
ωJ(S)≤ωA(S)+ωBJ(S)≤αω{S}+ϕωJ(S). (.)
Now, if α = , inequality (.) becomes ω{J(S)} ≤ ϕ(ω{J(S)}), which implies that ω{J(S)}= . Otherwise, by recalling the assumption thatϕ(r) < ( –α)rforr> , inequality (.) becomes
ωJ(S)<αω{S}+ ( –α)ωJ(S), i.e.,ωJ(S)<ω{S}.
In both cases,Jis shown to beω-condensing. Now the use of Theorem . achieves the
proof.
Remark . It should be noticed to the following particular cases:
() If we takeB= , then we return the above theorem back to [, Theorem .], which is an extension of the Darbo fixed-point theorem forω-contractive operators. () If we takeα= and the functionϕ(r) =βr(≤β< ) in the above theorem, we
obtain a result which was [, Theorem .].
() If we only take the functionϕ(r) =βr(≤β< –α) in the above theorem, we obtain the following Corollary ., which is a new fixed-point theorem for the sum of two operators.
() If we only takeα= in the above theorem, we obtain the following Corollary ., which is the new version of Krasnosel’skii-type fixed-point theorems.
Corollary . Let M be a nonempty, bounded, closed and convex subset of a Banach
space X.Suppose that A:M→X and B:X→X are two operators such that
(i) Ais a continuousω-α-contraction withα∈[, ),andAsatisfies(A), (ii) Bis a strict contraction withβ∈[, –α),andBsatisfies(A), (iii) (x=Bx+Ay,y∈M)⇒x∈M.
Then there is a point x∈M such that Ax+Bx=x.
Corollary . Let M be a nonempty, bounded, closed and convex subset of a Banach space X.Suppose that A:M→X and B:X→X are two operators such that
(i) Ais a continuous,A(M)is relatively weakly compact andAsatisfies(A), (ii) Bis anϕ-nonlinear contraction,andBsatisfies(A),
(iii) (x=Bx+Ay,y∈M)⇒x∈M.
Then there is a point x∈M such that Ax+Bx=x.
Remark . The above corollary is a variant and supplement of [, Theorem .], in which the author demands that the operatorsAandBare weakly sequentially continuous, andBis strictly contractive.
Now, on the basis of Corollary ., we prove the following fixed-point theorem for the sum of a weakly-strongly continuous operator and a nonexpansive operator.
Theorem . Let M be a nonempty, bounded, closed and convex subset of a Banach space X.Suppose that A:M→X and B:X→X are two operators such that
(i) Ais weakly-strongly continuous,andA(M)is relatively weakly compact, (ii) Bis nonexpansive andω-condensing,
(iii) I–Bis demiclosed,
(iv) ifλ∈(, )andx=λBx+Ayfor somey∈M,thenx∈M.
Then there is a point x∈M such that Ax+Bx=x.
Remark .
() Recall that an operatorT:D(T)⊂X→Xis said to be demiclosed if for any sequence(xn)n∈NinD(T)thatxnxandTxn→y, thenx∈D(T)andTx=y.
() The assumption (iv) in the above theorem was first introduced in [], it is slight different with [, Corollary .] and [, Theorem .].
() In [, Theorem .] and [, Theorem .], the following condition is required: if
(xn)n∈Nis a sequence ofMsuch that(xn–Txn)n∈Nis weakly convergent, then
(xn)n∈Nhas a weakly convergent subsequence. In the above theorem, we replaced it
with theω-condensing ofB.
Proof of Theorem . For eachλ∈(, ), the operatorsAandλBfulfill the conditions of Corollary . and, therefore, there is a pointxλ∈Msuch thatxλ=λBxλ+Axλ. Now choose a sequence (λn)n∈N⊂(, ) such thatλn→. Consequently, there exists a sequence
(xn)n∈N⊂Msuch that
xn=λnBxn+Axn.
LetS={xn:n∈N}. We claim thatSis relatively weakly compact. Suppose that it is not
the case, by assumption (i) and (ii), we have
ω{S}=ω{xn:n∈N}=ω{λnBxn+Axn:n∈N}
≤λnω
B(S)+ωA(S)=λnω
B(S)<ω{S}.
This contradiction tells us that the sequence (xn)n∈Nhas a weakly convergent subsequence,
and then (I–λnkB)xnk →Ax. Since (xnk)k∈N is contained in bounded setM, andBmaps Minto a bounded set (Bis nonexpansive), thenBxnkis norm bounded. Thus, we have
( –λnk)Bxnk →. Moreover, we have
(I–B)xnk–Ax≤(I–B)xnk– (I–λnkB)xnk+(I–λnkB)xnk–Ax
= ( –λnk)Bxnk+(I–λnkB)xnk–Ax→,
that is, (I–B)xnk→Ax. By assumption (iii), we have (I–B)x=Ax, and then the proof is
achieved.
If the Banach space X is reflexive, then B is always ω-condensing on M (see, e.g., [, p.]). Moreover, if we supposed that X is uniformly convex Banach space, then
I–B:M→Xis demiclosed (see,e.g., [, pp.-]). Thus, we obtain the following consequence.
Corollary . Let M be a nonempty,bounded,closed and convex subset of a uniformly convex Banach space X.Suppose that A:M→X and B:X→X are two operators such that
(i) Ais weakly-strongly continuous, (ii) Bis nonexpansive,
(iii) ifλ∈(, )andx=λBx+Ayfor somey∈M,thenx∈M.
Then there is a point x∈M such that Ax+Bx=x.
In order to use the above results on the whole space, we first prove the following result.
Theorem . Let X be a Banach space X.Assume that the operator T:X→X be con-tinuousω-condensing and satisfies(A).Then either
(a) equationx=Txhas a solution,or
(b) the set{x∈X:x=λT(x)}is unbounded for someλ∈(, ).
Proof Choose an arbitraryR> . Define for eachx∈X
ρ(x) =
⎧ ⎨ ⎩
x, x ≤R,
R
xx, x>R.
Clearly,ρ is a continuous retraction ofXonBR. Thus, we can define the mappingTρ: BR→BRbyTρx=ρ(Tx).
Since T and ρ are continuous, obviously Tρ is also continuous. Furthermore, since
T satisfies (A) and ρ is continuous, hence Tρ also satisfies (A). We next claim that
ω{Tρ(S)}<ω{S}for anyS⊂Brwithω{S}> .
Indeed,Tρ(Br) =ρ(T(Br)). For anyx∈Br, there are two possibilities:
() Tx ≤R; in this case,ρ(Tx) =Tx∈T(S)⊂co(T(S)∪ {}).
() Tx>R; in this case,ρ(Tx) = TxRTx=TxRTx+ ( –TxR)·∈co(T(S)∪ {}).
Now, by using the properties of the measure of weak noncompactness and properties of
T, we have that
ωTρ(S)
≤ωT(S)<ω{S},
as claimed, that is,Tρis aω-condensing.
The above argument shows thatTρ:Br→Bris under the conditions of Theorem . and
thus we have that there existsx∈Brsuch thatx=Tρx. Indeed, we obtain the following results:
(a) ifTx∈BR, thenx=Tρx=ρ(Tx) =Tx, that is,Thas a fixed point; otherwise,
(b) ifTx∈/BR, thenx=Tρx=ρ(Tx) =TxRTx, that is,xis a solution of the
equationx=λTxforλ= R
Tx∈(, )andx=R.
Consequently, if there is nox∈BRsuch thatx=Txfor anyR> , then the above
ar-guments show that the set of solutions of equation x=λT(x) is unbounded for some
λ∈(, ).
We are now in a position to prove the main result on the whole space.
Theorem . Let X be a Banach space.Suppose that A,B:X→X are two operators such
that
(i) Ais a continuousω-α-contraction withα∈[, ),andAsatisfies(A),
(ii) Bis anϕ-nonlinear contraction withϕ(r) < ( –α)rforr> ,andBsatisfies(A), (iii) functionϕsatisfieslimr→+∞[r–ϕ(r)] = +∞.
Then,either
(a) the equationx=Ax+Bxhas a solution,or
(b) the set{x∈X:x=λB(x/λ) +λA(x)}is unbounded for someλ∈(, ).
Remark . Obviously, assumption (iii) in the above theorem is unnecessary whenever
α= .
Proof of Theorem. As in the proof of Theorem ., it can be seen that the compound operatorJ:= (I–B)–Ais well defined fromXintoX. Clearly,Jis continuous andJsatisfies (A) by Lemma ..
Let us prove thatJmaps bounded set into a bounded set. For any bounded setSsuch thatu,v∈(I–B)–A(S), there existx,y∈Ssuch thatu= (I–B)–Axandv= (I–B)–Ay, that is,u–Bu=Axandv–Bv=Ay. Thus, by assumption (ii) and the boundness ofA(S), we have
u–v–ϕu–v≤ Ax–Ay ≤diamA(S)< +∞. (.)
Suppose thatJ(S) = (I–B)–A(S) is unbounded,i.e., there exist sequences (u
n)n∈N and
(vn)n∈N such that un–vn →+∞, and then un–vn–ϕ(un–vn)→+∞ by
as-sumption (iii). This is a contradiction withdiam(A(S)) < +∞in (.) and, therefore,J(S) is bounded.
It is similar to that of Theorem . to prove thatω{J(S)}<ω{S}for every bounded setS
(a) the equationx=Ax+Bxhas a solution which is the solution of equationx=Jx, or (b) the set{x∈X:x=λB(x/λ) +λA(x)}={x∈X:x=λ(I–B)–Ax}is unbounded for
someλ∈(, ).
Corollary . Let X be a Banach space.Suppose that A,B:X→X are two operators such that
(i) Ais a continuousω-α-contraction withα∈[, ),andAsatisfies(A), (ii) Bis a strict contraction withβ∈[, –α),andBsatisfies(A).
Then,either
(a) the equationx=Ax+Bxhas a solution,or
(b) the set{x∈X:x=λB(x/λ) +λA(x)}is unbounded for someλ∈(, ).
Corollary . Let X be a Banach space.Suppose that A,B:X→X are two operators such that
(i) Ais a continuous,A(M)is relatively weakly compact,andAsatisfies(A), (ii) Bis anϕ-nonlinear contraction,andBsatisfies(A),
(iii) functionϕsatisfieslimr→+∞[r–ϕ(r)] = +∞.
Then,either
(a) the equationx=Ax+Bxhas a solution,or
(b) the set{x∈X:x=λB(x/λ) +λA(x)}is unbounded for someλ∈(, ).
4 Application to Hammerstein integral equations inL1space
Letbe a domain ofRn. A functionf:×X→Yis said to be a Carathéodory function
if
(i) for any fixedx∈X, the functiont→f(t,x)is measurable fromtoY; (ii) for almost anyt∈, the functionf(t,·) :X→Y is continuous.
Letm() be the set of all measurable functionsψ:→X. Iff is a Carathéodory func-tion, thenf defines an operatorNf :m()→m() byNf(ψ)(t)→f(t,ψ(t)). This operator
is called the Nemytskii operator associated tof (or the superposition operator). Regarding its continuity and weak compactness, we have the following lemma.
Lemma . Let X, Y be two finite dimensional Banach spaces.If f :×X→Y is a Carathéodory function,then the Nemytskii operatorNf maps 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
+()denotes the positive cone of the space L() (see[]or[]).
With the conditions of Lemma ., the operatorNf is obviously continuous and maps
bounded sets ofL(;X) into bounded sets ofL(;Y).
Lemma .(see [, Lemma .]) Letbe a bounded domain inRn.If f :×X→Y is a
Carathéodory function andNf maps L(;X)into L(;Y),thenNf satisfies(A).
Remark . Although Nemytskii operatorNf satisfies (A), generally it is not weakly
inLspaces (see, for instance, [, Theorem .]). The question of considering the weak sequential continuity of the Nemytskii operator acting from spaceLpto spaceLq(≤p,q<
∞) is discussed in [] and the answer is shown to be negative at least forp= .
Next, we give an example of application for Theorem . in the Banach space of inte-grable functionL(;X).
Example . We will study now the existence of solutions for the following variant of Hammerstein’s integral equation
ψ(t) =gt,ψ(s)+μ
k(t,s)fs,ψ(s)ds, (.)
inL(;X), the space of Lebesgue integrable functions on a measurable subsetofRn
with values in a finite dimensional Banach spaceX. Here,f is a nonlinear function andkis measurable, whilegis a function satisfyingϕ-nonlinear contractive condition inL(;X).
First, observe that the above problem may be written in the form
ψ=Aψ+Bψ,
whereBis the Nemytskii operator associated to the functiong(i.e.,B=Ng) fromL(;X)
intoL(;X) by
Bψ(t) =gt,ψ(t),
andA=μLNf is the product of the Nemytskii operator associated tof and the linear
integral operatorμLwhereμ∈CandLis defined fromL(;Y) intoL(;X) by
(Lψ)(t) =
k(t,s)ψ(s)ds. (.)
Let us now introduce the following assumptions.
Assumptions .
(a) f is a Carathéodory function andNf acts fromL(;X)intoL(;Y);
(b) b|μ|L< (the constantbwas introduced in Lemma .);
(c) g:×X→Xis a measurable function withg(·, )∈L(;X), and there exists a
continuous and nondecreasing functionϕ:R+→R+such that
g(t,x) –g(t,y)Xdt≤ϕx–yL(;X)
for allx,y∈X,
whereϕ(r) < ( –b|μ|L)rforr> andlimr→+∞[r–ϕ(r)] = +∞;
(d) the functionk:×→L(Y,X)is strongly measurable and the linear operatorL
defined by (.) mapsL(;Y)intoL(;X);
(f ) the solution of the integral equation
ψ(t) =λg
t, λψ(t)
+λμ
k(t,s)fs,ψ(s)ds
is bounded forλ∈(, ).
Remark . () Clearly, the condition thatlimr→+∞[r–ϕ(r)] = +∞in assumption (c) is unnecessary wheneverb|μ|L = ;
() It should be noted that assumptions (d) and (e) lead to the estimate
k(t,s)x(s)ds≤ ρL∞(;L(Y,X))xL(;Y),
and so
Lψ=
k(t,s)ψ(s)ds
X
dt≤ ρL(;L∞)ψL(;Y).
This shows that the linear operator L is continuous, hence weakly continuous from
L(;Y) intoL(;X) and thatL ≤ ρ
L(;L∞). () By assumption (c), we get
g(t,u)Xdt≤
g(t, )Xdt+ϕuX
<
g(t, )Xdt+ –b|λ|LuX,
for every u∈X. This shows that the Nemytskii operator Ng is continuous and maps
bounded sets ofL(;X) into bounded sets ofL(;X). According to Lemma ., the op-eratorBsatisfies (A).
Now we are in a position to state our main result.
Theorem . Let X and Y be two finite dimensional Banach spaces andbe a bounded domain ofRn.Assume that the conditions(a)-(f)are satisfied,then the problem(.)has
at least one solution in L(;X).
Proof Let us first observe that Lemma . implies that there area∈L
+() andb> such that
f(t,x)Y≤a(t) +bxX.
So, for any bounded subsetSofL(;X), we have
S
Nfψ(t)Ydt≤
S
a(t)dt+b
S
ψ(t)Xdt.
According to (.), this leadsω{Nf(S)} ≤bω{S}. Thus, we have
ωA(S)=ωμLNf(S)
≤b|μ|Lω(S),
On the other hand, clearlyAis continuous (see Lemma . and Remark .()). Now we check thatAsatisfies the condition (A). For this end, let (xn)n∈Nbe a weakly convergent
sequence ofL(;X). Using the fact thatNfsatisfies (A), and then (Nfxn)n∈Nhas a weakly
convergent subsequence, say (Nfxnk)k∈N. Moreover, the continuity of the linear operator Limplies its weak continuity onL(;Y). Thus, the sequence (LN
fxnk)k∈N,i.e.(Axnk)k∈N
converges pointwisely for a.e.t∈. Using Vitali’s convergence theorem, we conclude that (Axnk)k∈Nconverges strongly inL
(;X). Therefore,Asatisfies (A).
Letx(t),y(t)∈L(;X). It follows from assumption (c) that
Bx–ByL(;X)=
gt,x(t)–gt,y(t)Xdt≤ϕx(t) –y(t)L(;X)
.
So,Bisϕ-nonlinear contractive onL(;X) and from Remark .(),Bsatisfies (A). The above arguments show thatAandBsatisfy the conditions of Theorem ., and
assumption (f ) allows us to affirm that equation (.) has a solution.
Remark . Equation (.) was respectively considered in [] and [] under different
assumptions. However, our results relax some assumptions of [, Theorem .] and [, Theorem .].
Competing interests
The author declares that he has no competing interests. Acknowledgements
The author is grateful to the referee for the careful reading of the manuscript. They brought to my attention a proof of Theorem 3.1, which had been published in [13] and [14]; also, the remarks motivated the author to make several other valuable improvements.
Received: 20 September 2012 Accepted: 4 April 2013 Published: 18 April 2013 References
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