R E S E A R C H
Open Access
Solvability of integrodifferential problems
via fixed point theory in
b
-metric spaces
Monica Cosentino
1, Mohamed Jleli
2, Bessem Samet
2*and Calogero Vetro
1*Correspondence:
2Department of Mathematics,
College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
Full list of author information is available at the end of the article
Abstract
The purpose of this paper is to study the existence of solutions set of
integrodifferential problems in Banach spaces. We obtain our results by using fixed point theorems for multivalued mappings, under new contractive conditions, in the setting of completeb-metric spaces. Also, we present a data dependence theorem for the solutions set of fixed point problems.
MSC: 47H10; 34A60
Keywords: b-metric space; differential inclusion; fixed point; multivalued mapping
1 Introduction
Measure theory is a classical topic in mathematical analysis which is usually studied in the setting of real and complex numbers and functions. Indeed, measures have applications in the foundations of integration, probability and ergodic theories. On the other hand, the-ory of multivalued mappings has an important role in various branches of mathematics because of its applications in optimal control problems involving integrodifferential inclu-sions. As a matter of fact, the theory of integrodifferential equations and inclusions has undergone rapid development over the last decades (see, for instance, [] and the refer-ences therein). Indeed, this theory has increased its significance in modern applied math-ematical models of real processes arising in many engineering and scientific disciplines such as physics, biology, economics, signal processing and data fitting.
Notice that in the literature there are many papers focusing on the solution of differential problems approached via fixed point theory (see, for example, [–] and the references therein). On the other hand, it is well known that metric spaces and their generalizations furnish an useful tool for the study of multivalued mappings. In this regard, Nadler [] was the first author who combined the ideas of contractions and multivalued mappings by providing a fixed point existence result.
Theorem .([]) Let(X,d)be a complete metric space and let T:X→CB(X)be a mul-tivalued mapping satisfying H(Tx,Ty)≤kd(x,y)for all x,y∈X,where k is a constant such that k∈(, ),and CB(X)denotes the family of non-empty,closed and bounded subsets of X.Then T has a fixed point,that is,there exists a point u∈X such that u∈Tu.
Later on, many generalizations, extensions and applications of this theorem have ap-peared in the literature (see, for instance, [–]). In this literature review, we start from
looking at the paper of Wardowski [] who introduced a new concept of contraction, calledF-contraction. Consequently, Wardowski proved fixed point theorems generalizing the Banach-Caccioppoli fixed point theorem (of which Theorem . is the multivalued ver-sion) in a new way than in the previous known theorems of the same class. Subsequently, Sgroi and Vetro [] extended Wardowski’s ideas to the case of multivalued mappings and studied the solution of certain functional and integral equations under a suitable set of hypotheses. Yet another inspiration for our work comes from Feng and Liu’s paper [] providing useful tools to establish both global and local fixed point theorems. Finally, we recall that the concept of metric space has been generalized in many directions to include measurements in a much more general sense. Here, we focus our attention on the notion ofb-metric spaces, which are metric spaces satisfying a relaxed form of triangle inequality, see Czerwik [] and Bakhtin []. Several researchers followed the idea of Czerwik and proved interesting results [–].
In this paper, we study the existence of solutions for certain integral problems of Fred-holm type in Banach spaces. Also, we present a data dependence theorem for the solutions set of fixed point problems. We obtain our results by using fixed point theorems for mul-tivalued mappings, under new contractive conditions, in the setting of completeb-metric spaces. Clearly, the presented theorems extend well-known results in the literature to b-metric spaces.
2 Preliminaries
In this section, we collect some basic definitions, lemmas and notations which will be used throughout the paper (see [, , , ] and the references therein). LetR+denote the set of all nonnegative real numbers andNdenote the set of positive integers.
Definition . LetXbe a non-empty set and lets≥ be a given real number. A func-tiond:X×X→R+is said to be ab-metric if and only if for allx,y,z∈Xthe following conditions are satisfied:
() d(x,y) = if and only ifx=y; () d(x,y) =d(y,x);
() d(x,z)≤s[d(x,y) +d(y,z)].
Then the triplet (X,d,s) is called ab-metric space.
Clearly, a (standard) metric space is also ab-metric space, but the converse is not always true.
Example . LetX= [, ] andd:X×X→R+ be defined byd(x,y) =|x–y| for all
x,y∈X. Clearly, (X,d, ) is ab-metric space that is not a metric space.
Again, let (X,d,s) be ab-metric space. The following notions are natural deductions from the corresponding metric versions:
(i) A sequence{xn} ⊆Xconverges tox∈Xiflimn→+∞d(xn,x) = .
(ii) A sequence{xn} ⊆Xis said to be a Cauchy sequence if, for every givenε> , there exists a positive integern(ε)such thatd(xm,xn) <εfor allm,n≥n(ε).
(iii) Ab-metric space(X,d,s)is said to be complete if and only if each Cauchy sequence converges to somex∈X.
Example .([]) Letp∈(, ). Then the setlp(R) :={{x n} ⊂R:
∞
n=|xn|p<∞} en-dowed with the functionald:lp(R)×lp(R)→Rgiven by
d{xn},{yn}
= ∞
n=
|xn–yn|p /p
for all{xn},{yn} ∈lp(R) is ab-metric space withs= /p.
Example .([]) Letp∈(, ). Then the spaceLp([, ]) of all real functionsf : [, ]→
Rsuch that|f(t)|pdt<∞endowed with the functionald:Lp([, ])×Lp([, ])→R given by
d(f,g) =
f(t) –g(t)pdt /p
for allf,g∈Lp([, ]) is ab-metric space withs= /p.
Now, we give a brief background for multivalued mappings defined in ab-metric space (X,d,s). ForA,B∈CB(X), define the functionH:CB(X)×CB(X)→R+by
H(A,B) =maxδ(A,B),δ(B,A),
where
δ(A,B) =supd(a,B),a∈A, δ(B,A) =supd(b,A),b∈B
with
d(a,C) =infd(a,x),x∈C.
Note thatHis called the Hausdorffb-metric induced by theb-metricd. We recall the following properties from [, , ].
Lemma . Let(X,d,s)be a b-metric space.For any A,B,C∈CB(X)and any x,y∈X,we
have the following:
(i) d(x,B)≤d(x,b)for anyb∈B; (ii) δ(A,B)≤H(A,B);
(iii) d(x,B)≤H(A,B)for anyx∈A; (iv) H(A,A) = ;
(v) H(A,B) =H(B,A);
(vi) H(A,C)≤s[H(A,B) +H(B,C)]; (vii) d(x,A)≤s[d(x,y) +d(y,A)].
Lemma . Let(X,d,s)be a b-metric space.For A∈CB(X)and x∈X,we have
d(x,A) = ⇐⇒ x∈A=A,
where A denotes the closure of the set A.
We conclude this section with two useful lemmas.
Lemma . Let(X,d,s)be a b-metric space and let{xn}be a sequence in X.Iflimn→+∞xn= y andlimn→+∞xn=z,then y=z.
Lemma . Let(X,d,s)be a b-metric space and let{xn}be a sequence in X such that
d(xn,xn+)≤λd(xn–,xn)
for someλ∈(,s–)and each n∈N.Then{x
n}is a Cauchy sequence in X.
3 Fixed point theory inb-metric spaces
3.1 Wardowski type theorem
We study the existence of fixed points for multivalued mappings by adapting the ideas in [] to theb-metric setting. The motivation of this research is to solve certain classes of integrodifferential problems. First, inspired by Wardowski [], we give the following definitions.
Definition . Lets≥ be a real number. We denote byFsthe family of all functions F:R+→Rwith the following properties:
(F) Fis strictly increasing;
(F) for each sequence{αn} ⊂R+of positive numberslimn→+∞αn= if and only if
limn→+∞F(αn) = –∞;
(F) for each sequence{αn} ⊂R+of positive numbers withlimn→+∞αn= , there exists k∈(, )such thatlimn→+∞(αn)kF(αn) = ;
(F) for each sequence{αn} ⊂R+of positive numbers such thatτ+F(sαn)≤F(αn–)
for alln∈Nand someτ∈R+, thenτ+F(snαn)≤F(sn–αn–)for alln∈N.
Example . LetF:R+→Rbe defined byF(x) =x+lnx. Clearly,F satisfies (F)-(F).
Here we show only (F).
Assume that, for alln∈Nand someτ∈R+, we haveτ+sαn+ln(sαn)≤αn–+lnαn–.
Sincex+lnxis an increasing function, thensαn<αn–. Thus
sn–– sαn+lnsn–≤
sn–– αn–+lnsn–
implies that
τ+snαn+lnsnαn=τ+sαn+
sn–– sαn+lnsn–+lnsαn
≤αn–+
sn–– αn–+lnsn–+lnαn–
=sn–αn–+ln
sn–αn–
and hence (F) holds true. Note that alsoF:R+→Rdefined byF(x) =lnxsatisfies
(F)-(F).
Definition . Let (X,d,s) be ab-metric space. A multivalued mappingT:X→CB(X)
is called anF-contraction of Nadler type if there existF∈Fsandτ ∈R+such that
τ+FsH(Tx,Ty)≤Fd(x,y)
for allx,y∈XwithTx=Ty.
Now, we are ready to state and prove our first main theorem.
Theorem . Let(X,d,s)be a complete b-metric space and let T:X→CB(X).Assume
that there exists a continuous from the right function F∈Fsandτ∈R+such that
τ+FsH(Tx,Ty)≤Fd(x,y) (.)
for all x,y∈X,Tx=Ty.Then T has a fixed point.
Proof Letx∈Xbe an arbitrary point ofXand choosex∈Tx. Clearly, ifx∈Tx, we
deduce thatx is a fixed point ofT and so we can conclude the proof. Now, we assume
thatx∈/Txand henceTx=Tx. SinceF∈Fsis continuous from the right, there exists a real numberh> such that
FhsH(Tx,Tx)
<FsH(Tx,Tx)
+τ.
Next, fromd(x,Tx) <hH(Tx,Tx), we deduce that there existsx∈Tx(obviously,x=
x) such thatd(x,x)≤hH(Tx,Tx). Therefore, we can write
Fsd(x,x)
≤FshH(Tx,Tx)
<FsH(Tx,Tx)
+τ,
which implies
τ+Fsd(x,x)
≤τ+FsH(Tx,Tx)
+τ
≤Fd(x,x)
+τ.
Consequently, we get
τ+Fsd(x,x)
≤Fd(x,x)
.
Iterating this procedure, we construct a sequence{xn} ⊂Xsuch thatxn∈/Txn,xn+∈Txn and
τ+Fsd(xn+,xn+)
≤Fd(xn,xn+)
In order to simplify the reading of calculations, letdn=d(xn,xn+) > for alln∈N∪ {}.
It follows by (.) and property (F) that
τ+Fsnd(xn,xn+)
≤Fsn–d(xn–,xn)
for alln∈N∪ {}. (.) Thus, by (.), we have
Fsndn
≤Fsn–dn–
–τ≤ · · · ≤F(d) –nτ for alln∈N, (.)
and so
lim
n→+∞F
sndn
= –∞,
which in view of property (F) gives
lim
n→+∞s nd
n= .
Now, by property (F) there existsk∈(, ) such that
lim
n→+∞
sndn
k Fsndn
= .
By (.), for alln∈N, we get
sndn k
Fsndn
–sndn k
F(d)≤
sndn
k
F(d) –nτ
–sndn
k F(d)
= –nτsndn k
≤. (.)
Passing to limit asn→+∞in (.), we obtain
lim
n→+∞n
sndn k
=
and hencelimn→+∞n/ksnd
n= . Now, the last limit implies that the series +∞
n=sndnis convergent and hence {xn} is a Cauchy sequence. Since (X,d,s) is a completeb-metric space, then there existsu∈Xsuch thatlimn→+∞xn=u. Finally, we prove thatuis a fixed point ofT, that is,u∈Tu.
Firstly, we observe that if there exists an increasing sequence{nk} ⊂Nsuch thatxnk∈Tu
for allk∈N, sinceTuis closed andlimk→+∞xnk=u, we deduceu∈Tuand hence the proof
is completed. Then we assume that there existsn∈Nsuch thatxn∈/Tufor alln∈Nwith n≥n. It follows thatTxn–=Tufor alln≥n.
Now, using (.) withx=xnandy=u, we obtain
τ+FsH(Txn,Tu)
≤Fd(xn,u)
,
which implies
τ+Fd(xn+,Tu)
≤τ+FsH(Txn,Tu)
≤Fd(xn,u)
SinceFis strictly increasing andτ ∈R+, we obtain
d(xn+,Tu) <d(xn,u).
Also, we have
d(u,Tu)≤sd(u,xn+) +d(xn+,Tu)
≤sd(u,xn+) +sd(xn,u),
and passing to limit asn→+∞in the previous inequality, we get
d(u,Tu)≤,
which impliesd(u,Tu) = . Finally, sinceTuis closed, we obtain thatu∈Tu, that is,uis a
fixed point ofT.
As an application of Theorem . we get the following proof of Nadler’s fixed point theorem inb-metric spaces [].
Theorem . Let(X,d,s)be a complete b-metric space and let T:X→CB(X).Assume that there exists k∈(, )such that
sH(Tx,Ty)≤kd(x,y) (.)
for all x,y∈X.Then T has a fixed point.
Proof Letτ ∈R+be such thatk=e–τ. From (.), for allx,y∈XwithTx=Ty, we get
FsH(Tx,Ty)≤–τ+Fd(x,y),
that is,
τ+FsH(Tx,Ty)≤Fd(x,y),
whereF(x) =lnx. Thus we can apply Theorem . to deduce thatThas a fixed point.
Remark . LetCL(X) be the family of non-empty and closed subsets ofX. Notice that Theorems . and . hold also in the case of a multivalued mappingT:X→CL(X).
3.2 Feng-Liu type theorems
Let (X,d,s) be ab-metric space and letT :X→CL(X) be a multivalued mapping. Let
Fix(T) :={x∈X:x∈Tx}denote the fixed point set ofT. Also, define the functionfT:X→
RasfTx=d(x,Tx). Then, for a positive constantα∈(, ) and eachx∈X, define the set
Iαx:=y∈Tx:αd(x,y)≤d(x,Tx).
Definition . Let (X,d,s) be ab-metric space and letT:X→CL(X) be a multivalued mapping. A function f :X→Ris calledT-lower semicontinuous if, for each{xn} ⊂X withxn+∈Txnandlimn→+∞xn=x∈X, we have
fx≤lim inf
n→+∞fxn.
Definition . LetT:X→CL(X) be a multivalued mapping. The graph ofTis the subset
{(x,y) :x∈X,y∈Tx}ofX×X; we denote the graph ofT by G(T). ThenT is a closed multivalued mapping if the graphG(T) is a closed subset of (X×X,d∗), where the metric d∗is given byd∗((x,y), (u,v)) =d(x,u) +d(y,v) for all (x,y), (u,v)∈X×X.
Now, we state and prove the following theorem.
Theorem . Let(X,d,s)be a complete b-metric space and let T:X→CL(X)be a mul-tivalued mapping.Suppose that there exists r∈(,s–α)withα∈(, )such that for any x∈X there is y∈Ix
αsatisfying
d(y,Ty)≤rd(x,y). (.)
Then T has a fixed point in X provided that one of the following conditions holds: (i) fTisT-lower semicontinuous,
(ii) T is closed.
Proof SinceTxis a non-empty closed set for anyx∈X,Ix
αis non-empty for any constant
α∈(, ). Now, for a fixed pointx∈X, there existsx∈Iαxsuch that
d(x,Tx)≤rd(x,x).
Ifxis not a fixed point ofT, we choosex∈Iαxsuch that
d(x,Tx)≤rd(x,x).
Again, ifxis not a fixed point ofT(and so on), by iterating this procedure, we can get an
iterative sequence{xn}, wherexn+∈Iαxnand
d(xn+,Txn+)≤rd(xn,xn+) for alln∈N∪ {}. (.)
On the other hand,xn+∈Iαxnimplies
The next step of the proof is to show that the sequence{xn}is a Cauchy sequence. Using (.) and (.), we get
d(xn+,xn+)≤
r
αd(xn,xn+) for alln∈N∪ {}. (.)
Sincer/α<s–, by Lemma . we deduce that{x
n}is a Cauchy sequence and so, by com-pleteness of theb-metric space (X,d,s),{xn}converges to someu∈X. Now we claim that xis a fixed point ofT. Therefore, we distinguish two cases.
Case: Suppose that (i) holds true. Again, by (.) and (.), we get
d(xn+,Txn+)≤
r
αd(xn,Txn) for alln∈N∪ {},
which implies
d(xn,Txn)≤ r
α
n
d(x,Tx) for alln∈N∪ {}.
Consequently,
lim inf
n→+∞fTxn=n→lim+∞fTxn=n→lim+∞d(xn,Txn) = .
Sincexn+∈Txn,fTisT-lower semicontinuous andlimn→+∞xn=u, we have
fTu=d(u,Tu) = .
SinceTuis closed, we get thatu∈Tu, that is,uis a fixed point ofT. Case: If (ii) holds true, then fromxn+∈Txnfor alln∈N∪ {}and
lim
n→+∞d ∗(x
n,xn+), (u,u)
= lim
n→+∞
d(xn,u) +d(xn+,u)
= ,
we get that (u,u)∈Gr(T) and henceu∈Tu. Thusuis a fixed point ofT.
This completes the proof.
Now, we show that Theorem . is a generalization of the following version of Nadler’s fixed point theorem inb-metric spaces.
Theorem . Let(X,d,s)be a complete b-metric space and let T:X→CL(X)be a mul-tivalued mapping such that for all x,y∈X,we have H(Tx,Ty)≤rd(x,y),where r∈(,s–), then T has a fixed point.
Proof We have to show that all the hypotheses of Theorem . hold true. Firstly, we prove thatTsatisfies condition (.) of Theorem .. Indeed, for allx∈Xandy∈Tx, we write
d(y,Ty)≤H(Tx,Ty)≤rd(x,y)
and hence the assertion holds trivially for eachx∈Xandy∈Iαxwithα∈(, ) such that r<αs–. It would remain to show thatf
semicontinuous. Indeed, let{xn} ⊂X be a sequence withxn+ ∈Txnandlimn→+∞xn= x∈X. Clearly, we have
d(x,Tx)≤sd(x,xn+) +H(Txn,Tx)
≤sd(x,xn+) +rd(xn,x)
,
and hence, passing to limit asn→+∞, we getfTx= . This implies that
fTx≤lim inf n→+∞fTxn.
This completes the proof.
Finally, we give a local version of Theorem ..
Theorem . Let(X,d,s)be a complete b-metric space,x∈X,R> and let T:X→
CL(X)be a multivalued mapping.Suppose that there exists r∈(,s–α)withα∈(, )
such that for any x∈B(x,R)there is y∈Iαxsatisfying
d(y,Ty)≤rd(x,y).
If d(x,Tx)≤αs( –srα)R,then T has a fixed point in B(x,R)provided one of the following
conditions holds:
(i) fTisT-lower semicontinuous, (ii) T is closed.
Proof Proceeding as in the proof of Theorem ., we construct an iterative sequence{xn} with initial pointx, withxn+∈Ixαn and satisfying the conditions (.)-(.) for alln∈
N∪ {}. From (.) andd(x,Tx)≤αs( –srα)R, we obtain
d(xn,xn+)≤
r
α
n d(x,x)
≤ rn
αn+d(x,Tx)
≤ r
α
n s –
sr
α
R
for alln∈N∪ {}.
This impliesxn∈B(x,R). Indeed, we have
d(x,xn)≤ n
k=
skd(xk–,xk)
≤
n
k=
sk r
α
k–
s –
sr
α
R
< –sr
α
R
+∞
k=
sr
α
k–
=R,
and soxn∈B(x,R). Thus, following the proof of Theorem ., it is easy to show thatT
4 Existence of solution for integral inclusions of Fredholm type
In this section, we study the solvability of integral inclusions of Fredholm type. Precisely, we present an existence result of solution under general conditions on multivalued oper-ators. For more on the solution of integral inclusions and related problems, the reader is referred to [, , ] and the references therein.
Now, we consider the following integral inclusion of Fredholm type:
x(t)∈f(t) +
Gt,s,x(s)ds for allt∈[, ], (.)
whereG: [, ]×[, ]×R→Kcv(R) is a multivalued operator, whereKcvdenotes the family of non-empty compact and convex subsets ofR.
LetI= [, ] and letC(I,R) be the space of all continuous functionsf:I→R. It is well known that such a space with the metric given by
d(x,y) =sup
t∈I
x(t) –y(t)=(x–y)∞ for allx,y∈C(I,R)
is a completeb-metric space withs= .
Adapting an idea in [], we prove the following theorem.
Theorem . Suppose that the following conditions hold:
(i) for eachx∈C(I,R),the multivalued operatorG:I×I×R→Kcv(R)is such that G(t,s,x(s))is lower semicontinuous inI×I;
(ii) f∈C(I,R);
(iii) there existsl(t,·)∈L(I),for eacht∈Iandsupt∈Il(t,s)ds≤
k
withk∈(, ), such that
HG(t,s,x),G(t,s,y)≤l(t,s)x(s) –y(s)
for allt,s∈Iand for allx,y∈R.
Then the integral inclusion(.)has at least one solution in C(I,R). Proof Let us define the multivalued operatorT:C(I,R)→CL(C(I,R)) by
Tx(t) =
v∈C(I,R) such thatv(t)∈f(t) +
Gt,s,x(s)ds,t∈I
for eachx∈C(I,R). Letx∈C(I,R) and denoteGx(t,s) :=G(t,s,x(s)),t,s∈I. For the mul-tivalued operatorGx:I×I→Kcv(R), by Michael’s selection theorem, we get that there exists a continuous operatorgx:I×I→Rsuch thatgx(t,s)∈Gx(t,s) for allt,s∈I. This implies thatf(t) +gx(t,s)ds∈Txand soTxis a non-empty set. It is an easy matter to show thatTxis closed, and so details are omitted (see also []). This implies thatTxis closed in (C(I,R),d).
we get
HGt,s,x(s),Gt,s,y(s)≤l(t,s)x(s) –y(s)
for allt,s∈Iand for allx,y∈R. Consequently, there existsw(t,s)∈Gy(t,s) such that gx(t,s) –w(t,s)≤l(t,s)x(s) –y(s)
for allt,s∈I. Now, we can consider the multivalued operatorSdefined by
S(t,s) =Gy(t,s)∩
u∈Rsuch thatgx(t,s) –u≤l(t,s)x(s) –y(s)
for allt,s∈I. Taking into account the fact that the multivalued operatorGis lower semi-continuous, it follows that there exists a continuous operator gy:I×I→Rsuch that gy(t,s)∈S(t,s) for allt,s∈I. We have
z(t) =f(t) +
gy(t,s)ds∈f(t) +
Gt,s,y(s)ds, t∈I
and
v(t) –z(t)≤
gx(t,s) –gy(t,s)ds
≤
l(t,s)x(s) –y(s)ds
≤
l(t,s)x(s) –y(s)ds
≤
l(t,s)(x–y)
∞ds
≤(x–y)∞
l(t,s)ds
for allt∈I. Thus,d(v,z)≤kd(x,y). Interchanging the roles ofxandy, we obtain that
H(Tx,Ty)≤kd(x,y)
for allx,y∈C(I,R). Thus, all the conditions of Theorem . are satisfied and hence we
deduce the existence of a solution of (.).
Remark . Consider the following differential inclusion:
x(t)∈Gt,s,x(s), t,s∈I, (.)
Example . For allx∈C(I,R) andt,s∈I, letG(t,s,x(s)) ={ν∈R:g(t,s,x(s))≤ν≤
g(t,s,x(s))}, whereg(t,s,x(s)) is upper semicontinuous inI×Iandg(t,s,x(s)) is lower
semicontinuous inI×I. Consider the following differential inclusion:
x(t)∈Gt,s,x(s), t,s∈I
and assume that there existsl(t,·)∈L(I), for eacht∈Iandsup
t∈I
l(t,s)ds≤
√
, such
that
maxg
t,s,x(s)–g
t,s,y(s)≤l(t,s)x(s) –y(s)
for allt,s∈Iand for allx,y∈R.
Clearly, the multivalued operatorGis compact and convex valued. Thus, all the hypothe-ses of Theorem . are satisfied withf(t) = , and hence the above two-point boundary value problem has at least one solution.
5 Stability of solutions set for fixed point problems
We study data dependence of solutions set for fixed point problems by using the technique presented in Section .. Indeed, in view of Theorem ., we prove a data dependence theorem of the fixed points set for two multivalued mappings.
Theorem . Let (X,d,s) be a complete b-metric space and let S,T : X→CL(X) be two multivalued mappings such that supx∈XH(Sx,Tx) < +∞. Suppose that there exists r∈(,s–α)withα∈(, )such that for any x∈X there is y∈Ix
αsatisfying
d(y,Sy)≤rd(x,y) and d(y,Ty)≤rd(x,y).
Then we have
HFix(S),Fix(T)≤ s
α–rssupx∈X
H(Sx,Tx)
provided that one of the following conditions holds:
(i) fSandfTare,respectively,S-lower andT-lower semicontinuous, (ii) SandT are closed.
Proof By Theorem . we deduce thatFix(S) andFix(T) are non-empty sets. Also, notice thatFix(S) andFix(T) are closed. Indeed, for instance, let{un} ⊂Fix(S) be a sequence such thatlimn→+∞un=u. Then, ifSis closed, the conclusion follows easily. On the other hand, iffSx:=d(x,Sx) isS-lower semicontinuous, then we have
≤d(u,Su) =fSu≤lim inf
n→+∞fSun=lim infn→+∞d(un,Sun) = .
It follows thatu∈Su, that is,u∈Fix(S). The same holds forFix(T).
Let us considerx∈Fix(S). Hence, there existsx∈Iαxwithd(x,Tx)≤rd(x,x). Now,
sinceαd(x,x)≤d(x,Tx), we obtain
Iterating this process, we can construct an iterative sequence{xn}such that (i) x∈Fix(S),
(ii) d(xn,Txn)≤(rα–)nd(x,Tx),
(iii) d(xn,xn+)≤α–(rα–)nd(x,Tx).
From (iii), we deduce that{xn}is a Cauchy sequence and therefore it converges to an ele-mentu∈X. From (ii), by following the same lines of proof as in Theorem ., we get that u∈Fix(T). Again, ifm>nfrom
d(xn,xm)≤
s(rα–)n
α–rs d(x,Tx),
lettingm→+∞, we deduce
d(xn,u)≤
s(rα–)n
α–rs d(x,Tx)
for eachn∈N∪ {}. Then, forn= , we get d(x,u)≤
s
α–rsd(x,Tx)
≤ s
α–rsH(Sx,Tx)
≤ s
α–rssupx∈X
H(Sx,Tx).
In a similar way we can prove that, for eachy∈Fix(T), there existsv∈Fix(S) such that
d(y,v)≤
s
α–rssupx∈X
H(Sx,Tx).
Thus, the proof is complete.
Building on Theorem . and dealing with a sequence of multivalued mappings, we ob-tain the following result.
Theorem . Let(X,d,s)be a complete b-metric space and let Ti:X→CL(X)be a se-quence of multivalued mappings.Suppose that there exists r∈(,s–α)withα∈(, )such
that for any x∈X there is y∈Ix
αsatisfying d(y,Tiy)≤rd(x,y)andlimi→+∞H(Tix,Tx) =
uniformly,where i∈N∪ {}.Then we have
lim
i→+∞H
Fix(Ti),Fix(T)
= .
Proof Letε> be arbitrary and chooseN∈Nsuch that
sup
x∈X
H(Tix,Tx) <
α–rs s ε
fori≥Nand for eachx∈X. Consequently, by Theorem . we get
HFix(Ti),Fix(T)
<ε
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Author details
1Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi, 34, Palermo, 90123, Italy. 2Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia.
Authors’ information
Calogero Vetro is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project No. IRG14-04.
Received: 3 January 2015 Accepted: 22 April 2015 References
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