R E S E A R C H
Open Access
The existence of fixed point theorems for
partial
q-set-valued quasi-contractions in
b-metric spaces and related results
Poom Kumam
1and Wutiphol Sintunavarat
2**Correspondence:
[email protected]; [email protected]
2Department of Mathematics and
Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand
Full list of author information is available at the end of the article
Abstract
In this paper, we present a new type of set-valued mappings called partial
q-set-valued quasi-contraction mappings and give results as regards fixed points for such mappings inb-metric spaces. By providing some examples, we show that our results are real generalizations of the main results of Aydiet al.(Fixed Point Theory Appl. 2012:88, 2012) and many results in the literature. We also consider fixed point results for single-valued mapping, fixed point results for set-valued mapping in b-metric space endowed with an arbitrary binary relation, and fixed point results in a b-metric space endowed with a graph. By using our result, we establish the existence of solution for the following an integral equations:x(c) =
φ
(c) +abK(c,r,x(r))dr, where b>a≥0,x∈C[a,b] (the set of continuous real functions defined on [a,b]⊆R),φ
: [a,b]→R, andK: [a,b]×[a,b]×R→Rare given mappings.MSC: 47H10; 54H25
Keywords:
α-admissible mappings; binary relations; fixed points;
b-metric spaces; q-set-valuedα-quasi-contraction mappings
1 Introduction
The Banach contraction principle is a very popular tool of mathematics in solving many problems in several branches of mathematics since it can be observed easily and comfort-ably. In , Czerwik [] introduced the concept ofb-metric spaces and also presented the fixed point theorem for contraction mappings inb-metric spaces, that is, we have a generalization of the Banach contraction principle in metric spaces. Afterward, many mathematicians studied fixed point theorems for single-valued and set-valued mappings inb-metric spaces (see [–] and references therein).
In , Aydiet al.[] extended the concept ofq-set-valued quasi-contraction mappings in metric spaces due to Amini-Harandi [] tob-metric spaces. They also established the fixed point results forq-set-valued quasi-contraction mappings inb-metric spaces. Re-cently, Sintunavaratet al.[] introduced some set-valued mappings calledq-set-valued
α-quasi-contraction mappings and obtained fixed point results for such mappings inb -metric spaces which are generalization of the results of Aydiet al.[], Amini-Harandi [] and many works in the literature.
Inspired and motivated by several results in the literature, we introduce the class of par-tialq-set-valued quasi-contraction mappings which is the wider class of many classes in
this field. As regards this class, we study and obtain fixed point results inb-metric spaces. These results extend, unify and generalize several well-known comparable results in the existing literature. As an application of our results, we prove the fixed point theorems for a single-valued mapping and give an example to show the generality of our result. We also study the fixed point results in ab-metric space endowed with an arbitrary binary rela-tion and endowed with a graph. As applicarela-tions, we apply our result to the proof of the existence of a solution for the following an integral equation:
x(c) =φ(c) +
b
a
Kc,r,x(r)dr, (.)
whereb>a≥,x∈C[a,b] (the set of continuous real functions defined on [a,b]⊆R),
φ: [a,b]→R, andK: [a,b]×[a,b]×R→Rare given mappings.
2 Preliminaries
In this section, we give some notations and basic knowledge in nonlinear analysis andb -metric spaces. Throughout this paper,R,R+, andNdenote the set of real numbers, the
set of nonnegative real numbers, and the set of positive integers, respectively.
Definition .([]) LetXbe a nonempty set ands≥ be a given real number. A func-tionald:X×X→R+is called ab-metric if, for allx,y,z∈X, the following conditions are
satisfied:
(B) d(x,y) = if and only ifx=y;
(B) d(x,y) =d(y,x);
(B) d(x,z)≤s[d(x,y) +d(y,z)].
A pair (X,d) is called ab-metric space with coefficients.
Remark . The result is obtained that any metric space is ab-metric space withs= . Thus the class ofb-metric spaces is larger than the class of metric spaces.
Some examples of b-metric spaces are given by Berinde [], Czerwik [], Heinonen []. Some well-known examples of ab-metric which show that theb-metric space is a real generalization of metric space are the following.
Example . The set of real numbers together with the functionald:R×R→R+,
d(x,y) :=|x–y|,
for allx,y∈R, is ab-metric space with coefficients= . However, we find thatdis not a metric onXsince the ordinary triangle inequality is not satisfied. Indeed,
d(, ) >d(, ) +d(, ).
s= p–. It is easy to see that conditions (B
) and (B) are satisfied. If <p<∞, then the
convexity of the functionf(x) =xp(x> ) implies the following inequality:
a+b
p
≤
ap+bp,
that is,
(a+b)p≤p–ap+bp
holds. Therefore, for eachx,y,z∈X, we get
ρ(x,y) =d(x,y)p
≤d(x,z) +d(z,y)p
≤p–d(x,z)p+d(z,y)p = p–ρ(x,z) +ρ(z,y).
Consequently, condition (B) is also satisfied and thusρis ab-metric onX.
Example . The setlp(R) with <p< , where
lp(R) :=
{xn} ⊆R
∞
n=
|xn|p<∞
,
together with the functionald:lp(R)×lp(R)→R+,
d(x,y) :=
∞
n=
|xn–yn|p
p
,
for eachx={xn},y={yn} ∈lp(R), is ab-metric space with coefficients=
p > . We see that
the above result also holds for the general caselp(X) with <p< , whereXis a Banach
space.
Example . Letpbe a given real number in the interval (, ). The spaceLp[, ] of all
real functionsx(t),t∈[, ] such that |x(t)|pdt< , together with the functional d:
Lp[, ]×Lp[, ]→R+,
d(x,y) :=
x(t) –y(t) pdt /p
, for eachx,y∈Lp[, ],
is ab-metric space with constants= p.
Example . LetX={, , }and a functionald:X×X→R+be defined by
d(, ) =d(, ) =d(, ) = ,
and
d(, ) =d(, ) =m,
wheremis given real number such thatm≥. It easy to see that
d(x,y)≤m
d(x,z) +d(z,y),
for all x,y,z∈X. Therefore, (X,d) is ab-metric space with coefficients=m/. We find that the ordinary triangle inequality does not hold ifm> and then (X,d) is not a metric space.
Next, we give the concepts of convergence, compactness, closedness, and completeness in ab-metric space.
Definition .([]) Let (X,d) be ab-metric space. The sequence{xn}inXis called:
() convergentif and only if there existsx∈Xsuch thatd(xn,x)→asn→ ∞. In this
case, we writelimn→∞xn=x.
() Cauchyif and only ifd(xn,xm)→asm,n→ ∞.
Remark . In ab-metric space (X,d) the following assertions hold:
() a convergent sequence has a unique limit; () each convergent sequence is Cauchy;
() in general a functionalb-metricd:X×X→R+for coefficients> is not jointly
continuous in all its variables.
The following example is an example of ab-metric which is not continuous.
Example .(see []) LetX=N∪ {∞}and a functionald:X×X→R+be defined by
d(x,y) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
, x=y,
y, x=∞andy=∞,
x, x=∞andy=∞,
| x–
y|, xandyare even,
, xandyare odd andx=y, , otherwise.
It is easy to see that conditions (B) and (B) are satisfied. Also, for eachx,y,z∈X, we have
d(x,z)≤d(x,y) +d(y,z).
Therefore, (X,d) is ab-metric space onXwith coefficients= .
Next, we show thatdis not continuous. Letxn= nfor eachn∈N. It is easy to see that
d(xn,∞) =d(n,∞) =
n→, asn→ ∞,
Definition . Theb-metric space (X,d) iscompleteif every Cauchy sequence inX con-verges.
Definition .([]) LetY be a nonempty subset of ab-metric spaceX. The closureY
ofYis the set of limits of all convergent sequences of points inY,i.e.,
Y:=x∈X: there exists a sequence{xn}inYsuch that lim n→∞xn=x
.
Definition .([]) Let (X,d) be ab-metric space. A subsetY⊆Xis called:
() closedif and only if for each sequence{xn}inYwhich converges to an elementx, we
havex∈Y(i.e.Y=Y);
() compactif and only if for every sequence of element inYthere exists a subsequence that converges to an element inY;
() boundedif and only ifδ(Y) :=sup{d(a,b)|a,b∈Y}<∞.
Throughout this paper, we use the following notations of collection of subsets of ab -metric space (X,d):
P(X) :={Y|Y⊆X};
P(X) :=Y∈P(X)|Y=∅;
Pb(X) :=
Y∈P(X)|Yis bounded;
Pcp(X) :=
Y∈P(X)|Yis compact;
Pcl(X) :=
Y∈P(X)|Yis closed;
Pb,cl(X) :=Pb(X)∩Pcl(X).
Next, we give the concept of generalized functionals on ab-metric space (X,d).
Definition . Let (X,d) be ab-metric space.
() The functionalD:P(X)×P(X)→R∪ {+∞}is said to bea gap functionalif and only if it is defined by
D(A,B) =
⎧ ⎪ ⎨ ⎪ ⎩
inf{d(a,b)|a∈A,b∈B}, A=∅ =B, , A=∅=B,
+∞, otherwise.
In particular, ifx∈Xthend(x,B) :=D({x},B).
() The functionalρ:P(X)×P(X)→R∪ {+∞}is said to be anexcess generalized functionalif and only if it is defined by
ρ(A,B) =
⎧ ⎪ ⎨ ⎪ ⎩
() The functionalH:P(X)×P(X)→R∪ {+∞}is said to be aPompeiu-Hausdorff generalized functionalif and only if it is defined by
H(A,B) =
⎧ ⎪ ⎨ ⎪ ⎩
max{ρ(A,B),ρ(B,A)}, A=∅ =B, , A=∅, +∞, otherwise.
Remark . Forb-metric space (X,d), the following assertions hold:
() (Pcp(X),H)is a completeb-metric space provided(X,d)is a completeb-metric
space;
() for eachA,B∈P(X)andx∈A, we have d(x,B)≤ρ(A,B)≤H(A,B);
() forx∈XandB∈P(X), we get
d(x,B)≤d(x,b),
for allb∈B.
The following lemmas are useful for the proofs in the main result.
Lemma .([]) Let(X,d)be a b-metric space.Then d(x,A)≤sd(x,B) +H(B,A),
for all x∈X and A,B∈P(X).In particular,we have
d(x,A)≤sd(x,y) +d(y,A),
for all x,y∈X and A∈P(X).
Lemma .([]) Let(X,d)be a b-metric space and A,B∈Pb,cl(X).Then for each>
and,for all b∈B,there exists a∈A such that d(a,b)≤H(A,B) +.
Lemma .([]) Let(X,d)be a b-metric space.For A∈Pb,cl(X)and x∈X,we have
d(x,A) = ⇒ x∈A.
Lemma .([]) Let(X,d)be a b-metric space with coefficient s≥and{xn}be a
se-quence in X such that
d(xn+,xn+)≤γd(xn,xn+),
for all n∈N,where≤γ < .Then{xn}is a Cauchy sequence in X provided that sγ < .
Definition .([]) LetXbe a nonempty set,t:X→Xandα:X×X→[,∞). We say thattisα-admissible if
forx,y∈Xfor whichα(x,y)≥ ⇒ α(tx,ty)≥.
They proved the fixed point results for single-valued mapping as regards this concept and also showed that these results can be utilized to derive fixed point theorems in par-tially ordered spaces. As an application, they obtain the existence of solutions for ordinary differential equations.
Afterward, Aslet al. [] and Mohammadiet al.[] introduced the concept of α∗ -admissibility andα-admissibility for set-valued mappings as follows.
Definition .([, ]) LetXbe a nonempty set,T:X→X, where Xis a collection
of nonempty subsets ofXandα:X×X→[,∞). We say that
() Tisα∗-admissible if
forx,y∈Xfor whichα(x,y)≥ ⇒ α∗(Tx,Ty)≥,
whereα∗(Tx,Ty) :=inf{α(a,b)|a∈Tx,b∈Ty}.
() Tisα-admissible if for eachx∈Xandy∈Txwithα(x,y)≥, we haveα(y,z)≥, for allz∈Ty.
Remark . IfTisα∗-admissible, thenTis alsoα-admissible mapping.
In recent investigations, the fixed point results for single-valued and set-valued map-pings via the concepts of being α-admissible and α∗-admissible occupies a prominent place in many aspects (see [–] and references therein).
3 Fixed point theorems for partialq-set-valued quasi-contraction mappings
In this section, we introduce the partialq-set-valued quasi-contraction mapping and ob-tain the theorem of the existence of a fixed point for such a mapping inb-metric spaces.
Throughout this paper, for the nonempty set Xand the given mappingα:X×X→
[,∞), we use the following notation:
α:=
(x,y)∈X×X:α(x,y)≥.
Definition . Let (X,d) be ab-metric space andα:X×X→[,∞) be a given map-ping. The set-valued mappingT:X→Pb,cl(X) is said to be apartial q-set-valued
quasi-contractionif, for all (x,y)∈X×X, (x,y)∈
α
⇒ H(Tx,Ty)≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx),
(.)
where ≤q< .
Theorem . Let (X,d) be a complete b-metric space with coefficient s≥, α :X× X→[,∞)be a given mapping and T :X→Pb,cl(X)be a partial q-set-valued
quasi-contraction.Suppose that the following conditions hold:
(i) Tisα-admissible;
(ii) there existx∈Xandx∈Txsuch that(x,x)∈
α;
(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈
α,for alln∈N,andxn→xas
n→ ∞,for somex∈X,then(xn,x)∈
α.
If q<s+s,then T has a fixed point in X,that is,there exists u∈X such that u∈Tu.
Proof Forx,y∈X, we obtain
maxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)=
if and only ifx=yis a fixed point ofT. Therefore, we suppose that
maxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)> , for allx,y∈X.
Now, we will set
ε:=
s+s–q
and β:=q+ε=
s+s+q
.
It follows fromq<s+sthatε> and <β<s+s.
Starting fromxandx∈Txin (ii), by Lemma ., there existsx∈Txsuch that
d(x,x)≤H(Tx,Tx)
+εmaxd(x,x),d(x,Tx),d(x,Tx),d(x,Tx),d(x,Tx)
. (.) It follows from (x,x)∈
αthat
H(Tx,Tx)≤qmax
d(x,x),d(x,Tx),d(x,Tx),d(x,Tx),d(x,Tx)
. (.) From (.) and (.), we get
d(x,x)≤(q+ε)max
d(x,x),d(x,Tx),d(x,Tx),d(x,Tx),d(x,Tx)
=βmaxd(x,x),d(x,Tx),d(x,Tx),d(x,Tx),d(x,Tx)
.
SinceT isα-admissible,x∈X, andx∈Txsuch thatα(x,x)≥, we getα(x,x)≥
and so (x,x)∈
α. Using Lemma ., there existsx∈Txsuch that
d(x,x)≤H(Tx,Tx)
+εmaxd(x,x),d(x,Tx),d(x,Tx),d(x,Tx),d(x,Tx)
. (.) SinceTis a partialq-set-valued quasi-contraction and (x,x)∈
α, we obtain
H(Tx,Tx)≤qmax
d(x,x),d(x,Tx),d(x,Tx),d(x,Tx),d(x,Tx)
From (.) and (.), we have
d(x,x)≤(q+ε)max
d(x,x),d(x,Tx),d(x,Tx),d(x,Tx),d(x,Tx)
=βmaxd(x,x),d(x,Tx),d(x,Tx),d(x,Tx),d(x,Tx)
.
By induction, we can construct a sequence{xn}inXsuch that, for eachn∈N, we have
xn∈Txn–, (xn–,xn)∈
α
and
d(xn,xn+)
≤βmaxd(xn–,xn),d(xn–,Txn–),d(xn,Txn),d(xn–,Txn),d(xn,Txn–)
. (.) If there existsn∈Nsuch thatxn–=xn, thenxn∈Txnand then the proof is complete.
For the rest, we will assume thatxn–=xn, that is,d(xn–,xn) > , for alln∈N. Now we
obtain, for alln∈N,
d(xn,xn+)≤βmax
d(xn–,xn),d(xn–,Txn–),d(xn,Txn),d(xn–,Txn),d(xn,Txn–)
≤βmaxd(xn–,xn),d(xn–,xn),d(xn,xn+),d(xn–,xn+),d(xn,xn)
≤βmaxd(xn–,xn),d(xn,xn+),s
d(xn–,xn) +d(xn,xn+)
≤βsd(xn–,xn) +d(xn,xn+)
and hence
d(xn,xn+)≤γd(xn–,xn), (.)
whereγ :=–ββss.
Sinces≥,β=(s+s+q), andq<s+s, we get
γs< . (.)
From (.), (.), and Lemma ., we see that {xn}is a Cauchy sequence inX. By the
completeness ofX, there existsu∈Xsuch that
lim
n→∞d(xn,u) = . (.)
Next, we will prove thatd(u,Tu) = . By the condition (iii), we have (xn,u)∈
α, for all
n∈N. From Lemma . and (.), for eachn∈N, we get
d(u,Tu)≤sd(u,xn+) +d(xn+,Tu)
≤sd(u,xn+) +H(Txn,Tu)
≤sd(u,xn+) +qmax
d(xn,u),d(xn,Txn),d(u,Tu),d(xn,Tu),d(u,Txn)
≤sd(u,xn+) +qmax
d(xn,u),d(xn,xn+),d(u,Tu),d(xn,Tu),d(u,xn+)
≤sd(u,xn+) +qmax
d(xn,u),s
d(xn,u) +d(u,xn+)
,d(u,Tu),
sd(xn,u) +d(u,Tu)
,d(u,xn+)
.
Lettingn→ ∞in the above inequality, we have
d(u,Tu)≤qsd(u,Tu). (.)
It follows fromq<
s+s thatqs< . From (.), we getd(u,Tu) = . Using Lemma .,
we haveu∈Tu, that is,uis a fixed point ofT. This completes the proof.
Theorem . Let (X,d) be a complete b-metric space with coefficient s≥, α :X× X→[,∞)be a given mapping and T :X→Pb,cl(X)be a partial q-set-valued
quasi-contraction.Suppose that the following conditions hold:
(i) Tisα∗-admissible;
(ii) there existx∈Xandx∈Txsuch that(x,x)∈
α;
(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈
α,for alln∈N,andxn→xas
n→ ∞,for somex∈X,then(xn,x)∈
α.
If we set q<s+s,then T has a fixed point in X,that is,there exists u∈X such that u∈Tu.
Proof We can prove this result by using Theorem . and Remark ..
Corollary .(Theorems ., . in []) Let(X,d)be a complete b-metric space with coefficient s≥,α:X×X→[,∞)be a given mapping and T:X→Pb,cl(X)be a
q-set-valuedα-quasi-contraction,that is,for all x,y∈X,we have
α(x,y)H(Tx,Ty)≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx), (.)
where≤q< .Suppose that the following conditions hold:
(i) Tisα-admissible(orα∗-admissible);
(ii) there existx∈Xandx∈Txsuch that(x,x)∈
α;
(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈
α,for alln∈N,andxn→xas
n→ ∞,for somex∈X,then(xn,x)∈
α.
If q<s+s,then T has a fixed point in X,that is,there exists u∈X such that u∈Tu.
Proof We will show that aq-set-valuedα-quasi-contraction is a partialq-set-valued quasi-contraction. Assume that (x,y)∈αand soα(x,y)≥. SinceTis aq-set-valuedα -quasi-contraction, we get
H(Tx,Ty)≤α(x,y)H(Tx,Ty)
≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).
Corollary . Let(X,d)be a complete b-metric space with coefficient s≥,α:X×X→
[,∞)be a given mapping and let T:X→Pb,cl(X)satisfy
H(Tx,Ty) +α(x,y)≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)+, (.)
for all x,y∈X,where≤q< and≥.Suppose that the following conditions hold:
(i) Tisα-admissible(orα∗-admissible);
(ii) there existx∈Xandx∈Txsuch that(x,x)∈
α;
(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈
α,for alln∈N,andxn→xas
n→ ∞,for somex∈X,then(xn,x)∈
α.
If q<s+s,then T has a fixed point in X,that is,there exists u∈X such that u∈Tu.
Proof We will show thatTis a partialq-set-valued quasi-contraction. Suppose that (x,y)∈
αand thenα(x,y)≥. From (.), we get
H(Tx,Ty) +≤H(Tx,Ty) +α(x,y)
≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)+,
that is,
H(Tx,Ty)≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).
This implies thatTis a partialq-set-valued quasi-contraction. By Theorem . (or Theo-rem .), we get the desired result.
Corollary . Let(X,d)be a complete b-metric space with coefficient s≥,α:X×X→
[,∞)be a given mapping and T:X→Pb,cl(X)satisfies
α(x,y) – +H(Tx,Ty)≤qmax{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}, (.)
for all x,y∈X,where≤q< and> .Suppose that the following conditions hold:
(i) Tisα-admissible(orα∗-admissible);
(ii) there existx∈Xandx∈Txsuch that(x,x)∈
α;
(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈
α,for alln∈N,andxn→xas
n→ ∞,for somex∈X,then(xn,x)∈
α.
If q<s+s,then T has a fixed point in X,that is,there exists u∈X such that u∈Tu.
Proof We will show thatTis a partialq-set-valued quasi-contraction. Suppose that (x,y)∈
αand thenα(x,y)≥. From (.), we get
H(Tx,Ty)≤α(x,y) – +H(Tx,Ty)
≤qmax{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}.
It follows from> that
This implies thatTis a partialq-set-valued quasi-contraction. By Theorem . (or Theo-rem .), we get the desired result.
Corollary .(Theorem . in []) Let(X,d)be a complete b-metric space with coefficient s≥and T:X→Pb,cl(X)be a q-set-valued quasi-contraction.If q<s+s,then T has a
fixed point in X,that is,there exists u∈X such that u∈Tu.
Proof Setα(x,y) = , for allx,y∈X. By Theorem . (or Theorem .), we obtain the
desired result.
Remark . If we takes= (it corresponds to the case of metric spaces), then the con-dition ofqin Theorem . becomesq<. Therefore, Theorems . and . are general-ization of several known fixed point results in metric spaces. Also Theorem . is a gen-eralization of Theorem . and . of Sintunavaratet al.[], Theorem . of Aydiet al.
[], main results of Amini-Harandi [], Daffer and Kaneko [], Rouhani and Moradi [], and Singhet al.[].
The following example shows that Theorem . properly generalizes Theorem . of Aydiet al.[].
Example . LetX=Rand the functionald:X×X→R+defined by
d(x,y) :=|x–y|,
for allx,y∈X. Clearly, (X,d) is a completeb-metric space with coefficients= . Define set-valued mappingT:X→Pb,cl(X) by
Tx=
⎧ ⎪ ⎨ ⎪ ⎩
[x,max{x, –}], x∈(–∞, ), [,x], x∈[, ], [min{x, },x], x∈(,∞), andα:X×X→[,∞) by
α(x,y) =
ln(x+y+e), x,y∈[, ], , otherwise. We obtain
H(T,T) = and
maxd(, ),d(,T),d(,T),d(,T),d(,T)= . Therefore,
for all ≤q< . This implies that the contraction condition of Theorem .. of Aydiet al.
[] is not true for this case. Therefore, Theorem . cannot be used to claim the existence of fixed point ofT.
Next, we show that Theorem . can be applied for this case. First of all, we show that
T is a partialq-set-valued quasi-contraction mapping, whereq= . Assume that (x,y)∈
α=
(x,y)∈X×X:α(x,y)≥= [, ]×[, ]. Then we have
H(Tx,Ty) = x –
y
= |x–y|
=qd(x,y)
≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).
This shows thatTis a partialq-set-valued quasi-contraction mapping. Also we have
q= <
=
s+s.
It is easy to see thatT is anα-admissible mapping. We find that there existx= and
x= .∈Txfor which (x,x)∈
α. Further, for any sequence{xn}inXwithxn→xas
n→ ∞, for somex∈X, and (xn,xn+)∈
α, for alln∈N, we see that (xn,x)∈
α, for all
n∈N.
Therefore, all hypotheses of Theorem . are satisfied and soThas a fixed point. In this case,T have infinitely many fixed points.
4 Consequences
4.1 Fixed point results of single-valued mappings
In this section, we give the fixed point result for single-valued mappings. Before presenting our results, we introduce the new concept of a partialq-single-valued quasi-contraction mapping.
Definition . Let (X,d) be ab-metric space andα:X×X→[,∞) be a mapping. The single-valued mappingt:X→Xis said to be apartial q-single-valued quasi-contraction
if
(x,y)∈
α ⇒ d(tx,ty)≤qmax
d(x,y),d(x,tx),d(y,ty),d(x,ty),d(y,tx), (.) where ≤q< .
Next, we give the fixed point result for partialq-single-valued quasi-contraction map-ping.
Theorem . Let(X,d)be a complete b-metric space with coefficient s≥,α:X×X→
[,∞)be a given mapping and t:X→X be a partial q-single-valued quasi-contraction.
(i) tisα-admissible;
(ii) there existsx∈Xsuch that(x,tx)∈
α;
(iii) if{xn}is a sequence inXsuch that(xn,xn+)∈
α,for alln∈N,andxn→xas
n→ ∞,for somex∈X,then(xn,x)∈α.
If q<s+s,then t has a fixed point in X,that is,there exists u∈X such that u=tu.
Proof It follows by applying Theorem . or Theorem ..
Remark . Theorem . is an extension of Corollary . of Sintunavarat et al.[], Corollary . of Aydiet al.[], and the result of Ćirić [].
Example . LetX=Rand the functionald:X×X→R+defined by
d(x,y) :=|x–y|,
for allx,y∈X. Clearly, (X,d) is a completeb-metric space with coefficients= . Define single-valued mappingt:X→Xby
tx=
⎧ ⎪ ⎨ ⎪ ⎩
max{x, –}, x∈(–∞, ),
x
, x∈[, ],
x, x∈(,∞), andα:X×X→[,∞) by
α(x,y) =
, x,y∈[, ], ., otherwise. We obtain
d(t,t) = and
maxd(, ),d(,t),d(,t),d(,t),d(,t)= . Therefore,
d(t,t) >qmaxd(, ),d(,t),d(,t),d(,t),d(,t),
for all ≤q< . This implies that the contraction condition of Corollary . of Aydiet al.
[] is not true for this case. Therefore, Corollary . of Aydiet al.[] cannot be used to claim the existence of fixed point oft.
Next, we show that Theorem . can be applying for this case. First of all, we show thattis a partialq-single-valued quasi-contraction mapping, whereq=. Assume that (x,y)∈α={(x,y)∈X×X:α(x,y)≥}= [, ]×[, ]. We obtain
d(tx,ty) = x –
y
= |x–y|
=qd(x,y)
≤qmaxd(x,y),d(x,tx),d(y,ty),d(x,ty),d(y,tx).
This shows thattis a partialq-single-valued quasi-contraction mapping. Also we have
q= <
=
s+s.
It is easy to see thattis anα-admissible mapping.
We find that there existsx= . such that (x,tx) = (., .)∈α. Further, for any
sequence{xn}inXwith xn→xasn→ ∞, for somex∈X, and (xn,xn+)∈
α, for all
n∈N, we obtain (xn,x)∈
α, for alln∈N, since [, ] is closed.
Therefore, all hypotheses of Theorem . are satisfied and sothas a fixed point, that is, a point ∈X.
4.2 Fixed point results onb-metric space endowed with an arbitrary binary
relation
In this section, we give the fixed point results on ab-metric space endowed with an arbi-trary binary relation. Before presenting our results, we give the following definitions.
Definition . Let (X,d) be ab-metric space andRbe a binary relation overX. We say thatT:X→Pb,cl(X) is a weakly preserving mapping if for eachx∈Xandy∈TxwithxRy,
we haveyRz, for allz∈Ty.
Definition . Let (X,d) be ab-metric space andRbe a binary relation over X. The set-valued mappingT:X→Pb,cl(X) is said to be aq-set-valued quasi-contraction with
respect toRif, for allx,y∈X, we have
xRy ⇒ H(Tx,Ty)≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx), (.)
where ≤q< .
Theorem . Let(X,d)be a complete b-metric space with coefficient s≥,Rbe a binary relation over X,and T:X→Pb,cl(X)be a q-set-valued quasi-contraction with respect toR.
Suppose that the following conditions hold:
(i) Tis a weakly preserving mapping;
(ii) there existx∈Xandx∈Txsuch thatxRx;
(iii) if{xn}is a sequence inXsuch thatxnRxn+,for alln∈N,andxn→xasn→ ∞,for somex∈X,thenxnRx.
If q<s+s,then T has a fixed point in X,that is,there exists u∈X such that u∈Tu.
Proof Consider the mappingα:X×X→[,∞) defined by
α(x,y) =
ifxRy;
From condition (ii), we get α(x,x)≥ and so (x,x)∈
α. It follows fromT being a
preserving mapping thatT is anα-admissible mapping. SinceT is aq-set-valued quasi-contraction with respect toR, we have, for allx,y∈X,
(x,y)∈
α
⇒ H(Tx,Ty)≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).
(.)
This implies thatT is a partialq-set-valued quasi-contraction mapping. Now all the hy-potheses of Theorem . are satisfied and so the existence of the fixed point ofTfollows
from Theorem ..
Next, we give some special case of Theorem . in partially orderedb-metric spaces. Before we study the next results, we give the following definitions.
Definition . LetXbe a nonempty set. Then (X,d,) is called a partially orderedb -metric space if (X,d) is ab-metric space and (X,) is a partially ordered space.
Definition . Let (X,d,) be a partially orderedb-metric space. We say thatT:X→ Pb,cl(X) is a weakly preserving mapping withif for eachx∈Xandy∈Txwithxy, we
haveyz, for allz∈Ty.
Definition . Let (X,d,) be a partially orderedb-metric space. The set-valued map-pingT:X→Pb,cl(X) is said to be aq-set-valued quasi-contraction with respect toif,
for allx,y∈X, we have
xy ⇒ H(Tx,Ty)≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx), (.) where ≤q< .
Corollary . Let(X,d,)be a complete partially ordered b-metric space with coefficient s≥and T:X→Pb,cl(X)be a q-set-valued quasi-contraction with respect to.Suppose
that the following conditions hold:
(i) Tis a weakly preserving mapping with; (ii) there existx∈Xandx∈Txsuch thatxx;
(iii) if{xn}is a sequence inXsuch thatxnxn+,for alln∈N,andxn→xasn→ ∞, for somex∈X,thenxnx.
If we set q<
s+s,then T has a fixed point in X,that is,there exists u∈X such that u∈Tu.
Proof The result follows from Theorem . by considering the binary relation.
4.3 Fixed point results onb-metric spaces endowed with a graph
Throughout this section, let (X,d) be ab-metric space. A set{(x,x) :x∈X}is called a diagonal of the Cartesian productX×Xand is denoted by. Consider a directed graphG
In this section, we give the fixed point results for set-valued mappings in ab-metric space endowed with a graph. Before presenting our results, we will introduce new definitions in ab-metric space endowed with a graph.
Definition . Let (X,d) be a b-metric space endowed with a graphGandT :X→ Pb,cl(X) be set-valued mapping. We say thatTweakly preserves the edges ofGif for each
x∈Xandy∈Txwith (x,y)∈E(G) implies (y,z)∈E(G), for allz∈Ty.
Definition . Let (X,d) be ab-metric space endowed with a graph G. A set-valued mappingT:X→Pb,cl(X) is said to be aq-G-set-valued quasi-contraction if, for allx,y∈X,
we have
(x,y)∈E(G)
⇒ H(Tx,Ty)≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx),
(.)
where ≤q< .
Example . LetXbe a nonempty set. Any mappingT :X→Pb,cl(X) defined byTx=
{a}, wherea∈X, is aq-G-set-valued quasi-contraction for any graphGwithV(G) =X.
Example . LetXbe a nonempty set. Any mappingT:X→Pb,cl(X) is trivially aq-G
-set-valued quasi-contraction, whereG= (V(G),E(G)) = (X,).
Definition . Let (X,d) be ab-metric space endowed with a graphG. We say thatX
hasG-regular property if givenx∈Xand sequence{xn}inXsuch thatxn→xasn→ ∞
and (xn,xn+)∈E(G), for alln∈N, then (xn,u)∈E(G), for alln∈N.
Here, we give a fixed point result for set-valued mappings in ab-metric space endowed with a graph.
Theorem . Let(X,d)be a complete b-metric space with coefficient s≥and endowed with a graph G and let T:X→Pb,cl(X)be a q-G-set-valued quasi-contraction.Suppose
that the following conditions hold:
(i) Tweakly preserves edges ofG;
(ii) there existx∈Xandx∈Txsuch that(x,x)∈E(G);
(iii) XhasG-regular property. If q<
s+s,then T has a fixed point in X,that is,there exists u∈X such that u∈Tu.
Proof Consider the mappingα:X×X→[,∞) defined by
α(x,y) =
, (x,y)∈E(G);
, otherwise. (.)
SinceTis aq-G-set-valued quasi-contraction, we have, for allx,y∈X, (x,y)∈
α
⇒ H(Tx,Ty)≤qmaxd(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).
This implies thatTis a partialq-set-valued quasi-contraction.
By construction ofαand condition (i), we find thatT isα-admissible. From condition (ii) and the construction ofα, we getα(x,x)≥ and thus (x,x)∈
α. UsingG-regular
property ofX, the result is obtained that the condition (iii) in Theorem . holds. Now all the hypotheses of Theorem . are satisfied and so the existence of the fixed point ofT
follows from Theorem ..
5 Existence of a solution for an integral equation
In this section, we prove the existence theorem for a solution of the following integral equation by using Theorem .:
x(c) =φ(c) +
b
a
Kc,r,x(r)dr, (.)
whereb>a≥,x∈C[a,b] (the set of continuous real functions defined on [a,b]⊆R),
φ: [a,b]→R, andK: [a,b]×[a,b]×R→Rare given mappings.
Theorem . Suppose that the following hypotheses hold:
(I) K: [a,b]×[a,b]×R→Ris continuous;
(I) there existsp≥satisfies the following condition for eachr,c∈[a,b]andx,y∈Xwith
x(w)≤y(w),for allw∈[a,b]:
Kc,r,x(r)–Kc,r,y(r) ≤ξ(c,r) x(r) –y(r) ,
whereξ: [a,b]×[a,b]→[,∞)is a continuous function satisfying
sup
c∈[a,b]
b
a
ξ(c,r)pdr
≤
(p–+ )(b–a)p–;
(I) there existsx∈Xsuch thatx(c)≤(tx)(c),for allc∈[a,b].
Then the integral equation(.)has a solution x∈X.
Proof LetX=C[a,b] and lett:X→Xbe a mapping defined by
(tx)(c) =
b
a
Kc,r,x(r)dr,
for allx∈Xandc∈[a,b]. Clearly,Xwith theb-metricd:X×X→R+given by
d(x,y) = sup
c∈[a,b]
x(c) –y(c) p,
for allx,y∈X, is a completeb-metric space with coefficients= p–.
Define a mappingα:X×X→[,∞) by
α(x,y) =
It is easy to see thattis anα-admissible mapping. From (I), we have (x,tx)∈
α. Also
we find that condition (iii) in Theorem . holds (see []).
Next, we show thattis a partial q-single-valued quasi-contraction mapping withq=
p–+ < p–(p–+) =s+s. Let ≤p<∞with p +p = . Now, letx,y∈Xbe such that
(x,y)∈α, that is,x(c)≤y(c), for allc∈[a,b]. From (I), (I), and the Hölder inequality,
for eachs∈[a,b] we have
(tx)(s) – (ty)(s) p≤
b
a
Kc,r,x(r)–Kc,r,y(r) dr p
≤
b
a
qdr
p b
a
|Kc,r,x(r)–Kc,r,y(r)|pdr
pp
≤(b–a)
p p
b
a
ξ(c,r)p x(r) –y(r) pdr
= (b–a)
p p
b
a
ξ(c,r)pd(x,y)dr
≤(b–a)
p p
b
a
ξ(c,r)pd(x,y)dr
= (b–a)p–
b
a
ξ(c,r)pdrd(x,y)
≤
p–+ max
d(x,y),d(x,tx),d(y,ty),d(x,ty),d(y,tx) =qmaxd(x,y),d(x,tx),d(y,ty),d(x,ty),d(y,tx).
This shows that
d(tx,ty)≤qmaxd(x,y),d(x,tx),d(y,ty),d(x,ty),d(y,tx).
Therefore, by using Theorem ., we see thatthas a fixed point, that is, there existsx∈ Xsuch thatxis a fixed point oft. This implies thatxis a solution for (.) because the existence of a solution of (.) is equivalent to the existence of a fixed point of t. This
completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok,
10140, Thailand.2Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University
Rangsit Center, Pathumthani, 12121, Thailand.
Acknowledgements
The second author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript.
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