R E S E A R C H
Open Access
On new fixed point results for
(
α
,
ψ
,
ξ
)
-contractive multi-valued mappings
on
α
-complete metric spaces and their
consequences
Marwan Amin Kutbi
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
The purpose of this paper is to establish new fixed point results for multi-valued mappings satisfying an (
α
,ψ
,ξ
)-contractive condition, and via Bianchini-Grandolfi gauge functions, onα
-complete metric spaces. Our results unify, generalize, and complement various results from the literature. We also give examples which support our main result while previous results in the literature are not applicable. Some of the fixed point results in metric spaces endowed with an arbitrary binary relation and endowed with a graph are given here to illustrate the usability of the obtained results.MSC: 47H10; 54H25
Keywords: (
α
,ψ
,ξ
)-contractive multi-valued mappings;α
-complete metric spaces;α
-admissible multi-valued mappings;α
∗-admissible multi-valued mappings; Bianchini-Grandolfi gauge functions1 Introduction and preliminaries
Throughout this paper, we denote byN,R+, and Rthe sets of positive integers,
non-negative real numbers, and real numbers, respectively.
We recollect some essential notations, required definitions, and primary results coher-ent with the literature. For a nonempty setX, we denote byN(X) the class of all nonempty subsets ofX. Let (X,d) be a metric space, we denote byCL(X) the class of all nonempty closed subsets ofX, byCB(X) the class of all nonempty closed bounded subsets ofX. For
A,B∈CL(X), let the functionalH:CL(X)×CL(X)→R+∪ {∞}be defined by
H(A,B) =
max{supa∈Ad(a,B),supb∈Bd(b,A)}, if the maximum exists,
∞, otherwise
for everyA,B∈CL(X), whered(a,B) =inf{d(a,b) :b∈B}is the distance fromatoB⊆X. Such a functional is called the generalized Pompeiu-Hausdorff metric induced byd.
In this paper, we denote bythe class of functionsψ: [,∞)→[,∞) satisfying the following conditions:
(ψ) ψis a nondecreasing function;
(ψ)
∞
n=ψn(t) <∞, for allt> , whereψnis thenth iterate ofψ.
These functions are known in the literature as Bianchini-Grandolfi gauge functions in some sources (seee.g.[–]).
Remark . For eachψ∈, we see that the following assertions hold: . limn→∞ψn(t) = , for allt> .
. ψ(t) <tfor eacht> ; . ψ() = ;
Example . The functionψ: [,∞)→[,∞) defined byψ(t) =kt, wherek∈[, ), is a Bianchini-Grandolfi gauge function.
Example . The functionψ: [,∞)→[,∞) defined by
ψ(t) =
⎧ ⎪ ⎨ ⎪ ⎩
t, ≤t< ,
t, t= ,
t, t> ,
is a Bianchini-Grandolfi gauge function.
In [], Sametet al.introduced the concepts of anα-admissible mapping and anα-ψ -contractive mapping as follows.
Definition .([]) LetTbe a self mapping on a nonempty setXandα:X×X→[,∞) be a mapping. We say thatT isα-admissibleif the following condition holds:
x,y∈X, α(x,y)≥ ⇒ α(Tx,Ty)≥.
Definition .([]) Let (X,d) be a metric space andT:X→Xbe a given mapping. We say thatT is anα-ψ-contractive mapping if there exist two functionsα:X×X→[,∞) andψ∈such that
α(x,y)d(Tx,Ty)≤ψd(x,y)
for allx,y∈X.
One also proved some fixed point theorems for such mappings on complete metric spaces and showed that these results can be utilized to derive fixed point theorems in partially ordered metric spaces.
Afterwards, Aslet al.[] introduced the concept of anα∗-admissible mapping which is
a multi-valued version of theα-admissible mapping provided in [].
Definition .([]) LetXbe a nonempty set,T:X→N(X) andα:X×X→[,∞) be two mappings. We say thatT isα∗-admissible if the following condition holds:
x,y∈X, α(x,y)≥ ⇒ α∗(Tx,Ty)≥,
They extended theα-ψ-contractive condition of Sametet al.[] from a single-valued version to a multi-valued version as follows.
Definition .([]) Let (X,d) be a metric space,T:X→CL(X) be a multi-valued map-ping andα:X×X→[,∞) be a given mapping. We say thatT is anα-ψ-contractive multi-valued mapping if there existsψ∈such that
α(x,y)H(Tx,Ty)≤ψd(x,y) for allx,y∈X.
Aslet al.[] also established a fixed point result for multi-valued mappings on complete metric spaces satisfying anα-ψ-contractive condition.
Recently, Aliet al.[] introduced the notion of (α,ψ,ξ)-contractive multi-valued map-pings, whereξ ∈andis the family of functions ξ : [,∞)→[,∞) satisfying the following conditions:
(ξ) ξ is continuous;
(ξ) ξ is nondecreasing on[,∞);
(ξ) ξ(t) = if and only ift= ; (ξ) ξ is subadditive.
Remark . From (ξ) and (ξ), we haveξ(t) > , for allt∈(,∞).
Example . Letξ: [,∞)→[,∞) be a mapping which is defined by
ξ(t) =
t
φ(s)ds
for eacht∈[,∞), whereφ: [,∞)→[,∞) is a Lebesgue integrable mapping which is summable on each compact subset of [,∞) and satisfies the following conditions:
• for each> , we haveφ(t)dt> ; • for eacha,b> , we have
a+b
φ(t)dt≤
a
φ(t)dt+
b
φ(t)dt.
Thenξ∈.
Lemma . Let(X,d)be a metric space.Ifξ∈,then(X,ξ◦d)is a metric space.
Lemma .([]) Let(X,d)be a metric space,ξ∈,and B∈CL(X).If there exists x∈X such thatξ(d(x,B)) > ,then there exists y∈B such that
ξd(x,y) <qξd(x,B) ,
where q> .
α:X×X→[,∞) such that
x,y∈X, α(x,y)≥ ⇒ ξH(Tx,Ty) ≤ψξM(x,y) , (.) whereM(x,y) =max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty)+d(y,Tx)}.
In the case whenψ∈is strictly increasing, the (α,ψ,ξ)-contractive mapping is called a strictly (α,ψ,ξ)-contractive mapping.
Aliet al.[] also prove fixed point results for (α,ψ,ξ)-contractive multi-valued mapping on complete metric spaces.
Question Is it possible to prove fixed point results for (α,ψ,ξ)-contractive multi-valued mappingTunder some weaker condition forT?
Question Is it possible to prove fixed point results for (α,ψ,ξ)-contractive multi-valued mapping in some space which is more general than complete metric spaces?
Question Is it possible to find some consequences or applications of the fixed point results?
On the other hand, Mohammadiet al.[] extended the concept of an α∗-admissible mapping toα-admissible as follows.
Definition .([]) LetXbe a nonempty set,T:X→N(X) andα:X×X→[,∞) be two given mappings. We say thatT isα-admissible whenever for eachx∈Xandy∈Tx
withα(x,y)≥, we haveα(y,z)≥, for allz∈Ty.
Remark . It is clear thatα∗-admissible mapping is alsoα-admissible, but the converse may not be true as shown in Example of [].
Recently, Hussainet al.[] introduced the concept ofα-completeness for metric space which is a weaker than the concept of completeness.
Definition .([]) Let (X,d) be a metric space andα:X×X→[,∞) be a mapping. The metric spaceXis said to beα-completeif and only if every Cauchy sequence{xn}in
Xwithα(xn,xn+)≥, for alln∈N, converges inX.
Remark . IfXis complete metric space, thenXis alsoα-complete metric space. But the converse is not true.
Example . LetX= (,∞) and the metricd:X×X→Rdefined byd(x,y) =|x–y|, for allx,y∈X. Defineα:X×X→[,∞) by
α(x,y) =
exxy+y, x,y∈[, ],
, otherwise.
thenxn∈[, ], for alln∈N. Since [, ] is a closed subset ofR, we see that ([, ],d) is a complete metric space and then there existsx∗∈[, ] such thatxn→x∗asn→ ∞.
In this paper, we establish new fixed point results for (α,ψ,ξ)-contractive multi-valued mappings onα-complete metric spaces by using the idea ofα-admissible multi-valued mapping due to Mohammadiet al.[]. These results are real generalization of main results of Aliet al.[] and many results in literature. We furnish some interesting examples which support our main theorems while results of Aliet al.[] are not applicable. We also obtain fixed point results in metric space endowed with an arbitrary binary relation and fixed point results in metric space endowed with graph.
2 Main results
First, we introduce the concept of α-continuity for multi-valued mappings in metric spaces.
Definition . Let (X,d) be a metric space,α:X×X→[,∞) andT:X→CL(X) be two given mappings. We sayT is anα-continuous multi-valued mapping on (CL(X),H) if, for all sequences{xn}withxn
d
→x∈Xasn→ ∞, andα(xn,xn+)≥, for alln∈N, we
haveTxn H
→Txasn→ ∞, that is,
lim
n→∞d(xn,x) = and α(xn,xn+)≥ for alln∈N ⇒ nlim→∞H(Txn,Tx) = .
Note that the continuity ofTimplies theα-continuity ofT, for all mappingsα. In gen-eral, the converse is not true (see in Example .).
Example . LetX= [,∞),λ∈[, ] and the metricd:X×X→Rdefined byd(x,y) =
|x–y|, for allx,y∈X. DefineT:X→CL(X) andα:X×X→[,∞) by
Tx=
{λx}, x∈[, ], {x}, x> and
α(x,y)
cosh(x+y), x,y∈[, ],
tanh(x+y), otherwise.
Clearly,Tis not a continuous multi-valued mapping on (CL(X),H). Indeed, for sequence
{xn}={ +n}inX, we see thatxn= +n d
→, butTxn={ +n} H
→ {} ={λ}=T. Next, we show thatTis anα-continue multi-valued mapping on (CL(X),H). Let{xn}be a sequence inXsuch thatxn
d
→x∈Xasn→ ∞andα(xn,xn+)≥, for alln∈N. Then
we havex,xn∈[, ], for alln∈N. Therefore,Txn={λxn} H
→ {λx}=Tx. This shows that
T is anα-continuous multi-valued mapping on (CL(X),H). Now we give first main result in this paper.
Theorem . Let (X,d) be a metric space and T :X→CL(X) be a strictly (α,ψ,ξ) -contractive mapping.Suppose that the following conditions hold:
(S) Tis anα-admissible multi-valued mapping;
(S) there existxandx∈Txsuch thatα(x,x)≥;
(S) Tis anα-continuous multi-valued mapping.
Then T has a fixed point.
Proof Starting fromxandx∈Txin (S), we haveα(x,x)≥. Ifx=x, then we see
thatxis a fixed point ofT. Assume thatx=x. Ifx∈Tx, we obtain thatxis a fixed
point ofT. Then we have nothing to prove. So we letx∈/Tx. From (α,ψ,ξ)-contractive
condition, we get
ξH(Tx,Tx) ≤ψ
ξ
max
d(x,x),d(x,Tx),d(x,Tx),
d(x,Tx) +d(x,Tx)
=ψ
ξ
max
d(x,x),d(x,Tx),
d(x,Tx)
≤ψ
ξ
max
d(x,x),d(x,Tx),
d(x,x) +d(x,Tx)
=ψξmaxd(x,x),d(x,Tx)
. (.)
Ifmax{d(x,x),d(x,Tx)}=d(x,Tx), then we get
< ξd(x,Tx) ≤ξH(Tx,Tx)
≤ψξmaxd(x,x),d(x,Tx)
=ψξd(x,Tx)
, (.)
which is a contradiction. Therefore,max{d(x,x),d(x,Tx)}=d(x,x). From (.), we get
<ξd(x,Tx) ≤ξ
H(Tx,Tx) ≤ψ
ξd(x,x)
. (.)
For fixedq> by using Lemma ., there existsx∈Txsuch that
<ξd(x,x) <qξ
d(x,Tx) . (.)
From (.) and (.), we have
<ξd(x,x) <qψ
ξd(x,x)
. (.)
Sinceψis strictly increasing function, we have
<ψξd(x,x)
<ψqψξd(x,x)
. (.)
Ifx=xorx∈Tx, then we find thatxis a fixed point ofTand thus we have nothing
to prove. Therefore, we may assume thatx=xandx∈/Tx. Sincex∈Tx,x∈Tx, α(x,x)≥, andT is anα-admissible multi-valued mapping, we haveα(x,x)≥.
Ap-plying from (α,ψ,ξ)-contractive condition, we have
ξH(Tx,Tx) ≤ψ ξ max
d(x,x),d(x,Tx),d(x,Tx),
d(x,Tx) +d(x,Tx)
=ψ ξ max
d(x,x),d(x,Tx),
d(x,Tx)
≤ψ ξ max
d(x,x),d(x,Tx),
d(x,x) +d(x,Tx)
=ψξmaxd(x,x),d(x,Tx)
. (.)
Suppose thatmax{d(x,x),d(x,Tx)}=d(x,Tx). From (.), we get
< ξd(x,Tx) ≤ξH(Tx,Tx)
≤ψξmaxd(x,x),d(x,Tx)
=ψξd(x,Tx)
, (.)
which is a contradiction. Therefore, we may letmax{d(x,x),d(x,Tx)}=d(x,x). From
(.), we have
<ξd(x,Tx) ≤ξ
H(Tx,Tx) ≤ψ
ξd(x,x)
. (.)
By using Lemma . withq, there existsx∈Txsuch that
<ξd(x,x) <qξ
d(x,Tx) . (.)
From (.) and (.), we get
<ξd(x,x) <qψ
ξd(x,x)
=ψqψξd(x,x)
. (.)
It follows fromψbeing a strictly increasing function that
<ψξd(x,x)
<ψqψξd(x,x)
. (.)
Continuing this process, we can construct a sequence{xn}inXsuch thatxn=xn+∈Txn,
α(xn,xn+)≥ (.)
and
<ξd(xn+,xn+) <ψn
qψξd(x,x)
(.)
Letm,n∈Nsuch thatm>n. By the triangle inequality, we have
ξd(xm,xn) ≤ m–
i=n
ξd(xi,xi+) <
m–
i=n
ψi–qψξd(x,x)
.
Sinceψ∈, we havelimn,m→∞ξ(d(xm,xn)) = . Using (ξ), we getlimn,m→∞d(xm,xn) = . This implies that{xn}is a Cauchy sequence in (X,d). From (.) and theα-completeness of (X,d), there existsx∗∈Xsuch thatxn
d
→x∗asn→ ∞. Byα-continuity of the multi-valued mappingT, we get
lim
n→∞H(Txn,Tx) = . (.)
Now we obtain
dx∗,Tx∗ = lim
n→∞d(xn+,Tx)≤nlim→∞H(Txn,Tx) = .
Therefore,x∗∈Tx∗and henceThas a fixed point. This completes the proof.
Corollary . Let(X,d)be a metric space and T :X→CL(X)be a strictly (α,ψ,ξ) -contractive mapping.Suppose that the following conditions hold:
(S) (X,d)is anα-complete metric space;
(S) Tis anα∗-admissible multi-valued mapping; (S) there existxandx∈Txsuch thatα(x,x)≥;
(S) T is anα-continuous multi-valued mapping.
Then T has a fixed point.
Corollary .(Theorem . in []) Let(X,d)be a complete metric space and T:X→ CL(X)be a strictly(α,ψ,ξ)-contractive mapping. Suppose that the following conditions hold:
(A) Tis anα∗-admissible multi-valued mapping;
(A) there existxandx∈Txsuch thatα(x,x)≥;
(A) Tis a continuous multi-valued mapping.
Then T has a fixed point.
Next, we give second main result in this work.
Theorem . Let (X,d) be a metric space and T:X→CL(X) be a strictly (α,ψ,ξ) -contractive mapping.Suppose that the following conditions hold:
(S) (X,d)is anα-complete metric space;
(S) Tis anα-admissible multi-valued mapping;
(S) there existxandx∈Txsuch thatα(x,x)≥;
(S) if{xn}is a sequence inXwithxn d
→x∈Xasn→ ∞andα(xn,xn+)≥,for alln∈N,
then we haveα(xn,x)≥,for alln∈N.
Proof Following the proof of Theorem ., we know that{xn}is a Cauchy sequence inX such thatxn
d
→x∗asn→ ∞and
α(xn,xn+)≥ (.)
for alln∈N. From condition (S), we get
αxn,x∗ ≥ (.)
for alln∈N. By using the (α,ψ,ξ)-contractive condition ofT, we have
ξHTxn,Tx∗
≤ψ ξ max
dxn,x∗ ,d(xn,Txn),d
x∗,Tx∗ ,d(xn,Tx
∗) +d(x∗,Tx
n)
(.)
for alln∈N. Suppose thatd(x∗,Tx∗) > . Let:= d(x∗,Tx∗). Sincexn d
→x∗ asn→ ∞, we can findN∈Nsuch that
dx∗,xn <
d(x∗,Tx∗)
(.)
for alln≥N. Furthermore, we obtain that
dx∗,Txn ≤d
x∗,xn+ <
d(x∗,Tx∗)
(.)
for alln≥N. Also, as{xn}is a Cauchy sequence, there existsN∈Nsuch that
d(xn,Txn)≤d(xn,xn+) <
d(x∗,Tx∗)
(.)
for alln≥N. It follows fromd(xn,Tx∗)→d(x∗,Tx∗) asn→ ∞that we can findN∈N
such that
dxn,Tx∗ <
d(x∗,Tx∗)
(.)
for alln≥N. Using (.)-(.), we get
max
dxn,x∗ ,d(xn,Txn),d
x∗,Tx∗ ,d(xn,Tx
∗) +d(x∗,Tx
n)
=dx∗,Tx∗ (.)
for alln≥N:=max{N,N,N}. Forn≥N, by (.) and the triangle inequality, we have ξdx∗,Tx∗ ≤ξdx∗,xn+
+ξHTxn,Tx∗
≤ξdx∗,xn+ +ψ ξ max
dxn,x∗ ,d(xn,Txn),d
x∗,Tx∗ ,
d(xn,Tx∗) +d(x∗,Txn)
=ξdx∗,xn+
Lettingn→ ∞in the above inequality, we get
ξdx∗,Tx∗ ≤ψξdx∗,Tx∗ .
This implies thatξ(d(x∗,Tx∗)) = , which is a contradiction. Therefore,d(x∗,Tx∗) = , that
is,x∗∈Tx∗. This completes the proof.
Corollary . Let (X,d)be a metric space and T :X→CL(X)be a strictly (α,ψ,ξ) -contractive mapping.Suppose that the following conditions hold:
(S) (X,d)is anα-complete metric space;
(S) Tis anα∗-admissible multi-valued mapping; (S) there existxandx∈Txsuch thatα(x,x)≥;
(S) if{xn}is a sequence inXwithxn d
→x∈Xasn→ ∞andα(xn,xn+)≥,for alln∈N,
then we haveα(xn,x)≥,for alln∈N.
Then T has a fixed point.
Corollary .(Theorem . in []) Let(X,d)be a complete metric space and T:X→ CL(X)be a strictly(α,ψ,ξ)-contractive mapping. Suppose that the following conditions hold:
(A) Tis anα∗-admissible multi-valued mapping;
(A) there existxandx∈Txsuch thatα(x,x)≥;
(A) if{xn}is a sequence inXwithxn d
→x∈Xasn→ ∞andα(xn,xn+)≥,for alln∈N,
then we haveα(xn,x)≥,for alln∈N.
Then T has a fixed point.
Remark . Theorems . and . generalize many results in the following sense: • The condition (.) is weaker than some kinds of the contractive conditions such as
Banach’s contractive condition [], Kannan’s contractive condition [], Chatterjea’s contractive condition [], Nadler’s contractive condition [],etc.;
• the condition of beingα-admissible of a multi-valued mappingT is weaker than the condition of beingα∗-admissible ofT;
• for the existence of fixed point, we merely require thatα-continuity ofTand
α-completeness ofX, whereas other result demands stronger than these conditions. Consequently, Theorems . and . extend and improve the following results:
• Theorem . and Theorem . of Aliet al.[]; • Theorem . and Theorem . of Sametet al.[]; • Theorem . of Aslet al.[];
• Theorem . and Theorem . of Amiriet al.[]; • Theorem . of Salimiet al.[];
• Theorem . and Theorem . of Mohammadiet al.[].
Next, we give an example to show that our result is more general than the results of Ali
Example . LetX= (–, ) and the metricd:X×X→Rdefined byd(x,y) =|x–y|, for allx,y∈X. DefineT:X→CL(X) andα:X×X→[,∞) by
Tx=
⎧ ⎪ ⎨ ⎪ ⎩
[–,|x|], x∈(–, ), [,x], x∈[, ], [x+ , ], x∈(, ) and
α(x,y) =
exycosh(x+y), x,y∈[, ], tanh(ex–ey), otherwise.
Clearly, (X,d) is not complete metric space. Therefore, the results of Aliet al.[] are not applicable here.
Next, we show that by Theorem . can be guaranteed the existence of a fixed point ofT. Define functionsψ,ξ: [,∞)→[,∞) byψ(t) =t andξ(t) =√t, for allt∈[,∞). It is easy to see thatψ∈andξ∈.
Firstly, we will show thatT is a strictly (α,ψ,ξ)-contractive mapping. Forx,y∈Xand
α(x,y)≥, we havex,y∈[, ] and then
ξH(Tx,Ty) =
|x–y|
=
|x–y|
≤
M(x,y) =ψξM(x,y) .
It is to be observed thatψis strictly increasing function. Therefore,Tis a strictly (α,ψ,ξ )-contractive mapping.
Moreover, it is easy to see thatT is anα-admissible multi-valued mapping and there existsx= ∈Xandx= /∈Txsuch that
α(x,x) =α(, /)≥.
Also,Tis anα-continuous mapping.
Finally, for each sequence{xn}inXwithxn d
→x∈Xasn→ ∞andα(xn,xn+)≥, for
alln∈N, we haveα(xn,x)≥, for alln∈N. Thus the condition (S) in Theorem . holds.
Therefore, by using Theorem . or ., we getThas a fixed point inX. In this case,T
has infinitely fixed points such as –, –, and .
3 Consequences
3.1 Fixed point results in metric spaces endowed with an arbitrary binary relation
It has been pointed out in some studies that some results in metric spaces endowed with an arbitrary binary relation can be concluded from the fixed point results related with
Let (X,d) be a metric space andRbe a binary relation overX. Denote
S:=R∪R–,
i.e.,
x,y∈X, xSy ⇐⇒ xRy or yRx.
Definition . LetXbe a nonempty set andRbe a binary relation overX. A multi-valued mappingT:X→N(X) is said to be aweakly comparativeif for eachx∈Xandy∈Txwith
xSy, we haveySz, for allz∈Ty.
Definition . Let (X,d) be a metric space andRbe a binary relation overX. The met-ric spaceXis said to beS-completeif and only if every Cauchy sequence{xn}inXwith
xnSxn+, for alln∈N, converges inX.
Definition . Let (X,d) be a metric space andRbe a binary relation overX. We say that
T:X→CL(X) is aS-continuous mappingto (CL(X),H) if for givenx∈Xand sequence
{xn}with lim
n→∞d(xn,x) = and xnSxn+ for alln∈N ⇒ nlim→∞H(Txn,Tx) = .
Definition . Let (X,d) be a metric space andRbe a binary relation overX. A mapping
T:X→CL(X) is called an (S,ψ,ξ)-contractive mapping if there exist two functionsψ∈ andξ∈such that
x,y∈X, xSy ⇒ ξH(Tx,Ty) ≤ψξM(x,y) , (.) whereM(x,y) =max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty)+d(y,Tx)}.
In the case whenψ∈is strictly increasing, the (S,ψ,ξ)-contractive mapping is called a strictly (S,ψ,ξ)-contractive mapping.
Theorem . Let(X,d)be a metric space,Rbe a binary relation over X and T:X→ CL(X)be a strictly(S,ψ,ξ)-contractive mapping.Suppose that the following conditions hold:
(S) (X,d)is anS-complete metric space;
(S) Tis a weakly comparative mapping;
(S) there existxandx∈Txsuch thatxSx;
(S) Tis aS-continuous multi-valued mapping.
Then T has a fixed point.
Proof This result can be obtain from Theorem . by define a mappingα:X×X→[,∞) by
α(x,y) =
, x,y∈xSy, , otherwise.
By using Theorem ., we get the following result.
Theorem . Let(X,d)be a metric space,Rbe a binary relation over X and T:X→ CL(X)be a strictly(S,ψ,ξ)-contractive mapping.Suppose that the following conditions hold:
(S) (X,d)is anS-complete metric space;
(S) Tis a weakly comparative mapping;
(S) there existxandx∈Txsuch thatxSx;
(S) if{xn}is a sequence inXwithxn→x∈Xasn→ ∞andxnSxn+,for alln∈N,then
we havexnSx,for alln∈N.
Then T has a fixed point.
3.2 Fixed point results in metric spaces endowed with graph
In , Jachymski [] obtained a generalization of Banach’s contraction principle for mappings on a metric space endowed with a graph. Afterwards, Dinevari and Frigon [] extended some results of Jachymski [] to multi-valued mappings. For more fixed point results on a metric space with a graph, one can refer to [–].
In this section, we give fixed point results on a metric space endowed with a graph. Before presenting our results, we give the following notions and definitions.
Throughout this section, let (X,d) be a metric space. A set{(x,x) :x∈X} is called a diagonal of the Cartesian productX×Xand is denoted by. Consider a graphGsuch that the setV(G) of its vertices coincides withXand the setE(G) of its edges contains all loops,i.e.,⊆E(G). We assumeGhas no parallel edges, so we can identifyGwith the pair (V(G),E(G)). Moreover, we may treatGas a weighted graph by assigning to each edge the distance between its vertices.
Definition . LetXbe a nonempty set endowed with a graphGandT:X→N(X) be a multi-valued mapping, whereXis a nonempty setX. We say thatT weakly preserves edges
if for eachx∈Xandy∈Txwith (x,y)∈E(G), we have (y,z)∈E(G), for allz∈Ty.
Definition . Let (X,d) be a metric space endowed with a graphG. The metric spaceX
is said to beE(G)-completeif and only if every Cauchy sequence{xn}inXwith (xn,xn+)∈
E(G), for alln∈N, converges inX.
Definition . Let (X,d) be a metric space endowed with a graphG. We say thatT:X→ CL(X) is anE(G)-continuous mappingto (CL(X),H) if for givenx∈Xand sequence{xn} with
lim
n→∞d(xn,x) = and (xn,xn+)∈E(G) for alln∈N ⇒ nlim→∞H(Txn,Tx) = .
Definition . Let (X,d) be a metric space endowed with a graphG. A mappingT:X→ CL(X) is called an (E(G),ψ,ξ)-contractive mapping if there exist two functionsψ∈and
ξ∈such that
In the case whenψ∈ is strictly increasing, the (E(G),ψ,ξ)-contractive mapping is called a strictly (E(G),ψ,ξ)-contractive mapping.
Theorem . Let(X,d)be a metric space endowed with a graph G,and T:X→CL(X)
be a strictly(E(G),ψ,ξ)-contractive mapping.Suppose that the following conditions hold: (S) (X,d)is anE(G)-complete metric space;
(S) Tweakly preserves edges;
(S) there existxandx∈Txsuch that(x,x)∈E(G);
(S) Tis anE(G)-continuous multi-valued mapping.
Then T has a fixed point.
Proof This result can be obtained from Theorem . by defining a mappingα:X×X→
[,∞) by
α(x,y) =
, (x,y)∈E(G), , otherwise.
This completes the proof.
By using Theorem ., we get the following result.
Theorem . Let(X,d)be a metric space endowed with a graph G and T:X→CL(X)
be a strictly(S,ψ,ξ)-contractive mapping.Suppose that the following conditions hold: (S) (X,d)is anE(G)-complete metric space;
(S) Tweakly preserves edges;
(S) there existxandx∈Txsuch that(x,x)∈E(G);
(S) if{xn} is a sequence inX withxn→x∈Xasn→ ∞ and(xn,xn+)∈E(G),for all
n∈N,then we have(xn,x)∈E(G)for alln∈N.
Then T has a fixed point.
Remark .
. If we assumeGis such thatE(G) :=X×X, then clearlyGis connected and our Theorems . and . improve Nadler’s contraction principle [] and in the case of a single-valued mapping, we improve Banach’s contraction principle [], Kannan’s contraction theorem [], Chatterjea’s contraction theorem [], and Bianchini and Grandolfi’s fixed point theorem.
. Theorems . and . are partial some generalized fixed point results endowed with a graph of Jachymski [] and Dinevari and Frigon [].
. Theorems . and . are generalizations of fixed point results of Theorem . and Theorem . of Aliet al.[] in a graph version.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.2Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Center, Pathumthani, 12121, Thailand.
Acknowledgements
The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. 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.
Received: 24 June 2014 Accepted: 20 November 2014 Published:16 Jan 2015
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10.1186/1687-1812-2015-2
