R E S E A R C H
Open Access
Some fixed point theorems for
(α
,
θ
,
k
)
-contractive multi-valued mappings
with some applications
Adoon Pansuwan
1, Wutiphol Sintunavarat
1, Vahid Parvaneh
2*and Yeol Je Cho
3,4**Correspondence:
[email protected]; [email protected]
2Department of Mathematics,
Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran
3Department of Mathematics
Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea Full list of author information is available at the end of the article
Abstract
In this paper, we introduce the notion of (α,
θ
,k)-contraction multi-valued mappings and establish some fixed point results for such mappings by using some control functions due to Jleliet al.(J. Inequal. Appl. 2014:439, 2014) in metric spaces and furnish some interesting examples to illustrate our main results. Also, we give some fixed point results in metric spaces endowed with a graph. Our results generalize and extend recent results given by some authors.MSC: 47H09; 47H10
Keywords:
α-admissible multi-valued mapping;
α-complete metric spaces;
α-continuous function
1 Introduction and preliminaries
Now, we recall some notations and primary results which are needed in the sequel. Let (X,d) be a metric space. We denote byN(X) the class of all nonempty subsets ofX, byCL(X) the class of all nonempty closed subsets ofX, byCB(X) the class of all nonempty closed bounded subsets ofXand byK(X) the class of all nonempty compact subsets ofX. For anyA,B∈CL(X), let the mappingH:CL(X)×CL(X)→R+∪ {∞}defined by
H(A,B) =
⎧ ⎨ ⎩
max{supa∈Ad(a,B),supb∈Bd(b,A)}, if the maximum exists;
∞, otherwise,
be thegeneralized Pompeiu-Hausdorff metricinduced byd, whered(a,B) =inf{d(a,b) : b∈B}is the distance fromatoB⊆X.
In , Nadler [] extended Banach’s contraction principle to the class of multi-valued mappings in metric spaces as follows.
Theorem .[] Let(X,d)be a complete metric space and T:X→CB(X)be a multi-valued mapping such that
H(Tx,Ty)≤kd(x,y) (.)
for all x,y∈X,where k∈[, ).Then T has at least one fixed point.
Since Nadler’s fixed point theorem, a number of authors have published many interest-ing fixed point theorems in several ways (see [–] and references therein).
In , Sametet al.[] introduced the concept ofα-admissible mapping as follows.
Definition .[] LetTbe a self-mapping on a nonempty setXandα:X×X→[,∞) be a mapping. The mappingT is said to beα-admissibleif the following condition holds:
x,y∈X, α(x,y)≥ ⇒ α(Tx,Ty)≥.
They also proved some fixed point theorems for such mappings under the generalized contractive conditions in complete metric spaces and showed that these results can be utilized to derive some fixed point theorems in partially ordered metric spaces.
Afterward, Aslet al.[] introduced the concept of anα∗-admissible mapping which is the multi-valued version ofα-admissible single-valued mapping provided in [].
Definition .[] LetXbe a nonempty set,T:X→N(X) andα:X×X→[,∞) be two given mappings. The mappingTis said to beα∗-admissibleif the following condition
holds:
x,y∈X, α(x,y)≥ ⇒ α∗(Tx,Ty)≥,
whereα∗(Tx,Ty) :=inf{α(a,b) :a∈Tx,b∈Ty}.
Aslet al.[] also established a fixed point result for multi-valued mappings in complete metric spaces satisfying some generalized contractive condition.
In , Mohammadiet al.[] extended the concept of anα∗-admissible mapping to the class ofα-admissible mappings as follows.
Definition .[] LetXbe a nonempty set,T:X→N(X) andα:X×X→[,∞) be two given mappings. The mappingT is said to beα-admissiblewhenever, for eachx∈Xand y∈Txwithα(x,y)≥, we haveα(y,z)≥ for allz∈Ty.
Remark . It is clear that anα∗-admissible mapping is alsoα-admissible, but the con-verse may not be true.
Recently, Hussain et al.[] introduced the concept of the α-completeness of metric spaces which is a weaker than the concept of the completeness.
Definition .[] Let (X,d) be a metric space andα:X×X→[,∞) be a mapping. The metric space X is said to be α-completeif every Cauchy sequence {xn}in X with
α(xn,xn+)≥ for alln∈Nconverges inX.
Remark . IfXis a complete metric space, thenXis also anα-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. Define a mappingα:X×X→[,∞) by
α(x,y) =
⎧ ⎨ ⎩
exxy+y, ifx,y∈[, ],
It is easy to see that (X,d) is not a complete metric space, but (X,d) is anα-complete metric space. Indeed, if{xn}is a Cauchy sequence inX such thatα(xn,xn+)≥ for all n∈N, thenxn∈[, ] for alln∈N. Since [, ] is a closed subset ofR, it follows that
([, ],d) is a complete metric space and so there existsx∗∈[, ] such thatxn→x∗as
n→ ∞.
Recently, Kutbi and Sintunavarat [] introduced the concept of the α-continuity for multi-valued mappings in metric spaces as follows.
Definition .[] Let (X,d) be a metric space,α:X×X→[,∞) andT:X→CL(X) be two given mappings. The mappingT:X→CL(X) is called anα-continuous multi-valued mapping if, for all sequence{xn}withxn
d
→x∈Xasn→ ∞andα(xn,xn+)≥ for all n∈N, we haveTxn
H
→Txasn→ ∞, that is, for alln∈N,
lim
n→∞d(xn,x) = , α(xn,xn+)≥ ⇒ nlim→∞H(Txn,Tx) = .
Note that the continuity ofTimplies theα-continuity ofTfor all mappingsα. In general, the converse is not true (see Example .).
Example .[] LetX= [,∞),λ∈[, ] and the metricd:X×X→Rdefined by d(x,y) =|x–y|for allx,y∈X. Define two mappingsT:X→CL(X) andα:X×X→[,∞) by
Tx=
⎧ ⎨ ⎩
{λx}, ifx∈[, ],
{x}, ifx>
and
α(x,y) =
⎧ ⎨ ⎩
cosh(x+y), ifx,y∈[, ],
tanh(x+y), otherwise.
Clearly,T is not a continue multi-valued mapping on (CL(X),H). Indeed, for a sequence
{xn}inXdefined byxn= +nfor eachn≥, 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 thatTis
anα-continue multi-valued mapping on (CL(X),H).
2 The main results
Recently, Jleliet al.[] introduced the classof all functionsθ: (,∞)→(,∞) satisfy-ing the followsatisfy-ing conditions:
(θ) θ is non-decreasing;
(θ) for each sequence{tn} ⊆(,∞),limn→∞θ(tn) = if and only iflimn→∞tn= ;
(θ) there existr∈(, )and∈(,∞]such thatlimt→+θ(t)–
tr =; (θ) θ is continuous;
and proved the following result.
Theorem .(Corollary . of []) Let(X,d)be a complete metric space and T:X→X be a given mapping.Suppose that there existθ∈and k∈(, )such that
x,y∈X, d(Tx,Ty)= ⇒ θd(Tx,Ty)≤θd(x,y)k. (.)
Then T has a unique fixed point.
Observe that Banach’s contraction principle follows immediately from the above theo-rem (see []).
In this section, we introduce the concept of (α,θ,k)-contraction multi-valued mappings and prove fixed point results for such mappings without the assumption of the complete-ness of domain of mappings and the continuity of mappings.
Definition . Let (X,d) be a metric space. A multi-valued mappingT:X→K(X) is said to be an (α,θ,k)-contractionif there existα:X×X→[,∞),θ ∈, andk∈(, ) such that
x,y∈X, H(Tx,Ty)= ⇒ α(x,y)θH(Tx,Ty)≤θM(x,y)k, (.)
where
M(x,y) =max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
.
Now, we give the first main result in this paper.
Theorem . Let(X,d)be a metric space and T:X→K(X)be an(α,θ,k)-contraction mapping.Suppose that the following conditions hold:
(S) (X,d)is anα-complete metric space;
(S) T is 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 in X.
Proof Starting fromxandx∈Txin (S), then we haveα(x,x)≥. Ifx=x, thenx
have nothing to prove. Letx∈/Tx, that is,d(x,Tx) > . SinceH(Tx,Tx)≥d(x,Tx) >
, it follows from the (α,θ,k)-contractive condition that
θH(Tx,Tx)
≤α(x,x)θ
H(Tx,Tx)
≤θ max
d(x,x),d(x,Tx),d(x,Tx),
d(x,Tx) +d(x,Tx)
k
=θ max
d(x,x),d(x,Tx),
d(x,Tx)
k
≤θ max
d(x,x),d(x,Tx),
d(x,x) +d(x,Tx)
k
=θmaxd(x,x),d(x,Tx)
k
. (.)
Ifmax{d(x,x),d(x,Tx)}=d(x,Tx), then we have
< θd(x,Tx)
≤θH(Tx,Tx)
≤θmaxd(x,x),d(x,Tx)
k
=θd(x,Tx)
k
, (.)
which is a contradiction. Therefore,max{d(x,x),d(x,Tx)}=d(x,x). From (.), it
fol-lows that
<θd(x,Tx)
≤θH(Tx,Tx)
≤θd(x,x)
k
. (.)
SinceTxis compact, there existsx∈Txsuch that
d(x,x) =d(x,Tx). (.)
From (.) and (.), it follows that
<θd(x,x)
<θd(x,x)
k
. (.)
Ifx=xorx∈Tx, then it follows thatxis a fixed point ofTand so 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 the (α,θ,k)-contractive condition, we have
θH(Tx,Tx)
≤α(x,x)θ
H(Tx,Tx)
≤θ max
d(x,x),d(x,Tx),d(x,Tx),
d(x,Tx) +d(x,Tx)
=θ max
d(x,x),d(x,Tx),
d(x,Tx)
k
≤θ max
d(x,x),d(x,Tx),
d(x,x) +d(x,Tx)
k
=θmaxd(x,x),d(x,Tx)
k
. (.)
Suppose thatmax{d(x,x),d(x,Tx)}=d(x,Tx). From (.), it follows that
< θd(x,Tx)
≤θH(Tx,Tx)
≤θmaxd(x,x),d(x,Tx)
k
(.)
=θd(x,Tx)
k
,
which is a contradiction. Therefore, we may letmax{d(x,x),d(x,Tx)}=d(x,x). From
(.), it follows that
<θd(x,Tx)
≤θH(Tx,Tx)
≤θd(x,x)
k
. (.)
SinceTxis compact, there existsx∈Txsuch that
d(x,x) =d(x,Tx). (.)
From (.) and (.), it follows that
<θd(x,x)
≤θd(x,x)
k
≤θd(x,x)
k
. (.)
Continuing this process, we can construct a sequence{xn}inXsuch that
xn=xn+∈Txn, (.)
α(xn,xn+)≥ (.)
and
<θd(xn+,xn+)
≤θd(x,x)
kn
(.)
for alln∈N∪ {}. This shows thatlimn→∞θ(d(xn,xn+)) = and so
lim
n→∞d(xn,xn+) = (.)
by our assumptions aboutθ. From similar arguments as in the proof of Theorem . of [], it follows that there existn∈Nandr∈(, ) such that
d(xn,xn+)≤
for alln≥n. Now, form>n>nwe have
d(xn,xm)≤ m–
i=n
d(xi,xi+)≤
m–
i=n
ir
.
Since <r< ,∞i=n
ir converges. Therefore,d(xn,xm)→ asm,n→ ∞. Thus we proved
that{xn}is a Cauchy sequence inX. From (.) and theα-completeness of (X,d), there
existsx∗∈Xsuch thatxn→x∗asn→ ∞.
By theα-continuity of multi-valued mappingT, we have
lim
n→∞H
Txn,Tx∗
= , (.)
which implies that
dx∗,Tx∗= lim
n→∞d
xn+,Tx∗
≤ lim
n→∞H
Txn,Tx∗
= .
Therefore,x∗∈Tx∗and henceThas a fixed point. This completes the proof.
Next, we give the second main result in this paper.
Theorem . Let(X,d)be a metric space and T:X→K(X)be an(α,θ,k)-contraction 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→x∈Xasn→ ∞andα(xn,xn+)≥for alln∈N,
then we have
θH(Txn,Tx)
≤θM(xn,x)
k
, (.)
where
M(xn,x) =max
d(xn,x),d(xn,Txn),d(x,Tx),
d(xn,Tx) +d(x,Txn)
for alln∈N.
Then T has a fixed point in X.
Proof Following the proof of Theorem ., we know that{xn}is a Cauchy sequence inX
such thatxn→x∗asn→ ∞and
α(xn,xn+)≥ (.)
for alln∈N. Suppose thatd(x∗,Tx∗) > . By using (.), we have
θdxn+,Tx∗
≤θHTxn,Tx∗
≤θ max
dxn,x∗
,d(xn,Txn),d
x∗,Tx∗,d(xn,Tx
∗) +d(x∗,Tx
n)
k
≤θ max
dxn,x∗
,d(xn,xn+),d
x∗,Tx∗,d(xn,Tx
∗) +d(x∗,x
n+)
k
(.)
for alln∈N. Lettingn→ ∞in (.), we have
θdx∗,Tx∗≤θdx∗,Tx∗k.
This implies that θ(d(x∗,Tx∗)) = , which is a contradiction. Therefore, we haved(x∗,
Tx∗) = , that is,x∗∈Tx∗. This completes the proof.
Remark . From Remark ., the conclusion in Theorems . and . are still hold if we replace condition (S) by the following condition:
(S) Tis anα∗-admissible multi-valued mapping.
Now, we give an example to illustrate Theorem ..
Example . LetX= (–, ) and the metricd:X×X→Rdefined byd(x,y) =|x–y| for allx,y∈X. DefineT:X→K(X) andα:X×X→[,∞) by
Tx=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
[–|x|,|x|], ifx∈(–, ),
[,x], ifx∈[, ],
[x+ , ], ifx∈(, )
and
α(x,y) =
⎧ ⎨ ⎩
, ifx,y∈[, ],
, otherwise.
Clearly, (X,d) is not a complete metric space. Many fixed point results are not applicable here.
Next, we show that Theorem . can be guarantee the existence of fixed point of T. Define a functionθ: (,∞)→(,∞) by
θ(t) =e
√
tet
for allt∈(,∞). It is easy to see thatθ∈(see also []).
Firstly, we show thatTis an (α,θ,k)-contraction multi-valued mappings withk= . For
allx,y∈[, ] withH(Tx,Ty)= , we havex=yand then
α(x,y)θH(Tx,Ty)=θ |x–y|
=e
|x–y|
e
|x–y|
≤e
=e
√
d(x,y)ed(x,y)
≤e
√
M(x,y)eM(x,y)
=θM(x,y)k,
that is, the condition (.) holds. Therefore,Tis an (α,θ,k)-contraction multi-valued map-pings withk=. Moreover, it is easy to see thatTis anα-admissible multi-valued mapping and there existsx= ∈Xandx= /∈Txsuch that
α(x,x) =α(, /)≥.
Finally, for each sequence{xn}inXwithxn→x∈Xasn→ ∞andα(xn,xn+)≥ for
alln∈N, we havex,xn∈[, ] for alln∈N. Then we obtain
θH(Txn,Tx)
=θ |xn–x|
=e
|xn–x|
e
|xn–x|
≤e
√
|xn–x|e|xn–x|
=e
√
d(xn,x)ed(xn,x)
≤e
√
M(xn,x)eM(xn,x)
=θM(xn,x)
k
for alln∈N. Thus the condition (S) in Theorem . holds. Therefore, by using
Theo-rem ., it follows thatT has a fixed point inX. In this case,T has infinitely fixed points such as –, –, and .
3 Some applications
In , Jachymski [] obtained a generalization of Banach’s contraction principle for mappings on a metric space endowed with a graph. Afterward, Dinevari and Frigon [] extended the results of Jachymski [] to multi-valued mappings.
In this section, we give the existence of fixed point theorems on a metric space endowed with graph. The following notions and definitions are needed.
Let (X,d) be a metric space. A set{(x,x) :x∈X}is called adiagonalof the Cartesian productX×X, which 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 assume thatGhas no parallel edges and 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 set. The mappingT preserves edges weaklyif, 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 space Xis said to beE(G)-completeif every Cauchy sequence{xn}inXwith (xn,xn+)∈E(G) for
Definition .[] Let (X,d) be a metric space endowed with a graphG. A mappingT : X→CL(X) is called anE(G)-continuous mappingto (CL(X),H) if, for anyx∈Xand any sequence{xn}withlimn→∞d(xn,x) = and (xn,xn+)∈E(G) for alln∈N, we have
lim
n→∞H(Txn,Tx) = .
Definition . Let (X,d) be a metric space endowed with a graphG. A mappingT:X→ K(X) is called an (E(G),θ,k)-contraction multi-valued mappingif there exist a function θ∈andk∈(, ) such that
x,y∈X, (x,y)∈E(G), H(Tx,Ty)= ⇒ θH(Tx,Ty)≤θM(x,y)k, (.)
where
M(x,y) =max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx)
.
Theorem . Let(X,d)be a metric space endowed with a graph G and T:X→K(X) be a(E(G),θ,k)-contraction multi-valued mapping.Suppose that the following conditions hold:
(S) (X,d)is anE(G)-complete metric space;
(S) Tpreserves edges weakly;
(S) there existxandx∈Txsuch that(x,x)∈E(G);
(S) Tis anE(G)-continuous multi-valued mapping. Then T has a fixed point in X.
Proof This result can be obtained from Theorem . if we define a mappingα:X×X→ [,∞) by
α(x,y) =
⎧ ⎨ ⎩
, if (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→K(X) be a(E(G),θ,k)-contraction multi-valued mapping.Suppose that the following conditions hold:
(S) (X,d)is anE(G)-complete metric space;
(S) Tpreserves edges weakly;
(S) there existxandx∈Txsuch that(x,x)∈E(G);
(S) if{xn}is a sequence inXwithxn→x∈Xasn→ ∞and(xn,xn+)∈E(G)for alln∈N,
then we have
θH(Txn,Tx)
≤θM(xn,x)
k
where
M(xn,x) =max
d(xn,x),d(xn,Txn),d(x,Tx),
d(xn,Tx) +d(x,Txn)
for alln∈N.
Then T has a fixed point in X.
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 and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Center,
Pathumthani, 12121, Thailand.2Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University,
Gilan-E-Gharb, Iran.3Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju,
660-701, Korea.4Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.
Acknowledgements
The first and second authors gratefully acknowledge the financial support provided by Thammasat University under the TU Research Scholar, Contract No. 1/7/2557. Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning
(2014R1A2A2A01002100).
Received: 8 April 2015 Accepted: 16 July 2015
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