# Fixed Point Theorems for Ćirić-Berinde Type Contractive Multivalued Mappings

Authors: Seong-Hoon Cho
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

We give a Ćirić-Berinde type contractive condition for multivalued mappings and analyze the existence of fixed point for these mappings.


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## Body

## 1. Introduction and Preliminaries

In 2012, Samet et al. [1] introduced the notions of α-ψ-contractive mapping and α-admissible mappings in metric spaces and obtained corresponding fixed point results, which are generalizations of ordered fixed point results (see [1]). Since then, by using their idea, some authors investigated fixed point results in the field. Asl et al. [2] extended some of results in [1] to multivalued mappings by introducing the notions of α*-ψ-contractive mapping and α*-admissible mapping.Recently, Salimi et al. [3] modified the notions of α-ψ-contractive mapping and α-admissible mappings by introducing another function η. And then, they gave generalizations of the results of Samet et al. [1] and Karapınar and Samet [4]. Hussain et al. [5] extended these modified notions to multivalued mappings. That is, they introduced the notion of α-η-contractive multifunctions and gave fixed point results for these multifunctions.Very recently, Ali et al. [6] generalized and extended the notion of α-ψ-contractive mapping by introducing the notion of (α,ψ,ξ)-contractive multivalued mappings and obtained fixed point theorems for these mappings in complete metric spaces.The purpose of this paper is to introduce the notion of Ćirić-Berinde type contractive multivalued mappings and to generalize and extend the notion ofα-η-contractive multifunctions and to establish fixed point theorems for Ćirić-Berinde type contractive multivalued mappings.Let(X,d) be a metric space. We denote by CB(X) the class of nonempty closed and bounded subsets of X and by CL(X) the class of nonempty closed subsets of X. Let H(·,·) be the generalized Hausdorff distance on CL(X); that is, for all A,B∈CL(X), (1)HA,B=max⁡sup⁡a∈Ad(a,B),sup⁡b∈Bd(b,A),hifthemaximumexists,∞,otherwise,where d(a,B)=inf⁡{d(a,b):b∈B} is the distance from point a to subset B.ForA,B∈CL(X), let D(A,B)=sup⁡x∈Ainf⁡y∈Bd(x,y).Then, we haveD(A,B)≤H(A,B) for all A,B∈CL(X).From now on, we denote by(2)Mx,y=max⁡⁡d(x,y),d(x,Tx),d(y,Ty),12hhhh12{d(x,Ty)+d(y,Tx)}for a multivalued map T:X→CL(X) and x,y∈X.We denote byΞ the class of all functions ξ:[0,∞)→[0,∞) such that(1)
ξis continuous;(2)
ξ is nondecreasing on [0,∞);(3)
ξ(t)=0 if and only if t=0;(4)
ξ is subadditive.Also, we denote byΨ the family of all nondecreasing functions ψ:[0,∞)→[0,∞) such that ∑n=1∞‍ψn(t)<∞ for each t>0, where ψn is the nth iterate of ψ.Note that ifψ∈Ψ, then ψ(0)=0 and 0<ψ(t)<t for all t>0.Let(X,d) be a metric space, and let α:X×X→[0,∞) be a function.We consider the following conditions:(1)
for any sequence{xn} in X with α(xn,xn+1)≥1 for all n∈N and limn→∞xn=x, we have (3)α(xn,x)≥1∀n∈N;(2)
for any sequence{xn} in X with α(xn,xn+1)≥1 for all n∈N and a cluster point x of {xn}, we have (4)limn→∞inf⁡α(xn,x)≥1;(3)
for any sequence{xn} in X with α(xn,xn+1)≥1 for all n∈N and a cluster point x of {xn}, there exists a subsequence {xn(k)} of {xn} such that (5)α(xn(k),x)≥1∀k∈N.Remark 1.
(1) implies (2) and (2) implies (3).Note that if(X,d) is a metric space and ξ∈Ξ, then (X,ξ∘d) is a metric space.Let(X,d) be a metric space, and let T:X→CL(X) be a multivalued mapping. Then, we say that(1)
T is called  α*-admissible [2] if (6)α(x,y)≥1impliesα*(Tx,Ty)≥1,whereα*(Tx,Ty)=inf⁡{α(a,b):a∈Tx,b∈Ty};(2)
T is called α-admissible [7] if, for each x∈X and y∈Tx with α(x,y)≥1, we have α(y,z)≥1 for all z∈Ty.Lemma 2.
Let(X,d) be a metric space, and let T:X→CL(X) be a multivalued mapping. If T is α*-admissible, then it is α-admissible.Proof.
Suppose thatT is an α*-admissible mapping.
Letx∈X and y∈Tx be such that α(x,y)≥1.
Letz∈Ty be given.
SinceT is α*-admissible, α(y,z)≥α*(Tx,Ty)≥1.Lemma 3.
Let(X,d) be a metric space, and let ξ∈Ξ and B∈CL(X).
Ifa∈X and ξ(d(a,B))<c, then there exists b∈B such that ξ(d(a,b))<c.Proof.
Letϵ=c-ξ(d(a,B)).
Sinceξ(d(a,B))<c and ξ∘d is metric on X, there exists b∈B such that ξ(d(a,b))<ξ(d(a,B))+ϵ by definition of infimum. Hence, ξ(d(a,b))<c.Let(X,d) be a metric space.A functionf:X→[0,∞) is calledupper semicontinuous if, for each x∈X and {xn}⊂X with limn→∞xn=x, we have limn→∞f(xn)≤f(x).A functionf:X→[0,∞) is calledlower semicontinuous if, for each x∈X and {xn}⊂X with limn→∞xn=x, we have f(x)≤limn→∞f(xn).For a multivalued mapT:X→CL(X), let fT:X→[0,∞) be a function defined by fT(x)=d(x,Tx).


## 2. Fixed Point Theorems

In this section, we establish fixed point theorems for Ćirić-Berinde type contractive multivalued mappings.Theorem 4.
Let(X,d) be a complete metric space, and let α:X×X→[0,∞) be a function. Suppose that a multivalued mapping T:X→CL(X) is α-admissible.
Assume that, for allx,y∈X,α(x,y)≥1 implies (7)ξHTx,Ty≤ψξMx,y+Lξdy,Tx,where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing.
Also, suppose that the following are satisfied:(1)
there existsx0∈X and x1∈Tx0 such that α(x0,x1)≥1;(2)
eitherT is continuous or fT is lower semicontinuous.
ThenT has a fixed point in X.Proof.
Letx0∈X and x1∈Tx0 be such that α(x0,x1)≥1. Let c be a real number with ξ(d(x0,x1))<ξ(c).
Ifx0=x1, then x1 is a fixed point.
Letx0≠x1.
Ifx1∈Tx1, then x1 is a fixed point. Let x1∉Tx1. Then d(x1,Tx1)>0.
From (7) we obtain(8)0<ξdx1,Tx1≤ξ(H(Tx0,Tx1))≤ψξmax⁡⁡d(x0,x1),d(x0,Tx0),d(x1,Tx1),12hhhhhhhhh12dx0,Tx1+dx1,Tx0+Lξdx1,Tx0≤ψξmax⁡⁡d(x0,x1),d(x0,x1),d(x1,Tx1),12hhhhhhhhh12dx0,Tx1+dx1,x1+Lξ(d(x1,x1))≤ψξmax⁡⁡d(x0,x1),d(x0,x1),d(x1,Tx1),12hhhhhhhhh12dx0,x1+dx1,Tx1=ψξmax⁡dx0,x1,dx1,Tx1.
Ifmax⁡{d(x0,x1),d(x1,Tx1)}=d(x1,Tx1), then we have 0<ξ(d(x1,Tx1))≤ψ(ξ(d(x1,Tx1)))<ξ(d(x1,Tx1)), which is a contradiction.
Thus,max⁡{d(x0,x1),d(x1,Tx1)}=d(x0,x1), and hence we have (9)0<ξ(d(x1,Tx1))≤ψ(ξ(d(x0,x1)))<ψ(ξ(c)).
Hence, there existsx2∈Tx1 such that (10)ξ(d(x1,x2))<ψ(ξ(c)).
SinceT is α-admissible, from condition (1) and x2∈Tx1, we have (11)α(x1,x2)≥1.
Ifx2∈Tx2, then x2 is a fixed point. Let x2∉Tx2.
Thend(x2,Tx2)>0, and so ξ(d(x2,Tx2))>0.
From (7) we obtain(12)0<ξ(d(x2,Tx2))≤ξ(H(Tx1,Tx2))≤ψξmax⁡⁡d(x1,x2),d(x1,Tx1),d(x2,Tx2),12hhhhhhhhh12{d(x1,Tx2)+d(x2,Tx1)}+Lξ(d(x2,Tx1))≤ψξmax⁡⁡d(x1,x2),d(x1,x2),d(x2,Tx2),12hhhhhhhhh12{d(x1,Tx2)+d(x2,x2)}+Lξ(d(x2,x2))≤ψξmax⁡⁡d(x1,x2),d(x1,x2),d(x2,Tx2),12hhhhhhhhh12dx1,x2+dx2,Tx2=ψξmax⁡dx1,x2,dx2,Tx2.
Ifmax⁡{d(x1,x2),d(x2,Tx2)}=d(x2,Tx2), then we have ξ(d(x2,Tx2))≤ψ(ξ(d(x2,Tx2)))<ξ(d(x2,Tx2)), which is a contradiction.
Thus,max⁡{d(x1,x2),d(x2,Tx2)}=d(x1,x2), and hence we have (13)ξ(d(x2,Tx2))≤ψ(ξ(d(x1,x2)))<ψ2(ξ(c)).
Hence, there existsx3∈Tx2 such that (14)ξ(d(x2,x3))<ψ2(ξ(c)).
SinceT is α-admissible, from x2∈Tx1 and α(x1,x2)≥1, we have (15)α(x2,x3)≥1.
By induction, we obtain a sequence{xn}⊂X such that, for all n∈N∪{0}, (16)α(xn,xn+1)≥1,xn+1∈Txn,xn≠xn+1,ξ(d(xn,xn+1))<ψn(ξ(c)).
Letϵ>0 be given.
Since∑n=0∞‍ψn(ξ(c))<∞, there exists N∈N such that (17)∑n≥N‍ψn(ξ(c))<ξ(ϵ).
For allm>n≥N, we have(18)ξdxn,xm≤∑k=nm-1‍ψkξc<∑n≥N‍ψn(ξ(c))<ξ(ϵ)which implies d(xn,xm)<ϵ for all m>n≥N. Hence, {xn} is a Cauchy sequence in X.
It follows from the completeness ofX that there exists (19)x*=limn→∞xn∈X.
Suppose thatT is continuous.
We have(20)dx*,Tx*≤dx*,xn+1+dxn+1,Tx*≤d(x*,xn+1)+H(xn,Tx*).
By lettingn→∞ in the above inequality, we obtain d(x*,Tx*)=0, and so x*∈Tx*.
Assume thatfT is lower semicontinuous.
Then,fT(x*)≤limn→∞fT(xn). Hence, d(x*,Tx*)≤limn→∞d(xn,Txn)≤limn→∞d(xn,xn+1)=0. Thus, x*∈Tx*.Corollary 5.
Let(X,d) be a complete metric space, and let α:X×X→[0,∞) be a function. Suppose that T:X→CL(X) is an α-admissible mapping.
Assume that, for allx,y∈X, (21)ξαx,yHTx,Ty≤ψξMx,y+Lξdy,Tx,where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing.
Also, suppose that conditions (1) and (2) of Theorem4 are satisfied.
ThenT has a fixed point in X.Remark 6.
If we haveξ(t)=t for all t≥0, L=0, and T is continuous, then Corollary 5 reduces to Theorem 3.4 of [7].Let(X,⪯) be an ordered set and A,B⊂X. We say that A⪯B whenever, for each a∈A, there exists b∈B such that a⪯b.Corollary 7.
Let(X,⪯,d) be a complete ordered metric space. Suppose that a multivalued mapping T:X→CL(X) satisfies (22)ξ(H(Tx,Ty))≤ψ(ξ(M(x,y)))+Lξ(d(y,Tx))for all x,y∈X with Tx⪯Ty (resp., Ty⪯Tx), where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing.
Assume that, for eachx∈X and y∈Tx with Tx⪯Ty (resp., Ty⪯Tx), we have Ty⪯Tz (resp., Tz⪯Ty) for all z∈Ty.
Also, suppose that the following are satisfied:(1)
there existsx0∈X and x1∈Tx0 such that Tx0⪯Tx1 (resp., Tx1⪯Tx0);(2)
eitherT is continuous or fT is lower semicontinuous.
ThenT has a fixed point in X.Remark 8.
If we haveξ(t)=t for all t≥0, L=0, and T is continuous, then Corollary 7 reduces to Corollary 3.6 of [7].From Theorem4 we obtain the following result.Corollary 9.
Let(X,d) be a complete metric space, and let α:X×X→[0,∞) be a function. Suppose that T:X→CL(X) is an α*-admissible mapping.
Assume that, for allx,y∈X,α(x,y)≥1 implies (23)ξHTx,Ty≤ψξMx,y+Lξdy,Tx,where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing.
Also, suppose that conditions (1) and (2) of Theorem4 are satisfied.
ThenT has a fixed point in X.Remark 10.
If we haveL=0 in Corollary 9, then Corollary 9 reduces to Theorem 2.5 of [6].Corollary 11.
Let(X,d) be a complete metric space, and let α:X×X→[0,∞) be a function. Suppose that T:X→CL(X) is an α*-admissible mapping.
Assume that, for allx,y∈X, (24)ξαx,yHTx,Ty≤ψξMx,y+Lξdy,Tx,where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing.
Also, suppose that conditions (1) and (2) of Theorem4 are satisfied.
ThenT has a fixed point in X.Remark 12.
In Corollary11, let ξ(t)=t for all t≥0 and α(x,y)=1 for all x,y∈X and ψ(t)=kt for all t≥0, where k∈[0,1). If T is single valued map, then Corollary 11 reduces to Theorem 2.2 of [8].Theorem 13.
Let(X,d) be a complete metric space, and let α:X×X→[0,∞) be a function. Suppose that a multivalued mapping T:X→CL(X) is α-admissible.
Assume that, for allx,y∈X,α(x,y)≥1 implies (25)ξHTx,Ty≤ψξMx,y+Lξdy,Tx,where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing and upper semicontinuous function. Also, suppose that the following are satisfied: (1)
there existsx0∈X and x1∈Tx0 such that α(x0,x1)≥1;(2)
for a sequence{xn} in X with α(xn,xn+1)≥1 for all n∈N∪{0} and a cluster point x of {xn}, there exists a subsequence {xn(k)} of {xn} such that, for all k∈N∪{0}, (26)α(xn(k),x)≥1.
ThenT has a fixed point in X.Proof.
Following the proof of Theorem4, we obtain a sequence {xn}⊂X with limn→∞xn=x*∈X such that, for all n∈N∪{0},(27)xn+1∈Txn,xn≠xn+1,α(xn,xn+1)≥1.
From (2) there exists a subsequence{xn(k)} of {xn} such that (28)α(xn(k),x*)≥1.
Thus, we have(29)ξdxnk+1,Tx*=ξHTxnk,Tx*≤ψ(ξ(M(xnk,x*)))+Lξdx*,xnk+1,where (30)Mxnk,x*=max⁡⁡d(xnk,x*),d(xnk,xnk+1),d(x*,Tx*),12hhhhh12dxnk,Tx*+dx*,xnk+1.
We have(31)limk→∞M(xn(k),x*)=d(x*,Tx*),and so (32)limk→∞ξ(M(xn(k),x*))=ξ(d(x*,Tx*)).
Suppose thatd(x*,Tx*)≠0.
Sinceψ is upper semicontinuous, (33)limk→∞ψ(ξ(M(xn(k),x*)))≤ψ(ξ(d(x*,Tx*))).
Lettingk→∞ in inequality (29) and using continuity of ξ, we obtain(34)0<ξdx*,Tx*≤limk→∞ψξMxnk,x*+limk→∞Lξdx*,xnk+1≤ψ(ξ(d(x*,Tx*)))<ξ(d(x*,Tx*))which is a contradiction. Hence, d(x*,Tx*)=0, and hence x* is a fixed point of T.The following example shows that upper semicontinuity ofψ cannot be dropped in Theorem 13.Example 14.
LetX=[0,∞), and let d(x,y)=|x-y| for all x,y≥0.
Define a mappingT:X→CL(X) by (35)Tx={12,1}(x=0),{34x}(0<x≤1),{2x}(x>1).
Letξ(t)=t for all t≥0, and let(36)ψ(t)=45t(t≥1),34t(0≤t<1).
Then,ξ∈Ξ, and ψ∈Ψ and ψ is a strictly increasing function.
Letα,η:X×X→[0,∞) be defined by(37)α(x,y)=4(0≤x,y≤1),0otherwise.
Obviously, condition (2) of Theorem13 is satisfied. Condition (1) of Theorem 13 is satisfied with x0=1/4.
We show that (7) is satisfied.
Letx,y∈X be such that α(x,y)≥1.
Then,0≤x,y≤1.
Ifx=y, then obviously (7) is satisfied.
Letx≠y.
Ifx=0 and 0<y≤1, then we obtain(38)ξHTx,Ty=H12,1,34y≤14≤ψ(d(x,Tx))≤ψ(ξ(M(x,y))).
Let0<x≤1 and 0<y≤1.
Then, we have(39)ξHTx,Ty=dTx,Ty=d34x,34y=34x-y=ψdx,y≤ψ(ξ(M(x,y))).
Thus, (7) is satisfied.
We now show thatT is α-admissible.
Letx∈X be given, and let y∈Tx be such that α(x,y)≥1.
Then,0≤x, y≤1.
Obviously,α(y,z)≥1 for all z∈Ty whenever 0<y≤1.
Ify=0, then Ty={1/2,1}. Hence, for all z∈Ty, α(y,z)≥1.
Hence,T is α-admissible. Thus, all hypotheses of Theorem 13 are satisfied. However, T has no fixed points.Note thatψ is not upper semicontinuous.Corollary 15.
Let(X,d) be a complete metric space, and let α:X×X→[0,∞) be a function. Suppose that T:X→CL(X) is an α-admissible mapping.
Assume that, for allx,y∈X, (40)ξαx,yHTx,Ty≤ψξMx,y+Lξdy,Tx,where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing and upper semicontinuous function. Also, suppose that conditions (1) and (2) of Theorem 13 are satisfied.
ThenT has a fixed point in X.Corollary 16.
Let(X,⪯,d) be a complete ordered metric space. Suppose that a multivalued mapping T:X→CL(X) satisfies (41)ξ(H(Tx,Ty))≤ψ(ξ(M(x,y)))+Lξ(d(y,Tx))for all x,y∈X with Tx⪯Ty (resp., Ty⪯Tx), where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing and upper semicontinuous function.
Assume that, for eachx∈X and y∈Tx with Tx⪯Ty (resp., Ty⪯Tx), we have Ty⪯Tz (resp., Tz⪯Ty) for all z∈Ty.
Also, suppose that the following are satisfied:(1)
there existsx0∈X and x1∈Tx0 such that Tx0⪯Tx1 (resp., Tx1⪯Tx0);(2)
for a sequence{xn} in X with xn⪯xn+1 (resp., xn+1⪯xn) for all n∈N∪{0} and a cluster point x of {xn}, there exists a subsequence {xn(k)} of {xn} such that, for all k∈N∪{0}, (42)xn(k)⪯x(resp.,x⪯xn(k)).
ThenT has a fixed point in X.Remark 17.
Corollary16 is a generalization and extension of the result of [9] to multivalued mappings.Corollary 18.
Let(X,d) be a complete metric space, and let α:X×X→[0,∞) be a function. Suppose that a multivalued mapping T:X→CL(X) is α*-admissible.
Assume that, for allx,y∈X, α(x,y)≥1 implies (43)ξHTx,Ty≤ψξMx,y+Lξdy,Tx,where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing and upper semicontinuous function.
Also, suppose that conditions (1) and (2) of Theorem13 are satisfied.
ThenT has a fixed point in X.Remark 19.
By takingL=0 in Corollary 18 and by applying Remark 1, Corollary 18 reduces to Theorem 2.6 of [6].Corollary 20.
Let(X,d) be a complete metric space, and let α:X×X→[0,∞) be a function. Suppose that T:X→CL(X) is an α*-admissible mapping.
Assume that, for allx,y∈X,(44)ξαx,yHTx,Ty≤ψξMx,y+Lξdy,Tx,where L≥0, ξ∈Ξ, and ψ∈Ψ is strictly increasing and upper semicontinuous function.
Also, suppose that conditions (1) and (2) of Theorem13 are satisfied.
ThenT has a fixed point in X.



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