# An Approach to Existence of Fixed Points of Generalized Contractive Multivalued Mappings of Integral Type via Admissible Mapping

Authors: Muhammad Usman Ali; Tayyab Kamran; Erdal Karapınar
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We investigate the existence of a fixed point of certain contractive multivalued mappings of integral type by using the admissible mapping. Our results generalize the several results on the topic in the literature involving Branciari, and Feng and Liu. We also construct some examples to illustrate our results.


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## Body

## 1. Preliminaries and Introduction

Fixed point theory is one of the most celebrated research areas that has an application potential not only in nonlinear but also in several branches of mathematics. As a consequence of this fact, several fixed point results have been reported. It is not easy to know, manage, and use all results of this reich theory to get an application. To overcome such problems and clarify the literature, several authors have suggested a more general construction in a way that a number of existing results turn into a consequence of the proposed one. One of the examples of this trend is the investigations of fixed point of certain operator by using theα-admissible mapping introduced Samet et al. [1]. This paper has been appreciated by several authors and this trend has been supported by reporting several interesting results; see for example [2–12].In this paper, we define (α*,ψ)-contractive multivalued mappings of integral type and discuss the existence of a fixed point of such mappings. Our construction and hence results improve, extend, and generalize several results including Branciari [13] and Feng and Liu [14].In what follows, we recall some basic definitions, notions, notations, and fundamental results for the sake of completeness. LetΨ be a family of nondecreasing functions, ψ:[0,∞)→[0,∞) such that ∑n=1∞ψn(t)<∞for each t>0, where ψn is the nth iterate of ψ. It is known that, for each ψ∈Ψ, we have ψ(t)<t for all t>0 and ψ(0)=0fort=0[1]. We denote byΦthe set of all Lebesgue integrable mappings,ϕ:[0,∞)→[0,∞)which is summable on each compact subset of[0,∞)and∫0ϵ‍ϕ(t)dt>0, for eachϵ>0.Let(X,d)be a metric space. We denote byN(X)the space of all nonempty subsets ofX, byB(X)the space of all nonempty bounded subsets ofX, and byCL(X)the space of all nonempty closed subsets ofX. ForA∈N(X)andx∈X,
(1)d(x,A)=inf⁡{d(x,a):a∈A}.
For everyA,B∈B(X),
(2)δ(A,B)=sup⁡{d(a,b):a∈A,b∈B}.
We denoteδ(A,B)byδ(x,B)whenA={x}. If, forx0∈X, there exists a sequence{xn}n∈NinXsuch thatxn∈Gxn-1, thenO(G,x0)={x0,x1,x2,…}is said to be an orbit ofG:X→CL(X)atx0. A mappingf:X→R isGorbitally lower semicontinuous atx, if{xn}is a sequence inO(G,x0)andxn→ximpliesf(x)≤liminf⁡nf(xn). Branciari [13] extended the Banach contraction principle [15] in the following way.Theorem 1.
Let(X,d)be a complete metric space and letG:X→Xbe a mapping such that
(3)∫0d(Tx,Ty)ϕ(t)dt≤c∫0d(x,y)ϕ(t)dt
for eachx,y∈X, wherec∈[0,1)andϕ∈Φ. ThenGhas a unique fixed point.Since then many authors used integral type contractive conditions to prove fixed point theorems in different settings; see for example [12, 16–22]. Feng and Liu [14] extended the result of Branciari [13] to multivalued mappings as follows.Theorem 2 (see [14]).
Let(X,d)be a complete metric space and letG:X→CL(X)be a mapping. Assume that for eachx∈Xandy∈Gx, there existsz∈Gysuch that
(4)∫0d(y,z)ϕ(t)dt≤ψ(∫0d(x,y)ϕ(t)dt),
whereψ∈Ψandϕ∈Φ. ThenGhas a fixed point inXprovidedf(ξ)=d(ξ,Gξ)is lower semicontinuous, withξ∈X.Definition 3 (see [3]).
Let(X,d)be a metric space andα:X×X→[0,∞)be a mapping. A mappingG:X→CL(X)isα*-admissible ifα(x,y)≥1⇒α*(Gx,Gy)≥1, whereα*(Gx,Gy)=inf⁡{α(a,b):a∈Gx,b∈Gy}.Definition 4 (see [3]).
Let(X,d)be a metric space. A mappingG:X→CL(X)is calledα*-ψ-contractive if there exist two functionsα:X×X→[0,∞)andψ∈Ψsuch that
(5)α*(Gx,Gy)H(Gx,Gy)≤ψ(d(x,y))
for allx,y∈X.Theorem 5 (see [3]).
Let(X,d)be a complete metric space, letα:X×X→[0,∞)be a function, letψ∈Ψbe a strictly increasing map, and letGbe a closed-valuedα*-admissible andα*-ψ-contractive multifunction onX. Suppose that there existx0∈Xandx1∈Gx0such thatα(x0,x1)≥1. Assume that if{xn}is a sequence inXsuch thatα(xn,xn+1)≥1for allnandxn→x, thenα(xn,x)≥1for alln. ThenGhas a fixed point.Definition 6 (see [2]).
Let(X,d)be a metric space and letG:X→CL(X)be a mapping. We say thatGis a generalized(α*,ψ)-contractive if there existsψ∈Ψsuch that
(6)α*(Gx,Gy)d(y,Gy)≤ψ(d(x,y))
for eachx∈Xandy∈Gx, whereα*(Gx,Gy)=inf⁡{α(a,b):a∈Gx,b∈Gy}.Theorem 7 (see [2]).
Let(X,d)be a complete metric space and letG:X→B(X)be a mapping such that for eachx∈Xandy∈Gx, we have
(7)α*(Gx,Gy)δ(y,Gy)≤ψ(d(x,y)),
whereψ∈Ψ. Assume that there existx0∈Xandx1∈Gx0such thatα(x0,x1)≥1. MoreoverGis anα*-admissible mapping. Then there exists an orbit{xn}ofGatx0andx∈Xsuch thatlim⁡n→∞xn=x. Moreover,{x}=Gxif and only iff(ξ)=δ(ξ,Gξ)is lower semicontinuous atx.


## 2. Main Results

In this section, we state and proof our main results. We first give the definition of the following notion.Definition 8.
Let(X,d)be a metric space. We say thatG:X→CL(X)is an integral type(α*,ψ)-contractive mapping if there exist two functionsψ∈Ψandϕ∈Φsuch that for eachx∈Xandy∈Gx, there existsz∈Gysatisfying
(8)∫0α*(Gx,Gy)d(y,z)‍ϕ(t)dt≤ψ(∫0d(x,y)‍ϕ(t)dt),
whereα*(Gx,Gy)=inf⁡{α(a,b):a∈Gx,b∈Gy}.Example 9.
LetX=R be endowed with the usual metricd. DefineG:X→CL(X)by
(9)Gx={[x,∞)ifx≥0(-∞,6x]ifx<0,
andα:X×X→[0,∞)by
(10)α(x,y)={x+y+1ifx,y≥00otherwise.
Takeψ(t)=t/4andϕ(t)=2tfor allt≥0. Then, for eachx∈Xandy∈Gx, there existsz∈Gysuch that
(11)∫0α*(Gx,Gy)d(y,z)‍ϕ(t)dt≤ψ(∫0d(x,y)‍ϕ(t)dt).
HenceGis an integral type(α*,ψ)-contractive mapping. Note that (4) does not hold atx=-2.Definition 10.
We say thatϕ∈Φis an integral subadditive if, for eacha,b>0, we have
(12)∫0a+b‍ϕ(t)dt≤∫0a‍ϕ(t)dt+∫0b‍ϕ(t)dt.We denote byΦs the class of all integral subadditive functionsϕ∈Φ.Example 11.
Letϕ1(t)=(1/2)(t+1)-1/2for allt≥0, ϕ2(t)=(2/3)(t+1)-1/3for allt≥0,  andϕ3(t)=e-tfor allt≥0. Thenϕi∈Φs, wherei=1,2,3.Definition 12.
Let(X,d)be a metric space. We say thatG:X→CL(X)is a subintegral type(α*,ψ)-contractive if there exist two functionsψ∈Ψandϕ∈Φssuch that for eachx∈Xandy∈Gx, there existsz∈Gysatisfying
(13)∫0α*(Gx,Gy)d(y,z)‍ϕ(t)dt≤ψ(∫0d(x,y)‍ϕ(t)dt),
whereα*(Gx,Gy)=inf⁡{α(a,b):a∈Gx,b∈Gy}.Example 13.
LetX=R be endowed with the usual metricd. DefineG:X→CL(X)by
(14)Gx={[x4,x2]ifx≥0,[24x,12x]ifx<0,
andα:X×X→[0,∞)by
(15)α(x,y)={2ifx=y=0,0otherwise.
Takeψ(t)=t/3andϕ(t)=(2/3)(t+1)-1/3for allt≥0. Then, for eachx∈Xandy∈Gx, there existsz∈Gysuch that
(16)∫0α*(Gx,Gy)d(y,z)‍ϕ(t)dt≤ψ(∫0d(x,y)‍ϕ(t)dt).
HenceGis an subintegral type(α*,ψ)-contractive mapping.Theorem 14.
Let(X,d)be a complete metric space and letG:X→CL(X)be anα*-admissible subintegral type(α*,ψ)-contractive mapping. Assume that there existx0∈Xandx1∈Gx0such thatα(x0,x1)≥1. Then there exists an orbit{xn}ofGatx0andx∈Xsuch thatlim⁡n→∞xn=x. Moreover,xis a fixed point ofGif and only iff(ξ)=d(ξ,Gξ)isGorbitally lower semicontinuous atx.Proof.
By the hypothesis, there existx0∈Xandx1∈Gx0such thatα(x0,x1)≥1. SinceGisα*-admissible, thenα*(Gx0,Gx1)≥1. Forx0∈Xandx1∈Gx0, there existsx2∈Gx1such that
(17)∫0d(x1,x2)‍ϕ(t)dt≤∫0α*(Gx0,Gx1)d(x1,x2)‍ϕ(t)dt≤ψ(∫0d(x0,x1)‍ϕ(t)dt).
Sinceψis nondecreasing, we have
(18)ψ(∫0d(x1,x2)‍ϕ(t)dt)≤ψ2(∫0d(x0,x1)‍ϕ(t)dt).
Asα(x1,x2)≥1byα*-admissibility ofG, we haveα*(Gx1,Gx2)≥1. Forx1∈Xandx2∈Gx1, there existsx3∈Gx2such that
(19)∫0d(x2,x3)‍ϕ(t)dt≤∫0α*(Gx1,Gx2)d(x2,x3)‍ϕ(t)dt≤ψ(∫0d(x1,x2)‍ϕ(t)dt)≤ψ2(∫0d(x0,x1)‍ϕ(t)dt).
Sinceψis nondecreasing, we have
(20)ψ(∫0d(x2,x3)‍ϕ(t)dt)≤ψ3(∫0d(x0,x1)‍ϕ(t)dt).
By continuing the same process, we get a sequence{xn}inXsuch thatxn∈Gxn-1,α(xn-1,xn)≥1, and
(21)∫0d(xn,xn+1)‍ϕ(t)dt≤ψn(∫0d(x0,x1)‍ϕ(t)dt),foreachn∈N.
Lettingn→∞in above inequality, we have
(22)lim⁡n→∞∫0d(xn,xn+1)‍ϕ(t)dt=0.
Also, we have
(23)lim⁡n→∞∫0d(xn,Gxn)‍ϕ(t)dt=0,
which implies that
(24)lim⁡n→∞d(xn,Gxn)=0.
For anyn,p∈N, we have
(25)d(xn,xn+p)≤∑i=nn+p-1‍d(xi,xi+1).
Sinceϕ∈Φs, it can be shown by induction that
(26)∫0d(xn,xn+p)‍ϕ(t)dt≤∑i=nn+p-1‍∫0d(xi,xi+1)‍ϕ(t)dt.
From (21) and (26), we have
(27)∫0d(xn,xn+p)‍ϕ(t)dt≤∑i=nn+p-1ψi(∫0d(x0,x1)‍ϕ(t)dt).
Sinceψ∈Ψit follows that{xn}is Cauchy sequence inX. AsXis complete, there existsx*∈Xsuch thatxn→x*asn→∞. Supposef(ξ)=d(ξ,Gξ)isGorbitally lower semicontinuous atx*; then
(28)d(x*,Gx*)≤liminf⁡nf(xn)=liminf⁡nd(xn,Gxn)=0.
By closedness ofGit follows thatx*∈Gx*. Conversely, suppose thatx*is a fixed point ofGthenf(x*)=0≤liminf⁡nf(xn).Example 15.
LetX=R be endowed with the usual metricd. DefineG:X→CL(X)by
(29)Gx={[x,x+1]ifx≥0,(-∞,6x]ifx<0,
andα:X×X→[0,∞)by
(30)α(x,y)={x+y+1ifx,y≥0,0otherwise.
Takeψ(t)=t/2andϕ(t)=(1/2)(t+1)-1/2for allt≥0. Then, for eachx∈Xandy∈Gx, there existsz∈Gysuch that
(31)∫0α*(Gx,Gy)d(y,z)‍ϕ(t)dt≤ψ(∫0d(x,y)‍ϕ(t)dt).
HenceGis a subintegral type(α*,ψ)-contractive mapping. Clearly,Gisα*-admissible. Also, we havex0=1andx1=2∈Gx0such thatα(x0,x1)=4. Therefore, all the conditions of Theorem 14 are satisfied andGhas infinitely many fixed points. Note that Theorem 2 in Section 1 is not applicable here. For example, takex=-1andy=-6.Definition 16.
Let(X,d)be a metric space. We say thatG:X→B(X)is an integral type(α*,ψ,δ)-contractive mapping if there exist two functionsψ∈Ψandϕ∈Φsuch that
(32)∫0α*(Gx,Gy)δ(y,Gy)‍ϕ(t)dt≤ψ(∫0d(x,y)‍ϕ(t)dt)
for eachx∈Xandy∈Gx, whereα*(Gx,Gy)=inf⁡{α(a,b):a∈Gx,b∈Gy}.Definition 17.
Let(X,d)be a metric space. We say thatG:X→B(X)is a subintegral type(α*,ψ,δ)-contractive mapping if there exist two functionsψ∈Ψandϕ∈Φssuch that
(33)∫0α*(Gx,Gy)δ(y,Gy)‍ϕ(t)dt≤ψ(∫0d(x,y)‍ϕ(t)dt)
for eachx∈Xandy∈Gx, whereα*(Gx,Gy)=inf⁡{α(a,b):a∈Gx,b∈Gy}.Theorem 18.
Let(X,d)be a complete metric space and letG:X→B(X)be anα*-admissible subintegral type(α*,ψ,δ)-contractive mapping. Assume that there existx0∈Xandx1∈Gx0such thatα(x0,x1)≥1. Then there exists an orbit{xn}ofGatx0andx∈Xsuch thatlim⁡n→∞xn=x. Moreover,x∈Xsuch that{x}=Gxif and only iff(ξ)=δ(ξ,Gξ)isGorbitally lower semicontinuous atx.Proof.
By the hypothesis, there existx0∈Xandx1∈Gx0such thatα(x0,x1)≥1. SinceGisα*-admissible, thenα*(Gx0,Gx1)≥1. Forx0∈Xandx1∈Gx0, we have
(34)∫0α*(Gx0,Gx1)δ(x1,Gx1)‍ϕ(t)dt≤ψ(∫0d(x0,x1)‍ϕ(t)dt).
SinceGx1≠∅, then we havex2∈Gx1such that
(35)∫0d(x1,x2)‍ϕ(t)dt≤∫0α*(Gx0,Gx1)δ(x1,Gx1)‍ϕ(t)dt≤ψ(∫0d(x0,x1)‍ϕ(t)dt).
Sinceψis nondecreasing, we have
(36)ψ(∫0d(x1,x2)‍ϕ(t)dt)≤ψ2(∫0d(x0,x1)‍ϕ(t)dt).
Asα(x1,x2)≥1byα*-admissibility ofG, we haveα*(Gx1,Gx2)≥1. Thus, we havex3∈Gx2such that
(37)∫0d(x2,x3)‍ϕ(t)dt≤∫0α*(Gx1,Gx2)δ(x2,Gx2)‍ϕ(t)dt≤ψ(∫0d(x1,x2)‍ϕ(t)dt)≤ψ2(∫0d(x0,x1)‍ϕ(t)dt).
Sinceψis nondecreasing, we have
(38)ψ(∫0d(x2,x3)‍ϕ(t)dt)≤ψ3(∫0d(x0,x1)‍ϕ(t)dt).
By continuing the same process, we get a sequence{xn}inXsuch thatxn∈Gxn-1,α(xn-1,xn)≥1, and
(39)∫0d(xn,xn+1)‍ϕ(t)dt≤∫0δ(xn,Gxn)‍ϕ(t)dt≤ψn(∫0d(x0,x1)‍ϕ(t)dt),hihhhhhhforeachn∈N.
Lettingn→∞in above inequality, we have
(40)lim⁡n→∞∫0δ(xn,Gxn)‍ϕ(t)dt=0,
which implies that
(41)lim⁡n→∞δ(xn,Gxn)=0.
For anyn,p∈N, we have
(42)d(xn,xn+p)≤∑i=nn+p-1‍d(xi,xi+1).
Sinceϕ∈Φs, it can be shown by induction that
(43)∫0d(xn,xn+p)‍ϕ(t)dt≤∑i=nn+p-1‍∫0d(xi,xi+1)‍ϕ(t)dt.
From (39) and (43), we have
(44)∫0d(xn,xn+p)‍ϕ(t)dt≤∑i=nn+p-1‍ψi(∫0d(x0,x1)‍ϕ(t)dt).
Sinceψ∈Ψit follows that{xn}is Cauchy sequence inX. AsXis complete, there existsx*∈Xsuch thatxn→x*asn→∞. Supposef(ξ)=δ(ξ,Gξ)isGorbitally lower semicontinuous atx*; then
(45)δ(x*,Gx*)≤liminf⁡nf(xn)=liminf⁡nδ(xn,Gxn)=0.
Hence,{x*}=Gx*becauseδ(A,B)=0impliesA=B={a}. Conversely, suppose that{x*}=Gx*. Thenf(x*)=0≤liminf⁡nf(xn).Example 19.
LetX={1,3,5,7,9,…}be endowed with the usual metricd. DefineG:X→B(X)by
(46)Gx={{x-2,x+2}ifx≠1,{1}ifx=1,
andα:X×X→[0,∞)by
(47)α(x,y)={1ifx=y=1,14otherwise.
Takeψ(t)=t/2andϕ(t)=(2/3)(t+1)-1/3for allt≥0. Clearly,Gis anα*-admissible subintegral type(α*,ψ,δ)-contractive mapping. Also, we havex0=1andx1=1∈Gx0such thatα(x0,x1)=1. Therefore, all the conditions of Theorem 18 hold andGhas fixed points.Example 20.
LetX=R be endowed with the usual metricd. DefineG:X→B(X)by
(48)Gx={{⌊x⌋,⌈x⌉}ifx≥0,(⌊x⌋4,⌈x⌉2)ifx<0,
andα:X×X→[0,∞)by
(49)α(x,y)={1ifx,y≥0,0otherwise.
Takeψ(t)=t/4andϕ(t)=e-tfor allt≥0. Then it is easy to check that all the conditions of Theorem 18 hold. ThereforeGhas infinitely many fixed points.Remark 21.
Letϕ(t)=1for allt≥0; Theorem 18 reduces to Theorem 7 in Section 1.Remark 22.
Note that subadditivity of the integral was needed in the proofs of Theorems14 and 18 in order to obtain inequalities (26) and (43). It is natural to ask wether the conclusions of Theorems 14 and 18 are valid if we replace subintegral contractive conditions (13) and (33) by integral contractive conditions (8) and (32), respectively. Looking at our proofs, we can say that it will be true if the inequalities (26) and (43) hold. Here we would like to mention that many authors (see for example [14, 23]) while proving the results on integral contractions have not assumed that the integral is subadditive but indeed they used the subadditivity of the integral in the proofs of their results while obtaining the inequalities comparable to inequalities (26) and (43).


## 3. Application

In this section, we obtain some fixed point results for partially ordered metric spaces, as consequences of aforementioned results. Moreover, we apply our result to prove the existence of solution for an integral equation.LetAandBbe subsets of a partially ordered set. We say thatA⪯rB, if for eacha∈Aandb∈B, we havea⪯b.Theorem 23.
Let(X,⪯,d)be a complete ordered metric space and letG:X→CL(X)be a mapping such that for eachx∈Xandy∈Gxwithx⪯y, there existsz∈Gysatisfying
(50)∫0d(y,z)‍ϕ(t)dt≤ψ(∫0d(x,y)‍ϕ(t)dt),
whereψ∈Ψandϕ∈Φs. Assume that there existx0∈Xandx1∈Gx0such thatx0⪯x1. Also, assume thatx⪯yimpliesGx⪯rGy. Then there exists an orbit{xn}ofGatx0andx∈Xsuch thatlim⁡n→∞xn=x. Moreover,xis a fixed point ofGif and only iff(ξ)=d(ξ,Gξ)isGorbitally lower semicontinuous atx.Proof.
Defineα:X×X→[0,∞)by
(51)α(x,y)={1ifx⪯y,0otherwise.
By using hypothesis of corollary and definition ofα, we haveα(x0,x1)=1. Asx⪯yimpliesGx⪯rGy, by using the definitions ofαand⪯r, we have thatα(x,y)=1impliesα*(Gx,Gy)=1. Moreover, it is easy to check thatGis an integral type(α*,ψ)-contractive mapping. Therefore, by Theorem 14, there exists an orbit{xn}ofGatx0andx∈Xsuch thatlim⁡n→∞xn=x. Moreover,xis a fixed point ofGif and only iff(ξ)=d(ξ,Gξ)isGorbitally lower semicontinuous atx.ConsideringG:X→Xandϕ(t)=1for eacht≥0, Theorem 23 reduces to following result.Corollary 24.
Let(X,⪯,d)be a complete ordered metric space and letG:X→Xbe a nondecreasing mapping such that, for eachx∈Xwithx⪯Gx, we have
(52)d(Gx,G2x)≤ψ(d(x,Gx)),
whereψ∈Ψ. Assume that there existsx0∈Xsuch thatx0⪯Gx0. Then there exists an orbit{xn}ofGatx0andx∈Xsuch thatlim⁡n→∞xn=x. Moreover,xis a fixed point ofGif and only iff(ξ)=d(ξ,Gξ)isGorbitally lower semicontinuous atx.Consider an integral equation of the form(53)x(t)=∫ab‍K(t,s,x(s))ds,t∈[a,b],
whereK:[a,b]×[a,b]×R→R is continuous and nondecreasing.Theorem 25.
Assume that(i)
foru,v∈C([a,b],R), withu(t)≤v(t)for eacht∈[a,b], we have
(54)|K(t,s,u(t))-K(t,s,v(t))|≤ψ(d(u,v))(b-a)
for eacht,s∈[a,b], whereψ∈Ψ;(ii)
for eacht,s∈[a,b], there existsx0∈C([a,b],R) such that
(55)x0(t)≤∫ab‍K(t,s,x0(s))ds.Then there exists an iterative sequence{xn}, starting fromx0, andx∈C([a,b],R)  such thatlim⁡n→∞xn=x. Moreover,xis a solution of (53) if and only iff(ξ)=d(ξ,y)is lower semicontinuous atx, wherey(t)=∫ab‍K(t,s,ξ(s))ds.Proof.
It is easy to see thatX=C([a,b],R) is complete with respect to the metricd(x,y)=max⁡t∈[a,b]|x(t)-y(t)|. We define partial ordering onXas follows:x⪯yif and only ifx(t)≤y(t)for eacht∈[a,b]. DefineG:X→XbyGx=y, wherey(t)=∫ab‍K(t,s,x(s))ds, for eacht,s∈[a,b]. From (ii), we havex0⪯Gx0. Forx∈X, letGx=yandGy=z; that is, y(t)=∫ab‍K(t,s,x(s))dsandz(t)=∫ab‍K(t,s,y(s))ds, for eacht,s∈[a,b]. Then, for eachx∈Xwithx⪯Gx, we have
(56)d(Gx,G2x)=max⁡t∈[a,b]|y(t)-z(t)|=max⁡t∈[a,b]|∫ab‍K(t,s,x(s))dshhhh-∫ab‍K(t,s,y(s))ds|≤max⁡t∈[a,b]∫ab|K(t,s,x(s))-K(t,s,y(s))|ds≤ψ(d(x,Gx))(b-a)(b-a).
That isd(Gx,G2x)≤ψ(d(x,Gx)), for eachx∈Xwithx⪯Gx. Clearly,Gis nondecreasing. Therefore, all conditions of Corollary 24 hold and the conclusions follow from Corollary 24.



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