# Common Fixed Point Results via Set-Valued Generalized Weak Contraction with Directed Graph and Its Application

Authors: Muhammad Shoaib; Muhammad Sarwar; Kamal Shah; Nabil Mlaiki
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this manuscript, common fixed point results for set-valued mapping under generalizedψ,ϕ1 and ψ,ϕ2 weak contraction without using Hausdorff metric are studied endowing with a graph. To demonstrate the authenticity of the established result, a suitable example and application to integral inclusion are also discussed.


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## Body

## 1. Introduction

The basic fixed point (F.P) result of Banach [1] has set the basis of metric fixed point theory (FPT) in a complete metric space for contraction mappings. Due to their results, FPT of specific single-valued mappings is interesting in its benefit owning constructive proofs and applications in industrial fields such as physics, computer science, engineering, image processing, telecommunication, and economics.The theory of set-valued mapping has applications in convex optimization, control theory, economics and differential inclusions, and integral inclusion. Differential inclusions are also used to analyze modeling errors and analyze robustness to bounded perturbations to model physical phenomena such as to model differential games and Coulomb friction and impact [2, 3]. Following the Banach contraction principle (BCP), Nadler [4] presented the idea of set-valued contractions and established that in complete metric space, a set-valued contraction possesses a F.P. Consequently, many authors generalized Nadler F.P theorem in various ways.Connecting FPT and graph theory, Echenique [5] provided a proof of Tarski fixed point result by using graphs. Espinola and Kirk [6] in 2006 applied fixed point results in graph theory. Recently, two fundamental results have appeared for FPT with a graph. The first result was given by Jachymski [7] for single-valued mappings and consequently Beg et al. [8] continued Jachymski result for set-valued mappings. Subsequently, Beg et al. [8] extended some results in [9] for set-valued mappings. Recently, Bojor [10] succeeded Jachymsky idea for Kannan contractions applying a new postulate called the weak T-connectivity of the graph. Fallahi and Aghanians formulated Chatterjea contractions using graphs in metric spaces endowed with a graph and investigated the existence of F.Ps. Kutbi and Sintunavarat established F.P analysis for set-valued o with graph approach by the generalized Hausdorff distance. Abbas and Nazir [11] got few results for pair of power graphic contraction on a metric space along with a graph. On metric space along with a graph, Abbas et al. [12] established the presence of common F.Ps of set-valued F- contraction mappings. For more details, see [13, 14].LetQ≠∅, △ represent the diagonal of Q×Q and G=vG,eG be a directed graph (with no parallel edges). The set vG denoted vertices coincides with Q, and the set eG is the edges of the graph furthermore △⊆eG. e∗G denotes the set all edges which is not loops, i.e., e∗G=eG−△. If path exist between any two vertices, then a graph G is connected. It is weakly connected if G˜ is connected; here, G˜ is an undirected form of the graph G.A clique,C, in an undirected graph G=vG,eG is a subset of the vertices, C⊆vG, such that every two distinct vertices are adjacent. This is equivalent to the condition that the induced subgraph of G induced by C is a complete graph.Jachymski and Jozwik [15] used the property P stated that for any sequence ϑn in Q; if ϑn⟶ϑ and ϑn,ϑn+1∈eG, then ϑn,ϑ∈eG.A mappingT:Q⟶CLQ is said to be upper semicontinuous if for ϑn∈Q and ηn∈Tϑn with ϑn⟶ϑ0 and ηn⟶η0 implies η0∈Tϑ0.AssumeT1,T2:Q⟶CLQ. Set(1)QT1,T2=r∈Q:r,ϑr∈eGwhereϑr∈T1r∩T2r.Definition 1.
Consider a metric spaceQ,d, G=vG,eG be a graph such that vG=Q, and let T:Q⟶CLQ. If m1,n1∈eG,u1,v1∈eG for all u1∈Tm1 and v1∈Tn1, then T is said to be graph-preserving.
Motivated from above in the present work, we give common F.P results for generalizedψ,ϕ1 and ψ,ϕ2 weak contraction, in metric spaces (endowed with a graph). To demonstrate the authenticity of our result, we give suitable example. Also, we discuss application to integral inclusion.


## 2. Fixed Point Results

In our main work, we used the following three classes:(2)Ψ=ψ:ℝ+⟶ℝ+is non−decreasing,continuous mapping such thatψς=0if and only ifς=0,Φ=ϕ:ℝ+⟶ℝ+non−decreasinglimn⟶∞ϕςn=0⇒limn⟶∞ςn=0,Ψ1=ψ1:ℝ5⟶ℝ+iψ1is continuous and non−decreasing in each coordinateiiψ1ς1,ς2,ς3,ς4,ς5=0impliesς1=ς2=ς3=ς4=ς5=0andψ1ς,ς,ς,ς,ς≤ςforς>0.Definition 2.
Consider a completeQ,d metric space. Assume T1,T2:Q⟶CLQ be two set valued mapping. Assume that for each vertex r in G and for each ϑr∈Tir,i=1,2, we have r,ϑr∈eG. A pair T1,T2 is said to be(i)
A graphicψ,ϕ1 contraction if for any r,t∈Q with r,t∈eG and ϑr∈Tir, there exists ϑt∈Tjt for i≠j and i,j=1,2 and ϑr,ϑt∈e∗G and(3)ψdϑr,ϑt≤ψΛψ1r,t;ϑr,ϑt−ϕΛψ1r,t;ϑr,ϑt,hold whereψ∈Ψ, ϕ∈Φ and ψ1∈Ψ1 and(4)Λψ1r,t;ϑr,ϑt=ψ1dr,t,dr,ϑr,dt,ϑt,dr,ϑt+dt,ϑr2,dt,ϑr+dr,ϑt2,∀r,t∈Q.(ii)
A graphicψ,ϕ2 contraction if any r,t∈Q with r,t∈eG and ϑr∈Tir, there exists ϑt∈Tjt for i,j=1,2, i≠j; furthermore, ϑr,ϑt∈e∗G and(5)ψdTr,Tt≤ψΛψ2r,t;ϑr,ϑt−ϕΛψ2r,t;ϑr,ϑt,hold where ψ∈Ψ, ϕ∈Φ, and ψ2∈Ψ1 and(6)Λψ2r,t;ϑr,ϑt=ψ1αdr,t+βdr,ϑr+γdt,ϑt+σ1dr,ϑt+dt,ϑr2+σ2dt,ϑr+dr,ϑt2.
∀r,t∈Q and α,β,γ,σ1,σ2≥0 with α+β+γ+σ1+σ2+σ3≤1.Theorem 1.
Consider a completeQ,d metric space endowed with a directed graph G such that vG=Q and eG⊇Δ. If mapping T1,T2:Q⟶CLQ form a graphic ψ,ϕ1 contraction, then the following statement holds:(i)
FxT1≠∅ or FxT2≠∅⇔FxT1=FxT1≠∅(ii)
QT1,T2≠∅ provided that FxT1∩FxT2≠∅(iii)
IfG is weakly connected and QT1,T2≠∅, then FxT1=FxT2≠∅ provided that either(a)
T1 or T2 is upper semicontinuous or(b)
G has property P and either T1 or T2 is bounded(iv)
FxT1∩FxT2 is a clique of G˜⇔FxT1∩FxT2 which is singleton.Proof.
To prove (i), letr∗∈T1r∗. Assume r∗∉T2r∗, then since T1,T2 form a graphic ψ,ϕ1 contraction, there exists an r∈T2r with r∗,r∈e∗G such that(7)ψdr∗,r≤ψΛψ1r∗,r∗;r∗,r−ϕΛψ1r∗,r∗;r∗,r,where(8)Λψ1r∗,r∗;r∗,r=ψ1dr∗,r∗,dr∗,r∗,dr∗,r,dr,r∗+dr∗,r∗2,dr∗,r∗+dr∗,r2≤ψ1dr∗,r∗,dr∗,r∗,dr∗,r,dr∗,r,dr∗,r.
Using property (i) ofψ1, we have(9)Λψ1r∗,r∗;r∗,r≤dr∗,r∗,which is contradiction. Hence, r∗∈Tr∗ and so FxT1⊆FxT2. Similarly, FxT2⊆FxT1; therefore, FxT1=FxT2.
To prove (ii), letFxT1∩FxT2≠∅, then there exists r∈Q such that r∈T1r∩T2r. As eG⊇Δ, we conclude that QT1,T2≠∅.
To prove (iii), letr0 is an arbitrary point of Q. If r0∈T1r0 or r0∈T1r0, then the proof is completed. So, we assume that r0∉Tir0 for i∈1,2. Now, for i,j∈1,2i≠j if r1∈Tir0, then there exists r2∈Tjr1 with r1,r2∈e∗G such that(10)ψdr1,r2≤ψΛψ1r0,r1;r1,r2−ϕΛψ1r0,r1;r1,r2,where,(11)Λψ1r0,r1;r1,r2=ψ1dr0,r1,dr0,r1,dr1,r2,dr0,r2+dr1,r12,dr1,r1+dr0,r22,Λψ1r0,r1;r1,r2=ψ1dr0,r1,dr0,r1,dr1,r2,dr0,r2+dr1,r12,dr0,r22.
Ifdr0,r1≤dr1,r2, then by simple calculation, we get(12)Λψ1r0,r1;r1,r2≤dr1,r2,ψdr1,r2≤ψdr1,r2−ϕdr1,r2,which gives contradiction; therefore,(13)Λψ1r0,r1;r1,r2=dr0,r1,and further we have(14)ψdr1,r2≤ψdr0,r1−ϕdr0,r1.
Similarly, for the pointr2∈Tjr1, there exists r3∈Tir1 with r2,r3∈e∗G such that(15)ψdr2,r3≤ψΛψ1r1,r2;r2,r3−ϕΛψ1r1,r2;r2,r3,where(16)Λψ1r1,r2;r2,r3=ψ1dr1,r2,r1,r2,dr2,r3,dr1,r3+dr2,r22,dr2,r2+dr1,r32,Λψ1r1,r2;r2,r3=ψ1dr1,r2,r1,r2,dr2,r3,dr1,r32,dr1,r31+dr1,r2.
Ifdr1,r2≤dr2,r3, then(17)Λψ1r1,r2;r2,r3≤dr2,r3,and then(18)ψdr2,r3≤ψdr2,r3−ϕdr2,r3,which gives contradiction; therefore(19)Λψ1r1,r2;r2,r3=dr1,r2,ψdr2,r3≤ψdr1,r2−ϕdr1,r2.
Continuing this way, forr2γ∈Tjr2γ−1, there exist r2γ+1∈Tjr2γ such that r2γ,,r2γ+1∈e∗G such that(20)ψdr2γ,r2γ+1≤ψΛψ1r2γ−1,r2γ;r2γ,r2γ+1−ϕΛψ1r2γ−1,r2γ;r2γ,r2γ+1,where(21)Λψ1r2γ−1,r2γ;r2γ,r2γ+1≤dr2γ−1,r2γ.
Therefore,(22)ψdr2γ,r2γ+1≤ψdr2γ−1,r2γ−ϕdr2γ−1,r2γ.
In similar patternr for r2γ+1∈Tjr2γ, there exist r2γ+2∈Tjr2γ+1 such that r2γ+1,,r2γ+2∈e∗G such that(23)ψdr2γ+1,r2γ+2≤ψΛψ1r2γ,r2γ+1;r2γ+1,r2γ+2−ϕΛψ1r2γ,r2γ+1;r2γ+1,r2γ+2,where(24)Λψ1r2γ,r2γ+1;r2γ+1,r2γ+2≤dr2γ−1,r2γ.
Therefore,(25)ψdr2γ,r2γ+1≤ψdr2γ−1,r2γ−ϕdr2γ−1,r2γ.
Hence, we obtain a sequencerγ in Q such that for rγ∈Tjrγ−1, there exist rγ+1∈Tjrγ such that rγ,,rγ+1∈e∗G, and(26)ψdrγ,rγ+1≤ψdrγ−1,rγ−ϕdrγ−1,rγ.
Letdγ=drγ,rγ+1, then the above equation implies that dγ+1≤dγ for all γ≥1. Since dγ is a decreasing positive real sequence, there exists ξ≥0, such that(27)limγ⟶∞dγ=ξ.
We shall show thatξ=0; by applying limit, we have(28)ψξ≤ψξ−ϕξ<ψξ,which is a contradiction; therefore, ξ=0 which implies that(29)limγ⟶∞dγ=0.
Now, we want show thatrγ is Cauchy. Suppose that rγ is not Cauchy. Then, there exist ϵ>0 and subsequences rγκ and rδκ of rγ with γκ>δκ>κ such that(30)drδκ,rγκ≥ϵ,∀κ∈ℕ.
Moreover, one can chooseγκ corresponding to δκ such that it is the smallest possible integer with γκ>δκ holding (30); then,(31)drδκ,rγκ−1<ϵ,∀κ∈ℕ,(32)ψdrδκ,rγκ≤ψΛψ1rγκ−1,rδκ;rδκ,rγκ−ϕΛψ1rγκ−1,rδκ;rδκ,rγκ,Λψ1rγκ−1,rδκ;rδκ,rγκ=ψ1drγκ−1,rδκ,drγκ−1,rδκ,drδκ,rγκdrγκ−1,rγκ+drδκ,rδκ2,drδκ,rδκ+drγκ−1,rγκ2.
Applying limitκ⟶∞, we get(33)limκ⟶∞Λψ1rγκ−1,rδκ;rδκ,rγκ≤ϵ.
Taking limit of (32) using (33), (29), and lower semicontinuity of ϕ, we have(34)ψϵ≤ψϵ−ϕϵ<ψϵ,which is contradiction; therefore, rγ is a Cauchy sequence. Now, if Ti is upper semicontinuous (USC), then as r2γ∈Q, r2γ+1∈Tir2γ with s2γ⟶r∗ and r2γ+1⟶r∗ as γ⟶∞ which implies that r∗∈Tir∗. Using (i), we get r∗∈Tir∗=Tjr∗. Similarly, the result holds when Tj is upper semicontinuous (USC).
Assume thatF is continuous. Since s2γ converges to r∗ as γ⟶∞ and r2γ,r2γ+1∈eG, we have r2γ,r∗∈e∗G. For r2γ∈Tjr2γ−1, there exists ϑγ∈Tir∗ such that r2γ,ϑγ∈e∗G. As rγ is bounded, limsupγ⟶∞ϑγ=ϑ∗ and liminfγ⟶∞ϑγ=ϑ∗, both exist. Assume that ϑ∗≠r∗. Since T1,T2 form a graphic F1− contraction,(35)ψdr2n,r3≤ψΛψ1r2γ−1,r∗;r2γ,ϑγ−ϕΛψ1r2γ−1,r∗;r2γ,ϑγ,where(36)Λψ1r2γ−1,r∗;r2γ,ϑγ=ψ1dr2γ−1,r∗,dr2γ−1,r2γ,dr∗,ϑγ,dr2γ,ϑγ+dr2,r22,dr2,r2+dr2γ,ϑγ2.
By applying limit and after simple calculation, we get contradiction; hence,ϑ∗=r∗. Similarly, taking the liminf gives ϑ∗=r∗. Since ϑγ∈Tir∗ for all γ≥1 and Tir∗ is closed set, it follows that r∗Tir∗. Now, from (i), we get r∗∈Tir∗ and hence FxT1=FxT2.
To prove (iv), assume the setFxT1∩FxT2 is a clique of G˜. We have shown that FxT1∩FxT2 is singleton set. Assume on contrary that there exist ϑ and v such that ϑ,v∈FxT1∩FxT2 but ϑ≠v. As ϑ,v∈e∗G and T1 and T2 form a graphic F1− contraction, so for ϑs,ϑt∈e∗G which implies(37)ψdϑ,v≤ψΛψ1ϑ,v;ϑ,v−ϕΛψ1ϑ,v;ϑ,v,where(38)Λψ1ϑ,v;ϑ,v=ψ1dϑ,v,dϑ,ϑ,dv,v,dr2γ,ϑγ+dr2,r22,dr2,r2+dr2γ,ϑγ2,which is a contradiction. Hence, ϑ=v. Conversely, if FxT1∩FxT2 is singleton, then it follows FxT1∩FxT2 is a clique of G˜.
By the same technique, it is essay to prove the following result.Theorem 2.
Consider a completeQ,d metric space endowed with a directed graph G such that vG=Q and eG⊇Δ. If mapping T1,T2:Q⟶CLQ make a graphic ψ,ϕ2 contraction, then the following statement holds:(i)
FxT1≠∅ or FxT2≠∅⇔FxT1=FxT1≠∅;(ii)
QT1,T2≠∅ provided that FxT1∩FxT2≠∅;(iii)
G is weakly connected and QT1,T2≠∅, then FxT1=FxT2≠∅ provided that either(a)
T1 or T2 is upper semicontinuous(USC) or(b)
G has property P and either T1 or T2 is bounded;(iv)
FxT1∩FxT2 is a clique of G˜⇔FxT1∩FxT2 which is singleton.
Now, we give example which satisfying Theorem1.Example 1.
LetQ=rγ=γγ+1/2:γ∈N=vG(39)eG=r,tt=2r,∀r,t∈vG,e∗G=r,tt≠2r,∀r,t∈vG.
LetvG be endowed with usual metric space. Let T1,T2:Q×Q⟶PQ, ϕ:ℝ+⟶ℝ+, and ψ:ℝ+⟶ℝ+ define by(40)T1r=r2,T2r=r2,ifr=r2,r1,rγ,ifr=rγ,ϕt=t4,ψt=t2,ψ1ς1,ς2,ς3,ς4,ς5=maxς1,ς2,ς3,ς4,ς5.
Forϑr,ϑt∈eG, we discuss the following cases:Case 1.1. If r=r1, t=rδ for δ>2, then for ϑr=r2∈T1r, there exist ϑt=rδ−1∈T2t such that(41)ψdϑr,ϑt=dϑr,ϑt2=dr2,rδ−12=δ2−δ−24≤δ2+δ−24≤Λψ1r,t;ϑr,ϑt2≤ψΛψ1r,t;ϑr,ϑt−ϕΛψ1r,t;ϑr,ϑt.Case 1.2. If r=rγ and t=rγ+1 for γ>1, then for ϑr=r2∈T1r, there exist ϑt=rγ−1∈T2t such that(42)ψdϑs,ϑt=dϑr,ϑt2=dr2,rγ−12=δ2−δ−24≤δ2+δ−24≤Λψ1r,t;ϑr,ϑt2≤ψΛψ1r,t;ϑr,ϑt−ϕΛψ1r,t;ϑr,ϑt.Case 1.3. If r=rγ and t=rδ for δ>γ>1, then for ϑr=r2∈T1r, there exist ϑt=rγ−1∈T2t such that(43)ψdϑr,ϑt=dϑr,ϑt2=dr2,rγ−12=γ2−γ−24≤γ2+γ−24≤Λψ1r,t;ϑr,ϑt2≤ψΛψ1r,t;ϑr,ϑt−ϕΛψ1r,t;ϑr,ϑt.Now, forr,t∈Q,ϑr∈T2r, there exist ϑt∈T1r such that ϑr,ϑt∈e∗G, we consider the following cases.Case 1.4. If r=rγ and t=r1 for γ>1, then for ϑr=rγ−1∈T2r, there exist ϑt=r2∈T1t such that(44)ψdϑr,ϑt=dϑr,ϑt2=drγ−1,r22=γ2−γ−24≤γ2+γ−24≤Λψ1r,t;ϑr,ϑt2≤ψΛψ1r,t;ϑr,ϑt−ϕΛψ1r,t;ϑr,ϑt.Case 1.5. If r=rγ and t=rδ for δ>γ>1, then for ϑr=rγ−1∈T2r, there exist ϑt=r2∈T1t such that(45)ψdϑr,ϑt=dϑr,ϑt2=drγ−1,r22=γ2−γ−24≤γ2+γ−24≤Λψ1r,t;ϑr,ϑt2≤ψΛψ1r,t;ϑr,ϑt−ϕΛψ1r,t;ϑr,ϑt.


## 3. Application to Integral Inclusion

Set-valued (SV) F.P results are explored extensively and have interesting application in integral and differential inclusion [16–19]. In this section, we derive sufficient conditions for the solutions of Fredholm-type integral inclusion. Set of all continuous function defined on ς,ξ is denoted by W=Cς,ξ. Define d:W×W⟶R+, by dr,t=supt∈ς,ξr−t, which is a complete metric space on W. Assume that the metric is endowed with a graph G. Consider the integral inclusion(46)rζ∈ϕζ+∫ςξKζ,η,rηdη,ζ∈ς,ξ,where K:ς,ξ×ς,ξ×R⟶PcvR, where PcvR is the family (collection) of nonempty compact and convex subsets of R. For each r∈Cς,ξ,R, the operator K.,.r is lower semicontinuous. Further, the function ϕ:ς,ξ⟶R is continuous.DefineT:Cς,ξ,R⟶CLCς,ξ,R by,(47)Trζ=ϑ∈Cς,ξ,R:ϑ∈ϕζ+∫ςξKζ,η,rηdη,r∈Cς,ξ,R, and set Kr:Kζ,η,rη,ζ,η∈ς,ξ. Now, for K:ς,ξ×ς,ξ×R⟶PcvR, by Michael’s selection theorem, there exist a continuous operator ms:ς,ξ×ς,ξ⟶R with mr∈Kr(48)ϕζ+∫ςξKζ,η,rηdη∈Trζ.This implies thatTr≠∅. Therefore, Tr is closed [18].Theorem 3.
Assume that the following assumptions holds:A1 There exist a continuous function G:ς,ξ×ς,ξ⟶0,∞ such that(49)HKζ,η,r,Kζ,η,t≤Gζ,ηΛψ1r,t;ϑr,ϑt,for eachζ,η∈ς,ξ.A2supζ∈ς,ξ∫ςξGζ,η≤1/2.A3 Assume that for every vertex r in G and for every ϑr∈Tir,i=1,2, we have r,ϑr∈eG.A4 If any r,t∈Q with r,t∈eG and ϑr∈Tir, there exists ϑt∈Tjt for i,j=1,2 and i≠j such that ϑr,ϑt∈e∗G.
Then, the integral inclusion (46) has a solution.Proof.
Letr,t∈W be such that t∈Ts. Then, we have mrζ,η∈Krζ,η for ζ,η∈ς,ξ such that uζ=ϕζ+∫ςξmrζ,η,sηdη. On other side, hypothesis ensures that there exist vζ,η∈Λtζ,η such that(50)mrζ,η−vζ,η≤Gζ,ηΛψ1r,t;ϑr,ϑt.
Consider the set valued operatorS defined by(51)Sζ,η=Λtζ,η∩w∈R:mrζ,η−vζ,η≤Gζ,ηΛψ1r,t;ϑr,ϑt.
Since the operatorT is lower semicontinuous, there exists mt:ς,ξ×ς,ξ⟶R such that mtζ,r∈Sζ,η for each ζ,η∈ς,ξ; thus,(52)ϑrζ=ϕζ+∫ςξmrζ,η,rηdη∈ϕζ+∫ςξKζ,η,rηdη,ζ∈ς,ξ.
Now, we have(53)dϑrζ,ϑtζ=supζ∈ς,ξϑrζ−ϑtζ≤supζ∈ς,ξ∫ςξmrζ,η,rη−mζ,η,tηdη≤supζ∈ς,ξ∫ςξGζ,ηΛψ1r,t;ϑr,ϑtdη≤Λψ1r,t;ϑr,ϑtsupζ∈ς,ξ∫ςξGζ,ηdη≤Λψ1r,t;ϑr,ϑt2,and we have(54)ψdϑrζ,ϑtζ≤ψΛψ1r,t;ϑr,ϑt−ϕΛψ1r,t;ϑr,ϑt.
By takingψζ=ζ,ϕζ=ζ/2. From Theorem 1, the integral inclusion (46) has a solution.


## 4. Conclusion

Common fixed point results for single-valued mappings are used to solve nonlinear integral, functional, and matrix equations, etc. However, the studies of the set-valued mapping generalize the concept of single valued mappings. Such studies have been applied to prove the existence of solution for integral and differential inclusion and existence of equilibria in game theory. The result of Abbas et al. [12] generalized many results for set-valued mapping related with F-contraction while our results generalized many results related with weak-contraction; for detail, Alber and Guerre-Delabriere [20] established results for single valued mapping and relax contraction condition in form of weak contraction. Then, Doric [21] proved common fixed point for generalized ψ,ϕ-weak contractions. Dutta and Choudhury [22] generalized the result in metric space. Loung and Thuan [23] and Rhoades [24] also used weak contraction in this line and established fixed point results in metric spaces. Similarly Zhang and Song [25] further explored the results for generalized ϕ-weak contractions. Our results generalized the mention results and many more in this direction for set-valued mapping induced with graph without using Hausdorff metric.



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