# Multivalued Fixed Point Results for New GeneralizedF-Dominated Contractive Mappings on Dislocated Metric Space with Application

Authors: Tahair Rasham; Abdullah Shoaib; Badriah A. S. Alamri; Muhammad Arshad
Journal: Journal of Function Spaces
 (2018)

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## Abstract

The purpose of this paper is to find out fixed point results for semi-α⁎-dominated multivalued mappings fulfilling a new generalized locally F-dominated multivalued contractive condition on a closed ball in complete dislocated metric space. Example and application both are given to show the novelty of our results. Our results merge, extend, and infer many results.


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## Body

## 1. Introduction and Preliminaries

Fixed point theory has a foundational role in functional analysis. Banach [1] established the fundamental fixed point theorem, which has played a significant role in different fields of applied mathematics. Due to its significance, a large number of authors have proven many interesting multiplications of his result (see [1–33]). Many authors introduced fixed point theorems in complete dislocated metric space. The idea of dislocated topology has been applied in the field of logic programming semantics (see [14]).Wardowski [33] introduced new type of contraction called F-contraction and showed a new generalized fixed point theorem. He generalized many previous fixed point results in different directions. Many other useful results on F-contractions can be seen in [3–5, 11, 12, 15, 18, 22, 24–27]. In this paper, we recalled the concept of F-contraction to obtain some common fixed point results for semi-α∗-dominated multivalued mappings on proximinal sets satisfying a new type rational F-contraction in the context of complete dislocated metric spaces. We have also given an example in which the mapping is not α∗-admissible but it fulfills the condition of α∗-dominated.Definition 1 (see [14]).
LetZ´ be a nonempty set. A mapping dl:Z´×Z´→[0,∞) is called a dislocated metric (or simply dl-metric) if the following conditions hold, for any tˇ,sˇ,zˇ∈Z´:(i)
Ifdl(tˇ,sˇ)=0, then tˇ=sˇ.(ii)
dltˇ,sˇ=dlsˇ,tˇ.(iii)
dl(tˇ,sˇ)≤dl(tˇ,zˇ)+dl(zˇ,sˇ).Thendl is called a dislocated metric on Z´, and the pair (Z´,dl) is called dislocated metric space or dl metric space. It is clear that if dl(tˇ,sˇ)=0, then, from (i), tˇ=sˇ. But if tˇ=sˇ, dl(tˇ,sˇ) may not be 0. We use DMS instead of dislocated metric space.Definition 2 (see [14]).
Let(Z´,dl) be a DMS:(i)
A sequence{cnˇ} in (Z´,dl) is called a Cauchy sequence if given ε>0; there corresponds nˇ0∈N such that for all nˇ,m≥nˇ0 one has dl(cm,cnˇ)<ε or limnˇ,m→∞dl(cnˇ,cm)=0.(ii)
A sequence{cnˇ} is said to be dislocated—converges (for short dl -converges) to l if limnˇ→∞dl(cnˇ,l)=0. In this case l is called a dl-limit of {cnˇ}.(iii)
(Z´,dl) is called complete if every Cauchy sequence in Z´ converges to a point l∈Z´ such that dl(l,l)=0.Definition 3 (see [29]).
LetH be a nonempty subset of DMS Z´ and let i∈Z´. An element v0∈H is called a best approximation in H if (1)dli,H=dli,v0,wheredli,H=infy∈Hdli,y.If eachi∈Z´ has at least one best approximation in H, then H is called a proximinal set. We denote P(Z´) as the set of all closed proximinal subsets of Z´.Definition 4 (see [29]).
The functionHdl:P(Z´)×P(Z´)→R+, defined by (2)HdlN,M=max⁡supnˇ∈N⁡dlnˇ,M,supm∈M⁡dlN,m,is called dislocated Hausdorff metric on P(Z´).Definition 5 (see [29]).
LetSˇ:Z´→P(Z´) be a multivalued mapping and α:Z´×Z´→[0,+∞). Let K⊆Z´; we say that Sˇ is semi-α⋆-admissible on K, whenever α(i,j)≥1 implies that α⋆(Sˇi,Sˇj)≥1, for all i,j∈K, where α⋆(Sˇi,Sˇj)=inf⁡{α(u,v):u∈Sˇi,v∈Sˇj}. If K=Z´, then we say that Sˇ is α⋆-admissible on Z´.Definition 6.
Let(Z´,dl) be a DMS. Let Sˇ:Z´→P(Z´) be multivalued mapping and α:Z´×Z´→[0,+∞). Let A⊆Z´, and we say that Sˇ is semi-α∗-dominated on H, whenever α∗(i,Sˇi)≥1 for all i∈H, where α∗(i,Sˇi)=inf⁡{α(i,l):l∈Sˇi}. If H=Z´, then we say that Sˇ is α∗-dominated on Z´. If Sˇ:Z´→Z´ is a self-mapping, then Sˇ is semi-α-dominated on H, whenever α(i,Sˇi)≥1 for all i∈H.Definition 7 (see [33]).
Let(Z´,d) be a metric space. A mapping H:Z´→Z´ is said to be an F-contraction if there exists τ>0 such that(3)dHj,Hk>0⟹τ+FdHj,Hk≤Fdj,k∀j,k∈Z´,where F:R+→R is a mapping satisfying the following conditions:(F1)F is strictly increasing; that is, for all j,k∈R+, such that j<k, F(j)<F(k).(F2) For each sequence{αnˇ}nˇ=1∞ of positive numbers, limnˇ→∞αnˇ=0 if and only if limnˇ→∞F(αnˇ)=-∞.(F3) There existsk∈(0,1) such that lim⁡α→0+αkF(α)=0.Lemma 8.
Let(Z´,dl) be a DMS. Let (P(Z´),Hdl) be a dislocated Hausdorff metric space on P(Z´). Then, for all G,H∈P(Z´) and for each g∈G, there exists hg∈H satisfying dl(g,H)=dl(g,hg); then Hdl(G,H)≥dl(g,hg).Example 9 (see [14]).
IfZ´=R+∪{0}, then dl(j,k)=j+k defines a dislocated metric dl on Z´.Example 10 (see [33]).
The family ofF is not empty:(1)
F(cˇ)=ln⁡(cˇ);cˇ>0.(2)
F(cˇ)=cˇ+ln⁡(cˇ);cˇ>0.(3)
F(cˇ)=-1/cˇ;cˇ>0.Example 11.
LetZ´=R. Define the mapping α:Z´×Z´→[0,∞) by (4)αj,k=1ifj>k12otherwise.Define the multivalued mappings Sˇ,Tˇ:Z´→P(Z´) by (5)Sˇj=j-4,j-3ifj∈Z´,Tˇk=k-2,k-1ifk∈Z´.Suppose that j=3 and k=2.5. As 3>2.5, then α(3,2.5)≥1. Now, α⋆(Sˇ3,Tˇ2.5)=inf⁡{α(a,b):a∈Sˇ3,b∈Tˇ2.5}=1/2≱1, and this means that the pair (Sˇ,Tˇ) is not α⋆-admissible. Also, α⋆(Sˇ3,Sˇ2)≱1 and α⋆(Tˇ3,Tˇ2)≱1. This implies that Sˇ and Tˇ are not α⋆-admissible individually. Now, α⋆(j,Sˇj)=inf⁡{α(j,b):b∈Sˇj}≥1, for all j∈Z´. Hence Sˇ is α⋆-dominated mapping. Similarly, α⋆(k,Tˇk)=inf⁡{α(k,b):b∈Tˇk}≥1. Hence it is clear that Sˇ and Tˇ are α⋆-dominated but not α⋆-admissible.


## 2. Main Result

Let(Z´,dl) be a DMS, let c0∈Z´, and let Sˇ,Tˇ:Z´→P(Z´) be the multifunctions on Z´. Let c1∈Sˇc0 be an element such that dl(c0,Sˇc0)=dl(c0,c1). Let c2∈Tˇc1 be such that dl(c1,Tˇc1)=dl(c1,c2). Let c3∈Sˇc2 be such that dl(c2,Sˇc2)=dl(c2,c3). Continuing this method, we get a sequence cnˇ of points in Z´ such that c2nˇ+1∈Sˇc2nˇ and c2nˇ+2∈Tˇc2nˇ+1, where nˇ=0,1,2,…. Also dl(c2nˇ,Sˇc2nˇ)=dl(c2nˇ,c2nˇ+1), and dl(c2nˇ+1,Tˇc2nˇ+1)=dl(c2nˇ+1,c2nˇ+2). We denote this iterative sequence by {TˇSˇ(cnˇ)}. We say that {TˇSˇ(cnˇ)} is a sequence in Z´ generated by c0.Theorem 12.
Let(Z´,dl) be a complete DMS. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0,c0∈Bdl(c0,r)¯⊆Z´ and let Sˇ,Tˇ:Z´→P(Z´) be the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(6)maxτ+α∗e˘,Sˇe˘FHdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FHdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η3dle˘,Tˇy^+η4dl2e˘,Sˇe˘·dly^,Tˇy^1+dl2e˘,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{TˇSˇ(cnˇ)} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η2,η3,η4>0 and η1+η2+2η3+η4<1 and(7)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η2+η3)/(1-η3-η4) and η3+η4≠1. Then {TˇSˇ(cnˇ)} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {TˇSˇ(cnˇ)}→u¯∈Bdl(c0,rˇ)¯. Also if inequality (6) holds for u¯ and either α(cnˇ,u¯)≥1 or α(u¯,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point u¯ in Bdl(c0,rˇ)¯.Proof.
Consider a sequence{TˇSˇ(cnˇ)}. From (7), we get (8)dlc0,c1≤dlc0,Sˇc0≤rˇ.It follows that (9)c1∈Bdlc0,rˇ¯.
Letc2,…,cj∈Bdl(c0,rˇ)¯ for some j∈N. If j=2ı`+1, where ı`=1,2,…,(j-1)/2, then since Sˇ,Tˇ:Z´→P(Z´) are semi-α∗-dominated mappings on Bdl(c0,rˇ)¯, α∗(c2ı`,Sˇc2ı`)≥1 and α∗(c2ı`+1,Tˇc2ı`+1)≥1. As α∗(c2ı`,Sˇc2ı`)≥1, this implies that inf⁡{α(c2ı`,b):b∈Sˇc2ı`}≥1. Also c2ı`+1∈Sˇc2ı`, so α(c2ı`,c2ı`+1)≥1. Now, by using Lemma 8, we have (10)τ+Fdlc2ı`+1,c2ı`+2≤τ+FHdlSˇc2ı`,Tˇc2ı`+1≤max⁡τ+α∗c2ı`,Sˇc2ı`FHdlSˇc2ı`,Tˇc2ı`+1,τ+α∗c2ı`+1,Tˇc2ı`+1FHdlTˇc2ı`+1,Sˇc2ı`≤Fη1dlc2ı`,c2ı`+1+η2dlc2ı`,Sˇc2ı`+η3dlc2ı`,Tˇc2ı`+1+η4dl2c2ı`,Sˇc2ı`·dlc2ı`+1,Tˇc2ı`+11+dl2c2ı`,c2ı`+1≤Fη1dlc2ı`,c2ı`+1+η2dlc2ı`,c2ı`+1+η3dlc2ı`,c2ı`+1+η3dlc2ı`+1,c2ı`+2+η4dl2c2ı`,c2ı`+1·dlc2ı`+1,c2ı`+21+dl2c2ı`,c2ı`+1≤Fη1+η2+η3dlc2ı`,c2ı`+1+η3+η4dlc2ı`+1,c2ı`+2-τ;this implies that (11)Fdlc2ı`+1,c2ı`+2≤Fη1+η2+η3dlc2ı`,c2ı`+1+η3+η4dlc2ı`+1,c2ı`+2,for all j∈N. As F is strictly increasing, we have (12)dlc2ı`+1,c2ı`+2<η1+η2+η3dlc2ı`,c2ı`+1+η3+η4dlc2ı`+1,c2ı`+2,which implies that(13)1-η3-η4dlc2ı`+1,c2ı`+2<η1+η2+η3dlc2ı`,c2ı`+1dlc2ı`+1,c2ı`+2<η1+η2+η31-η3-η4dlc2ı`,c2ı`+1.Here λ=(η1+η2+η3)/(1-η3-η4)<1. Hence(14)dlc2ı`+1,c2ı`+2<λdlc2ı`,c2ı`+1<λ2dlc2ı`-1,c2ı`<⋯<λjdlc0,c1.Now, (15)dlc0,cj+1≤dlc0,c1+dlc1,c2+⋯+dlcj,cj+1≤dlc0,c11+λ+⋯+λj≤1-λrˇ1-λj+11-λ<rˇ.Thus, cj+1∈Bdl(c0,rˇ)¯. Hence cnˇ∈Bdl(c0,rˇ)¯, for all nˇ∈N. Continuing this process, we get (16)τ+Fdlcnˇ,cnˇ+1≤τ+FHdlSˇcnˇ-1,Tˇcnˇ≤max⁡τ+α∗cnˇ-1,Sˇcnˇ-1FHdlSˇcnˇ-1,Tˇcnˇ,τ+α∗cnˇ,TˇcnˇFHdlTˇcnˇ,Sˇcnˇ-1≤Fη1dlcnˇ-1,cnˇ+η2dlcnˇ-1,Sˇcnˇ-1+η3dlcnˇ-1,Tˇcnˇ+η4dl2cnˇ-1,Sˇcnˇ-1·dlcnˇ,Tˇcnˇ1+dl2cnˇ-1,cnˇ≤Fη1dlcnˇ-1,cnˇ+η2dlcnˇ-1,cnˇ+η3dlcnˇ-1,cnˇ+η3dlcnˇ,cnˇ-1+η4dl2cnˇ-1,cnˇ·dlcnˇ,cnˇ+11+dl2cnˇ-1,cnˇ≤Fη1+η2+η3dlcnˇ-1,cnˇ+η3+η4dlcnˇ,cnˇ+1-τ;this implies that (17)Fdlcnˇ,cnˇ+1≤Fη1+η2+η3dlcnˇ-1,cnˇ+η3+η4dlcnˇ,cnˇ+1,for all nˇ∈N. As F is strictly increasing, we have (18)dlcnˇ,cnˇ+1<η1+η2+η3dlcnˇ-1,cnˇ+η3+η4dlcnˇ,cnˇ+1,which implies that (19)1-η3-η4dlcnˇ,cnˇ+1<η1+η2+η3dlcnˇ-1,cnˇdlcnˇ,cnˇ+1<η1+η2+η31-η3-η4dlcnˇ-1,cnˇ.Here λ=(η1+η2+η3)/(1-η3-η4)<1. Hence(20)dlcnˇ,cnˇ+1<λdlcnˇ-1,cnˇ<dlcnˇ-1,cnˇ.Consequently, (21)τ+Fdlcnˇ,cnˇ+1≤Fdlcnˇ-1,cnˇ,which implies that (22)Fdlcnˇ,cnˇ+1≤Fdlcnˇ-1,cnˇ-τ⋮≤Fdlc0,c1-nˇτ.This implies that(23)Fdlcnˇ,cnˇ+1≤Fdlc0,c1-nˇτ.And so limnˇ→∞⁡F(dl(cnˇ,cnˇ+1))=-∞. By (F2), we find that(24)limnˇ→∞⁡Fdlcnˇ,cnˇ+1=0.We shall prove that {TˇSˇ(cnˇ)} is Cauchy in (Z´,dl). So it suffices to show that limnˇ→∞⁡dl(cnˇ,cm)=0. We argue by contradiction. Suppose that there exist ∈>0 and sequences (nˇ(q)) and (m(q)) of natural numbers such that(25)dlcnˇq,cnˇq+1≥∈,dlcnˇq+1,cmq<∈mq>nˇq>q,∀q∈N.By triangular inequality, we have(26)dlcnˇq,cmq≤dlcnˇq,cnˇq+1+dlcnˇq+1,cmq≤∈+dlcnˇq,cnˇq+1≤∈+dlcnˇq,cnˇq+1.From (24), there exist q1∈N such that, for all q≥q1, (27)dlcnˇq,Tˇcnˇq<∈.Combining (26) with (27) yields(28)dlcnˇq,cmq<2∈,∀q≥q1.As Sˇ,Tˇ:Z´→P(Z´) are semi-α∗-dominated mappings on Bdl(c0,rˇ)¯, α∗(cnˇ(q),Sˇcnˇ(q))≥1 and α∗(cm(q),Tˇcm(q))≥1, for all m,nˇ∈N. Now, by using Lemma 8 and condition (6), we get(29)F∈≤FHdlSˇcnˇq,Tˇcmq≤max⁡τ+α∗cnˇq,SˇcnˇqFHdlSˇcnˇq,Tˇcmq,τ+α∗cmq,TˇcmqFHdlTˇcmq,Sˇcnˇq≤Fη1dlcnˇq,cmq+η2dlcnˇq,Sˇcnˇq+η3dlcnˇq,Tˇcmq+η4dl2cnˇq,Sˇcnˇq·dlcmq,Tˇcmq1+dl2cnˇq,cmq-τ,F∈≤Fη1dlcnˇq,cmq+η2dlcnˇq,cnˇq+1+η3dlcnˇq,cmq+1+η4dl2cnˇq,cnˇq+1·dlcmq,cmq+11+dl2cnˇq,cmq-τ,F∈≤Fη1dlcnˇq,cmq+η2dlcnˇq,cnˇq+1+η3dlcnˇq,cmq+η3dlcmq,cmq+1+η4dl2cnˇq,cnˇq+1·dlcmq,cmq+11+dl2cnˇq,cmq-τ,which implies that (30)F∈≤F2η1∈+η2∈+3η3∈+η4∈-τ.As 2η1+η2+3η3+η4<1, we get (31)2η1∈+η2∈+3η3∈+η4∈<∈,we deduce that (32)F∈<F∈;which is a contradiction. Thus, {TˇSˇ(cnˇ)} is a Cauchy sequence in (Bdl(c0,rˇ)¯,dl). Since (Bdl(c0,rˇ)¯,dl) is a complete metric space, so there exist u¯∈Bdl(c0,rˇ)¯ such that {TˇSˇ(cnˇ)}→u¯ as nˇ→∞; then(33)limnˇ→∞⁡dlcnˇ,u¯=0.Since α∗(u¯,Tˇu¯)≥1 and α∗(c2nˇ,Sˇc2nˇ)≥1, by using Lemma 8 and inequality (6), we have (34)Fdlc2nˇ+1,Tˇu¯≤FHdlSˇc2nˇ,Tˇu¯≤maxτ+α∗c2nˇ,Sˇc2nˇFHdlSˇc2nˇ,Tˇu¯,τ+α∗u¯,Tˇu¯FHdlTˇu¯,Sˇc2nˇ≤Fη1dlc2nˇ,u¯+η2dlc2nˇ,Sˇc2nˇ+η3dlc2nˇ,Tˇu¯+η4dl2c2nˇ,Sˇc2nˇ·dlu¯,Tˇu¯1+dl2c2nˇ,u¯-τ≤Fη1dlc2nˇ,u¯+η2dlc2nˇ,Sˇc2nˇ+η3dlc2nˇ,u¯+η3u¯,Tˇu¯+η4dl2c2nˇ,Sˇc2nˇ·dlu¯,Tˇu¯1+dl2c2nˇ,u¯-τ.By using (33), we get (35)Fdlu¯,Tˇu¯≤Fη3dlu¯,Tˇu¯-τ,which implies that (36)dlu¯,Tˇu¯<η3dlu¯,Tˇu¯<dlu¯,Tˇu¯,which is a contradiction; hence dl(u¯,Tˇu¯)=0 or u¯∈Tˇu¯. Similarly, by using Lemma 8, inequality (6), and the inequality (37)dlu¯,Sˇu¯≤dlu¯,c2nˇ+2+dlc2nˇ+2,Sˇu¯,we can show that dl(u¯,Sˇu¯)=0.u¯∈Sˇu¯. Hence, Sˇ and Tˇ have a common fixed point u¯ in Bdl(c0,rˇ)¯. Now, (38)dlu¯,u¯≤dlu¯,Tˇu¯+dlTˇu¯,u¯≤0.This implies that dl(u¯,u¯)=0.Example 13.
LetZ´=Q+∪{0} and let dl:Z´×Z´→Z´ be the complete DMS on Z´ defined by (39)dli,j=i+j∀i,j∈Z´.Define the multivalued mapping, Sˇ,Tˇ:Z´×Z´→P(Z´), by (40)Sˇz´=z´3,23z´ifz´∈0,7∩Z´z´,z´+1ifz´∈7,∞∩Z´,Tˇz´=z´4,34z´ifz´∈0,7∩Z´z´+1,z´+3ifz´∈7,∞∩Z´.Considering that x0=1,rˇ=8, Bdl(x0,rˇ)¯=[0,7]∩Z´. So we obtain a sequence {TˇSˇ(xnˇ)}={1,1/3,1/12,1/36,…} in Z´ generated by x0. Also, Bdl(x0,rˇ)¯∩{TˇSˇ(xnˇ)}={1,1/3,1/12,1/36,…}. Let (41)αx,y=1ifx,y∈0,7∩Z´32otherwise,τ∈(0,12/95], η1=1/5, η2=1/10, η3=1/15, η4=1/30, and λ=4/9. (42)dlx0,Sˇx0=43<1-498=1-λrˇ.Now, if x,y∈Bdl(x0,rˇ)¯∩{TˇSˇ(xnˇ)}, then we have the following cases.Case 1.
Ifmax⁡{τ+α∗(x,Sˇx)F(Hdl(Sˇx,Tˇy)),τ+α∗(y,Tˇy)F(Hdl(Tˇy,Sˇx))}=τ+α∗(x,Sˇx)F(Hdl(Sˇx,Tˇy)), then we consider only (43)HdlSˇx,Tˇy=max⁡supa∈Sˇx⁡dla,Tˇy,supb∈Tˇy⁡dlSˇx,b=max⁡supa∈Sˇx⁡dla,y4,3y4,supb∈Tˇy⁡dlx3,2x3,b=⁡max⁡dl2x3,y4,3y4,dlx3,2x3,3y4=max⁡dl2x3,y4,dlx3,3y4⁡=max⁡2x3+y4,x3+3y4<15dlx,y+110dlx,x3,23x+115dlx,y4,34y+130dl2x,x/3,2/3x·dly,y/4,3/4y1+dl2x,y=15x+y+2x15+4x+y60+5x2y2541+x+y2.Thus, (44)HdlSˇx,Tˇy<η1dlx,y+η2dlx,Sˇx+η3dlx,Tˇy+η4dl2x,Sˇx·dly,Tˇy1+dl2x,y,which implies that (45)τ+ln⁡HdlSˇx,Tˇy≤ln⁡η1dlx,y+η2dlx,Sˇx+η3dlx,Tˇy+η4dl2x,Sˇx·dly,Tˇy1+dl2x,y.That is, if F(cˇ)=ln⁡(cˇ), then (46)τ+α∗x,SˇxFHdlSˇx,Tˇy≤Fη1dlx,y+η2dlx,Sˇx+η3dlx,Tˇy+η4dl2x,Sˇx·dly,Tˇy1+dl2x,y.Case 2.
Ifmax⁡{τ+α∗(x,Sˇx)F(Hdl(Sˇx,Tˇy)),τ+α∗(y,Tˇy)F(Hdl(Tˇy,Sˇx))}=τ+α∗(y,Tˇy)F(HdlTˇy,Sˇx, then by using the similar arguments of Case 1 we can get the same results. Now, if x=8, y=9∈(7,∞)∩Z´, then (47)τ+α∗y,TˇyFHdlSˇx,Tˇy>Fη1dlx,y+η2dlx,Sˇx+η3dlx,Tˇy+η4dl2x,Sˇx·dly,Tˇy1+dl2x,y.And consequently condition (6) does not hold on Z´. Thus the mappings Sˇ and Tˇ satisfy all the conditions of Theorem 12 on closed ball rather than on whole space.If we takeSˇ=Tˇ in Theorem 12, then we are left with the result.Corollary 14.
Let(Z´,dl) be a complete DMS. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0, let c0∈Bdl(c0,rˇ)¯⊆Z´, and let Sˇ:Z´→P(Z´) be the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(48)max⁡τ+α∗e˘,Sˇe˘FHdlSˇe˘,Sˇy^,τ+α∗y^,Sˇy^FHdlSˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η3dle˘,Sˇy^+η4dl2e˘,Sˇe˘·dly^,Sˇy^1+dl2x,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{SˇSˇ(cnˇ)} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η2,η3,η4>0 and η1+η2+2η3+η4<1 and (49)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η2+η3)/(1-η3-η4) and η3+η4≠1. Then {TˇSˇ(cnˇ)} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {SˇSˇ(cnˇ)}→u¯∈Bdl(c0,rˇ)¯. Also if inequality (48) holds for e˘ and either α(cnˇ,u¯)≥1 or α(u¯,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point u¯ in Bdl(c0,rˇ)¯.If we takeη2=0 in Theorem 12, then we are left with the result.Corollary 15.
Let(Z´,dl) be a complete DMS. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0, let c0∈Bdl(c0,rˇ)¯⊆Z´, and let Sˇ,Tˇ:Z´→P(Z´) be the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(50)max⁡τ+α∗e˘,Sˇe˘FHdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FHdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η3dle˘,Tˇy^+η4dl2e˘,Sˇe˘·dly^,Tˇy^1+dl2e˘,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{TˇSˇ(cnˇ)} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η3,η4>0 and η1+2η3+η4<1 and (51)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η3)/(1-η3-η4) and η3+η4≠1. Then {TˇSˇ(cnˇ)} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {TˇSˇ(cnˇ)}→u¯∈Bdl(c0,rˇ)¯. Also if inequality (50) holds for u¯ and either α(cnˇ,u¯)≥1 or α(u¯,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point u¯ in Bdl(c0,rˇ)¯.If we takeη3=0 in Theorem 12, then we are left with the result.Corollary 16.
LetZ´,dl be a complete DMS. Suppose that there exists a function α:Z´×Z´→0,∞. Let rˇ>0, let c0∈Bdlc0,rˇ¯⊆Z´, and let Sˇ,Tˇ:Z´→PZ´ be the semi-α∗-dominated mappings on Bdlc0,rˇ¯. Assume that, for some τ>0,(52)maxτ+α∗e˘,Sˇe˘FHdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FHdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η4dl2e˘,Sˇe˘·dly^,Tˇy^1+dl2e˘,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{TˇSˇ(cnˇ)} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η2,η4>0 and η1+η2+η4<1 and (53)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η2)/(1-η4) and 1-η4≠0. Then {TˇSˇ(cnˇ)} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {TˇSˇ(cnˇ)}→u¯∈Bdl(c0,rˇ)¯. Also if inequality (52) holds for u¯ and either α(cnˇ,u¯)≥1 or α(u¯,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point u¯ in Bdl(c0,rˇ)¯.If we takeη4=0 in Theorem 12, then we are left only with the result.Corollary 17.
Let(Z´,dl) be a complete DMS. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0, let c0∈Bdl(c0,rˇ)¯⊆Z´, and let Sˇ,Tˇ:Z´→P(Z´) be the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(54)maxτ+α∗e˘,Sˇe˘FHdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FHdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η3dle˘,Tˇy^for all e˘,y^∈Bdl(c0,rˇ)¯∩{TˇSˇ(cnˇ)} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η2,η3>0 and η1+η2+2η3<1 and (55)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η2+η3)/(1-η3) and 1-η3≠0. Then {TˇSˇ(cnˇ)} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {TˇSˇ(cnˇ)}→u¯∈Bdl(u¯0,rˇ)¯. Also if inequality (54) holds for u¯ and either α(cnˇ,u¯)≥1 or α(u¯,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point u¯ in Bdl(c0,rˇ)¯.


## 3. Fixed Point Results for Graphic Contractions

In this section, we present an application of Theorem12 in graph theory. Jachymski [21],proved the result concerning for contraction mappings on metric space with a graph. Hussain et al. [16] introduced the fixed points theorem for ψb-graphic contraction and gave an application to system of integral equations. A graph G˘ is a connected graph if there must exist a path among any two different vertices (for details, see [13, 32]).Definition 18.
LetZ´ be a nonempty set and let G˘=(V(G˘),E(G˘)) be a graph such that V(G˘)=Z´, and Sˇ:Z´→P(Z´) is said to be multigraph-dominated if (z´,u`)∈E(G˘), for all u`∈Sˇz´.Theorem 19.
Let(Z´,dl) be a complete DMS endowed with a graph G˘. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0, let c0∈Bdlc0,rˇ¯,Sˇ,Tˇ:Z´→P(Z´), and assume a sequence {TˇSˇ(cnˇ)} in Z´ generated by c0, with (c0,c1)∈E(G˘). Assume that the following hold:
(i)Sˇ and Tˇ are multigraph-dominated for all e˘,y^∈Bdl(c0,rˇ)¯∩TˇSˇcnˇ.
(ii) There exist someη1,η2,η3,η4,τ>0, and(56)maxτ+α∗e˘,Sˇe˘FHdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FHdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η3dle˘,Tˇy^+η4dl2e˘,Sˇe˘·dly^,Tˇy^1+dl2e˘,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{TˇSˇ(cnˇ)} and (e˘,y^)∈E(G˘) or y^,e˘∈EG˘.
(iii)∑ı`=0nˇλı`(dl(c0,Sˇc0))≤rˇ for all nˇ∈N∪{0}.Then,{TˇSˇ(cnˇ)} is a sequence in Bdl(c0,rˇ)¯, (cnˇ,cnˇ+1)∈E(G˘), and {TˇSˇ(cnˇ)}→m∗. Also, if inequality (56) holds for m∗ and (cnˇ,m∗)∈E(G˘) or (m∗,cnˇ)∈E(G˘) for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point m∗ in Bdl(c0,rˇ)¯ and dl(m∗,m∗)=0.Proof.
Defineα:Z´×Z´→[0,∞) by (57)αe˘,y^=1,ife˘∈Bdlc0,rˇ¯,e˘,y^∈EG˘ory^,e˘∈EG˘0,otherwise.As Sˇ and Tˇ are semigraph-dominated on Bdl(c0,rˇ)¯, then, for e˘∈Bdl(c0,rˇ)¯, (e˘,y^)∈E(G˘) for all y^∈Sˇe˘ and (e˘,y^)∈E(G˘) for all y^∈Tˇe˘. So, α(e˘,y^)=1 for all y^∈Sˇe˘ and α(e˘,y^)=1 for all y^∈Tˇe˘. This implies that inf⁡{α(e˘,y^):y^∈Sˇe˘}=1 and inf⁡{α(e˘,y^):y^∈Tˇe˘}=1. Hence α∗(e˘,Sˇe˘)=1, α∗(e˘,Tˇe˘)=1 for all e˘∈Bdl(c0,rˇ)¯. So, Sˇ,Tˇ:Z´→P(Z´) are the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Moreover, inequality (56) can be written as (58)maxτ+α∗e˘,Sˇe˘FHdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FHdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η3dle˘,Tˇy^+η4dl2e˘,Sˇe˘·dly^,Tˇy^1+dl2e˘,y^for all elements e˘,y^ in Bdl(c0,rˇ)¯∩{TˇSˇ(xnˇ)} with either α(e˘,y^)≥1 or α(y^,e˘)≥1. Also, (iii) holds. Then, by Theorem 12, we have that {TˇSˇ(cnˇ)} is a sequence in Bdl(c0,rˇ)¯ and {TˇSˇ(cnˇ)}→m∗∈Bdl(c0,rˇ)¯. Now, cnˇ,m∗∈Bdl(c0,rˇ)¯ and either (cnˇ,m∗)∈E(G˘) or (m∗,cnˇ)∈E(G˘) implies that either α(cnˇ,m∗)≥1 or α(m∗,cnˇ)≥1. So, all the conditions of Theorem 12 are satisfied. Hence, by Theorem 12, Sˇ and Tˇ have a common fixed point m∗ in Bdl(c0,rˇ)¯ and dl(m∗,m∗)=0.


## 4. Fixed Point Results for Single Valued Mapping

In this section, we discuss some new fixed point results for single valued mapping in complete DMS.Theorem 20.
Let(Z´,dl) be a complete DMS. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0, let c0∈Bdl(c0,rˇ)¯⊆Z´, and let Sˇ,Tˇ:Z´→Z´ be the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(59)maxτ+α∗e˘,Sˇe˘FdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η3dle˘,Tˇy^+η4dl2e˘,Sˇe˘·dly^,Tˇy^1+dl2e˘,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{cnˇ} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η2,η3,η4>0 and η1+η2+2η3+η4<1 and (60)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η2+η3)/(1-η3-η4) and η3+η4≠1. Then {cnˇ} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {cnˇ}→v∈Bdl(c0,rˇ)¯. Also if inequality (59) holds for v and either α(cnˇ,v)≥1 or α(v,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point v in Bdl(c0,rˇ)¯.Proof.
The proof of the above Theorem is the same as Theorem12.If we takeSˇ=Tˇ in Theorem 20, then we have the following result.Corollary 21.
Let(Z´,dl) be a complete DMS. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0, let c0∈Bdl(c0,rˇ)¯⊆Z´, and let Sˇ:Z´→Z´ be the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(61)maxτ+α∗e˘,Sˇe˘FdlSˇe˘,Sˇy^,τ+α∗y^,Sˇy^FdlSˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η3dle˘,Sˇy^+η4dl2e˘,Sˇe˘·dly^,Sˇy^1+dl2e˘,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{cnˇ} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η2,η3,η4>0 and η1+η2+η3+η4<1 and (62)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η2)/(1-η3-η4) and η3+η4≠1. Then {cnˇ} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {cnˇ}→v∈Bdl(c0,rˇ)¯. Also if inequality (61) holds for v and either α(cnˇ,v)≥1 or α(v,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point v in Bdl(c0,rˇ)¯.If we takeη2=0 in Theorem 20, then we are left with the result.Corollary 22.
Let(Z´,dl) be a complete DMS. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0, let c0∈Bdl(c0,rˇ)¯⊆Z´, and let Sˇ,Tˇ:Z´→Z´ be the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(63)maxτ+α∗e˘,Sˇe˘FdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η3dle˘,Tˇy^+η4dl2e˘,Sˇe˘·dly^,Tˇy^1+dl2e˘,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{cnˇ} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η3,η4>0 and η1+2η3+η4<1 and (64)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η3)/(1-η3-η4) and η3+η4≠1. Then {cnˇ} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {cnˇ}→v∈Bdl(c0,rˇ)¯. Also if inequality (63) holds for v and either α(cnˇ,v)≥1 or α(v,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point v in Bdl(c0,rˇ)¯.If we takeη3=0 in Theorem 20, then we are left with the result.Corollary 23.
Let(Z´,dl) be a complete DMS. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0, let c0∈Bdl(c0,rˇ)¯⊆Z´, and let Sˇ,Tˇ:Z´→Z´ be the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(65)maxτ+α∗e˘,Sˇe˘FdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η4dl2e˘,Sˇe˘·dly^,Tˇy^1+dl2e˘,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{cnˇ} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η2,η4>0 and η1+η2+η4<1 and (66)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η2)/(1-η4) and 1-η4≠0. Then {cnˇ} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {cnˇ}→v∈Bdl(c0,rˇ)¯. Also if inequality (65) holds for v and either α(cnˇ,v)≥1 or α(v,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point v in Bdl(c0,rˇ)¯.If we takeη4=0 in Theorem 20, then we are left with the result.Corollary 24.
Let(Z´,dl) be a complete DMS. Suppose that there exists a function α:Z´×Z´→[0,∞). Let rˇ>0, let c0∈Bdl(c0,rˇ)¯⊆Z´, and let Sˇ,Tˇ:Z´→Z´ be the semi-α∗-dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(67)maxτ+α∗e˘,Sˇe˘FdlSˇe˘,Tˇy^,τ+α∗y^,Tˇy^FdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η3dle˘,Tˇy^for all e˘,y^∈Bdl(e˘0,rˇ)¯∩{cnˇ} with either α(e˘,y^)≥1 or α(y^,e˘)≥1, where η1,η2,η3>0 and η1+η2+2η3<1 and (68)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η2+η3)/(1-η3) and 1-η3≠0. Then {cnˇ} is a sequence in Bdl(c0,rˇ)¯, α(cnˇ,cnˇ+1)≥1 for all nˇ∈N∪{0}, and {cnˇ}→v∈Bdl(c0,rˇ)¯. Also if inequality (67) holds for v and either α(cnˇ,v)≥1 or α(v,cnˇ)≥1 for all nˇ∈N∪{0}, then Sˇ and Tˇ have common fixed point v in Bdl(c0,rˇ)¯.


## 5. Application to the Systems of Integral Equations

Theorem 25.
Let(Z´,dl) be a complete DMS. Let rˇ>0, let c0∈Bdl(c0,rˇ)¯⊆Z´, and let Sˇ,Tˇ:Z´→Z´ be the dominated mappings on Bdl(c0,rˇ)¯. Assume that, for some τ>0,(69)maxτ+FdlSˇe˘,Tˇy^,τ+FdlTˇy^,Sˇe˘≤Fη1dle˘,y^+η2dle˘,Sˇe˘+η3dle˘,Tˇy^+η4dl2e˘,Sˇe˘·dly^,Tˇy^1+dl2e˘,y^for all e˘,y^∈Bdl(c0,rˇ)¯∩{cnˇ} with η1,η2,η3,η4>0, where η1+η2+2η3+η4<1 and(70)dlc0,Sˇc0≤1-λrˇ,where λ=(η1+η2+η3)/(1-η3-η4) and η3+η4≠1. Then {cnˇ} is a sequence in Bdl(c0,rˇ)¯, for all nˇ∈N∪{0} and {cnˇ}→v∈Bdl(c0,rˇ)¯. Then Sˇ and Tˇ have common fixed point v in Bdl(c0,rˇ)¯.Proof.
The proof of the above theorem is the same as the theorem proven in the previous section. In this section, we discuss the application of fixed point Theorem25 in form of Volterra type integral equations.(71)u˘k=∫0kH1k,h,u˘hdh,cˇk=∫0kH2k,h,cˇhdhfor all k∈[0,1]. We find the solution of (71). Let Z´=C´([0,1],R+) be the set of all continuous functions on [0,1], endowed with the complete dislocated metric. For u˘∈C´([0,1],R+), define supremum norm as u˘τ=supk∈[0,1]{u˘(k)e-τk}, where τ>0 is taken arbitrarily. Then define (72)dτu˘,cˇ=supk∈0,1u˘k+cˇke-τk=u˘+cˇτfor all u˘,cˇ∈C´([0,1],R+), and, with these settings, (C´([0,1],R+),dτ) becomes a complete DMS.Now we prove the following theorem to ensure the existence of solution of integral equations.Theorem 26.
Assume that the following conditions are satisfied:(i)
H1,H2:[0,1]×[0,1]×C´([0,1],R+)→R.(ii)
Define(73)Sˇu˘k=∫0kH1k,h,u˘hdh,Tˇcˇk=∫0kH2k,h,cˇhdh.Suppose that there exists τ>0, such that (74)H1k,h,u˘+H2k,h,cˇ≤τNu˘,cˇτNu˘,cˇτ+12for all k,h∈[0,1] and u˘,cˇ∈C´([0,1],R), where (75)Nu˘,cˇ=η1u˘k+cˇk+η2u˘k+Sˇu˘k+η3u˘k+Tˇcˇk+η4u˘k+Sˇu˘k2·cˇk+Tˇcˇk1+u˘k+cˇk2,where η1, η2, η3,η4≥0 and η1+η2+2η3+η4<1. Then integral equations (71) have a solution.Proof.
By assumption (ii),(76)Sˇu˘k+Tˇcˇk=∫0kH1k,h,u˘h+H2k,h,cˇhdh,≤∫0kττNu˘,cˇτ+12Nu˘,cˇe-τheτhdh,≤∫0kττNu˘,cˇτ+12Nu˘,cˇτeτhdh,≤τNu˘,cˇττNu˘,cˇτ+12∫0keτhdh,≤Nu˘,cˇττNu˘,cˇτ+12eτk.This implies that (77)Sˇu˘k+Tˇcˇke-τk≤Nu˘,cˇττNu˘,cˇτ+12,Sˇu˘k+Tˇcˇkτ≤Nu˘,cˇττNu˘,cˇτ+12,τNu˘,cˇτ+1Nu˘,cˇτ≤1Sˇu˘k+Tˇcˇkτ,τ+1Nu˘,cˇτ≤1Sˇu˘k+Tˇcˇkτ,which further implies that (78)τ-1Sˇu˘k+Tˇcˇkτ≤-1Nu˘,cˇτ.So all the conditions of Theorem 25 are satisfied for (cˇ)=-1/cˇ;cˇ>0; and dτ(u˘,cˇ)=u˘+cˇτ. Hence integral equations given in (71) have a unique common solution.



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*Source: 4808764-2018-02-01.xml*