# New Types ofF-Contraction for Multivalued Mappings and Related Fixed Point Results in Abstract Spaces

Authors: Shaif Saleh Alshoraify; Abdullah Shoaib; Muhammad Arshad
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this article, we establish fixed point results for a pair of multivalued mappings satisfying generalized contraction on a sequence in dislocatedb-quasi metric spaces and Fρs⁎ Khan type contraction on a sequence in b-quasi metric spaces. An example and an application have been discussed. Our results modify and generalize many existing results in literature.


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## Body

## 1. Introduction and Preliminaries

A pointv is said to be a fixed point of a multivalued/self-mapping E, if v∈Ev/v=Ev. Fixed point theory has a large number of applications, for example, [1–4]. Czewick [5] initiated the study of fixed point in b-metric spaces. Many authors used the concept of b-metric spaces to prove the existence and the uniqueness of a fixed point for several contraction mappings [6–9]. Furthermore, dislocated quasi-metric spaces [10–13] generalized abstract spaces such as dislocated metric spaces [14] and quasi-metric spaces [15–17]. Recently, Klin-eam and Suanoom [18] introduced the concept of dislocated b-quasi metric spaces. Fixed point results in complete dislocated b-quasi metric spaces can be seen in [19, 20].Wardowski [21] generalized many fixed point results in a beautiful way by introducing F-contraction (see also [6, 22–30]). Nadler [31] extended Banach’s contraction mapping principle to a fundamental fixed point theorem for multivalued mappings. Since then, an interesting and rich fixed point theory for such mappings was developed in many directions; see [32–36]. The results of single valued mappings can be generalized by using multivalued mappings. Results for multivalued mappings have applications in engineering, Nash equilibria, and game theory [37–40]. Rasham et al. [41] obtained fixed point results for a pair of multivalued F-contractive mappings, which are extensions of some multivalued fixed point results.This paper introduces new types ofF-contractions on a sequence and generalizes many recent results. An example has been given to show how our results are valid when the others fail. An application has been given to obtain a solution of a system of integral equations.Definition 1 (see [18]).
LetY be a nonempty set and s≥1 a real number. A mapping dqb:Y×Y→[0,∞) is called a dislocated quasi b-metric (or simply dqb-metric), if the following conditions hold for any x,y,z∈Y:(a)
Ifdqb(x,y)=dqb(y,x)=0, then x=y;(b)
dqb(x,y)≤sdqb(x,z)+dqb(z,y).
The pair(Y,dqb) is called a dislocated quasi b-metric space (in short dislocated b-quasi-metric space).
The following remarks can be observed:
(a) Ifs=1, then a dislocated b-quasi-metric space becomes a dislocated quasi-metric space [12];
(b) ifs=1 and x=y implies dqb(x,y)=dqb(y,x)=0, then (Y,dqb) becomes a quasi-metric space [17];
(c) ifdqb(x,y)=dqb(y,x) and x=y implies dqb(x,y)=0, then (Y,dqb) becomes a b-metric space [9].Example 2 (see [20], let Y=R+ and p>1).
Definedqb:Y×Y→R+ by dqb(x,y)=x-y+x for x,y∈X. Then (Y,dqb) is a dqb-metric space with s=2p>1. But it is not a quasi b-metric space. Also it is not a dislocated b-metric space. It is obvious that (Y,dqb) is neither b-metric space nor dislocated quasi-metric space.Definition 3 (see [11]).
Let(Y,dqb) be a dislocated b-quasi-metric space. Let {yn} be a sequence in (Y,dqb), and then
(a){yn} is called Cauchy if ∀ε>0, ∃n0∈N such that ∀n>m≥n0 (respectively ∀m>n≥n0), dqb(ym,yn)<ε.
(b){yn} dislocated quasi b-converges (for short dqb -converges) to y∈Y, if limn→∞dqb(yn,y)=limn→∞dqb(y,yn)=0 or for any ε>0, there exists n0∈N, such that for all n>n0, dqb(y,yn)<ε and dqb(yn,y)<ε. In this case y is called a dqb-limit of {yn}.
(c)(Y,dqb) is called complete if every Cauchy sequence in Y converges to a point y∈Y.Definition 4 (see [12]).
Let(Y,dqb) be a dislocated b-quasi metric space. Let K be a nonempty subset of Y and let x∈Y. An element y0∈K is called a best approximation in K if (1)dqbx,K=dqbx,y0,wheredqbx,K=infy∈K⁡dqbx,yanddqbK,x=dqby0,x,wheredqbK,x=infy∈K⁡dqby,x.If each x∈Y has at least one best approximation in K, then K is called a proximinal set.It is clear that ifdqb(x,K)=dqb(K,x)=0, then x∈K. But if x∈K, then dqb(x,K) or dqb(K,x) may not equal zero. We denote P(Y) by the set of all proximinal subsets of Y.Definition 5 (see [12]).
The functionHdq:P(Y)×P(Y)→R+, defined by (2)HdqbA,B=max⁡supa∈A⁡dqa,B,supb∈B⁡dqbA,bis called dislocated quasi Hausdorff b metric on P(Y). Also (P(Y),Hdqb) is known as dislocated quasi Hausdorff b-metric space, where P(Y) is the proximinal subset of Y.Ali et al. [6] extended the family of mapping F defined by [21] to the family FS of all functions F:R+→R such that(F1)F is strictly increasing, that is, for all x,y∈R+ such that x<y implies F(x)<F(y);(F2) for each sequence{ϑn}n=1∞ of positive numbers, limn→∞ϑn=0 if and only if limn→∞F(ϑn)=-∞;(F3) there existsk∈(0,1) such that limϑ→0+ϑkF(ϑ)=0.(F4) For each sequenceϑn of positive real numbers and such that τ+F(sϑn)≤F(ϑn-1) for each n∈N, and some τ>0, we have τ+F(snϑn)≤F(sn-1ϑn-1), for each n∈N.Lemma 6.
Let(Y,dqb,s) be a dislocated b-quasi-metric space. Let (P(Y),Hdqb) be the dislocated quasi Hausdorff b-metric space on P(Y). Then, for all A,B∈P(Y) and for each a∈A, there exists ba∈B, such that Hdqb(A,B)≥dqb(a,ba) and Hdq(B,A)≥dqb(ba,a), where dqb(a,B)=dqb(a,ba) and dqb(B,a)=dqb(ba,a).Lemma 7 (see [6]).
Let(Y,db,s) be a b-metric space and let yn be any sequence in Y for which there exist τ>0 and F∈FS such that τ+F(sdqb(yn,yn+1))≤F(dqb(yn-1,yn)), n∈N. Then yn is a Cauchy sequence in Y.Lemma 8.
Let(X,dqb,s) be a dislocated b-quasi metric space, and let xn be any sequence in X for which there exist τ>0 and F∈FS such that(3)τ+Fsmax⁡dqbyn,yn+1,dqbyn+1,yn≤Fmax⁡dqbyn-1,yn,dqbyn,yn-1for each n∈N. Then yn is a Cauchy sequence in X.Proof.
Letϑn=max{dqb(yn,yn+1),dqb(yn+1,yn)}, for each n∈N. Thus, by (3) and property (F4), we get (4)τ+Fsnϑn≤Fsn-1ϑn-1,n∈N.Following similar arguments as given in [6], we obtain yn is a Cauchy sequence in X.


## 2. Main Result

Let(Y,dqb) be a dislocated b-quasi metric space, y0∈Y and S,T:Y→P(Y) be multifunctions on Y. Let y1∈Sy0 be an element such that dqb(y0,Sy0)=dqb(y0,y1), dqb(Sy0,y0)=dqb(y1,y0). Let y2∈Ty1 be such that dqb(y1,Ty1)=dqb(y1,y2), dqb(Ty1,y1)=dqb(y2,y1). Let y3∈Sy2 be such that dqb(y2,Sy2)=dqb(y2,y3) and so on. Thus, we construct a sequence yn of points in Y such that y2n+1∈Sy2n and y2n+2∈Ty2n+1, with dqb(y2n,Sy2n)=dqb(y2n,y2n+1), dqb(Sy2n,y2n)=dqb(y2n+1,y2n), and dqb(y2n+1,Ty2n+1)=dqb(y2n+1,y2n+2), dqb(Ty2n+1,y2n+1)=dqb(y2n+2,y2n+1), where n=0,1,2,⋯. We denote this iterative sequence by {TS(yn)}. We say that {TS(yn)} is a sequence in Y generated by y0. If T=S, then we say that {XT(yn)} is a sequence in Y generated by y0.Let us introduce the following definition:Definition 9.
Let(Y,dqb,s) be a dislocated b-quasi-metric space and S,T:Y→P(Y) be two multivalued mappings. The pair S,T is called a DQF-contraction, if there exists F∈FS and τ,a>0 such that for every two consecutive points x,y belonging to the range of an iterative sequence {TS(yn)} with max{Hdqb(Sx,Ty),Hdqb(Ty,Sx),Dqb(x,y),Dqb(y,x)}>0, we have(5)τ+max⁡FsHdqbSx,Ty,FsHdqbTy,Sx≤min⁡FDqbx,y,FDqby,xwhere(6)Dqbx,y=max⁡dqbx,y,dqbx,Sx.dqby,Tya+max⁡dqbx,y,dqby,x,dqbx,Sx,dqby,Ty,And we now prove the following main result.Theorem 10.
Let(Y,dqb,s) be a complete dislocated b-quasi-metric with s≥1 and S,T be a DQF-contraction. Then {TS(yn)}→u∈Y. Also, if (5) holds for each x,y∈{u}, then S and T have a common fixed point u in Y and dqb(u,u)=0.Proof.
Let{TS(yp)} be the iterative sequence in Y generated by a point y0∈Y. If (7)max⁡HdqbSy2p′,Ty2p′+1,HdqbTy2p′+1,Sy2p′,Dqby2p′,y2p′+1,Dqby2p′+1,y2p′≯0for some p′∈N∪{0}, then (8)HdqbSy2p′,Ty2p′+1=HdqbTy2p′+1,Sy2p′=Dqby2p′,y2p′+1=Dqby2p′+1,y2p′=0Clearly, if Dqb(y2p′,y2p′+1)=0, then dqb(y2p′,y2p′+1)=0. Also Dqb(y2p′+1,y2p′)=0 implies dqb(y2p′+1,y2p′)=0. So, y2p′=y2p′+1 and y2p′∈Sy2p′. Now, Hdqb(Sy2p′,Ty2p′+1)=0 implies dqb(y2p′+1,Ty2p′+1)=0 and Hdqb(Ty2p′+1,Sy2p′)=0 implies dqb(Ty2p′+1,y2p′+1)=0. So, y2p′+1∈Ty2p′+1 and y2p′ is a common fixed point of S and T. So the proof is completed in this case. Now, let (9)max⁡HdqbSy2p,Ty2p+1,HdqbTy2p+1,Sy2p,Dqby2p,y2p+1,Dqby2p+1,y2p>0,for all p∈N∪{0}. By Lemma 6, we have(10)dqby2p,y2p+1≤HdqbTy2p-1,Sy2p,dqby2p+1,y2p≤HdqbSy2p,Ty2p-1,and(11)dqby2p+1,y2p+2≤HdqbSy2p,Ty2p+1,dqby2p+2,y2p+1≤HdqbTy2p+1,Sy2p.From (11), (F1) and using condition (5), we get (12)Fsdqby2p+1,y2p+2≤FsHdqbSy2p,Ty2p+1≤max⁡FsHdqbSy2p,Ty2p+1,FsHdqbTy2p+1,Sy2p≤min⁡FDqby2p,y2p+1,FDqby2p+1,y2p-τ≤FDqby2p,y2p+1-τ,From (6), we have (13)Dqby2p,y2p+1=max⁡dqby2p,y2p+1,dqby2p,Sy2p.dqby2p+1,Ty2p+1a+maxdqby2p,y2p+1,dqby2p+1,y2p,dqby2p,Sy2p,dqby2p+1,Ty2p+1=max⁡dqby2p,y2p+1,dqby2p,y2p+1.dqby2p+1,y2p+2a+maxdqby2p,y2p+1,dqby2p+1,y2p,dqby2p,y2p+1,dqby2p+1,y2p+2≤max⁡dqby2p,y2p+1,dqby2p+1,y2p+2.If max{dqb(y2p,y2p+1),dqb(y2p+1,y2p+2)}=dqb(y2p+1,y2p+2), then (14)Fsdqby2p+1,y2p+2≤Fdqby2p+1,y2p+2-τ,which is a contradiction due to (F1) and s≥1. Therefore,(15)Fsdqby2p+1,y2p+2≤Fdqby2p,y2p+1-τ,(16)Fsdqby2p+1,y2p+2≤Fmax⁡dqby2p,y2p+1,dqby2p+1,y2p-τ.
From (11), (F1) and using condition (5), we get (17)Fsdqby2p+2,y2p+1≤FsHdqbTy2p+1,Sy2p≤max⁡FsHdqbSy2p,Ty2p+1,FsHdqbTy2p+1,Sy2p≤min⁡FDqby2p,y2p+1,FDqby2p+1,y2p-τ≤FDqby2p,y2p+1-τ=Fmax⁡dqby2p,y2p+1,dqby2p+1,y2p+2-τBy using (15) and (F1), we get (18)Fsdqby2p+2,y2p+1≤Fmax⁡dqby2p,y2p+1,1sdqby2p,y2p+1-τ=Fdqby2p,y2p+1-τ≤Fmax⁡dqby2p,y2p+1,dqby2p+1,y2p-τ.(19)Fsdqby2p+2,y2p+1≤Fmax⁡dqby2p,y2p+1,dqby2p+1,y2p-τ.Combining (16) and (19), we get(20)max⁡Fsdqby2p+2,y2p+1,Fsdqby2p+1,y2p+2≤Fmax⁡dqby2p,y2p+1,dqby2p+1,y2p-τ.By using (10) and (5), we have (21)Fsdqby2p,y2p+1≤FsHdqbTy2p-1,Sy2p≤max⁡FsHdqbSy2p,Ty2p-1,FsHdqbTy2p-1,Sy2p≤min⁡FDqby2p-1,y2p,FDqby2p,y2p-1-τ≤FDqby2p,y2p-1-τ.From (6), we have (22)Dqby2p,y2p-1=max⁡dqby2p,y2p-1,dqby2p,y2p+1.dqby2p-1,y2pa+maxdqby2p,y2p-1,dqby2p-1,y2p,dqby2p,y2p+1,dqby2p-1,y2p≤max⁡dqby2p,y2p-1,dqby2p-1,y2p,dqby2p,y2p+1.If max{dqb(y2p,y2p-1),dqb(y2p-1,y2p),dqb(y2p,y2p+1)}=dqb(y2p,y2p+1), then we obtain (23)Fsdqby2p,y2p+1≤Fdqby2p,y2p+1-τ,which is a contradiction due to (F1). Therefore,(24)Fsdqby2p,y2p+1≤Fmax⁡dqby2p-1,y2p,dqby2p,y2p-1-τ.By using (10) and (5), we have (25)Fsdqby2p+1,y2p≤FsHdqbSy2p,Ty2p-1≤FDqby2p,y2p-1-τ≤Fmax⁡dqby2p,y2p-1,dqby2p-1,y2p,dqby2p,y2p+1-τ.From (24), dqb(y2p,y2p+1)<max{dqb(y2p-1,y2p),dqb(y2p,y2p-1)}, so(26)Fsdqby2p+1,y2p≤Fmax⁡dqby2p,y2p-1,dqby2p-1,y2p-τ.Combining (24) and (26), we get(27)max⁡Fsdqby2p,y2p+1,Fsdqby2p+1,y2p≤max⁡dqby2p,y2p-1,dqby2p-1,y2p-τ.Combining (20) and (27), we get(28)τ+Fsmax⁡dqbyn,yn+1,dqbyn+1,yn≤Fmax⁡dqbyn-1,yn,dqbyn,yn-1By Lemma 8, {TS(yn)} is a Cauchy sequence in (Y,dqb). Since (Y,dqb) is a complete dislocated b-quasi-metric space, so there exists u∈Y such that {TS(yn)}→u; that is,(29)limn→∞⁡dqbyn,u=limn→∞⁡dqbu,yn=0.Now, suppose dqb(u,Tu)>0, and then Dqb(y2n,u)>0, so (30)max⁡HdqbSy2n,Tu,HdqbTu,Sy2n,Dqby2n,u,Dqbu,y2n>0.By using Lemma 6 and (5), we have (31)τ+Fsdqby2n+1,Tu≤τ+max⁡FsHdqbSy2n,Tu,FsHdqbTu,Sy2n≤min⁡FDqby2n,u,FDqbu,y2n≤FDqby2n,u.Since F is strictly increasing, we have (32)sdqby2n+1,Tu<Dqby2n,u.Taking limn→∞⁡ on both sides, we get(33)limn→∞⁡sdqby2n+1,Tu<limn→∞⁡Dqby2n,uFrom (6) (34)Dqby2n,u=max⁡dqby2n,u,dqby2n,y2n+1.dqbu,Tua+max⁡dqby2n,u,dqbu,y2n,dqby2n,y2n+1,dqbu,Tu.Taking limit as n→∞, and by using (29), we get(35)limn→∞⁡Dqby2n,u=dqbu,Tu.Using inequality (35) in (33), we get(36)limn→∞⁡sdqby2n+1,Tu<dqbu,Tu.Now, (37)dqbu,Tu≤sdqbu,y2n+1+sdqby2n+1,Tu.Taking limit as n→∞,(38)dqbu,Tu≤slimn→∞⁡dqbu,y2n+1+limn→∞⁡sdqby2n+1,Tu.Using inequalities (29) and (36) in (38), we get (39)dqbu,Tu<dqbu,Tu.This is a contradiction, so dqb(u,Tu)=0. Now, suppose dqb(Tu,u)>0, and then there exists n0∈N such that dqb(Tu,y2n+1)>0 for all n≥n0. By Lemma 6  dqb(Tu,y2n+1)≤Hdqb(Tu,Sy2n), so (40)max⁡HdqbSy2n,Tu,HdqbTu,Sy2n,Dqby2n,u,Dqbu,y2n>0.for all n≥n0. Following similar arguments as above, we get(41)limn→∞⁡sdqbTu,y2n+1<dqbu,Tu=0.Now, (42)dqbTu,u≤sdqbTu,y2n+1+sdqby2n+1,u.Taking limit as n→∞, and using inequalities (29) and (41), we get (43)dqbTu,u≤0which is a contradiction, so dqb(Tu,u)=0. Hence u∈Tu. Similarly by using (29), Lemma 6, and the inequality (44)τ+dqbSu,y2n+2≤τ+FHdqbSu,Ty2n+1,we can show that dqb(Su,u)=0. Similarly, dqb(u,Su)=0. Hence, the pair (S,T) has a common fixed point u in (Y,dqb). Now, (45)dqbu,u≤dqbu,Tu+dqbTu,u≤0.This implies that dqb(u,u)=0. Hence the proof is completed.Now, let us introduce the following example.Example 11.
LetY={0}∪Q+ and dqb(x,y)=x+2y2 if x≠y, and dqb(x,y)=0, if x=y. Then (Y,dqb) is a dislocated b-quasi-metric space with s=2. Define the mappings S,T:Y→P(Y) as follows: (46)Sy=14y,25y∩Q+,for  ally∈0,7,74,712,748,…,y+1,y+4∩Q+,otherwise.Ty=13y,38y∩Q+,for  ally∈0,7,74,712,748,…,y+3,y+6∩Q+,otherwise.Case 1.
Ifτ+max{F(sHdqb(Sx,Ty)),F(sHdqb(Tx,Sy))}=τ+F(sHdqb(Sx,Ty))≤min{F(Dqb(x,y)),F(Dqb(y,x))} holds. Define the function F:R+→R by F(x)=ln(x) for all x∈R+ and τ>0. As x,y∈Y, τ=ln(1.2) and by taking y0=7, we define the sequence {TS(yn)}={7,7/4,7/12,7/48,⋯} in Y generated by y0=7. Also, {TS(yn)}→0. Now, if x,y∈{TS(yn)}∪{0}, we have (47)sHdqbSx,Ty=2Hdqb14x,25x,13y,38y=2max⁡supa∈Sx⁡dqba,13y,38y,supb∈Ty⁡dqb14x,25x,b=2max⁡dqb2x5,y3,dqbx4,38y=2max⁡2x5+2y32,x4+34y2.Also (48)Dqbx,y=max⁡dqbx,y,dqbx,x/4,2x/5.dqby,y/3,3y/81+maxdqbx,y,dqby,x,dqbx,x4,2x5,dqby,y3,3y8=max⁡dqbx,y,dqbx,x/4.dqby,y/31+max⁡dqbx,y,dqby,x,dqbx,x4,dqby,y3=max⁡x+2y2,5xy241+x+2y2,3x22,5y32=x+2y2.Case (i). If max2x/5+2y/32,x/4+3/4y2=x/4+3/4y2, and τ=ln1.2, then we have (49)3x+3y2≤20x+2y265x4+34y2≤x+2y2ln1.2+ln⁡x4+34y2≤ln⁡x+2y2.This implies that (50)τ+F(sHdqbSx,Ty≤FDqbx,y.Case (ii). Similarly, if max2x/5+2y/32,x/4+3/4y2=2x/5+2y/32, and τ=ln1.2, then we have (51)483x+5y2≤1125x+2y2652x5+2y32≤x+2y2ln1.2+ln⁡2x5+2y32≤ln⁡x+2y2.Hence, (52)τ+F(sHdqbSx,Ty≤FDqbx,y.Case 2.
Ifmax{τ+F(sHdqb(Sx,Ty)),τ+F(sHdqb(Tx,Sy))}=τ+F(sHdqb(Tx,Sy)) holds. (53)sHdqbTx,Sy=2max⁡supb∈Tx⁡dqbb,Sy,supa∈Sy⁡dqbTx,a=2max⁡supb∈Tx⁡dqbb,14y,25y,supa∈Sy⁡dqb13x,38x,a=2max⁡dqb3x8,y4,dqbx3,2y5=2max⁡3x8+2y42,x3+4y52,where (54)Dqby,x=max⁡dqby,x,dqbx,x/4,2x/5.dqby,y/3,3y/81+max⁡dqbx,y,dqby,x,dqbx,x4,2x5,dqby,y3,3y8=max⁡dqby,x,dqbx,x/4.dqby,y/31+max⁡dqbx,y,dqby,x,dqbx,x4,dqby,y3Dqby,x=max⁡y+2x2,5xy241+y+2x2,3x22,5y32=y+2x2.Case (i). If max3x/8+2y/42,x/3+4y/52=x/3+4y/52, and τ=ln1.2, then we have (55)125x+12y2≤1125y+2x265x3+4y52≤y+2x2ln1.2+ln⁡x3+4y52≤ln⁡y+2x2,so (56)τ+FsHdqbTx,Sy≤FDqby,x.Case (ii). Similarly, if max3x/8+2y/42,x/3+4y/52=3x/8+2y/42, and τ=ln1.2, then we have (57)123x+4y2≤320y+2x2653x8+2y42≤y+2x2ln1.2+ln⁡3x8+2y42≤ln⁡y+2x2.Hence, (58)τ+FsHdqbTx,Sy≤FDqby,x.Now, if x,y∉{TS(yn)}, then the contraction does not hold. Hence all the hypotheses of Theorem 10 are satisfied so S and T have a common fixed point.If we takeS=T in Theorem 10, then we obtain the following theorem.Theorem 12.
Let(Y,dqb) be a complete dislocated b-quasi-metric space with s≥1 and S:Y→P(Y) be a multivalued mapping such that for every two consecutive points x,y belonging to the range of an iterative sequence {S(yn)} with Dqb(x,y)>0, F∈FS, τ,a>0(59)τ+FsHqbSx,Sy≤FDqbx,y,where (60)Dqbx,y=max⁡dqbx,y,dqbx,Sx.dqby,Sya+dqbx,y,dqbx,Sx,dqby,Sy.Then {S(yn)}→u∈Y. Moreover, if (59) also holds for u, then S has a fixed point u in Y and dqb(u,u)=0.Remark 13.
By setting the different values ofDqb(x,y) in (6), we can obtain different results on multivalued F-contractions as corollaries of Theorem 10.


## 3.Fρs∗ -Khan Type Contraction in Quasi b-Metric Spaces

Piri et al. [42] extended the results of Khan [43] and Fisher [44] by introducing a new general contractive condition with rational expressions. Recently, Piri et al. [30] improved some fixed point results of Fk-Khan type self-mapping on complete metric spaces. In this section, we introduce a new type of contraction satisfying an inequality of rational expressions and prove a new fixed point theorem concerning this type of contraction. Our result is real generalization of Khan fixed point theorem; we introduced Fρs∗-Khan type multivalued for two mappings in b-quasi-metric space. We start this section with the following definitions.Definition 14.
LetY be a nonempty set, s≥1, and ρs:X×X→[0,+∞) be a mapping such that ρsx,y≥s and ρsy,x≥s, implying x=y. Let M⊆Y define ρs∗x,M=infρsx,a,a∈M and ρs∗M,y=infρsb,y,b∈M. Let S,T:Y→PY be the multivalued mappings; then the pair (S,T) is said to be ρs∗-Alt multivalued mapping; if x∈Y, then (61)aρs∗x,Sx≥s,qbx,Sx=qbx,yandqbSx,x=qby,ximpliesρs∗Sy,y≥s,bρs∗Sx,x≥s,qbx,Tx=qbx,yandqbTx,x=qby,ximpliesρs∗y,Sy≥s.Definition 15 (see [30]).
Let(X,d) be a metric space. A mapping T:X→X is said to be F-Khan type contraction if there exists τ∈0,∞ and F∈Fk such that (62)τ+FdTx,Ty≤Fdx,Tx.dx,Ty+dy,Ty.dy,Txmax⁡dx,Ty,dy,Tx,for all x,y∈X, and if maxd(x,Ty),d(y,Tx)≠0, then Tx≠Ty and if maxd(x,Ty),d(y,Tx)=0, then Tx=Ty.Definition 16.
Let(Y,qb,s) be a b-quasi-metric space and (S,T) be a pair of ρs∗ multivalued mappings. Then S,T is called Fρs∗ Khan type contraction, if there exists F∈FS and τ>0 such that for every two consecutive points x,y belonging to the range of an iterative sequence {TS(yn)} with ρs∗Sy,y≥s, ρs∗x,Sx≥s, and max{Hqb(Sx,Ty),Hqb(Ty,Sx),qb(x,y),qb(y,x)}>0, we have(63)τ+max⁡FsHqbSx,Ty,FsHqbTy,Sx≤min⁡FQbx,y,FQby,x,where(64)Qbx,y=qbx,Sxqbx,Ty+qby,Tyqby,Sxmax⁡qbx,Ty,qby,Sx.Theorem 17.
Let(Y,qb,s) be acompleteb-quasi-metric space with s≥1. Let ρs:Y×Y→[0,+∞) and S,T be a pair of Fρs∗ Khan type contractions and the set G(S)=x:ρs∗x,Sx≥s is closed and contained y0. Then {TS(yn)}→u∈Y. Also, if (63) holds for each x,y∈{u}, then S and T have a common fixed point u in Y and qb(u,u)=0.Proof.
Asy0 is an arbitrary element of GS, from condition of the theorem ρs∗y0,Sy0≥s. Let {TS(yn)} be the iterative sequence in Y generated by a point y0∈Y. Let y2p′,y2p′+1 be elements of this sequence. Clearly, if (65)max⁡HqbSy2p′,Ty2p′+1,HqbTy2p′+1,Sy2p′,qby2p′,y2p′+1,qby2p′+1,y2p′≯0,for some p′∈N∪{0}, then (66)HqbSy2p′,Ty2p′+1=HqbTy2p′+1,Sy2p′=qby2p′,y2p′+1=qby2p′+1,y2p′=0.As qb(y2p′,y2p′+1)=qb(y2p′+1,y2p′)=0, so y2p′=y2p′+1 and y2p′∈Sy2p′. Now, Hqb(Sy2p′,Ty2p′+1)=0 implies qb(y2p′+1,Ty2p′+1)=0 and Hqb(Ty2p′+1,Sy2p′)=0 implies qb(Ty2p′+1,y2p′+1)=0. So, y2p′+1∈Ty2p′+1 and y2p′ is a common fixed point of S and T. So the proof is done. In order to find common fixed point of both S and T, when (67)max⁡HqbSy2p,Ty2p+1,HqbTy2p+1,Sy2p,qby2p,y2p+1,qby2p+1,y2p>0,for all p∈0∪N. Since ρs∗y0,Sy0≥s, qby0,Sy0=qby0,y1 and qbSy0,y0=qby1,y0. As (S,T) is ρs∗ multivalued mapping, ρs∗Sy1,y1≥s. Now, ρs∗Sy1,y1≥s, qby1,Ty1=qby1,y2 and qbTy1,y1=qby2,y1 implies that ρs∗y2,Sy2≥s. By induction we deduce that ρs∗y2p,Sy2p≥s and ρs∗Sy2p+1,y2p+1≥s, for all p=0,1,2,⋯. Now, by Lemma 6, we have(68)qby2p,y2p+1≤HqbTy2p-1,Sy2p,qby2p+1,y2p≤HqbSy2p,Ty2p-1and(69)qby2p+1,y2p+2≤HqbSy2p,Ty2p+1,qby2p+2,y2p+1≤HqbTy2p+1,Sy2p.As s≥1, then (69) implies (70)Fsqby2p+1,y2p+2≤FsHqbSy2p,Ty2p+1≤max⁡FsHqbSy2p,Ty2p+1,FsHqbTy2p+1,Sy2p.As y2p,y2p+1∈{TS(yn)}, ρs∗y2p,Sy2p≥s and ρs∗Sy2p+1,y2p+1≥s, then by using the condition (63), we get (71)Fsqby2p+1,y2p+2≤min⁡FQby2p,y2p+1,FQby2p+1,y2p-τ≤FQby2p,y2p+1-τ.From (64), we get (72)Qby2p,y2p+1=qby2p,Sy2pqby2p,Ty2p+1+qby2p+1,Ty2p+1qby2p+1,Sy2pmax⁡qby2p,Ty2p+1,qbSy2p,y2p+1=qby2p,y2p+1.qby2p,Ty2p+1+qby2p+1,y2p+2×0max⁡qby2p,Ty2p+1,0=qby2p,y2p+1.Therefore,(73)Fsqby2p+1,y2p+2≤Fqby2p,y2p+1-τand this implies(74)Fsqby2p+1,y2p+2≤Fmax⁡qby2p,y2p+1,qby2p+1,y2p-τ.
Ass≥1, then (69) implies (75)Fsqby2p+2,y2p+1≤FsHqbTy2p+1,Sy2p≤max⁡FsHqbTy2p+1,Sy2p,FsHqbSy2p,Ty2p+1As y2p+1,y2p∈{TS(yn)}, ρs∗Sy2p+1,,y2p+1≥s and ρs∗y2p,Sy2p≥s, then using condition (63), we get (76)Fsqby2p+2,y2p+1≤min⁡FQby2p,y2p+1,FQby2p+1,y2p-τ≤FQby2p,y2p+1-τ=Fqby2p,y2p+1-τ.Therefore,(77)Fsqby2p+2,y2p+1≤Fmax⁡qby2p,y2p+1,qby2p+1,y2p-τ.Combining (74) and (77), we get (78)max⁡Fsqby2p+1,y2p+2,Fsqby2p+2,y2p+1≤Fmax⁡qby2p,y2p+1,qby2p+1,y2p-τ.As s≥1, then (68) implies (79)Fsqby2p,y2p+1≤sHqbTy2p-1,Sy2p≤max⁡FsHqbSy2p,Ty2p-1,FsHqbTy2p-1,Sy2pAs y2p,y2p-1∈{TS(yn)}, ρs∗y2p,,Sy2p≥s and ρs∗Sy2p-1,y2p-1≥s, then by using condition (63), we get (80)Fsqby2p,y2p+1≤min⁡FQby2p,y2p-1,FQby2p-1,y2p-τ≤FQby2p,y2p-1-τ.Fsqby2p,y2p+1≤Fqby2p,Sy2p.qby2p,Ty2p-1+qby2p-1,Ty2p-1.qby2p-1,Sy2pmax⁡qby2p,Ty2p-1,qby2p-1,Sy2p-τ≤Fqby2p,y2p+1.qby2p,y2p+qby2p-1,y2p.qby2p-1,Sy2pmax⁡0,qby2p-1,Sy2p-τ≤Fqby2p-1,y2p-τ.Therefore,(81)Fsqby2p,y2p+1≤Fmax⁡qby2p-1,y2p,qby2p,y2p-1-τ.Similarly, by using (63), (64), and (68), we get(82)Fsqby2p+1,y2p≤Fmax⁡qby2p-1,y2p,qby2p,y2p-1-τ.Combining (81) and (82), we get (83)τ+Fsmax⁡qby2p,y2p+1Fqby2p+1,y2p≤Fmax⁡qby2p-1,y2p,qby2p,y2p-1.Combining (78) and (83), we get(84)τ+Fsmax⁡qbyn,yn+1,qbyn+1,yn≤Fmax⁡qbyn-1,yn,qbyn,yn-1.By Lemma 8, {TS(yn)} is a Cauchy sequence in (Y,qb).ρs∗y2p,,Sy2p≥s for all p∈N. So {y2p} is a subsequence of {TS(yn)} contained in G(S). As G(S) is closed, there exists u∈G(S) such that {y2p}→u, that is,(85)limn→∞⁡qbyn,u=limn→∞⁡qbu,yn=0.Also (86)ρs∗u,Su≥s.Now, we show that u is a fixed point for S. We claim that qb(Su,u)=qbu,Su=0. On the contrary, we assume that qbu,Su>0. Now(87)qbu,Su≤sqbu,y2n+qby2n,Su.So, there exists n0∈N such that qb(y2n,Su)>0 for all n≥n0. By Lemma 6, we have 0<qb(y2n,Su)≤Hqb(Ty2n-1,Su) for all n≥n0, so (88)max⁡HqbTy2n-1,Su,HqbSu,Ty2n-1,qbu,y2n-1,qby2n-1,u>0,for all n≥n0. By Lemma 6, and s≥1, we get (89)τ+Fsqby2n,Su≤τ+Fsmax⁡HqbTy2n-1,Su,HqbSu,Ty2n-1.Now, ρs∗u,Su≥s and ρs∗Sy2n-1,y2n-1≥s, and then by (64), we get (90)τ+Fsqby2n,Su≤FQby2n-1,u.Since F is strictly increasing, we have(91)sqby2n,Su<Qby2n-1,u.Taking limit as n→∞, on both sides of inequality (91), we get(92)limn→∞⁡sqby2n,Su<limn→∞⁡Qby2n-1,uSince qbu,Ty2n-1≤qbu,y2n, taking limit as n→∞, on both sides, we get(93)limn→∞⁡qbu,Ty2n-1=0By (64), we have (94)Qby2n-1,u=qby2n-1,Ty2n-1qby2n-1,Su+qbu,Suqbu,Ty2n-1max⁡qby2n-1,Su,qbu,Ty2n-1Taking limit as n→∞ and using inequality (93), we have (95)limn→∞⁡Qby2n-1,u=limn→∞⁡qby2n-1,y2n=0.Now, inequality (92) implies (96)limn→∞⁡sqby2n,Su<0.Taking limit as n→∞ on both sides of inequality (87) and using the above inequality, we have (97)qbu,Su<0.So our assumption is wrong and qb(u,Su)=0. Now assume that qb(Su,u)>0, and then there exists n1∈N such that qb(Su,y2n)>0 for all n≥n1. By Lemma 6  qb(Su,y2n)≤Hqb(Su,Ty2n-1), so (98)max⁡HqbTy2n-1,Su,HqbSu,Ty2n-1,qbu,y2n-1,qby2n-1,u>0,for all n≥n1. Following similar arguments as above, we get(99)limn→∞⁡sqbSu,y2n<0.Now,(100)qbSu,u≤sqbSu,y2n+sqby2n,u.Taking limit as n→∞, on both sides of inequality (100) and using (85) and (99), we get (101)qbSu,u<0which is a contradiction, so qb(Su,u)=0. Hence u∈Su. As ρs∗u,Su≥s and qbu,Su=qbSu,u=qb0,0, then Definition 14 implies (102)ρs∗Su,u≥s.Now, we show that u is a fixed point for T. We claim that qbu,Tu=0. On the contrary, we assume that qbu,Tu>0, and then there exists n2∈N such that qb(y2n+1,Tu)>0 for all n≥n2. By Lemma 6, 0<qb(y2n+1,Tu)≤Hqb(Sy2n,Tu), so (103)max⁡HqbSy2n,Tu,HqbTu,Sy2n,qby2n,u,qbu,y2n>0,for all n≥n2. By Lemma 6, and s≥1, we get (104)τ+Fsqby2n+1,Tu≤τ+max⁡FsHqbSy2n,Tu,FsHqbTu,Sy2n.Now, ρs∗y2n,Sy2n≥s and ρs∗Su,u≥s, and then by (64), we get (105)τ+Fsqby2n+1,Tu≤FQby2n,u.Since F is strictly increasing, we have(106)sqby2n+1,Tu<Qby2n,u.Taking limit n→∞, on both sides of inequality (106), we get(107)limn→∞⁡sqby2n+1,Tu<limn→∞⁡Qby2n,u.Since qbu,Sy2n≤qbu,y2n+1, taking limit n→∞, on both sides, we get(108)limn→∞⁡qbu,Sy2n=0By using (64), we get(109)Qby2n,u=qby2n,y2n+1qby2n,Tu+qbu,Tuqbu,Sy2nmax⁡qby2n,Tu,qbu,Sy2n.Taking limit as n→∞ and using inequality (108), we have (110)limn→∞⁡Qby2n,u=limn→∞⁡qby2n,y2n+1=0.Now, inequality (107) implies(111)limn→∞⁡sqby2n+1,Tu<0Now (112)qbu,Tu≤sqbu,y2n+1+sqby2n+1,Tu.Taking limit as n→∞,(113)qbu,Tu≤slimn→∞⁡qbu,2n+1+limn→∞⁡sqby2n+1,Tu.Using inequalities (85) and (111) in (113), we get (114)qbu,Tu<0.This is a contradiction, so qb(u,Tu)=0. Now assume that qb(Tu,u)>0, and then there exists n3∈N such that qb(Tu,y2n+1)>0 for all n≥n3. By Lemma 6  qb(Tu,y2n+1)≤Hqb(Tu,Sy2n), so (115)max⁡HqbSy2n,Tu,HqbTu,Sy2n,qy2n,u,qbu,y2n>0.for all n≥n3. Following similar arguments as above, we get (116)qbTu,u<0.So qb(Tu,u)=0. Hence u∈Tu. As ρs∗Su,u≥s and qbu,Tu=qbTu,u=qb0,0, then Definition 14 implies (117)ρs∗u,Su≥s.Hence, the pair (S,T) has a common fixed point u in (Y,qb). Hence the proof is completed.Corollary 18 (see [30]).
Let(X,d) be a complete metric space and T:X→X be an F-Khan contraction. Then, T has a unique fixed point x∗∈X and for every x∈X the sequence Tnx converges to x∗.


## 4. Single Valued Result with Application to System of Integral Equations

LetS,T:Y→Y be two self-mappings and x0∈Y. Let x1=Sx0, x2=Tx1, x3=Sx2 and so on. In this way, we construct a sequence xn in X such that (118)x2p+1=Sx2pandx2p+2=Tx2p+1,wherep=0,1,2,….We say that {TS(xn)} is a sequence in Y generated by x0.The following result is obtained by replacing the multivalued mappings with the single valued mappings in Theorem10. Our result generalizes Theorem 24 in [41]. Also, we prove uniqueness of common fixed point in our result.Theorem 19.
Let(Y,dqb) be a complete dislocated b-quasi-metric space with constant s≥1 and S,T:Y→Y be two self-mappings. If there exists F∈FS and τ,a>0 such that for every two consecutive points x,y belonging to the range of an iterative sequence {TS(yn)} with max{dqb(Sx,Ty),dqb(Ty,Sx),Dqb(x,y),Dqb(y,x)}>0, we have(119)τ+max⁡FsdqbSx,Ty,FsdqbTy,Sx≤min⁡FDqbx,y,FDqby,x,where(120)Dqbx,y=max⁡dqbx,y,dqbx,Sx.dqby,Tya+maxdqbx,y,dqby,x,dqbx,Sx,dqby,Ty,then {TS(yn)→u∈X. Also, if u satisfies (119), then S and T have a unique common fixed point u in X and dqb(u,u)=0.Proof.
Now, we have to prove uniqueness only. Letx∗ be another common fixed point of S,T. Suppose dqb(Su,Tx∗)>0. Then, we have (121)τ+FsdqbSu,Tx∗≤Fmax⁡dqbu,x∗,dqbu,Su.dqbx∗,Tx∗1+max⁡dqbu,x∗,dqbx∗,u,dqbu,Su,dqbx∗,Tx∗,which implies that (122)sdqbu,x∗<dqbu,x∗which is contradiction. Then dqb(Su,Tx∗)=0. Also (123)τ+FsdqbSx∗,Tu≤Fmax⁡dqbx∗,u,dqbx∗,Sx∗.dqbu,Tu1+maxdqbx∗,u,dqbu,x∗,dqbx∗,Sx∗,dqbu,Tu,And then, we get dqb(Sx∗,Tu)=0. So, x∗=u. Now, we deduce the following main result.Corollary 20.
Let(Y,dqb) be a complete dislocated b metric space with constant s≥1 and S,T:Y→Y be two self-mappings. If there exists F∈FS and τ,a>0 such that for every two consecutive points x,y belonging to the range of an iterative sequence {TS(yn)} with max{dqb(Sx,Ty),Dqb(x,y)}>0, we have(124)τ+FsdqbSx,Ty≤FDqbx,y,where (125)Dqbx,y=max⁡dqbx,y,dqbx,Sx.dqby,Tya+dqbx,y,dqbx,Sx,dqby,Ty,then {TS(yn)→u∈X. Also, if u satisfies (124), then S and T have a unique common fixed point u in X and dqb(u,u)=0.LetF be the set of all functions F:R+→R defined by [21]. Then, we have the following new result.Corollary 21.
Let(Y,dqb) be a complete dislocated quasi-metric space and S,T:Y→Y be two self-mappings. If there exists F∈F and τ,a>0 such that for every two consecutive points x,y belonging to the range of an iterative sequence {TS(yn)} with max{dqb(Sx,Ty),dqb(Ty,Sx),Dqb(x,y),Dqb(y,x)}>0, we have(126)τ+max⁡FdqbSx,Ty,FdqbTy,Sx≤min⁡FDqbx,y,FDqby,x,where (127)Dqbx,y=max⁡dqbx,y,dqbx,Sx.dqby,Tya+max⁡dqbx,y,dqby,x,dqbx,Sx,dqby,Ty,then {TS(yn)→u∈X. Also, if u satisfies (126), then S and T have a unique common fixed point u in X and dqb(u,u)=0.Now, as an application, we discuss the application of Theorem19 to find solution of the system of Volterra type integral equations. Consider the following integral equations:(128)ut=∫0tK1t,s,usds,(129)vt=∫0tK2t,s,vsdsfor all t∈[0,1]. We find the solution of (128) and (129). Let X=C([0,1],R+) be the set of all continuous functions on [0,1], endowed with the complete dislocated b-quasi-metric. For u∈C([0,1],R+), define supremum norm as uτ=supt∈[0,1]{u(t)e-τt}, where τ>0 is taken arbitrarily. Then define (130)dτu,v=supt∈0,1⁡ut+2vte-τt2=u+2vτ2for all u,v∈C([0,1],R+), and with these settings, (C([0,1],R+),dτ) becomes a dislocated b-quasi-metric space.Now we prove the following theorem to ensure the existence of solution of integral equations.Theorem 22.
Assume the following conditions are satisfied:
(i)K1,K2:[0,1]×[0,1]×R+→R+ and f,g:[0,1]→R+ are continuous.
(ii) Define(131)Sut=∫0tK1t,s,usds,Tvt=∫0tK2t,s,vsds.Suppose there exist τ>1, such that(132)max⁡K1t,s,u+2K2t,s,v,K2t,s,v+2K1t,s,u≤τe2τs-τmin⁡Mu,v,Mv,u,for all t,s∈[0,1] and u,v∈C([0,1],R), where (133)Mu,v=max⁡u+2v2,u+2Su2v+2Tv2a+max⁡u+2v2,v+2u2,u+2Su2,v+2Tv2.Then integral equations (128) and (129) have a unique solution.Proof.
By assumption (ii) and (132), we have (134)max⁡Su+2Tv,Tv+2Su=max⁡∫0tK1t,s,u+2K2t,s,vds,∫0tK2t,s,v+2K1t,s,uds≤∫0tτe2τs-τmin⁡Mu,v,Mv,udsmax⁡Su+2Tv,Tv+2Su2≤τe-τmin⁡Mu,v,Mv,u∫0te2τsds≤12e-τmin⁡Mu,v,Mv,ue2τt.This implies (135)max⁡Su+2Tv,Tv+2Sue-τt2≤12e-τmin⁡Mu,v,Mv,u.That is, (136)2max⁡Su+2Tv,Tv+2Suτ2≤e-τmin⁡Mu,v,Mv,u,which further implies (137)τ+2ln⁡max⁡Su+2Tv,Tv+2Suτ2≤ln⁡min⁡Mu,v,Mv,u,τ+max⁡sln⁡Su+2Tvτ2,sln⁡Tv+2Suτ2≤ln⁡min⁡Mu,v,Mv,u.So, all the conditions of Theorem 19 are satisfied for (a)=ln⁡a, dτ(u,v)=u+2vτ2, s=2. Hence integral equations given in (128)and (129) have a common unique solution.Remark 23.
By setting different values ofM(u,v) in (132), we can obtain different weak contractive inequalities and results as corollaries of Theorem 22.


## 5. Conclusion

In this work, we have discussed the notion of dislocatedb-quasi-metric space and given an application to find the solutions of the nonlinear integral equations in such spaces. New results in b-quasi-metric, quasi-metric, quasi dislocated metric, dislocated metric, and metric can be obtained as corollaries of our theorems, which are still not present in the literature. The notions of ρs∗-Alt multivalued mapping and Fρs∗ Khan type contraction on a sequence have been introduced. Our observation is that the fixed points of mappings which are contractive only on a sequence can be ensured by the fixed point results. Our results extend the results given in [41, 45].



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