# Fixed Point Theorems for Set-ValuedL-Contractions in Branciari Distance Spaces

Authors: Seong-Hoon Cho
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

In this paper, the notion of set-valuedL-contractions is introduced, and a new fixed point theorem for such contractions is established. An example to illustrate main theorem is given.


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## Body

## 1. Introduction and Preliminaries

Branciari [1] introduced a Branciari distance by replacing the triangle inequality in a metric with the rectangular inequality as follows.A mapd:X×X⟶0,∞, where X is a nonempty set, is said to be Branciari distance on X if and only if it satisfies the following conditions:For allx,y∈X and for all distinct points u,v∈X, each of them different from x and y is as follows:(d1)dx,y=0 if and only if x=y(d2)dx,y=dy,x(d3)dx,y≤dx,u+du,v+dv,yThe pairX,d is called a Branciari distance space, whenever d is a Branciari distance on X.In many papers, for example, [2–8], it is called generalized metric space, Branciari metric space, or rectangular metric space. However, these names do not reflect and indicate the meaning well of the notion of Branciari distance spaces because Branciari distance can not reduce to the standard metric. Further it is well known that a Branciari distance space X,d does not have a topology which is compatible with d (see [8]). For these reasons, we rename and use it as Branciari distance space.Branciari [1] extended the Banach contraction principle to Branciari distance space.After that, a lot of authors, for example, [2–15] and references therein, obtained fixed point results in such spaces. Jain et al. [16] obtained fixed point results in extended Branciari b-distance spaces [17] by defining the notion of certain contractive conditions, and they gave an application to nonlinear fractional differential equations.Branciari [1] investigated the existence of fixed points with the following two conditions:
(i)
The topology of a Branciari distance space is a Hausdorff topological space(ii)
Any Branciari distance is continuous in each coordinatesHowever, it is known that the above two conditions are not correct (see [14, 15]).Sarma et al. [15] and Samet [14] (see also [3, 4, 8, 18]) show that Branciari distances have the following topological disadvantages.(B1) A Branciari distance does not need to be continuous in each coordinates(B2) A convergent sequence in Branciari distance spaces does not need to be Cauchy(B3) The topology of a Branciari distance space does not need to be a Hausdorff topological space(B4) An open ball does not need to be an open setNote that it follows from (B3) that the uniqueness of limits can not be guaranteed.In despite of the above toplogical feature, the existence of fixed points can be investigated without additional conditions such as continuity of Branciari distances or/and Hausdorffness of the topology of Branciari distance spaces. This is why researchers are interested in Branciari distance spaces.Letθ:0,∞⟶1,∞ be a function.Consider the following conditions:(θ1) θ is nondecreasing, i.e., θt1≤θt2, whenever t1≤t2(θ2) For any sequence tn of points in 0,∞(1)limn⟶∞θtn=1⟺limn⟶∞tn=0.(θ3) There exist r∈0,1 and l∈0,∞ such that
(2)limt⟶0+θt−1tr=l.(θ4) θ is continuous(θ5) θ is strictly increasing(θ6) θinfA=infθA for all A⊂0,∞ with infA>0(θ7) For any finite sequence tkk=1n⊂0,∞(3)θ∑k=1ntk≤∏k=1nθtk.Jleli and Samet [19] introduced the notion of θ-contractions and generalized the Banach contraction principle in the setting of Branciari distance spaces, where θ:0,∞⟶1,∞ is a function such that (θ1), (θ2), and (θ3) are satisfied.Since then, Jleli et al. [7] obtained a generalization of result of [19] with conditions (θ1), (θ2), (θ3), and (θ4). Arshad et al. [20] extended the result of [19] by using the notion of α-orbital admissible mapping with conditions (θ1), (θ2), and (θ3). Also, Ahmad et al. [21] extended the result of Jleli and Samet [19] to metric spaces by using conditions (θ1), (θ2), and (θ4). Durmaz and Altun [22] obtained a generalization of the result of Klim and Wardowski [23] by defining the concept of set-valued θ-contractions with control function θ satisfying (θ1), (θ2), (θ3), and (θ6). Abdeljawad et al. [17] introduced the concept of extended Branciari b-distance, and they extended the result of Jleli and Samet [19] to extended Branciari b-distance spaces with control function θ satisfying (θ2) and (θ3). Cho [24] introduced the notion of generalized set-valued weak θ-contractions in metric spaces and obtained fixed point results for such contractions with control function θ satisfying (θ1), (θ2), (θ4), and (θ7).Recently, Cho [25] introduced the concept of L-contractions in Branciari distance spaces and established a fixed point theorem for such contractions. He unified concepts of some contractions which exist in literature including θ-contractions.Very recently, Saleh et al. [26] extended the result of Cho [25] by introducing the concepts of generalized L-contractions in Branciari distance spaces. Aydi et al. [27] extended the result of Cho [25] to partial metric spaces.In the paper, we introduce notions of set-valuedL-contractions and set-valued L∗-contractions in Branciari distance spaces and prove the existence of fixed points for both type of contractions.Khojasteh et al. [28] introduced the notion of Z-contractions by using the concept of simulation functions and unified the some existing metric fixed point results. The authors of [29–32] gave generalizations of simulation functions and obtained generalizations of results of [28]. Moreover, Demma et al. [33] and Yamaod and Sintunavarat [34] extended the results of [28] to b-metric spaces by using the notion of b-simulation functions and s-simulation functions, respectively.Letζ:0,∞×0,∞⟶R be a function. Consider the following conditions:(ζ1) ξ0,0=0(ζ2) ξt,s<s−t∀s,t>0(ζ2′) ξt,s<ψs−ψt∀s,t>0, where ψ:0,∞⟶0,∞ is continuous and strictly increasing function with ψ0=0(ζ3) For any sequence tn,sn⊂0,∞(4)limn⟶∞tn=limn⟶∞sn>0⟹limsupn⟶∞ζtn,sn<0.(ζ4) For any sequence tn,sn⊂0,∞ with tn<sn∀n=1,2,3,⋯(5)limn⟶∞tn=limn⟶∞sn>0⟹limsupn⟶∞ζtn,sn<0.(ζ5) If for any sequence tn,sn⊂0,∞(6)0<limn⟶∞tn≤liminfn⟶∞sn≤limsupn⟶∞sn≤blimn⟶∞tn<+∞,where b≥1, then we have
(7)limsupn⟶∞ζbtn,sn<0.(ζ6) If for any sequence tn,sn⊂0,∞(8)0<liminfn⟶∞tn≤slimsupn⟶∞sn≤s2liminfn⟶∞tn,0<liminfn⟶∞sn≤slimsupn⟶∞tn≤s2liminfn⟶∞sn,where s≥1, then we have
(9)limsupn⟶∞ζtn,sn<0.Letζ:0,∞×0,∞⟶R be a function.Then, we say that(1)
ζ:0,∞×0,∞⟶R is called a simulation function in the sense of Khojasteh et al. [28] if and only if (ζ1), (ζ2), and (ζ3) hold(2)
ζ:0,∞×0,∞⟶R is called a simulation function in the sense of Argoubi et al. [30] if and only if (ζ2) and (ζ3) hold(3)
ζ:0,∞×0,∞⟶R is called a simulation function in the sense of Roldán-López-de-Hierro et al. [32] if and only if (ζ1), (ζ2), and (ζ4) hold(4)
ζ:0,∞×0,∞⟶R is called a simulation function in the sense of Isik et al. [31] if and only if (ζ2) and (ζ4) hold(5)
ζ:0,∞×0,∞⟶R is called a ψ-simulation function [29] if and only if (ζ2′) and (ζ3) hold(6)
ζ:0,∞×0,∞⟶R is called a b-simulation function [33] if and only if (ζ2) and (ζ5) hold(7)
ζ:0,∞×0,∞⟶R is calld a s-simulation function [34] if and only if (ζ2) and (ζ6) holdDenoteZK(resp., ZA, Zψ) by the family of all simulation functions in the sense of Khojasteh et al. (resp., all simulation functions in the sense of Argoubi et al., all ψ-simulation functions).Note that every simulation function in the sense of Argoubi et al. is aψ-simulation function. In fact, let ζ∈ZA. If we take ψt=t,∀t≥0, then ζ∈Zψ.Proposition 1.
The following are satisfied.(1)
ZA⊂Zψ and ZA≠Zψ ([29])(2)
ZK⊂ZA and ZK≠ZA ([30])DenoteZR (resp., ZI) by the class of all simulations functions in the sense of Roldán-López-de-Hierro et al. (resp., in the sense of Isik et al.). Also, we denote by Zb and Zs the set of all b-simulation functions and s-simulation functions, respectively.Example 1 (see [28, 32, 35, 36]).
Letζi:0,∞×0,∞⟶ℝ,i=1,2,3,4,5,6,7 be a function defined as follows:
(1)
ζ1t,s=ks−t,∀s,t≥0, where 0≤k<1(2)
ζ2t,s=ϕs−t,∀s,t≥0, where ϕ:0,∞⟶0,∞ is a function such that ϕs<s,∀s>0 and limsupt⟶sϕt<s(3)
ζ3t,s=sϕs−t,∀s,t≥0, where ϕ:0,∞⟶0,1 is a function such that limsupt⟶s+ϕt<1,∀s>0(4)
ζ4t,s=s−ϕs−t,∀s,t≥0, where ϕ:0,∞⟶0,∞ is a function such that ∀s>0,liminft⟶sϕt>0 or ϕ:0,∞⟶0,∞ is continuous such that ϕt=0 if and only if t=0(5)
ζ5t,s=ψs−φt,∀t,s≥0 where ψ,φ:0,∞⟶0,∞ are continuous functions such that ψt=φt=0 if and only if t=0, ψt<t≤φt,∀t>0 and φ is increasing(6)
ζ6t,s=ηs−t,∀s,t≥0, where η:0,∞⟶0,∞ is upper semicontinuous with ηt<t,∀t>0 and ηt=0 if and only if t=0(7)
ζ7t,s=s−∫0tϕudu,∀s,t≥0, where ϕ:0,∞⟶0,∞ is a function such that for each t>0,∫0tϕudu exists and ∫0tϕudu>tThen,ζ1,ζ2,ζ3,ζ4,ζ5,ζ6,ζ7∈ZK.Example 2.
Letζ:0,∞×0,∞⟶ℝ be a function defined by
(10)ζt,s=s−t−1.
Then, (ζ2) is satisfied.
Lettn,sn be sequnces of points in 0,∞ such that
(11)tn<sn∀n=1,2,3,⋯andlimn⟶∞tn=limn⟶∞sn>0.
Then, we have(12)limsupn⟶∞ζtn,sn=limsupn⟶∞sn−tn−1=limsupn⟶∞sn−liminfn⟶∞tn−1<0.
Thus,ζ∈ZI, but ζ∉ZR, because ζ0,0≠0. Hence, ZI≠ZR.Example 3 (see [31]).
LetζI:0,∞×0,∞⟶ℝ be a function defined as
(13)ζIt,s=1ifs,t=0,0ors=t,2s−tifs<t,λs−totherwise,where 0<λ<0.Then,ζI∈ZI, but ζI∉ZA. Hence, ZA≠ZI. Also, we know that ζI∉ZK and ζI∉ZR.Note thatZK⊂ZR. The following examples show that ZK≠ZR.Example 4.
Letζ:0,∞×0,∞⟶ℝ be a function defined as
(14)ζt,s=0ifs,t=0,0,−1ifs=t,2s−tifs<t,λs−totherwise,where 0<λ<0. Then, (ζ1) and (ζ2) are satisfied.We show that (ζ5) is not satisfied.To show this, lettn,sn⊂0,∞ be two sequences such that
(15)limn⟶∞tn=limn⟶∞sn>0.We may assume thatsn<tn,∀n=1,2,3,⋯.Then(16)limsupn⟶∞ζtn,sn=0.Hence, (ζ5) is not satisfied. Thus, ζ∉ZK.We now show that (ζ6) is satisfied.Letan,bn⊂0,∞ be two sequences such that
(17)an<bn∀n=1,2,3,⋯andlimn⟶∞an=limn⟶∞bn=c>0.Then(18)limsupn⟶∞ζan,bn=λlimsupn⟶∞bn−liminfn⟶∞an=λc−c<0.Hence, (ζ6) holds. Thus, ζ∈ZR. Therefore, ZR≠ZK.Proposition 2.
The following inclusion relations are satisfied.(1)
ZK⊊ZA⊊ZI(2)
ZK⊊ZA⊊Zψ(3)
ZK⊊ZR⊊ZIProposition 3.
Ifζ∈Zs is decreasing in the first coordinate, then ζ∈Zb.Proof.
Letζ∈Zs. Then, (ζ2) and (ζ6) hold.
Assume thatζ∈Zs is decreasing in the first coordinate.
We show that (ζ5) holds.
Lettn,sn⊂0,∞ be two sequences such that
(19)0<limn⟶∞tn≤liminfn⟶∞sn≤limsupn⟶∞sn≤blimn⟶∞tn<+∞,where b>1.
From (19), we infer that
(20)0<liminfn⟶∞tn=limn⟶∞tn≤bliminfn⟶∞sn≤blimsupn⟶∞sn≤b2limtn=b2liminfn⟶∞tn,which implies
(21)0<liminfn⟶∞tn≤blimsupn⟶∞sn≤b2liminfn⟶∞tn.
It follows from (19) that
(22)0<limtn≤liminfn⟶∞sn≤blimsupn⟶∞tn,blimtn=bliminfn⟶∞tn≤b2liminfn⟶∞sn.
Thus, we obtain(23)0<liminfn⟶∞sn≤blimsupn⟶∞tn≤b2liminfn⟶∞sn.
By applying (21) and (23) to (ζ6), we have
(24)limsupn⟶∞ζtn,sn<0.
Sinceζ is decreasing in the first coordinate,
(25)limsupn⟶∞ζbtn,sn≤limsupn⟶∞ζtn,sn<0.
Hence, it follows from (19) and (25) that (ζ5) is satisfied. Thus, ζ∈Zb.Note that simulation functionsζi∈ZK,i=1,2,3,4,5,6,7, given in Example 1 are s-simulation and decreasing functions in the first coordinate. Hence, ζi∈Zb,i=1,2,3,4,5,6,7 (see also [33]).For more details and examples of simulation functions, we refer to [28, 33, 34, 36–40], and for ψ-simulation functions, we refer to [29, 41, 42].Now, we recall the concept ofL-simulation function and give the definition of L∗-simulation function.Letξ:1,∞×1,∞⟶R be a function. Consider the following conditions:(ξ1) ξ·,· is a decreasing function on the first coordinate(ξ2) ξ1,1=1(ξ3) ξrt,s=1/rξt,s,∀r>1(ξ4) ξt,s<s/t∀s,t>1(ξ5) For any sequence tn,sn⊂1,∞ with tn≤sn∀n=1,2,3,⋯(26)limn⟶∞tn=limn⟶∞sn>1⟹limsupn⟶∞ξtn,sn<1.Letξ:1,∞×1,∞⟶R be a function.Then, we say that(1)
ξ is L-simulation function [25] if and only if it satisfies conditions (ξ2), (ξ4), and (ξ5)(2)
ξ isL∗-simulation function if and only if it satisfies conditions (ξ1), (ξ2), (ξ3), (ξ4), and (ξ5)DenoteL∗ by the family of all L∗-simulation functions, and L by the class of all L-simulation functions.Note thatξt,t<1∀t>1, and L∗⊂L.Example 5 (see [25]).
Letξb,ξw,ξ:1,∞×1,∞⟶ℝ be functions defined as follows, respectively:
(1)
ξbt,s=sk/t∀t,s≥1 where k∈0,1(2)
ξwt,s=s/tϕs∀t,s≥1 where ϕ:1,∞⟶1,∞ is nondecreasing and lower semicontinuous such that ϕ−11=1(27)ξt,s1ifs,t=1,1,s2tifs<t,sλtotherwise,
∀s,t≥1, where λ∈0,1.Then,ξb,ξw,ξ∈L∗.Example 6.
Letξi:1,∞×1,∞⟶ℝ,i=1,2,3, be a function defined as follows:
(1)
ξ1t,s=ψs/φt,∀t,s≥1 where ψ,φ:1,∞⟶1,∞ are continuous functions such that ψt=φt=1 if and only if t=1, ψt<t≤φt,∀t>1 and φ is an increasing function(2)
ξ2t,s=ηs/t,∀s,t≥1, where η:1,∞⟶1,∞ is upper semicontinuous with ηt<t,∀t>1 and ηt=1 if and only if t=1(3)
ξ3t,s=s/∫0tϕudu,∀s,t≥1, where ϕ:0,∞⟶0,∞ is a function such that for each t>0,∫0tϕudu exists and ∫0tϕudu>t, and ∫01ϕudu=1
Then,ξ1,ξ3∈L, and ξ2∈L∗.
Note that ifφ:1,∞⟶1,∞ is satisfied condition
(28)φrt=rφt∀r>1,t≥1,then ξ1∈L∗.We recall the following definitions which are in [1].LetX,d be a Branciari distance space, xn⊂X be a sequence, and x∈X.Then, we say that(1)
xn is convergent to x (denoted by limn⟶∞xn=x) if and only if limn⟶∞dx,xn=0(2)
xn is Cauchy if and only if limn,m⟶∞dxn,xm=0(3)
X,d is complete if and only if every Cauchy sequence in X is convergent to some point in XLetX,d be a Branciari distance space.We denote byCLX the class of nonempty closed subsets of X. Let H·,· be the Hausdorff distance on CLX, i.e., for all A,B∈CLX,
(29)HA,B=maxsupa∈Ada,B,supb∈Bdb,A,ifthemaximumexists,∞,otherwise,where da,B=infda,b:b∈B is the distance from the point a to the subset B.ForA,B∈CLX, let DA,B=supx∈Adx,B.Then, we haveDA,B≤HA,B for all A,B∈CLX.Lemma 4 (see [43]).
LetX,d be a Branciari distance space, xn⊂X be a Cauchy sequence, and x,y∈X. If there exists a positive integer N such that
(1)
xn≠xm∀n,m>N(2)
xn≠x∀n>N(3)
xn≠y∀n>N(4)
limn⟶∞dxn,x=0=limn⟶∞dxn,y
thenx=y.Lemma 5.
IfX,d is a Branciari distance space, then CX⊂CLX, where CX is the class of nonempty compact subsets of X.Proof.
LetA∈CX, and let xn⊂A be a sequence such that
(30)limn⟶∞dxn,x=0,wherex∈X.
It follows from compactness ofA that there exists a convergent subsequence xnk of xn.
Let(31)limn⟶∞dxnk,a=0,fora∈A.
Since(32)limn⟶∞dxnk,x=0,from Lemma 4, x=a∈A. Hence, A∈CLX.Lemma 6.
LetX,d be a Branciari distance space, and let A,B∈CLX.
Ifa∈A and da,B<c, then there exists b∈B such that da,b<c.Proof.
Letε=c−da,B.
It follows from definition of infimum that there existsb∈B such that da,b<da,B+ε. Hence, da,b<c.


## 2. Fixed Point Theorems

We denote byΘ the class of all functions θ:0,∞⟶1,∞ such that conditions (θ2), (θ4), and (θ5) hold.LetX,d be a Branciari distance space.A set-valued mappingT:X⟶2X, where 2X is the family of all nonempty subsets of X, is called set-valued L-contraction with respect to ξ∈L if and only if for all x,y∈X with dx,y>0, and for all u∈Tx, there exists v∈Ty with du,v>0 such that
(33)ξθdu,v,θdx,y≥1,where θ∈Θ.Now, we prove our main result.Theorem 7.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set-valued L-contraction with respect to ξ∈L.Then,T has a fixed point.Proof.
Letx0∈X be a point, and let x1∈Tx0 be such that dx0,x1>0.
From (33), there exists x2∈Tx1 with dx1,x2>0 such that
(34)1≤ξθdx1,x2,θdx0,x1<θdx0,x1θdx1,x2,which implies
(35)θdx1,x2<θdx0,x1,and so
(36)dx1,x2<dx0,x1.
Again, from (33), there exists x3∈Tx2 with dx2,x3>0 such that
(37)1≤ξθdx2,x3,θdx1,x2<θdx1,x2θdx2,x3,which implies
(38)θdx2,x3<θdx1,x2,and so
(39)dx2,x3<dx1,x2.
Inductively, we can find a sequencexn⊂X such that, ∀n=1,2,3,⋯,(40)xn−1≠xn,xn∈Txn−1anddxn,xn+1<dxn−1,xn.
Sincedxn−1,xn is a decreasing sequence, there exists r≥0 such that
(41)limn⟶∞dxn−1,xn=r.
We now show thatr=0.
Assume thatr≠0.
Then, it follows fromθ2 that
(42)limn⟶∞θdxn−1,xn≠1,andsolimn⟶∞θdxn−1,xn>1.
Lettn=θdxn,xn+1 and sn=θdxn−1,xn∀n=1,2,3,⋯.
Then,tn≤sn∀n=1,2,3,⋯ and
(43)limn⟶∞tn=limn⟶∞sn>1.
It follows from (ξ5) that
(44)1≤limsupn⟶∞ξtn,sn<1,which is a contradiction.
Thus, we have(45)limn⟶∞dxn−1,xn=0,and so
(46)limn⟶∞θdxn−1,xn=1.
We now show thatxn is a Cauchy sequence.
On the contrary, assume thatxn is not a Cauchy sequence.
Then, there exists anε>0 for which we can find subsequences xmk and xnk of xn such that mk is the smallest index for which mk>nk>k∀k=1,2,3,⋯(47)dxmk,xnk≥εanddxmk−1,xnk<ε.
It follows from (47) and condition (d3) that
(48)ε≤dxmk,xnk≤dxnk,xmk−2+dxmk−2,xmk−1+dxmk−1,xmk<ε+dxmk−2,xmk−1+dxmk−1,xmk.
Lettingk⟶∞ in above inequality, we have
(49)limk⟶∞dxmk,xnk=ε=limk⟶∞dxmk−2,xnk.
From (33), there exists znk+1∈Txnk with dxmk+1,znk+1>0∀k=1,2,3,⋯ such that
(50)1≤ξθdxmk+1,znk+1,θdxmk,xnk<θdxmk,xnkθdxmk+1,znk+1,which implies
(51)θdxmk+1,znk+1<θdxmk,xnk,∀k=1,2,3,⋯.
So(52)dxmk+1,znk+1<dxmk,xnk,∀k=1,2,3,⋯.
Taking limit supremum in above inequality and using (49), we have
(53)limsupn⟶∞dxmk+1,znk+1≤ε.
We deduce that(54)dxmk,xnk≤dxmk,xmk+1+dxmk+1,Txnk+dTxnk,xnk≤dxmk,xmk+1+dxmk+1,znk+1+dxnk+1,xnk.
Taking limit infimum in above inequality and using (53), we have
(55)ε≤liminfn⟶∞dxmk+1,znk+1.
It follows from (53) and (55) that
(56)limn⟶∞dxmk+1,znk+1=ε.
Let(57)tk=θdxmk+1,znk+1andsk=θdxmk,xnk.
Then,tk≤sk∀k=1,2,3,⋯.
It follows from (49), (56), (θ2), and (θ4) that
(58)limk⟶∞sk=limk⟶∞tk>1.
It follows from (ξ5) that
(59)1≤limk⟶∞supξtk,sk<1,which is a contradiction.
Thus,xn is a Cauchy sequence.
SinceX is complete, there exists a point x∗∈X such that
(60)limn⟶∞dx∗,xn=0.
It follows from (33) that there exists yn∈Tx∗ with dxn+1,yn>0 such that
(61)1≤ξθdxn+1,yn,θdxn,x∗<θdxn,x∗θdxn+1,yn,which implies
(62)θdxn+1,yn<θdxn,x∗,and hence
(63)dxn+1,yn<dxn,x∗.
Thus, we have(64)limn⟶∞dxn+1,yn=0.
Since(65)dx∗,yn≤dx∗,xn+dxn,xn+1+dxn+1,yn,limn⟶∞dx∗,yn=0.
BecauseTx∗∈CLX and yn⊂Tx∗, x∗∈Tx∗.We give an example to illustrate Theorem7.Example 1.
LetX=1,2,3,4 and define d:X×X⟶0,∞ as follows:
(66)d1,2=d2,1=3,d2,3=d3,2=d1,3=d3,1=1,d1,4=d4,1=d2,4=d4,2=d3,4=d4,3=4,dx,x=0∀x∈X.Then,X,d is a complete Branciari distance space but not a metric space (see [9]).Define a mapT:X⟶CLX by
(67)Tx=2,3x=1,2,3,1,4x=4,and a function θ:0,∞⟶1,∞ by
(68)θt=et.We now show thatT is a set valued L-contraction with respect to ξw, where ξwt,s=s/tϕs∀t,s≥1,ϕs=1/2s+1/2∀s≥1.We consider the following cases.Case 1.
x=1 and y=2.
For2∈T1, there exists 3∈T2 with d2,3>0 such that
(69)ξwθd2,3,θd1,2=ξwθ1,θ3=θ3θ1ϕ3=e22>1,and for 3∈T1, there exists 2∈T2 with d3,2>0 such that
(70)ξwθd3,2,θd1,2=ξwθ1,θ3=θ3θ1ϕ3=e22>1.Case 2.
x=1 and y=3.
For2∈T1, there exists 3∈T3 with d2,3>0 such that
(71)ξwθd2,3,θd1,3=ξwθ1,θ1=θ1θ1ϕ1=1,and for 3∈T1, there exists 2∈T3 with d3,2>0 such that
(72)ξwθd3,2,θd1,3=ξwθ1,θ1=θ1θ1ϕ1=1.Case 3.
x=1 and y=4.
For2∈T1, there exists 1∈T4 with d2,1>0 such that
(73)ξwθd2,1,θd1,4=ξwθ3,θ4=θ4θ3ϕ4=e2.5>1,and for 3∈T1, there exists 1∈T4 with d3,1>0 such that
(74)ξwθd3,1,θd1,4=ξwθ1,θ4=θ4θ1ϕ4=e32.5>1.Case 4.
x=2 and y=3.
For2∈T2, there exists 3∈T3 with d2,3>0 such that
(75)ξθd2,3,θd2,3=ξθ1,θ1=θ1θ1ϕ1=1,and for 3∈T2, there exists 2∈T3 with d3,2>0 such that
(76)ξθd3,2,θd2,3=ξθ1,θ1=θ1θ1ϕ1=1.Case 5.
x=2 and y=4.
For2∈T3, there exists 1∈T4 with d2,1>0 such that
(77)ξθd2,1,θd3,4=ξθ3,θ4=θ4θ3ϕ4=e2.5>1,and for 3∈T3, there exists 1∈T4 with d3,1>0 such that
(78)ξθd3,1,θd3,4=ξθ1,θ4=θ4θ1ϕ4=e32.5>1.Case 6.
x=3 and y=4.
For2∈T3, there exists 1∈T4 with d2,1>0 such that
(79)ξθd2,1,θd3,4=ξθ3,θ4=θ4θ3ϕ4=e2.5>1,and for 3∈T3, there exists 1∈T4 with d3,1>0 such that
(80)ξθd3,1,θd3,4=ξθ1,θ4=θ4θ1ϕ4=e32.5>1.Hence,T is a set valued L-contraction with respect to ξwThus, all hypotheses of Theorem7 are satisfied, and T has fixed points 2, 3, and 4.Note that the Nadler’s fixed point theorem does not hold. Ifx=2 and y=4, then
(81)HT2,T4≤kd2,4,k∈0,1,so
(82)4k≥4,which is a contradiction.From Theorem7, we have the following corollary.Corollary 8.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(83)ξθDTx,Ty,θdx,y≥r,where ξ∈L∗,θ∈Θ and r>1.Then,T has a fixed point.Proof.
Since(84)infz∈Txdz,Ty>0,for each u∈Tx,du,Ty>0. Hence, we have
(85)θdu,Ty≤θDTx,Ty<rθDTx,Ty.
It follows from (θ5) that there exists the inverse function θ−1 of θ, and so from (85), we have
(86)du,Ty<θ−1rθDTx,Ty.
From Lemma6, there exists v∈Ty such that
(87)du,v<θ−1rθDTx,Ty.
Hence(88)θdu,v<rθDTx,Ty.
Thus, we have(89)ξθdu,v,θdx,y>ξrθDTx,Ty,θdx,y=1rξθDTx,Ty,θdx,y≥1rr=1.Thus, condition (33) holds and from Theorem 7, T has a fixed point.Corollary 9.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(90)ξθHTx,Ty,θdx,y≥r,where ξ∈L∗,θ∈Θ and r>1.Then,T has a fixed point.Proof.
It follows from (ξ1) and (90) that
(91)ξθDTx,Ty,θdx,y≥ξθHTx,Ty,θdx,y≥r,which implies
(92)ξθDTx,Ty,θdx,y≥r.Thus, from Corollary8, T has a fixed point.


## 3. Consequence

We have fixed point results by applying simulation functions given in Example5 and Example 6 to Theorem 7.By takingξ=ξb in Theorem 7, we obtain Corollary 10.Corollary 10.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map.
Suppose that for allx,y∈X with dx,y>0 and u∈Tx, there exists v∈Ty with du,v>0 such that
(93)θdu,v≤θdx,yk,where θ∈Θ and k∈0,1.Then,T has a fixed point.Remark 11.
Corollary10 is a generalization of Theorem 7 of [19] and Theorem 2.2 of [21] without condition (θ3) to set-valued maps.Corollary 12.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(94)θDTx,Ty<θdx,yk,where θ∈Θ,k∈0,1.Then,T has a fixed point.Proof.
Sinceinfz∈Txdz,Ty>0, we have that, for each u∈Tx,
(95)θdu,Ty≤θDTx,Ty<θdx,yk,and so
(96)du,Ty<θ−1θdx,yk.
Hence, there existsv∈Ty such that
(97)du,v<θ−1θdx,yk,andsoθdu,v<θdx,yk.
By Corollary10, T has a fixed point.Corollary 13.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(98)θHTx,Ty<θdx,yk,where θ∈Θ,k∈0,1.Then,T has a fixed point.Remark 14.
Corollary13 is a generalization of Theorem 2 of [44] to Branciari distance spaces with conditions:
(99)θ5andinfz∈Txdz,Ty>0,∀x,y∈X.By takingξ=ξw in Theorem 7, we obtain Corollary 15.Corollary 15.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map.
Suppose that for allx,y∈X and u∈Tx, there exists v∈Ty with du,v>0 such that
(100)θdu,v≤θdx,yϕθdx,y,where θ∈Θ and ϕ:1,∞⟶1,∞ is nondecreasing and lower semicontinuous such that ϕ−11=1.Then,T has a fixed point.Remark 16.
Corollary15 is a generalization of Corollary 2.7 [24] to Branciari distance spaces without condition (θ7).Corollary 17.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(101)θDTx,Ty<θdx,yϕθdx,y,where θ∈Θ and ϕ:1,∞⟶1,∞ is nondecreasing and lower semicontinuous such that ϕ−11=1.Then,T has a fixed point.Proof.
Suppose that condition (101) satisfied.
Sinceinfz∈Txdz,Ty>0, we have that, for each u∈Tx,
(102)θdu,Ty≤θDTx,Ty<θdx,yϕθdx,y.
Thus(103)du,Ty<θ−1θdx,yϕθdx,y,and so there exists v∈Ty such that
(104)θdu,v<θdx,yϕθdx,y.
By Corollary15, T has a fixed point.Corollary 18.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(105)θHTx,Ty<θdx,yϕθdx,y,where θ∈Θ and ϕ:1,∞⟶1,∞ is nondecreasing and lower semicontinuous such that ϕ−11=1.Then,T has a fixed point.Corollary 19.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map.
Suppose that for allx,y∈X with dx,y>0 and for all u∈Tx, there exists v∈Ty with du,v>0 such that
(106)du,v≤dx,y−φdx,y,where φ:0,∞⟶0,∞ is nondecreasing and lower semicontinuous such that φ−10=0.Then,T has a fixed point.By the same way as proof of Corollary15 in [24], we can prove Corollary 19.Corollary 20.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(107)DTx,Ty<dx,y−φdx,y,where φ:0,∞⟶0,∞ is nondecreasing and lower semicontinuous such that φ−10=0.Then,T has a fixed point.Corollary 21.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(108)HTx,Ty<dx,y−φdx,y,where φ:0,∞⟶0,∞ is nondecreasing and lower semicontinuous such that φ−10=0.Then,T has a fixed point.By takingθt=2−2/πarctan1/tα, where α∈0,1,t>0 in Corollary 15, we obtain the following result.Corollary 22.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map.
Suppose that for allx,y∈X and u∈Tx, there exists v∈Ty with du,v>0 such that
(109)2−2πarctan1du,vα≤2−2/πarctan1/dx,yαϕ2−2/πarctan1/dx,yα,where α∈0,1 and ϕ:1,∞⟶1,∞ is nondecreasing and lower semicontinuous such that ϕ−11=1.Then,T has a fixed point.Corollary 23.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(110)2−2πarctan1DTx,Tyα<2−2/πarctan1/dx,yαϕ2−2/πarctan1/dx,yα,
whereα∈0,1 and ϕ:1,∞⟶1,∞ is nondecreasing and lower semicontinuous such that ϕ−11=1.Then,T has a fixed point.Corollary 24.
LetX,d be a complete Branciari distance space, and let T:X⟶CLX be a set valued map such that for all x,y∈X with dx,y>0 and infz∈Txdz,Ty>0(111)2−2πarctan1HTx,Tyα<2−2/πarctan1/dx,yαϕ2−2/πarctan1/dx,yα,
whereα∈0,1 and ϕ:1,∞⟶1,∞ is nondecreasing and lower semicontinuous such that ϕ−11=1.Then,T has a fixed point.


## 4. Conclusion

One can unify and merge some existing fixed point theorems by usingL-simulation functions and L∗-simulation functions in Branciari distance spaces. One can obtain some concequence of the main theorem by applying L-simulation functions and L∗-simulation functions given in Example 1 and Example 2. Further, one can derive all the results of the paper in the setting of metric spaces.


## 5. Suggestion

We suggest that theb-simulation function can be extended in a similar way to the one in which the simulation function is extended to the L-simulation function. The main theorem can be extended and generalized to b-metric space, Branciari b-distance space, and extended Branciari b-distance space using certain extended simulation functions, and the existing fixed point theorem can be interpreted.



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*Source: 6384818-2021-11-29.xml*