# Periodic and Fixed Points for Caristi-TypeG-Contractions in Extended b-Gauge Spaces

Authors: Nosheen Zikria; Maria Samreen; Tayyab Kamran; Seher Sultan Yeşilkaya
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce extendedb-gauge spaces and the extended family of generalized extended pseudo-b-distances. Moreover, we define the sequential completeness and construct the Caristi-type G-contractions in the framework of extended b-gauge spaces. Furthermore, we develop periodic and fixed point results in this new setting endowed with a graph. The obtained results of this paper not only generalize but also unify and improve the existing results in the corresponding literature.


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## Body

## 1. Introduction and Preliminaries

The famous Caristi fixed point theorem [1] states that a self-mapping T on a complete metric space U,p possesses a fixed point w in U if
(1)pu,Tu≤fu−fTu,for all u∈U, where f:U⟶0,∞ is a lower semicontinuous function.Indeed, Caristi [1] observed these results when he searched for alternative proof of the outstanding fixed point theorem of Banach. It is known also Caristi-Kirk fixed point theorem [2]. In fact, Caristi’s theorem is equivalent to metric completeness [3]. For some other contributions to this topic, we refer to [4–10].In view of extending the concept of Banach contraction, BanachG-contraction was introduced by Jachymaski [11] in complete metric space accompanied with the graph G where the set of vertex matches with the metric space (see also [12–22]).The notion of metric space has been refined and extended in several distinct directions, by many authors [23–25]. Among all, the notion of gauge space was initiated by Dugundji [26] as a generalization of a metric space. In 1973, Reilly [27] studied quasi-gauge spaces and proved that it generalizes topological spaces, quasi uniform spaces, and quasi metric spaces. This notion was extended as b-gauge spaces by Ali et al. [28] in 2015. For further facts on gauge spaces, we recommend the reader to [29–36].In 2013, Wlodarczyk and Plebaniak [37] have given the notion of left (right) J-families of generalized pseudo distances in quasi-gauge spaces that generalizes the abovementioned distances and provides powerful and useful tools for finding solutions to various problems of nonlinear analysis.This paper is aimed at introducing extendedb-gauge spaces U,Qφ;Ω and the extended Jφ;Ω-family of generalized extended pseudo-b-distances generated by U,Qφ;Ω. Moreover, by using extended Jφ;Ω-family, we define the extended Jφ;Ω-sequential completeness and construct the Caristi-type G-contractions T:U⟶ClJφ;ΩU. Furthermore, we investigate periodic and fixed point results for these mappings in the new setting endowed with a graph, which generalizes and improves the existing results in the literature of fixed point theory.In what follows, we recollect some essential concepts and basic results which shall be used in the sequel. For a nonempty setU, we use the notation 2U to denote the set of all nonempty subsets of the space U. If T:U⟶2U is a multivalued map, then the sets of all fixed points are denoted by Fix T, that is, Fix T=u∈U:u∈Tu. In addition, the set of all periodic points of T is denoted by Per T, that is, Per T=u∈U:u∈TkuforsomekinN, where Tk=T∘T∘T∘⋯..∘T (k-times). A dynamic process of the system U,T starting at w0∈U is a sequence wm:m∈0∪N defined by ∀m∈Nwm∈Twm−1.One of the most interesting extension of a metric is the notion ofb-metric [38, 39].Definition 1.
LetU be a nonempty set and s≥1. A map q:U×U⟶0,∞ is b-metric, if it satisfies the following properties:
(a)
qe,f=0⇔e=f(b)
qe,f=qf,e(c)
qe,g≤sqe,f+qf,g
for alle,f,g∈U. Here, the pair U,q,s is called b-metric space.Indeed,b-metric is one of the most interesting and original generalizations of the notion metric. As it is seen obviously, in the case of s=1, the notions b-metric and standard metric coincide. On the other hand, despite the standard metric, b-metric is not continuous despite metric. Further, an open (closed) ball is not an open (closed) set. For more details on b-metric and interesting examples, we refer to, e.g., [40–49].In 2017, Kamran et al. [50] refined the notion of b-metric under the name “extended b-metric.”Definition 2.
SupposeU be a nonvoid set and let φ:U×U⟶1,∞. A map q:U×U⟶0,∞ is said to be an extended b-metric, if it satisfies the following properties:
(a)
qe,f=0⇔e=f(b)
qe,f=qf,e(c)
qe,g≤φe,gqe,f+qf,g, for all e,f,g∈U
For given extendedb-metric q on U, a pair U,q is called extended b-metric space.Definition 3.
LetU be a nonvoid set. The map q:U×U⟶0,∞ is called to be pseudo metric, if it satisfies the following properties:
(a)
qe,e=0(b)
qe,f=qf,e(c)
qe,g≤qe,f+qf,g, for all e,f,g∈U
The pairU,q is said to be pseudo metric space.In 2015, Ali et al. [28] have defined gauge spaces in the setting of bs-pseudo metrics called b-gauge spaces. In order to introduce extended b-gauge spaces, we start here with the introduction of the notion of extended pseudo-b-metric.Definition 4.
LetU be a nonempty set and φ:U×U⟶1,∞. A map q:U×U⟶0,∞ is an extended pseudo-b-metric, if it satisfies the following properties:
(a)
qe,e=0(b)
qe,f=qf,e(c)
qe,g≤φe,gqe,f+qf,g, for all e,f,g∈U
The pairU,q is called extended pseudo-b-metric space.Example 1.
LetU=0,1. Define q:U×U⟶0,∞ and φ:U×U⟶1,∞ for all e,f∈U as follows:
(2)qe,f=e−f2,φe,f=e+f+2,for all e,f,g∈U. Then, q is an extended pseudo-b-metric on U. Indeed, qe,e=0 and qe,f=qf,e. Further, qe,g≤φe,gqe,f+qf,g holds.Example 2.
LetU=e,f,g and φ:U×U⟶1,∞ with φe,f=∣e∣+∣f∣+2. Define q:U×U⟶0,∞ as follows:
(3)qe,e=0,qe,f=qf,e=1,qf,g=qg,f=12,andqe,g=qg,e=2,for all e,f,g∈U. Further, qe,g≤φe,gqe,f+qf,g holds. In conclusion, q is an extended pseudo-b-metric on U. Notice that 2=qe,g>3/2=qe,f+qf,g; thus, q is not a pseudo metric on U. This example shows that extended pseudo-b-metric is more general than pseudo metric.Definition 5.
Each familyQφ;Ω=qβ:β∈Ω of extended pseudo-b-metrics qβ:U×U⟶0,∞, β∈Ω, is called an extended b-gauge on U.Definition 6.
Let the familyQφ;Ω=qβ:β∈Ω be an extended b-gauge on U. The topology TQφ;Ω whose subbase is defined by the family BQφ;Ω=Be,εβ:e∈U,εβ>0,β∈Ω of all balls Be,εβ=f∈U:qβe,f<εβ is called the topology induced by Qφ;Ω on U.Definition 7.
SupposeU,T be a topological space and Qφ;Ω be an extended b-gauge on U such that T=TQφ;Ω. Then, the topological space is called to be an extended b-gauge space, which is denoted by U,Qφ;Ω.Remark 8.
(a) Each gauge space isbs-gauge space (for s =1), and each b-gauge space is an extended b-gauge space (for φβu,v=s, for each β∈Ω, where s≥1). Therefore, we can term extended b-gauge spaces as the largest general spaces. (b) We observe that if φβu,v=s, for each β∈Ω, where s≥1, the above definitions turn down to the corresponding definitions in b-gauge spaces, and if φβu,v=1 for each β∈Ω, the above definitions turn down to the corresponding definitions in gauge spaces.We now introduce the notion of extendedJφ;Ω-families of generalized extended pseudo-b-distances on U (extended Jφ;Ω-families is the generalization of extended quasi-b-gauges).Definition 9.
LetU,Qφ;Ω be an extended b-gauge space. The family Jφ;Ω=Jβ:β∈Ω where Jβ:U×U⟶0,∞,β∈Ω is said to be the extended Jφ;Ω-family of generalized extended pseudo-b-distances on U (for short, extended Jφ;Ω-family on U) if there exists φ=φββ∈Ω, where φβ:U×U⟶1,∞ such that for each β∈Ω and for all u,v,w∈U, the following hold:
(J1)Jβu,w≤φβu,wJβu,v+Jβv,w
(J2) For each sequencesum:m∈N and vm:m∈N in U fulfilling
(4)limm⟶∞supn>mJβum,un=0,(5)limm⟶∞Jβvm,um=0,the following holds:
(6)limm⟶∞qβvm,um=0.
We denoteJU,Qφ;Ω=Jφ;Ω:Jφ;Ω=Jβ:β∈Ω.We mention here some trivial properties of extendedJφ;Ω-families in the following remark.Remark 10.
LetU,Qφ;Ω be an extended b-gauge space. Then, for each β∈Ω and for all u,v,w∈U, the following hold:
(a)
Qφ;Ω∈JU,Qφ;Ω(b)
LetJφ;Ω∈JU,Qφ;Ω. If Jβv,v=0 and Jβu,v=Jβv,u, then Jβ is an extended pseudo-b-metric(c)
There exist examples ofJφ;Ω∈JU,Qφ;Ω which show that the maps Jβ are not an extended pseudo-b-metrics (see following Example 3)Example 3.
SupposeU=0,1⊂R. Let Qφ;Ω=q be the family of pseudo-b-metric where q:U×U⟶0,∞ be defined as in Example 1.
Let the setF=1/8,1⊂U. Let d∈0,∞Ω satisfies δF<d, where δF=supqe,f:e,f∈F. Let J:U×U⟶0,∞ and φ:U×U⟶1,∞ for all e,f∈U define as follows:
(7)Je,f=qe,fifF∩e,f=e,f,d=4ifF∩e,f≠e,f,and φe,f=e+f+2. Then, Jφ;Ω=J∈JU,Q.We observe thatJe,g≤φe,gJe,f+Jf,g for all e,f,g∈U; thus, condition J1 holds. Indeed, condition J1 will not hold in case if there exists some e,f,g∈U such that Je,g=d, Je,f=qe,f, Jf,g=qf,g, and φe,gqe,f+qf,g≤d. However, then this implies the existence of h∈e,g with h∉F, and on other hand, e,f,g∈F, which is impossible.Now suppose that (4) and (5) are satisfied by the sequences um:m∈N and vm:m∈N in U. Then, (5) implies
(8)∀0<ε<d∃m1∈N∀m≥m1Jvm,um<ε.By (8) and (7), we have
(9)∀m≥m1F∩vm,um=vm,um,∀0<ε<d∃m1∈N∀m≥m1qvm,um=Jvm,um<ε.Thus, (6) is satisfied by the sequences um:m∈N and vm:m∈N. Therefore, Jφ;Ω is an extended Jφ;Ω-family on U.Now, using extendedJφ;Ω-family on U, we establish the following concepts of extended Jφ;Ω-completeness in the extended b-gauge space U,Qφ;Ω.Definition 11.
LetU,Qφ;Ω be an extended b-gauge space. Let Jφ;Ω=Jβ:β∈Ω be the extended Jφ;Ω-family on U. A sequence um:m∈N is extended Jφ;Ω-cauchy sequence in U if, for all β∈Ω, limm⟶∞supn>mJβum,un=0.Definition 12.
LetU,Qφ;Ω be an extended b-gauge space. Let Jφ;Ω=Jβ:β∈Ω be the extended Jφ;Ω-family on U. The sequence um:m∈N is called to be extended Jφ;Ω-convergent to u∈U if limm⟶∞Jφ;Ωum=u, where
(10)limm⟶∞Jφ;Ωum=u⇔limm⟶∞Jβu,um=0=limm⟶∞Jβum,u.Definition 13.
LetU,Qφ;Ω be an extended b-gauge space. Let Jφ;Ω=Jβ:β∈Ω be the extended Jφ;Ω-family on U. If Sum:m∈NJφ;Ω≠∅, where Sum:m∈NJφ;Ω=u∈U:limm⟶∞Jφ;Ωum=u. Then, the sequence um:m∈N in U is extended Jφ;Ω-convergent in U.Definition 14.
LetU,Qφ;Ω be an extended b-gauge space. Let Jφ;Ω=Jβ:β∈Ω be the extended Jφ;Ω-family on U. The space U,Qφ;Ω is called Jφ;Ω-sequentially complete extended b-gauge space, if every extended Jφ;Ω-Cauchy in U is an extended Jφ;Ω-convergent in U.Remark 15.
LetU,Qφ;Ω be an extended b-gauge space.
(a)
For each subsequencevm:m∈N of um:m∈N, where um:m∈N is an extended Jφ;Ω-convergent in U, we have Sum:m∈NJφ;Ω⊂Svm:m∈NJφ;Ω(b)
We observe that ifφβu,v=s for all β∈Ω, where s≥1 and Jφ;Ω=Qφ;Ω, the above definitions of completeness reduce to the corresponding definitions in b-gauge spaces (see [28])Definition 16.
LetU,Qφ;Ω be an extended b-gauge space. The map Tk:U⟶U, where k∈N is called to be an extended Qφ;Ω-closed map on U if for each sequence xm:m∈N in TkU, which is extended Qφ;Ω-converging in U, i.e., Sxm:m∈NQφ;Ω≠∅ and its subsequences vm:m∈N and um:m∈N satisfy ∀m∈Nvm∈Tkum has the property that there exists w∈Sxm:m∈NQφ;Ω such that w∈Tkw.Definition 17.
LetU,Qφ;Ω be an extended b-gauge space, and let Jφ;Ω=Jβ:β∈Ω be the extended Jφ;Ω-family on U. A set Y∈2U is a Jφ;Ω-closed in U if Y=clUJφ;ΩY, where clUJφ;ΩY, is the Jφ;Ω-closure in U, which indicates the set of all x∈U for which there exists a sequence xm:m∈N in Y which Jφ;Ω-converges to x.
DefineClJφ;ΩU=Y∈2U:Y=clUJφ;ΩY. Thus, ClJφ;ΩU denotes the class of all nonempty Jφ;Ω-closed subsets of U.Definition 18.
LetU,Qφ;Ω be an extended b-gauge space, let Jφ;Ω=Jβ:β∈Ω be the extended Jφ;Ω-family on U, and let, for each β∈Ω, u∈U and for all V∈ClJφ;ΩU,
(11)Jβu,V=infJβu,z:z∈V.
Define onClJφ;ΩU the distance DβJφ;Ω of Hausdorff type, where DβJφ;Ω:ClJφ;ΩU×ClJφ;ΩU⟶0,∞,β∈Ω as follows:
(12)DβJφ;ΩU,V=maxsupu∈UJβu,V,supv∈VJβv,U,ifthemaximumexists;∞,otherwise
for eachβ∈Ω and for all U,V∈ClJφ;ΩU.
In this paper,Ω is a directed set and U,Qφ;Ω be an extended b-gauge space enriched with the graph G=V,E where the set of vertices coincides with set U and the set of edges E contains v,v:v∈V. Also, G is such that no two edges are parallel.


## 2. Periodic and Fixed Point Theorems

Our main results for multivalued mappings are now given below.Theorem 19.
LetU,Qφ;Ω be an extended b-gauge space. Let Jφ;Ω=Jβ:β∈Ω, where Jβ:U×U⟶0,∞, be the extended Jφ;Ω-family on U such that U,Qφ;Ω is extended Jφ;Ω-sequentially complete. Let T:U⟶ClJφ;ΩU be a multivalued edge preserving map and ϕβ:U⟶0,∞, β∈Ω be a lower semicontinuous function such that for each u∈U and v∈Tu where u,v∈E, we have, for all β∈Ω,
(13)Jβv,Tv≤ϕβu−ϕβv.
Assume, moreover, that the following conditions hold:(i)
There existz0∈U and z1∈Tz0 such that z0,z1∈E(ii)
For eachrβ:rβ>1β∈Ω and u∈U, there exists v∈Tu such that
(14)Jβu,v≤rβJβu,Tu,for all β∈Ω. Then, the following statements hold:(I)
For anyz0∈U, zm:m∈0∪N is extended Qφ;Ω-convergent sequence in U; thus, ∀z0∈USzm:m∈0∪NQφ;Ω≠∅(II)
Furthermore, assume thatTk for some k∈N is an extended Qφ;Ω-closed map on U. Then,
(a1) FixTk≠∅
(a2) ∀z0∈U∃z∈FixTkz∈Szm:m∈0∪NQφ;ΩProof.
(I) We first show thatzm:m∈0∪N is an extended Jφ;Ω-Cauchy sequence in U.By assumption (i), there existsz0∈U and z1∈Tz0 such that z0,z1∈E. Now using (13), we can write, for each β∈Ω,
(15)Jβz1,Tz1≤ϕβz0−ϕβz1.Now by using assumption (ii) and (15), we have rβ>1 for each β∈Ω and z2∈Tz1 such that
(16)Jβz1,z2≤rβJβz1,Tz1≤rβϕβz0−ϕβz1.AsT is edge preserving, we can write z1,z2∈E. Proceeding in the same manner, we have a sequence zm:m∈0∪N such that zm,zm+1∈E and for each m∈N and for all β∈Ω, we have
(17)Jβzm,zm+1≤rβJβzm,Tzm≤rβϕβzm−1−ϕβzm.This implies that the sequenceϕβzm is a nonincreasing sequence; hence, there exits lβ≥0 such that ϕβzm⟶lβ as m⟶∞. Now for m,p∈N and each β∈Ω, we have
(18)Jβzm,zm+p≤φβzm,zm+pJβzm,zm+1+φβzm,zm+pφβzm+1,zm+pJβzm+1,zm+2+φβzm,zm+pφβzm+1,zm+pφβzm+2,zm+pJβzm+2,zm+3+⋯+φβzm,zm+pφβzm+1,zm+p⋯φβzm+p−1,zm+pJβzm+p−1,zm+p≤φβzm,zm+prβϕβzm−1−ϕβzm+φβzm,zm+pφβzm+1,zm+prβϕβzm−ϕβzm+1+φβzm,zm+pφβzm+1,zm+pφβzm+2,zm+prβϕβzm+1−ϕβzm+2+⋯+φβzm,zm+pφβzm+1,zm+p⋯φβzm+p−1,zm+prβϕβzm+p−2−ϕβzm+p−1.Lettingm⟶∞, we have ϕβzm⟶lβ. This implies that zm:m∈0∪N is an extended Jφ;Ω-cauchy sequence in U, i.e., for all β∈Ω and for each z0∈U,
(19)∀ε>0∃k∈N∀n,m∈N;n>m≥kJβzm,zn<ε.Now, sinceU,Qφ;Ω is extended Jφ;Ω-sequentially complete b-gauge space, we have zm:m∈0∪N extended Jφ;Ω-convergent in U, i.e., for all z∈Szm:m∈0∪NJφ;Ω, we have, for all β∈Ω and for each ε>0,
(20)∃k∈N∀m∈N;m≥kJβz,zm<ε.Thus, from (19) and (20), fixing z∈Szm:m∈0∪NJφ;Ω, defining um=zm:m∈0∪N and vm=z:m∈0∪N, and applying J2 to these sequences, we get, for all β∈Ω and for each ε>0,
(21)∃k∈N∀m∈N;m≥kqβz,zm<ε.This impliesSzm:m∈0∪NQφ;Ω≠∅.(II) To provea1, let z0∈U be arbitrary and fixed. Since Szm:m∈0∪NQφ;Ω≠∅ and we have
(22)zm+1k∈Tkzmk,form∈0∪N,thus defining zm=zm−1+k:m∈N, we can write
(23)zm:m∈N⊂TkU,Szm:m∈0∪NQφ;Ω=Szm:m∈0∪NQφ;Ω≠∅.Also, its subsequences(24)ym=zm+1k:m∈N⊂TkU,xm=zmk:m∈N⊂TkU,satisfy, for all m∈N,
(25)ym=Tkxmand are extended Qφ;Ω-convergent to each point z∈Szm:m∈0∪NQφ;Ω. Now, using the fact below,
(26)Szm:m∈NQφ;Ω⊂Sym:m∈NQφ;ΩandSzm:m∈NQφ;Ω⊂Sxm:m∈NQφ;Ω.And the supposition thatTk for some k∈N is an extended Qφ;Ω-closed map on U, we have
(27)∃z∈Szm:m∈0∪NQφ;Ω=Szm:m∈0∪NQφ;Ωz∈Tkz.Thus,a1 holds. The assertion a2 follows from a1 and the fact that Szm:m∈0∪NQφ;Ω≠∅. Hence, the theorem is proved.☐Theorem 20.
LetU,Qφ;Ω be an extended b-gauge space. Let Jφ;Ω=Jβ:β∈Ω, where Jβ:U×U⟶0,∞, be the extended Jφ;Ω-family on U such that U,Qφ;Ω is extended Jφ;Ω-sequentially complete. Let T:U⟶ClJφ;ΩU be a multivalued edge preserving map and ϕβ:U⟶0,∞, β∈Ω be a lower semicontinuous function such that for each u∈U and v∈Tu where u,v∈E, we have, for each β∈Ω,
(28)Jβu,v≤ϕβu−ϕβv.
Assume, moreover, that the following condition holds:(i)
There existz0∈U and z1∈Tz0 such that z0,z1∈E
Then, the following statements hold:(I)
For anyz0∈U, zm:m∈0∪N is extended Qφ;Ω-convergent sequence in U; thus, ∀z0∈USzm:m∈0∪NQφ;Ω≠∅(II)
Furthermore, assume thatTk for some k∈N is an extended Qφ;Ω-closed map on U. Then,
(b1) FixTk≠∅
(b2) ∀z0∈U∃z∈FixTkz∈Szm:m∈0∪NQφ;ΩProof.
(I) We first show thatzm:m∈0∪N is an extended Jφ;Ω-Cauchy sequence in U. By assumption (i), there exist z0∈U and z1∈Tz0 such that z0,z1∈E. Now using (28), we can write, for each β∈Ω,(29)Jβz0,z1≤ϕβz0−ϕβz1.
AsT is edge preserving, we can write z1,z2∈E. Proceeding in the same manner, we have a sequence zm:m∈0∪N such that zm,zm+1∈E and for each m∈N and for all β∈Ω, we have
(30)Jβzm,zm+1≤ϕβzm−ϕβzm+1.
This implies that the sequenceϕβzm is a nonincreasing sequence; hence, there exits lβ≥0 such that ϕβzm⟶lβ as m⟶∞. Now for m,p∈N and each β∈Ω, we have
(31)Jβzm,zm+p≤φβzm,zm+pJβzm,zm+1+φβzm,zm+pφβzm+1,zm+pJβzm+1,zm+2+φβzm,zm+pφβzm+1,zm+pφβzm+2,zm+pJβzm+2,zm+3+⋯+φβzm,zm+pφβzm+1,zm+p⋯φβzm+p−1,zm+pJβzm+p−1,zm+p≤φβzm,zm+pϕβzm−ϕβzm+1+φβzm,zm+pφβzm+1,zm+pϕβzm+1−ϕβzm+2+φβzm,zm+pφβzm+1,zm+pφβzm+2,zm+pϕβzm+2−ϕβzm+3+⋯+φβzm,zm+pφβzm+1,zm+p⋯φβzm+p−1,zm+pϕβzm+p−1−ϕβzm+p.
Lettingm⟶∞, we have ϕβzm⟶lβ. This implies that zm:m∈0∪N is an extended Jφ;Ω-cauchy sequence in U, i.e., for all β∈Ω and for each z0∈U,
(32)∀ε>0∃k∈N∀n,m∈N;n>m≥kJβzm,zn<ε.
Now, sinceU,Qφ;Ω is extended Jφ;Ω-sequentially complete b-gauge space, we have zm:m∈0∪N extended Jφ;Ω-convergent in U, i.e., for all z∈Szm:m∈0∪NJφ;Ω, we have, for all β∈Ω and for each ε>0,
(33)∃k∈N∀m∈N;m≥kJβz,zm<ε.
Thus, from (32) and (33), fixing z∈Szm:m∈0∪NJφ;Ω, defining um=zm:m∈0∪N and vm=z:m∈0∪N, and applying J2 to these sequences, we get, for all β∈Ω and for each ε>0,
(34)∃k∈N∀m∈N;m≥kqβz,zm<ε.
This impliesSzm:m∈0∪NQφ;Ω≠∅.
(II) To proveb1, let z0∈U be arbitrary and fixed. Since Szm:m∈0∪NQφ;Ω≠∅ and we have
(35)zm+1k∈Tkzmk,form∈0∪N,thus defining zm=zm−1+k:m∈N, we can write
(36)zm:m∈N⊂TkU,Szm:m∈0∪NQφ;Ω=Szm:m∈0∪NQφ;Ω≠∅.
Also, its subsequences(37)ym=zm+1k:m∈N⊂TkU,xm=zmk:m∈N⊂TkU,satisfy, for all m∈N,
(38)ym=Tkxm,and are extended Qφ;Ω-convergent to each point z∈Szm:m∈0∪NQφ;Ω. Now, using the fact below,
(39)Szm:m∈NQφ;Ω⊂Sym:m∈NQφ;ΩandSzm:m∈NQφ;Ω⊂Sxm:m∈NQφ;Ω.
And the supposition thatTk for some k∈N is an extended Qφ;Ω-closed map on U, we have
(40)∃z∈Szm:m∈0∪NQφ;Ω=Szm:m∈0∪NQφ;Ωz∈Tkz.
Thus,b1 holds. The assertion b2 follows from b1 and the fact that Szm:m∈0∪NQφ;Ω≠∅. Hence, the theorem is proved.☐Theorem 21.
LetU,Qφ;Ω be an extended b-gauge space. Let Jφ;Ω=Jβ:β∈Ω, where Jβ:U×U⟶0,∞, be the extended Jφ;Ω-family on U such that U,Qφ;Ω is extended Jφ;Ω-sequentially complete. Let T:U⟶ClJφ;ΩU be a multivalued edge preserving map and ψβ:U⟶0,∞, β∈Ω be a upper semicontinuous function such that for each u∈U and v∈Tu where u,v∈E, we have, for each β∈Ω,
(41)Jβv,Tv≤ψβu−ψβv.
Assume, moreover, that the following conditions hold:(i)
There existz0∈U and z1∈Tz0 such that z0,z1∈E(ii)
For eachrβ:rβ>1β∈Ω and x∈U, there exists y∈Tx such that for each β∈Ω,(42)Jβx,y≤rβJβx,Tx.
Then, the following statements hold:(I)
For anyz0∈U, zm:m∈0∪N is extended Qφ;Ω-convergent sequence in U; thus, ∀z0∈USzm:m∈0∪NQφ;Ω≠∅(II)
Furthermore, assume thatTk for some k∈N is an extended Qφ;Ω-closed map on U. Then,
(c1) FixTk≠∅
(c2) ∀z0∈U∃z∈FixTkz∈Szm:m∈0∪NQφ;ΩProof.
(I) We first show thatzm:m∈0∪N is an extended Jφ;Ω-Cauchy sequence in U. By assumption (i), there exist z0∈U and z1∈Tz0 such that z0,z1∈E. Now using (41), we can write, for each β∈Ω,
(43)Jβz1,Tz1≤ψβz0−ψβz1.
Now by using assumption (ii) and (43), we have rβ>1 for each β∈Ω and z2∈Tz1 such that
(44)Jβz1,z2≤rβJβz1,Tz1≤rβψβz0−ψβz1.
AsT is edge preserving, we can write z1,z2∈E. Proceeding as above, we have a sequence zm:m∈0∪N such that zm,zm+1∈E, and for each m∈N and for all β∈Ω, we have
(45)Jβzm,zm+1≤rβJβzm,Tzm≤rβψβzm−1−ψβzm.
This implies that the sequenceψβzm is a nonincreasing sequence; hence, there exits lβ≥0 such that ψβzm⟶lβ as m⟶∞. Now for m,p∈N and each β∈Ω, we have
(46)Jβzm,zm+p≤φβzm,zm+pJβzm,zm+1+φβzm,zm+pφβzm+1,zm+pJβzm+1,zm+2+φβzm,zm+pφβzm+1,zm+pφβzm+2,zm+pJβzm+2,zm+3+⋯+φβzm,zm+pφβzm+1,zm+p⋯φβzm+p−1,zm+pJβzm+p−1,zm+p≤φβzm,zm+prβψβzm−1−ψβzm+φβzm,zm+pφβzm+1,zm+prβψβzm−ψβzm+1+φβzm,zm+pφβzm+1,zm+pφβzm+2,zm+prβψβzm+1−ψβzm+2+⋯+φβzm,zm+pφβzm+1,zm+p⋯φβzm+p−1,zm+prβψβzm+p−2−ψβzm+p−1.
Lettingm⟶∞, we have ψβzm⟶lβ. This implies that zm:m∈0∪N is an extended Jφ;Ω-cauchy sequence in U, i.e., for all β∈Ω and for each z0∈U,
(47)∀ε>0∃k∈N∀n,m∈N;n>m≥kJβzm,zn<ε.
Now, sinceU,Qφ;Ω is extended Jφ;Ω-sequentially complete b-gauge space, we have zm:m∈0∪N extended Jφ;Ω-convergent in U, i.e., for all z∈Szm:m∈0∪NJφ;Ω, we have, for all β∈Ω and for each ε>0,
(48)∃k∈N∀m∈N;m≥kJβz,zm<ε.
Thus, from (47) and (48), fixing z∈Szm:m∈0∪NJφ;Ω, defining um=zm:m∈0∪N and vm=z:m∈0∪N, and applying J2 to these sequences, we get, for all β∈Ω and for each ε>0,
(49)∃k∈N∀m∈N;m≥kqβz,zm<ε.
This impliesSzm:m∈0∪NQφ;Ω≠∅.
(II) To provec1, let z0∈U be arbitrary and fixed. Since Szm:m∈0∪NQφ;Ω≠∅ and we have
(50)zm+1k∈Tkzmk,form∈0∪N,
thus definingzm=zm−1+k:m∈N, we can write
(51)zm:m∈N⊂TkU,Szm:m∈0∪NQφ;Ω=Szm:m∈0∪NQφ;Ω≠∅.
Also, its subsequences(52)ym=zm+1k:m∈N⊂TkU,xm=zmk:m∈N⊂TkU,satisfy, for all m∈N,
(53)ym=Tkxm,and are extended Qφ;Ω-convergent to each point z∈Szm:m∈0∪NQφ;Ω. Now, using the fact below,
(54)Szm:m∈NQφ;Ω⊂Sym:m∈NQφ;ΩandSzm:m∈NQφ;Ω⊂Sxm:m∈NQφ;Ω.
And the supposition thatTk for some k∈N is an extended Qφ;Ω-closed map on U, we have
(55)∃z∈Szm:m∈0∪NQφ;Ω=Szm:m∈0∪NQφ;Ωz∈Tkz.
Thus,c1 holds. The assertion c2 follows from c1 and the fact that Szm:m∈0∪NQφ;Ω≠∅. Hence, the theorem is proved.☐Remark 22.
(a) The fixed point results concerning Caristi-type contractions in gauge space in [51] require the completeness of the space U,d. Therefore, our main theorems for Caristi-type G-contractions in the extended b-gauge space are a new generalization of the results in [51] in which assumptions are weaker and assertions are stronger.
(b) Our results for Caristi-typeG-contractions in extended b-gauge space tell about periodic points as well, hence improve the results in [51]
(c) We observe that by taking∀β∈Ωφβu,v=s≥1 in this paper, we obtain the results in b-gauge space.



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*Source: 1865172-2021-06-11.xml*