# Fixed Point Results for Multivalued Contractive Mappings Endowed with Graphic Structure

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

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## Abstract

The aim of this paper is to introduce the notion of admissible multivalued mappings and to set up fixed point results for such mappings fulfilling generalized locally Ćirić type rational-contractive conditions on a closed ball in complete dislocatedb-metric space. Example and application have been given to demonstrate the novelty of our results. Our results combine, extend, and infer several comparable results in the existing literature.


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## Body

## 1. Introduction and Preliminaries

FP theory plays a fundamental role in functional analysis. Banach proved significant result for contraction mappings. After that, a huge number of FP theorems have been established by various authors and they made different generalizations of the Banach’s result. Shoaib et al. [1], discussed the result related to α∗-ψ-Ćirić type multivalued mappings on an intersection of a closed ball and a sequence with graph. Further FP results on a closed ball can be seen ([2–12]).Boriceanu [13] proved FP results for multivalued generalized contraction on a set with two b-metrics. After this Aydi et al. [14] established FP theorem for set-valued quasi contraction in b-metric spaces. Nawab et al. [15] established the new idea of dislocated b-metric space as a conception of metric space and proved some common FP results for four mappings fulfilling the generalized weak contractive conditions in partially ordered dislocated b-metric space.Nadler [16] initiated the study of FP theorems for the multivalued mappings (see also [17]). Several results on multivalued mappings have been observed (see [18–20]). Asl et al. [21] gave the idea of α∗-ψ contractive multifunctions and α∗-admissible mapping and got some FP conclusions for these multifunctions (see also [22, 23]). In 1974, Ćirić [24] introduced quasi contraction. Reference [25] established some new common fixed points of generalized rational-contractive mappings in dislocated metric spaces with applications. In this paper, the concept of new Ćirić type rational multivalued contraction has been introduced. Now we prove new type of result for a different multivalued rational expression studied by Rasham et al. [6]. Common FP results for such contraction on a closed ball in complete dislocated b-metric space have been established. Example and application have been given. We give the following definitions and results which will be needed in the sequel.Definition 1 (see [15]).
LetM be a nonempty set and let db:M×M→[0,∞) be a function, called a dislocated b-metric (or simply db-metric), if for any g,q,z˙∈M and t≥1 the following conditions hold:(i)
ifdb(g,q)=0, then g=q;(ii)
db(g,q)=db(q,g);(iii)
db(g,q)≤t[db(g,z˙)+db(z˙,q)].
The pair(M,db) is called a dislocated b-metric space. It should be noted that the class of db metric spaces is effectively larger than that of dl metric spaces, since db is a dl metric when t=1.
It is clear that ifdb(g,q)=0, then from (i), g=q. But if g=q, db(g,q) may not be 0. For g∈M and ε>0,B(g,ε)¯=q∈M:dbg,q≤ε is a closed ball in (M,db). We use D.B.M space instead of dislocated b-metric space.Example 2.
IfM=R+∪0, then db(g,q)=(g+q)2 defines a D.B.Mdb on M.Definition 3 (see [15]).
Let(M,db) be a D.B.M space.
(i) A sequence{gn} in (M,db) is called Cauchy sequence if, given ε>0, there corresponds n0∈N such that for all n,m≥n0 we have db(gm,gn)<ε or limn,m→∞db(gn,gm)=0.
(ii) A sequence{gn} dislocated b-converges (for short db-converges) to g if limn→∞db(gn,g)=0. In this case g is called a db-limit of {gn}.
(iii)(M,db) is called complete if every Cauchy sequence in M converges to a point g∈M such that db(g,g)=0.Definition 4.
LetH^ be a nonempty subset of D.B.M space M and let g∈M. An element q0∈H^ is called a best approximation in H^ if (1)dbg,H^=dbg,q0,wheredbg,H^=infq∈H^dbg,q.
If eachg∈M has at least one best approximation in H^, then H^ is called a proximinal set.
We denote byP(M) the set of all proximinal subsets of M. Let Ψ, where t≥1, denote the family of all nondecreasing functions ψ:[0,+∞)→[0,+∞) such that ∑k=1+∞tkψk(u)<+∞ and tψ(u)<u for all u>0, where ψk is the kth iterate of ψ. Also tn+1ψn+1(u)=tntψ(ψn(u))<tnψn(u).Definition 5 (see [26]).
LetB,A:M→P(M) be the closed valued multifunctions and β:M×M→[0,+∞) be a function. We say that the pair (B,A) is β⋆-admissible if for all g,q∈M(2)βg,q≥1⇒β⋆Bg,Aq≥1,andβ⋆Ag,Bq≥1,where β⋆(Ag,Bq)=infβa¯,b:a¯∈Ag,b∈Bq. When B=A, then we obtain the definition of α∗-admissible mapping given in [21].Definition 6.
Let(M,db) be a D.B.M space, B:M→P(M) be multivalued mapping, and α:M×M→[0,+∞). Let A¯⊆M, and we say that the B is semi-α∗-admissible on A¯, whenever α(g,q)≥1 implies that α∗(Bg,Bq)≥1 for all g,q∈A¯, where α∗(Bg,Bq)=infαa¯,b:a¯∈Bg,b∈Bq. If A¯=M, then we say that the B is α∗-admissible on M.Definition 7.
The functionHdb:P(M)×P(M)→R+, defined by (3)HdbA¯,B=max⁡supa¯∈A¯dba¯,B,supb∈BdbA¯,b,is called dislocated Hausdorff b-metric on P(M).Lemma 8.
Let(M,db) be a D.B.M space. Let (P(M),Hdb) is a dislocated Hausdorff b-metric space on P(M). Then for all A¯,B∈P(M) and for each a¯∈A¯ there exists ba¯∈B satisfying db(a¯,B)=db(a¯,ba¯); then Hdb(A¯,B)≥db(a¯,ba¯).


## 2. Main Result

Let(M,db) be a D.B.M space, and let g0∈M and B:M→P(M) be the multifunctions on M. Then there exist g1∈Bg0 such that db(g0,Bg0)=db(g0,g1). Let g2∈Bg1 be such that db(g1,Bg1)=db(g1,g2). Continuing this process, we construct a sequence gn of points in M such that gn+1∈Bgn,db(gn,Bgn)=db(gn,gn+1). We denote this iterative sequence by {MB(gn)}. We say that {MB(gn)} is a sequence in M generated by g0.Theorem 9.
Let(M,db) be a complete D.B.M space, r´>0,g0∈Bdb(g0,r´)¯, and B:M→P(M) be a semi-α∗-admissible multifunction on Bdb(g0,r´)¯; {MB(gn)} is a sequence in M generated by g0,α(g0,g1)≥1. Assume that, for some ψ∈Ψ and (4)Dbg,q=max⁡dbg,q,dbg,Bg.dbq,Bqa¯+dbg,q,dbg,Bg,dbq,Bqwhere a¯>0, the following hold:(5)α∗Bg,BqHdbBg,Bq≤ψDbg,q∀g,q∈Bdbg0,r´¯∩MBgn(6)∑i=0nBi+1ψidbg0,g1≤r´∀n∈N∪0.Then, {MB(gn)} is a sequence in Bdb(g0,r´)¯, α(gn,gn+1)≥1, and {MB(gn)}→g∗∈Bdb(g0,r´)¯. Also if α(gn,g∗)≥1 or α(g∗,gn)≥1, for all n∈N∪0 and the inequality (5) holds for all g,q∈Bdbg0,r´¯∩MBgn∪g∗, then B has a common fixed point g∗ in Bdb(g0,r´)¯.Proof.
Consider a sequence{MB(gn)} generated by g0. Then, we have gn∈Bgn-1, and db(gn-1,Bgn-1)=db(gn-1,gn), for all n∈N. By Lemma 8, we have db(gn,gn+1)≤Hdb(Bgn-1,Bgn) for all n∈N. If g0=g1, then g0 is a fixed point in Bdb(g0,r´)¯ of B. Let g0≠g1. From (6), we have (7)dbg0,g1≤∑i=0nψidbg0,g1≤r´.It follows that (8)g1∈Bdbg0,r´¯.If g1=g2, then g1 is a fixed point in Bdb(g0,r´)¯ of B. Let g1≠g2. Since α(g0,g1)≥1 and B is semi-α∗-admissible multifunction on Bdbg0,r´¯, then α∗(Bg0,Bg1)≥1. As α∗(Bg0,Bg1)≥1,g1∈Bg0 and g2∈Bg1, so α(g1,g2)≥1. Let g2,⋯,gj∈Bdb(g0,r´)¯ for some j∈N. As α∗(Bg1,Bg2)≥1, we have α(g2,g3)≥1, which further implies α∗(Bg2,Bg3)≥1. Continuing this process, we have α∗(Bgj-1,Bgj)≥1. Now, by using Lemma 8(9)dbgj,gj+1≤HdbBgj-1,Bgj≤α∗Bgj-1,BgjHdbBgj-1,Bgj≤ψDbgj-1,gj=ψmax⁡dbgj-1,gj,dbgj-1,Bgj-1.dbgj,Bgja¯+dbgj-1,gj,dbgj-1,Bgj-1,dbgj,Bgj=ψmax⁡dbgj-1,gj,dbgj-1,gj.dbgj,gj+1a¯+dbgj-1,gj,dbgj-1,gj,dbgj,gj+1=ψmax⁡dbgj-1,gj,dbgj,gj+1.If maxdb(gj-1,gj),db(gj,gj+1)=db(gj,gj+1), then db(gj,gj+1)≤ψ(db(gj,gj+1)). This is contradiction to the fact that ψ(u)<u for all u>0. Hence, we obtain maxdb(gj-1,gj),db(gj,gj+1)=db(gj-1,gj). Therefore, we have(10)dbgj,gj+1≤ψdbgj-1,gj≤⋯≤ψjdbg0,g1.Now, by using triangular inequality and by (10), we have (11)dbg0,gj+1≤tdbg0,g1+t2dbg1,g2+⋯+tj+1dbgj,gj+1≤tdbg0,g1+t2ψdbg0,g1+⋯+tj+1ψjdbg0,g1≤∑i=0jti+1ψidbg0,g1≤r´.Thus gj+1∈Bdb(g0,r´)¯. Hence, by induction, gn∈Bdb(g0,r´)¯. As α∗(Bgj-1,Bgj)≥1,gj∈Bgj,gj+1∈Bgj, then we have α(gj,gj+1)≥1. Also B is semi-α∗-admissible multifunction on Bdb(g0,r´)¯, and therefore α∗(Bgj,Bgj+1)≥1. This further implies that α(gj+1,gj+2)≥1. Continuing this process, we have α(gn,gn+1)≥1 for all n∈N. Now, inequality (10) can be written as(12)dbgn,gn+1≤ψndbg0,g1∀n∈N.Fix ∈>0 and let k1(∈)∈N, such that (13)
∑
k
≥
k
1
∈
t
k
ψ
k
d
b
g
0
,
g
1
<
∈
.Let n,m∈N with m>n>k1(∈). Now, we have (14)
d
b
g
n
,
g
m
≤
∑
k
=
n
m
-
1
d
b
g
k
,
g
k
+
1
≤
∑
k
=
n
m
-
1
t
k
ψ
k
d
b
g
0
,
g
1
,
by
12
d
b
g
n
,
g
m
≤
∑
k
≥
k
1
∈
t
k
ψ
k
d
b
g
0
,
g
1
<
∈
.Thus, {MB(gn)} is a Cauchy sequence in (Bdb(g0,r´)¯,db). As every closed ball in a complete D.B.M space is complete, there exist g∗∈Bdb(g0,r´)¯ such that {MB(gn)}→g∗, and(15)limn→∞dbgn,g∗=0.By assumption, we have α(gn,g∗)≥1 for all n∈N∪0. Thus, α∗(Bgn,Bg∗)≥1. Now, we have (16)dbg∗,Bg∗≤tdbg∗,gn+1+tdbgn+1,Bg∗≤tdbg∗,gn+1+tHdbBgn,Bg∗by  Lemma  8≤tdbg∗,gn+1+tα∗Bgn,Bg∗HdbBgn,Bg∗≤tdbg∗,gn+1+tψmax⁡dbgn,g∗,dbgn,Bgn.dbg∗,Bg∗a¯+dbgn,g∗,dbgn,Bgn,dbg∗,Bg∗≤tdbg∗,gn+1+tψmax⁡dbgn,g∗,dbgn,gn+1.dbg∗,Bg∗a¯+dbgn,g∗,dbgn,gn+1,dbg∗,Bg∗.Letting n→∞ and by using inequality (15), we obtain (1-t)db(g∗,Bg∗)≤0. So (1-t)≠0, and then db(g∗,Bg∗)=0. Hence g∗∈Bg∗. So B has a fixed point in Bdbg0,r´.¯
LetM be a nonempty set. Then (M,⪯,db) is called a preordered D.B.M space if db is called D.B.M on M. Let (M,⪯,db) be a preordered D.B.M space and H,K⊆M. We say that H⪯r´K whenever for each a¯∈H there exist b∈K such that a¯⪯b. Also, we say that H⪯r´K whenever, for each a¯∈H and b∈K, we have a¯⪯b.Corollary 10.
Let(M,⪯,db) be a preordered complete D.B.M space, r´>0,g0∈Bdb(g0,r´)¯, and B:M→P(M) be a multifunction on Bdb(g0,r´)¯; {MB(gn)} is a sequence generated by g0, with g0⪯g1. Assume that, for some ψ∈Ψ and (17)Dbg,q=max⁡dbg,q,dbg,Bg.dbq,Bqa¯+dbg,q,dbg,Bg,dbq,Bqwhere a¯>0, the following hold:(18)HdbBg,Bq≤ψDbg,q∀g,q∈Bdbg0,r´¯∩MBgnwithg⪯q(19)and∑i=0nti+1ψidbg0,g1≤r´∀n∈N∪0.If g,q∈Bdb(g0,r´)¯, such that g⪯q implies Bg⪯r´Bq. Then {MB(gn)} is a sequence in Bdb(g0,r´)¯, gn⪯gn+1, and {MB(gn)}→g∗∈Bdb(g0,r´)¯. Also if g∗⪯gn or gn⪯g∗, for all n∈N∪0, and inequality (18) holds for all g,q∈Bdbg0,r´¯∩MBgn∪g∗. Then g∗ is a fixed point of B in Bdb(g0,r´)¯.Corollary 11.
Let(M,⪯,db) be a preordered complete D.B.M space, r´>0,g0∈Bdb(g0,r´)¯, and B:M→P(M) be a multifunction on Bdb(g0,r´)¯; {MB(gn)} is a sequence generated by g0, with g0⪯g1. Assume that, for some k∈0,1 and (20)Dbg,q=max⁡dbg,q,dbg,Bg.dbq,Bqa¯+dbg,q,dbg,Bg,dbq,Bqwhere a¯>0, the following hold:(21)HdbBg,Bq≤kDbg,q∀g,q∈Bdbg0,r´¯∩MBgnwithg⪯q(22)and∑i=0nti+1kidbg0,g1≤r´∀n∈N∪0.If g,q∈Bdb(g0,r´)¯, such that g⪯q implies Bg⪯r´Bq. Then {MB(gn)} is a sequence in Bdb(g0,r´)¯, gn⪯gn+1, and {MB(gn)}→g∗∈Bdb(g0,r´)¯. Also if g∗⪯gn or gn⪯g∗, for all n∈N∪0, and inequality (21) holds for all g,q∈Bdbg0,r´¯∩MBgn∪g∗. Then g∗ is a fixed point of B in Bdb(g0,r´)¯.Corollary 12.
Let(M,⪯,dl) be a preordered D.M space, r´>0,g0∈Bdl(g0,r´)¯, and B:M→P(M) be a multifunction on Bdb(g0,r´)¯; {MB(gn)} is a sequence generated by g0, with g0⪯g1. Assume that, for some ψ∈Ψ and (23)Dlg,q=maxdlg,q,dlg,Bg.dlq,Bqa¯+dlg,q,dlg,Bg,dlq,Bqwhere a¯>0, the following hold:(24)HdlBg,Bq≤ψDlg,q∀g,q∈Bdlg0,r´¯∩MBgnwithg⪯q(25)and∑i=0nψidlg0,g1≤r´∀n∈N∪0.If g,q∈Bdl(g0,r´)¯, such that g⪯q implies Bg⪯r´Bq. Then {MB(gn)} is a sequence in Bdl(g0,r´)¯, gn⪯gn+1, and {MB(gn)}→g∗∈Bdl(g0,r´)¯. Also if g∗⪯gn or gn⪯g∗, for all n∈N∪0, and inequality (24) holds for all g,q∈Bdlg0,r´¯∩MBgn∪g∗. Then g∗ is a fixed point of B in Bdl(g0,r´)¯.Corollary 13.
Let(M,⪯,dl) be a preordered complete D.M space, r´>0,g0∈Bdl(g0,r´)¯, and B:M→P(M) be a multifunction on Bdb(g0,r´)¯; {MB(gn)} is a sequence generated by g0, with g0⪯g1. Assume that, for some k∈[0,1) and (26)Dlg,q=maxdlg,q,dlg,Bg.dlq,Bqa¯+dlg,q,dlg,Bg,dlq,Bqwhere a¯>0, the following hold:(27)HdlBg,Bq≤kDlg,q∀g,q∈Bdlg0,r´¯∩MBgnwithg⪯q(28)and∑i=0nkidlg0,g1≤r´∀n∈N∪0.If g,q∈Bdl(g0,r´)¯, such that g⪯q implies Bg⪯r´Bq. Then {MB(gn)} is a sequence in Bdl(g0,r´)¯, gn⪯gn+1, and {MB(gn)}→g∗∈Bdl(g0,r´)¯. Also if g∗⪯gn or gn⪯g∗, for all n∈N∪0, and inequality (27) holds for all g,q∈Bdlg0,r´¯∩MBgn∪g∗. Then g∗ is a fixed point of B in Bdlg0,r´¯.Example 14.
LetM=Q+∪0 and let db:M×M→M be the D.B.M space on M defined by (29)dbg,q=g+q2∀g,q∈Mwith parameter t>1. Define the multivalued mappings, B:M×M→P(M) by (30)Bg=g3,23gifg∈0,9∩Mg,g+1ifg∈9,∞∩M.Considering g0=1,r´=100, and a¯=1,b=2, then Bdb(g0,r´)¯=[0,9]∩M. Now db(g0,Bg0)=db(1,B1)=db(1,1/3)=16/9. So we obtain a sequence {MB(gn)}={1,1/3,1/9,1/27,….} in M generated by g0. Let t=1.2,ψ(t)=4t/5, and then tψ(t)<t. Define (31)αg,q=1ifg,q∈0,9∩M32otherwise.Now, (32)α∗B10,B11HdbB10,B11=32484>ψDbg,q=45484.So the contractive condition does not hold on M. Now, for all g,q∈Bdbg0,r´¯∩MBgn, we have (33)α∗Bg,BqHdbBg,Bq=1max⁡supa¯∈Bgdba¯,Bq,supb∈BqdbBg,b=max⁡supa¯∈Bgdba¯,q3,2q3,supb∈Bqdbg3,2g3,b=max⁡supa¯∈Bgdb2g3,q3,2q3,supb∈Bqdbg3,2g3,2q3=max⁡db2g3,q3,dbg3,2q3=max⁡2g+q29,g+2q29≤ψmax⁡g+q2,256g2q2811+g+q2,16g29,16q29=ψdbg,q.So the contractive condition holds on Bdb(c˙0,r´)¯∩MBgn. As t=1.2>1, then (34)∑i=0nti+1ψidbg0,g1=169×65∑i=0n2425i<100=r´.Hence, all the conditions of Theorem 9 are satisfied. Now, we have that {MB(gn)} is a sequence in Bdb(g0,r´)¯,α(gn,gn+1)≥1 and {MB(gn)}→0∈Bdb(g0,r´)¯. Moreover, 0 is a fixed point of B.


## 3. Fixed Point Results For Graphic Contractions

In this section, we present an application of Theorem9 in graph theory. Jachymski [27] proved the result concerning contraction mappings on metric space with a graph. Hussain et al. [28] discussed the fixed points theorem for graphic contraction and gave an application. A graph G˙ is affix if there is a way among any two vertices (see for details [29, 30]).Definition 15.
LetM be a nonempty set and G˙=(V(G˙),Q(G)) be a graph such that V(G˙)=M, and B:M→P(M) is said to be multigraph preserving if (g,q)∈Q(G), and then (w,p)∈Q(G) for all w∈Bg and p∈Bq.Theorem 16.
Let(M,db) be a complete D.B.M space endowed with a graph G˙. Suppose there exists a function α:M×M→[0,∞). Let, r´>0,g0∈Bdb(g0,r´)¯, B:M→P(M), and let for a sequence {MB(gn)} in M generated by g0, with (g0,g1)∈Q(G). Suppose that the following are satisfy:
(i)B is a graph preserving for all g,q∈Bdbg0,r´¯∩MBgn;
(ii) there existsψ∈Ψ and (35)Dbg,q=max⁡dbg,q,dlg,Bg.dlq,Bqa¯+dlg,q,dbg,Bg,dbq,Bqwhere a¯>0 such that(36)HdbBg,Bq≤ψDbg,q,for all g,q∈Bdbg0,r´¯∩MBgn and (g,q)∈Q(G);
(iii)∑i=0nti+1ψidbg0,Bg0≤r´ for all n∈N∪0 and t>1.
Then,{MB(gn)} is a sequence in Bdb(g0,r´)¯,(gn,gn+1)∈Q(G) and {MB(gn)}→g∗. Also, if and inequality (36) holds for g∗ and (gn,g∗)∈Q(G) or (g∗,gn)∈Q(G) for all n∈N∪0, then B has a fixed point g∗ in Bdb(g0,r´)¯.Proof.
Define,α:M×M→[0,∞) by (37)αg,q=1,ifg∈Bdbg0,r´¯,g,q∈QG0,otherwise.As {MB(gn)} is a sequence in M generated by g0 with (g0,g1)∈Q(G), we have α(g0,g1)≥1. Let α(g,q)≥1, and then (g,q)∈Q(G). From (i), we have (w,p)∈Q(G) for all w∈Bg and p∈Bq. This implies that α(w,p)=1 for all w∈Bg and p∈Bq. This implies that infαw,p:w∈Bg,p∈Bq=1. So, B:M→P(M) is a semi-α∗-admissible multifunction on Bdb(g0,r´)¯. Moreover, inequality (36) can be written as (38)α∗Bg,BqHdbBg,Bq≤ψDbg,q,for all elements g,q in Bdb(g0,r´)¯∩MBgn with either α(g,q)≥1 or α(q,g)≥1. Also, (iii) holds. Then, by Theorem 9, we have that {MB(gn)} is a sequence in Bdb(g0,r´)¯ and {MB(gn)}→g∗∈Bdb(g0,r´)¯. Now, gn,g∗∈Bdb(g0,r´)¯ and either (gn,g∗)∈Q(G) or (g∗,gn)∈Q(G) for all n∈N∪0, and inequality (36) holds for all g,q∈Bdbg0,r´¯∩MBgn∪g∗. Then we have α(gn,g∗)≥1 or α(g∗,gn)≥1 for all n∈N∪0 and inequality (5) holds for all g,q∈Bdbg0,r´¯∩MBgn∪g∗. So, all the conditions of Theorem 9 are satisfied. Hence, by Theorem 9, B has a common fixed point g∗ in Bdb(g0,r´)¯ and db(g∗,g∗)=0.


## 4. Fixed Point Results for Single-Valued Mapping

In this section we discussed some fixed point results for self-mapping in completeD.B.M space. Let (M,db) be a D.B.M space, g0∈M, and B:M→M be a mapping. Let g1=Bg0, g2=Bg1. Continuing this process, we construct a sequence gn of points in M such that gn+1=Bgn. We denote this iterative sequence by gn. We say that {gn} is a sequence in M generated by g0.Theorem 17.
Let(M,db) be a complete D.B.M space, r´>0,g0∈Bdb(g0,r´)¯, and B:M→M be a semi-α-admissible function on Bdb(g0,r´)¯; {gn} is a sequence in M generated by g0,α(g0,g1)≥1. Assume that, for some ψ∈Ψ and (39)Dbg,q=maxdbg,q,dbg,Bg.dbq,Bqa¯+dbg,q,dbg,Bg,dbq,Bqwhere a¯>0, the following hold:(40)αBg,BqHdbBg,Bq≤ψDbg,q∀g,q∈Bdbg0,r´¯∩gn(41)∑i=0nBi+1ψidbg0,g1≤r´∀n∈N∪0.Then, {gn} is a sequence in Bdb(g0,r´)¯, α(gn,gn+1)≥1, and {gn}→g∗∈Bdb(g0,r´)¯. Also if α(gn,g∗)≥1 or α(g∗,gn)≥1, for all n∈N∪{0}, and inequality (40) holds for all g,q∈Bdbg0,r´¯∩gn∪g∗. Then B has a common fixed point g∗ in Bdb(g0,r´)¯.Proof.
The proof of the above theorem is similar to Theorem17.Corollary 18.
Let(M,⪯,db) be a preordered complete D.B.M space, r´>0,g0∈Bdb(g0,r´)¯, and B:M→M be a self-mapping on Bdb(g0,r´)¯; {gn} is a sequence generated by g0, with g0⪯g1. Assume that, for some k∈[0,1) and (42)Dbg,q=max⁡dbg,q,dbg,Bg.dbq,Bqa¯+dbg,q,dbg,Bg,dbq,Bqwhere a¯>0, the following hold:(43)HdbBg,Bq≤kDbg,q∀g,q∈Bdbg0,r´¯∩gnwithg⪯q(44)and∑i=0nti+1kidbg0,g1≤r´∀n∈N∪0.If g,q∈Bdb(g0,r´)¯, such that g⪯q implies Bg⪯r´Bq. Then {gn} is a sequence in Bdb(g0,r´)¯, gn⪯gn+1, and {gn}→g∗∈Bdb(g0,r´)¯. Also if g∗⪯gn or gn⪯g∗, for all n∈N∪0, and inequality (43) holds for all g,q∈Bdbg0,r´¯∩gn∪g∗. Then g∗ is a fixed point of B in Bdb(g0,r´)¯.Corollary 19.
Let(M,⪯,dl) be a preordered complete D.M space, r´>0,g0∈Bdl(g0,r´)¯, and B:M→M be a self-mapping on Bdl(g0,r´)¯; {gn} is a sequence generated by g0, with g0⪯g1. Assume that, for some ψ∈Ψ and (45)Dlg,q=max⁡dlg,q,dlg,Bg.dlq,Bqa¯+dlg,q,dlg,Bg,dlq,Bqwhere a¯>0, the following hold:(46)HdlBg,Bq≤ψDlg,q∀g,q∈Bdlg0,r´¯∩gnwithg⪯q(47)and∑i=0nψidlg0,g1≤r´∀n∈N∪0.If g,q∈Bdl(g0,r´)¯, such that g⪯q implies Bg⪯r´Bq. Then {gn} is a sequence in Bdl(g0,r´)¯, gn⪯gn+1, and {gn}→g∗∈Bdl(g0,r´)¯. Also if g∗⪯gn or gn⪯g∗, for all n∈N∪0, and inequality (46) holds for all g,q∈Bdlg0,r´¯∩gn∪g∗. Then g∗ is a fixed point of B in Bdl(g0,r´)¯.
Recall that if(M,⪯) is a preordered set and A:M→M is such that for g,q∈M, with g⪯q implying Ag⪯Aq, then the mapping A is said to be nondecreasing.Corollary 20.
Let(M,⪯,dl) be a preordered complete D.M space, r´>0,g0 be an arbitrary point in Bdl(g0,r´)¯, B:M→M be a self-mapping on Bdl(g0,r´)¯, and {gn} be a Picard sequence in M with initial guess g0, with g0⪯g1. For some k∈[0,1) and (48)Dlg,q=max⁡dlg,q,dlg,Bg.dlq,Bqa¯+dlg,q,dlg,Bg,dlq,Bqwhere a¯>0, the following hold:(49)dlBg,Bq≤kDlg,q∀g,q∈Bdlg0,r´¯∩gnwithg⪯q(50)and∑i=0jkidlg0,g1≤r´∀j∈N∪0.Then, {gn} is a sequence in Bdl(g0,r´)¯, such that gn⪯gn+1 and {gn}→g∗∈Bdl(g0,r´)¯. Also if g∗⪯gn or gn⪯g∗, for all n∈N∪0, and inequality (49) holds for all g,q∈Bdlg0,r´¯∩gn∪g∗. Then g∗ is a fixed point of B in Bdl(g0,r´)¯.



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*Source: 5816364-2018-12-19.xml*