# Common Fixed Point Results for a Pair of Multivalued Mappings in Complex-Valuedb-Metric Spaces

Authors: Olawale Kazeem Oyewole; Kazeem Olalekan Aremu; Lateef Olakunle Jolaoso
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

Several fixed point results for the existence of common fixed points of multivalued contractive mappings have been established in complex-valued metric space. In this paper, we study the existence of common fixed points for a pair of multivalued contractive mappings satisfying some rational inequalities in the framework of complex-valuedb-metric spaces. The contractive condition used in this paper generalizes many contractive conditions used by other authors in the literature. Employing our results, we check the existence solution to the Riemann-Liouville equation.


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## Body

## 1. Introduction

Fixed point theory is a well-researched area of mathematics; in particular, results concerning fixed points of contractive type mappings are found useful for determining the existence and uniqueness of solutions of various mathematical models. In this field, Banach [1] introduced the notion of contraction mapping in a complete metric space and gave a fixed point result for finding the fixed point of the contraction mapping. Later in 1969, Kannan [2] gave another contractive type mapping that demonstrated the fixed point theorem. However, in the Kannan contraction result, the continuity property required for the result of Banach was shown to be not necessary. Other authors have also studied several contraction mappings with differing properties (see, for instance, Chaterjea [3]). Since then, the theory of fixed points has been developed regarding results on finding fixed points of self and nonself mappings which are single-valued in a metric space.Moreover, the study of fixed points for multivalued type contractive mappings was pioneered by Nadler [4] and further studied by Markin [5]. Since then, many researchers have generalized and extended various fixed point results from single-valued contractive mappings to multivalued contractive type mappings. For more literature concerning such extensions and generalizations, see, for instance, [6–12] and other references therein.On the other hand, the axiomatic development of metric spaces was started by M. Fréchet, a French mathematician in the year 1906. The importance of metric spaces in the natural growth of functional analysis is huge. Several authors have drawn inspirations from the impact of this natural idea to mathematics and functional analysis in particular. Therefore, there have been several generalizations of this notion in the forms of rectangular metric spaces, semimetric spaces, quasimetric spaces, quasisemimetric spaces,D-metric spaces, cone metric spaces, and more recently the graphical rectangular b-metric spaces. We refer the reader to the following references for surveys on these generalizations [1, 13–18].One of these generalizations in the last decade is that of Azam et al. [19, 20]. They introduced the notion of complex-valued metric spaces, and some fixed point theorems for mappings with some rational inequalities were established. The central and core idea is to define rational expressions which are not well posed in the cone metric spaces, and thus, such results of analysis cannot be extended to cone metric spaces but to complex-valued metric spaces. Complex-valued metric spaces find interesting applications in many branches of mathematics such as algebraic geometry and number theory as well as in field of studies such as physics, thermodynamics, and electrical engineering.Furthermore, the idea ofb-metric was introduced in 1989 by Bakhtin [21]. Based on this presentation, Rao et al. [22] introduced the concept of fixed point theorems on complex-valued b-metric spaces which is a natural generalization of the complex-valued metric spaces. The relationship between the complex-valued b-metric and cone metric space is well known. Inspired by [22], many authors have proven the existence of fixed points of different mappings satisfying rational inequalities in the framework of complex-valued b-metric spaces (see [19, 23] for more details).In 2016, Singh et al. [24] introduced a contractive type mapping satisfying some rational inequalities. They obtained the existence of common fixed point for a pair of single-valued mappings satisfying more general contraction conditions in the framework of complex-valued metric spaces. Since then, fixed point theory has been the center of extensive research for many authors (see, e.g., [25, 26]).Our motivation in this work is in twofolds: we extend the results of [19, 24, 27] from complex-valued metric spaces to complex-valued b-metric spaces; we also generalize the contraction mappings used therein by introducing a multivalued contraction mapping satisfying a general condition in complex-valued b-metric spaces. Our result also improves and strengthens the results of [20, 22, 28, 29] and many other related results in the literature.


## 2. Preliminaries

In this section, we give some basic definitions and results which will be useful in establishing our main result. Letℂ be the set of complex numbers and z1,z2∈ℂ. Also, we define partial order ≺ and ⪯ on ℂ as follows:
(i)
z1≺z2 if and only if Rez1<Rez2 and Imz1<Imz2(ii)
z1⪯z2 if and only if Rez1≤Rez2 and Imz1≤Imz2Definition 1.
Let
X
be a nonempty set and
τ
≥
1
be a real number. A function
d
c
:
X
×
X
→
ℂ
is called complex-valued
b
-metric, if for all
x
,
y
,
z
∈
X
,
the following conditions hold:
(i)
0
⪯
d
c
x
,
y and dcx,y=0, if and only if x=y
(ii)
d
c
x
,
y
=
d
c
y
,
x
(iii)
d
x
,
y
⪯
τ
d
c
x
,
z
+
d
c
z
,
y.
Then, a setX satisfying such metric dc written in pair as X,dc is called a complex-valued b-metric space.Example 2 (see [23]).
LetX=0,1, define a mapping dc:X×X→ℂ by dc=x−y2+ix−y2 for all x,y∈X. Then, X,dc is a complex-valued b-metric space with τ=2.Definition 3 (see [19]).
Let
X
,
d
c
be a complex-valued
b
-metric space, and then, a point
x
∈
X
is
(i)
an interior point of a setA⊆X, if there exists 0≺r∈ℂ such that Nx,r=y∈X:dcx,y≺r⊆A
(ii)
the limit of a setA⊆X, if for every 0≺r∈ℂ,Nx,r∩Dx≠∅
(iii)
A
⊆
X which is called an open set if every element A is an interior point ADefinition 4 (see [22]).
Let
x
n
be a sequence in a complex-valued
b
-metric space
X
,
d
c
and
x
∈
X
,
then
(i)
x is the limit point of xn if for every c∈ℂ with 0≺c, there exits n0∈ℚ such that dcxn,x≺c for all n>n0, we write limn→∞xn=x
(ii)
if for everyc∈ℂ with 0≺c, there exists n0∈ℚ such that dcxn,xn+m≺c for all n>n0 and n,m∈ℚ, then xn is a cauchy sequence in X,dc
(iii)
X
,
d
c is complete if every cauchy sequence is convergent in X,dcLemma 5 (see [28]).
Let
X
,
d
c
be a complex-valued
b
-metric space and
x
n
be a sequence in
X
,
d
c
.
Then,
x
n
converges to
x
∈
X
if and only if
∣
d
c
x
n
,
x
∣
→
0
as
n
→
∞
.Lemma 6 (see [28]).
Let
X
,
d
c
be a complex-valued
b
-metric space and
x
n
be a sequence in
X
,
d
c
.
Then,
x
n
is a Cauchy sequence if and only if
∣
d
c
x
n
,
x
n
+
m
∣
→
0
as
n
→
∞
.We denote byCBX the family of closed and bounded subsets of the set X.Definition 7 (see [19]).
Let
X
,
d
c
be a complex-valued
b
-metric space and
x
n
be a sequence in
X
,
d
c
.
Denote
s
u
=
z
∈
ℂ
:
u
⪯
z
and
(1)
s
x
,
B
=
⋃
b
∈
B
s
d
a
,
b
=
⋃
b
∈
B
z
∈
ℂ
:
d
a
,
b
⪯
z
,for a∈X and B∈CBX. For A,B∈CBX, we denote
(2)sA,B=⋂a∈Asa,B∩⋂b∈Bsb,A.Remark 8.
Let
X
,
d
c
be a complex-valued
b
-metric space with
τ
=
1
.
If
ℂ
=
ℝ
,
then
X
,
d
c
is a metric space. Moreover, for
A
,
B
∈
C
B
X
,
H
A
,
B
=
inf
s
A
,
B
is the Hausdorff metric induced by
d
c
.Definition 9 (see [22]).
Let
X
,
d
c
be a complex-valued
b
-metric space.
(i)
LetT:X→CBX be a multivalued mapping. For x∈X and A∈CBX, define
(3)
W
x
A
=
d
c
x
,
a
:
a
∈
A
.
Thus, forx,y∈X,
(4)WxTy=dcx,u:u∈Ty.(ii)
A mappingF:X→2ℂ is said to be bounded from below if for each x∈X there exists zx∈ℂ such that zx⪯v for all v∈Fx(iii)
For a multivalued mappingB:X→CBX, we say that B has a lower bound property on X,dc, if for any x∈X the mapping Fx:X→2ℂ defined by Fxv=WxFv is bounded below. This implies that for x,v∈X, there exists an element lxBv∈ℂ such that lxBv⪯a for all a∈WxBv, where lxBv is the lower bound of B associated with x,v(iv)
The multivalued mappingB:X→CBX is said to have the greatest lower bound property (g.l.b. property) on X,dc if a greatest lower bound WxBv exists in ℂ for every x,v∈X. We denote by dcx,Bv the g.l.b. of WxBv. That is,(5)dx,Bv=infdx,u:u∈BvDefinition 10 (see [23]).
Let
X
,
d
c
be a complex-valued
b
-metric space and
S
,
T
:
X
→
C
B
X
be multivalued mappings.
(i)
A pointx∈X is called a fixed point T if x∈Tx
(ii)
A pointx∈X is called a common fixed point of S and T if x∈Sx and x∈Tx


## 3. Main Result

In this section, we state and prove our main findings in the sequel.Theorem 11.
Let
X
,
d
c
be a complete complex-valued
b
-metric space and let
S
,
T
:
X
→
C
B
X
be multivalued mappings with g.l.b. property. Let
λ
,
μ
,
γ
,
δ
:
X
×
X
×
X
→
0
,
1
be mappings such that
∀
x
,
y
∈
X
,
(6)
λ
x
2
n
,
y
,
a
≤
λ
x
0
,
y
,
a
,
λ
x
,
y
2
n
,
a
≤
λ
x
,
y
0
,
a
,
(7)
μ
x
2
n
,
y
,
a
≤
μ
x
0
,
y
,
a
,
μ
x
,
y
2
n
,
a
≤
μ
x
,
y
0
,
a
,
(8)
γ
x
2
n
,
y
,
a
≤
γ
x
0
,
y
,
a
,
γ
x
,
y
2
n
,
a
≤
γ
x
,
y
0
,
a
,
(9)
δ
x
2
n
,
y
,
a
≤
δ
x
0
,
y
,
a
,
δ
x
,
y
2
n
,
a
≤
δ
x
,
y
0
,
a
,
(10)
λ
x
,
y
,
a
d
c
x
,
y
+
μ
x
,
y
,
a
d
c
x
,
S
x
d
c
y
,
T
y
1
+
d
c
x
,
y
+
γ
x
,
y
,
a
d
c
y
,
S
x
d
c
x
,
T
y
1
+
d
c
x
,
y
+
δ
x
,
y
,
a
d
c
x
,
S
x
d
c
x
,
T
y
+
d
c
y
,
T
y
d
c
y
,
S
x
1
+
d
c
x
,
T
y
+
d
c
y
,
S
x
∈
s
S
x
,
T
y
,
(11)
τ
λ
x
,
y
,
a
+
μ
x
,
y
,
a
+
γ
x
,
y
,
a
+
δ
x
,
y
,
a
<
1
.Then,S and T have a unique common fixed point.Proof.
Letx0 be an arbitrary point in X, then Tx0≠∅. Pick x1∈Tx0, set x=x0 and y=x1 in (6). Then, we get
(12)λx0,x1,adcx0,x1+μx0,x1,adcx0,Sx0dcx1,Tx11+dcx0,x1+γx0,x1,adcx1,Sx0dcx0,Tx11+dcx0,x1+δx0,x1,adcx0,Sx0dcx0,Tx1+dcx1,Tx1dcx1,Sx01+dcx0,Tx1+dcx1,Sx0∈sSx0,Tx1.
This implies(13)λx0,x1,adcx0,x1+μx0,x1,adcx0,Sx0dcx1,Tx11+dcx0,x1+γx0,x1,adcx1,Sx0dcx0,Tx11+dcx0,x1+δx0,x1,adcx0,Sx0dcx0,Tx1+dcx1,Tx1dcx1,Sx01+dcx0,Tx1+dcx1,Sx0∈⋂p∈Sx0sp,Tx1,λx0,x1,adcx0,x1+μx0,x1,adcx0,Sx0dcx1,Tx11+dcx0,x1+γx0,x1,adcx1,Sx0dcx0,Tx11+dcx0,x1δx0,x1,adcx0,Sx0dcx0,Tx1+dcx1,Tx1dcx1,Sx01+dcx0,Tx1+dcx1,Sx0∈sp,Tx1,∀p∈Sx0.
Sincex1∈Sx0, we get
(14)λx0,x1,adcx0,x1+μx0,x1,adcx0,Sx0dcx1,Tx11+dcx0,x1+γx0,x1,adcx1,Sx0dcx0,Tx11+dcx0,x1+δx0,x1,adcx0,Sx0dcx0,Tx1+dcx1,Tx1dcx1,Sx01+dcx0,Tx1+dcx1,Sx0∈sx1,Tx1,λx0,x1,adcx0,x1+μx0,x1,adcx0,Sx0dcx1,Tx11+dcx0,x1+γx0,x1,adcx1,Sx0dcx0,Tx11+dcx0,x1+δx0,x1,adcx0,Sx0dcx0,Tx1+dcx1,Tx1dcx1,Sx01+dcx0,Tx1+dcx1,Sx0∈⋃q∈Tx1sdcx1,q.
Thus, there exists somex2∈Tx1 such that
(15)λx0,x1,adcx0,x1+μx0,x1,adcx0,Sx0dcx1,Tx11+dcx0,x1+γx0,x1,adcx1,Sx0dcx0,Tx11+dcx0,x1+δx0,x1,adcx0,Sx0dcx0,Tx1+dcx1,Tx1dcx1,Sx01+dcx0,Tx1+dcx1,Sx0∈⋃q∈Tx1sdcx1,x2.
By using the g.l.b. property ofS and T, we obtain that
(16)dcx1,x2⪯λx0,x1,adcx0,x1+μx0,x1,adcx0,x1dcx1,x21+dcx0,x1+γx0,x1,adcx1,x1dcx0,x21+dcx0,x1+δx0,x1,adcx0,x1dcx0,x2+dcx1,x2dcx1,x11+dcx0,x2+dcx1,x1,which implies that
(17)dcx1,x2≤λx0,x1,adcx0,x1+μx0,x1,adcx0,x1dcx1,x21+dcx0,x1+δx0,x1,adcx0,x1dcx0,x21+dcx0,x2,∣dcx1,x2∣=λx0,x1,a∣dcx0,x1∣+μx0,x1,a∣dcx0,x1∣∣1+dcx0,x1∣∣dcx1,x2∣+δx0,x1,a∣dcx0,x2∣∣1+dcx0,x2∣dcx0,x1.
That is,(18)dcx1,x2≤λx0,x1,adcx0,x1+μx0,x1,adcx1,x2+δx0,x1,adcx0,x1,from which we get
(19)∣dcx1,x2∣≤λx0,x1,a+δx0,x1,a1−μx0,x1,adcx0,x1.
Letρ=λx0,x1,a+δx0,x1,a/1−μx0,x1,a. Clearly, ρ<1; then, we can inductively define a sequence xn∈X such that ∣dcxn,xn+m∣≤ρn∣dcx0,x1∣, for n=0,1,⋯,x2n+1∈Sx2n, and x2n+2∈Tx2n+1. Now, for m>n and the fact that X is complex-valued b-metric space, we get
(20)∣dcxn,xm∣≤τ∣dcxn,xn+1∣+τ2∣dcxn+1,xn+2∣+⋯+τm∣dcxm−1,xm∣≤τρndcx0,x1+τ2ρn+1dcx0,x1+⋯+τmρm−1dcx0,x1≤τρn1+τρ+τρ2+⋯+τρm−1dcx0,x1.
That is,(21)∣dcxn,xm∣≤τρn1−τρdcx0,x1.
Lettingm,n→∞ in (21), we get ∣dcxn,xm∣→0. This implies by Lemma 6 that xn is a cauchy sequence in X. Since X is complete, there exists u∈X such that xn→u as n→∞.
Next, we show thatu∈Su and u∈Tu. Again, from (6), we have
(22)λx2n,u,adcx2n,u+μx2n,u,adcx2n,Sx2ndcu,Tu1+dcx2n,u+γx2n,u,adcu,Sx2ndcx2n,Tu1+dcx2n,u+δx2n,u,adcx2n,Sx2ndcx2n,Tu+dcu,Tudcu,Sx2n1+dcx2n,Tu+dcu,Sx2n∈sSx2n,Tu.
This implies(23)λx2n,u,adcx2n,u+μx2n,u,adcx2n,Sx2ndcu,Tu1+dcx2n,u+γx2n,u,adcu,Sx2ndcx2n,Tu1+dcx2n,u+δx2n,u,adcx2n,Sx2ndcx2n,Tu+dcu,Tudcu,Sx2n1+dcx2n,Tu+dcu,Sx2n∈⋂p∈Sxsp,Tu,λx2n,u,adcx2n,u+μx2n,u,adcx2n,Sx2ndcu,Tu1+dcx2n,u+γx2n,u,adcu,Sx2ndcx2n,Tu1+dcx2n,u+δx2n,u,adcx2n,Sx2ndcx2n,Tu+dcu,Tudcu,Sx2n1+dcx2n,Tu+dcu,Sx2n∈sp,Tu,∀p∈Sx2n.
Now, sincex2n+1∈Sx2n, we have
(24)λx2n,u,adcx2n,u+μx2n,u,adcx2n,Sx2ndcu,Tu1+dcx2n,u+γx2n,u,adcu,Sx2ndcx2n,Tu1+dcx2n,u+δx2n,u,adcx2n,Sx2ndcx2n,Tu+dcu,Tudcu,Sx2n1+dcx2n,Tu+dcu,Sx2n∈sx2n+1,Tu.
From which we obtain that(25)λx2n,u,adcx2n,u+μx2n,u,adcx2n,Sx2ndcu,Tu1+dcx2n,u+γx2n,u,adcu,Sx2ndcx2n,Tu1+dcx2n,u+δx2n,u,adcx2n,Sx2ndcx2n,Tu+dcu,Tudcu,Sx2n1+dcx2n,Tu+dcu,Sx2n∈sx2n+1,Tu=⋃q∈Tusdx2n+1,q.
This implies that there exists someun∈Tu such that
(26)λx2n,u,adcx2n,u+μx2n,u,adcx2n,Sx2ndcu,Tu1+dcx2n,u+γx2n,u,adcu,Sx2ndcx2n,Tu1+dcx2n,u+δx2n,u,adcx2n,Sx2ndcx2n,Tu+dcu,Tudcu,Sx2n1+dcx2n,Tu+dcu,Sx2nsdx2n+1,un.
That is,(27)dcx2n+1,un⪯λx2n,u,adcx2n,u+μx2n,u,adcx2n,Sx2ndcu,Tu1+dcx2n,u+γx2n,u,adcu,Sx2ndcx2n,Tu1+dcx2n,u+δx2n,u,adcx2n,Sx2ndcx2n,Tu+dcu,Tudcu,Sx2n1+dcx2n,Tu+dcu,Sx2n.
By using the g.l.b. property ofS and T, we get
(28)dcx2n+1,un⪯λx2n,u,adcx2n,u+μx2n,u,adcx2n,x2n+1dcu,un1+dcx2n,un+γx2n,u,adcu,x2n+1dcx2n,un1+dcx2n,u+δx2n,u,adcx2n,x2n+1dcx2n,un+dcu,undcu,x2n+11+dcx2n,un+dcu,x2n+1.
Fromdcu,un⪯τdcu,x2n+1+dcx2n+1,un, we have
(29)dcu,un⪯τdcu,x2n+1+τλx2n,u,adcx2n,u+τμx2n,u,adcx2n,x2n+1dcu,un1+dcx2n,u+τγx2n,u,adcu,x2n+1dcx2n,un1+dcx2n,u+τδx2n,u,adcx2n,x2n+1dcx2n,un+dcu,undcu,x2n+11+dcx2n,un+dcu,x2n+1⪯τdcu,x2n+1+τλx0,u,adcx2n,u+τμx0,u,adcx2n,x2n+1dcu,un1+dcx2n,u+τγx0,u,adcu,x2n+1dcx2n,un1+dcx2n,u+τδx0,u,adcx2n,x2n+1dcx2n,un+dcu,undcu,x2n+11+dcx2n,un+dcu,x2n+1.
Therefore, we get(30)dcu,un≤τdcu,x2n+1+τλx0,u,adcx2n,u+τμx0,u,adcx2n,x2n+1dcu,un1+dcx2n,u+τγx0,u,adcu,x2n+1dcx2n,un1+dcx2n,u+τδx0,u,adcx2n,x2n+1dcx2n,un+dcu,undcu,x2n+11+dcx2n,un+dcu,x2n+1.By lettingn→∞ in the above inequality, we get ∣dcu,un∣→0 as n→∞. By Lemma 5, we get that limn→∞un=u. Since Tu is closed, we have u∈Tu. Proceeding in a similar fashion, it follows easily that u∈Su. Therefore, S and T have a common fixed point.Corollary 12.
Let
X
,
d
c
be a complete complex-valued
b
-metric space and let
S
,
T
:
X
→
C
B
X
be multivalued mappings with g.l.b. property. Let
λ
,
μ
,
γ
,
δ
:
X
×
X
→
0
,
1
be mappings such that
∀
x
,
y
∈
X:
(31)λx2n,y≤λx0,y,λx,y2n≤λx,y0,μx2n,y≤μx0,y,μx,y2n≤μx,y0,γx2n,y≤γx0,y,γx,y2n≤γx,y0,δx2n,y≤δx0,y,δx,y2n≤δx,y0,λx,ydcx,y+μx,ydcx,Sxdcy,Ty1+dcx,y+γx,ydcy,Sxdcx,Ty1+dcx,y+δx,ydcx,Sxdcx,Ty+dcy,Tydcy,Sx1+dcx,Ty+dcy,Sx∈sSx,Ty,τλx,y+μx,y+γx,y+δx,y<1.Then,S and T have a unique common fixed point.Proof.
The proof follows by settingλx,y,a=λx,y,μx,y,a=μx,y,γx,y,a=γx,y, and δx,y,a=δx,y in the proof of Theorem 11.By settingμ=γ=0 in Theorem 11, we have the following corollary.Corollary 13.
Let
X
,
d
c
be a complete complex valued
b
-metric space and let
S
,
T
:
X
→
C
B
X
be multivalued mappings with g.l.b. property. Let
λ
,
δ
:
X
×
X
×
X
→
0
,
1
be mappings such that
∀
x
,
y
∈
X
,
(32)
λ
x
2
n
,
y
,
a
≤
λ
x
0
,
y
,
a
,
λ
x
,
y
2
n
,
a
≤
λ
x
,
y
0
,
a
,
δ
x
2
n
,
y
,
a
≤
δ
x
0
,
y
,
a
,
δ
x
,
y
2
n
,
a
≤
δ
x
,
y
0
,
a
,
λ
x
,
y
,
a
d
c
x
,
y
+
δ
x
,
y
,
a
d
c
x
,
S
x
d
c
x
,
T
y
+
d
c
y
,
T
y
d
c
y
,
S
x
1
+
d
c
x
,
T
y
+
d
c
y
,
S
x
∈
s
S
x
,
T
y
,
τ
λ
x
,
y
,
a
+
δ
x
,
y
,
a
<
1
.Then,S and T have a unique common fixed point.Also, by settingγ=δ=0 in Theorem 11, we have the following consequence.Corollary 14.
Let
X
,
d
c
be a complete complex-valued
b
-metric space and let
S
,
T
:
X
→
C
B
X
be multivalued mappings with g.l.b. property. Let
λ
,
μ
,
γ
,
δ
:
X
×
X
×
X
→
0
,
1
be mappings such that
∀
x
,
y
∈
X
,
(33)
λ
x
2
n
,
y
,
a
≤
λ
x
0
,
y
,
a
,
λ
x
,
y
2
n
,
a
≤
λ
x
,
y
0
,
a
,
μ
x
2
n
,
y
,
a
≤
μ
x
0
,
y
,
a
,
μ
x
,
y
2
n
,
a
≤
μ
x
,
y
0
,
a
.
λ
x
,
y
,
a
d
c
x
,
y
+
μ
x
,
y
,
a
d
c
x
,
S
x
d
c
y
,
T
y
1
+
d
c
x
,
y
∈
s
S
x
,
T
y
,
τ
λ
x
,
y
,
a
+
μ
x
,
y
,
a
<
1
.Then,S and T have a unique common fixed point.Corollary 15.
Let
X
,
d
c
be a complete complex-valued
b
-metric space and let
S
,
T
:
X
→
C
B
X
be multivalued mappings with g.l.b property. Let
λ
,
μ
:
X
×
X
×
X
→
0
,
1
be mappings such that
∀
x
,
y
∈
X
,
(34)
λ
x
2
n
,
y
,
a
≤
λ
x
0
,
y
,
a
,
λ
x
,
y
2
n
,
a
≤
λ
x
,
y
0
,
a
,
μ
x
2
n
,
y
,
a
≤
μ
x
0
,
y
,
a
,
μ
x
,
y
2
n
,
a
≤
μ
x
,
y
0
,
a
,
γ
x
2
n
,
y
,
a
≤
γ
x
0
,
y
,
a
,
γ
x
,
y
2
n
,
a
≤
γ
x
,
y
0
,
a
,
δ
x
2
n
,
y
,
a
≤
δ
x
0
,
y
,
a
,
δ
x
,
y
2
n
,
a
≤
δ
x
,
y
0
,
a
,
λ
x
,
y
,
a
d
c
x
,
y
+
μ
x
,
y
,
a
d
c
x
,
S
x
d
c
y
,
T
y
1
+
d
c
x
,
y
+
γ
x
,
y
,
a
d
c
y
,
S
x
d
c
x
,
T
y
1
+
d
c
x
,
y
+
δ
x
,
y
,
a
d
c
x
,
S
x
d
c
x
,
T
y
+
d
c
y
,
T
y
d
c
y
,
S
x
1
+
d
c
x
,
T
y
+
d
c
y
,
S
x
∈
s
S
x
,
T
y
,
τ
λ
x
,
y
,
a
+
μ
x
,
y
,
a
+
γ
x
,
y
,
a
+
δ
x
,
y
,
a
<
1
.Then,S and T have a unique common fixed point.By puttingS=T in our main theorem, we obtain the following corollary.Corollary 16.
Let
X
,
d
c
be a complete complex-valued
b
-metric space and let
T
:
X
→
C
B
X
be multivalued mappings with g.l.b. property. Let
λ
,
μ
,
γ
,
δ
:
X
×
X
×
X
→
0
,
1
be mappings such that
∀
x
,
y
∈
X
,
(i)
λ
x
2
n
,
y
,
a
≤
λ
x
0
,
y
,
a
,
μ
x
2
n
,
y
,
a
≤
μ
x
0
,
y
,
a
,
γ
x
2
n
,
y
,
a
≤
γ
x
0
,
y
,
a
,
(35)
δ
x
2
n
,
y
,
a
≤
δ
x
0
,
y
,
a
(ii)
λ
x
,
y
,
a
d
c
x
,
y
+
μ
x
,
y
,
a
d
c
x
,
T
x
d
c
y
,
T
y
/
1
+
d
c
x
,
y
+
γ
x
,
y
,
a
d
c
y
,
T
x
d
c
x
,
T
y
/
1
+
d
c
x
,
y
+
δ
x
,
y
,
a
d
c
x
,
T
x
d
c
x
,
T
y
+
d
c
y
,
T
y
d
c
y
,
T
x
/
1
+
d
c
x
,
T
y
+
d
c
y
,
T
x
∈
s
T
x
,
T
y
(iii)
τ
λ
x
,
y
,
a
+
μ
x
,
y
,
a
+
γ
x
,
y
,
a
+
δ
x
,
y
,
a
<
1Then,T has a unique fixed point in X.Corollary 17.
Let
X
,
d
c
be a complete complex-valued
b
-metric space and
T
:
X
→
X
.
Let
λ
:
X
×
X
×
X
→
0
,
1
be mappings such that
∀
x
,
y
∈
X
,
(i)
λ
x
,
y
,
a
≤
b
,
b
∈
ℝ
(ii)
d
c
T
x
,
T
y
⪯
λ
x
,
y
,
a
d
c
x
,
y
(iii)
τ
λ
x
,
y
,
a
<
1Then,T has a unique fixed point in X.Example 18.
Let
X
=
0
,
1
.
Define
d
c
:
X
×
X
→
ℂ
by
d
c
x
,
y
=
∣
x
−
y
∣
e
i
θ
,
where
tan
θ
=
y
/
x
.
Then,
X
,
d
c
is a complete complex-valued
b
-metric space. Consider the mappings
S
,
T
:
X
→
C
B
X
be defined by
(36)
S
x
=
t
∈
X
:
0
≤
t
≤
x
7
,
T
x
=
t
∈
X
:
0
≤
t
≤
x
3
.
Observe that by settingx=y=0 in Theorem 11, the contractive condition becomes trivial. Now, consider the case for which x and y are nonzeros, we obtain by using the definition of dc that
(37)dcx,y=x−yeiθ,dcx,Sx=∣x−x7∣eiθ,dcy,Ty=y−y3eiθ,dcy,Sx=∣y−x7∣eiθ,dcx,Ty=x−y3eiθ,dcSx,Ty=x7−y3eiθ.
Furthermore, for allx,y∈X and fixed a=1/5∈X. Define the functions λ,μ,γ,δ:X×X×X→0,1 by λx,y,a=x/7+y/4+a,μx,y,a=xya/14,γx,y,a=2xya2/14, and δx,y,a=3xya3/14. Choose τ∈1,2, it is clear that τλx,y,a+μx,y,a+γx,y,a+δx,y,a<1 for each x,y∈X and fixed a=1/5∈X. Now, define
(38)xn=1n+1,n=0,1,2,3,⋯,then,
(39)λx2n,y,a=172n+1+y5+15≤1+y5+15,n=0,1,2,⋯=λx0,y,a.
Also, consider(40)λx,x2n,a=x7+152n+1+15≤x7+1+15,n=0,1,2,⋯=λx,x0,a.
Similarly, we can show that(41)μx2n,y,a≤μx0,y,a,μx,y2n,a≤μx,y0,a,γx2n,y,a≤γx0,y,a,γx,y2n,a≤μx,y0,a,δx2n,y,a≤δx0,y,a,δx,y2n,a≤δx,y0,a.
To conclude the example, we only need to show that∣dcSx,Ty∣≤τλx,y,a∣dcx,y∣. Indeed, for x,y∈X and a=1/2∈X,
(42)x7−y3eiθ≤27∣x−y∣eiθ≤25∣x−y∣eiθ≤2x7+y5+15∣x−y∣eiθ,∀x,y∈X=τλx,y,adcx,y,with τ∈1,2 and a=1/5∈X. Thus, ∣dcSx,Ty∣≤τλx,y,a∣dcx,y∣, for all x,y∈X and for a=1/5∈X.Therefore, all conditions of Theorem11 are satisfied and 0 is a common fixed point of S and T.


## 4. Application

### 4.1. Application to Riemann-Liouville Equation

In this section, we establish the existence of a solution of a Riemann-Liouville of the form(43)RLc1Ituxt=Γu∫c1tt−su−1xsds,Ru>0,where u∈ℂ,xt∈X=C0,1,ℝ and t,s∈0,1 which is the fractional integral. Let X=C0,1,ℝ be the space of continuous function and dc:X×X→ℂ be defined as
(44)dcu,v=maxt∈0,1∥ut−vt∥1+a2eitan−1a2,with τ=2. It is well known that X,dc is a complete complex-valued b-metric space. Define T:X→X by
(45)Txt=Γu∫c1tt−su−1xsds.Theorem 19.
Let
X
=
C
0
,
1
,
ℝ
,
for all
x
,
y
∈
X
and
a
∈
ℝ
,
let
λ
:
X
×
X
×
X
→
0
,
1
be given by
λ
x
,
y
,
a
≤
1
/
16
and suppose that
(46)
ma
x
t
∈
0
,
1
1
Γ
u
+
1
t
−
s
u
−
1
t
−
c
1
u
∣
t
−
s
u
−
1
∣
2
⪯
1
64
,then Equation (43) has a solution.Proof.
It is well-known thatx∈X is a fixed point of T if and only if x is a solution of problem (43). Note that at some point. Now, observe that for all u,v∈X, we have that
(47)dcTx,Ty=maxt∈0,1∣Txt−Tyt∣1+a2eitan−1a2=maxt∈0,1∣1Γu∫c1tt−su−1xs−1Γu∫c1tt−su−1ys∣1+a2eitan−1a2≾maxt∈0,11Γu∣∫c1tt−su−1ds∣∣xs−ys∣1+a2eitan−1a2⪯maxt∈0,11Γu∫c1t∣t−su−1ds∣∣xs−ys∣1+a2eitan−1a2⪯maxt∈0,11Γut−su−1∣t−su−1∣∫c1t∣t−su−1ds∣xs−ys1+a2eitan−1a2.
Letq=t−s, we get
(48)dcTx,Ty⪯maxt∈0,11Γut−su−1∣t−su−1∣∫t−c10qu−1dq∣xsys∣1+a2eitan−1a2·⪯maxt∈0,11Γut−su−1∣t−su−1∣quut−c10∣xs−ys∣1+a2eitan−1a2=maxt∈0,11Γut−su−1∣t−su−1∣t−c1uu∣xs−ys∣1+a2eitan−1a2⪯maxt∈0,11Γu+1t−su−1t−c1u∣t−su−1∣∣xs−ys∣1+a2eitan−1a2=maxt∈0,1∣xs−ys∣1+a2eitan−1a2·maxt∈0,11Γu+1t−su−1t−c1u∣t−su−1∣2⪯dx,y·164,that is,
(49)dcTx,Ty≤116dcx,y.
Thus, we have thatdTx,Ty⪯λdx,y. Clearly, all conditions of Corollary 17 are satisfied and guarantee the existence of the fixed point x∈X. Thus, x is the solution of the integral Equation (43).


## 4.1. Application to Riemann-Liouville Equation

In this section, we establish the existence of a solution of a Riemann-Liouville of the form(43)RLc1Ituxt=Γu∫c1tt−su−1xsds,Ru>0,where u∈ℂ,xt∈X=C0,1,ℝ and t,s∈0,1 which is the fractional integral. Let X=C0,1,ℝ be the space of continuous function and dc:X×X→ℂ be defined as
(44)dcu,v=maxt∈0,1∥ut−vt∥1+a2eitan−1a2,with τ=2. It is well known that X,dc is a complete complex-valued b-metric space. Define T:X→X by
(45)Txt=Γu∫c1tt−su−1xsds.Theorem 19.
Let
X
=
C
0
,
1
,
ℝ
,
for all
x
,
y
∈
X
and
a
∈
ℝ
,
let
λ
:
X
×
X
×
X
→
0
,
1
be given by
λ
x
,
y
,
a
≤
1
/
16
and suppose that
(46)
ma
x
t
∈
0
,
1
1
Γ
u
+
1
t
−
s
u
−
1
t
−
c
1
u
∣
t
−
s
u
−
1
∣
2
⪯
1
64
,then Equation (43) has a solution.Proof.
It is well-known thatx∈X is a fixed point of T if and only if x is a solution of problem (43). Note that at some point. Now, observe that for all u,v∈X, we have that
(47)dcTx,Ty=maxt∈0,1∣Txt−Tyt∣1+a2eitan−1a2=maxt∈0,1∣1Γu∫c1tt−su−1xs−1Γu∫c1tt−su−1ys∣1+a2eitan−1a2≾maxt∈0,11Γu∣∫c1tt−su−1ds∣∣xs−ys∣1+a2eitan−1a2⪯maxt∈0,11Γu∫c1t∣t−su−1ds∣∣xs−ys∣1+a2eitan−1a2⪯maxt∈0,11Γut−su−1∣t−su−1∣∫c1t∣t−su−1ds∣xs−ys1+a2eitan−1a2.
Letq=t−s, we get
(48)dcTx,Ty⪯maxt∈0,11Γut−su−1∣t−su−1∣∫t−c10qu−1dq∣xsys∣1+a2eitan−1a2·⪯maxt∈0,11Γut−su−1∣t−su−1∣quut−c10∣xs−ys∣1+a2eitan−1a2=maxt∈0,11Γut−su−1∣t−su−1∣t−c1uu∣xs−ys∣1+a2eitan−1a2⪯maxt∈0,11Γu+1t−su−1t−c1u∣t−su−1∣∣xs−ys∣1+a2eitan−1a2=maxt∈0,1∣xs−ys∣1+a2eitan−1a2·maxt∈0,11Γu+1t−su−1t−c1u∣t−su−1∣2⪯dx,y·164,that is,
(49)dcTx,Ty≤116dcx,y.
Thus, we have thatdTx,Ty⪯λdx,y. Clearly, all conditions of Corollary 17 are satisfied and guarantee the existence of the fixed point x∈X. Thus, x is the solution of the integral Equation (43).



---

*Source: 9738971-2020-09-29.xml*