# A New Class of Contraction inb-Metric Spaces and Applications

Authors: Preeti Kaushik; Sanjay Kumar; Kenan Tas
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

A novel class ofα-β-contraction for a pair of mappings is introduced in the setting of b-metric spaces. Existence and uniqueness of coincidence and common fixed points for such kind of mappings are investigated. Results are supported with relevant examples. At the end, results are applied to find the solution of an integral equation.


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## Body

## 1. Introduction and Prelims

Fixed point theorems in metric spaces and generalized metric spaces provide a tool to solve many problems and have applications in nonlinear analysis and in many other fields. Most of the problems of applied mathematics reduce to solve a given equality which in turn may be reduced to find the fixed points of a certain mapping or the common fixed points of pairs of mappings. In order to solve particular problems, researchers tried to generalize various contraction conditions, auxiliary mappings, and metric spaces. Here, in this paper, we will present common fixed point theorems forα-β-contractive mappings in the framework ofb-metric space. Therefore, to have a clear understanding of the paper, we will discuss theb-metric space, Geraghty type mappings, and α-admissible mappings step by step.Theb-metric space or metric type space was introduced by Czerwik [1] in 1993. In this interesting paper, Czerwik [1] generalized the Banach contraction principle in the context of completeb-metric spaces. After that many researchers reported the existence and uniqueness of fixed points of various operators in the setting ofb-metric spaces (see, e.g., [2–13] and some references therein).Let us have a look on definitions, examples, and properties ofb-metric space.Definition 1 (see [1, 4]).
LetX be a nonempty space, and let s≥1 be a given real number. A functional d:X×X→[0,∞) is said to beb-metric if the following conditions hold good:(1)
d(x,y) = 0 if and only if x=y.(2)
d(x,y)=d(y,x).(3)
d(x,z)≤s[d(x,y)+d(y,z)], for all x,y, and z∈X.Ifd satisfies all the aboveb-metric axioms, then the pair (X,d) is called ab-metric or metric type space.The class ofb-metric spaces is larger than that of metric spaces, since ab-metric is a metric when s=1. All the metric spaces areb-metric spaces but not vice versa. We can illustrate this with the help of the following example.Example 2.
LetX={x1,x2,x3,x4} and d(x1,x2)=k≥2, d(x1,x3) = d(x1,x4) = d(x2,x3) = d(x2,x4) = d(x3,x4)=1, d(xi,xj)=d(xj,xi) for i,j=1,2,3,4 and d(xi,xi)=0, i=1,2,3,4.
Then(1)dxi,xj≤k2dxi,xn+dxn,xjforn,i,j=1,2,3,4,and if k>2, the ordinary triangle inequality does not hold.Definition 3 (see [1]).
Let(X,d) be ab-metric space.(a)
A sequence{xn} in X is calledb-convergent if and only if there exists x∈X such that d(xn,x)→0 as n→∞.(b)
The sequence{xn} in X is said to beb-Cauchy if and only if d(xn,xm)→0, asn, m→∞.
Theb-metric space (X,d) is calledb-complete if everyb-Cauchy sequence in X isb-convergent.The following lemmas are useful to prove our results.Lemma 4 (see [9]).
Let(X,d) be ab-metric space and let {xn} be a sequence in X such that(2)limn→∞⁡dxn,xn+1=0.
If{xn} is not ab-Cauchy sequence, then there exist ε>0 and two sequences {m(k)} and {n(k)} of positive integers such that the four sequences(3)dxmk,xnk,dxmk,xnk+1,dxmk+1,xnk,dxmk+1,xnk+1exist and the following hold:(4)ε≤limk→∞⁡inf⁡dxmk,xnk≤limk→∞⁡sup⁡dxmk,xnk≤εs,(5)εs≤limk→∞⁡inf⁡dxmk,xnk+1≤limk→∞⁡sup⁡dxmk,xnk+1≤εs2,(6)εs≤limk→∞⁡inf⁡dxmk+1,xnk≤limk→∞⁡sup⁡dxmk+1,xnk≤εs2,(7)εs2≤limk→∞⁡inf⁡dxmk+1,xnk+1≤limk→∞⁡sup⁡dxmk+1,xnk+1≤εs3.Lemma 5 (see [9]).
Let(X,d) be ab-metric space and let {xn} and {yn} beb-convergent to x and y, respectively. Then one has(8)1s2dx,y≤limn→∞⁡inf⁡dxn,yn≤limn→∞⁡sup⁡dxn,yn≤s2dx,y.
In particular, ifx=y, then one has limn→∞⁡d(xn,yn)=0. Moreover, for each z∈X, one has(9)1sdx,z≤limn→∞⁡inf⁡dxn,z≤limn→∞⁡sup⁡dxn,z≤sdx,z.Next, we address briefly the concept of Geraghty type mappings.In 1973, Geraghty [14] generalized the Banach contraction principle in the setting of complete metric spaces by considering an auxiliary function. This function is known as Geraghty type function or mapping. Later on, many authors [11, 15–17] characterized the result of Geraghty in the context of various metric spaces and proved many interesting results. The definition of this new class of mapping is as follows.Definition 6.
LetB denote the class of real functions β:[0,+∞)→[0,1) satisfying the condition(10)βtn⟶1impliestn⟶0.
For instance, consider the function given byβt=e-2t for t>0 and β(0)∈[0,1); here β∈B.In 1973, Geraghty generalized the Banach contraction principle in the following form.Theorem 7 (see [14]).
Let(X,d) be a complete metric space, and let F:X→X be a self-map.
Suppose that there existsβ∈B such that (11)dFx,Fy≤βdx,ydx,yholds for all x,y∈X. Then F has a unique fixed point z∈X and for each x∈X the Picard sequence {Fnx} converges to z when n→∞.In 2011, Dukic et al. [16] introduced the Geraghty type β functions inb-metric space as follows.Definition 8 (see [16]).
Let(X,d) be ab-metric space with given s>1. Consider the class Bs of real functions β:[0,+∞)→[0,1/s) satisfying the property(12)βtn⟶1simpliestn⟶0.
An example of a function inBs is given by β(t)=1/se-t for t>0 and β(0)∈[0,1/s).Lastly, we discuss theα-admissible mappings, examples, and alpha-admissibility for a pair of mappings.The concept ofα-admissible mappings was introduced by Samet et al. [18] in 2012. They established some fixed point theorems for such mappings in complete metric spaces and showed some examples and applications to ordinary differential equations. Since then, many researchers extended the idea and generalized fixed point results for single-valued and multivalued α-admissible mappings in various abstract spaces (see, e.g., [2, 3, 11, 19, 20] and more references in the literature). The following definitions and examples reveal the basics of α-admissible mappings.Definition 9 (see [18]).
A self-mappingF:X→X defined on a nonempty set X is α-admissible if, for all x,y∈X, one has (13)αx,y≥1⟹αFx,Fy≥1,where α:X×X→[0,∞) is a given function under consideration.Example 10.
LetX=[0,∞) and assume that F:X→X and α:X×X→[0,∞) by Fx=√x, for all x∈X, and(14)αx,y=ex-y;x≥y0;x<y.
ThenF is α-admissible map.In 2014, a new notion ofh-α-admissible mapping was introduced by Rosa and Vetro [20]. The definition of this notion is as follows.Definition 11 (see [20]).
LetF,h:X→X be two self-mappings defined on a nonempty set X. And consider the map α:X×X→[0,∞). Then mappingF is calledh-α-admissible if, for all x,y∈X,(15)αhx,hy≥1,impliesαFx,Fy≥1.Example 12.
LetX=[0,∞). We will define the mapping α:X×X→[0,∞) by(16)αx,y=1;x≥y0;x<yand consider the mappings F,h:X→X by F(x)=ex and h(x)=x2 for all x∈X. Then, the mappingF ish-α-admissible mapping.Allahyari et al. [2] defined the α-regular space with respect to some self-mapping defined on space as given below.Definition 13 (see [2]).
Let(X,d) be ab-metric space. Suppose that h:X→X and α:X×X→[0,∞) are two operators. X is α-regular with respect to h if, for every sequence {xn}∈X, α(hxn,hxn+1)≥1 for all n∈N and hxn→hx∈h(X) as n→∞; then there exists a subsequence {hxn(k)} of {hxn} such that, for all k∈N, α(hxn(k),hx)≥1.Definition 14 (see [11]).
LetX be a nonempty set. Then the map α:X×X→[0,∞) is called transitive if for u,v,w∈X one has(17)αu,v≥1,αv,w≥1⇓αu,w≥1.In 2014, Sintunavarat [11] proved the fixed point theorem for generalized α-β-contraction mapping in metric space and established Ulam-Hyers stability and well-posedness via fixed point results. The purpose of this paper is to define α-β-contraction mapping for a pair of mappings and to prove coincidence point and common fixed point theorems in the context ofb-metric space. We will provide suitable examples to support our results. At the end, we will discuss some applications in the context of integral equations.


## 2. Coincidence and Common Fixed Point Theorems

First of all, we will present some definitions and results which will make our results easy to understand.Definition 15.
LetF and g be two self-mappings defined on a nonempty set X. Then a point x∈X is called coincidence point of F and g if Fx=gx. Moreover, if x=Fx=gx, then x is called common fixed point ofF and g.Definition 16.
Two self-mappingsF and g defined on a nonempty set X are weakly compatible if the maps commute at each coincidence point; that is, if Fx=gx, for some x∈X, then Fgx=gFx.In 2014, Sintunavarat [11] defined the generalized α-β-contraction mapping in metric spaces as follows.Definition 17 (see [11]).
LetF be a self-mapping on a nonempty setX and there exist two functions α:X×X→[0,∞) and β∈B. We say thatF is α-β-contraction mapping if the following condition holds:(18)αx,y-1+δ∗dFx,Fy≤δβdx,ydx,y,for all x,y∈X, where 1<δ≤δ∗.Now, we will introduce our notions in the setting ofb-metric space.Definition 18.
LetF and g be two self-mappings defined onb-metric space (X,d) with given s>1 and there exist two functions α:X×X→[0,∞) and β∈Bs. We say thatF is α-β (b)-contraction with respect to g if the following condition holds:(19)αgx,gy-1+δdFx,Fy≤δβdgx,gydgx,gy,for all x,y∈X, where 1<δ.Next, we present the coincidence and common fixed point theorems for a new class of contraction inb-metric space.Theorem 19.
Let(X,d) be a completeb-metric space and let F,g:X→X be two self-mappings such that F(x)⊆g(x) and one of these two subsets ofX is complete. Suppose thatF is α-β (b)-contraction with respect to g mapping satisfying the following conditions:(i)
F is g-α-admissible and α is transitive mapping.(ii)
There existsx0∈X such that α(gx0,Fx0)≥1.(iii)
X is α-regular with respect to g.
ThenF and g have a coincidence point.Moreover, ifF and g are weakly compatible and hypothesis has one more additional assumption,(A1)
eitherα(u,v)≥1 or α(v,u)≥1 whenever Fu=gu and Fv=gv,thenF and g have a unique common fixed point.Proof.
Letx0∈X such that α(gx0,Fx0)≥1. In order to prove thatF and g have a point of coincidence and using F(x)⊆g(x), define two sequences {xn} and {yn} inX such that(20)yn=Fxn=gxn+1∀n∈N.
Now, ifyn=yn+1 for any n∈N, then gxn+1=yn=yn+1=Fxn+1 andF and g have a point of coincidence.
Without loss of generality, one can suppose thatyn≠yn+1 for each n∈N.
Hence,F is g-α-admissible and α(gx0,gx1)=α(gx0,Fx0)≥1. Similarly, α(gx1,gx2)=α(Fx0,Fx1)≥1. By induction, we can easily deduce (21)αgxn-1,gxn≥1∀n∈N.
Step
1. We shall show that limn→∞⁡d(yn+1,yn)=0. Now, it follows from (19) and (21) for each n∈N that(22)δdyn+1,yn=δdFxn+1,Fxn≤αgxn+1,gxn-1+δdFxn+1,Fxn≤δβdgxn+1,gxndgxn+1,gxn.
This implies that(23)dyn+1,yn≤βdgxn+1,gxndgxn+1,gxn≤1sdyn,yn-1∀n∈N.
Further, from (23), it follows that(24)limn→∞⁡dyn+1,yn=0.
Step
2. Next, we show that {yn} is a Cauchy sequence inb-metric space X. On the contrary, assume that {yn} is not a Cauchy sequence. Then there exists ε>0 and subsequences of integers np and mp with np>mp≥0 such that(25)dymp,ynp>ε,dymp,ynp-1<ε,∀p∈N.
Sincenp>mp and α is transitive mapping, we can deduce easily, by using triangle inequality,(26)αgxmp+1,gxnp≥1∀p∈N.
Now, using (19) and (26), we can consider(27)δdymp+1,ynp=δdFxmp+1,Fxnp≤αgxmp+1,gxnp-1+δdFxmp+1,Fxnp≤δβdgxmp+1,gxnpdgxmp+1,gxnp.
This gives(28)dymp+1,ynp≤βdgxmp+1,gxnpdgxmp+1,gxnp≤βdymp,ynp-1dymp,ynp-1≤βdymp,ynp-1ε(29)dymp+1,ynpε≤βdymp,ynp-1<1swith np>mp>p for all p∈N.
Hence, by Lemma4 and (29), we have(30)1s=1εεs≤limp→∞⁡inf⁡1εdymp+1,ynp≤limp→∞⁡sup⁡1εdymp+1,ynp≤limp→∞⁡inf⁡βdymp,ynp-1≤limp→∞⁡sup⁡βdymp,ynp-1≤1s;that is,(31)limp→∞⁡βdymp,ynp-1=1s.
Also,β∈Bs implies that limp→∞⁡d(ymp,ynp-1)=0. However, this is not possible, as, using (4) of Lemma 4, we get that(32)εs≤1sdymp,ynp≤dymp,ynp-1+dynp-1,ynp⟶0,as p→∞, which leads to contradicting (25).
Therefore,{yn} is ab-Cauchy sequence and by hypothesis we can suppose that g(X) is to be complete subspace ofX (proof can be derived in a similar manner when F(X) is supposed to be complete). Thenb-completeness of g(X) implies that {yn}=Fxn={gxn+1}b-converges to a point u∈g(X), where u=gz for some z∈X, or in other words(33)limn→∞⁡Fxn=limn→∞⁡gxn=gzfor  somez∈X.
Step 3. Next, we will prove that Fz=gz. SinceX is α-regular with respect to g and using (33), we have(34)αgxnp,gz≥1∀p∈N.
To showFz=gz, we consider(35)δdFz,gz≤δsdFz,Fxnp+sdgxnp+1,gz≤δsdFz,Fxnp×δsdgxnp+1,gz≤αgz,gxnp-1+δsdFz,Fxnp×δsdgxnp+1,gz≤δsβdgz,gxnpdgz,gxnp×δsdgxnp+1,gz.
This in turns implies that(36)dFz,gz≤sβdgz,gxnpdgz,gxnp+sdgxnp+1,gz1sdFz,gz≤βdgz,gxnpdgz,gxnp+dgxnp+1,gz<1sdgz,gxnp+dgxnp+1,gz⟶0asn⟶∞.
Thus,Fz=gz=u is a point of coincidence ofF and g.
Step 4. Next, we prove thatF and g have a common fixed point. Firstly, we claim that if Fu=gu and Fv=gv, then gu=gv. By hypotheses, α(u,v)≥1 or α(v,u)≥1. Suppose that α(u,v)≥1 and consider that(37)δdgu,gv=δdFu,Fv≤αgu,gv-1+δdFu,Fv≤δβdgu,gvdgu,gv.
Then we get that(38)dgu,gv≤βdgu,gvdgu,gvdgu,gv≤dgu,gv,which is a contradiction.
Therefore, we conclude thatgu=gv. If we take α(v,u)≥1, then we also get the same result.
Secondly, IfF and g are weakly compatible, then, for any u∈X, if u=Fv=gv (becauseF and g have a point of coincidence, proven earlier in Step 3), then, using weak compatibility ofF and g, we get(39)Fu=Fgv=gFv=gu.
Thusu is a coincidence point ofF and g; then, using the result in first part, gu=gv, which leads to Fu=gu=gv=Fv=u.
Therefore,u is a common fixed point ofF and g.
We can prove the uniqueness of the common fixed point ofF and g by making use of condition (19) and assumption (A1) of hypothesis. The proof is very simple; therefore we do not go through details.If we considerg as identity mapping in the above theorem, we deduce the following corollary.Corollary 20.
Let(X,d) be a completeb-metric space and let F:X→X be continuous α-admissible mapping satisfying the following condition:(40)αx,y-1+δdFx,Fy≤δβdx,ydx,y,for all x,y∈X and β∈Bs, where 1<δ.
Ifα is transitive mapping and there exists x0∈X such that α(x0,Fx0)≥1, thenF has a fixed point.Takingαx,y=1 and δ=1 in Corollary 20, we get the following variant of Geraghty theorem.Corollary 21.
Let(X,d) be a completeb-metric space and let F:X→X be self-mapping satisfying the following condition:(41)dFx,Fy≤βdx,ydx,y,for all x,y∈X and β∈Bs. ThenF has a unique fixed point z∈X and, for each x∈X, the Picard sequence Fnx converges to z when n→∞.The following lemma derived from [7] is very useful to prove our next theorem.Lemma 22 (see [7]).
Let{yn} be a sequence in a metric type space orb-metric space (X,d) such that(42)dyn+1,yn≤λdyn,yn-1.for some λ,0<λ<1/s and each n=1,2,…. Then {yn} is a Cauchy sequence inX.Theorem 23.
Let(X,d) be a completeb-metric space and be α-regular with respect to g. And let F,g:X→X be two self-mappings, whereF is g-α-admissible and α is transitive, and, for x0∈X, one has α(gx0,Fx0)≥1. If F(x)⊆g(x) and one of these two subsets of X is complete, suppose that there exists λ∈[0,1/s) such that (43)αgx,gy-1+δdFx,Fy≤δλMF,g;x,yfor all x,y∈X and 1<δ.
Consider(44)MF,g;x,y=max⁡dgx,gy,dgx,Fx,dgy,Fy,dgx,Fy+dgy,Fx2s.
ThenF and g have a coincidence point.Moreover, ifF and g are weakly compatible and hypothesis has one more additional assumption,(A1)
eitherα(u,v)≥1 or α(v,u)≥1 whenever Fu=gu and Fv=gv,thenF and g have a unique common fixed point.Proof.
Letx0∈X be arbitrary and, using condition F(X)⊆g(X), we can easily construct a Jungck sequence {yn} satisfying (45)yn=Fxn=gxn+1∀n∈N.
Now, ifyn=yn+1 for any n∈N, then gxn+1=yn=yn+1=Fxn+1 andF and g have a point of coincidence.
Without loss of generality, we assume thatyn≠yn+1 for each n∈N.
Hence,F is g-α-admissible and α(gx0,gx1)=α(gx0,Fx0)≥1. Similarly, α(gx1,gx2)=α(Fx0,Fx1)≥1. By induction, we can easily deduce (46)αgxn-1,gxn≥1∀n∈N.Step 1. We shall show that {yn} is ab-Cauchy sequence. Now, taking (43) and (46), we obtain, for each n∈N,(47)δdyn+1,yn=δdFxn+1,Fxn≤αgxn+1,gxn-1+δdFxn+1,Fxn≤δλMF,g;xn+1,xn.
This implies that(48)dyn+1,yn≤λMF,g;xn+1,xnor(49)dyn+1,yn≤λmax⁡dgxn+1,gxn,dgxn+1,Fxn+1,dgxn,Fxn,dgxn+1,Fxn+dgxn,Fxn+12s≤λmax⁡dyn,yn-1,dyn,yn+1,dyn-1,yn,dyn,yn+dyn-1,yn+12s≤λmax⁡dyn,yn-1,dyn-1,yn+dyn,yn+12.
Now, if maxdyn,yn-1,(dyn-1,yn+dyn,yn+1)/2 = (dyn-1,yn+dyn,yn+1)/2, then dyn,yn-1< (dyn-1,yn+dyn,yn+1)/2<d(yn+1,yn).
Then (49) implies that (50)dyn+1,yn≤λdyn+1,yn,which is impossible as λ<1.
Therefore, we deduce that maxdyn,yn-1,(dyn-1,yn+dyn,yn+1)/2=dyn,yn-1.
This in turn implies that(51)dyn+1,yn≤λdyn,yn-1.
Using Lemma22, we obtain that {yn} is ab-Cauchy sequence and by hypothesis we can suppose that g(X) is to be complete subspace of X (the proof when F(X) is similar). Thenb-completeness of g(X) implies that {yn}=Fxn={gxn+1}b-converges to a point u∈g(X), where u=gz for some z∈X, or in other words(52)limn→∞⁡Fxn=limn→∞⁡gxn=gzfor  somez∈X.Step2. Next, we will prove that Fz=gz. Since X is α-regular with respect to g and using (52), we have(53)αgxnp,gz≥1∀p∈N.
To showFz=gz, we consider(54)δdFxnp,Fz≤αgxnp,gz-1+δdFxnp,Fz≤δλMF,g;xnp,z.
This implies that(55)dFxnp,Fz≤λMF,g;xnp,zor(56)dFxnp,Fz≤λmax⁡dgxnp,gz,dgxnp,Fxnp,dgz,Fz,dgxnp,Fz+dgz,Fxnp2s.
Also,Fxn(p)→gz and gxn(p)→gz as n→∞, implying that (57)dgxnp,Fxnp=sdgxnp,gz+sdgz,Fxnp⟶0asn⟶∞.
Then we have only two cases.
Case
1
(58)
d
F
x
n
p
,
F
z
≤
λ
d
g
z
,
F
z
≤
λ
s
d
g
z
,
F
x
n
p
+
d
F
x
n
p
,
F
z
1
-
λ
s
d
F
x
n
p
,
F
z
≤
λ
s
d
g
z
,
F
x
n
p
⟶
0
as
n
⟶
∞
.
Since1-λs > 0, it follows that Fxn(p)→Fz.
Case
2
(59)
d
F
x
n
p
,
F
z
≤
λ
d
g
x
n
p
,
F
z
2
s
≤
λ
s
d
g
x
n
p
,
F
x
n
p
+
d
F
x
n
p
,
F
z
2
s
1
-
λ
2
d
F
x
n
p
,
F
z
≤
λ
2
d
g
x
n
p
,
F
x
n
p
⟶
0
as
n
⟶
∞
.
Since1-λ/2>0, it follows that Fxn(p)→Fz.
Uniqueness of the limit of a sequence implies thatFz=gz.
Using conditions (A1) and weak compatibility ofF and g of hypothesis, we can easily prove thatF and g have a unique common fixed point.From the above theorem, one can deduce the following corollary easily.Corollary 24.
Let(X,d) be a completeb-metric space and F:X→X is continuous and α-admissible. Suppose that there exists λ∈[0,1/s) such that (60)αx,y-1+δdFx,Fy≤δλMx,yfor all x,y∈X and 1<δ.
Consider(61)Mx,y=max⁡dx,y,dx,Fx,dy,Fy,dx,Fy+dy,Fx2s.
Ifα is a transitive mapping and there exists x0∈X such that α(x0,Fx0)≥1, then F has a fixed point.Takingαx,y=1 and δ=1, we get the following variant of Corollary 3.12 of [7].Corollary 25.
Let(X,d) be a completeb-metric space and let F:X→X be self-mapping. Suppose that there exists λ∈[0,1/s) such that, for all x,y∈X,(62)dFx,Fy≤λMx,y,where (63)Mx,y=max⁡dx,y,dx,Fx,dy,Fy,dx,Fy+dy,Fx2s.
ThenF has a unique fixed point z∈X.In what follows, we furnish illustrative examples wherein one demonstrates Theorems19 and 23 on the existence and uniqueness of a common fixed point.Example 26.
LetX={0,1,3} be ab-metric space with metricd given by d(x,y)=(x-y)2 with s=2. Consider the mappings F,g:X→X defined by F(0)=1,F(1)=1, and F(3)=1, and g by g(0)=1, g(1)=1, and g(3)=3. Let us take β∈Bs, as β(t)=1/2 for t>0 and β(0)∈[0,1/2).
Andα:X×X→[0,∞) by(64)αx,y=1,whenx,y≥00,otherwise;then one can examine easily that(65)dF0,F1=d1,1=0,βdg0,g1dg0,g1=βd1,1d1,1=0dF0,F3=d1,1=0,βdg0,g3dg0,g3=βd1,3d1,3=2dF1,F3=d1,1=0,βdg1,g3dg1,g3=βd1,3d1,3=2.
Thus, in all cases, we get(66)αgx,gy-1+δdFx,Fy≤δβdgx,gydgx,gyfor all x,y∈X, where 1<δ.Also we can check thatF and g satisfy all the other assumptions of Theorem 19 and thus the pairF and g has a unique common fixed point x=1.Example 27.
LetX={0,1,3} be ab-metric space with metricd given by d(x,y)=(x-y)2 with s=2. Consider the mappings F,g:X→X defined by F(0)=0, F(1)=1, and F(3)=0 and g by g(0)=3, g(1)=1, and g(3)=0. Let us take λ∈[0,1/2).
Andα:X×X→[0,∞) by (67)αx,y=1,whenx,y≥00,otherwise.
Now, we verify Theorem23 by considering the following cases.Case 1 (take points 0 and 1).
(68)
d
F
0
,
F
1
=
d
0,1
=
1
d
g
0
,
g
1
=
d
3,1
=
4
d
g
0
,
F
0
=
d
3,0
=
9
d
g
1
,
F
1
=
d
1,1
=
0
d
g
0
,
F
1
+
d
g
1
,
F
0
2
·
2
=
d
3,1
+
d
1,0
4
=
4
+
1
4
=
5
4
.
M
F
,
g
;
0,1
=
max
⁡
d
g
0
,
g
1
,
d
g
0
,
F
0
,
d
g
1
,
F
1
,
d
g
0
,
F
1
+
d
g
1
,
F
0
4
=
4,9
,
0
,
5
4
=
9Case 2 (take points 1 and 3).
(69)
d
F
1
,
F
3
=
d
1,1
=
0
,
M
F
,
g
;
1,3
=
max
⁡
d
g
1
,
g
3
,
d
g
1
,
F
1
,
d
g
3
,
F
3
,
d
g
1
,
F
3
+
d
g
3
,
F
1
4
=
1,0
,
0
,
2
4
=
1
.Case 3 (take points 0 and 3).
(70)
d
F
0
,
F
3
=
d
0,0
=
0
,
M
F
,
g
;
0,3
=
max
⁡
d
g
0
,
g
3
,
d
g
0
,
F
0
,
d
g
3
,
F
3
,
d
g
0
,
F
3
+
d
g
3
,
F
0
4
=
9,9
,
0
,
9
4
=
9
.Thus, in all cases,F and g satisfy all the assumptions of Theorem 23 and thus the pairF and g has a unique common fixed point x=1.


## 3. An Application to an Integral Equation

This section deals with the applications of results proven in the previous section. Here, we will investigate the solution of integral equation through our results.Consider the following integral equation:(71)xt=∫01Kt,r,xrdr,where K:[0,1]×[0,1]×R→R.LetX=C[0,1] be the set of continuous real functions defined on [0,1]. Define theb-metric by d(x(t),y(t))=maxt∈[0,1]⁡(x(t)+y(t))p for all x,y∈X.Considerp>1. Then (X,d) is a completeb-metric space with the constant s=2p-1.F
(
x
(
t
)
)
=
∫
0
1
K
(
t
,
r
,
x
r
)
d
r for all x∈X and for all t∈[0,1]. Then the existence of a solution to (71) is equivalent to the existence of a fixed point ofF. Now, we prove the following result.Theorem 28.
Let us suppose that the following hypotheses hold:(i)
K:[0,1]×[0,1]×R→R is continuous.(ii)
For allt,r∈[0,1], there exists a continuous operator ξ:[0,1]×[0,1]→R such that(72)Kt,r,xr+Kt,r,yr<λ1/pξt,rxr+yr,supt∈0,1⁡∫01ξt,rdr≤1,where0<λ<1/s.(iii)
There exists a functionμ:R2→R such that, for all t∈I and for all a,b∈R with μ(a,b)≥0, we have(73)μx1t,∫01Kt,r,xrdr≥0forx∈CI,∀t∈I,μxt,yt≥0,μ∫01Kt,r,xr,∫01Kt,r,yrdr≥0,∀t∈I,∀x,y∈CI.
Then the integral equation (71) has a unique solution x∈X.Proof.
We defineα:C(I)×C(I)→[0,∞) by(74)αx,y=1,whenμxt,yt≥00,otherwise.
Then, for allx,y∈C(I), we have α(x,y)=1 and α(y,z)=1, implying that α(x,z)=1 for all x,y,z∈C(I), which proves that α is a transitive mapping.
Ifα(x,y)=1 for all x,y∈C(I), then μ(x(t),y(t))≥0. From (iii), we have μ(Fx(t),Fy(t))≥0, and so α(Fx,Fy)=1. Thus,F is α-admissible.
From (iii), there existsx1∈C(I) such that α(x1,Fx1)=1.
Also we observe from (72) that(75)Fxt+Fytp=∫01Kt,r,xrdr+∫01Kt,r,yrdrp≤∫01Kt,r,xrdr+∫01Kt,r,yrdrp≤∫01Kt,r,xr+Kt,r,yrdrp≤∫01λ1/pξt,rxr+yrdrp=∫01λ1/pξt,rxr+yrp1/pdrp≤∫01λ1/pξt,rd1/pxr,yrdrp=λdxt,yt∫01ξt,rdrp≤λdxt,yt≤λMxt,yt,which in turn implies that [αx,y-1+δ]d(Fx,Fy)≤δλM(x,y).
Now, all the conditions of Corollary24 hold andF has a unique fixed point x∈X, which means that x is the unique solution for the integral equation (71).


## 4. Conclusion

To conclude, we can assert that our results are novel, interesting, and generalized while considering the alpha-admissible Geraghty type mappings. These results extend, improve, unify, and generalize many theorems based on alpha-admissible mappings inb-metric spaces. Examples are presented in a simplest form and illustrate the theorems. Application to integral equation adds value to our research.



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*Source: 9718535-2017-06-11.xml*