# On Newχ-Fixed Point Results for λ−Υ,χ-Contractions in Complete Metric Spaces with Applications

Authors: Babak Mohammadi; Wutiphol Sintunavarat; Vahid Parvaneh
Journal: Journal of Mathematics
 (2021)

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## Abstract

The main aim of this work is to introduce the new concept ofλ−Υ,χ-contraction self-mappings and prove the existence of χ-fixed points for such mappings in metric spaces. Our results generalize and improve some results in existing literature. Moreover, some fixed point results in partial metric spaces can be derived from our χ-fixed points results. Finally, the existence of solutions of nonlinear integral equations is investigated via the theoretical results in this work.


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## Body

## 1. Introduction and Preliminaries

One of the most famous metrical fixed point theorem is the Banach contraction principle (BCP) which is the classical tool for solving several nonlinear problems. Based on the noncomplexity and the usefulness of this principle, many mathematicians have improved, extended, and generalized it into several directions. For instance, in [1], on the basis of the probabilistic metric space and the S-metric space, Hu and Gu introduced the concept of the probabilistic metric space, which is called the Menger probabilistic S-metric space. They also proved some fixed point theorems in the framework of Menger probabilistic S-metric spaces. In [2], using the notion of the cyclic representation of a nonempty set with respect to a pair of mappings, Mohanta and Biswas obtained coincidence points and common fixed points of a pair of self-mappings satisfying a type of contraction condition involving comparison functions and (w)-comparison functions in partial metric spaces.Many researchers attempted to introduce the new idea on generalizations of a metric space and then they investigated fixed point results in new spaces.In 1994, partial metric spaces were introduced initially by Matthews [3]. One of the important points in this space is the possibility of being nonzero the self-distance.Definition 1.
(see [3]). Let Π be a nonempty set. A mapping W:Π×Π⟶0,∞ is called a partial metric if and only if(p1)WX,X=WY,Y=WX,Y⇔X=Y,(p2)WX,X≤WX,Y,(p3)WX,Y=WY,X,(p4)WX,Y≤WX,Z+WZ,Y−WZ,Z,for any X,Y,Z∈Π. Moreover, the pair Π,W will be a partial metric space.
Note that any metric space is a partial metric space but the reverse is not true, in general. An example of a partial metric space is the pair0,∞,W, where WX,Y=maxX,Y for all X,Y∈0,∞. We see that WX,X may not to be zero for some X∈Π. For further examples of a partial metric, we refer to [3].Definition 2.
(see [3]). Let Π,W be a partial metric space.(1)
Xn⊆Π is said to be converging to a point X∈Π if and only if limn⟶∞WXn,X=WX,X.(2)
Xn⊆Π is called a Cauchy sequence if and only if limn,m⟶∞WXn,Xm exists and is finite.(3)
Π,W is said to be complete if and only if every Cauchy sequence Xn⊆Π converges to some point X∈Π such that limn,m⟶∞WXn,Xm=limn⟶∞WXn,X=WX,X.Remark 1.
(see [3]). If Π,W is a partial metric space, then the pair Π,dW is a metric space where dW:Π×Π⟶0,∞ is defined by dWX,Y=2WX,Y−WX,X−WY,Y for all X,Y∈Π.Lemma 1 (see [3]).
LetΠ,W be a partial metric space.(i)
Xn is Cauchy in Π,W if and only if Xn is Cauchy in Π,dW.(ii)
The partial metric spaceΠ,W is complete if and only if the metric space Π,dW is complete.(iii)
For eachXn⊆Π and X∈Π, limn⟶∞dWXn,X=0⇔limn,m⟶∞WXn,Xm=limn⟶∞WXn,X=WX,X.According to the published work of Matthews [3], fixed point results in partial metric spaces have been investigated widely by many mathematicians. In 2014, the new concepts of χ-fixed points, χ-Picard mappings, and weakly χ-Picard mappings have been introduced by Jleli et al. [4]. Several χ-fixed point results for mappings satisfying the generalized Banach contractive condition based on the idea of new control function are proved in [4]. Moreover, they also claimed that some fixed point results in partial metric spaces can be derived from these χ-fixed point results in metric spaces. Next, we recall the definitions of χ-fixed points, χ-Picard mappings, and weakly χ-Picard mappings. Before presenting these definitions, some notations are needed.LetΠ be a nonempty set, χ:Π⟶0,∞ be a given function, and Γ:Π⟶Π be a mapping.Throughout this paper, unless otherwise specified, the set of all fixed points ofΓ is denoted by FΓ=X∈Π|ΓX=X and the set of all zeros of χ is denoted by Zχ=X∈Π|χX=0.Definition 3.
(see [4]). Let Π be a nonempty set, χ:Π⟶0,∞ be a given function, and Γ:Π⟶Π be a mapping. Z∈Π is called a χ-fixed point of Γ if and only if Z is a fixed point of Γ such that χZ=0, that is, Z∈FΓ∩Zχ.Definition 4.
(see [4]). Let Π be a nonempty set and χ:Π⟶0,∞ be a given function. A mapping Γ:Π⟶Π is called a χ-Picard mapping if the following conditions hold:(i)
FΓ∩Zχ=Z(ii)
ΓnX⟶Z as n⟶∞ for any X∈ΠDefinition 5.
(see [4]). Let Π be a nonempty set and χ:Π⟶0,∞ be a given function. A mapping Γ:Π⟶Π is called a weakly χ-Picard mapping if the following conditions hold:(i)
Γ has at least one χ-fixed point(ii)
The sequenceΓnX converges to some χ-fixed point of Γ for any X∈Π
A new control functionΥ:0,∞3⟶0,∞ has been introduced by Jleli et al. [4] where(Y1)
maxa,b≤Υa,b,c, for all a,b,c∈0,∞(Y2)
Υ0,0,0=0(Y3)
Υ is continuous
Throughout this paper, unless otherwise is specified, the class of all functions satisfying the propertiesΥ1−Υ3 is denoted by Υ¯.Example 1.
(see [4]). Suppose that the mappings Υ1,Υ2,Υ3:0,∞3⟶0,∞ are defined by Υ1a,b,c=a+b+c,Υ2a,b,c=maxa,b+c,Υ3a,b,c=a+a2+b+c for all a,b,c∈0,∞. Then, Υ1,Υ2,Υ3∈Υ¯.
Using the notion of control functions inΥ, Jleli et al. [4] introduced the ideas of Υ,χ-contractions and Υ,χ-weak contractions and proved existence of χ-fixed point for such mappings as follows.Definition 6.
(see [4]). Let Π,d be a metric space, χ:Π⟶0,∞ be a given function, and Υ∈Υ¯. A mapping Γ:Π⟶Π is called an Υ,χ-contraction if and only if there is k∈0,∞ such that ΥdΓX,ΓY,χΓX,χΓY≤kΥdX,Y,χX,χY, for all X,Y∈Π.Definition 7.
(see [4]). Let Π,d be a metric space, χ:Π⟶0,∞ be a given function, and Υ∈Υ¯. A mapping Γ:Π⟶Π is called an Υ,χ-weak contraction if and only if there are k∈0,∞ and L≥0 such that ΥdΓX,ΓY,χΓX,χΓY≤kΥdX,Y,χX,χY+LΥdY,ΓX,χY,χΓX−Υ0,χY,χΓX for all X,Y∈Π.Theorem 1 (see [4]).
LetΠ,d be a complete metric space, χ:Π⟶0,∞ be a given function, and Υ∈Υ¯. Assume that(H1)χ is lower semicontinuous(H2)Γ:Π⟶Π is an Υ,χ-contraction mapping
Then, the following assertions hold:(i)
FΓ⊆Zχ(ii)
Γ is a χ-Picard mapping(iii)
IfX∈Π and Z is a χ-fixed point of Γ, then dΓnX,Z≤kn/1−kΥdΓX,X,χΓX,χX, for all n∈ℕTheorem 2 (see [4]).
LetΠ,d be a complete metric space, χ:Π⟶0,∞ be a given function, and Υ∈Υ¯. Assume that(H1)χ is lower semicontinuous(H2)Γ:Π⟶Π is an Υ,χ-weak contraction mapping
Then, the following assertions hold:(i)
FΓ⊆Zχ(ii)
Γ is a weakly χ-Picard mapping(iii)
IfX∈Π and Z is a χ-fixed point of Γ, then dΓnX,Z≤kn/1−kΥdX,ΓX,χX,χΓX, for all n∈ℕ
Nowadays, many authors have extended the Banach contractive condition in the BCP into many ways by using various types of the control functions. In 2014, Jleli and Samet [5] presented the new idea of a control function and proved the fixed point results for mappings involving this new control function. Here, we restate the idea of the control function proposed in Jleli and Samet [5] and give the main work in [5] which is the main inspiration in this paper.LetΛ be the set of all functions λ:0,∞⟶1,∞ so that(i)
λ is non-decreasing(ii)
For each sequenceιn⊂0,∞, limn⟶∞λιn=1 if and only if limn⟶∞ιn=0(iii)
There existr∈0,1 and l∈0,∞ such that limt⟶0+λt−1/tr=lTheorem 3 (see [5]).
LetΠ,d be a complete metric space and Γ:Π⟶Π be a given mapping. Suppose that there exist λ∈Λ and k∈0,1 such that for all X,Y∈Π with ΓX≠ΓY, one has λdΓX,ΓY≤λdX,Yk. Then, Γ possesses a unique fixed point.Recall thatχ:Π⟶0,∞ is lower semicontinuous at x0 if liminfx⟶x0χx≥χx0.Note that there is no discussion so far on the combination of several ideas of contraction mappings in the literature. The goal of this work is to present the new concept of aλ-Υ,χ-contraction self-mappings. The existence results of χ-fixed points for such contraction mappings in metric spaces are provided. The main results of Jleli and Samet [4] and Jleli et al. [5] are particular cases of our main results. Furthermore, we give some fixed point results in partial metric spaces which can be derived from our χ-fixed points results. Finally, we apply the theoretical results in this work to prove the existence of solutions of nonlinear integral equations.


## 2. Main Results

To present the main result in this paper, we start with the following definition which is larger than the idea of many contraction mappings in the literature.Definition 8.
LetΠ,d be a metric space, χ:Π⟶0,∞ be a given function, Υ∈Υ¯, and λ∈Λ. A mapping Γ:Π⟶Π is called a λ-Υ,χ-contraction if and only if there exists k∈0,1 such that(1)λΥdΓX,ΓY,χΓX,χΓY≤λΥdX,Y,χX,χYk,for all X,Y∈Π.
Now, we present the main results in this paper.Theorem 4.
LetΠ,d be a complete metric space, χ:Π⟶0,∞ be a given function, Υ∈Υ¯, and λ∈Λ. Assume that(i)
χ is lower semicontinuous(ii)
Γ:Π⟶Π is an λ-Υ,χ-contraction
Then, the following assertions hold:(i)
FΓ⊆Zχ(ii)
Γ is a χ-Picard mappingProof.
Suppose thatG∈Π is a fixed point of Γ. Appling (1) with X=Y=G, we obtain λΥ0,χG,χG≤λΥ0,χG,χGk. This implies λΥ0,χG,χG=1 and so Υ0,χG,χG=0. Then, χG≤Υ0,χG,χG=0 which implies χG=0. Thus, we have proved i. Now, let X be an arbitrary point. From (1), we obtain(2)λdΓnX,Γn+1X≤λΥdΓnX,Γn+1X,χΓnX,χΓn+1X,≤λΥdΓn−1X,ΓnX,χΓn−1X,χΓnXk⋮≤λΥdX,ΓX,χX,χΓXkn,for all n∈ℕ. If n⟶∞ in the above inequality, we obtain limn⟶∞λdΓnX,Γn+1X=1 and so limn⟶∞dΓnX,Γn+1X=0. Thus, there exist r∈0,1 and l∈0,∞ such that(3)limn⟶∞λdΓnX,Γn+1X−1dΓnX,Γn+1Xr=l.
Similar to the proof of Theorem 2.1 in [5], we deduce ΓnX is a Cauchy sequence. Since Π,d is complete, there exists Z∈Π such that ΓnX⟶Z as n⟶∞. From (2), we obtain 1≤λχΓnX≤λΥdX,ΓX,χX,χΓXkn for all n∈ℕ. If n⟶∞ in the above inequality, we obtain limn⟶∞λχΓnX=1 and so limn⟶∞χΓnX=0. Since χ is lower semicontinuous, we obtain χZ=0. Again, using (1), we obtain(4)1≤limn⟶∞λdΓn+1X,ΓZ≤limn⟶∞λΥdΓnX,Z,χΓnX,χZk=λΥ0,0,0k=1.
Thus,dZ,ΓZ=0, that is, Z is a fixed point of Γ. Therefore, Z is also a χ-fixed point of Γ.
To show the uniqueness of fixed point, letZ,Z′ be two χ-fixed points of Γ. Applying (1) for X=Z,Y=Z′, we get λΥdZ,Z′,0,0≤λΥdZ,Z′,0,0k. This implies λΥdZ,Z′,0,0=1 and so ΥdZ,Z′,0,0=0. Therefore, dZ,Z′≤ΥdZ,Z′,0,0=0 which gives us dZ,Z′=0. Thus, Z=Z′. Therefore, we have proved ii.
TakingΥa,b,c=a+b+c and χ≡0 in the above theorem, we have the following.Corollary 1.
LetΠ,d be a complete metric space and λ∈Λ. Assume that(i)
There existsk∈0,1 such that(5)λΥdΓX,ΓY≤λΥdX,Yk,for allX,Y∈Π.Then, the following assertions hold:(i)
FΓ⊆Zχ(ii)
Γ is a χ-Picard mappingTakingλt=t for all t≥0 in the above corollary, we obtain the BCP.Next, we present the second idea of the new mappings satisfying the generalized contractive condition which is similar to the first idea and then we prove the existence of aχ-fixed point result for this mapping.Definition 9.
LetΠ,d be a metric space, χ:Π⟶0,∞ be a given function, Υ∈Υ¯, and λ∈Λ. A mapping Γ:Π⟶Π is called a λ-Υ,χ-weak contraction if and only if there exist k∈0,1 and L≥0 such that(6)λΥdΓX,ΓY,χΓX,χΓY≤λΥdX,Y,χX,χYkλΥdY,ΓX,χY,χΓX−Υ0,χY,χΓXL,for all X,Y∈Π.Theorem 5.
LetΠ,d be a complete metric space, χ:Π⟶0,∞ be a given function, Υ∈Υ¯, and λ∈Λ. Assume that(i)
χ is lower semicontinuous(ii)
Γ:Π⟶Π is a λ-Υ,χ-weak contraction
Then, the following assertions hold:(i)
FΓ⊆Zχ(ii)
Γ is a weakly χ-Picard mappingProof.
Suppose thatG∈Π is a fixed point of Γ. Appling (6) with X=Y=G, we obtain(7)λΥ0,χG,χG≤λΥ0,χG,χGkλΥ0,χG,χG−ΥΥ0,χG,χGL=λΥ0,χG,χGk.
This implies thatλΥ0,χG,χG=1 and so Υ0,χG,χG=0. Then, χG≤Υ0,χG,χG=0 which implies χG=0. Thus, we have proved i. Now, let X be an arbitrary point. From (6), we obtain(8)λdΓnX,Γn+1X≤λΥdΓnX,Γn+1X,χΓnX,χΓn+1X,≤λΥdΓn−1X,ΓnX,χΓn−1X,χΓnXkλΥdΓnX,ΓnX,χΓnX,χΓnX−Υ0,χΓnX,χΓnXL=λΥdΓn−1X,ΓnX,χΓn−1X,χΓnXk⋮≤λΥdX,ΓX,χX,χΓXkn.
Taking the limit asn⟶∞ in the above inequality, it gives us limn⟶∞λdΓnX,Γn+1X=1 and so limn⟶∞dΓnX,Γn+1X=0. Thus, there exist r∈0,1 and l∈0,∞ such that(9)limn⟶∞λdΓnX,Γn+1X−1dΓnX,Γn+1Xr=l.
Similar to proof of Theorem 2.1 in [5], we deduce ΓnX is a Cauchy sequence. Since Π,d is complete, there exists Z∈Π such that ΓnX⟶Z as n⟶∞. From (8), we obtain. 1≤λχΓnX≤λΥdX,ΓX,χX,χΓXkn.
Takingn⟶∞, we obtain limn⟶∞λχΓnX=1 and so limn⟶∞χΓnX=0. Since χ is lower semicontinuous, we obtain χZ=0. Again using (11), we obtain(10)1≤limn⟶∞λdΓn+1X,ΓZ≤limn⟶∞λΥdΓnX,Z,χΓnX,χZkλΥdΓnX,Z,χΓnX,χZ−Υ0,χΓnX,χZL=λΥ0,0,0k=1.
Therefore,λdZ,ΓZ=1 which implies dZ,ΓZ=0, that is, Z is a fixed point of Γ.
TakingΥa,b,c=a+b+c and χ≡0 in the above theorem, we have the following.Corollary 2.
LetΠ,d be a complete metric space and λ∈Λ. Assume that(i)
There existk∈0,1 and L≥0 such that(11)λΥdΓX,ΓY,χΓX,χΓY≤λΥdX,YkλΥdY,ΓXL,for all X,Y∈Π.
Then, the following assertions hold:(i)
FΓ⊆Zχ(ii)
Γ is a χ-Picard mappingRemark 2.
Note that the advantage of Corollary2 is that we can choose the power L=0 to obtain Corollary 1. That is, Corollary 2 is more general than Corollary 1. Also, by taking different functions λ, we can obtain many contractive conditions in Theorems 4 and 5.
Next illustrative example is furnished which demonstrates the validity of the hypotheses and degree of utility of Theorem6 while previous results in the literature are not applicable.Example 2.
LetΠ=τn|n∈ℕ∪0, where τn=nn+1/2 for all n∈ℕ and τ0=0. Obviously, Π,d is a complete metric space with the metric d:Π×Π⟶0,∞ defined by dX,Y=X−Y for all X,Y∈Π. Define a mapping Γ:Π⟶Π by Γτn=τn−1 for all n∈ℕ and Γτ0=τ0=0. Then, Γ is not a Banach contraction mapping, since(12)limn⟶∞dΓτn,Γτ1dτn,τ1=limn⟶∞τn−1−τ0τn−τ1=limn⟶∞nn−1/2nn+1/2−1=1.
Therefore, the BCP cannot be applied in this example.
Now, we define a functionλ∈Λ by λt=etet for all t∈0,∞ and a function Υ∈Υ¯ by Υa,b,c=a+b+c for all a,b,c∈0,∞. Also, we define a function χ:Π⟶0,∞ by χX=X for all X∈Π. We shall show that Γ is a λ-Υ,χ-contraction mapping. For any m,n∈ℕ with n>m, we have(13)dΓτn,Γτm+Γτn+Γτmdτn,τm+τn+τmedΓτn,Γτm+Γτn+Γτm−dτn,τm+τn+τm=nn−1/2−mm−1/2+nn−1/2+mm−1/2nn+1/2−mm+1/2+nn+1/2+mm+1/2enn−1/2−mm−1/2+nn−1/2+mm−1/2−nn+1/2−mm+1/2+nn+1/2+mm+1/2=nn−1nn+1e−2n≤e−2.
Puttingk=e−1, the above inequality is equivalent to(14)dΓτn,Γτm+Γτn+ΓτmedΓτn,Γτm+Γτn+Γτm≤k2dτn,τm+τn+τmedτn,τm+τn+τm,or equivalently(15)edΓτn,Γτm+Γτn+ΓτmedΓτn,Γτm+Γτn+Γτm≤ekdτn,τm+τn+τmedτn,τm+τn+τm.
Therefore, we obtain(16)λΥdΓτn,Γτm,χΓτn,χΓτm≤λΥdτn,τm,χτn,χτmk.
Then, all hypotheses of Theorem7 hold and so Γ has a unique χ-fixed point. Here, τ0=0 is the unique χ-fixed point of Γ.


## 3. Applications of Theoretical Results

In this section, we give two applications of our main results in the previous section. These applications consist of two parts. The first part is related to the fixed point results in partial metric spaces. The second part shows the application of theoretical results to solve the nonlinear integral equation.Theorem 6.
LetΠ,W be a complete partial metric space and Γ:Π⟶Π be a mapping such that(17)λWΓX,ΓY≤λWX,Yk,for all X,Y∈Π, where k∈0,1. Then, Γ has a unique fixed point Z. Moreover, χZ=0 implies WZ,Z=0.Proof.
Define a metricdW:Π×Π⟶0,∞ by(18)dWX,Y=2WX,Y−WX,X−WY,Y,for all X,Y∈Π. In addition, we define a new metric d:Π×Π⟶0,∞ by(19)dX,Y=dWX,Y/2,for all X,Y∈Π. Also, we set a function χ:Π⟶0,∞ and a function Υ∈Υ¯ by(20)χX=WX,X2,for allX∈Π,Υa,b,c=a+b+c,for alla,b,c∈0,∞.
Then, from (17), we have(21)λdΓX,ΓY+WΓX,ΓX+WΓY,ΓY≤λdX,Y+WX,X+WY,Yk,for all X,Y∈Π. It yields that(22)λΥdΓX,ΓY,χΓX,χΓY≤λΥdX,Y,χX,χYk,for all X,Y∈Π. By Theorem 7, Γ has a unique χ-fixed point Z. It implies that Γ has a unique fixed point Z∈Π. Moreover, χZ=0 implies WZ,Z=0.
Based on the proof of the above theorem and Theorem5, we get the following result.Theorem 7.
LetΠ,W be a compete partial metric space and Γ:Π⟶Π be a mapping such that(23)λWΓX,ΓY≤λWX,YkλWY,ΓX−WY,Y−WΓX,ΓXL,for all X,Y∈Π, where k∈0,1. Then, Γ has a fixed point Z. Moreover, WZ,Z=0.Next, we will consider the following nonlinear integral equation:(24)Xt=ϕt+∫abQt,s,Xsds,where a,b∈ℝ, X∈Ca,b (the set of all continuous functions from a,b to ℝ), and ϕ:a,b⟶ℝ and Q:a,b×a,b×ℝ⟶ℝ are two given functions.Theorem 8.
Consider integral equation (24). Suppose that the following conditions hold:(i)
Q:a,b×a,b×ℝ⟶ℝ is continuous and(25)λ∫abQt,s,αds−Qt,s,βds≤∫abλQt,s,α−Qt,s,βds,for allt,s∈a,b and for all α,β∈ℝ.(ii)
There existλ∈Λ and k∈0,1 such that(26)λQt,s,α−Qt,s,β≤λα−βkb−a,for all t,s∈a,b and for all α,β∈ℝ.Then, integral equation (24) has a unique solution.Proof.
LetΠ=Ca,b. Define the metric d on Π by dX,Y=supt∈a,bXt−Yt for all X,Y∈Π. Then, Π,d is a complete metric space. Consider a mapping Γ:Π⟶Π defined by ΓXt=ϕt+∫atQt,s,Xsds for all X∈Π. Define the control function Υ:0,∞3⟶0,∞ by Υa,b,c=a+b+c for all a,b,c∈0,∞. Also, define χ:Π⟶0,∞ by χX=0 for all X∈Π. Let X,Y∈Π and t∈a,b. Then, we have(27)λΓXt−ΓYt=λ∫atQt,s,Xsds−∫atQt,s,Ysds≤∫abλQt,s,Xs−Qt,s,Ysds≤∫abλXs−Yskb−ads≤1b−a∫abλdX,Ykds≤λdX,Yk.
SinceχX=0 for all X∈Π, we get(28)λdΓX,ΓY+χΓX+χΓY≤λdX,Y+χX+χYk.
Therefore,(29)λΥdΓX,ΓY,χΓX,χΓY≤λΥdX,Y,χX,χYk.
Thus,Γ:Π⟶Π is a λ-Υ,χ-contraction mapping. By Theorem 7, Γ has a unique χ-fixed point X∈Π, that is, ΓXt=Xt for all t∈a,b and χX=0 which means that integral equation (24) has a unique solution.


## 4. Conclusions

In this paper, we obtained some fixed point results first in a metric space and then in a partial metric space as results. The famous Banach contraction principle is a special case of our results. There are other terms such asdX,ΓY, dY,ΓY, and dX,ΓX which we can consider in future research. But, certainly, we should also work with other control functions. For more details in this direction, the readers can refer to [6].



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