# Stability Results for Enriched Contraction Mappings in Convex Metric Spaces

Authors: Rekha Panicker; Rahul Shukla
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this paper, we obtain some stability results of fixed point sets for a sequence of enriched contraction mappings in the setting of convex metric spaces. In particular, two types of convergence of mappings, namely,G-convergence and H-convergence are considered. We also illustrate our results by an application to an initial value problem for an ordinary differential equation.


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## Body

## 1. Introduction

Stability is a term that refers to the limiting characteristics of a system. The stability of fixed points is the study of the relationship between the convergence of a sequence of mappings on a metric space or a Banach space and the sequence of its fixed points. If the fixed point sets of a sequence of mappings converge to the set of fixed points of their limit mappings, the sequence is said to be stable. We recommend the references [1–8] for some interesting results on the stability of fixed point sets of families of mappings in various settings. In this connection, see also the paper by Reich and Shafrir [9]. Pointwise and uniform convergence play a significant role in the study of fixed point stability. The preceding concepts, however, do not hold true when the domain of definition of all mappings in the study is neither the same space nor a unique nonempty subset of it. Barbet and Nachi [10] overcame this problem by introducing two new ideas of convergence called G-convergence and H-convergence, which they used to derive stability results in metric spaces. These results generalize the corresponding results of Bonsall [2], Fraser and Nadler [4], and Nadler [6]. The Barbet-Nachi work is unique in the sense that it redefines pointwise and uniform convergence for operators defined on the subsets of a space rather than the entire metric space. G-convergence generalized pointwise convergence, while H-convergence extended uniform convergence. Afterwards, many authors generalized these results in different settings for various type of mappings (see, for example, [11–15]).On the other hand, Takahashi [16] considered the concept of convex metric space and looked at a few fixed point theorems for nonexpansive mappings in this space. He observed that the concept of fixed point theorems in Banach spaces can be extended to convex metric spaces. Subsequently, Ciric [17], Naimpally et al. [18], and many others studied fixed point theorems for different classes of mappings in convex metric spaces as well as a subclass of it (compare [19, 20]). In addition, Berinde and Păcurar [21, 22] obtained some fixed point theorems for a class of enriched contraction mappings.In this paper, we study some convergence results in the sense of Barbet and Nachi [10] for a sequence of enriched contraction mappings and their fixed points in convex metric spaces as well as a subclass of it.


## 2. Preliminaries

We present some definitions, notions, and facts from the literature in this section. Throughout this paper, we use the notationℕ to denote the set of real numbers and ℕ¯ to denote ℕ∪∞.Definition 1 (see [16]).
LetB,d be a metric space. A continuous function W:B2×0,1⟶B is said to be a convex structure on B if, for all ϑ,ν∈B and any λ∈0,1,
(1)du,Wϑ,ν,λ≤λdu,ϑ+1−λdu,ν,
for anyu∈B.A metric spaceB,d endowed with a convex structure W is called a convex metric space and is usually denoted by B,d,W.Obviously, any linear normed space and each of its convex subsets are convex metric spaces, with the natural convex structure(2)Wϑ,ν,λ=λϑ+1−λν,for allϑ,ν∈B and λ∈0,1. However, the converse is not true: there are a number of examples of convex metric spaces that cannot be nested in any Banach space [16]. Let B,d,W be a convex metric space and ξ:B⟶B be a self mapping. Fixξ denotes the set of all fixed points of ξ, that is,
(3)Fixξ=ϑ∈Bξϑ=ϑ.LetB,d be a metric space and 0,l⊂ℝ. A mapping c:0,l⟶B is called as geodesic path from ϑ to ν if
(4)c0=ϑ,cl=ν,dct,ct′=t−t′,for everyt,t′∈0,l. The image c0,l of c forms a geodesic joining ϑ and ν. It is noted that the geodesic segment joining ϑ and ν is not necessarily unique. In some results of this paper, we shall deal with a subclass of convex metric spaces called W-hyperbolic metric space (or W-hyperbolic space) in the sense of Kohlenbach [23].Definition 2 (see [23]).
The tripletB,d,W is called a hyperbolic metric space if B,d is a metric space and the function W:B×B×0,1⟶B satisfies the following conditions for all ϑ,ν,z,w∈B and μ,η∈0,1:
(W1)dz,Wϑ,ν,μ≤1−μdz,ϑ+μdz,ν,
(W2)dWϑ,ν,μ,Wϑ,ν,η=μ−ηdϑ,ν,
(W3)Wϑ,ν,μ=Wν,ϑ,1−μ,
(W4)dWϑ,z,μ,Wν,w,μ≤1−μdϑ,ν+μdz,w.Remark 3.
IfWϑ,ν,μ=1−μϑ+μν for all ϑ,ν∈B,μ∈0,1, then it can be seen that all normed linear spaces are included in these spaces.Remark 4.
If conditions (W1)–(W3) are satisfied, thenB,d,W is the hyperbolic type space considered by Goebel and Kirk [24].We shall write(5)Wϑ,ν,μ≔1−μϑ⊕μν,to indicate the pointWϑ,ν,μ in a given hyperbolic metric space.


## 3.b,θ-Enriched Contraction Mapping

First, we recall the following definitions and results from Berinde and Păcurar [21] which play an important role in this paper.Definition 5.
LetB,. be a Banach space. A mapping ξ:B⟶B is said to be a b,θ-enriched contraction mapping if there exist b∈0,∞ and θ∈0,b+1 such that for all ϑ,ν∈B,
(6)bϑ−ν+ξϑ−ξν≤θϑ−ν.Remark 6.
It is pointed out in [21] that every contraction mapping ξ is a 0,θ-enriched mapping. The class of quasicontraction mapping and the class of b,θ-enriched contraction mappings are independent in nature. We present some examples to support our claim.Example 1 (see [21]).
LetC=0,1⊂ℝ with the usual metric and ξ:C⟶C be a mapping defined as
ξϑ=1−ϑforallϑ∈C.Then,Fξ=1/2. ξ is a b,1−b-enriched contraction mapping for any b∈0,1. At ϑ=0.4 and ν=0.6,
(7)dξϑ,ξν=0.2>kmax0.2,0.2,0.2,0,0=kmax0.4−0.6,0.4−0.6,0.6−0.4,0.4−0.4,0.6,0.6=kmaxdϑ,ν,dϑ,ξϑ,dν,ξν,dϑ,ξν,dν,ξϑ,for anyk∈0,1. Thus, ξ is not a quasicontraction mapping.Example 2.
LetK=0,1⊂ℝ with the usual metric and ξ:K⟶K be a mapping defined as
(8)ξϑ=14,ifϑ≠1,12,ifϑ=1.Then,Fξ=1/4. Now, we show that ξ is a quasicontraction mapping. If ϑ≠1 and ν=1, then
(9)dξϑ,ξν=14−12=14≤k34≤kmax1−ϑ,14−ϑ,1−12,12−ϑ,1−14=kmaxdϑ,ν,dϑ,ξϑ,dν,ξν,dϑ,ξν,dν,ξϑ,for anyk≥1/3. On the other hand, at ϑ=0.9 and ν=1,
(10)bϑ−ν+ξϑ−ξν=b0.9−1+14−12=b−0.1−14>b+10.1>θϑ−ν.ξ is not a b-enriched contraction mapping for any b∈0,∞.Now, we consider Definition5 in convex spaces:Definition 7 (see [6]).
LetB,d,W be a convex metric space. A mapping ξ:B⟶B is said to be an enriched contraction mapping if there exist c∈0,1 and λ∈0,1 such that
(11)dWϑ,ξϑ,λ,Wν,ξν,λ≤cdϑ,νforallϑ,ν∈B.Define the mappingS:B⟶B by Sϑ=Wϑ,ξϑ,λ=1−λϑ+λξϑ. Then, S is a Banach contraction, and hence, it has a fixed point in complete metric spaces.


## 4. Barbet-Nachi-Type Convergence

Following the conditions developed by Barbet and Nachi [10], we define the following conceptions of convergence for enriched contraction mappings in convex metric spaces, which generalize the notions of pointwise and uniform convergence.LetBnn∈ℕ¯ be a family of nonempty subsets of the convex metric space B and ξn:Bn⟶Bn∈ℕ¯ a family of enriched contraction mappings [10]. Then, ξ∞ is said to be a G-limit of the sequence ξnn∈ℕ, or equivalently, ξnn∈ℕ¯ satisfies the property G if the following condition holds:G:Grξ∞⊂liminfGrξn: for every ϑ∈B∞, there exists a sequence ϑn in ∏n∈ℕBn such that
(12)limndϑn,ϑ=0,limndξnϑn,ξ∞ϑ=0,whereGrξ stands for the graph of ξ.Further,ξ∞ is a H-limit of the sequence of enriched contraction mappings ξnn∈ℕ, or equivalently, ξnn∈ℕ¯ satisfies the property H if the following condition holds [10]:H: For all sequences ϑn in ∏n∈ℕBn, there exists a sequence νn in B∞ such that
(13)limndϑn,νn=0,limndξnϑn,ξ∞νn=0.Lemma 8.
LetB,d,W be a convex metric space, Bnn∈ℕ¯ a family of nonempty subsets of B, and ξn:Bn⟶B a sequence of enriched contraction mappings. Define the mapping Sn:Bn⟶B by
(14)Snϑn=Wϑn,ξnϑn,λ,
whereϑn∈∏n∈ℕBn. Then, for any λ∈0,1, Fixξn=FixSn.Proof.
Letan∈Fixξn. That is, ξnan=an.
(15)dan,Snan=dan,Wan,ξnan,λ≤λdan,an+1−λdan,ξnan=0.
That is,an∈FixSn.
Conversely, assumean∈FixSn. This means that dan,Snan=0, and hence, we have
(16)dan,Wan,ξnan,λ=0⇒1−λdan,ξnan=0.
But1−λ≠0, which implies dan,ξnan=0.
That is,an∈Fixξn. Hence Fixξn=FixSn.


## 5. Stability Results forG-Convergence

The following stability result improves and generalizes the results of Fraser and Nadler [4] and Nachi [25].Theorem 9.
LetB,d,W be a convex metric space and Bnn∈ℕ¯ a family of nonempty subsets of B. ξn:Bn⟶Bn∈ℕ¯ is a family of enriched contraction mappings satisfying the property G. If for all n∈ℕ¯, ϑn is a fixed point of ξn, then the sequence ϑnn∈ℕ converges to ϑ∞.Proof.
Sinceξn satisfies property G, ∀ϑ∞∈B∞, there exists a sequence νn in ∏n∈ℕBn such that νn⟶ϑ∞ and ξnνn⟶ξ∞ϑ∞. Thus,
(17)dϑn,ϑ∞=dξnϑn,ξ∞ϑ∞.
ButFixξn=FixSn. Therefore,
(18)dϑn,ϑ∞=dSnϑn,ξ∞ϑ∞≤dSnϑn,Snνn+dSnνn,ξnνn+dξnνn,ξ∞ϑ∞≤cdϑn,νn+dWνn,ξnνn,λ,ξnνn+dξnνn,ξ∞ϑ∞≤cdϑn,ϑ∞+dνn,ϑ∞+λdνn,ξnνn+dξnνn,ξ∞ϑ∞,1−cdϑn,ϑ∞≤cdνn,ϑ∞+λdνn,ξ∞ϑ∞+dξnνn,ξ∞ϑ∞+dξnνn,ξ∞ϑ∞.
Taking the limit asn⟶∞, we obtain
(19)limn⟶∞1−cdϑn,ϑ∞≤0.
Since0≤c<1, we get limn⟶∞dϑn,ϑ∞=0.Corollary 10.
LetB,d,W be a convex metric space and ξn:B⟶Bn∈ℕ¯ a family of enriched contraction mappings satisfying the property G. If for all n∈ℕ¯, ϑn is a fixed point of ξn, then the sequence ϑnn∈ℕ converges to ϑ∞.Proof.
By takingBn=B for n∈ℕ¯ in Theorem 9, we find the desired conclusion easily.The following theorem proves the existence of a fixed point for aG-limit of a sequence of enriched contraction mappings.Theorem 11.
LetB,d,W be a convex metric space and Bnn∈ℕ¯ a family of nonempty subsets of B. Suppose that ξn:Bn⟶Bn∈ℕ¯ is a sequence of enriched contraction mappings satisfying the property G. If ϑn∈Bn is a fixed point of ξn for each n∈ℕ and the sequence ϑn admits a subsequence converging to ϑ∞∈B∞, then ϑ∞ is a fixed point of ξ∞.Proof.
Letϑnk be a subsequence of ϑn such that limk⟶∞ϑnk=ϑ∞∈B∞. By the property G, there exists a sequence νn in ∏n∈ℕBn such that dνn,ϑ∞=0 and dξnνn,ξ∞ϑ∞=0. For any k∈ℕ, we have
(20)dϑ∞,ξ∞ϑ∞≤dϑ∞,ϑnk+dξnkϑnk,ξnkνnk+dξnkνnk,ξ∞ϑ∞.
Sinceξnk is an enriched contraction mapping,
(21)dSnkϑnk,Snkνnk≤cdϑnk,νnk.
Butξnkϑnk=ϑnk=Snkϑnk. Then, we have
(22)dξnkϑnk,ξnkνnk=dSnkϑnk,ξnkνnk≤dSnkϑnk,Snkνnk+dSnkνnk,ξnkνnk≤cdϑnk,νnk+dWνnk,ξnkνnk,λ,ξnkνnk=cdϑnk,νnk+λdνnk,ξnkνnk≤cdϑnk,ϑ∞+dνnk,ϑ∞+λdνnk,ξ∞ϑ∞+dξnkνnk,ξ∞ϑ∞.
Then, from (20), we have
(23)dϑ∞,ξ∞ϑ∞≤dϑ∞,ϑnk+cdϑnk,ϑ∞+dvnk,ϑ∞+λdvnk,ξ∞ϑ∞+dξnkvnk,ξ∞ϑ∞+dξnkvnk,ξ∞ϑ∞.
Lettingk⟶∞, we deduce that
(24)dϑ∞,ξ∞ϑ∞≤λlimk⟶∞dνnk,ξ∞ϑ∞≤λlimk⟶∞dνnk,ϑ∞+dϑ∞,ξ∞ϑ∞,
which implies(25)dϑ∞,ξ∞ϑ∞≤λdϑ∞,ξ∞ϑ∞.
That is,(26)1−λdϑ∞,ξ∞ϑ∞≤0.
Butλ∈0,1. Hence ξ∞ϑ∞=ϑ∞.Proposition 12.
LetB,d,W be a hyperbolic metric space, Bnn∈ℕ¯ a family of nonempty subsets of B, and ξn:Bn⟶Bn∈ℕ¯ a family of mappings satisfying the property G, such that, for any n∈ℕ, ξn is an enriched contraction mapping. Then, ξ∞ is also an enriched contraction mapping.Proof.
Letϑ and ν be two elements in B∞. By the property G, there exist two sequences ϑn and νn in ∏n∈ℕBn converging, respectively, to ϑ and ν such that the sequences ξnϑn and ξnνn converge to ξ∞ϑ and ξ∞ν, respectively. For λ=1/b+1, one can set
(27)S∞ϑ=Wϑ,ξ∞ϑ,λ,
for allϑ∈B∞. For any n∈ℕ, we have
(28)dS∞ϑ,S∞ν≤dS∞ϑ,Snϑn+dSnϑn,Snνn+dSnνn,S∞ν≤dS∞ϑ,Snϑn+cdϑn,νn+dSnνn,S∞ν.
Now, we prove the following claim:(29)dS∞ϑ,Snϑn⟶0anddSnνn,S∞ν⟶0asn⟶∞.
Givenϑn⟶ϑ and ξnϑn⟶ξ∞ϑ as n⟶∞. Then,
(30)dS∞ϑ,Snϑn=dWϑ,ξ∞ϑ,λ,Wϑn,ξnϑn,λ≤1−λdϑ,ϑn+λdξ∞ϑ,ξnϑn⟶0asn⟶∞.
Further,νn⟶ν and ξnνn⟶ξ∞ν as n⟶∞. Thus,
(31)dSnνn,S∞ν=dWνn,ξnνn,λ,Wν,ξ∞ν,λ≤1−λdνn,ν+λdξnνn,ξ∞ν⟶0asn⟶∞.
Lettingn⟶∞ in (28), (30), and (31), we have
(32)dS∞ϑ,S∞ν≤cdϑ,ν,
and the conclusion holds.The following result in Proposition 4 of [10] follows from Proposition 12.Corollary 13.
LetB be a metric space, Bnn∈ℕ¯ a family of nonempty subsets of B, and ξn:Bn⟶Bn∈ℕ¯ a family of mappings satisfying property G such that, for any n∈ℕ, ξn is a kn-contraction. Then, ξ∞ is a k-contraction where kn is a bounded (resp.convergent) sequence with ≕supnknresp.limnkn.


## 6. Stability Results forH-Convergence

Now, we present another stability result using theH-convergence.Theorem 14.
LetB,d;W be a hyperbolic metric space and Bnn∈ℕ¯ a family of nonempty subsets of B. Let ξn:Bn⟶Bn∈ℕ¯ be a family of mappings satisfying the property H such that ξ∞ is an enriched contraction mapping. If for each n∈ℕ¯, ϑn∈Bn is a fixed point of ξn, then the sequence ϑnn∈ℕ converges to ϑ∞.Proof.
By propertyH, there exists a sequence νn in B∞ such that dϑn,νn⟶0 and dξnϑn,ξ∞νn⟶0. ξ∞ is an enriched contraction which implies
(33)dWϑ,ξ∞ϑ,λ,Wν,ξ∞ν,λ≤cdϑ,νforallϑ,ν∈B∞.
Givenϑn is a fixed point of ξn for each n∈ℕ¯. Then, ϑn is a fixed point of Sn as well. That is, Snϑn=ϑn for each n∈ℕ¯. Also,
(34)dSnϑn,S∞νn=d1−λϑn+λξnϑn,1−λνn+λξ∞νn≤1−λdϑn,νn+λξnϑn,ξ∞νn.
By lettingn⟶∞, we see that dSnϑn,S∞νn=0.
Now, by using the triangle inequality, we get(35)dϑn,ϑ∞=dSnϑn,S∞ϑ∞≤dSnϑn,S∞νn+dS∞νn,S∞ϑ∞.
Asξ∞ is an enriched contraction mapping, S∞ is a contraction mapping. Hence, by letting n⟶∞, we obtain
(36)limn⟶∞dϑn,ϑ∞≤limn⟶∞dS∞νn,S∞ϑ∞≤limn⟶∞cdνn,ϑ∞≤limn⟶∞cdϑn,νn+dϑn,ϑ∞=climn⟶∞dϑn,ϑ∞.
Thus,(37)1−climn⟶∞dϑn,ϑ∞=0,
and hence, the conclusion follows.Corollary 15.
LetB,d,W be a hyperbolic metric space and ξn:B⟶Bn∈ℕ¯ be a family of mappings satisfying the property H converging to the enriched contraction mapping ξ∞:B⟶B. If for any n∈ℕ¯, ϑn is a fixed point of ξn, then the sequence ϑnn∈ℕ converges to ϑ∞.Proof.
By takingBn=B for all n∈ℕ¯ in Theorem 14, we obtain the desired result immediately.


## 7. Application

We consider the following application of our results. The following theorem is motivated due to Nadler [6].Theorem 16.
LetD be an open subset of ℝ2, a,b∈D, K∈ℝ, and K>0. Suppose that ξi is a sequence of real valued continuous functions defined on D such that ξiϑ,ν≤K for all ϑ,ν∈D with a G-limit ξ, a continuous function on D.
The setC=ϑ,ν:ϑ−a≤pandν−b≤Kϑ−a is a subset of D with p>0.
LetKi∈ℝ and Ki>0 for all i∈ℕ∪0; Ki be a bounded sequence. For all ϑ,ν,ϑ,z∈D, we have ξiϑ,ν−ξiϑ,z≤Kiν−z where 0<Kip<1 for all i∈ℕ∪0.Then, the sequenceνi converges uniformly on I=a−p,a+p to ν0, where for each i∈ℕ∪0,νi is the unique solution on I of the initial value problem
(38)ν′ϑ=ξiϑ,νϑ;νa=b.Proof.
LetB be the set of all real valued continuous functions on I with graph lying in C and with Lipschitz constant K. Then, B with the supremum norm . is a compact Banach space. For each g∈B, define
(39)Aigϑ=b+∫aϑξit,gtdt∀ϑ∈I.
For eachi∈ℕ∪0,
(40)bgt−ht+Aigϑ−Aihϑ=bgt−ht+∫aϑξit,gtdt−∫aϑξit,htdt≤bgt−ht+Ki∫aϑgt−htdt≤bsupt∈a,ϑgt−ht+Kisupt∈a,ϑgt−ht∫aϑdt≤bg−h+Kipg−h=θig−h,
whereθi=b+Kip∈0,b+1 and Ai is a b,θi contraction mapping from B to B for all i∈ℕ∪0. By Proposition 12, A is also an enriched contraction mapping on B. For each ϑ∈I,g∈B, i∈ℕ∪0,
(41)Aigϑ−Agϑ=∫aϑξit,gt−ξit,gtdt.
Sinceξ is the G-limit of ξi, the sequence of integrands converges to zero and is uniformly bounded by 2K. The Lebesgue-bounded convergence theorem guarantees that the sequence of integrals on the R.H.S. goes to 0 as i⟶∞. Therefore Ag is the G-limit of Aig on I. Now, by Proposition 4 of [26], the G-limit is equivalent to the pointwise limit. It is easy to see that Aig is uniformly continuous on I for each i∈ℕ∪0, and hence, the sequence Aig is equicontinuous on the compact set I. Therefore, the sequence Aig converges uniformly to Ag on I. Hence, the sequence Ai converges pointwise to A on B. Since B is a compact Banach space, the sequence νi has a convergent subsequence νij converging to ν. By Theorem 11, ν is a fixed point of A. From Theorem 9, the sequence νi, where νi is the unique fixed point of Ai for each i∈ℕ∪0, converges to the fixed point ν of A. The result follows since these fixed points are the unique solutions of the initial value problem.



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