Generalized Enriched Nonexpansive Mappings and Their Fixed Point Theorems

ThispaperintroducesanovelcategoryofnonlinearmappingsandprovidesseveraltheoremsontheirexistenceandconvergenceinBanachspaces,subjecttovariousassumptions.Moreover,weobtainconvergencetheoremsconcerningiteratesof α -Krasnosel ’ ski ĭ mapping associated with the newly de ﬁ ned class of mappings. Further, we present that α -Krasnosel ’ ski ĭ mapping associated with b -enriched quasinonexpansive mapping is asymptotically regular. Furthermore, some new convergence theorems concerning b -enriched quasino-nexpansive mappings have been proved


Introduction
Consider a Banach space ðX; k:jjÞ and a nonempty subset B ⊆ X.A mapping Ψ : B → B is nonexpansive if kΨ ðϑÞ − Ψ ðϱÞjj ≤ kϑ − ϱjj for all ϑ; ϱ 2 B. A point z 2 B is a fixed point of Ψ if Ψ ðzÞ ¼ z.While nonexpansive mappings may not have fixed points in a general Banach space.Browder [1], Göhde [2], and Kirk [3] independently proved fixed point theorems for nonexpansive mappings with certain geometric properties, such as uniform convexity or normal structure.Since then, numerous authors have obtained various extensions and generalizations of nonexpansive mappings and their results.Some of these notable extensions and generalizations are summarized in Pant et al.'s [4] study.
In 2008, Suzuki [5] introduced a novel category of nonexpansive mappings called mappings that fulfill condition (C) and derived significant fixed point results for this category.García-Falset et al. [6] further generalized condition (C) into the class of mappings satisfying condition (E).
Definition 1 [6].Let B be a nonempty subset of a Banach space X: A mapping Ψ : B → B is said to satisfy condition ðE μ Þ on B if there exists μ ≥ 1 such that for all ϑ; ϱ 2 B : We say that Ψ satisfies condition ðEÞ on B whenever Ψ satisfies ðE μ Þ for some μ ≥ 1: This class of mappings properly contains many important classes of generalized nonexpansive mappings, see Pant et al's [7] study.
A novel category of nonlinear mappings was introduced by Berinde [8] in a recent publication.
Definition 2 [8].Let ðX; k:jjÞ be a Banach space.A mapping Ψ : X → X is said to be b-enriched nonexpansive mapping if 9 b 2 ½0; 1Þ such that: In the recent years, a number of papers have appeared in the literature dealing with the fixed point theorems for enriched nonexpansive type mappings [9][10][11].It was proven that any enriched contraction mapping defined on a Banach space has a unique fixed point, which can be approximated by means of the Krasnoselskij iterative scheme.In Berinde's [12] study (also [13]), it was demonstrated that every nonexpansive mapping Ψ is a zero-enriched mapping.Building in this work, Shukla and Pant [14] recently extended the category of enriched nonexpansive mappings in the vein of Suzuki and introduced the subsequent class of mappings.Definition 3. Let ðX; k:jjÞ be a Banach space and B a nonempty subset of X: A mapping Ψ : B → B is said to be Suzuki-enriched nonexpansive mapping if there exists b 2 ½0; 1Þ such that for all ϑ; ϱ 2 B: It can be seen that every Suzuki-nonexpansive mapping Ψ is a Suzuki-enriched nonexpansive mapping with b ¼ 0: Motivated by García-Falset et al. [6], we generalize Suzukienriched nonexpansive mappings and consider a new class of mappings known as (E)-enriched nonexpansive mappings.In fact, we introduce a class of mapping, which contains both the Suzuki-enriched nonexpansive mappings and the class of mappings satisfying condition (E).Indeed the class of b-enriched nonexpansive mappings and that of mappings satisfying condition (E) are independent in nature.A couple of examples below illustrate these facts.
Example 2 [5].Let B ¼ ½0; 3 ⊂ R with the usual norm.Define Ψ : B → B by the following equation: Then, Ψ satisfies condition (E).However at ϑ ¼ 2:5 and ϱ ¼ 3; Ψ is not b-enriched nonexpansive mapping for any b 2 ½0; 1Þ: This paper is organized as follows: Section 2 deals with some preliminary results which are utilized throughout this paper.In Section 3, we coined a new class of mappings, namely, (E)-enriched nonexpansive mapping.We show that (E)-enriched nonexpansive mapping properly contains some nonlinear mappings and present an illustrative example.Section 4 is devoted to some existence and convergence theorems concerning (E)-enriched nonexpansive mapping.In Section 5, we present some new developments of enriched quasinonexpansive mapping.Particularly, an (E)-enriched nonexpansive mapping with fixed point is an enriched quasinonexpansive mapping.We discuss convergence of the iterates of α-Krasnosel'skiĭ mapping associated with enriched quasinonexpansive mapping.
Then for any d >0, ε>0, and ζ; ϱ 2 X with kζjj ≤ d; kϱjj ≤ d; kζ − ϱjj ≥ ε, there exists a δ>0 such that: Definition 4 [16].A Banach space X satisfies Opial property if for every weakly convergent sequence fζ n g with weak limit ζ 2 X, it holds: for all ϱ 2 X with ζ ≠ ϱ: Definition 5 [17].Suppose B is a nonempty subset of a Banach space X.Let ζ be an element in X such that there exists a point ϱ in B satisfying the following condition: for any z 2 B, kϱ − ζjj ≤ kz − ζjj.In this case, we refer to ϱ as a metric projection of ζ onto B and denote it by P B ðζÞ.If P B ðζÞ exists and is uniquely determined for all x 2 X, then we call the mapping P B : X → B, the metric projection onto B.
Definition 6 [17].A mapping Ψ : B → B is said to be quasinonexpansive if: The fact that a nonexpansive mapping with a fixed point is quasinonexpansive is widely recognized.However, it should be noted that the converse may not hold.
Proposition 1 [6].Let B be a nonempty subset of a Banach space X: If Ψ : B → B is a mapping satisfying condition ðEÞ with FðΨ Þ ≠ ; then Ψ is quasinonexpansive.

2
Abstract and Applied Analysis Definition 7 [18].Let B be a nonempty convex subset of a Banach space X and Ψ : B → B be a mapping.A mapping Ψ α : B → B is said to be an α-Krasnosel'skiĭ mapping associated with Ψ if there exists α 2 ð0; 1Þ such that: for all ζ 2 B: Definition 8 [19].Let B be a nonempty subset of a Banach space X: A mapping Ψ : B → B is called asymptotically regular if: Lemma 1 (Browder [20]; demiclosedness principle Lemma 2 (Berinde [8]).Let B be a nonempty convex subset of a Banach space X, and Ψ : B → B be a mapping.Define S : B → B as follows: for all ζ 2 B and λ 2 ð0; 1Þ: Then, FðSÞ ¼ FðΨ Þ:

(E)-Enriched Nonexpansive Mapping
This section presents a novel category of mappings, which we describe as follows.
Definition 9. Let ðX; k:jjÞ be a Banach space and B be a nonempty subset of X: We define a mapping Ψ : B → X an (E)-enriched nonexpansive mapping if 9 b 2 ½0; 1Þ and M 2 ½1; 1Þ such that: It can be seen that every mapping Ψ satisfying condition (E) is an (E)-enriched nonexpansive mapping with b ¼ 0: Remark 1.If B is a nonempty subset of X and Ψ : B → X is (E)-enriched nonexpansive, and there exists a sequence fζ n g in B such that kζ n − Ψ ðζ n Þjj → 0. Such a sequence is called almost fixed point sequence (a.f.p.s.) for Ψ .
Proposition 2. Let Ψ : B → X be a Suzuki-enriched nonexpansive mapping with any b 2 ½0; 1Þ: Then, Ψ is an (E)enriched nonexpansive mapping for any b 2 ½0; 1Þ and M ¼ 2b þ 3: Proof.We assume that Ψ is a Suzuki-enriched nonexpansive mapping.Then, 1 2ðbþ1Þ kζ − Ψ ðζÞjj ≤ kζ − Ψ ðζÞjj implies the following equation: Now, we show that either: Arguing by contradiction, we suppose that: By the triangle inequality, we get the following equation: By Equation ( 13), we get the following equation: Abstract and Applied Analysis which is a contradiction.Consequently, Equation ( 14) holds.Therefore, from Equation ( 14), we have either: In the first case, we have the following equation: In the other case, we have the following equation: Therefore in both the cases, we get the following equation: The following example shows that (E)-enriched nonexpansive mapping properly contains the class of Suzukienriched nonexpansive mappings.
Example 3. Let R be the set of real numbers equipped with the standard norm and B ¼ ½0; 4 a subset of R. Let Ψ : B → B be a mapping defined by the following equation: First, we show that Ψ is (E)-enriched nonexpansive mapping.For this, we consider the following nontrivial cases: Case (1).If ζ ≤ 3 and ϱ ¼ 4, then the following equation is obtained: Case (2).If ζ >3 and ϱ ¼ 4, then the following equation is obtained: Moreover, for ζ ¼ 3 and ϱ ¼ 4, we get the following equations: Thus, Ψ is not a Suzuki-enriched nonexpansive mapping with any b 2 ½0; 1Þ:

Some Fixed Point Theorems
In this section, we present some new fixed point theorems for (E)-enriched nonexpansive mappings.
ð0; 1Þ and put b ¼ 1−μ μ in Equation (28), then the above inequality is equivalent to the following equations: Define the mapping S as follows: Thus, the following equation is obtained: Then, from Equation (30), we get the following equation: for all ζ; ϱ 2 B: Thus, S is a mapping satisfying condition (E).From Equation (32), it follows that if fζ n g is an a.f.p.s. for Ψ then fζ n g is an a.f.p.s. for S: Thus, all the assumptions of [  Proof.In view of Theorem 1 and [6, Theorem 3], one can complete the proof.

□
In the next theorem, we present the convergence of iterates of α-Krasnosel'skiĭ mapping associated with (E)-enriched nonexpansive mapping.
Proof.Using the same technique as in Theorem 1, one can define a mapping S : X → X as follows: and S is a mapping satisfying condition (E).For a given ζ 0 2 X and γ 2 ð0; 1Þ, one can define a sequence fζ n g as follows: Using the definition of S, we have the following equation Since fζ n g ¼ fΨ n α ðζ 0 Þg strongly converges to a point ζ † in X: Thus, all assumptions of [21, Theorem 1] are satisfied, and ζ † 2 FðSÞ: But FðSÞ ¼ FðΨ Þ.This completes the proof.□ Theorem 6.Let B be a nonempty closed convex subset of a uniformly convex Banach space X and Ψ : B → B an (E)enriched nonexpansive mapping with FðΨ Þ ¼ fζ † g: Assume that the mapping I − Ψ is demiclosed at zero.Then for each ζ 0 2 B, the sequence of iterates fΨ n α ðζ 0 Þg converges weakly to ζ † ; where Ψ α is the α-Krasnosel'skiĭ mapping associated with Ψ and α 2 ð0;

Enriched Quasinonexpansive Mapping
In this section, we present some new convergence results for b-enriched quasinonexpansive mapping.Shukla and Pant [14] introduced the following new class of mappings.Definition 10.Let ðX; k:jjÞ be a Banach space and B a nonempty subset of X.A mapping Ψ : B → B is said to be b-enriched quasinonexpansive mapping if there exists b 2 ½0; 1Þ such that for all ζ 2 B and ϱ 2 FðΨ Þ ≠ ;: It is noted that every quasinonexpansive mapping is a zero-enriched quasinonexpansive mapping and every b-enriched nonexpansive mapping with a fixed point is b-enriched quasinonexpansive mapping.
Thus, Ψ is a b-enriched quasinonexpansive mapping.□ The following example demonstrates that converse of the above proposition does not hold.
Proof.By the definition of b-enriched quasinonexpansive mapping, we have the following equation: 43), then the above inequality is equivalent to the following equation: Define the mapping S as follows: From Lemma 2, FðSÞ ¼ FðΨ Þ.Then, from Equation (44), we get the following equation: for all ζ 2 B and ϱ 2 FðSÞ: Thus, S : B → B is a quasinonexpansive mapping.And, we obtain the following equation: Let ϱ 0 2 B: For each n 2 N ∪ f0g; define ϱ nþ1 ¼ Ψ α ðϱ n Þ: Thus, the following equations are obtained: and In order to prove that Ψ α is asymptotically regular, it suffices to prove that lim n→1 kΨ ðϱ n Þ − ϱ n jj ¼ 0: Since FðΨ Þ ≠ ;, let ζ 0 2 FðΨ Þ: Since S is a quasinonexpansive mapping and FðSÞ ¼ FðΨ Þ: for all n 2 N ∪ f0g: From Equation (45), the following equations are obtained: Now: From Equation (52), the following equation is obtained: Abstract and Applied Analysis Take β ¼ α μ , then β 2 ð0; 1Þ: From Equation (50), the following equation is obtained: Therefore, the sequence fkζ 0 − ϱ n jjg is bounded by u 0 ¼ kζ 0 − ϱ 0 jj: If ϱ n 0 ¼ ζ 0 for any n 0 2 N then from Equation (56), ϱ n → ζ 0 as n → 1: If ϱ n ≠ ζ 0 for all n 2 N; take the following conditions: If β ≤ 1 2 and from Equation (55), we have the following equation: Since X is a uniformly convex Banach space, by using the definition of uniformly convex space with kz n jj ≤ 1; kz 0 n jj ≤ 1, and the following equation: Noting that modulus of convexity δðεÞ is a nondecreasing function of ε, we obtain the following equation: From Equations ( 60) and (62), the following equation is obtained: Using induction in the above inequality, it follows that: We shall prove that lim n→1 kSðϱ n Þ − ϱ n jj ¼ 0: Arguing by contradiction, consider that kSðϱ n Þ − ϱ n jj does not converge to zero.Then, there exists a subsequence fϱ n k g of fϱ n g such that kSðϱ n k Þ − ϱ n k jj converges to η>0: Since δð⋅Þ 2 ½0; 1 is nondecreasing and β ≤ 1 2 ; we have 1 for all k 2 N: From Equation (63) and for sufficiently large k; we have the following equation: and by the uniform convexity of X; we get the following equation: Using induction in the above inequality, we get the following equation: Using the similar argument as in the previous case, it can be easily shown that kSðϱ n Þ − ϱ n jj → 0 as n → 1: Therefore, in both cases, kSðϱ n Þ − ϱ n jj → 0 as n → 1: From Equation (47), kΨ ðϱ n Þ − ϱ n jj → 0 as n → 1, Ψ α is asymptotically regular and this completes the proof.
and S is a quasinonexpansive mapping.For given ζ 0 2 B, β 2 ð0; 1Þ, we can define a sequence fζ n g as follows: From Theorem 6, the sequence fζ n g converges to some ζ * 2 FðSÞ: Using the definition of S, we have the following equation: Take α ¼ β bþ1 2 0; ½ 1 bþ1 Þ, then, we obtain the following equation: Hence, fζ n g ¼ fΨ n α ðζ 0 Þg strongly converges to a fixed point of S. But FðSÞ ¼ FðΨ Þ.This completes the proof.□ Theorem 11.Let B be a nonempty closed convex subset of a uniformly convex Banach space X: Let S : B → B be a quasinonexpansive mapping with FðSÞ ≠ ; and P the metric projection from X into FðSÞ: Then for each ζ 2 B, the sequence fPS n ðζÞg converges to some ϱ 2 FðSÞ: Proof.Following the same line of proof of [7,Theorem 6], one can complete the proof.

Theorem 2 .
Let ðX; k:jjÞ be a Banach space and B a nonempty subset of X: Let Ψ : B → X be a mapping.If:(a) Ψ is an (E)-enriched nonexpansive mapping on B; (b) there exists an a.f.p.s.fζ n g for Ψ in B such that fζ n g converges weakly to a point z in B, and (c) ðX; k:jjÞ satisfies the Opial property.Then, Ψ ðzÞ ¼ z: Proof.By the definition of mapping Ψ , we have the following equation: Abstract and Applied Analysis(a) Ψ is an (E)-enriched nonexpansive mapping on B; and (b) inf fkζ − Ψ ðζÞjj : ζ 2 Bg ¼ 0, then Ψ admits a fixed point.

for all ζ 2 ½ 1 2nπþ π 2 and ϱ n ¼ 1 2nπ for all n 2 N
− 1; 1 and Ψ is a b-enriched quasinonexpansive mapping with b ¼ 0:On the other hand, let ζ n ¼ ; we get the following equation:

□ Remark 2 .
The above theorem is a generalization of[18,   Theorem 1] for a more general class of mappings.Theorem 9. Let B be a nonempty closed subset of a Banach space X and S : B → B a quasinonexpansive mapping.Assume that X is strictly convex and B is a convex compact subset of X: If S is continuous then for any ζ 0 2 B; α 2 ð0; 1Þ; the α-Krasnosel'skiĭ process fS n α ðζ 0 Þg converges to some ζ * 2 FðSÞ: Proof.Following the same line of proof of [7, Theorem 5], one can complete the proof.□ Theorem 10.Let B and X be the same as in Theorem 6.Let Ψ : B → B be a b-enriched quasinonexpansive mapping with FðΨ Þ ≠ ;: If Ψ is continuous then for any ζ 0 2 B; α 2 ð0; 1 bþ1 Þ; the α-Krasnosel'skiĭ process fΨ n α ðζ 0 Þg converges to some ζ * 2 FðΨ Þ: Proof.Following the same proof technique as in Theorem 4, we can define a mapping S : B → B as follows:

□ Theorem 12 .□
Let B; X, and P be same as in Theorem 8. Let Ψ : B → B be a b-enriched quasinonexpansive mapping with FðΨ Þ ≠ ;: Then for each ζ 2 B, α ¼ 1 bþ1 ; the α-Krasnosel'skiĭ process fΨ n α ðζÞg converges to some ϱ 2 FðΨ Þ:Proof.In view of Theorems 4 and 8, one can complete the proof.Abstract and Applied Analysis 9 ). Suppose B is a nonempty subset of a Banach space X that satisfies the Opial property.
Let Ψ : B → B be a mapping that satisfies condition (E).Consider a sequence ζ n in B such that ζ n weakly converges to ζ, and lim n→1 kζ n − Ψ ðζ n Þjj ¼ 0: Then, we have Ψ ðζÞ ¼ ζ, which implies that I − Ψ is demiclosed at zero.Proof.The proof directly follows from [6, Theorem 1].□ Using the same technique as in Theorem 1, one can define a mapping S : B → B such that S is a mapping satisfying condition (E).In fact, the demiclosedness of I − Ψ and I − Ψ α at zeros are equivalent.Further, demiclosedness of I − Ψ and I − S at zeros are equivalent.Keeping [21, Theorem 2.2] in mind, one can complete the proof.Let B be a nonempty closed convex subset of a uniformly convex Banach space X which has the Opial property.Let Ψ : B → B be an (E)-enriched nonexpansive mapping with FðΨ Þ ≠ ;: Then for each ζ 0 2 B and α 2 ð0; 1 bþ1 Þ, the sequence of iterates fΨ n α ðζ 0 Þg converges weakly to a fixed point of Ψ :Proof.From [21, Theorem 2.2] and Theorem 6, one can complete the proof.