# On Fixed Point Convergence Results for a General Class of Nonlinear Mappings with a Supportive Application

Authors: Fayyaz Ahmad; Kifayat Ullah; Junaid Ahmad; Om Kalthum S. K. Mohamed; Awad A. Bakery
Journal: Journal of Mathematics
 (2023)

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## Abstract

In this article, we considered the class of generalizedα,β-nonexpansive (GABN) mappings that properly includes all nonexpansive, Suzuki nonexpansive (SN), generalized α-nonexpansive (GAN), and Reich–Suzuki nonexpansive (RSN) mappings. We used the iterative scheme JA for finding fixed points of these mappings in a Banach space setting. We provided both weak and strong convergence results under some mild conditions on the mapping, domain, and on the parameters involved in our iterative scheme. To support these results numerically, we constructed a new example of GABN mappings and proved that the JA iterative scheme converges to its fixed point. Moreover, we proved that JA iterative scheme converges faster to the fixed point corresponding to the some other iterative schemes of the literature. Eventually, we carried out an application of our main outcome to solve a split feasibility of problems (SFPs) in the setting of GABN mappings. Thus, our results were new in the literature and improved well-known results of the literature.


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## Body

## 1. Introduction

We know that some time many analytical methods fail to find exact solution of the problems, therefore, fixed point theory recommends some alternate techniques for solving these problems. First, we expressed the solution of the problem in the fixed point of a certain map (the map may be contraction, nonexpansive, or generalized nonexpansive [1]). In this situation, an existence of a solution and the existence of a fixed point have the same meanings. We used some suitable iterative methods to find approximate unique fixed point (see, e.g., [2] and others). The Banach Contraction Principle (BCP) indicates, among other things, that if the operator is a contraction and the subset is a closed subset of a Banach space, there may be a single fixed point. As an operator P on a subset V of a Banach space B is called a contraction if, for all v,w∈V, we have the following equation:(1)Pv−Pw≤αv−w,where α∈0,1.A nonexpansive mappingP is a mapping that satisfies equation (1) for α=1. Actually, the Picard iteration method vr+1=Pvr was suggested by the BCP proof to determine the estimated value of the unique fixed point of the contraction P. If V is closed bounded and convex and B is a uniformly convex Banach space (UCBS), then P has a fixed point (see, e.g., Kirk [3], Browder [4], and Gohde [5]). In applied sciences, nonexpansive mappings have a key role for solving fixed points problems. Therefore, we tried to use some extensions of these mappings.SupposeP is a self-map, that is, P:V⟶V and v,w∈V, where V is any subset of a Banach space, then P is called as follows:(a)
SN [6] if 1/2v−Pv≤v−w⇒Pv−Pw≤v−w(b)
GAN [7] (GAN, for short) if 1/2v−Pv≤v−w⇒Pv−Pw≤αw−Pv+αv−Pw+1−2αv−w(c)
RSN [8] (RSN, for short) if 1/2v−Pv≤v−w⇒Pv−Pw≤βv−Pv+βw−Pw+1−2βv−wThe classes of RSN and GAN self-maps properly include the class of SN self-maps. It is very natural to ask that whether there exists any class of mappings that includes all the classes of mappings mentioned above.To answer the above question in affirmative, Ullah et al. [1] presented a new class of nonlinear mappings that includes all the above mappings.Definition 1.
(see [1]). A self-map P on a subset V of a Banach space is said to be GABN when, for all v,w∈V, one can find two real constants α,β∈0,1 such that α+β<1 such that(2)12v−Pv≤v−w⇒Pv−Pw≤αv−Pw+αw−Pv+βv−Pv+βw−Pw+1−2α−2βv−w.
The class of GABN mappings includes all these mappings, and thus, the concept of GABN mappings is more difficult but more important than the other mappings mentioned above. The purpose of this work is to carry out some new fixed point convergence results under an effective iterative scheme. Using these results, we show that a SFP can be solved. Eventually, our main results will be supported by a numerical example. Consequently, our results were new in the fixed point literature and improved the well-known results of many authors.


## 2. Preliminaries

The purpose of this section is to present some earlier results and definitions that will be used in our main outcome.Definition 2.
(see [9]). If vr denotes any weakly convergent sequence in B, then a Banach space B is said to be with Opial’s condition if, for all v′∈B−v, we have the following equation:(3)liminfr⟶∞vr−v<liminfr⟶∞vr−v′,where v is the weak limit of V.Definition 3.
(see [10, 11]). Suppose vr denotes any bounded sequence in a closed convex subset V of a UCBS B. In this case, one denotes and defines the asymptotic radius of vr on the set V by rV,vr=inflimsupr⟶∞‖vr−v:v∈V, while the asymptotic center of vr on the set V is denoted and defined as AV,vr=v∈V:limsupr⟶∞vr−v=rV,vr. Moreover, the set AV,vr contains only one element.
Now, we present some propositions and lemmas to understand the characterize of GABN maps.Proposition 1.
LetV be a nonempty subset of a Banach space B and P be a self-map. Then,(a)
WhenP is SN, then it follows that P is generalized (0,0)-nonexpansive(b)
WhenP is GAN, then it follows that P is generalized α,0-nonexpansive(c)
WhenP is β–RSN, then it follows that P is generalized 0,β-nonexpansive
The following key results are from [1].Lemma 1.
Suppose thatP form a GABN self-map on a subset V of a Banach space B. Then,(i)
IfP admits at least one fixed point, then Pq−Pv≤q−v for all q∈FP and for all v∈V.(ii)
By choosingvandw in V, the following estimate holds:(4)v−Pw≤3+α+β1−α−βv−Pv+v−w.(iii)
IfB satisfies Opial’s condition, then one has the following equation:(5)vr⊆V,vr⇀q,vr−Pvr⟶0⇒Pq=q.The following lemma is well-known in the literature and can be found in [12].Lemma 2.
SupposeB be a UCBS and 0<a≤kr≤b<1 for all m≥1. Considering wr and sr sequences in UCBS Bsatisfying the conditions limsupr⟶∞wr≤γ, limsupr⟶∞sr≤γ, and limr⟶∞krwr+1−krsr=γ for any real number γ≥0, we have limr⟶∞wr−sr=0.


## 3. Convergence Theorems in UCBS

Unlike contractions, the Picard iteration is not suitable for nonexpansive mappings [13]. There are simple examples in the literature that shows that if a nonexpansive mapping has a unique fixed point, then it is possible that the Picard may not converge to the fixed point. Thus, in the case of nonexpansive mappings, we studied iterative schemes that are more general than the Picard iteration. This section thus employs the JA iterative scheme of Abdeljawad et al. [14] to provide certain weak and strong convergence results for the class of generalized α,β-nonexpansive mapping in the linear setting of a UCBS. For αr,βr,γr∈0,1, we collect some iterative processes of the literature as follows.In 1953, Mann [15] suggested the following iteration:(6)v1=v∈V,vr+1=1−αrvr+αrPvr.In 1974, Ishikawa [16] generalized Mann iteration (6) to the setting of two steps as follows:(7)v1=v∈V,wr=1−βrvr+βrPvr,vr+1=1−αrvr+αrPwr.In 2000, Noor [16] generalized Ishikawa iteration (7) to the setting of three steps as follows:(8)v1=v∈V,sr=1−γrvr+γrPvr,wr=1−βrvr+βrPsr,vr+1=1−αrvr+αrPwr.In 2007, Agarwal et al. [17] suggested a new two-step iteration called S-iteration process as follows:(9)v1=v∈V,wr=1−βrvr+βrPvr,vr+1=1−αrPvr+αrPwm.In 2014, Abbas and Nazir [18] introduced the following new three-step iteration as follows:(10)v1=v∈V,sr=1−γrvr+γrPvr,wr=1−βrPvr+βrPwr,vr+1=1−αrPwr+αrPsr.In 2016, Thakur et al. [19] proposed the following new iteration:(11)v1=v∈V,sr=1−βrvr+βrPvr,wr=P1−αrvr+αrsr,vr+1=Pwr.In 2018, Ullah and Arshad [20] suggested M-iteration as follows:(12)v1=v∈V,sr=1−αrvr+αrPvr,wr=Psr,vr+1=Pwr.Recently, Abdeljawad et al. [14] constructed a novel iteration as follows:(13)v1=v∈V,sr=1−βrvr+βrPvr,wr=Psr,vr+1=P1−αrPvr+αrPwr.When we compute a fixed point of a certain nonlinear self-map, we try to use a faster iterative scheme in order to obtain a high accurate result in fewer steps of iterations. Agarwal et al. [17] used iterative scheme (9) for contractions and nonexpansive mappings and proved scheme (9) in this case is better than the Mann iteration (6). However, we note that Agarwal iteration (9) is a two-step iteration and it is known that three-step iteration is better than the one- and two-step iterations. Thus, in [18], Abbas and Nazir constructed a three-step iteration and proved that this new three-step iterative scheme converges better from the Agarwal iteration for nonexpansive mappings. After this, Thakur et al. [19] constructed iterative scheme (11) and proved that this scheme is better than the Abbas iteration for SN mappings. Thakur et al. [19] results were recently improved by Ullahn and Arshad [20] and Abdeljwad et al. [14] by constructing iterative schemes (12) and (13), respectively. In this article, we improved the results in [14, 19, 20] to the general context of GABN mappings.Lemma 3.
Suppose thatP is a GABN self-map on a convex closed subset V of a UCBS. If the fixed point set FP is nonempty and vr is produced from JA iteration (13), then, subsequently, limr⟶∞‖vr−q‖ exists for each choice of q in FP.Proof.
Fixing any pointq∈FP and then applying Lemma 1 (i), one has the following equation:(14)sr−q=1−βrvr+βrPvr−q≤1−βrvr+βrPvr−q≤1−βrvr−q+βrPvr−q≤1−βrvr−q+βrvr−q≤vr−q,wr−q=Psr−q≤sr−q,which implies that(15)vr+1−q=P1−αrPvr+αrPwr−q≤1−αrPvr+αrPwr−q≤1−αrPvr−q+αrPwr−q≤1−αrvr−q+αrwr−q≤1−αrvr−q+αrsr−q≤1−αrvr−q+αrvr−q≤vr−q.
Hence,vr−q is bounded below and nonincreasing, so limm⟶∞vr−q exists for every q∈FP.Theorem 1.
Suppose thatP is a GABN self-map on a convex closed subset V of a UCBS. Let vr be produced from the JA iteration (13). Then, subsequently, the fixed point set FP is nonempty if and only if vr is bounded in V and satisfies the equation limr⟶∞Pvr−vr=0.Proof.
LetFP≠∅ and q∈FP. Then, by Lemma 3, limr⟶∞vr−q exists and vr is bounded. We put(16)limr⟶∞vr−q=γ.
By the proof of Lemma3 together with equation (16), we obtained the following equation:(17)limsupr⟶∞sr−q≤limsupr⟶∞vr−q=γ.
By Lemma1 (i), we get the following equation:(18)limsupr⟶∞Pvr−q≤limsupr⟶∞vr−q=γ.
Also, by the proof of Lemma3, we get the following equation:(19)vr+1−q≤1−αrvr−q+αrsr−q.
Thus,(20)vr+1−q−vr−q≤vr+1−q−vr−qαr≤sr−q−vr−q.
So, we obtainedvr+1−q≤sr−q. Therefore,(21)γ≤liminfr⟶∞sr−q.
From equations (19) and (21), we get the following equation:(22)γ=limr⟶∞sr−q.
From (22), we get the following equation:(23)γ=limr⟶∞sr−q=limr⟶∞1−βrvr+βrPvr−q≤limr⟶∞1−βrvr+βrPvr−q=limr⟶∞1−βrvr−q+βrPvr−q≤limr⟶∞1−βrvr−q+limr⟶∞βrPvr−q≤limr⟶∞1−βrvr−q+limr⟶∞βrvr−q≤γ.
Thus,(24)γ=limr⟶∞sr−q=limr⟶∞1−βrvr−q+βrPvr−q.
Since0<βr<1 for all r≥1, so using equation (16), (18), and (24) together with Lemma 1, we get the following equation:(25)limr⟶∞Pvr−vr=0.
Conversely, suppose thatvr is bounded and limr⟶∞Pvr−vr=0 and q∈AV,vr. By Lemma 1 (ii), we have the following equation:(26)rPq,vr=limsupr⟶∞vr−Pq≤3+α+β1−α−βlimsupr⟶∞Pvr−vr+limsupr⟶∞vr−q=limsupr⟶∞vr−q=rq,vr.
So,Pq∈AV,vr. As B is a UCBS, therefore, AV,vr consists of a single point. Hence, Pq=q, that is, the fixed point set of P is nonempty.Theorem 2.
Suppose thatP is a GABN self-map on a convex subset V of a UCBS. If the fixed point set FP is nonempty and vr is produced from JA iteration (13), then, subsequently, vk converges (strongly) to some fixed point of P if the set V is compact.Proof.
Since the subsetV is convex and compact, we have a subsequence of the iterative sequence vr that we may denote by vrk such that vrk⟶q, where q is a point in V. We prove that q is a fixed point of P and strong limit of the iterative sequence vr and that vrk⟶Tq. For this, we used Theorem 1 and, hence, we get limr⟶∞Pvrk−vrk=0. Hence, applying Lemma 1 (ii), one has the following equation:(27)vrk−Pq≤3+α+β1−α−βvrk−Pvrk+vrk−q⟶0.
Thus,q=Pq, so q become a fixed point of P. Using Lemma 3, limr⟶∞vr−q exists. It now clear that q is the strong limit of the iterative sequence vr. Hence, the proof is finished.
The next result does not require the compactness condition. This result is valid in general Banach space setting.Theorem 3.
Suppose thatP is a GABN self-map on a convex closed subset V of a Banach space. If the fixed point set FP is nonempty and vr is produced from JA iteration (13), then, subsequently, vk converges (strongly) to some fixed point of P if liminfr⟶∞distvr,FP=0.Definition 4.
(see [21]). Suppose that P is a self-map on a convex closed subset V of a UCBS. Then, P is said to be with condition (I) if there is a function K with K0=0 and Ku>0 for all u∈0,∞ and all v∈V imply that v−Pv≥Kdistv,FP for all v∈V.
Now, the strong convergence using condition (I) ofP is as follows.Theorem 4.
Suppose thatP is a GABN self-map on a convex closed subset V of a UCBS. If the fixed point set FP is nonempty and vr is produced from the JA iteration (13), then, subsequently, vk converges (strongly) to some fixed point of P if P is with condition (I).Proof.
It follows from Theorem1 that(28)liminfr⟶∞Pvr−vr=0.
AsP satisfies condition I, so(29)vr−Pvr≥kdistvr,FP.
From equation (28), we get the following equation:(30)liminfr⟶∞Kdistvr,FP=0.
Using the properties ofK, we get the following equation:(31)liminfr⟶∞distvr,FP=0.
Subsequently, we carry out all the requirements for Theorem3, and hence, vr converges strong to some fixed point of P.
The last result of this section is the following weak convergence that is obtained under Opial’s condition of the underlying domain.Theorem 5.
Suppose thatP form a GABN self-map on a convex closed subset V of a UCBS B. If the fixed point set FP is nonempty and vr is produced from JA iteration (13), then, subsequently, vk converges (weakly) to some fixed point of P if B is with Opial’s condition.Proof.
We notice thatB is reflexive because UCBS is always reflexive. Now, since the sequence of iterates vr is bounded in V and so it has a convergent (weakly) subsequence, we denote it here by vrj and its weak limit is denoted here by v1. However, using Theorem 1, one has limj⟶∞Pvrj−vrj=0. Thus, applying Lemma 1 (iii), the point v1 becomes the fixed point for P. We now show that v1 is a weak limit for vk and this will finish the proof. For this purpose, we assume on the contrary that v1 is not a weak limit for vk and so we can find a new subsequence, namely, vrk of vr such that vrk is a weak convergent to some v2 that is different from v1. As shown earlier, we can show that v2 is a fixed point of P. Now, using Opial’s condition of P, it follows that(32)limr⟶∞vr−v1=limj⟶∞vrj−v1<limj⟶∞vrj−v2=limr⟶∞vr−v2=limk⟶∞vrk−v2<limk⟶∞vrk−v1=limr⟶∞vr−v1.
The above strict inequality suggests a contradiction becausev2≠v1. Thus, we must accept that the sequence of iterates vr converges weakly to v1.


## 4. Example

Now, we offer a numerical example of GABN self-map as follows.Example 1.
We construct a self-mapP:0,∞⟶0,∞ using the rule Pv=2v/3 if 1/3<v<∞ and Pv=0 when 0≤v≤1/3. The first target is to show that P is GABN on its domain. We set α=β=1/4. Then, we have three cases.
If0≤v,w≤1/3, then we have the following equation:(33)14v−Pw+14w−Pv+14v−Pv+14w−Pw≥0=Pv−Pw.
If1/3<v,w<∞, then we have the following equation:(34)14v−Pw+14w−Pv+14v−Pv+14w−Pw=14v−2w3+14w−2v3+14v−2v3+14w−2w3≥147v3−7w3+14v3−w3≥148v3−8w3=23v−w=Pv−Pw.
If1/3<v<∞ and 0≤w≤1/3, then we have the following equation:(35)14v−Pw+14w−Pv+14v−Pv+14w−Pw=14v+14w−2v3+14v−2v3+14w=14v+14w−2v3+14v3+14w≥148v3=23v=Pv−Pw.
Hence,P is GABN with α=β=1/4. However, for v=1/3 and w=2/3, we have 1/2v−Pv<v−w. However,(i)
Pv−Pw>v−w(ii)
Pv−Pw>1/4v−Pw+1/4w−Pv+1−21/4v−w(iii)
Pv−Pw>1/4v−Pv+1/4w−Pw+1−21/4v−w
From (i)–(iii), we see thatP does not belong to the classes of SN, GAN, and RSN mappings. We now set αr=0.90 and βr=0.65 and compare the high accuracy our JA iteration in Table 1 and Figure 1. Clearly, our JA iterative scheme requires less steps to reach the fixed point.Table 1
Sequences generated by various iterations for Example1.
rJA (13)M (12)Thakur (11)S (9)11010101022.41646090543.40740740742.95802469143.86250000030.58392833071.16104252400.87499100755.655555555400.39561448960.25882450043.19853086450001.80894689960001.02305996870000.57859724980000.32722888890000100000110000120000Figure 1
Various iteration schemes vs. our JA iterative scheme.
(a)(b)


## 5. Application to Split Feasibility Problems

Now, we consider Hilbert spaces, namely,B1 and B2 with C⊆B1 and Q⊆B2, compact and convex. If the operator Y:ℬ1⟶ℬ2 bounded and linear, the SFP [22] is defined as follows:(36)Searchq∗∈C:Yq∗∈Q.In our study, we target SFP (36) and assume that its solution set S is nonempty. In [23], it is shown that any point q∗∈C solves our SFP (36) iff q∗ solves essentially the following equation:(37)v=PCI−ηY∗I−PQYv,where the set PC (resp. the set PQ) is well-known in the literature and denotes the nearest point projection onto the set C (resp. onto the set Q) and η>0 and the mapping Y∗ denotes the adjoint operator of Y. Using the concept of nonexpansive self-maps, Byrne [24] was the first who proved that if η>0 with 0<η<2/ξ, then the self-map(38)P=PCI−ηY∗I−PQY,is eventually nonexpansive, and its CQ iterative scheme is given by the following formula:(39)vr+1=PCI−ηY∗I−PQYvr,r≥0,which converges (weakly) to some point of S.It is not always an easy task to improve a weak convergence result to the setting of strong convergence. In previous literatures, authors used nonexpansive mappings for establishing a weak converges for SFPs. Our alternative approach in this article is to provide a strong convergence for SFPs using the concept of GABN mappings and new general projection type iterative scheme.Theorem 6.
We now assume SFP (36) such that its solution set is nonempty and 0<η<2/ξ and PCI−ηY∗I−PQY is a GABN self-map. In this case, the sequence of iterates vr (13) converges (strongly) to some solution of SFP (36).Proof.
We takeP=PCI−ηY∗I−PQY, that is, the map P is generalized α,β-nonexpansive. In the view of Theorem 2, vr converges in the strong sense to some point of FP. Since FP=S, it follows that vr converges strongly to some solution of SFP (36). This finishes the proof.


## 6. Conclusions

The following new results are obtained in this research:(a)
The JA iterative scheme is used for finding fixed points of GABN mappings(b)
Both weak and strong convergence results are obtained under possible mild assumptions(c)
We constructed a new example of GABN mappings(d)
We proved that JA iterative scheme is more effective than the other schemes of the literature(e)
We carried out an application of our work to solve a SFP in the setting of GABN mappings(f)
Accordingly, we improved and extended some recent and old results due to various authors in [14, 19, 20] from nonexpansive, SN, and GAN mappings to the general class of GABN mappings [25–27].



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