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Every non-convex pair $(C, D)$ ( C , D ) may not have proximal normal structure even in a Hilbert space. In this article, we use cyclic relatively nonexpansive maps with respect to orbits to show the presence of best proximity points in $C\cup D$ C ∪ D , where $C\cup D$ C ∪ D is a cyclic T-regular set and $(C, D)$ ( C , D ) is a non-empty, non-convex proximal pair in a real Hilbert space. Moreover, we show the presence of best proximity points and fixed points for non-cyclic relatively nonexpansive maps with respect to orbits defined on $C\cup D$ C ∪ D , where C and D are T-regular sets in a uniformly convex Banach space satisfying $T(C)\subseteq C$ T ( C ) ⊆ C , $T(D)\subseteq D$ T ( D ) ⊆ D wherein the convergence of Kranoselskii’s iteration process is also discussed.

In this research, we prove strong and weak convergence results for a class of mappings which is much more general than that of Suzuki nonexpansive mappings on Banach space through the Picard–Krasnoselskii hybrid iteration process. Using a numerical example, we prove that the Picard–Krasnoselskii hybrid iteration process converges faster than both of the Picard and Krasnoselskii iteration processes. Our results are the extension and improvement of many well-known results of the literature.

The purpose of this paper is to introduce the modified Agarwal-O’Regan-Sahu iteration process (S-iteration) for finding endpoints of multivalued nonexpansive mappings in the setting of Banach spaces. Under suitable conditions, some weak and strong convergence results of the iterative sequence generated by the proposed process are proved. Our results especially improve and unify some recent results of Panyanak (J. Fixed Point Theory Appl. (2018)). At the end of the paper, we offer an example to illustrate the main results.

The purpose of this paper is to modify Abbas iteration for two G-nonexpansive mappings. We perform the convergence analysis of newly defined iteration process in uniformly convex Banach space. To compare the rate of convergence of the proposed method with well known Ishikawa iteration, modified S-iteration and Thianwan new iteration processes, a numerical example is also constructed.

In this paper, we prove that a three-step iteration process is stable for contractive-like mappings. It is also proved analytically and numerically that the considered process converges faster than some remarkable iterative processes for contractive-like mappings. Furthermore, some convergence results are proved for the mappings satisfying Suzuki’s condition (C) in uniformly convex Banach spaces. A couple of nontrivial numerical examples are presented to support the main results and the visualization is showed by Matlab. Finally, by utilizing our main result the solution of a nonlinear fractional differential equation is approximated.

In this paper, we establish weak and strong convergence theorems for mean nonexpansive maps in Banach spaces under the Picard–Mann hybrid iteration process. We also construct an example of mean nonexpansive mappings and show that it exceeds the class of nonexpansive mappings. To show the numerical accuracy of our main outcome, we show that Picard–Mann hybrid iteration process of this example is more effective than all of the Picard, Mann, and Ishikawa iterative processes.

In this paper, we prove the strong convergence theorems for nearly nonexpansive mappings, using the modified Picard-Mann hybrid iteration process in the context of uniformly convex Banach space., Comment: 7 pages

The purpose of this research work is to prove some weak and strong convergence results for maps satisfying (E)-condition through three-step Thakur (J. Inequal. Appl.2014, 2014:328.) iterative process in Banach spaces. We also present a new example of maps satisfying (E)-condition, and prove that its three-step Thakur iterative process is more efficient than the other well-known three-step iterative processes. At the end of the paper, we apply our results for finding solutions of split feasibility problems. The presented research work updates some of the results of the current literature. This research was funded by Basque Government Grant IT1207-1.

In this paper, from some confusions in examples on the convergence of certain iteration processes in the literature, we introduce the coordinate-convexity to study the convergence of a new three-step iteration process in Banach spaces with directed graphs. We prove some comparison results of the rate of convergence of the proposed iteration process and certain iteration processes for G-contractive mappings. We also prove the weak and strong convergence results for three G-nonexpansive mappings, and give three numerical examples to illustrate the obtained results.

‎In this paper we propose a new iteration process‎, ‎called the $K^{ast }$ iteration process‎, ‎for approximation of fixed‎ ‎points‎. ‎We show that our iteration process is faster than the existing well-known iteration processes using numerical examples‎. ‎Stability of the $K^{ast‎}‎$ iteration process is also discussed‎. ‎Finally we prove some weak and strong convergence theorems for Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces‎. ‎Our results are the extension‎, ‎improvement and generalization of many well-known results in the literature of iterations in‎ ‎fixed point theory‎.

In this work, we introduce a new iteration process for solving the fixed point problem of a finite family of nonexpansive mappings. We then prove, in Banach spaces, the strong convergence theorem under some mild conditions. Finally, we give some numerical results to support our main result.

In this paper, we present a new modified iteration process in the setting of uniformly convex Banach space. The newly obtained iteration process can be used to approximate a common fixed point of three nonexpansive mappings. We have obtained strong and weak convergence results for three nonexpansive mappings. Additionally, we have provided an example to support the theoretical proof. In the process, several relevant results are improved and generalized.

In this paper, we introduce a three step iteration method and show that this method can be used to approximate fixed point of weak contraction mappings. Furthermore, we prove that this iteration method is equivalent to Mann iterative scheme and converges faster than Picard-S iterative scheme for the class of weak contraction mappings. We also present tables and three graphics to support this result. Finally, we prove a data dependence result for weak contraction mappings using this three step iterative scheme.

Let K be a closed convex subset of X, and let T : K → K be a self-mapping with the set FT of fixed points such that ‖Tx − ρ‖ ≤ δ‖x − ρ‖ for all x ∈ K, ρ ∈ FT and some δ ∈ (0, 1). We introduce a new iteration process called Picard-hybrid iteration and show that this iteration process converges to the unique fixed point of T. It is also shown that our iteration process converges more rapidly than the Picard–Mann and Picard iteration processes. Our result improves a recent result of S. H. Khan and some other results.

In this paper, we introduce a new accelerated iteration for finding a fixed point of monotone generalizedα-nonexpansive mapping in an ordered Banach space. We establish some weak and strong convergence theorems of fixed point for monotone generalizedα-nonexpansive mapping in a uniformly convex Banach space with a partial order. Further, we provide a numerical example to illustrate the convergence behavior and effectiveness of the proposed iteration process.

In this paper, we introduce a new modified iteration process for approximating common endpoint of a multivalued α-nonexpansive mapping and a multivalued mapping satisfying condition (E′) in uniformly convex hyperbolic spaces. As a by-product, we improve the main results of Panyanak (2018) with a new and faster algorithm for two multivalued mappings. Moreover, we give a numerical example to substantiate our results. Our work is new and holds simultaneously in uniformly convex Banach spaces as well as CAT(0) spaces.

In the present article, we establish a semilocal convergence theorem for the S-iteration process of Newton---Kantorovich like in Banach space setting for solving nonlinear operator equations and discuss its semilocal convergence analysis. We apply our result to solve the Fredholm-integral equations.

In this paper, we prove a weak convergence theorem of the implicit iteration process for the semigroups of Lipschitz pseudocontractive mappings in uniformly convex Banach spaces with the Opial property.

In this paper, we introduce a new iteration process for solving fixed point problem of Bregman strongly nonexpansive mappings, and then we study a strong convergence theorem for a common fixed point of Bregman strongly nonexpansive mappings in the framework of reflexive Banach spaces.

Mann&rsquo, s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fr&eacute, chet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann&rsquo, s iteration process and its variants.

In this paper, we prove some weak and strong convergence results for generalized [Formula: see text]-nonexpansive mappings using [Formula: see text] iteration process in the framework of Banach spaces. This generalizes former results proved by Ullah and Arshad [Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat 32(1) (2018) 187–196].

In Banach space, the composite implicit iterative process for uniformly L- Lipschitzian asymptotically hemi-pesudocontractive mappings are studied, and the sufficient and necessary conditions of strong convergence for the composite implicit iterative process are obtained. AMS subject classifications: 47H09, 47J05, 47J25

In this paper, we study convergence of modified Ishikawa iteration process involving a pair of hybrid mappings satisfying a generalized nonexpansive condition on Banach spaces. In process, several relevant results are generalized and improved.

In this paper, weak and strong convergence theorems of three step iteration process with errors are established for two weakly inward and non-self asymptotically nonexpansive mappings in Banach spaces. the results obtained in this paper extend and improve the several recent results in this area.

The purpose of this paper is to establish a strong convergence of an implicit iteration process to a common fixed point for a finite family of Lipschitz $\phi-$uniformly pseudocontractive mappings in real Banach spaces. The results presented here improve and extend the corresponding results in [2, 4, 6] and the consecutive remark explains in details about the facts.

In this article, A demiclosed principle is proved for generalized asymptotically nonexpansive mapping in a Banach space. Additionally, it proves strong convergence of a composite implicit iteration scheme to a common fixed point for two finite families of generalized asymptotically quasi-nonexpansive mappings in a nonempty closed convex subset of a uniformly convex Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. The results of this article improve and extend the corresponding results of Xu and Ori [23], Zhou and Chang [26], Chang et al. [3], Yang and Yu [25], Shahzad and Zegeye [20].

We show that an (eventually) strongly increasing and positively homogeneous mapping T defined on a Banach space can be turned into an Edelstein contraction with respect to Hilbert’s projective metric. By applying the Edelstein contraction theorem, a nonlinear version of the famous Krein-Rutman theorem is presented, and a simple iteration process {Tkx/‖Tkx‖} (∀ x ∈ P+) is given for finding a positive eigenvector with positive eigenvalue of T. In particular, the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein contraction with respect to Hilbert’s projective metric. As a result, the nonlinear Perron-Frobenius property of a nonnegative tensor A is reached easily.

The purpose of this paper is to introduce more general modified two step and three step Ishikawa iterative with errors for local strongly pseudo contractive and local strongly accretive mappings which is much more general than the important class of strongly pseudo contractive and strongly accretive mappings. Also, we study the convergence of this iteration for locally mappings in the framework of Banach space. The results presented in this paper improve, generalize of the results of Mogbademn and OlaleauRafq, Yuguang and Fang and others.

We introduce a new implicit random iteration process generated by a finite family of asymptotically quasi-nonexpansive-type mappings and study necessary and sufficient conditions for the convergence of this process in a uniformly convex Banach space. The results presented in this paper extend and improve the recent ones announced by Plubtieng et al. (2007), Beg and Thakur (2009), and Saluja and Nashine (2012).

Let $K$ be a nonempty, closed, and convex subset of a real uniformly convex Banach space $E$ . Let $\left\{{T}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ and $\left\{{S}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ be two infinite families of asymptotically nonexpansive mappings from $K$ to itself with $F:=\left\{x\in K:{T}_{\lambda }x=x={S}_{\lambda }x,\lambda \in \mathrm{\Lambda }\right\}\ne \varnothing $ . For an arbitrary initial point ${x}_{0}\in K$ , $\left\{{x}_{n}\right\}$ is defined as follows: ${x}_{n}={\alpha }_{n}{x}_{n-1}+{\beta }_{n}\left({T}_{n-1}^{*}{\right)}^{{m}_{n-1}}{x}_{n-1}+{\gamma }_{n}\left({T}_{n}^{*}{\right)}^{{m}_{n}}{y}_{n}$ , ${y}_{n}={\alpha }_{n}^{\prime }{x}_{n}+{\beta }_{n}^{\prime }\left({S}_{n-1}^{*}{\right)}^{{m}_{n-1}}{x}_{n-1}+{\gamma }_{n}^{\prime }\left({S}_{n}^{*}{\right)}^{{m}_{n}}{x}_{n}$ , $n=1,2,3,\dots $ , where ${T}_{n}^{*}={T}_{{\lambda }_{{i}_{n}}}$ and ${S}_{n}^{*}={S}_{{\lambda }_{{i}_{n}}}$ with ${i}_{n}$ and ${m}_{n}$ satisfying the positive integer equation: $n=i+\left(m-1\right)m/2$ , $m\ge i$ ; $\left\{{T}_{{\lambda }_{i}}{\right\}}_{i=1}^{\infty }$ and $\left\{{S}_{{\lambda }_{i}}{\right\}}_{i=1}^{\infty }$ are two countable subsets of $\left\{{T}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ and $\left\{{S}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }},$ respectively; $\left\{{\alpha }_{n}\right\}$ , $\left\{{\beta }_{n}\right\}$ , $\left\{{\gamma }_{n}\right\}$ , $\left\{{\alpha }_{n}^{\prime }\right\}$ , $\left\{{\beta }_{n}^{\prime }\right\}$ , and $\left\{{\gamma }_{n}^{\prime }\right\}$ are sequences in $\left[\delta ,1-\delta \right]$ for some $\delta \in \left(0,1\right)$ , satisfying ${\alpha }_{n}+{\beta }_{n}+{\gamma }_{n}=1$ and ${\alpha }_{n}^{\prime }+{\beta }_{n}^{\prime }+{\gamma }_{n}^{\prime }=1$ . Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings $\left\{{T}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ and $\left\{{S}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.

In this paper, we prove strong and weak convergence results of modified Ishikawa iterative process to common fixed points of two nearly lipschitzian mappings in the framework of hyperbolic spaces.Our results extend and improve results of (4) and (11) from uniformly convex Banach spaces and CAT (0) spaces to hyperbolic spaces.

In this paper we propose a new iteration process, called AK iteration process, for approximation of fixed points for contraction mappings. We show that our iteration process is faster than the leading Vatan Two-step iteration process for contraction mappings. Numerical examples are given to support the analytic proofs. Stability of AK iteration process and data dependence result for contraction mappings by employing AK iteration process are also discussed.

This work is concerned with the strong convergence of an approximating common fixed point sequence of a finite family of multi-valued mappings in a uniformly convex and smooth Banach space using an implicit iteration scheme introduced by Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773].

In this paper, we introduce an iteration process for approximating common fixed points of two nonself asymptotically nonexpansive map- pings in Banach spaces. Our process contains Mann iteration process and some other processes for nonself mappings but is independent of Ishikawa iteration process. We prove some weak and strong convergence theorems for this iteration process. Our results generalize and improve some results in contemporary literature.

The aim of this article is to study an implicit iteration processwith errors for a nite family of non-Lipschitzian asymptotically non-expansive mappings in the intermediate sense in Banach spaces. Alsowe establish some strong convergence theorems and a weak convergencetheorem for said scheme to converge to a common xed point for non-Lipschitzian asymptotically nonexpansive mappings in the intermediatesense. The results presented in this paper extend and improve the corre-sponding results of [1], [3]-[8], [10]-[11], [13]-[14], [16] and many others. 1. Introduction and preliminariesLet K be a nonempty subset of a real Banach space E. Let T: K !Kbe a mapping. We use F(T) to denote the set of xed points of T, that is,F(T) = fx2K: Tx= xg. Recall the following concepts.(1) Tis nonexpansive ifkTx Tyk kx yk; (1)for all x;y2K.(2) Tis asymptotically nonexpansive if there exists a sequence fa n gin [1;1)with a n !1 as n!1such thatkT n x T n yk a n kx yk; (2)for all x;y2Kand n 1.(3) Tis uniformly L-Lipschitzian if there exists a constant L>0 such thatkT

The purpose of this paper is to study the strong convergence of a non-implicit iteration process with errors for asymptoticallyI-nonexpansive mappings in the intermediate sense in the framework of Banach spaces. The results presented in this paper extend and improve the corresponding results recently announced.

Let C be a closed convex subset of a Banach space E. Let {T(t):t⩾0} be a strongly continuous semigroup of nonexpansive mappings on C such that ∩t⩾0F(T(t))≠0. Let {αn} and {tn} be sequences of real numbers satisfying appropriate conditions, then for arbitrary x0∈C, the Mann type implicit iteration process {xn} given by xn=αnxn−1+(1−αn)T(tn)xn,n⩾0, weakly (strongly) converges to an element of ∩t⩾0F(T(t)).

The purpose of this paper is to study the strong and weak convergence of a finite step iteration process with errors for two finite families of asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in the paper improve and extend some results in Sitthikul and Saejung (2009) [9].

In this paper, we study a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces. We prove some weak and strong convergence theorems for this new iteration process. The results of this paper improve and extend the corresponding results of Chidume et al. (2003) [10], Osilike and Aniagbosor (2000) [3], Schu (1991) [4], Takahashi and Kim (1998) [9], Tian et al. (2007) [18], Wang (2006) [11], Yang (2007) [17] and others.

Strong convergence of two iterative schemes is proved to approach some zero of multivalued accretive operators in a Banach space. The first one is a regularization method for Rockafellar's proximal point algorithm of the resolvent and the second one is a kind of Halpern type iteration process of the resolvent. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The composite implicit iteration process introduced by Su and Li [J. Math. Anal. Appl. 320 (2006) 882-891] is modified. A strong convergence theorem for approximation of common fixed points of finite family of k-strictly asymptotically pseudo-contractive mappings is proved in Banach spaces using the modified iteration process.

Some convergence theorems for approximating to a common fixed point of an infinite family of strictly pseudocontractive mappings of Browder-Petryshyn type are proved in the setting of Banach spaces by using a new composite implicit iterative process with errors. The results presented in the paper generalize and improve the main results of Bai and Kim [1], Gu [4], Osilike [5], Su and Li [7], and Xu and Ori [8].