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Fixed point theory is a branch of mathematics that studies solutions that remain unchanged under a given transformation or operator, and it has numerous applications in fields such as mathematics, economics, computer science, engineering, and physics. In the present article, we offer a quicker iteration technique, the D * * iteration technique, for approximating fixed points in generalized α -nonexpansive mappings and nearly contracted mappings. In uniformly convex Banach spaces, we develop weak and strong convergence results for the D * * iteration approach to the fixed points of generalized α -nonexpansive mappings. In order to demonstrate the effectiveness of our recommended iteration strategy, we provide comprehensive analytical, numerical, and graphical explanations. Here, we also demonstrate the stability consequences of the new iteration technique. We approximately solve a fractional Volterra–Fredholm integro-differential problem as an application of our major findings. Our findings amend and expand upon some previously published results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two modified Tseng's extragradient algorithms with a new generalized adaptive stepsize for solving monotone variational inequalities (VI) in reflexive Banach spaces. The advantage of our methods is that stepsizes do not require prior knowledge of the Lipschitz constant of the cost mapping. Based on Bregman projection-type methods, we prove weak and strong convergence of the proposed algorithms to a solution of VI. Some numerical experiments to show the efficiency of our methods including a comparison with related methods are provided. [ABSTRACT FROM AUTHOR]

In this article, we enquire for some weak and strong convergence results for a class of mappings satisfying Condition (E) via Picard Normal S-iterative algorithm (PNSA) in the setting of uniformly convex Banach space (UCBS). Eventually, we furnish an example to substantial our attained findings and numerically compare PNSA with that of some other well-known iterative algorithms. Additionally, we discuss the existence of the solution of a nonlinear functional integral equation as an application of our result. [ABSTRACT FROM AUTHOR]

In this paper, we show a counterexample to the new iterative scheme introduced by Rezapour et al. in "A new modified iterative scheme for finding common fixed points in Banach spaces: application in variational inequality problems" [2]. We propose a modified iteration to conclude the convergence result. Moreover, some of our results are established under a weaker assumption. [ABSTRACT FROM AUTHOR]

We give a multi-valued version of M iterative process [K. Ullah and M. Arshad, Numerical reckoning fixed points Suzuki's generalized nonexpansive mappings via new iteration process, Filomat 32 (2018) 187–196] in the setting of Banach spaces. After this, we prove some weak and strong convergence results for the class of multi-valued quasi-nonexpansive mappings (which contains the class of multi-valued nonexpansive mappings) with proper numerical examples using multi-valued version of M-iteration process. These results generalize several well-known comparable results in the literature. [ABSTRACT FROM AUTHOR]

We study a perturbed inertial Krasnoselskii-Mann-type algorithm and prove that the algorithm is an approximate fixed point sequence for Lipschitz pseudocontractive maps in arbitrary real Banach spaces. Strong convergence results are then established for our inertial iteration scheme for approximation of fixed points of Lipschitz pseudocontractive maps and solutions of certain important accretive-type operator equations in certain real Banach spaces. Implementation of our algorithm is illustrated using numerical examples in both finite and infinite dimensional Banach spaces. Our results improve rate of convergence and extend several related recent results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iteration scheme, named as the S**-iteration scheme, for approximation of fixed points of Suzuki's generalized non-expansive mappings. We also put forward some weak and strong convergence theorems for Suzuki's generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. We prove the stability of our instigated scheme and give numerical examples to show that this scheme is faster than Agarwal, Abbas, Thakur and Ullah iteration schemes. Our results comprehend, improve and consolidate many results in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of Bregman nonexpansive sequences and mappings in a Banach space, prove an ergodic theorem and a fixed point theorem, and some weak convergence theorems for such sequences and maps with rather mild assumptions on the Banach space. These results extend our previous results for ø-nonexpansive sequences and give a partial answer to the open problem raised in 1981 by Djafari Rouhani for nonexpansive sequences in a Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of ϕ -nonexpansive sequences and maps in a Banach space E, and prove an ergodic theorem, a fixed point theorem, and some weak and strong convergence theorems for such sequences and maps in E, with rather mild assumptions on the Banach space E, weaker than those assumed by previous authors for nonexpansive maps. These results give a partial answer to the still open problem raised in 1981 by Djafari Rouhani for nonexpansive sequences in a Banach space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose two modified forward-backward splitting algorithms for zeros of the sum of a maximal monotone operator and a Bregman inverse strongly monotone operator in reflexive Banach spaces. We prove weak and strong convergence theorems of the generated sequences by the proposed methods under some suitable conditions. We apply our results to study the variational inequality problem and the equilibrium problem. Finally, a numerical example is given to illustrate the proposed methods. The results presented in this paper improve and generalize many known results in recent literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a new hybrid iterative method for solving the generalized mixed equilibrium problems and the fixed point problems of quasi- φ-asymptotically nonexpansive multivalued mapping in uniformly smooth and uniformly convex Banach space. We prove weak and strong convergence theorem. Our results generalize and improve many recent announced results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we present a concept of semigroups of enriched Kannan mappings, which is based on enriching contractive type mappings that have been effectively used in many recent studies. We investigate some weak and strong convergence theorems for the Mann iterative process to approximate common fixed points of enriched Kannan semigroups in uniformly convex Banach spaces. In the end, by using MATLAB, we present a numerical example to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iteration process which is faster than the leading; S [J. Nonlinear Convex Anal. 8, no. 1 (2007), 61{79], Thakur et al. [App. Math. Comp. 275 (2016), 147{155] and M [Filomat 32, no. 1 (2018), 187{196] iterations for numerical reckoning fixed points. Using this new iteration process, some fixed point convergence results for generalized ff-nonexpansive mappings in the setting of uniformly convex Banach spaces are proved. At the end of paper, we offer a numerical example to compare the rate of convergence of the proposed iteration process with the leading iteration processes. [ABSTRACT FROM AUTHOR]

Let C be a nonempty subset of a Banach space X. A mapping T: C → C is said to satisfy(Bγ,μ) condition if there exists γ ∈ [0,1] and μ ∈ [0, ½] satisfying 2μ ≤ γ such that for each x,y ∈ C, γ||x-Tx||≤||x-y||+μ||y-Ty|| implies ||Tx-Ty||≤(1-γ)||x-y||+μ(||x-Ty||+||y-Tx||). In this paper, we obtain some convergence theorems for such mappings using M iterative process in uniformly convex Banach space setting. Our results extend and improve many results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the iterations of strongly (asymptotic) quasi ø-nonexpansive mappings in Banach spaces. First, we prove weak convergence of the generated sequence to a common fixed point of an infinite family of strongly asymptotic quasi ø-nonexpansive mappings. Next we prove strong convergence of the generated sequence by an additional assumption. In the sequel, invoke of Halpern regularization method, we prove strong convergence of the generated sequence to a common fixed point of the family of mappings without any extra conditions. Finally, we give some applications of our main results in convex minimization and equilibrium problems and present numerical examples to illustrate and support them. [ABSTRACT FROM AUTHOR]

In this paper, we prove some weak and strong convergence results for generalized α -nonexpansive mappings using M 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]. [ABSTRACT FROM AUTHOR]

Representation formulas for faces and support functions of the values of maximal monotone operators are established in two cases: either the operators are defined on reflexive and locally uniformly convex real Banach spaces with locally uniformly convex duals, or their domains have nonempty interiors on real Banach spaces. Faces and support functions are characterized by the limit values of the minimal-norm selections of maximal monotone operators in the first case while in the second case they are represented by the limit values of any selection of maximal monotone operators. These obtained formulas are applied to study the structure of maximal monotone operators: the local unique determination from their minimal-norm selections, the local and global decompositions, and the unique determination on dense subsets of their domains. [ABSTRACT FROM AUTHOR]

In this paper, we study to approximate solutions of quasi-equilibrium problems in Banach spaces, which generalizes equilibrium problems and quasi-variational inequalities. In fact, we prove weak convergence of the sequence generated by the proximal point method to a solution of the quasi-equilibrium problem. Then we show that the generated sequence converges strongly to a solution of the quasi-equilibrium problem. [ABSTRACT FROM AUTHOR]

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In this paper, we present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward-Forward algorithm (PFFSA) and the Forward-Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field. [ABSTRACT FROM AUTHOR]

In this paper we propose a new iteration process, called the K * 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 * 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. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the split equality common fixed point problems of quasi-nonexpansive multi-valued mappings in the setting of Banach spaces. For solving this kind of problems, some new iterative algorithms are proposed. Under suitable conditions, some weak and strong convergence theorems for the sequences generated by the proposed algorithm are proved. The results presented in this paper are new which also improve and extend some recent results announced by some authors. [ABSTRACT FROM AUTHOR]

In this article, we prove several new ergodic, weak, and strong convergence theorems for solutions to the following general second-order difference equationwhereAis a maximal monotone operator in a real Hilbert spaceHand {cn} and {θn} are positive real sequences. We do not assumeA−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set ofAbeing nonempty. These theorems provide new approximation results for zeroes of monotone operators, as well as significantly unify and extend previously known results by assuming much weaker conditions on the coefficients {cn} and {θn}. [ABSTRACT FROM AUTHOR]

We provide a new two-step iteration scheme of mixed type for two asymptotically nonexpansive self mappings in the intermediate sense and two asymptotically nonexpansive non-self mappings in the intermediate sense and establish some strong and weak convergence theorems for mentioned scheme and mappings in uniformly convex Banach spaces. Our results extend and generalize the corresponding results of Chidume et al. [C. E. Chidume, E. U. Ofoedu, H. Zegeye, J. Math. Anal. Appl., 280 (2003), 364-374] and [C. E. Chidume, N. Shahzad, H. Zegeye, Numer. Funct. Anal. Optim., 25 (2004), 239-257], Guo et al. [W. Guo, W. Guo, Appl. Math. Lett., 24 (2011), 2181-2185] and [W. Guo, Y. J. Cho, W. Guo, Fixed Point Theory Appl., 2012 (2012), 15 pages], Saluja [G. S. Saluja, J. Indian Math. Soc. (N.S.), 81 (2014), 369-385], Schu [J. Schu, Bull. Austral. Math. Soc., 43 (1991), 153-159], Tan and Xu [K. K. Tan, H. K. Xu, J. Math. Anal. Appl., 178 (1993), 301-308], Wang [L. Wang, J. Math. Anal. Appl., 323 (2006), 550-557], Wei and Guo [S. I. Wei, W. Guo, Commun. Math. Res., 31 (2015), 149-160] and [S. Wei, W. Guo, J. Math. Study, 48 (2015), 256-264]. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study modified two-step iteration process for a pair of quasi-nonexpansive mapping and asymptotically quasi-nonexpansive mapping in the intermediate sense. We also establish some strong convergence theorems and a weak convergence theorem for this iteration scheme and mappings in the framework of real Banach spaces. The results presented in this paper extend, improve and generalize some previous results from the existing literature (see for example [1, 3, 5-7, 9-13]). [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove strong convergence theorem by a modified three step iterative process with errors for three nonexpansive mappings in the frame work of uniformly smooth Banach spaces. The main feature of this scheme is that its special cases can handle both strong convergence like Halpern type and weak convergence like Ishikawa type iteration schemes. Our result extend and generalize the result of S. H. Khan [10], Kim and Xu [12] and many other authors. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak and strong convergence of finite step iteration process with errors for two finite families of asymptotically quasi-nonexpansive mappings in the framework of uniformly convex Banach spaces. Also, we establish some strong convergence theorems using semi-compactness and condition (B) which are weaker than completely continuous condition and some weak convergence theorems using Opial's condition and Kadec-Klee property. The results established in this paper improve and extend some results in [1, 17] and many others from the existing literature. [ABSTRACT FROM AUTHOR]

Based on the results proposed by Li and Song [Modified proximal-point algorithm for maximal monotone operators in Banach spaces], J. Optim. Theory appl. 138 (2008) 45-64.], we modify and generate our new iterative scheme for solving generalized mixed equilibrium problems and finding zeroes of maximal monotone operators in a Banach space under the appropriate conditions. We also prove strong and weak convergence theorems of this proximal point algorithm and give an example with numerical test which corresponding to our main results. Furthermore, we also consider the convex minimization problem and the problem of finding a zero point of an α-inverse strongly monotone operator as its appplications. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a new class of mappings called asymptotically nonspreading mappings and establish weak and strong convergence theorems of the iterative sequences generated by these mappings in a real Banach space. We modify Halpern’s iterations for finding a fixed point of an asymptotically nonspreading mapping and provide an affirmative answer to an open problem posed by Kurokawa and Takahashi in their final remark of (Kurokawa and Takahashi in Nonlinear Anal. 73:1562-1568, 2010) for nonspreading mappings. Furthermore, we investigate the approximation of common fixed points of asymptotically nonspreading mappings and nonexpansive mappings and derive a strong convergence theorem by a new hybrid method for these mappings. Our results improve and generalize many known results in the current literature. MSC: 47H10, 37C25. [ABSTRACT FROM AUTHOR]

Let H be a real Hilbert space and let A : D(A) ⊂ H → H be a (possibly multivalued) maximal monotone operator. We are concerned with the difference equation Δun + cnAun+1 ∍ fn, n = 0, 1, ..., where (cn) ⊂ (0,+∞), ( fn) ⊂ H are p-periodic sequences for a positive integer p. We investigate the existence of periodic solutions to this equation as well as the weak or strong convergence of solutions to p-periodic solutions. The first result of this paper (Theorem 1) is a discrete analogue of the 1977 result by Baillon and Haraux (on the periodic forcing problem for the continuous counterpart of the above equation) and was essentially stated by Djafari Rouhani and Khatibzadeh in a recent paper [5]. Here we provide a simpler proof of this result that is based on old existing results due to Browder and Petryshyn [4] and Opial (see, e.g., [6], p.5). A strong convergence result is also given and some examples are discussed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study the weak and strong convergence of two-step implicit iteration process with errors to a common fixed point for a finite family of asymptotically nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), Zhou and Chang (2002) and Gu and Lu (2006). [ABSTRACT FROM AUTHOR]

In this paper, we establish some weak and strong convergence theorems of modified two-step iteration process for two asymptotically quasi-nonexpansive mappings in the intermediate sense to converge to common fixed points in the setting of real uniformly convex Banach spaces. The results presented in this paper extend, improve and generalize some previous results from the existing literature. [ABSTRACT FROM AUTHOR]

Abstract: We firstly study the inclusion problem of for an H-accretive operator introduced in [Yaping Fang, Nanjing Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004) 647–653] in Banach spaces and obtain strong convergence theorems and weak convergence theorems. Simultaneously, we analyze the relations between m-accretive operators and H-accretive operators and give some numerical examples to explain main results. The main results presented in this paper mainly extend and improve the results of [Tomas Dominguez Benavides, Genaro Lopez Acedo, Hong-Kun Xu, Iterative solutions for zeros of accretive operators, Math. Nachr. 248–249 (2003) 62–71] and [Hong-Kun Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643] from m-accretive operators to H-accretive operators. [Copyright &y& Elsevier]

The purpose of this paper is to establish weak and strong convergence theorems of three-step iterative algorithm with errors to approximate a common fixed point for non-Lipschitzian asymptotically quasi-nonexpansive mappings in the intermediate sense in the setup of Banach spaces. The results presented in this paper extend and improve the corresponding results of [1, 7, 8, 14, 15, 16, 17, 18, 21] and many others. [ABSTRACT FROM AUTHOR]

This paper deals with a method for approximating a solution of the following problem: find hierarchically a common fixed point of a finite family of nonself nonexpansive mappings with respect to a nonexpansive self mapping on a closed convex subset of a smooth and reflexive Banach space X, which admits a weakly sequentially continuous duality mapping. First, we prove a weak convergence theorem which extends and improves one recent result proved by Yao and Liou (see Inverse Problems 24 (), doi:). Secondly, when the self mapping is a contraction, we prove, under different restrictions on parameters, a strong convergence result which generalize some recent results in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss some properties of iterates generated by a strict pseudo-contraction or a finite family of strict pseudo-contractions in a real -uniformly smooth Banach space. We prove the weak convergence and strong convergence of pseudo-contractive mappings. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349], Acedo and Xu [G.L. Acedo, H.K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67 (2007) 2258–2271] and Zhou [H. Zhou, Convergence theorems for -strict pseudo-contractions in 2-uniformly smooth Banach spaces, Nonlinear Anal. 69 (2008) 3160–3173]. [Copyright &y& Elsevier]

The purpose of this paper is to introduce and study two hybrid proximal-point algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of solutions to the equation 0∈ Tx for a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space X. Strong and weak convergence results of these two hybrid proximal-point algorithms are established, respectively. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss some properties of iterates generated by a strict pseudo-contraction or a finite family of strict pseudo-contractions in a real 2-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 324 (2007) 336–349]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in . This result extends a recent result of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] both from nonexpansive mappings to -strict pseudo-contractions and from Hilbert spaces to 2-uniformly smooth Banach spaces. [Copyright &y& Elsevier]

In a real HILBERT space H, starting from an arbitrary initial point xo ϵ H, an iterative process is defined as follows: xn+1 = anxn + (1-an)Tfλn+1 yn, yn = bnxn + (1 - bn)Tgβn xn, n ≥ 0, where Tfλn+1 x= Tx - λn+1μff(Tx), (Tgβn x = Tx - βnμgg(Tx), (∀ x ϵ H), T : H → H a nonexpansive mapping with F(T) ≠ Ø ; and f (resp. g) : H → H an ηf (resp. ηg)-strongly monotone and kf (resp. kg)-Lipschitzian mapping, {an} ⊂ (0, 1), {λn} ⊂ (0, 1) and {βn} ⊂ [0, 1). Under some suitable conditions, several convergence results of the sequence {xn} are shown. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove some strong and weak convergence theorems using a modified iterative process for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space. This will improve and generalize the corresponding results in the existing literature. Finally, we will state that our theorems can be generalized to the case of finitely many mappings. [Copyright &y& Elsevier]

We introduce three-step iterative schemes with errors for two and three nonexpansive maps and establish weak and strong convergence theorems for these schemes. Mann-type and Ishikawa-type convergence results are included in the analysis of these new iteration schemes. The results presented in this paper substantially improve and extend the results due to [S.H. Khan, H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal. 8 (2005) 1295–1301], [N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005) 1031–1039], [W. Takahashi, T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5 (1995) 45–58], [K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993) 301–308] and [H.F. Senter, W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974) 375–380]. [Copyright &y& Elsevier]

A few weak and strong convergence theorems of the modified three-step iterative sequence with errors and the modified Ishikawa iterative sequence with errors for asymptotically non-expansive mappings in any non-empty closed convex subsets of uniformly convex Banach spaces are established. The results presented in this paper substantially extend the results due to Chang (2001), Osilike and Aniagbosor (2000), Rhoades (1994) and Schu (1991). [ABSTRACT FROM AUTHOR]

In this paper, we provide and study the concept of multi-valued generalized (α , β) -nonexpansive mappings, which is the multi-valued version of the recently developed generalized (α , β) -nonexpansive mappings. We establish some elementary properties and fixed point existence results for these mappings. Moreover, a multi-valued version of the M-iterative scheme is proposed for approximating fixed points of these mappings in the weak and strong senses. Using an example, we also show that M-iterative scheme converges faster as compared to many other schemes for this class of mappings. [ABSTRACT FROM AUTHOR]

In this paper, we establish some strong and weak convergence results of modified S-iteration process for two generalized asymptotically quasi-nonexpansive non-self mappings in the framework of Banach spaces. Our results extend and generalize the previous works from the current existing literature. [ABSTRACT FROM AUTHOR]