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In this paper, using the modified hybrid Picard-Mann iteration process, we establish Δ-convergence and strong convergence theorems for total asymptotically nonexpansive mappings on a $CAT(0)$ space. Results established in the paper extend and improve a number of results in the literature. A numerical example is also given to examine the fastness of the proposed iteration process under different control conditions and initial points. [ABSTRACT FROM AUTHOR]

In this paper, we prove strong and weak convergence theorems for a mapping defined on a bounded, closed and convex subset of a uniformly convex Banach space, satisfying the RCSC condition. This condition was introduced by Karapınar (Dynamical Systems and Methods, 2012). We first establish the demiclosed principle for the mapping satisfying the RCSC condition. Then, using this principle, we establish the weak and strong convergence theorems. Results in the paper extend and improve a number of important results in this literature such as Khan and Suzuki (Nonlinear Anal. 80:211-215, 2013) and Reich (J. Math. Anal. Appl. 67:274-276, 1979). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new class of probabilistic normed spaces and to study fixed point problems in this class of probabilistic normed spaces. This paper includes the following two contents: (1) The definition of a new class of probabilistic normed spaces, the so-called S-probabilistic normed spaces, is given. In order to study the fixed point problems, some relevant properties of S-probabilistic normed spaces are discussed and some basic useful results are obtained; (2) The notion of probabilistic weak convergence is firstly presented in this paper. Therefore the probabilistic weak and strong convergence theorems of fixed points for nonexpansive mappings, asymptotically nonexpansive mappings and strongly pseudocontractive mappings are also proved by using the new methods and techniques. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iteration algorithm for finding a split common fixed point of an asymptotically nonexpansive mapping in the frameworks of two real Banach spaces. Under some suitable conditions imposed on the sequences of parameters, some strong convergence theorems are proved, which also solve some variational inequalities that are closely related to optimization problems. The results here generalize and improve the main results of other authors. [ABSTRACT FROM AUTHOR]

In this paper, we consider the class of monotone ρ-nonexpansive semigroups and give existence and convergence results for common fixed points. First, we prove that the set of common fixed points is nonempty in uniformly convex modular spaces and modular spaces. Then we introduce an iteration algorithm to approximate a common fixed point for the same class of semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iteration method for finding a split common fixed point of asymptotically nonexpansive semigroups in the setting of two Banach spaces, and we obtain some weak and strong convergence theorems of the iteration scheme proposed. The results presented in the paper are new and improve and extend some recent corresponding results. [ABSTRACT FROM AUTHOR]

In this paper, based on some geometrical properties of CAT p (0) spaces, for p ≥ 2 , we obtain two fixed point results for monotone multivalued nonexpansive mappings and proximally monotone nonexpansive mappings. Which under some assumptions, reduce to coincide and generalize a fixed point result for monotone nonexpansive mappings. This work is a continuity of the previous work of Ran and Reurings, Nieto and Rodríguez-López done for monotone contraction mappings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of generalized nonexpansive mappings. Some new fixed point theorems for these mappings are obtained. [ABSTRACT FROM AUTHOR]

This paper considers the fixed point problem for a nonexpansive mapping on a real Hilbert space and proposes novel line search fixed point algorithms to accelerate the search. The termination conditions for the line search are based on the well-known Wolfe conditions that are used to ensure the convergence and stability of unconstrained optimization algorithms. The directions to search for fixed points are generated by using the ideas of the steepest descent direction and conventional nonlinear conjugate gradient directions for unconstrained optimization. We perform convergence as well as convergence rate analyses on the algorithms for solving the fixed point problem under certain assumptions. The main contribution of this paper is to make a concrete response to an issue of constrained smooth convex optimization; that is, whether or not we can devise nonlinear conjugate gradient algorithms to solve constrained smooth convex optimization problems. We show that the proposed fixed point algorithms include ones with nonlinear conjugate gradient directions which can solve constrained smooth convex optimization problems. To illustrate the practicality of the algorithms, we apply them to concrete constrained smooth convex optimization problems, such as constrained quadratic programming problems and generalized convex feasibility problems, and numerically compare them with previous algorithms based on the Krasnosel'skiĭ-Mann fixed point algorithm. The results show that the proposed algorithms dramatically reduce the running time and iterations needed to find optimal solutions to the concrete optimization problems compared with the previous algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we prove Browder's convergence theorem for G-nonexpansive mappings in a Hilbert space with a directed graph. Moreover, we also prove strong convergence of the Halpern iteration process to a fixed point of G-nonexpansive mappings in a Hilbert space endowed with a directed graph. The main results obtained in this paper extend and generalize many well-known results in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present accelerations of the Mann and CQ algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the Mann and CQ algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated Mann algorithm by numerically comparing with the Mann algorithm. A numerical example is provided to illustrate that the acceleration of the CQ algorithm is ineffective. [ABSTRACT FROM AUTHOR]

Let the set $C \subset L_{1}([0,1])$ be nonempty, convex and compact for the convergence almost everywhere and $T: C \rightarrow C$ be a monotone nonexpansive mapping. In this paper, we study the behavior of the Krasnoselskii-Ishikawa iteration sequence $\{f_{n}\}$ defined by $f_{n+1} = \lambda f_{n} + (1-\lambda)T(f_{n})$, $n = 1, 2, \ldots$ , $\lambda\in(0,1)$. Then we prove a fixed point theorem for these mappings. Our result is new and was never investigated. [ABSTRACT FROM AUTHOR]

In this paper, we prove the weak and strong convergence of a sequence $\{x_{n}\}$ generated by the Ishikawa iteration to some common fixed points of two G-nonexpansive mappings defined on a Banach space endowed with a graph. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iterative algorithm to approximate fixed points of total asymptotically nonexpansive mappings in $\operatorname{CAT}(0)$ spaces. We also provide two examples to illustrate the convergence behavior of the proposed algorithm and numerically compare the convergence of the proposed iteration scheme with the existing schemes. [ABSTRACT FROM AUTHOR]

In this paper, we prove some existence theorems of fixed points of a monotone nonexpansive mapping T in a Banach space E with the partial order '≤', where a such mapping may be discontinuous. In particular, in finite dimensional spaces, such a mapping T has a fixed point in E if and only if the sequence $\{T^{n}0\}$ is bounded in E. In order to find a fixed point of such a mapping T, we prove the weak convergence of the Mann iteration scheme under the condition $\sum_{n=1}^{\infty}\beta_{n}(1-\beta_{n})=\infty$, which entails $\beta _{n}=\frac{1}{n+1}$ as a special case. [ABSTRACT FROM AUTHOR]

In this paper, a new modified proximal point algorithm involving fixed point iterates of asymptotically nonexpansive mappings in $\operatorname {CAT}(0)$ spaces is proposed and the existence of a sequence generated by our iterative process converging to a minimizer of a convex function and a common fixed point of asymptotically nonexpansive mappings is proved. [ABSTRACT FROM AUTHOR]

In this paper, using a multistep iterative scheme, we establish strong and Δ-convergence theorems for finite families of total asymptotically quasi-nonexpansive mappings in uniformly convex hyperbolic spaces. We then establish Δ- and polar convergence theorems for finite families of total asymptotically nonexpansive mappings in CAT(0) spaces. These new theorems are extensions, improvements, and generalizations of some recently announced results by many authors. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a weak convergence algorithm, called the linesearch algorithm, for solving a split equilibrium problem and nonexpansive mapping (SEPNM) in real Hilbert spaces, in which the first bifunction is pseudomonotone with respect to its solution set, the second bifunction is monotone, and fixed point mappings are nonexpansive. In this algorithm, we combine the extragradient method incorporated with the Armijo linesearch rule for solving equilibrium problems and the Mann method for finding a fixed point of an nonexpansive mapping. We then combine the proposed algorithm with hybrid cutting technique to get a strong convergence algorithm for SEPNM. Special cases of these algorithms are also given. [ABSTRACT FROM AUTHOR]

Let X be a Banach space or a complete hyperbolic metric space. Let C be a nonempty, bounded, closed, and convex subset of X and $T: C \rightarrow C$ be a monotone nonexpansive mapping. In this paper, we show that if X is a Banach space which is uniformly convex in every direction or a uniformly convex hyperbolic metric space, then T has a fixed point. This is the analog to Browder and Göhde's fixed point theorem for monotone nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In this paper we deal with stationary points (also known as endpoints) of nonexpansive set-valued mappings and show that the existence of such points under certain conditions follows as a consequence of the existence of approximate stationary sequences. In particular we provide abstract extensions of well-known fixed point theorems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new modified proximal point algorithm involving fixed point iterates of nonexpansive mappings in $\operatorname{CAT}(0)$ spaces and prove that the sequence generated by our iterative process converges to a minimizer of a convex function and a fixed point of mappings. [ABSTRACT FROM AUTHOR]

In this paper, we prove some strong and △-convergence theorems for a finite family of multivalued quasi-nonexpansive mappings satisfying condition $(E)$ in $\operatorname{CAT}(\kappa)$ spaces. Our results extend the corresponding results of Abkar and Eslamian (Nonlinear Anal. 75:1895-1903, 2012), Panyanak (Fixed Point Theory Appl. 2014:1, 2014), Shahzad and Zegeye (Nonlinear Anal. 71:838-844, 2009) and many others. [ABSTRACT FROM AUTHOR]

For the purpose of this paper, we use the method different from the relaxed extragradient method for finding a common element of the set of fixed points of a quasi-nonexpansive mapping, the set of solutions of equilibrium problems, and the set of solutions of a modified system of variational inequalities without demiclosed condition of W and $W_{\omega}:= (1-\omega )I+\omega W$, where W is a quasi-nonexpansive mapping and $\omega\in (0,\frac{1}{2} )$ in the framework of Hilbert space. By using our main result, we obtain a strong convergence theorem involving a finite family of nonspreading mappings and another corollary. Moreover, we give a numerical example to encourage our main theorem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iterative sequence for solving common problems which consist of split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems of the generated sequence $\{x_{n}\}$ by the shrinking projection method. Our results improve and extend the previous results given in the literature. [ABSTRACT FROM AUTHOR]

The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces are established. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. The results presented in this paper extend and improve the main results of Refs. (Moudafi in J. Math. Anal. Appl. 241:46-55, 2000; Xu et al. in Fixed Point Theory Appl. 2015:41, 2015). Moreover, applications to a more general system of variational inequalities, the constrained convex minimization problem and K-mapping are included. [ABSTRACT FROM AUTHOR]

Let $(X,\|\cdot\|)$ be a Banach space. Let C be a nonempty, bounded, closed, and convex subset of X and $T: C \rightarrow C$ be a monotone nonexpansive mapping. In this paper, it is shown that a technique of Mann which is defined by is fruitful in finding a fixed point of monotone nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In the paper, we introduce a new hybrid algorithm which is not based on the modification to weak convergence algorithms. The strong convergence theorem of the proposed algorithm is presented. Finally, the numerical experiments suggest that the new algorithm could be faster than Nakajo and Takahashi's algorithm in J. Math. Anal. Appl. 279:372-379, 2003. [ABSTRACT FROM AUTHOR]

In this paper, we establish coupled best proximity point theorems for multivalued mappings. Our results extend some recent results by Ali et al. (Abstr. Appl. Anal. 2014:181598, 2014) as well as other results in the literature. We also give examples to support our main results. [ABSTRACT FROM AUTHOR]

In this paper, a new modified Ishikawa iterative algorithm with errors by a shrinking projection method for generalized mixed equilibrium problems and a countable family of uniformly Bregman totally quasi- D-asymptotically nonexpansive mappings is introduced and investigated in the framework of a real Banach space. Strong convergence of the sequence generated by the proposed algorithm is derived under some suitable assumptions. These results are new and develop some recent results in this field. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to construct iterative methods for finding the fixed points of nonexpansive mappings. We present a modified semi-implicit midpoint rule with the viscosity technique. We prove that the suggested method converges strongly to a special fixed point of nonexpansive mappings under some different control conditions. Some applications are also included. [ABSTRACT FROM AUTHOR]

In this paper, we study strong and △-convergence for a newly defined two-step iteration process involving two asymptotically nonexpansive mappings in the intermediate sense which is wider than the class of asymptotically nonexpansive mappings in the setting of $\operatorname{CAT}(0)$ spaces. Our results generalize, unify and extend many known results from the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of fixed points of monotone nonexpansive mappings defined in Banach spaces endowed with a graph. This work is a continuity of the previous results of Ran and Reurings, Nieto et al., and Jachimsky done for contraction mappings defined in metric spaces endowed with a graph. [ABSTRACT FROM AUTHOR]

In this paper, we prove the demiclosed principle for total asymptotically nonexpansive nonself mappings in hyperbolic spaces. Then we obtain convergence theorems of the mixed Agarwal-O'Regan-Sahu type iteration for total asymptotically nonexpansive nonself mappings. Our results extend some results in the literature. MSC: 47H09, 49M05. [ABSTRACT FROM AUTHOR]

In this paper, we discuss sufficient and necessary conditions for the existence of best proximity points for non-self- α-nonexpansive mappings in Banach spaces. We obtain convergence results under some assumptions, and we prove the existence of common best proximity points for a family of non-self- α-nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate common solutions to a family of mixed equilibrium problems with a relaxed η- α-monotone mapping and a nonlinear operator equation involving an infinite family of asymptotically quasi- ϕ-nonexpansive mappings in the intermediate sense. Strong convergence theorems of common solutions are established in a strictly convex and uniformly smooth Banach space. These results extend many important recent ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we prove strong convergence theorems for Noor-type iteration schemes involving quasi-nonexpansive multivalued mappings in the framework of $\operatorname{CAT}(0)$ spaces. The results we obtain are generalizations of Panyanak (Nonlinear Anal. 70:1547-1556, 2009), Sastry and Babu (Czechoslov. Math. J. 55:817-826, 2005), Shazhad and Zegeye (Nonlinear Anal. 71:838-844, 2009), Song and Wang (Comput. Math. Appl. 55:2999-3002, 2008; Nonlinear Anal. 70:1547-1556, 2009) and many others in the sense of Noor-type iteration process in the setting of $\operatorname{CAT}(0)$ spaces. [ABSTRACT FROM AUTHOR]

In the paper, first, we introduce a new hybrid projection algorithm and present its strong convergence theorem. Next, we analyze different hybrid algorithms in computing and conclude that our proposed algorithm has an advantage. Finally, the numerical experiments validate the efficiency and advantages of the new algorithm. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex real Banach space E with dual space E∗. In this paper, a Krasnoselskii-type subgradient extragradient iterative algorithm is constructed and used to approximate a common element of solutions of variational inequality problems and fixed points of a countable family of relatively nonexpansive maps. The theorems proved are improvement of the results of Censor et al. (J. Optim. Theory Appl. 148:318-335, 2011). [ABSTRACT FROM AUTHOR]

The purpose of this work is to introduce and study an iterative method to approximate solutions of a hierarchical fixed point problem and a variational inequality problem involving a finite family of nonexpansive mappings on a real Hilbert space. Further, we prove that the sequence generated by the proposed iterative method converges to a solution of the hierarchical fixed point problem for a finite family of nonexpansive mappings which is the unique solution of the variational inequality problem. The results presented in this paper are the extension and generalization of some previously known results in this area. An example which satisfies all the conditions of the iterative method and the convergence result is given. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new iteration method, called an InerSP (an inertial S-iteration process), by combining a modified S-iteration process with the inertial extrapolation. This strategy is for speeding up the convergence of the algorithm. We then prove the convergence theorems of a sequence generated by our new method for finding a common fixed point of nonexpansive mappings in a Banach space. We also present numerical examples to illustrate that the acceleration of our algorithm is effective. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of generalized nonexpansive mappings which is wider than the class of mappings satisfying (C) condition. Different properties and some fixed point results for these mappings are obtained here. The convergence of some iteration schemes to the fixed point is also discussed with suitable examples. [ABSTRACT FROM AUTHOR]

In this paper, some strong and Δ-convergence results are proved for Suzuki generalized nonexpansive mappings in the setting of CAT(0) spaces using the K iteration process. We also give an example to show the efficiency of the K iteration process. Our results are the extension, improvement and generalization of many well-known results in the literature of fixed point theory in CAT(0) spaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose two strongly convergent algorithms which combines diagonal subgradient method, projection method and proximal method to solve split equilibrium problems and split common fixed point problems of nonexpansive mappings in a real Hilbert space: fixed point set constrained split equilibrium problems (FPSCSEPs) in real Hilbert spaces. The computations of first algorthim requires prior knowledge of operator norm. To estimate the norm of an operator is not always easy, and if it is not easy to estimate the norm of an operator, we purpose another iterative algorithm with a way of selecting the step-sizes such that the implementation of the algorithm does not need any prior information as regards the operator norm. The strong convergence properties of the algorithms are established under mild assumptions on equilibrium bifunctions. We also report some applications and numerical results to compare and illustrate the convergence of the proposed algorithms. [ABSTRACT FROM AUTHOR]

An inertial iterative algorithm for approximating a point in the set of zeros of a maximal monotone operator which is also a common fixed point of a countable family of relatively nonexpansive operators is studied. Strong convergence theorem is proved in a uniformly convex and uniformly smooth real Banach space. This theorem extends, generalizes and complements several recent important results. Furthermore, the theorem is applied to convex optimization problems and to J-fixed point problems. Finally, some numerical examples are presented to show the effect of the inertial term in the convergence of the sequence of the algorithm. [ABSTRACT FROM AUTHOR]

Let E be a real Banach space with dual space E ∗ . A new class of relatively weakJ-nonexpansive maps, T : E → E ∗ , is introduced and studied. An algorithm to approximate a common element of J-fixed points for a countable family of relatively weak J-nonexpansive maps and zeros of a countable family of inverse strongly monotone maps in a 2-uniformly convex and uniformly smooth real Banach space is constructed. Furthermore, assuming existence, the sequence of the algorithm is proved to converge strongly. Finally, a numerical example is given to illustrate the convergence of the sequence generated by the algorithm. [ABSTRACT FROM AUTHOR]

We consider a new type of monotone nonexpansive mappings in an ordered Banach space X with partial order ⪯. This new class of nonlinear mappings properly contains nonexpansive, firmly-nonexpansive and Suzuki-type generalized nonexpansive mappings and partially extends α-nonexpansive mappings. We obtain some existence theorems and weak and strong convergence theorems for the Mann iteration. Some useful examples are presented to illustrate the facts. [ABSTRACT FROM AUTHOR]

We discuss Panayanak's results on the existence of an endpoint of a multivalued nonexpansive mapping. We show that all of his results can be extended and some can be established in a wider class of mappings. Out of his three open questions, two of them are solved in affirmative. [ABSTRACT FROM AUTHOR]

We establish Δ-convergence results of a sequence generated by the modified S-iteration process for two nonexpansive mappings in complete $\operatorname {CAT}(\kappa)$ spaces. Some numerical examples are also provided to compare with the Ishikawa-type iteration process. Our main result extends the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

We first present the concepts of proximal contraction and proximal nonexpansive mappings on star-shaped sets in probabilistic Banach (Menger) spaces. We derive some results about the best proximity points for these mappings in probabilistic Banach (Menger) spaces. Next, we bring some examples that defend our main results. [ABSTRACT FROM AUTHOR]