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Let,andbe real Hilbert spaces, letandbe two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common fixed-point problem:Section.Displaywhereandare two firmly quasi-nonexpansive operators with nonempty fixed-point setsand. Note that, by takingand, we recover the split common fixed-point problem originally introduced by Cesnor and Segal. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. The weak convergence result of algorithm is obtained and some applied nonlinear analysis examples are stated. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to obtain strong convergence result for approximation of solution to constrained convex minimization problem using a new iterative scheme in a real Hilbert space. Furthermore, we give numerical analysis of our iterative scheme. [ABSTRACT FROM AUTHOR]

Let A and B be nonempty subsets of a Banach space X and T : A → B be a non-self mapping. An approximate sequence of best proximity points for the mapping T is a sequence {xn} in A such that limn→∞ || xn − T xn || → dist(A, B). In the current paper, we survey the existence of approximate best proximity point sequences for single and multivalued non-self mappings in strictly convex Banach spaces. We also introduce a geometric notion on a nonempty and convex pair of subsets of a Banach space, called semi-Opial condition, and establish some new best proximity point theorems. [ABSTRACT FROM AUTHOR]

In this paper, new numerical algorithms are introduced for finding the solution of a variational inequality problem whose constraint set is the common elements of the set of fixed points of a demicontractive mapping and the set of solutions of an equilibrium problem for a monotone mapping in a real Hilbert space. The strong convergence of the iterates generated by these algorithms is obtained by combining a viscosity approximation method with an extragradient method. First, this is done when the basic iteration comes directly from the extragradient method, under a Lipschitz-type condition on the equilibrium function. Then, it is shown that this rather strong condition can be omitted when an Armijo-backtracking linesearch is incorporated into the extragradient iteration. The particular case of variational inequality problems is also examined. [ABSTRACT FROM AUTHOR]

In this paper, we propose hybrid implicit and explicit viscosity iterative algorithms for solving general hierarchical fixed-point problems for a countable family of non-expansive mappings in uniformly smooth Banach spaces. These hybrid viscosity algorithms are based on the well-known viscosity approximation method and hybrid steepest-descent method. We obtain some strong convergence theorems under suitable conditions. Our results extend, improve, supplement and develop the recent results in the literature. [ABSTRACT FROM PUBLISHER]

Many convex optimization problems in a Hilbert space ? can be written as the following variational inequality problemVIP (?,C): Findsuch thatfor allz ? C, whereC ? ? is closed convex and ?: ? ? ? is monotone. We consider a special case ofVIP (?,C), whereandUi: ? ? ? are quasi-nonexpansive operators having a common fixed point,i ? I: = {1, 2,…,m}. A standard method forVIP (?,C) is the projected gradient methoduk+1 = PC(uk ? ??uk) which generates sequences converging to a unique solution ofVIP (?,C) if ? is strongly monotone and Lipschitz continuous. Unfortunately, the method cannot be applied for, because, in general,PCucannot be computed explicitly,u? ?. Lions in 1977 and Bauschke in 1996 considered a special case of, where ? = Id ?a, for somea? ?,Uiare firmly nonexpansive or nonexpansive, respectively, and studied the convergence properties of the following method:uk+1 = Ui kuk ? ?k?Ui kuk, where ?k ? 0 andis a cyclic control, i.e.,ik = k(mod m) +1 for allk ? 0 (see [1, 22]). We apply this method in case ? is strongly monotone and Lipschitz continuous,Uiare quasi-nonexpansive andis almost-cyclic. We present the method in a more general formwhereTk: ? ? ?,k ? 0, are quasi-nonexpansive,and Fix Tkapproximate Fix Tin some sense. A special case of the method withTk = Tfor allk ? 0 was studied in Yamada and Ogura [32] and by Yamada in [31], (in the latter paper,Twas supposed to be nonexpansive). We give sufficient conditions for the convergence of (1) as well as present examples of methods which satisfy these conditions. [ABSTRACT FROM AUTHOR]

In this paper, we extend Darbo's fixed point theorem in Banach space via the concept of the class of operators O(f;.) and obtain a tripled fixed point theorem. The main tool applied in our investigation is the technique of measures of noncompactness. Finally, as an application of our obtained results, we analyze the existence of solutions for a system of integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we study an operator generated by a finite family of generalized nonexpansive mappings in a Banach space. We first prove that the set of fixed points of this operator is identical to the set of all common fixed points of the mappings. Next, using this operator, we construct an iterative sequence to approximate a common fixed point of the family of generalized nonexpansive mappings. We finally apply our results to solve the feasibility problem in Banach spaces. [ABSTRACT FROM AUTHOR]

Let E be a real 2-uniformly convex Banach space with topological dual E ∗ . We prove weak convergence of the forward-reflected-backward splitting method to a solution of inclusion of sum of two monotone operators. Moreover, we give a variant of the method in which the step size does not depend on the Lipschitz constant of one of the operators. Finally, our results extend and complement several existing results in the literature. We also give some numerical illustrations of the proposed method in comparison with other method in the literature to further demonstrate the applicability and efficiency of our method. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to introduce an iterative scheme for solving multiple-set split feasiblity problems inp-uniformly convex Banach spaces which are also uniformly smooth using Bregman distance techniques. We further obtain a strong convergence result for approximating solutions of multiple-set split feasiblity problems in the framework ofp-uniformly convex Banach spaces which are also uniformly smooth. [ABSTRACT FROM AUTHOR]

We introduce new self-adaptive algorithms for solving the split common zero point problem with multiple output sets in Hilbert space. We also apply our main results to solving split feasibility problems with multiple output sets. [ABSTRACT FROM AUTHOR]

In this article, we pursue two goals. First, a new iterative scheme based on the resolvent operator method for finding a common element of the set of solutions of a generalized variational inclusion problem and the set of fixed points of a total asymptotically nonexpansive mapping in a real Banach space is constructed. Under some parameters controlling conditions, the strong convergence of the sequence generated by our proposed iterative algorithm to a common element of the above-mentioned two sets is proved. Our second purpose is to investigate and analyze the concept of H(.,.)-accretive operator that appeared in the literature and to point out some comments concerning it. Several new examples are also provided. [ABSTRACT FROM AUTHOR]

We consider a new type of cyclic (noncyclic) mappings, called diametrically relatively nonexpansive maps which contains properly the class of cyclic (noncyclic) relatively nonexpansive mappings. For such mappings we obtain existence results of best proximity points (pairs) in the framework of Busemann convex spaces and generalize the recent conclusions in this direction. We also present a characterization of proximal normal structure in term of best proximity points (pairs) for diametrically relatively nonexpansive mappings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the random Jungck-Mann-type and the random Jungck-Ishikawa-type iterative processes. We prove some convergence and stability results for these random iterative processes for certain random operators. Furthermore, we apply our results to study random non-linear integral equation of the Hammerstein type. Our results generalize, extend and unify several well-known deterministic results in the literature. Moreover, our results generalize recent results of Okeke and Abbas and Okeke and Kim. [ABSTRACT FROM AUTHOR]

In order to discover the minimum-norm solution of the pseudomonotone variational inequality problem in a real Hilbert space, we provide two variants of the inertial extragradient approach with a novel generalized adaptive step size. Two of the suggested algorithms make use of the projection and contraction methods. We demonstrate several strong convergence findings without requiring the prior knowledge of the Lipschitz constant of the mapping. Finally, we give a number of numerical examples that highlight the benefits and effectiveness of the suggested algorithms and how they may be used to solve the optimal control problem. [ABSTRACT FROM AUTHOR]

We study the split monotone variational inclusion problem in two real Hilbert spaces. Combining the inertial and relaxation techniques with the proximal contraction algorithm, we propose two new methods for solving this problem without the usual co-coerciveness assumption on the associated operators. [ABSTRACT FROM AUTHOR]

We study the asymptotic behaviour of the trajectory of a nonautonomous evolution equation governed by a quasi-nonexpansive operator in Hilbert spaces. We prove the weak convergence of the trajectory to a fixed point of the operator by relying on Lyapunov analysis. Under a metric subregularity condition, we further derive a flexible global exponential-type rate for the distance of the trajectory to the set of fixed points. The results obtained are applied to analyse the asymptotic behaviour of the trajectory of an adaptive Douglas–Rachford dynamical system, which is applied for finding a zero of the sum of two operators, one of which is strongly monotone while the other one is weakly monotone. [ABSTRACT FROM AUTHOR]

Using a product space approach, we propose and study several iterative methods for solving certain types of split equality problems in Hilbert spaces. [ABSTRACT FROM AUTHOR]

Let the VI indicate a variational inclusion, the CFPP denote a common fixed point problem of countably many nonexpansive mappings, and the SVI represent a system of variational inequalities. We introduce a composite viscosity implicit method for solving the VI and CFPP with the SVI constraint in the framework of uniformly convex and q-uniformly smooth Banach space where 1 < q ≤ 2. Moreover, we prove the strong convergence of the sequences generated by the proposed implicit method to a solution of a certain hierarchical variational inequality (HVI). In addition, our results are also applied for solving the fixed point problem (FPP) of nonexpansive mapping, variational inequality problem, convex minimization problem and split feasibility problem in Hilbert spaces. [ABSTRACT FROM AUTHOR]

Recently, the author, together with L. Leuştean and A. Nicolae, introduced the notion of jointly firmly nonexpansive families of mappings in order to investigate in an abstract manner the convergence of proximal methods. Here, we further the study of this concept, by giving a characterization in terms of the classical resolvent identity, by improving on the rate of convergence previously obtained for the uniform case, and by giving a treatment of the asymptotic behaviour at infinity of such families. [ABSTRACT FROM AUTHOR]

We establish several necessary and sufficient conditions for the existence and uniqueness of Hermitian positive definite (HPD) solutions to the general matrix equation X p = A + ∑ i = 1 m M i ∗ (B + X − 1 ) − 1 M i , where p, m are positive integers, Mi (i = 1, 2, ..., m) are n × n nonsingular complex matrices, A and B are HPD matrices, and then give three algorithms to get the unique solution. Moreover, we give two numerical examples to illustrate the effectiveness of the theoretical results and the behavior of the considered methods. [ABSTRACT FROM AUTHOR]

In this article, we present an inertial subgradient extragradient-type method that uses a non-monotone step size rule to find a numerical solution to equilibrium problems in real Hilbert spaces. The presented iterative scheme is based on an extragradient subgradient method and an inertial-type scheme. In fact, the proposed iterative scheme is effective in terms of performance, and the key advantage derives directly from the use of the variable step size rule, which is revised by each iteration on the basis of the Lipschitz-type constants as well as certain prior iterations. We obtain a weak convergence theorem for a new method by using mild conditions on a bifunction. Applications of the main results are given to solve various nonlinear problems. Several numerical findings are given in order to illustrate the numerical behaviour of the proposed method and to compare it to others. [ABSTRACT FROM AUTHOR]

A new inclusion problem is introduced using generalized Cayley operator and we call it Cayley inclusion problem. We also study its corresponding resolvent equation problem. By using a generalized resolvent operator and generalized Yosida approximation operator, first we establish a fixed point formulation for Cayley inclusion problem. An algorithm is defined to find the solution of Cayley inclusion problem. An existence and convergence result is proved. Secondly, we have shown the equivalence of Cayley inclusion problem with a resolvent equation. We define an iterative algorithm with some of its equivalent forms for solving resolvent equation problem. A numerical example is constructed and a convergence graph is shown by using MATLAB program. [ABSTRACT FROM AUTHOR]

We present several fixed point and convergence results for nonexpansive set-valued mappings which map a closed subset of a complete metric space into the space. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present a new iterative technique for approximating solutions of split generalized equilibrium problem and common fixed points of multivalued demicontractive mappings satisfying the gate conditions in real Hilbert spaces. Unlike the earlier results in this direction, we obtain a strong convergence result using an Armijo line search rule for determining the best appropriate step size for the next iteration. Also, our algorithm is designed in such a way that it does not require a projection onto the feasible set. We further give an analysis of the convergence rate of our method which is shown to be O (1 / t). Finally, the performances and comparisons with some existing methods are presented through numerical experiments. This result extends, generalizes and improves many of the existing related results in a unified way. [ABSTRACT FROM AUTHOR]

In this work, we introduce a modified extragradient method for solving the fixed point problem of a nonexpansive mapping and the variational inclusion problem for two accretive operators in the framework of Banach spaces. We then prove its strong convergence under certain assumptions imposed on the parameters. As applications, we apply our main result to the variational inequality problem, split feasibility problem and the LASSO problem. [ABSTRACT FROM AUTHOR]

We study perturbations of a hybrid steepest descent method for locating common fixed points of an arbitrary pool { T λ } of nonexpansive mappings. The difficulty of handling a possibly uncountable family is settled down to dealing with a countable family of auxiliary mappings { S n } associated to { T λ } , in the sense that the approximate fixed points of { S n } provide the common fixed points of { T λ }. Algorithms with strong convergence for solving the associated variational inequality problems are presented. Applications to convex minimization problems and convex feasibility problems are provided, together with numerical examples for comparisons of our algorithms with the existing ones. [ABSTRACT FROM AUTHOR]

We study the split common fixed point problem with multiple output sets in Hilbert spaces. In order to solve this problem, we propose two new algorithms and establish two strong convergence theorems for both of them. Moreover, using these methods, we also remove the assumptions imposed on the norm of the transfer operators. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate a descent-type iterative algorithm for finding a common element of fixed-point sets of a finite family of nonexpansive mappings and zero-point sets of pseudomonotone mappings in Hilbert spaces. With suitable assumptions, we derive necessary and sufficient conditions for the strong convergence of the algorithm. Numerical examples and applications to signal processing problems are provided to support the main results. [ABSTRACT FROM AUTHOR]

Best proximity results for cyclic and noncyclic F G -contractive operators in (strictly convex) Banach spaces are obtained via the measure of noncompactness. The work extends some of recent investigations in this direction. The obtained results are applied to investigation of optimum solutions for a system of second order differential equations with two initial conditions followed by an example. [ABSTRACT FROM AUTHOR]

Using proof-theoretical techniques, we analyze a proof by Hong-Kun Xu regarding a result of strong convergence for the Halpern type proximal point algorithm. We obtain a rate of metastability (in the sense of Terence Tao) and also a rate of asymptotic regularity for the iteration. Furthermore, our final quantitative result bypasses the sequential weak compactness argument present in the original proof. This elimination is reflected in the extraction of primitive recursive quantitative information. This work follows from recent results in Proof Mining regarding the removal of sequential weak compactness arguments. [ABSTRACT FROM AUTHOR]

For a certain class of mean nonexpansive self-mappings of complete metric spaces, we show that all the iterates of each mapping in this class converge, uniformly on bounded subsets, to its unique fixed point. [ABSTRACT FROM AUTHOR]

We study the multiple-set split common null point problem in two Hilbert spaces. In order to solve this problem, we propose two new parallel algorithms and establish strong convergence theorems for both of them. Our schemes combine the hybrid and shrinking projection methods with the proximal point algorithm. [ABSTRACT FROM AUTHOR]

The influence of errors on the convergence of infinite products of weak quasi-contraction mappings in b-metric spaces is explored. An example demonstrating the necessity of convergence of the sequence of computational errors to zero is also provided. Moreover, we discuss weak ergodic theorems in the setting of b-metric spaces. [ABSTRACT FROM AUTHOR]

In this article, we combine the viscosity approximation method and proximal operator to propose modified proximal gradient methods for solving the unconstrained convex optimization problems. The bounded perturbation resilience of these methods is investigated in the general Hilbert space H. Under reasonable parameter conditions, we prove that our algorithms strongly converge to the unique solution of a variational inequality problem. Our results improve and extend the corresponding results reported by many authors recently. [ABSTRACT FROM AUTHOR]

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set S. We assume that S = C ∩ A − 1 (Q) is the nonempty solution set of a (multiple-set) split convex feasibility problem, where C and Q are both closed and convex subsets of two real Hilbert spaces H 1 and H 2 , respectively, and the operator A acting between them is linear. We consider a modification of the gradient projection method the main idea of which is to replace at each step the metric projection onto S by another metric projection onto a half-space which contains S. We propose three variants of a method for constructing the above-mentioned half-spaces by employing the multiple-set and the split structure of the set S. For the split part we make use of the Landweber transform. [ABSTRACT FROM AUTHOR]

We propose and study new iterative methods for solving the generalized split common null point problem in Hilbert spaces. [ABSTRACT FROM AUTHOR]

We introduce three new parallel algorithms for solving the split common fixed point problem in Hilbert spaces. Those iterative methods for solving split common fixed point problems which involve step sizes that depend on the norm of a given bounded linear operator are often not easy to implement because one has to compute the norm of this operator. Therefore, in addition to two such algorithms, we also propose a new iterative method involving step sizes which are selected in such a way that the implementation of the method does not require the computation or estimation of the norm of the given operator. Several corollaries of our main results regarding the solution of multiple-set split feasibility, split common null point, split variational inequality and split mixed equilibrium problems are also presented. [ABSTRACT FROM AUTHOR]

Recently, Chang et al. (2015) constructed an algorithm that converges weakly to the solution of the split equality fixed point problem for quasi-pseudo-contractive mappings under some suitable conditions. They also showed that strong convergence is obtained in the case when the quasi-pseudo-contractive mappings are semi-compact. In this article, we construct an algorithm for quasi-pseudo-contractive mappings that always converge strongly to some solution of the split equality fixed point problem under mild conditions. We mention that we do not require the quasi-pseudo-contractive mappings to be semi-compact to obtain strong convergence. The algorithm does not require any prior knowledge of operator norms. The result of this article provides a unified framework for this type of problems. [ABSTRACT FROM AUTHOR]

Accretive and nonexpansive operators are investigated based on two iterative algorithms. Strong and weak convergence theorems of common solutions are established in the framework of uniformly convex and q-uniformly smooth Banach spaces. [ABSTRACT FROM AUTHOR]

We propose two modified Tseng's extragradient methods (also known as Forward–Backward–Forward methods) for solving non-Lipschitzian and pseudo-monotone variational inequalities in real Hilbert spaces. Under mild and standard conditions, we obtain the weak and strong convergence of the proposed methods. Numerical examples for illustrating the behaviour of the proposed methods are also presented [ABSTRACT FROM AUTHOR]

A new𝒮-generated Ishikawa iteration with errors is proposed for a pair of quasi-nonexpansive mapping and uniformlyL-Lipschitzian asymptotically pseudo-contractive mapping in real Banach spaces. We show that the proposed iterative scheme converges strongly to a common solution of quasi-nonexpansive mapping and uniformlyL-Lipschitzian asymptotically pseudo-contractive mapping in real Banach spaces. A comparison table is prepared using a numeric example which shows that the proposed iterative algorithm is faster than some known iterative algorithms. [ABSTRACT FROM AUTHOR]

This article uses the shrinking projection method introduced by Takahashi, Kubota and Takeuchi to propose an iteration algorithm for a countable family of Bregman multi-valued quasi-nonexpansive mappings in order to have the strong convergence under a limit condition in the framework of reflexive Banach spaces. We apply our results to a zero point problem of maximal monotone mappings and equilibrium problems in reflexive Banach spaces. The results presented in the article improve and extend the corresponding results of that found in the literature. [ABSTRACT FROM AUTHOR]

In this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani. [ABSTRACT FROM AUTHOR]

In this article, we provide a general iterative method for solving an equilibrium and a constrained convex minimization problem. By using the idea of regularized gradient-projection algorithm (RGPA), we find a common element, which is also a solution of a variational inequality problem. Then the strong convergence theorems are obtained under suitable conditions. [ABSTRACT FROM AUTHOR]

We study the convergence properties of an iterative method for a variational inequality defined on a solution set of the split common fixed point problem. The method involves Landweber-type operators related to the problem as well as their extrapolations in an almost cyclic way. The evaluation of these extrapolations does not require prior knowledge of the matrix norm. We prove the strong convergence under the assumption that the operators employed in the method are approximately shrinking. [ABSTRACT FROM PUBLISHER]

We define uniform paracontraction for an arbitrary set ofmatrices and show that an infinite product of matrices drawn from a uniformly paracontracting set is convergent. Moreover, if the uniformly paracontracting set is finite and the matrices are drawn in a regulated way, the infinite product is exponentially convergent. [ABSTRACT FROM PUBLISHER]

We introduce a new hybrid projection method in mathematical programming for finding a common element of the set of common fixed points of a countable family of relatively quasi-nonexpansive mappings, the set of the variational inequality for an-inverse-strongly monotone operator, the set of solutions of the mixed equilibrium problem and a zero of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Furthermore, we give some numerical examples which support our main theorem in the last part. [ABSTRACT FROM AUTHOR]

In this note, by means of the technique of measures of weak noncom- pactness, we establish a generalized form of fixed point theorem for the sum ofT+Sin weak topology setups of a metrizable locally convex space, whereSis not weakly compact,I−Tallows to be noninvertible, andTis not necessarily continuous. The obtained results unify and significantly extend a lot of previously known extensions of Krasnoselskii fixed-point theorems. The analysis presented here reveals the essential characteristics of the Krasnoselskii type fixed-point theorem in weak topology settings. [ABSTRACT FROM AUTHOR]

We introduce an iterative process which converges strongly to a common minimum-norm solution of a variational inequality problem for an-inverse strongly monotone mapping and a fixed point of relatively non-expansive mapping in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of non-linear operators. [ABSTRACT FROM AUTHOR]