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In this paper, we proposed two different methods for solving pseudomonotone and strongly pseudomonotone equilibrium problems. We can examine these methods as an extension and improvement of the Popov's extragradient method. We replaced the second minimization problem onto a closed convex set in the Popov's extragradient method, with a half-space minimization problem that is updated on each iteration and also formulates a useful method for determining the appropriate stepsize on each iteration. The weak convergence theorem of the first method and strong convergence theorem for the second method is well-established based on a standard assumption on a cost bifunction. We also consider various numerical examples to support our well-established convergence results, and we can see that the proposed methods depict a significant improvement in terms of the number of iterations and execution time. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and analyze two inertial algorithms with self-adaptive stepsizes for solving variational inequalities in reflexive Banach spaces. Our algorithms are based on inertial hybrid and shrinking projection methods. Knowledge of the Lipschitz constant of the cost operator is not required. Under appropriate conditions, the strong convergence of the algorithms is established. We also present several numerical experiments which bring out the efficiency and the advantages of the proposed algorithms. Our work provides extensions of many known results from Hilbert spaces to reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

A large number of nonlinear and optimization problems can be reduced to altering point problems. This paper aims to introduce the parallel normal S-iteration technique and study its convergence rates for solving such problems in infinite-dimensional Hilbert spaces under practical assumptions. We place particular emphasis on the parallel splitting method for the sum of two maximal monotone operators and that can apply for solving a class of convex composite minimization problems. Moreover, we present applications of our iterative methods to some nonlinear problems, such as a system of variational inequalities and a system of inclusion problems. Finally, to demonstrate the applicability of the altering point technique, the performances of our proposed parallel normal S-iteration methods are presented through numerical experiments in signal recovery problems. [ABSTRACT FROM AUTHOR]

Let,,be real Hilbert spaces, let,be two bounded linear operators. The multiple-set split equality common fixed-point problem (MSECFP) under consideration in this paper is to(Section.Display) whereare integers,andare quasi-nonexpansive mappings with nonempty fixed-point sets. Note that, the above problem (1) allows asymmetric and partial relations between the variablesand. Ifand, then the MSECFP (1) reduces to the multiple-set split common fixed-point problem proposed by Censor et al. In this paper, we introduce mixed cyclic and simultaneous iterative algorithms for the MSECFP (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. Weak convergence results are given. [ABSTRACT FROM AUTHOR]

In this paper, using a new shrinking projection method, we study the strong convergence theorem for finding a common point of the set of common fixed points of a finite family of Bregman demigeneralized mappings, common zero points of a finite family of Bregman inverse strongly monotone mappings and zero points of maximal monotone mapping in reflexive Banach space. Using this result, we get a new strong convergence theorem in Banach space. Our result extends and improves important recent results presented by many authors. [ABSTRACT FROM AUTHOR]

Halpern's iteration method, discovered by Halpern in 1967, is an iterative algorithm for finding fixed points of a nonexpansive mapping in Hilbert and Banach spaces. Since many optimization problems can be cast into fixed point problems of nonexpansive mappings, Halpern's method plays an important role in optimization methods. This paper discusses recent advances in convergence and rate of convergence results of Halpern's method, and applications in optimization problems, including variational inequalities, monotone inclusions, Douglas-Rachford splitting method, and minimax problems. [ABSTRACT FROM AUTHOR]

Let A and B be two nonempty subsets of a normed linear space X. A mapping T : A ∪ B → A ∪ B is said to be noncyclic if T (A) ⊆ A and T (B) ⊆ B. In the current paper, we consider the problem of finding the best proximity pair for the noncyclic mapping T, that is, two fixed points of T which achieve the minimum distance between the sets A and B. We do it from some different approaches. The common condition on these results is relatively nonexpansivity of the mapping T. At the first conclusion, we obtain the existence of best proximity pairs in the setting of uniformly convex in every direction Banach spaces where the pair (A, B) is nonconvex. Then we conclude a similar result by replacing the geometric property of Opial's property of the Banach space and adding another assumption on the mapping T, called condition (C). We also show that the same result is true when X is a 2-uniformly convex Banach space. In the setting of k-uniformly convex Banach spaces, we prove that every nonempty, and convex pair of subsets has a geometric notion of proximal normal structure and then, we deduce the existence of best proximity pairs for relatively nonexpansive mappings in such spaces. [ABSTRACT FROM AUTHOR]

In this paper, we show a convergence theorem to a common fixed point for finitely many mappings by the Halpern's iterative scheme using balanced mappings on an admissible complete CAT(1) space. [ABSTRACT FROM AUTHOR]

Let X be a uniformly smooth and 2-uniformly convex real Banach space with dual space X ∗ . In this paper, a Halpern-type subgradient extragradient algorithm is constructed. The sequence, generated by the algorithm, converges strongly to a common solution of variational inequality and two convex minimization problems. This result is obtained as an application of a Halpern-type subgradient extragradient algorithm, for approximating a common solution of variational inequality and J-fixed points of two continuous J-pseudocontractions. The theorem proved complements, improves and unifies many recent results in the literature. Finally, numerical experiments are given to illustrate the convergence of the sequence generated by the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, △ -convergence and strong convergence theorems for asymptotically quasi-nonexpansive sequences in Hadamard spaces are derived. The results extend and improve recent results in the literature and some of them seem new even in Hilbert spaces. Applications to convex minimization and common fixed point problems are discussed. [ABSTRACT FROM AUTHOR]

This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski–Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'ski–Mann iteration, with which the algorithm is proved to achieve the optimal convergence rate. Then we present an approximation to the optimal relaxation parameter sequence since the optimal relaxation parameter sequence involves fixed points of the operators and can't be used in actual computing. Thirdly, we apply our results to the operators splitting method, such as forward–backward splitting methods and Douglas–Rachford splitting method. Finally, numerical experiments are provided to illustrate that Krasnosel'ski–Mann iteration and the relaxed projected method with our proposed relaxation parameter sequence behave better than others. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find a common element in the intersection of the set of zeros of the inclusion problem of sum of two monotone mappings and the set of fixed points of a Bregman quasi nonexpansive mapping in a reflexive Banach space by using Bregman distance and shrinking projection method. Under suitable conditions, some strong convergence theorems are proved. As applications, we utilize our results to study the convex minimization problem, variational inequality problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new modified proximal point algorithm for a finite family of non-expansive mappings in the framework of CAT(0) spaces. We establish Δ-convergence and strong convergence theorems under some mild conditions. Our results extend some known results which appeared in the literature. [ABSTRACT FROM PUBLISHER]

In this paper, we consider a common solution of three problems in Hilbert spaces including the split generalized equilibrium problem, the variational inequality problem and fixed point problem. For finding the solution, we present a new iterative method and prove the strongly convergence theorem under mild conditions. Moreover, some numerical examples are given in the last section. [ABSTRACT FROM PUBLISHER]

The content of this paper is twofold. First, it aims to provide some new Newton-like methods for solving the root-finding problem in the complex plane. Moreover a convergence test for the resulted methods is phrased and proved. The pseudo-Newton method of Kalantari for finding the maximum modulus of complex polynomials arises as particular case of the newly proposed procedures. Secondly, a recently introduced Thakur iterative process is used in connection with the newly described methods. Its stability and data dependence is subject to analysis. Ultimately, an illustrative analysis regarding some modified Thakur iteration procedures, is obtained via polynomiographic techniques. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two inertial accelerated algorithms for solving the split common fixed-point problem of directed operators in real Hilbert space. The proposed iterative algorithms combine the primal-dual method and the inertial method with the self-adaptive stepsizes such that the implementation of our algorithms does not need any prior information about bounded linear operator norms. Under suitable conditions, the weak and strong convergence results of the algorithms are obtained. Numerical results which involve image restoration problems are reported to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the concept of the convex graphical rectangular b-metric space ( G R b C M S) by means of the convex structure. Moreover, the definitions of G-contraction mappings, G-quasi-contraction mappings, T-Mann sequences and TB-Mann sequences are presented in G R b C M S. Then we obtain strong convergence theorems for these mappings in G R b C M S under some suitable conditions. Furthermore, an example for supporting our main result is shown. [ABSTRACT FROM AUTHOR]

The variational inequality problem plays an important role in nonlinear analysis and optimization. It is a generalization of the nonlinear complementarity problem. For a variational inequality problem in a Hilbert space, the extragradient algorithm with inertial effects has been studied. For a variational inequality problem in a Banach space, Nakajo introduced Haugazeau's hybrid method and Liu introduced the Halpern subgradient extragradient method. In this paper, we construct a new inertial iterative method for solving variational inequality problems in Banach spaces based on the work we mentioned above. We propose a strong convergence theorem. As applications, our result can be used to solve constrained convex minimization problems. [ABSTRACT FROM AUTHOR]

A convergence rate of Mann iteration has been studied by many mathematicians. Recently, its rate was improved in a Hilbert space. In this paper, we consider a convergence rate in a geodesic space and generalize the known result. [ABSTRACT FROM AUTHOR]

The variable Lebesgue spaces are a generalization of the classical Lebesgue spaces, replacing the constant exponent p with a variable exponent function p (⋅). Beside their intrinsic interest, they are also very important for their applications to partial differential equations and variational integrals with non-standard growth conditions. In this paper, we study some properties related to the existence of a fixed point for several classes of mappings defined on these spaces, considering both the modular and its induced norm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new system of variational inclusion problems called system of generalized set-valued variational inclusion problems involving α β - H ((. ,.) , (. ,.)) -mixed accretive mapping in q-uniformly smooth Banach spaces. By means of proximal-point mapping method, we prove the existence of solution of this system of variational inclusions. Further, we propose an iterative algorithm for finding the approximate solution of the system. Furthermore, we prove that the sequences generated by the iterative algorithm converge strongly to a solution of the system. [ABSTRACT FROM AUTHOR]

In this paper, we introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set. [ABSTRACT FROM AUTHOR]

Kohlenbach and Leuştean have shown in 2010 that any asymptotically nonexpansive self-mapping of a bounded nonempty UCW-hyperbolic space has a fixed point. In this paper, we adapt a construction due to Moloney in order to provide a sequence that converges strongly to such a fixed point. [ABSTRACT FROM AUTHOR]

In this paper, we propose an over-relaxed proximal scaled gradient algorithm for solving the non-smooth composite optimization problem in Hilbert space. We prove the strong convergence and the bounded perturbation resilience of this algorithm under some mild conditions. We also list the superiorized version of the algorithm. We apply the algorithms to the l 1 − l 2 problem to illustrate the performance and validity. The numerical examples show that the over-relaxed algorithm can achieve fewer iteration steps and less running times in comparison to the under-relaxed algorithm, and that the superiorization algorithm has the minimum iterative steps among the three algorithms. [ABSTRACT FROM AUTHOR]

In recent years, the Forward-Backward algorithm (FBA) received much attention due to its various applications in image recovery, signal processing, and machine learning. In this paper, we consider the FBA in the setting of Banach spaces that are uniformly convex and q-uniformly smooth. We introduce two viscosity FBA, and one of them with weakly contractive mapping, which generalizes many previous results on the viscosity approximation method with fixed contraction. Moreover, we establish their strong convergences under more general conditions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mapping given by a finite family of multivalued monotone mappings in a CAT(0) space. We further propose a modified Halpern-type algorithm for the mapping and prove a strong convergence theorem for approximating a common solution of finite family of monotone inclusion problems in a complete CAT(0) space. We also applied our results to solve a finite family of minimization problems in a complete CAT(0) space. Some non-trivial examples of monotone mappings in complete CAT(0) space setting were also studied and a numerical example of our algorithm to further show its applicability is also presented. Our numerical experiment shows that our algorithm converges faster than that proposed in [42] by Takahashi and Shimoji, Mathematical and Computer Modelling32 (2000), 1463–1471. [ABSTRACT FROM AUTHOR]

In this paper, we study the triple hierarchical variational inequality problem (THVIP, for short) with the hierarchical variational inequality constraint for finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Based on Korpelevich's extragradient method, hybrid steepest-descent method, Mann's implicit iteration method and Halpern's iteration method, a general implicit hybrid iterative algorithm for solving this THVIP is proposed. Under mild conditions, the strong convergence of the iterative sequences generated by the algorithm is established. Our results improve and extend the corresponding results in the earlier and recent literature. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in solving minimization problem and common fixed point problem of finite family consisting asymptotically nonexpansive mappings in Hadamard spaces. For finding common solutions of their problems, we introduce a new modified proximal point algorithm involving convex combination technique. Under suitable conditions, Δ-convergence and strong convergence theorems are proved. Furthermore, we provide some numerical examples which show the efficiency and implementation of the proposed algorithm as well as show computational overview by comparing it with previously known algorithms. [ABSTRACT FROM AUTHOR]

In this paper, using the hybrid method defined by Nakajo and Takahashi, we first obtain a strong convergence theorem for two noncommutative generic skew generalized nonspreading mappings in a Banach space. Next, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove another strong convergence theorem for the mappings in a Banach space. Using these results, we get new strong convergence theorems by the hybrid method and the shrinking projection method in a Hilbert space and a Banach space. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new type of generalized mixed ordered quasi-variational inclusion problem in the setting of real ordered Hilbert spaces. Further, we propose a perturbed three-step iterative algorithm for solving generalized mixed ordered quasi-variational inclusion problem. By using the relaxed resolvent operator method with XOR technique, we prove the existence of solution of generalized mixed ordered quasi-variational inclusion problem and discuss the convergence analysis and stability of the proposed algorithm. The iterative algorithm and results presented in this paper generalize, improve and significantly refine many previously known results of this area. Moreover, we give a numerical example to illustrate and show the convergence of the proposed algorithm in support of our main result. [ABSTRACT FROM AUTHOR]

In this paper, an improvement of the extragradient algorithm is considered for finding a solution of pseudo-monotone equilibrium problems in real Hilbert spaces. We replace the second minimization problem onto a closed convex set in the extragradient method, with a subgradient projection onto some constructible half-space and propose a new method of getting the step-size. A convergence theorem for the generated sequence is proved and applied for finding a solution of variational inequality problem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce an iterative method for approximating a point in the set of zeros of the sum of two monotone mappings, which is also a solution of a fixed point problem for a Bregman strongly nonexpansive mapping in a real reflexive Banach space. With our iterative technique, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a variational inclusion problem for sum of two monotone mappings and the set of solutions of a fixed point problem for Bregman strongly nonexpansive mapping. We give applications of our result to convex minimization problem, convex feasibility problem, variational inequality problem, and equilibrium problem. Our result complements and extends some recent results in literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish strong and Δ-convergence theorems of the modified hybrid-CR three steps iteration with perturbations for total asymptotically non-expansive mapping in CAT (0) spaces. Our results improve and extend the corresponding results from the current literature. We also provide three examples to illustrate the convergence behaviour of the proposed algorithm and numerically compare the convergence of the proposed iteration scheme with the existing schemes. [ABSTRACT FROM AUTHOR]

Ranking fuzzy numbers is a key tool in decision making and other fuzzy analysis. Several strategies have been proposed for this task. However, none of them is universally accepted because counter-intuitive examples appear in all cases. Very recently, a novel methodology for ordering fuzzy numbers that satisfy several human-according properties has been introduced. This procedure overcomes some drawbacks of previous techniques and it is not based on a ranking index. In this paper, due to its feasible application in computation, we show how the novel fuzzy ranking can be particularly applied to trapezoidal (and triangular) fuzzy numbers by analysing all possible cases. Hence, new properties appear. Finally, a comparison (with other existing methods) study is carried out to show the reasonability of the obtained orderings and to illustrate the advantages of the proposed methodology. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new modified proximal point algorithm for nonexpansive mappings in non-positive curvature metric spaces and also we prove the sequence generated by the proposed algorithms converges to a common solution between minimization problem and fixed point problem. Moreover, we give some numerical examples to illustrate our main results, that is, our algorithm is more efficient than the algorithm of Cholamjiak et al. and others. [ABSTRACT FROM AUTHOR]

In this paper, a Halpern-type proximal point algorithm for approximating a common solution of monotone inclusion problem, minimization problem (MP) and fixed point problem is introduced. Using our algorithm, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a monotone inclusion problem, the set of solutions of an MP and the set of solutions of a fixed point problem for non-expansive mappings in complete CAT(0) spaces. [ABSTRACT FROM AUTHOR]

In this paper, we suggest two new iterative methods for finding an element of the solution set of split variational inclusion problem in real Hilbert spaces. Under suitable conditions, we present weak and strong convergence theorems for these methods. We also apply the proposed algorithms to study the split feasibility problem. Finally, we give some numerical results which show that our proposed algorithms are efficient and implementable from the numerical point of view. [ABSTRACT FROM AUTHOR]

In this paper we apply the Douglas–Rachford (DR) method to solve the problem of finding a point in the intersection of the interior of a closed convex cone and a closed convex set in an infinite-dimensional Hilbert space. For this purpose, we propose two variants of the DR method which can find a point in the intersection in a finite number of iterations. In order to analyse the finite termination of the methods, we use some properties of the metric projection and a result regarding the rate of convergence of fixed point iterations. As applications of the results, we propose the methods for solving the conic and semidefinite feasibility problems, which terminate at a solution in a finite number of iterations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a hybrid viscosity iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of common fixed points of one finite family of nonexpansive mappings, and another infinite family of nonexpansive mappings in a real Hilbert space. This hybrid viscosity iterative algorithm is based on viscosity approximation method, Mann’s iterative method, projection method, strongly positive bounded linear Operator, and-mapping approaches. We study the strong convergence of the proposed algorithm to a common element under appropriate assumptions, which also solves an optimization problem. The result presented in this paper improves and extends some known results in the literature. [ABSTRACT FROM AUTHOR]

Let K be a nonempty closed convex subset of a real Hilbert space H1. Let be N multivalued strictly pseudocontractive mappings. In this paper, we introduce a new hybrid subgradient method for finding a common point in the set of common fixed points of a finite family of multivalued strictly pseudocontractions, the solution set of class of pseumonotone equilibrium problem and solution set of split feasibility problem. Weak and strong convergence of the iterative sequence are established. Our result is generalization and improvement of several recent results. [ABSTRACT FROM AUTHOR]

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP's with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space. [ABSTRACT FROM AUTHOR]

Proximal splitting algorithms for monotone inclusions (and convex optimization problems) in Hilbert spaces share the common feature to guarantee for the generated sequences in general weak convergence to a solution. In order to achieve strong convergence, one usually needs to impose more restrictive properties for the involved operators, like strong monotonicity (respectively, strong convexity for optimization problems). In this paper, we propose a modified Krasnosel'skiĭ–Mann algorithm in connection with the determination of a fixed point of a nonexpansive mapping and show strong convergence of the iteratively generated sequence to the minimal norm solution of the problem. Relying on this, we derive a forward–backward and a Douglas–Rachford algorithm, both endowed with Tikhonov regularization terms, which generate iterates that strongly converge to the minimal norm solution of the set of zeros of the sum of two maximally monotone operators. Furthermore, we formulate strong convergent primal–dual algorithms of forward–backward and Douglas–Rachford-type for highly structured monotone inclusion problems involving parallel-sums and compositions with linear operators. The resulting iterative schemes are particularized to the solving of convex minimization problems. The theoretical results are illustrated by numerical experiments on the split feasibility problem in infinite dimensional spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present two new iterative algorithms to find the minimum norm fixed point of nonexpansive nonself-mappings in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding ones announced by Yao and Xu [Yao Y, Xu HK. Iterative methods for finding minimum norm fixed point of mappings with applications. Optimization. 2011;60:645–658] and many others. Some applications to convex minimization and split feast problems(SFP) are also included. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce a one-step iterative scheme for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mappings in a real uniformly convex Banach space. We establish weak and strong convergence theorems of the proposed iterative scheme under some control conditions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the weak and strong convergence of an implicit iteration process to a common fixed point for two finite families of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Xu and Ori, Numer. Funct. Anal. Optim. 2001; 22:767-773, Zhou and Chang, Numer. Fund. Anal. Optim. 2002; 23:911-921, Chidume and Shahzad, Nonlinear. Anal. 2005; 62: 1149-1156. [ABSTRACT FROM AUTHOR]

The aim of this paper is to generalize the results of Yamada et al. [Yamada, I., Ogura, N., Yamashita, Y., Sakaniwa, K. (1998). Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces. Numer. Funct. Anal. Optimiz. 19(l):165-190], and to provide complementary results to those of Deutsch and Yamada [Deutsch, F., Yamada, I. (1998). Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19(1&2):33-56] in which they consider the minimization of some function d over a closed convex set F, the nonempty intersection of N fixed point sets. We start by considering a quadratic function θ and providing a relaxation of conditions of Theorem 1 of Yamada et al. (1998) to obtain a sequence of fixed points of certain contraction maps, converging to the unique minimizer of θ over F. We then extend Theorem 2 and obtain a complementary result to Theorem 3 of Yamada et al. (1998) by replacing the condition lim n → ∞ (λn - λn+1)/λ²n+1 = 0 on the parameters by the more general condition lim n → ∞ λn / λn+1 = 1. We next look at minimizing a more general function θ than a quadratic function which was proposed by Deutsch and Yamada (1998) and show that the sequence of fixed points of certain maps converge to the unique minimizer of 9 over F. Finally, we prove a complementary result to that of Deutsch and Yamada (1998) by using the alternate condition on the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we consider, in a finite dimensional real Hilbert space , the variational inequality problem VIP : find , where is nonexpansive mapping with bounded and is paramonotone and Lipschitzian over . The nonstrictly convex minimization over the bounded fixed point set of a nonexpansive mapping is a typical example of such a variational inequality problem. We show that the hybrid steepest descent method, of which convergence properties were examined in some cases for example (Yamada, I. (2000). Convex projection algorithm from POCS to Hybrid steepest descent method. The Journal of the IEICE (in Japanese) 83:616–623; Yamada, I. (2001). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S., eds. Inherently Parallel Algorithm for Feasibility and Optimization. Elsevier; Ogura, N., Yamada, I. (2002). Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping. Numer. Funct. Anal. Optim. 23:113–137), is still applicable to the case where and T satisfy the above conditions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the generalized-D-gap function for an equilibrium problem. We give sufficient conditions under which a critical point of the generalized-D-gap function is a solution of the equilibrium problem, and develop a descent scheme for locating these critical points. Using the generalized-D-gap function, we construct an error bound for the equilibrium problem. We support our approach with concrete examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the split common null point problem with resolvents of maximal monotone operators in Banach spaces. Then, using the shrinking projection method, we prove a strong convergence theorem for finding a solution of the split common null point problem in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we present an iterative scheme for Bregman strongly nonexpansive mappings in the framework of Banach spaces. Furthermore, we prove the strong convergence theorem for finding common fixed points with the set of solutions of an equilibrium problem. [ABSTRACT FROM AUTHOR]