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The paper introduces and analyzes the convergence of a new iterative algorithm for approximating solutions of equilibrium problems involving strongly pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. The algorithm uses a stepsize sequence which is non-increasing, diminishing, and non-summable. This leads to the main advantage of the algorithm, namely that the construction of solution approximations and the proof of its convergence are done without the prior knowledge of the modulus of strong pseudomonotonicity and Lipschitz-type constants of bifunctions. The strongly convergent theorem is established under suitable assumptions. The paper also discusses the assumptions used in the formulation of the convergent theorem. Several numerical results are reported to illustrate the behavior of the algorithm with different sequences of stepsizes and also to compare it with others. [ABSTRACT FROM AUTHOR]

The paper proposes a new extragradient algorithm for solving strongly pseudomonotone equilibrium problems which satisfy a Lipschitz-type condition recently introduced by Mastroeni in auxiliary problem principle. The main novelty of the paper is that the algorithm generates the strongly convergent sequences in Hilbert spaces without the prior knowledge of Lipschitz-type constants and any hybrid method. Several numerical experiments on a test problem are also presented to illustrate the convergence of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the varying stepsize gradient projection algorithm (GPA) for solving the split equality problem (SEP) in Hilbert spaces, and study its linear convergence. In particular, we introduce a notion of bounded linear regularity property for the SEP, and use it to establish the linear convergence property for the varying stepsize GPA. We provide some mild sufficient conditions to ensure the bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize GPA. To the best of our knowledge, this is the first work to study the linear convergence for the SEP. [ABSTRACT FROM AUTHOR]

In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature. [ABSTRACT FROM AUTHOR]

Split variational inclusion problem is an important problem, and it is a generalization of the split feasibility problem. In this paper, we present feasible algorithms for the split variational inclusion problems in Hilbert spaces, and provide convergence theorems for these algorithms. As application, we study the split feasibility problem in real Hilbert spaces. Final, numerical results are given for our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak and strong convergence of the proximal point algorithm for equilibrium problems of pseudo-monotone type in Hilbert spaces. We prove the weak convergence of the generated sequence to a common solution of two equilibrium problems and some strong convergence results with additional assumptions on pseudo-monotone bifunctions. Then we study a regularization of Halpern-type and prove the strong convergence of the generated sequence to an equilibrium point of two pseudo-monotone bifunctions without any additional assumption on bifunctions. Finally, some examples of pseudo-monotone bifunctions from pseudo-monotone operators and Nash-Cournot oligopolistic equilibrium models are also presented. Our results extend some similar results in the literature for monotone and pseudo-monotone equilibrium problems and also the related results for variational inequalities associated with monotone and pseudo-monotone operators. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two finite families of demicontractive mappings. The strong convergence theorem of the proposed algorithm is established under some suitable control conditions in a real Hilbert space. Our result generalizes several recent results in the current literature. [ABSTRACT FROM AUTHOR]

A new convex feasibility problem, the split equality problem (SEP), has been proposed by Moudafi and Byrne. The SEP was solved through the ACQA and ARCQA algorithms. In this paper the SEPs are extended to infinite-dimensional SEPs in Hilbert spaces and we established the strong convergence of a proposed algorithm to a solution of general split equality problems (GSEPs). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan 2010 and many others. [ABSTRACT FROM AUTHOR]

The gradient projection algorithm plays an important role in solving constrained convex minimization problems. In general, the gradient projection algorithm has only weak convergence in infinite-dimensional Hilbert spaces. Recently, H. K. Xu (2011) provided two modified gradient projection algorithms which have strong convergence. Motivated by Xu's work, in the present paper, we suggest three more simpler variant gradient projection methods so that strong convergence is guaranteed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we suggest and analyze two projection methods (one implicit and one explicit) for finding a particular solution of a mixed equilibrium problem in a real Hilbert space. Furthermore, we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the split equality problem (SEP) in Hilbert spaces. By converting it to a coupled fixed-point equation, we propose a new algorithm for solving the SEP. Whenever the convex sets involved are level sets of given convex functionals, we propose two new relaxed alternating algorithms for the SEP. The first relaxed algorithm is shown to be weakly convergent and the second strongly convergent. A new idea is introduced in order to prove strong convergence of the second relaxed algorithm, which gives an affirmative answer to Moudafi's question. Finally, preliminary numerical results show the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, in this paper, we introduce an inexact three-operator splitting algorithm to solve this type of monotone inclusion problem. The theoretical convergence properties of the proposed iterative algorithm are studied in general Hilbert spaces under mild conditions on the iterative parameters. As a corollary, we obtain general convergence results of the inexact forward-backward splitting algorithm and the inexact Douglas-Rachford splitting algorithm, which extend the existing results in the literature. [ABSTRACT FROM AUTHOR]