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The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

The main purpose of this paper is to propose a new modified contraction method for solving a certain class of split monotone variational inclusion problems in real Hilbert spaces. We prove that the sequence generated by the proposed method converges strongly to a solution of the aforementioned problem. Our strong convergence result is obtained when the underline operator is monotone and Lipschitz continuous, and the knowledge of its Lipschitz constant is not required. As application, we solved the split linear inverse problems, for which we also considered a special case of this problem, namely, the LASSO problem. We also give some numerical illustrations of the proposed method in comparison with other methods in the literature to further show the applicability and advantage of our results. The results obtained in this paper generalize and improve many recent results in this direction. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a new iterative algorithm to approximate the solution for a variational inequality problem in real Hilbert spaces. A strong convergence result for the above problem is established under certain mild conditions. Our proposed method requires the computation of only one projection onto the feasible set in each iteration. Some numerical examples are presented to support that our proposed method performs better than some known comparable methods for solving variational inequality problems.

A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, we introduce a new extragradient-like method to solve the problems of variational inequalities in real Hilbert space involving pseudomonotone operators. The method has a clear advantage because of a variable stepsize formula that is revised on each iteration based on the previous iterations. The key advantage of the method is that it works without the prior knowledge of the Lipschitz constant. Strong convergence of the method is proved under mild conditions. Several numerical experiments are reported to show the numerical behaviour of the method.

We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish some strong convergence theorems for our newly developed methods under certain restriction. Our results extend and improve several recently announced results. Furthermore, we give several numerical experiments to show that our proposed algorithms performs better in comparison with several existing methods.

In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second method are well established based on the mild condition on a bifunction. Such methods have the advantage of not involving any line search procedure or any knowledge of the Lipschitz-type constants of the bifunction. One practical reason is that the stepsize involving in these methods is updated based on some previous iterations or uses a stepsize sequence that is non-summable. We consider the well-known Nash–Cournot equilibrium models to support our well-established convergence results and see the advantage of the proposed methods over other well-known methods.

In this article, we introduce a new multi-step iteration for approximating a common fixed point of a finite class of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We prove a strong convergence theorem for the proposed iterative algorithm under certain hypotheses. Additionally, we also use our results for the solution of variational inequality problems and to find the zero points of maximal monotone operators. The theorems furnished in this work are new and well-established and generalize many well-known recent research works in this field. [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]

In this article, we introduce a new multi-step iteration for approximating a common fixed point of a finite class of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We prove a strong convergence theorem for the proposed iterative algorithm under certain hypotheses. Additionally, we also use our results for the solution of variational inequality problems and to find the zero points of maximal monotone operators. The theorems furnished in this work are new and well-established and generalize many well-known recent research works in this field.

We introduce an iterative process for finding an element of common solutions of finite family of equilibrium problems and finite family of variational inequality problems for continuous monotone mappings 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 PUBLISHER]

In this paper, we introduce an algorithm which converges strongly to a solution of a finite family of variational inequity problems for η-inverse strongly accretive mappings in the set of common fixed points of a finite family of A-strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. [ABSTRACT FROM AUTHOR]

Let $$C$$ be a nonempty closed convex subset of a real Hilbert space $$H$$. Let $$\{T_i\}^{\infty }_{i=1}:C\rightarrow H$$ be an infinite family of generalized asymptotically nonexpansive nonself mappings. By using a specific way of choosing the indexes of the involved mappings, we prove strong convergence of Mann's type iteration to a common fixed point of $$\{T_i\}^{\infty }_{i=1}$$ without the compactness assumption imposed either on $$T$$ or on $$C$$ provided that the interior of common fixed points is nonempty. The results extend previous results restricted to the situation of at most finite families of such mappings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an iterative process which converges strongly to a common solution of finite family of variational inequality problems for γ-inverse strongly monotone mappings and fixed point of two continuous quasi- $${\phi}$$-asymptotically nonexpansive mappings in Banach spaces. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings. [ABSTRACT FROM AUTHOR]

Abstract: Let be a nonempty, closed and convex subset of a real Hilbert space . Let , be a finite family of generalized asymptotically nonexpansive mappings. It is our purpose, in this paper to prove strong convergence of Mann’s type method to a common fixed point of provided that the interior of common fixed points is nonempty. No compactness assumption is imposed either on or on . As a consequence, it is proved that Mann’s method converges for a fixed point of nonexpansive mapping provided that interior of . The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings. [Copyright &y& Elsevier]

In this paper, we introduce an iterative process which converges strongly to a common solution of variational inequality problems for two monotone mappings in Banach spaces. Furthermore, our convergence theorem is applied to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings. [ABSTRACT FROM AUTHOR]

Abstract: We introduce an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems and common solutions of finite family of variational inequality problems for monotone mappings in Banach spaces. Our theorem extends and unifies most of the results that have been proved for this important class of nonlinear operators. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the fixed points of -asymptotically nonexpansive mapping, the set of solutions of the equilibrium problem and the set of solutions of the variational inequality for an inverse strongly monotone operator in the framework of Banach spaces. We show that the iterative scheme converges strongly to a common element of the above three sets under appropriate conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce an iterative process which converges strongly to a common element of set of common fixed points of countably infinite family of closed relatively quasi- nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for a -inverse strongly monotone mapping in Banach spaces. Our theorems improve, generalize, unify and extend several results recently announced. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce an iterative process which converges strongly to a common element of a set of common fixed points of finite family of closed relatively quasi-nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for an -inverse strongly monotone mapping in Banach spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we prove strong convergence theorems to a zero of monotone mapping and a fixed point of relatively weak nonexpansive mapping. Moreover, strong convergence theorems to a point which is a fixed point of relatively weak nonexpansive mapping and a solution of a certain variational problem are proved under appropriate conditions. [Copyright &y& Elsevier]

The aim of this paper is to propose a new iterative algorithm to approximate the solution for a variational inequality problem in real Hilbert spaces. A strong convergence result for the above problem is established under certain mild conditions. Our proposed method requires the computation of only one projection onto the feasible set in each iteration. Some numerical examples are presented to support that our proposed method performs better than some known comparable methods for solving variational inequality problems. [ABSTRACT FROM AUTHOR]

A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, we introduce a new extragradient-like method to solve the problems of variational inequalities in real Hilbert space involving pseudomonotone operators. The method has a clear advantage because of a variable stepsize formula that is revised on each iteration based on the previous iterations. The key advantage of the method is that it works without the prior knowledge of the Lipschitz constant. Strong convergence of the method is proved under mild conditions. Several numerical experiments are reported to show the numerical behaviour of the method. [ABSTRACT FROM AUTHOR]

We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish some strong convergence theorems for our newly developed methods under certain restriction. Our results extend and improve several recently announced results. Furthermore, we give several numerical experiments to show that our proposed algorithms performs better in comparison with several existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second method are well established based on the mild condition on a bifunction. Such methods have the advantage of not involving any line search procedure or any knowledge of the Lipschitz-type constants of the bifunction. One practical reason is that the stepsize involving in these methods is updated based on some previous iterations or uses a stepsize sequence that is non-summable. We consider the well-known Nash–Cournot equilibrium models to support our well-established convergence results and see the advantage of the proposed methods over other well-known methods. [ABSTRACT FROM AUTHOR]