A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem

Abstract

In this paper, we propose a new extragradient method consisting of the hybrid steepest descent method, a single projection method and an Armijo line searching the technique for approximating a solution of variational inequality problem and finding the fixed point of demicontractive mapping in a real Hilbert space. The essence of this algorithm is that a single projection is required in each iteration and the step size for the next iterate is determined in such a way that there is no need for a prior estimate of the Lipschitz constant of the underlying operator. We state and prove a strong convergence theorem for approximating common solutions of variational inequality and fixed points problem under some mild conditions on the control sequences. By casting the problem into an equivalent problem in a suitable product space, we are able to present a simultaneous algorithm for solving the split equality problem without prior knowledge of the operator norm. Finally, we give some numerical examples to show the efficiency of our algorithm over some other algorithms in the literature.

On θ-generalized demimetric mappings and monotone operators in Hadamard spaces

“…with ( ) ≠ ∅ F T is said to be θ-generalized demimetric (see also [7]), if 〈 − ( − )〉 ≥ ‖ − ‖ θ u v J u Tu u Tu , 2 (3) for all ∈ u C and ∈ ( ) v F T , where J is a duality mapping on E. Fixed point problems of nonlinear mappings have been a very attractive area of research in nonlinear analysis that has enjoyed a prosperous development due to its extensive applications in diverse mathematical fields such as inverse problems, signal processing, game theory and fuzzy theory (see [8][9][10][11][12][13][14][15][16][17][18][19][20] and references therein). The pioneer work of this study in Hadamard spaces is due to Kirk [21,22].…”

An inertial forward-backward splitting method for approximating solutions of certain optimization problems

“…holds for every y ∈ H with x = y. Recently, several iterative methods have been devised for solving equilibrium problems, variational inequality problems, monotone inclusion problems, minimization problems and their related optimization problems; see, e.g., [6,7,8,9,10,11,12,13,14,15,16]) and the references therein.…”

Strong convergence inertial projection and contraction method with self adaptive stepsize for pseudomonotone variational inequalities and fixed point problems

“…The authors also proved that the sequence {x n } generated by the SEGM converges weakly to a solution of the VIP (1.1) if the stepsize condition λ ∈ (0, 1 L ). Several modifications of the EGM and SEGM have been introduced by many authors; see for instance [12,22,[24][25][26][41][42][43]. Recently, He [17] modified the EGM and introduced a projection and contraction method (PCM) which requires only a single projection per each iteration as follows:…”