An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings

TL;DR: In this paper, an iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize was proposed.

Abstract: In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the framework of Hilbert spaces. First, we introduce a new iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize. Unlike in many existing subgradient extragradient techniques in literature, the two projections of our proposed algorithm are made onto some half-spaces. Furthermore, we prove a strong convergence theorem for approximating a common solution of the variational inequality and fixed point of an infinite family of nonexpansive mappings under some mild conditions. The main advantages of our method are: the self-adaptive stepsize which avoids the need to know a priori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which accelerates convergence rate of the algorithm. Second, we apply our theorem to solve generalised mixed equilibrium problem, zero point problems and convex minimization problem. Finally, we present some numerical examples to demonstrate the efficiency of our algorithm in comparison with other existing methods in literature. Our results improve and extend several existing works in the current literature in this direction.