Inertial hybrid gradient method with adaptive step size for variational inequality and fixed point problems of multivalued mappings in Banach spaces

We propose in this article a new inertial hybrid gradient method with self-adaptive step size for approximating a common solution of variational inequality and fixed point problems for an infinite family of relatively nonexpansive multivalued mappings in Banach spaces. Unlike in many existing hybrid gradient methods, here the projection onto the closed convex set is replaced with projection onto some half-space which can easily be implemented. We incorporate into the proposed algorithm inertial term and self-adaptive step size which help to accelerate rate of convergence of iterative schemes. Moreover, we prove a strong convergence theorem without the knowledge of the Lipschitz constant of the monotone operator and we apply our result to find a common solution of constrained convex minimization and fixed point problems in Banach spaces. Finally, we present a numerical example to demonstrate the efficiency of our algorithm in comparison with some recent iterative methods in the literature.