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In each iteration, the projection methods require computing at least one projection onto the closed convex set. However, projections onto a general closed convex set are not easily executed, a fact that might affect the efficiency and applicability of the projection methods. To overcome this drawback, we propose two iterative methods with self-adaptive step size that combines the Halpern method with a relaxed projection method for approximating a common solution of variational inequality and fixed point problems for an infinite family of multivalued relatively nonexpansive mappings in the setting of Banach spaces. The core of our algorithms is to replace every projection onto the closed convex set with a projection onto some half-space and this guarantees the easy implementation of our proposed methods. Moreover, the step size of each algorithm is self-adaptive. We prove strong convergence theorems without the knowledge of the Lipschitz constant of the monotone operator and we apply our results to finding a common solution of constrained convex minimization and fixed point problems in Banach spaces. Finally, we present some numerical examples in order to demonstrate the efficiency of our algorithms in comparison with some recent iterative methods.

mathematical and computational applications

Relaxed Projection Methods with Self-Adaptive Step Size for Solving Variational Inequality and Fixed Point Problems for an Infinite Family of Multivalued Relatively Nonexpansive Mappings in Banach Spaces