A modified inertial shrinking projection algorithm with adaptive step size for solving splitgeneralized equilibrium, monotone inclusion and fixed point problems

Abstract. In this paper, we study the common solution problem of split generalized equilibrium problem, monotone inclusion problem and common fixed point problem for a countable family of strict pseudo-contractive multivalued mappings. We propose a modified shrinking projection algorithm of inertial form with self-adaptive step sizes for finding a common solution of the aforementioned problem. The self-adaptive step size eliminates the difficulty of computing the operator norm while the inertial term accelerates the rate of convergence of the proposed algorithm. Moreover, unlike several of the existing results in the literature, the monotone inclusion problem considered is a more general problem involving the sum of Lipschitz continuous monotone operators and maximal monotone opera- tors, and knowledge of the Lipschitz constant is not required to implement our algorithm. Under some mild conditions, we establish strong convergence result for the proposed method. Finally, we present some applications and numerical experiments to illustrate the usefulness and applicability of our algorithm as well as comparing it with some related methods. Our results improve and extend corresponding results in the literature. Mathematics Subject Classification (2010): 65K15, 47J25, 65J15. Keywords: Split generalized equilibrium problem, monotone inclusion problem, inertial method, fixed point problem, strict pseudo-contractions, multivalued mappings.