Iterative approximation of common attractive points of $$(\alpha ,\beta )$$ ( α , β ) -generalized hybrid set-valued mappings

Abstract

In this paper, we introduce the concepts of the set of all the common attractive points (CAP(S, T)) and the set of all the common strongly attractive points (CSAP(S, T)) for the set-valued mappings S and T in a Hilbert space. Moreover, some fundamental properties related to the sets A(T), F(T) and CAP(S, T) are given. Furthermore, we generalize the Agarwal iteration for the case of two set-valued mappings and obtain a weak convergence theorem for two $$(\alpha ,\beta )$$ -generalized hybrid set-valued mappings in a Hilbert space. Moreover, an example of two $$(\alpha ,\beta )$$ -generalized hybrid set-valued mappings which have a common strongly attractive point is shown. Then we finish the work using the proposed algorithm to find a common element of the set of solutions of an equilibrium problem and the sets of fixed points of two $$(\alpha ,\beta )$$ -generalized hybrid set-valued mappings.