Proximal point algorithms based on<i>S</i>-iterative technique for nearly asymptotically quasi-nonexpansive mappings and applications

Abstract

In this paper, we combine theS-iteration process introduced by Agarwal et al. (J. Nonlinear Convex Anal.,8(1), 61-792007) with the proximal point algorithm introduced by Rockafellar (SIAM J. Control Optim.,14, 877-8981976) to propose a new modified proximal point algorithm based on theS-type iteration process for approximating a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces and prove the o-convergence of the proposed algorithm for solving common minimization problem and common fixed point problem. Our result generalizes, extends and unifies the corresponding results of Dhompongsa and Panyanak (Comput. Math. Appl.,56, 2572-25792008), Khan and Abbas (Comput. Math. Appl.,61, 109-1162011), Abbas et al. (Math. Comput. Modelling,55, 1418-14272012) and many more.