alpha$-nonexpansive mapping

Recently, the authors introduce a four-step iterative algorithm called the UD-iteration scheme (Udofia and Igbokwe [35]). Here we introduce the multivalued version of the UD-iteration scheme and show that it can be used to approximate the fixed points of multivalued contraction and multivalued generalized $\alpha$-nonexpansive mappings. we prove strong and weak convergence of the iteration scheme to the fixed point of multivalued generalized $\alpha$-nonexpansive mapping. We also prove that the scheme is $\varUpsilon$-stable and Data dependent. Convergence analysis shows that the multivalued UD-iteration scheme has a faster rate of convergence for multivalued contraction and multivalued generalized $\alpha$-nonexpansive mappings than some well-known existing iteration schemes in the literature.

In this paper, we introduce a new type of modified generalized $\alpha$-nonexpansive mapping and establish some fixed point properties and demiclosedness principle for this class of mappings in the framework of  uniformly convex Banach spaces. We further  propose a new iterative method for approximating a common fixed point of two modified generalized $\alpha$-nonexpansive mappings and present some weak and strong convergence theorems for these mappings in uniformly convex Banach spaces. In addition, we apply our result to solve a  convex-constrained minimization problem, variational inequality and split feasibility problem and present some numerical experiments in infinite dimensional spaces to establish the applicability and efficiency of our proposed algorithm. The obtained results in this paper improve and extend   some related results in the literature.