Modified viscosity type iteration for total asymptotically nonexpansive mappings in CAT(0) spaces and its application to optimization problems

Abstract

In this paper, we introduce a modified two-step viscosity iteration process for total asymptotically nonexpansive mappings in CAT(0) spaces. We prove strong convergence of the proposed iteration process to a fixed point of total asymptotically nonexpansive mappings in CAT(0) spaces, which also shows that the limit of the sequence generated by proposed iteration process solves the solution of the variational inequality. We also provide illustrating a numerical example for supporting our main results. Moreover, we show the existence of solutions of our consequently results for some applications.

Mann iteration for monotone nonexpansive mappings in ordered CAT(0) space with an application to integral equations

“…The fixed point theory in spaces is gaining attention of researchers, and many results have been obtained for single- and multivalued mappings in a space. For different aspects of fixed point theory in spaces, we refer to [15–24]. The following few results are necessary for our subsequent discussion.…”

A New Iterative Algorithm for Total Asymptotically Non-Expansive Mapping in CAT(0) Spaces

“…where {ςm} is real sequence in (0,1). After the development of S-iteration process,numerous researchers developed iteration processes, including Thakur et al [1], modified Picard S-hybrid iteration [7], modified Picard Mann hybrid [1]. In the last decades, so many researchers attract to approximate fixed point in CAT(0) space (see [8], [9]).…”

“…It is worth noting that the class of asymptotically non-expansive maps is correctly included within the class of asymptotically non-expansive mappings in the intermediate sense (see, e.g., [18]). Alber et al [19] initially established the concept of total asymptotically non-expansive mappings in 2006 (see also [20], [21], [18], [7]) defined as follows.…”

“…In 2012, Chang et al [22] demiclosedness principle for total asymptotically non-expansive mapping in CAT(0) space and established strong and Delta-convergence theorem for Mann iteration process. As a result, other researchers established strong and Delta-convergence theorem for the same mapping through modified Ishikawa [23], S-iteration process [24], modified Picard S-hybrid iteration [7], modified Picard Mann hybrid [1] in CAT(0) spaces. Recently, Izhar et al [2] presented A modified iteration process {ςm} and {νm}, which is defined below:…”

Hybrid algorithm with perturbations for total asymptotically non-expansive mappings in CAT(0) space

“…For convenience, from now on, we will use the notation (1 − t) ⊕ ty for unique point z satisfying (1). For details as regards spaces please see [9,11,12]. Some results are recalled here for CAT(0) space X.…”