Some Convergence Results for Non-Linear Maps in Banach Spaces

Abstract

A strong convergence theorem for asymptotically generalized Φ− hemicontractive map in real Banach space is proved using the iterative sequence generated by this map. The result of this paper extend and improve the very recent result of Kim et al., (2009) which itself is a generalization of many of the previous results.

A convergence theorem of multi-step iterative scheme for nonlinear maps

“…Furthermore, our result also improves and extends the corresponding results in [1,3]. For this, we need the following Lemmas.…”

Convergence of Multi-Step Iterative Sequence for Nonlinear Uniformly L-Lipschitzian Mappings

“…His result extends, improves and unifies a host of recent results. Although, Mogbademu and Xue [9] had earlier obtained a strong convergence theorem for asymptotically generalized Φ− hemicontractive map in real Banach spaces using the iterative sequence generated by this map.…”

“…His result extends, improves and unifies a host of recent results. Although, Mogbademu and Xue [9] had earlier obtained a strong convergence theorem for asymptotically generalized Φ− hemicontractive map in real Banach spaces using the iterative sequence generated by this map. More recently, Xue, Rafiq and Zhou [19] employed an analytical technique to prove the convergent of an Ishikawa and Mann iterations for nonlinear mappings in uniformly smooth real Banach spaces.…”

“…For the convergence analysis of the predictor-corrector and multistep iteration sequences for solving the variational inequalities and optimization problems (see Noor [11], Noor et al [12]). Lemma 1.1 [1,9]. Let E be real Banach Space and J : E → 2 E * be the normalized duality mapping.…”

Three-step iterations for total asymptotically nonexpansive mappings in CAT(0) spaces

“…are appropriate sequences in (0,1) is called modified three-step iterative sequence. Iteration scheme (5) is independent of modified Noor iteration [30], modified Ishikawa iteration and modified Mann iteration schemes.…”