Convergence and Data Dependence Results for AK Iterative Algorithm

Abstract

This paper concentrates on studying convergence and data dependence of AK iteration for the class of maps which was introduced by Berinde. Also, we will support our results with an example.

Convergence, data dependence and $T$-stability of $AK$ iteration procedure for contractive like operators

“…3x for x ∈ [1,3]. ThenT is an approximate operator of T with η = 1 3 . We take α n = 1 2 n and β n = 1 3 n for n = 0, 1, 2... so that lim…”

“…x 0 ∈ K z n = T ((1 − β n )x n + β n T x n ) y n = T ((1 − α n )z n + α n T z n ) x n+1 = Ty n (1) where {α n } ∞ n=0 and {β n } ∞ n=0 are real sequences in [0,1]. Ullah and Arshad [4] proved the convergence, data dependence and T -stability of AK iteration procedure under certain assumptions on α n s for contraction maps as follows.…”

“…Ertürk [1] proved that convergence and data dependence for AK iteration procedure with certain assumptions on α n s and β n s for the mapping T : K → K that satisfies the condition…”

“…Theorem 1.6. [1] Let T : K → K be an operator that satisfies the inequality (3). For any x 0 ∈ K, let {x n } ∞ n=0 be the iterative sequence defined by (1) where…”

“…Theorem 1.7. [1] Let T : K → K be an operator that satisfies the inequality (3),T : K → K be an approximate operator of T with η > 0. Let {α n } ∞ n=0 and {β n } ∞ n=0 be the sequences…”