Modified multi-step iterative processes for a finite family of asymptotically pseudocontrative maps

Abstract

Very recently, Mogbademu [6] answered in affirmative the open problem proposed in [9] by extending an iterative algorithm for two asymptotically pseudocontractive mappings to three mappings. He further asked if his results could be extended to the case of finite family of mappings. This sequel gives a positive answer to the open problem and equally extends so many results in this area of research.

On the Equivalence of Stochastic Fixed Point Iterations for Generalized φ-Contractive-Like Operators

“…Since (i) → (ii) and (ii) → (i), it is shown that the convergence of random implicit Jungck-Kirk-Mann iteration (14) is equivalent to the convergence of random implicit Jungck-Kirk-multistep iteration (11) when applied to a pair of random weakly compatible generalized -contractive-like conditions (19). This ends the proof.…”

“…Since the random implicit Jungck-Kirk-multistep iteration (11) generalizes other random implicit Jungck-Kirk-type iterations (12), (13), and (14), then Theorem 7 leads to the following corollaries.…”

“…as sequences satisfying (14) and (11), respectively, then the following are equivalent: (11) converges strongly to ( ).…”

“…Assume that and are random weakly compatible. Let ( ) be the random common and { ( , ( ))} ∞ =0 as sequences satisfying (14), (13), and (12) (12) converges strongly to .…”

“…The concept of employing various iterative schemes in approximating fixed points of contractive-like operators is very useful in fixed point theory and applications and other relevant fields like numerical analysis, operation research, and so forth (see [8][9][10][11][12][13][14][15]) This is due to the close relationship that exists between the problem of solving nonlinear equations and that of approximating fixed points of corresponding contractive-like operator.…”