International Journal of Scientific Engineering and Science Volume 5, Issue 7, pp. 5-8, 2021. ISSN (Online):2456-7361 5 http://ijses.com/ All rights reserved On the Strong Convergence and Stability Results of Some Modified Iterative Scheme for a General Class of Operators Alfred Olufemi Bosede 1 , Awe Gbemisola Sikirat 2 , Adegoke Stephen Olaniyan 3 1, 2, 3 Department of Mathematics, Lagos State University, Lagos, Nigeria Abstract— In this article, we introduce and establish the strong convergence and stability results of some modified iterative schemes for a general class of operators introduced by Bosede and Rhoades [4] in an arbitrary Banach space. Numerical example is given to prove that our results are significant refinement and improvement to results obtained by several authors in literature. Keywords— Fixed Point, Ishikawa Iterative Scheme, Mann Iterative Scheme, Noor Iterative Scheme, T-Stable, Zamfirescu Operators. I. INTRODUCTION Definition 1.1: Let (E, d) be a metric space and T: E → E a self map of E. Suppose F T = {p ∈E: T p = p} is the set of fixed points of T in E.Let ∈E, then the sequence * + defined by , n ≥ 0 (1) is called Picard Iterative Scheme. Definition 1.2: Let E be a Banach space and T: E → E a self map of E. For ∈E, the sequence * + defined by ( ) (2) where* + is a sequence in [0, 1) such that ∞ refers to as Mann Iterative scheme. Definition 1.3: Let E be a Banach space and T: E → E a self map of E. For ∈E, the sequence * + defined by \ ( ) ( ) (3) where sequences * + and * + ⊂ [0, 1) such that ∞ refers to as Ishikawa Iterative scheme. Definition 1.4: Let E be a Banach space and T: E → E a self map of E. For ∈E, the sequence * + defined by ( ) ( ) (4) ( ) where sequences* + , * + and * + ⊂ [0, 1) such that ∞ refers to as Noor Iterative scheme. Remark 1.5: Obviously in (4), if it reduces to (3). If , it also reduces to (2) and if , in (4) it is becomes Picards Iterative Scheme (1). Over the years, several authors have been modifying Iterative Processes for different classes of operators. In 2006, Rafiq A.[13] analyzed a modified three-step iterative scheme for solving nonlinear operators in Banach spaces. Olaleru J. and Mogbademu A.[10] proved the convergence results for modified Noor iteration when applied to three generalized strongly pseudocontrative maps defined on a Banach space. Results obtained generalized works of several authors. Okeke G. A. and Olaleru J. O.[9] introduced a new three step iterative scheme with errors to approximate the unique common fixed point of a family of three strongly pseudocontractive (accretive) mappings on Banach spaces which generalized and improved the result of Olaleru J. & Mogbademu A.[10] and corrected the results of Rafiq A.[14]. Modified Noor Iteration for non expansive semi groups with generalized contraction in Banach spaces was investigated by Pakkapan P. A and Rabian W. [13]. Akewe H. and Olaleru J. O.[2] established some strong convergence and stability results of multistep iterative scheme for a general class of operators. Their convergence results generalized and extended the results of Berinde [6], Bosede [3], Olaleru [9], Rafiq [14] among others. A modified mixed Ishikawa iteration for common fixed points of two asymptotically Quasi Pseudo contrative type non-self mapping was studied by Wang Y. and Suthep S. [16]. Mohammad A. and Mohammad Z. A.[8] established the strong convergence theorem of Noor iterative scheme for a class of Zamfirescu operators in arbitrary Banach spaces. Their results was an extension and generalization of the recent results of Xu B. L. et al. [18], Berinde V. [6], Zhou H. et al. [20] and many other authors. Withun P. and Suthep S. [17] proposed the following modified iterative scheme in which they proved the strong convergence of the proposed scheme to a fixed point of a weak contraction: ( ) ( ) (5) ( ) where sequences { }, { }, { }, { } and { } are sequences in [0, 1]. They showed that their proposed scheme converges faster than Mann, Ishikawa and Noor iterations. Motivated mostly by the work of Withun P. et al [17], we proposed the following iterative schemes: Algorithm 1.6: Let (E, d) be a complete metric space and T: E → E a self map. Let F T be the set of fixed point of T, that is,F T = {u ∈E: T u = u}. The sequence* + defined by , ( )- (6) where * + , * + and * + ⊂[0, 1) is called the modified Mann Iterative Scheme. If for ∈, the sequence* + defined by , ( )-