Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme

Abstract

The aim of this paper is to propose a new faster iterative scheme (called AA-iteration) to approximate the fixed point of (b,η)-enriched contraction mapping in the framework of Banach spaces. It is also proved that our iteration is stable and converges faster than many iterations existing in the literature. For validity of our proposed scheme, we presented some numerical examples. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in the uniformly convex Banach space. Finally, we approximate the solution of delay fractional differential equations using AA-iterative scheme.

Solving Fractional Differential Equations via Fixed Points of Chatterjea Maps

“…Many researchers have studied this topic because it has many applications. Related to this matter, we suggest the recent literature [29][30][31][32][33][34][35] and the references therein.…”

The AA-Viscosity Algorithm for Fixed-Point, Generalized Equilibrium and Variational Inclusion Problems

“…There have been several iterative methods proposed in the literature for the solution of nonlinear problems. For instance, in 2022, Abbas et al [31] proposed an iterative method known as the AA (Abbas-Asghar)-iteration.…”

“…It was shown that the AA-iteration method has a faster rate of convergence than other well-known iteration methods existing in the literature [31]. Note that the AA-iteration method has been successfully applied for obtaining the solutions of operator equations involving nonexpansive-type mappings; for instance, see [32][33][34].…”

An Algorithm for Nonlinear Problems Based on Fixed Point Methodologies With Applications

“…In 1974, Ishikawa [5] resolved that problem and proposed a two steps iterative method. Some other examples of commonly used iterative methods, to approximate the fixed points of nonexpansive mappings are by Noor [6], Agarwal [7], Abbas and Nazir [8], Thukar et al [9], Ullah and Arshad [10], Ullah et al [11], Saleem et al [12], Abbas et al [13], Ahamd et al [14], JK iteration (see, also [15] and many others) proved the convergence results for Suzuki-type generalized nonexpansive mappings.…”

“…Let {a s }, {b s } and {c s } are three sequences of real numbers in (0, 1) then Mann [4], Ishikawa [5], Noor [6], Agarwal [7], Abbas and Nazir [8], Thukar [9], Ullah and Arshad [10], Ullah et al [11] and Abbas et al [13] iterative methods are respectivley given below:…”