A Krasnoselskii–Ishikawa Iterative Algorithm for Monotone Reich Contractions in Partially Ordered Banach Spaces with an Application

Abstract

Iterative algorithms have been utilized for the computation of approximate solutions of stationary and evolutionary problems associated with differential equations. The aim of this article is to introduce concepts of monotone Reich and Chatterjea nonexpansive mappings on partially ordered Banach spaces. We describe sufficient conditions for the existence of an approximate fixed-point sequence (AFPS) and prove certain fixed-point results using the Krasnoselskii–Ishikawa iterative algorithm. Moreover, we present some interesting examples to highlight the superiority of our results. Lastly, we provide both weak and strong convergence results for such mappings and consider an application of our results to prove the existence of a solution to an initial value problem.

Solving Fractional Differential Equations via Fixed Points of Chatterjea Maps

“…Fixed point theory plays an important role in various branches of mathematics as well as in nonlinear functional analysis, and is very useful for solving many existence problems in nonlinear differential and integral equations with applications in engineering and behavioural sciences. Recently, many authors have provided the extended fixed point theorems for the different classes of contraction type mappings, such as Kannan, Reich, Chatterjea and Ćirić-Reich-Rus mappings (see [1][2][3][4][5][6][7][8][9][10]).…”

“…Reich [16] showed the generalized Banach's theorem and observed that Kannan's theorem is a particular case of it with a suitable selection of the constant. Reich type mappings and generalized nonexpansive mappings have been important research area on their own for many authors which has been applied in various spaces such as metric space, Banach space, and partially ordered Banach spaces (see [5,8,9,[17][18][19]).…”

Connected ϵ‐Chainable Sets and Existence Results

“…Edelstein [1] extended the contraction principle in the setting of the ϵ-chainable metric space using local contraction. Many researchers extend the local contraction in diferent ways and prove the contraction principle (see [6][7][8][9][10][11][12]). Te notion of fnitely chainable metric space was introduced by Atsuji [13].…”