Convergence of solutions for functional integro-differential equations with nonlinear boundary conditions

Abstract

This paper is concerned with the convergence of solutions for a class of functional integro-differential equations with nonlinear boundary conditions. New comparison principles are obtained. By using the comparison principles and quasilinearization method, we present two monotone iterative sequences uniformly and monotonically converging to the unique solution with rate of order 2. Meanwhile, an example is given to demonstrate applications of the result reported.

Convergence of Antiperiodic Boundary Value Problems for First-Order Integro-Differential Equations

“…It is well known that the method of quasilinearization (QSL) provides a powerful tool for obtaining convergence of approximate solutions of nonlinear problems [28,29]. e technique of upper and lower solutions coupled with the QSL have been applied successfully to obtain monotone sequences of approximate solutions converging uniformly and quadratically to the unique solution of integro-differential equations with antiperiodic boundary value conditions [30][31][32]. In terms of applications, it is important to pay attention to the high-order convergence of sequences of approximate solutions.…”

Rapid Convergence of Approximate Solutions for Fractional Differential Equations

“…However, to the best of our knowledge, there are few results of rapid convergence of fractional differential equations. Recently, Wang and others obtained the results on rapid convergence of solutions for various differential equations [29][30][31][32][33]. Inspired and motivated by [34,35], in the present paper, we will discuss the rapid convergence of approximate solutions of fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions with coupled lower and upper solutions, and construct sequences of approximate solutions that converge rapidly to the extremal solutions of (1) by using an improved quasilinearization method (rate of convergence k ≥ 2).…”