Monotone Iterative Solutions for Nonlinear Boundary Value Problems of Fractional Differential Equation

Abstract

By means of the method of quasi-lower and quasi-upper solutions and monotone iterative technique, we consider the nonlinear boundary value problems with Caputo fractional derivative and introduce two well-defined monotone sequences of quasi-lower and quasi-upper solutions which converge uniformly to the actual solution of the problem, and then the existence results of the solution for the problems are established. A numerical iterative scheme is introduced to obtain an accurate approximate solution and to give one example to demonstrate the accuracy and efficiency of the new approach.

On the existence and uniqueness of solutions for a class of non-linear fractional boundary value problems

“…The problem (1.1) with f ¼ fðt; yÞ and non-homogenous boundary conditions of Dirichlet type was studied by Al-Refai and Hajji (2011), where some existence and uniqueness results were established using the monotone iterative sequences of upper and lower solutions. In addition, the same problem (1.1) with fðt; yÞ ¼ f 0 ðt; yÞ þ f 1 ðt; yÞ þ f 2 ðt; yÞ was studied by Hu et al (2013) using quasi-lower and quasi-upper solutions and monotone iterative technique. The problem (1.1) with f ¼ fðt; yÞ and homogeneous boundary conditions of Dirichlet type and D d 0 þ is the standard Riemann-Liouville fractional derivative discussed by Bai and Lu (2005).…”