Generalized Monotone Iterative Technique for Caputo Fractional Differential Equation with Periodic Boundary Condition via Initial Value Problem

J. D. Ramírez,A. S. Vatsala

doi:10.1155/2012/842813

Abstract

We develop a generalized monotone method using coupled lower and upper solutions for Caputo fractional differential equations with periodic boundary conditions of order , where . We develop results which provide natural monotone sequences or intertwined monotone sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. However, these monotone iterates are solutions of linear initial value problems which are easier to compute.

Highlights

The study of fractional differential equations has acquired popularity in the last few decades due to its multiple applications, see 1–5 for more information
It was not until recently that a study on the existence of solutions by using upper and lower solutions, which is well established for ordinary differential equations in 6, has been done for fractional differential equations
We will use coupled lower and upper solutions combined with a generalized monotone method of initial value problems to prove the existence of coupled minimal and maximal periodic solutions

Summary

Introduction

The study of fractional differential equations has acquired popularity in the last few decades due to its multiple applications, see 1–5 for more information. We will use coupled lower and upper solutions combined with a generalized monotone method of initial value problems to prove the existence of coupled minimal and maximal periodic solutions. International Journal of Differential Equations value problems we have used a generalized monotone method of initial value problems. This idea was presented in 18 for integrodifferential equations. The advantage of this method, compared to what was developed in 3, 15 , is that it avoids computing the solution of the linear periodic boundary value problem using the Mittag-Leffler function at every step of the iterates. We modify the comparison theorem which does not require the Holder continuity condition as in 3

Preliminary Definitions and Comparison Results

Numerical Examples

Concluding Remarks