Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions

Abstract

Recently, Wang and Xie [T. Wang, F. Xie, J. Nonlinear Sci. Appl., 1 (2009), 206-212] developed monotone iterative method for Riemann-Liouville fractional differential equations with integral boundary conditions with the strong hypothesis of locally Hölder continuity and obtained existence and uniqueness of a solution for the problem. In this paper, we apply the comparison result without locally Hölder continuity due to Vasundhara Devi to develop monotone iterative method for the problem and obtain existence and uniqueness of a solution of the problem. c 2014 All rights reserved.Keywords: Fractional differential equations, existence and uniqueness, lower and upper solutions, integral boundary conditions. 2010 MSC: 34A12, 34C60, 34A45.

Existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations with fractional integral boundary conditions

“…Consider the following coupled system of nonlinear FDE 19) where 2, 3, 4) and a ik (t)(i = 1, 2; k = 1, 2, 3, 4, 5) are nonnegative functions. Obviously, Theorem 3.1 implies that FDE (3.19) has at least one solution.…”

“…Many books on fractional differential equations, fractional calculus have been published (see, for example [15,16,18,21,32]), and some recent papers on the topic, see [1,3,4,5,7,9,11,12,14,19,24,25,26,32] and the references therein. At the same time, there are many papers concerned with solvability of coupled systems of nonlinear fractional differential equations, because of their wide applications.…”

Undetermined coefficients for local fractional differential equations

“…Fractional differential equations have been of great interest for the last thirty years because of their applications in applied sciences, see [6,8] and [12]. The main definitions which are of wide use are the Riemann-Liouville definition and the Caputo definition, see [10,11].…”

Difference equations involving causal operators with nonlinear boundary conditions

“…As an important branch, boundary value problems (see [5,11,12,19]), especially, difference equations with nonlinear boundary value problems have drawn much attention. In 2008, Wang [15] investigated the first-order functional difference problems with nonlinear boundary value conditions, obtained the existence of extremal solutions by using the monotone iterative method.…”