tailieunhanh - Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 107384, 11

Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 107384, 11 pages doi: Research Article New Existence Results for Nonlinear Fractional Differential Equations with Three-Point Integral Boundary Conditions Bashir Ahmad,1 Sotiris K. Ntouyas,2 and Ahmed Alsaedi1 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, . Box 80203, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece Correspondence should be addressed to Bashir Ahmad, bashir qau@ Received 30 October 2010; Revised 12 December 2010; Accepted 12 December 2010 Academic Editor: Dumitru Baleanu Copyright q 2011 Bashir Ahmad et al. This is an open access article. | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 107384 11 pages doi 2011 107384 Research Article New Existence Results for Nonlinear Fractional Differential Equations with Three-Point Integral Boundary Conditions Bashir Ahmad 1 Sotiris K. Ntouyas 2 and Ahmed Alsaedi1 1 Department of Mathematics Faculty of Science King Abdulaziz University . Box 80203 Jeddah 21589 Saudi Arabia 2 Department of Mathematics University ofIoannina 451 10 Ioannina Greece Correspondence should be addressed to Bashir Ahmad bashir_qau@ Received 30 October 2010 Revised 12 December 2010 Accepted 12 December 2010 Academic Editor Dumitru Baleanu Copyright 2011 Bashir Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. This paper studies a boundary value problem of nonlinear fractional differential equations of order q e 1 2 with three-point integral boundary conditions. Some new existence and uniqueness results are obtained by using standard fixed point theorems and Leray-Schauder degree theory. Our results are new in the sense that the nonlocal parameter in three-point integral boundary conditions appears in the integral part of the conditions in contrast to the available literature on three-point boundary value problems which deals with the three-point boundary conditions restrictions on the solution or gradient of the solution of the problem. Some illustrative examples are also discussed. 1. Introduction In recent years boundary value problems for nonlinear fractional differential equations have been addressed by several researchers. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes see 1 . These characteristics of the fractional derivatives make the fractional-order models .