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RESEARCH

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Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions Bashir Ahmad1* and Sotiris K Ntouyas2 * Correspondence: [email protected] 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Full list of author information is available at the end of the article

Abstract This article studies a new class of nonlocal boundary value problems of nonlinear differential equations and inclusions of fractional order with strip conditions. We extend the idea of four-point nonlocal boundary conditions (x (0) = σ x (μ) , x (1) = ηx (v) , σ , η ∈ R, 0 < μ, v < 1) to nonlocal strip conditions of the form: x(0) = σ

β α



x(s)ds , x(1) = η

γ

δ

x(s)ds, 0 < α < β < γ < δ < 1 .

These strip conditions may be regarded as six-point boundary conditions. Some new existence and uniqueness results are obtained for this class of nonlocal problems by using standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also discussed. MSC 2000: 26A33; 34A12; 34A40. Keywords: fractional differential equations, fractional differential inclusions, nonlocal boundary conditions, fixed point theorems, Leray-Schauder degree

1 Introduction The subject of fractional calculus has recently evolved as an interesting and popular field of research. A variety of results on initial and boundary value problems of fractional order can easily be found in the recent literature on the topic. These results involve the theoretical development as well as the methods of solution for the fractional-order problems. It is mainly due to the extensive application of fractional calculus in the mathematical modeling of physical, engineering, and biological phenomena. For some recent results on the topic, see [1-19] and the references therein. In this article, we discuss the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations and inclusions of order q Î (1, 2] with nonlocal strip conditions. As a first problem, we consider the following boundary value problem of fractional differential equations ⎧ ⎪ ⎨

c q D x(t) = f (t, x(t), 0 < t < 1, 1 < q ≤ 2, β δ ⎪ ⎩ x(0) = σ x(s)ds, x(1) = η x(s)ds, 0 < α < β < γ < δ < 1, α

(1:1)

γ

© 2012 Ahmad and Ntouyas; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ahmad and Ntouyas Boundary Value Problems 2012, 2012:55 http://www.boundaryvalueproblems.com/content/2012/1/55

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where cDq denotes the Caputo fractional derivative of order q, f : [0, 1] × R → R is a given continuous function and s, h are appropriately chosen real numbers. The boundary conditions in the problem (1.1) can be regarded as six-point nonlocal boundary conditi