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On Nonlocal Boundary Value Problems for Nonlinear Integro-differential Equations of Arbitrary Fractional Order

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Abstract

In this paper, we prove the existence of solutions of a nonlocal boundary value problem for nonlinear integro-differential equations of fractional order given by

$$ \begin{array}{ll} ^cD^qx(t) = f(t,x(t),(\phi x)(t),(\psi x)(t)), \quad 0 < t < 1,\\x(0) = \beta x(\eta), x'(0) =0, x''(0) =0, \ldots, x^{(m-2)}(0) =0, x(1)= \alpha x(\eta), \end{array}$$

where \({q \in (m-1, m], m \in \mathbb{N}, m \ge 2}$, $0< \eta <1\) , and \({\phi x}\) and \({\psi x}\) are integral operators. The existence results are established by means of the contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also presented.

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  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia

    Bashir Ahmad

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Ahmad, B. On Nonlocal Boundary Value Problems for Nonlinear Integro-differential Equations of Arbitrary Fractional Order. Results. Math. 63, 183–194 (2013). https://doi.org/10.1007/s00025-011-0187-9

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  • DOI: https://doi.org/10.1007/s00025-011-0187-9

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