Results in Mathematics

, Volume 63, Issue 1–2, pp 183–194

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



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.

Mathematics Subject Classification (2000)

Primary 26A33 Secondary 34B15 


Nonlinear fractional differential equations nonlocal boundary conditions existence fixed point theorem 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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