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Solvability of a Three-Point Fractional Nonlinear Boundary Value Problem

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Abstract

In this paper we study the fractional boundary value problem

$$\begin{array}{ll}& ^{c}D_{0^{+}}^{q}u \left( t \right) =f(t, u(t)),\quad 0 < t <1 \\ & u\left( 0 \right) = \alpha u^{\prime} \left( 0 \right) ,\quad u \left( 1\right) =\beta u^{\prime } \left( \eta \right) ,\end{array}$$

where \({1 < q < 2, \alpha , \beta \in IR}\) and \({^{c}D_{0^{+}}^{q}}\) denotes the Caputo’s fractional derivative. Using Banach contraction principle and Leray–Schauder nonlinear alternative we prove the existence and uniqueness of solutions. Some examples are given to illustrate our results.

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Authors and Affiliations

  1. Laboratory of Advanced Materials, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000, Annaba, Algeria

    A. Guezane-Lakoud

  2. Laboratory LASEA, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000, Annaba, Algeria

    R. Khaldi

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  1. A. Guezane-Lakoud
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  2. R. Khaldi
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Correspondence to A. Guezane-Lakoud.

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Guezane-Lakoud, A., Khaldi, R. Solvability of a Three-Point Fractional Nonlinear Boundary Value Problem. Differ Equ Dyn Syst 20, 395–403 (2012). https://doi.org/10.1007/s12591-012-0125-7

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  • DOI: https://doi.org/10.1007/s12591-012-0125-7

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