Electron. J. Diff. Equ., Vol. 2012 (2012), No. 98, pp. 1-22.

Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions

Bashir Ahmad, Sotiris K. Ntouyas

Abstract:
We study boundary value problems of nonlinear fractional differential equations and inclusions of order $q \in (m-1, m]$, $m \ge 2$ with multi-strip boundary conditions. Multi-strip boundary conditions may be regarded as the generalization of multi-point boundary conditions. Our problem is new in the sense that we consider a nonlocal strip condition of the form:
$$ x(1)=\sum_{i=1}^{n-2}\alpha_i \int^{\eta_i}_{\zeta_i} x(s)ds, $$
which can be viewed as an extension of a multi-point nonlocal boundary condition:
$$ x(1)=\sum_{i=1}^{n-2}\alpha_i x(\eta_i). $$
In fact, the strip condition corresponds to a continuous distribution of the values of the unknown function on arbitrary finite segments $(\zeta_i,\eta_i)$ of the interval $[0,1]$ and the effect of these strips is accumulated at $x=1$. Such problems occur in the applied fields such as wave propagation and geophysics. Some new existence and uniqueness results are obtained by using a variety of fixed point theorems. Some illustrative examples are also discussed.

Submitted May 20, 2012. Published June 12, 2012.
Math Subject Classifications: 26A33, 34A60, 34B10, 34B15.
Key Words: Fractional differential inclusions; integral boundary conditions; existence; contraction principle; Krasnoselskii's fixed point theorem; Leray-Schauder degree; Leray-Schauder nonlinear alternative; nonlinear contractions.

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Bashir Ahmad
Department of Mathematics, Faculty of Science
King Abdulaziz University
P.O. Box 80203, Jeddah 21589, Saudi Arabia
email: bashir_qau@yahoo.com
Sotiris K. Ntouyas
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: sntouyas@uoi.gr

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