Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 98, pp. 1-22.
Title: Nonlinear fractional differential equations and inclusions
of arbitrary order and multi-strip boundary conditions
Authors: Bashir Ahmad (King Abdulaziz Univ., Jeddah, Saudi Arabia)
Sotiris K. Ntouyas (Univ. of Ioannina, Greece)
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.