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.