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

Ahmad, Bashir ; Ntouyas, Sotiris K.

Electronic journal of differential equations, Tome 2012 (2012)

Résumé

We study boundary value problems of nonlinear fractional differential equations and inclusions of order

$ 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.

Détail

Classification : 26A33, 34A60, 34B10, 34B15
Keywords: fractional differential inclusions, integral boundary conditions, existence, contraction principle, Krasnoselskii's fixed point theorem, Leray-Schauder degree, Leray-Schauder nonlinear alternative, nonlinear contractions