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ISSN: 2333-9721
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Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions

Keywords: 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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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.

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