Existence of solutions for a nonlinear fractional order differential equation

E. Kaufmann, University of Arkansas at Little Rock, Little Rock, AR, U.S.A.
Kouadio D. Yao, University of Arkansas at Little Rock, Little Rock, AR, U.S.A.

E. J. Qualitative Theory of Diff. Equ., No. 71. (2009), pp. 1-9.

Abstract: Let $D^\alpha$ denote the Riemann-Liouville fractional differential operator of order $\alpha$. Let $1 < \alpha < 2$ and $0 < \beta < \alpha$. Define the operator $L$ by $L = D^\alpha - a D^\beta$ where $a \in \mathbb{R}$. We give sufficient conditions for the existence of solutions of the nonlinear fractional boundary value problem
\begin{eqnarray*}
&&Lu(t) + f(t, u(t)) = 0, \quad 0 < t < 1,\\
&&u(0) = 0, \, u(1)= 0.
\end{eqnarray*}

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