Ismaiel Krim, Hamza Tabti

Nonlinear Fractional Differential Equations with Integral Boundary Conditions and Parameter Dependence

abstract:
This work is concerned with studying the existence of positive solutions for nonlinear fractional differential equations problems with integral boundary conditions and parameter dependence:
\begin{gather*}
D^{\beta}_{0^{+}}u(t) + f(t,u(t))=0, \;\;0<t<1, \\
u(0) =u'(0)=0, \;\;u(1)=\lambda\int\limits_{0}^{1}\psi(r)u(r)\,dr,
\end{gather*}
where $2<\beta\leq3$ and $\lambda>0$, $D^{\beta}_{0^{+}}$ is the Riemann-Liouville fractional derivative, $f$ is a continuous function and $\psi$ is a continuous function on $[0,1]$. Using the fixed point theorem on the cone, we show when this type of problem has at least one solution.

Mathematics Subject Classification: 26A33, 34B18

Key words and phrases: Fractional differential equations, Fixed point theorem, Riemann-Liouville fractional derivative, Positive solutions, Green's function, (superlinear) condition