Rim Bourguiba, Faten Toumi

Existence Results of a Singular Fractional Differential Equation with Perturbed Term

abstract:
The boundary value problem
\begin{gather*}
D^{\alpha}u(t)+\mu a(t) f(t,u(t))-q(t)=0, \\ u(0)=u^{\prime }(0)=\cdots=u^{(n-2)}(0)=0, \quad u(1)=\lambda\int\limits_{0}^{1}u(s)\,ds
\end{gather*}
is studied, where $\mu$ is a positive parameter, $f:[0,1]\times[0;+\infty)\to[0;+\infty)$ and $a:(0,1)\to [0,+\infty)$ are continuous functions, while $q:(0,1)\to [0,+\infty)$ is a measurable function. The case, where the function $a$ has singularities at the points $t=0$ and $t=1$, is admissible.

Conditions are found guaranteeing, respectively, the existence of at least one and at least two positive solutions. Examples are gives.

Mathematics Subject Classification: 34A08, 34B18, 35G60

Key words and phrases: Fractional differential equation, positive solution, integral boundary conditions, Green's function, dependence on a parameter, perturbed term