Rim Bourguiba, Faten Toumi
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