Eigenvalue problem for some class of nonlinear fractional differential equation (Q1947137)

In this paper, the authors study the eigenvalue problem for the following nonlinear fractional differential equation: \[ D_{0^{+}}^{\alpha }u(t)=\lambda f(u(t)),\,\,\,u(0)=u(1)={u}^{\prime}(1)=0, \] where \(D_{0^{+}}^{\alpha }\) is the Riemann-Liouville fractional derivative with \( 3<\alpha \leq 4,\) \(\lambda \) is a positive parameter and \(f:(0,+\infty )\rightarrow (0,+\infty )\) is a continuous function. Some sufficient conditions for nonexistence and existence of at least one or two positive solutions for above boundary value problem are obtained. For this, with the help of a corresponding fractional Green function, the above problem is reduced to an equivalent Fredholm integral equation and then Guo-Krasnosel'skij's fixed point theorem on cones is used. Finally, some examples are given to illustrate the main results.