Positive solutions for boundary value problems involving nonlinear fractional \(q\)-difference equations (Q2852266)

The author investigates the eigenvalue intervals of nonlinear boundary value problems involving fractional \(q\)-difference equations of the form NEWLINE\[NEWLINE\begin{aligned} (D^\alpha_{q}u)(t)+\lambda f(u(t))=0,\, 0<t<1, \\ u(0)=(D_{q}u)(0)= (D_{q}u)(1)=0, \end{aligned}NEWLINE\]NEWLINE where \(2< \alpha \leq 3,\,D\alpha_{q}\) is the Riemann-Liouville fractional \(q\)-derivative, \(\lambda\) is a positive parameter and \(f:[0,1] \times \mathbb{R} \rightarrow \mathbb{R}\) is a nonnegative continuous function. The results are obtained by means of the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones. Furthermore, sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. Examples are presented to illustrate the results.