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Eigenvalue problem for fractional difference equation with nonlocal conditions
Yongshun Zhao,Shurong Sun
Keywords:boundary value problem; fractional difference equation; positive solutions; eigenvalue problem; fixed-point theorem.
Abstract:
      In this work, we investigate a class of boundary value problem for fractional difference equation with nonlocal conditions $$ \begin{cases}\Delta^{\nu}u(t)+\lambda f(t+\nu-1,u(t+\nu-1))=0, t\in[0,b+1]_{\mathbb{N}_{0}}, \\u(\nu-2)=h(u), \Delta u(\nu+b)=g(u),\end{cases} $$ where $1<\nu\leq2$ is a real number, $f:[\nu-1,\nu+b]_{\mathbb{N}_{\nu-1}}\times\mathbb{R}\rightarrow(0,+\infty)$ is a continuous function, $g,h$ are given functionals, $b\geq2$ is an integer, $\lambda>0$ is a parameter. By upper and lower solutions method, we can present the existence result of positive solutions. The eigenvalue intervals to this problem are studied by the properties of the Green function and Guo-Krasnosel'skii fixed point theorem in cones.