A survey on existence of solution to a singular fractional difference boundary value problem

Document Type : Research Paper

Author

Department of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, Sari, Iran

10.22128/gadm.2024.764.1101

Abstract

In this paper, we deal with the existence of a positive solution for the following fractional discrete boundary-value problem

\begin{equation*}

\begin{cases}

_{T+1}\nabla_k^{\alpha}\left( ^{}_k\nabla_{0}^{\alpha}(u(k))\right)=\lambda f(k,u(k)), \ \ k \in [1,T]_{\mathbb{N}_{0}},\\

u(0)= u(T+1)=0,

\end{cases}

\end{equation*}

where $0< \alpha<1$ and $^{}_k\nabla_{0}^{\alpha}$ is the left nabla discrete fractional difference and $^{}_{T+1}\nabla_k^{\alpha}$ is the right nabla discrete fractional difference $f: [1,T]_{\mathbb{N}_{0}}\times (0,+\infty)\to\mathbb{R}$ may be singular at $t=0$ and may change sign.

and $\lambda>0$ is a parameter. The technical method is variational approach for differentiable functionals. An example is included to illustrate the main results.

Keywords