A survey on multiplicity results for fractional difference equations and variational method

In this paper, we deal with the existence and multiplicity solutions, for the following fractional  discrete boundary-value problem begin{equation*} begin{cases} _{T+۱}nabla_k^{alpha}left( ^{}_knabla_{۰}^{alpha}(u(k))right)+{^{}_knabla}_{۰}^{alpha}left( ^{}_{T+۱}nabla_k^{alpha}(u(k))right)=lambda f(k,u(k)), quad k in [۱,T]_{mathbb{N}_{۰}}, u(۰)= u(T+۱)=۰, end{cases} end{equation*} where ۰leq alphaleq۱ and ^{}_{۰}nabla_k^{alpha} is  the left nabla discrete fractional difference  and ^{}_knabla_{T+۱}^{alpha} is the right nabla discrete fractional difference  and   f: [۱,T]_{mathbb{N}_{۰}}timesmathbb{R}tomathbb{R} is a continuous function and lambda>۰ is a parameter. The technical approach is based on the critical point theory and some local minimum theorems for differentiable functionals. Several examples are included to illustrate the main results. textbf{MSC(۲۰۱۰):} ۲۶A۳۳; ۳۹A۱۰; ۳۹A۲۷. textbf{Keywords:}  Discrete fractional calculus, Discrete nonlinear boundary value problem, Non trivial solution, Variational methods, Critical point theory.

Keywords:

Discrete fractional calculus , Discrete nonlinear boundary value problem , Non trivial solution , Variational methods , Critical point theory.