A Survey on Existence of a Solution to Singular Fractional Difference Equation

In this paper, we deal with the existence of a positive solution for the following fractional discrete boundary-value problemT+۱∇αk(k∇α۰(UK)))=λƒ(K,U(K)), k∈[۱,T]N۰,u(۰)=u (T+۱)=۰,where ۰<α<۱ and k∇α۰ is the left nabla discrete fractional difference and T+۱∇αk is the right nabla discrete fractional difference ƒ:[۱,T]N۰×(۰,+∞)→R may be singular at t=۰  and may change sign and λ>۰ is a parameter. The technical method is variational approach for differentiable functionals. An example is included to illustrate the main results.

Keywords:

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