EXISTENCE OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENCE EQUATIONS WITH INITIAL CONDITIONS

In this paper, we use the well-known upper and lower solution method to prove the existence of solutions for a nonlinear nabla fractional difference equation with an initial condition on the finite discrete domain. We prove that the solution of the initial value problem stays between lower and upper solutions under some assumptions on the nonlinear term. We illustrate the main result of the paper with two examples where one can see the necessity of the well-order in the lower and upper solutions (i.e. a lower solution is less than or equal to an upper solution) to obtain a solution between the upper and lower solutions. As an application, we use the quasilinearization method to construct two sequences of upper and lower solutions which converge to the solution of the initial value problem.