Behavior of some higher order nonlinear rational partial difference equations

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Abstract In this paper we give the closed form expressions of some higher order nonlinear rational partial difference equations in the form X n , m = X n − r , m − r Ψ + ∏ i = 1 r X n − i , m − i where n , m ∈ N and the initial values X n, t , X t , m − r are real numbers with t ∈ { 0 , − 1 , − 2 , … , − r + 1 } such that ∏ j = 0 r − 1 X j − r + 1 , i + j − r + 1 ≠ − Ψ and ∏ j = 0 r − 1 X i + j − r + 2 , j − r + 1 ≠ − Ψ , i ∈ N 0 . We will use a new method to prove the results by using what we call ‘piecewise double mathematical induction’ which we introduce here for the first time as a generalization of many types of mathematical induction. As a direct consequences, we investigate and conclude the explicit solutions of some higher order ordinary difference equations.