Stability Analysis of a k-Delayed Rational Difference Equation with Exponential Response

In this paper, we study the asymptotic and boundedness behavior of the solutions of the difference equation xn+1 = αe−xn + βxn−1 (1) A + Bxn + Cxn−k n = 0, 1, 2, . . . , where α, β, A, B, C are positive real numbers, k ∈ {1, 2, . . . }, and initial conditions x−k , x−k+1, . . . , x−1, x0 are arbitrary positive real numbers. Key words: Equilibrium point, Local asymptotic stability, Global asymptotic stability, Positive solutions, Boundedness, Periodicity.