Abstract
Let k be a nonnegative integer and c a real number greater than or equal to 1. We present qualitative global behavior of solutions to a rational nonlinear higher-order difference equation zn+1=(czn+zn-k+c-1znzn-k)/(znzn-k+c), n≥0, with positive initial values z-k,z-k+1,⋯,z0, and show the global asymptotic stability of its positive equilibrium solution.
Highlights
Difference equations have wide applications in biology, computer science, digital signal processing, and economics
We present qualitative global behavior of solutions to a rational nonlinear higher-order difference equation zn+1 = (c(zn + zn−k) + (c − 1)znzn−k)/(znzn−k + c), n ≥ 0, with positive initial values z−k, z−k+1, ⋅ ⋅ ⋅, z0, and show the global asymptotic stability of its positive equilibrium solution
A general solution structure exists for linear difference equations [ ]
Summary
Difference equations have wide applications in biology, computer science, digital signal processing, and economics. We present qualitative global behavior of solutions to a rational nonlinear higher-order difference equation zn+1 = (c(zn + zn−k) + (c − 1)znzn−k)/(znzn−k + c), n ≥ 0, with positive initial values z−k, z−k+1, ⋅ ⋅ ⋅ , z0, and show the global asymptotic stability of its positive equilibrium solution.
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