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Abstract
We investigate the asymptotic behavior of the recursive difference equation yn+1 = (α+βy n )/(1+yn-1) when the parameters α < 0 and β ∈ ℝ. In particular, we establish the boundedness and the global stability of solutions for different ranges of the parameters α and β. We also give a summary of results and open questions on the more general recursive sequences yn+1 = (a + by n )/(A + Byn-1), when the parameters a, b, A, B ∈ ℝ and abAB ≠ 0.
Highlights
The monograph by Kulenovicand Ladas [10] presents a wealth of up-to-date results on the boundedness, global stability, and the periodicity of solutions of all rational difference equations of the form xn+1
The nonnegativity of the parameters and the initial conditions ensures the existence of the sequence {xn} for all positive integers n
We give a global stability result for solutions of (1.4) with initial conditions in the invariant interval obtained in the previous section
Summary
The monograph by Kulenovicand Ladas [10] presents a wealth of up-to-date results on the boundedness, global stability, and the periodicity of solutions of all rational difference equations of the form xn+1. The following theorem establishes the stability of the real fixed points of the rational recursion (1.4). When y = y2, we have that β > 1 + 2y2 and it is easy to check that the fixed point y2 is a repeller. We end this section with a theorem giving different bound estimates for positive solutions of recursion (2.1). We present a sequence of lemmas showing the instability of the unique fixed point y = (β − 1)/2.
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