On the Recursive Sequence
نویسنده
چکیده
For all values of the parameter γ, (1.1) has a unique positive equilibrium x̄ = (γ + 1)/2. When 0 < γ < 1, the positive equilibrium x̄ is locally asymptotically stable. In the case where γ = 1, the characteristic equation of the linearized equation about the positive equilibrium x̄ = 1 has three eigenvalues, one of which is −1, and the other two are 0 and 1/2. In addition, when γ = 1, (1.1) possesses infinitely many period-two solutions of the form {a,a/(2a− 1),a,a/(2a− 1), . . .} for all a > 1/2. When γ > 1, the equilibrium x̄ is hyperbolic. The investigation of (1.1) has been posed as an open problem in [1, 2]. In this paper, we will show that when 0 < γ < 1, the interval [γ,1] is an invariant interval for (1.1) and that every solution of (1.1) falling into this interval converges to the positive equilibrium x̄. Furthermore, we will show that when γ = 1, every positive solution {xn}n=−2 of (1.1) which is eventually bounded from below by 1/2 converges to a (not necessarily prime) period-two solution. Finally, when γ > 1, we will prove that (1.1) possesses unbounded solutions. We also pose some open questions for (1.1). We say that a solution {xn}n=−k of a difference equation is bounded and persists if there exist positive constants P and Q such that
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