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DEMONSTRATIO MATHEMATICAVol. XXXVIIINo 32005George L. Karakostas, Stevo StevicON THE RECURSIVE SEQUENCEXn+1 = α +f(xn,...,xn-k+1)Abstract. The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation of the formhXfi+i = α + 77—Γιη = 0,1,...is investigated, where a> 0, k € Ν and / : [0,oo)fc — (0,oo) is a continuous functionnondecreasing in each variable.1. IntroductionWe investigate the behavior of the (positive) solutions of a differenceequation of the form(1)χ η + ι = α + —;j{Xn>·r,· · ι %n—k+1)η = 0,1,...where a is a nonnegative real number. The motivation of this paper is thesecond order nonlinear rational recursive sequence(2)xn+i =—,η = 0,1,...Λ -τ Xjiwhere the parameters Α, Β, Η, A and the initial conditions x_i and XQ arepositive real numbers, discussed in [5]. By the change of variables xn = AynEq. (2) reduces to the difference equationcx\(3),,yn+i =p +qyn +τ—,ryn1 + Vn~l, „n = η 0 ι, 1 , . . .whereαβηa n dr =Ä ·2000 Mathematics Subject Classification: Primary 39A10, 39A11.Key words and phrases·. Equilibrium, positive solution, difference equation, boundedness, global attractivity, asymptotic periodicity.596G. L. Karakostas, S. StevicThe following trichotomy result was proved in [5]:THEOREM A. Consider Eq. (3). If it holds r =
Demonstratio Mathematica – de Gruyter
Published: Jul 1, 2005
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