On The Recursive Sequence Xn+1 = Alpha+(xn-1/xn^k) - ResearchGate

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ArticleOn the recursive sequence xn+1 = alpha+(xn-1/xn^k)

• January 2009
• Applied Mathematics Letters 22:91-95

• Source
• DBLP

Authors: A. E. Hamza

• Cairo University

Ahmed Morsy

• Prince Sattam bin Abdulaziz University

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Abstract

In this work we investigate the global behavior of the difference equation xn+1 = α + xn−1 x k n , n = 0, 1, 2, . . . where α ∈ (0,∞) and k ∈ (0,∞).

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On the nonautonomous difference equation x n+1 =A n +x n-1 p x n qArticle

• Feb 2011
• APPL MATH COMPUT

• Garyfalos Papaschinopoulos
• Christos Schinas
• Gesthimani Stefanidou

In this paper we study the asymptotic behavior and the periodicity of the positive solutions of the nonautonomous difference equation:ViewShow abstractDynamics of a higher order rational difference equationArticle

• Oct 2006
• APPL MATH COMPUT

• Mohammad Saleh
• Saida Abu Albaha

In this paper, we will investigate a nonlinear rational difference equation of higher order. Our concentration is on invariant intervals, periodic character, the character of semicycles and global asymptotic stability of all positive solutions of x(n+1) = beta x(n)+gamma x(n-k)/Bx(n)+Cx(n-k), n = 0, 1, ... It is worth to mention that our results solve the open problem proposed by Kulenvic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002]. (c) 2006 Elsevier Inc. All rights reserved.ViewShow abstractOn a max-type and a min-type difference equationArticleFull-text available

• Sep 2009
• APPL MATH COMPUT

• Elsayed M Elsayed
• Bratislav D Iricanin

This note shows that every positive solution to the following third order non–autonomous max-type difference equationwhen (An)n∈N0 is a three-periodic sequence of positive numbers, is periodic with period three. The same result was proved for the following min-type difference equationViewShow abstractOn the recursive sequence y n+1 =(p+y n-1 )/(qy n +y n-1 )ArticleFull-text available

• Nov 2000

• Witold Kosmala
• Mustafa R. S. Kulenović
• G. Ladas
• C. T. Teixeira

ViewOn the behavior of solutions of x n+1 =p+(x n-1 /x n )Article

• Jun 2004

• J. Feuer

Our goal in this article is to complete the study of the behavior of solutions of the equation in the title when the parameter p is positive and the initial conditions are arbitrary positive numbers. Our main focus is the case 0 1.ViewShow abstractAn introduction to difference equations. 2nd edArticle

• Jan 1999

• Saber Elaydi

ViewDynamics of Second Order Rational Difference Equations: With Open Problems and ConjecturesArticle

• Jan 2002

• Mustafa R. S. Kulenović
• G. Ladas

INTRODUCTION AND CLASSIFICATION OF EQUATION TYPES PRELIMINARY RESULTS Definitions of Stability and Linearized Stability Analysis The Stable Manifold Theorem in the Plane Global Asymptotic Stability of the Zero Equilibrium Global Attractivity of the Positive Equilibrium Limiting Solutions The Riccati Equation Semicycle Analysis LOCAL STABILITY, SEMICYCLES, PERIODICITY, AND INVARIANT INTERVALS Equilibrium Points Stability of the Zero Equilibrium Local Stability of the Positive Equilibrium When is Every Solution Periodic with the same Period? Existence of Prime Period Two Solutions Local Asymptotic Stability of a Two Cycle Convergence to Period Two Solutions when C=0 Invariant Intervals Open Problems and Conjectures (1,1)-TYPE EQUATIONS Introduction The Case a=g=A=B=0: xn+1= b xn/C xn-1 The Case a=b=A=C=0: xn+1=g xn-1/B xn Open Problems and Conjectures (1,2)-TYPE EQUATIONS Introduction The Case b=g=C=0: xn+1= a /(A+ B xn) The Case b=g=A=0: xn+1= a /(B xn+ C xn-1) The Case a=g=B=0: xn+1= b xn/(A + C xn-1) The Case a=g=A=0: xn+1= b xn/(B xn+ C xn-1) The Case a=b=C=0: xn+1= g xn-1/(A+ B xn) The Case a=b=A=0: xn+1= g xn-1/(B xn+ C xn-1) Open Problems and Conjectures (2,1)-TYPE EQUATIONS Introduction The Case g=A=B=0: xn+1=(a + b xn)/(C xn-1) The Case g=A=C=0: xn+1=(a + b xn)/B xn Open Problems and Conjectures (2,2)-TYPE EQUATIONS(2,2)- Type Equations Introduction The Case g=C=0: xn+1=(a + b xn)/(A+ B xn) The Case g=B=0: xn+1=(a + b xn)/(A + C xn-1) The Case g=A=0: xn+1=(a + b xn)/(B xn+ C xn-1) The Case b=C=0: xn+1=(a + g xn-1)/(A+ B xn) The Case b=A=0: xn+1=(a + g xn-1)/(B xn+ C xn-1) The Case a=C=0: xn+1=(b xn+ g xn-1)/(A+ B xn) The Case a=B=0: xn+1=(b xn+ g xn-1)/(A + C xn-1) The Case a=A=0: xn+1=(b xn+ g xn-1)/(B xn+ C xn-1) Open Problems and Conjectures (2,3)-TYPE EQUATIONS Introduction The Case g=0: xn+1=(a + b xn)/(A+ B xn+ C xn-1) The Case b=0: xn+1=(a + g xn-1)/(A+ B xn+ C xn-1) The Case a=0: xn+1=(b xn+ g xn-1)/(A+ B xn+ C xn-1) Open Problems and Conjectures (3,2)-TYPE EQUATIONS Introduction The Case C=0: xn+1=(a + b xn+ g xn-1)/(A+ B xn ) The Case B=0: xn+1=(a + b xn+ g xn-1)/(A+ C xn-1) The Case A=0: xn+1=(a + b xn+ g xn-1)/(B xn+ C xn-1) Open Problems and Conjectures THE (3,3)-TYPE EQUATION The (3,3)- Type Equation: xn+1=(a + b xn+ g xn-1 )/(A+ B xn+ C xn-1) Linearized Stability Analysis Invariant Intervals Convergence Results Open Problems and Conjectures APPENDIX: Global Attractivity for Higher Order Equations BIBLIOGRAPHYViewShow abstractOn the recursive sequence x n+1 =α+x n-1 /x nArticle

• Jan 1999

• Amal Amleh
• E.A. Grove
• G. Ladas
• Dimitrios Georgiou

ViewOn the Recursive Sequence x(n+1) = (a + b x(n-1) )/(g + x(n) )Article

• Jan 2000

• Mustafa R. S. Kulenović
• C. H. Gibbons
• G. Ladas

We show that the equation in the title with nonnegative parameters and nonnegative initial conditions exhibits a trichotomy character concerning periodicity, convergence, and boundedness which depends on whether the parameter is equal, less, or greater than the sum of the parameters and A.ViewShow abstractGlobal Behavior of Nonlinear Difference Equations of Higher Order with ApplicationBook

• Jan 1993

• Vlajko Kocic
• G. Ladas

Preface. 1. Introduction and Preliminaries. 2. Global Stability Results. 3. Rational Recursive Sequences. 4. Applications. 5. Periodic Cycles. 6. Open Problems and Conjectures. Appendix: A. The Riccati Difference Equation. B. A Generalized Contraction Principle. C. Global Behaviour of Systems of Nonlinear Difference Equations. Bibliography. Subject Index. Author Index.ViewShow abstractDifference Equations: An Introduction with ApplicationsArticle

• Jan 2001

• W. G Kelley
• Allan Peterson

ViewOn the recursive sequenceArticle

• Jan 2000
• J DIFFER EQU APPL

• Mustafa R. S. Kulenović
• G. Ladas
• N.R. Prokup

We investigate the global stability, and the periodicity of the recursive sequence where the parameters α,β and A and the initial condition x-1 and Xo are non negative real numbers.ViewShow abstract

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