Boundedness character of a class of difference equations

Abstract A complete picture regarding the boundedness character of positive solutions to the following difference equation x n = max { A , x n − 1 p x n − k p } , n ∈ N 0 , where k ≥ 2 and the parameters A and p are positive real numbers, is given. In particular, for the case p k − 1 ∈ ( 0 , k k / ( k − 1 ) k − 1 ) , we prove that all solutions to the equation are bounded. We also present corresponding results concerning the following closely related difference equation x n = A + x n − 1 p x n − k p , n ∈ N 0 .

[1]  Lothar Berg,et al.  On the Asymptotics of Nonlinear Difference Equations , 2002 .

[2]  Ali Gelisken,et al.  On a Max-Type Difference Equation , 2010 .

[3]  J. Feuer.  On the eventual periodicity of with a period-four parameter , 2006 .

[4]  W. T. Patula,et al.  A reciprocal difference equation with maximum , 2002 .

[5]  J. Feuer,et al.  On the eventual periodicity of xn+1 = max(1/xn, An/xn-1) with a period-five parameter , 2008, Comput. Math. Appl..

[6]  H. M. El-Owaidy,et al.  On asymptotic behaviour of the difference equation xn+1=α+(xn-k/xn) , 2004, Appl. Math. Comput..

[7]  G. Ladas,et al.  On the recursive sequence _{+1}=\frac{}_{}+\frac{1}_{-2} , 1998 .

[8]  Wan-Tong Li,et al.  On the recursive sequencexn+1=α-(xn/xn−1) , 2005 .

[9]  William T. Patula,et al.  On a Max Type Recurrence Relation with Periodic Coefficients , 2004 .

[10]  Stevo Stevi´c,et al.  ON THE RECURSIVE SEQUENCE $x_{n+1}=\displaystyle\frac{A}{\prod^k_{i=0}x_{n-i}}+\displaystyle\frac{1}{\prod^{2(k+1)}_{j=k+2}x_{n-j}}$ , 2003 .

[11]  Cengiz Çinar,et al.  On positive solutions of a reciprocal difference equation with minimum , 2005 .

[12]  Kenneth S. Berenhaut,et al.  The behaviour of the positive solutions of the difference equation , 2006 .

[13]  Stevo Stevic,et al.  Quantitative bounds for the recursive sequence yn = A + yn / (yn-k) , 2006, Appl. Math. Lett..

[14]  Stevo Stević,et al.  On the Behaviour of the Solutions of a Second-Order Difference Equation , 2007 .

[15]  T. Sun,et al.  On boundedness of the solutions of the difference equation xn , 2006 .

[16]  Stevo Stević,et al.  On the Recursive Sequence xn=1 , 2007 .

[17]  A. M. Ahmed,et al.  On asymptotic behaviour of the difference equation $$X_{N + 1} = \alpha + \frac{{X_{N - 1} ^P }}{{X_N ^P }}$$ , 2003 .

[18]  Stevo Stević,et al.  On the recursive sequence $$x_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}$$ , 2005 .

[19]  Stevo Stević,et al.  Asymptotic behavior of a class of nonlinear difference equations , 2006 .

[20]  Kenneth S. Berenhaut,et al.  The global attractivity of the rational difference equation _{}=1+\frac{_{-}}_{-} , 2007 .

[21]  C. Kent,et al.  On the Recursive Sequence x n+1= , 2003 .

[22]  Kenneth S. Berenhaut,et al.  Boundedness character of positive solutions of a max difference equation , 2006 .

[23]  G. Ladas,et al.  On the Recursive Sequencexn + 1 = α + xn − 1/xn☆ , 1999 .

[24]  Bratislav Iričanin,et al.  A Global Convergence Result for a Higher Order Difference Equation , 2007 .