On fourteen solvable systems of difference equations

Abstract In this paper, we mainly consider the systems of difference equations x n + 1 = 1 + p n q n , y n + 1 = 1 + r n s n , n ∈ N 0 , where each of the sequences p n , q n , r n and s n represents either the sequence x n or the sequence y n , with nonzero real initial values x 0 and y 0 . Then we solve fourteen out of sixteen possible systems. It is noteworthy to depict that the solutions are presented in terms of Fibonacci numbers for twelve systems of these fourteen systems.

[1]  Stevo Stevic,et al.  On a solvable system of difference equations of fourth order , 2013, Appl. Math. Comput..

[2]  Stevo Stevic,et al.  Solutions of a max-type system of difference equations , 2012, Appl. Math. Comput..

[3]  J. Diblík,et al.  On a Third-Order System of Difference Equations with Variable Coefficients , 2012 .

[4]  Stevo Stević,et al.  On the difference equation xn = xn-2/(bn + cnxn-1xn-2) , 2011, Appl. Math. Comput..

[5]  Stevo Stevic,et al.  On a system of difference equations with period two coefficients , 2011, Appl. Math. Comput..

[6]  Stevo Stevic,et al.  On some solvable systems of difference equations , 2012, Appl. Math. Comput..

[7]  Stevo Stevic,et al.  On a third-order system of difference equations , 2012, Appl. Math. Comput..

[8]  Garyfalos Papaschinopoulos,et al.  On the fuzzy difference equation xn+1 = A+B/xn , 2002, Soft Comput..

[9]  Necati Taskara,et al.  The periodicity and solutions of the rational difference equation with periodic coefficients , 2011, Comput. Math. Appl..

[10]  J. Diblík,et al.  On the Difference Equation ${x}_{n}={a}_{n}{x}_{n-k}/({b}_{n}+{c}_{n}{x}_{n-1}\cdots {x}_{n-k})$ , 2012 .

[11]  Louis Brand,et al.  A Sequence Defined by a Difference Equation , 1955 .

[12]  S. Stević.  More on a rational recurrence relation. , 2004 .

[13]  Stevo Stevic,et al.  On a solvable rational system of difference equations , 2012, Appl. Math. Comput..

[14]  Stevo Stevic,et al.  On the difference equation xn = xn-k / (b + cxn-1...xn-k) , 2012, Appl. Math. Comput..

[15]  I. Bajo,et al.  Global behaviour of a second-order nonlinear difference equation , 2009, 0905.3642.

[16]  On the solutions of two special types of Riccati difference equation via Fibonacci numbers , 2013 .

[17]  Stevo Stevic.  On a system of difference equations , 2011, Appl. Math. Comput..

[18]  Stevo Stevic,et al.  On some systems of difference equations , 2011, Appl. Math. Comput..

[19]  Thomas Koshy,et al.  Fibonacci and Lucas Numbers With Applications , 2018 .

[20]  H. Feshbach,et al.  Finite Difference Equations , 1959 .