On a system of two exponential type difference equations. (English) Zbl 1198.39028

The authors investigate global stability and asymptotic behavior of solutions of the system of two difference equations
\[ y_{n+1}=\biggl(1-\sum_{j=0}^{k-1}z_{n-j}\biggr)\left(1-e^{-By_n}\right),\quad z_{n+1}=\biggl(1-\sum_{j=1}^{k-1}y_{n-j}\biggr)\left(1-e^{-Cz_n}\right) \]
where \(B,C\) are positive real constants. The papers by S. Stević [Discrete Dyn. Nat. Soc. 2007, Article ID 87519, 10 p. (2007; Zbl 1180.39004)] and by D. C. Zhang and B. Shi [J. Math. Anal. Appl. 278, No. 1, 194–202 (2003; Zbl 1025.39013)], where one equation of the above form is investigated, serve as a motivation. A typical result of the paper is the following statement.
Theorem. Suppose that \(0<B\leq 1\), \(0<C\leq 1\) and the positive initial conditions satisfy \(\sum_{j=0}^{k-1}y_{-j}<1\), \(\sum_{j=0}^{k-1}z_{-j}<1\). Then the solution \((y_n,z_n)\) determined by these initial conditions tends to the zero equilibrium \((0,0)\).
A detailed analysis of the solvability of the nonlinear algebraic system describing the equilibria of the investigated system of the two difference equations play an important role in the proofs of the authors’ statements.


39A30 Stability theory for difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
39A10 Additive difference equations