General nonlinear Volterra difference equations with infinite delay are considered. A new explicit criterion for global exponential stability is given. Furthermore, we present a stability bound for equations subject to nonlinear perturbations. Two examples are given to illustrate the results obtained.

In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence $$x_{n+1}=\frac{a_0{x}_n+a_1{x}_{n-1}+\cdots +a_k{x}_{n-k}}{b_0{x}_n+b_1{x}_{n-1}+\cdots +b_k{x}_{n-k}},n=0,1,\cdots$$ where the parameters $a_i$ and $b_i$ for $i=0,1,\cdots ,k$ are positive real numbers and the initial conditions $x_{-k},x_{-k+1},\cdots ,x_0$ are arbitrary positive numbers.

The main objective of this paper is to study the boundedness character, the periodic character, the convergence and the global stability of positive solutions of the difference equation $$x_{n+1}=\left(A+\sum _{i=0}^k{\alpha }_i{x}_{n-i}\right)/\sum _{i=0}^k{\beta }_i{x}_{n-i},n=0,1,2,\cdots$$ where the coefficients $A$, ${\alpha }_i$, ${\beta }_i$ and the initial conditions $x_{-k},x_{-k+1},\cdots ,x_{-1},x_0$ are positive real numbers, while $k$ is a positive integer number.

The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation $$x_{n+1}=\frac{{\alpha }_0{x}_n+{\alpha }_1{x}_{n-l}+{\alpha }_2{x}_{n-k}}{{\beta }_0{x}_n+{\beta }_1{x}_{n-l}+{\beta }_2{x}_{n-k}},n=0,1,2,\cdots$$ where the coefficients ${\alpha }_i,{\beta }_i\in (0,\infty )$ for $i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $x_{-k},\cdots ,x_{-l},\cdots ,x_{-1},x_0$ are arbitrary positive real numbers such that $l<k$. Some numerical experiments are presented.