In this paper, we introduce an integral operator on the unit ball $\BB\subset\CC^n$ . The boundedness and compactness of the operator from the Zygmund space to the Bloch-type space ${\mathcal B}_\mu(\BB)$ or the little Bloch-type space ${\mathcal B}_{\mu,0}(\BB)$ are investigated.

Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.

We show that the difference equation where , has a positive solution converging to zero, by finding a finite asymptotic expansion of the solution. We also show that if and are arbitrarily given positive numbers, then there exists a solution of the equation such that the subsequences , have partial sums of exponential power series as finite asymptotic expansions. Finally, we sketch the case when q of the limits ( ) are vanishing, where we determine only heuristically the next term of the asymptotic expansion.

It is shown that the following system of difference equations $$z_{n+1}=\frac{z_{n}^{a}}{w_{n-1}^{b}},\qquad w_{n+1}=\frac{w_{n}^{c}}{z_{n-1}^{d}}, \quad n\in \mathbb {N}_{0}, $$ where $a,b,c,d\in \mathbb {Z}$ , $z_{-1}, z_{0}, w_{-1}, w_{0}\in \mathbb {C}$ , is solvable in closed form.

We show that the following system of difference equations where , , , and sequences , , and are real, can be solved in closed form. For the case when the sequences , , and are constant and , we apply obtained formulas in the investigation of the asymptotic behaviour of well-defined solutions of the system. We also find domain of undefinable solutions of the system. Our results considerably extend and improve some recent results in the literature.