We obtain solutions to some conjectures about the nonlinear difference equation $$x_{n+1}=\alpha +\beta x_{n-1}{\mathrm{e}}^{-x_n},n=0,1,\cdots ,\alpha ,\beta >0.$$ More precisely, we get not only a condition under which the equilibrium point of the above equation is globally asymptotically stable but also a condition under which the above equation has a unique positive cycle of prime period two. We also prove some further results.

Oscillatory properties of solutions to the system of first-order linear difference equations $$\begin{array}{cc}\hfill \Delta u_k& =q_k{v}_k\hfill \\ \hfill \Delta v_k& =-p_k{u}_{k+1},\hfill \end{array}$$ are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011).

The asymptotic and oscillatory behavior of solutions of Volterra summation equation and second order linear difference equation are studied.