Oscillatory Properties of Solutions of the Fourth Order Difference Equations with Quasidifferences

A class of fourth--order neutral type difference equations with quasidifferences and deviating arguments is considered. Our approach is based on studying the considered equation as a system of a four--dimensional difference system. The sufficient conditions under which the considered equation has no quickly oscillatory solutions are given

On the asymptotic behavior of solutions of linear differential equations

In the paper sufficient conditions for the difference equation $$\Delta x_n=\sum^r_{i=0}a_n^{(i)}x_{n+i}$$ to have a solution which tends to a constant, are given. Applying these conditions, an asymptotic formula for a solution of an $m$-th order equation is presented

On the asymptotically periodic solution of some linear difference equations

summary:For the linear difference equation $$x_{n+1} -a_n x_n = \sum _{i=0}^r a_n^{(i)}x_{n+i}, \;\;\; n \in N$$ sufficient conditions for the existence of an asymptotically periodic solutions are given

Conditions for having a diffeomorphism between two Banach spaces

We provide sufficient conditions for a mapping acting between two Banach spaces to be a diffeomorphism. Standart arguments allow us to obtain a local diffeomorphism. It is proved to be global using mountain pass geometry