arXiv subjects

Janusz Migda

Publications and source records attributed to Janusz Migda.

3 recordsLinked to original sources

arXiv2015

Qualitative approximation of solutions to difference equations

We present a new approach to the theory of asymptotic properties of solutions of difference equations. Usually, two sequences $x,y$ are called asymptotically equivalent if the sequence $x-y$ is convergent to zero i.e., $x-y\in c_0$, where $c_0$ denotes the space of all convergent to zero sequences. We replace the space $c_0$ by various subspaces of $c_0$. Our approach is based on using the iterated remainder operator. Moreover, we use the regional topology on the space of all real sequences and the `regional' version of the Schauder fixed point theorem.

math.CA↗

arXiv2014

Regional topology and approximative solutions of difference and differential equations

We introduce a topology, which we call the regional topology, on the space of all real functions on a given locally compact metric space. Next we obtain a new versions of Schauder's fixed point theorem and Ascoli's theorem. We use these theorems and the properties of the iterated remainder operator to establish conditions under which there exist solutions, with prescribed asymptotic behavior, of some difference and differential equations.

math.CA↗

arXiv2014

Asymptotically polynomial solutions of difference equations of neutral type

Asymptotic properties of solutions of difference equation of the form \[ Δ^m(x_n+u_nx_{n+k})=a_nf(n,x_{σ(n)})+b_n \] are studied. We give sufficient conditions under which all solutions, or all solutions with polynomial growth, or all nonoscillatory solutions are asymptotically polynomial. We use a new technique which allows us to control the degree of approximation.

math.CA↗