Asymptotically polynomial solutions of difference equations of neutral type

Asymptotic properties of solutions of difference equation of the form $$\Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n$$ 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

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

Asymptotic behavior of solutions to difference equations in Banach spaces

We investigate the asymptotic properties of solutions to higher order nonlinear difference equations in Banach spaces. We introduce a new technique based on a vector version of discrete L’Hospital’s rule, remainder operator, and the regional topology on the space of all sequences on a given Banach space. We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we are dealing with the problem of approximation of solutions. Our technique allows us to control the degree of approximation of solutions

Approximative solutions of difference equations

Asymptotic properties of solutions of difference equations of the form $$\Delta^m x_n=a_nf(n,x_{\sigma(n)})+b_n$$ are studied. Using the iterated remainder operator and fixed point theorems we obtain sufficient conditions under which for any solution $y$ of the equation $\Delta^my=b$ and for any real $s\leq 0$ there exists a solution $x$ of the above equation such that $\Delta^kx=\Delta^ky+\mathrm{o}(n^{s-k})$ for any nonnegative integer $k\leq m$. Using a discrete variant of the Bihari lemma and a certain new technique we give also sufficient conditions under which for a given real $s\leq m-1$ all solutions $x$ of the equation satisfy the condition $x=y+\mathrm{o}(n^s)$ where $y$ is a solution of the equation $\Delta^my=b$. Moreover, we give sufficient conditions under which for a given natural $k<m$ all solutions $x$ of the equation satisfy the condition $x=y+u$ for a certain solution $y$ of the equation $\Delta^my=b$ and a certain sequence $u$ such that $\Delta^ku=\mathrm{o}(1)$

Qualitative approximation of solutions to discrete Volterra equations

We present a new approach to the theory of asymptotic properties of solutions to discrete Volterra equations of the form \begin{equation*} \Delta^m x_n=b_n+\sum_{k=1}^{n}K(n,k)f(k,x_{\sigma(k)}). \end{equation*} Our method is based on using the iterated remainder operator and asymptotic difference pairs. This approach allows us to control the degree of approximation

Asymptotic behavior of solutions of discrete Volterra equations

We consider the nonlinear discrete Volterra equations of non-convolution type $$\Delta^m x_n=b_n+\sum\limits_{i=1}^{n}K(n,i)f\left(i,x_i\right), \quad n\geq 1.$$ We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior, especially asymptotically polynomial and asymptotically periodic solutions. We use $\operatorname{o}(n^s)$, for a given nonpositive real $s$, as a measure of approximation. We also give conditions under which all solutions are asymptotically polynomial

Asymptotic properties of solutions to difference equations of Emden-Fowler type

We study the higher order difference equations of the following form mxn = an f(xσ(n) ) + bn. We are interested in the asymptotic behavior of solutions x of the above equation. Assuming f is a power type function and ∆ myn = bn, we present sufficient conditions that guarantee the existence of a solution x such that xn = yn + o(n s where s ≤ 0 is fixed. We establish also conditions under which for a given solution x there exists a sequence y such that ∆ myn = bn and x has the above asymptotic behavior