In the paper we consider the difference equation of neutral type $${\Delta }^3[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,n\in \mathbb{N}\left(n_0\right),$$ where $p,q:\mathbb{N}\left(n_0\right)\to {ℝ}_{+}$; $\sigma ,\tau :\mathbb{N}\to \mathbb{Z}$, $\sigma$ is strictly increasing and $\underset{n\to \infty }{lim}\sigma \left(n\right)=\infty ;$ $\tau$ is nondecreasing and $\underset{n\to \infty }{lim}\tau \left(n\right)=\infty$, $f:ℝ\to ℝ$, $xf\left(x\right)>0$. We examine the following two cases: $$0<p\left(n\right)\le {\lambda }^{*}<1,\sigma \left(n\right)=n-k,\tau \left(n\right)=n-l,$$ and $$1<{\lambda }_{*}\le p\left(n\right),\sigma \left(n\right)=n+k,\tau \left(n\right)=n+l,$$ where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\to \infty$ with a weaker assumption on $q$ than the usual assumption $\sum _{i=n_0}^{\infty }q\left(i\right)=\infty$ that is used in literature.

In this paper we consider the first order difference equation in a Banach space $\Delta x_n={\sum }_{i=0}^{\infty }{a}_n^{i}f\left(x_{n+i}\right)$. We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation $\Delta x_n={\sum }_{i=0}^{\infty }{a}_n^{i}g\left(x_{n+i}\right)+{\sum }_{i=0}^{\infty }{b}_n^{i}h\left(x_{n+i}\right)+y_n$ and give conditions when this equation has solutions. In this note we extend the results from [8,9]. For example, in [9] the function f is a real Lipschitz function. We suppose that f has values in a Banach space and satisfies some conditions with respect to the measure...

This paper is concerned with the delay partial difference equation (1) $A_{m+1,n}+A_{m,n+1}-A_{m,n}+{\Sigma }_{i=1}^u{p}_i{A}_{m-k_i,n-l_i}=0$ where $p_i$ are real numbers, $k_i$ and $l_i$ are nonnegative integers, u is a positive integer. Sufficient and necessary conditions for all solutions of (1) to be oscillatory are obtained.

We consider the Cauchy problem in an unbounded region for equations of the type either $D_{t}z(t,x)=f(t,x,z(t,x),z_{(t,x)},D_{x}z(t,x))$ or $D_{t}z(t,x)=f(t,x,z(t,x),z,D_{x}z(t,x))$. We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.

In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation $$x_{n+1}=\frac{a{x}_n{x}_{n-1}}{-b{x}_n+c{x}_{n-2}},n\in {\mathbb{N}}_0$$ where $a$, $b$, $c$ are positive real numbers and the initial conditions $x_{-2}$, $x_{-1}$, $x_0$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a<c$ or $a>c$ with $(a-c)/b<1$. When $a>c$ with $(a-c)/b>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.

For nonlinear difference equations, it is difficult to obtain analytic solutions, especially when all the eigenvalues of the equation are of absolute value 1. We consider a second order nonlinear difference equation which can be transformed into the following simultaneous system of nonlinear difference equations: ⎧ x(t+1) = X(x(t),y(t)) ⎨ ⎩ y(t+1) = Y(x(t), y(t)) where $X(x,y)=\lambda ₁x+\mu y+{\sum }_{i+j\ge 2}{c}_{ij}{x}^i{y}^j$, $Y(x,y)=\lambda ₂y+{\sum }_{i+j\ge 2}{d}_{ij}{x}^i{y}^j$ satisfy some conditions. For these equations, we have obtained analytic solutions in the cases "|λ₁| ≠ 1 or |λ₂|...

The main result of the paper estimates the asymptotic behavior of local polynomial approximation for $L_p$ functions at a point via the behavior of μ-differences, a generalization of the kth difference. The result is applied to prove several new and extend classical results on pointwise differentiability of $L_p$ functions including Marcinkiewicz-Zygmund’s and M. Weiss’ theorems. In particular, we present a solution of the problem posed in the 30s by Marcinkiewicz and Zygmund.

We investigate the distribution of zeros and shared values of the difference operator on meromorphic functions. In particular, we show that if f is a transcendental meromorphic function of finite order with a small number of poles, c is a non-zero complex constant such that ${\Delta }_c^{k}f\ne 0$ for n ≥ 2, and a is a small function with respect to f, then $fⁿ{\Delta }_c^{k}f$ equals a (≠ 0,∞) at infinitely many points. Uniqueness of difference polynomials with the same 1-points or fixed points is also proved.

We consider the equation $$x^{\text{'}\text{'}}+a^2\left(t\right)x=0,a\left(t\right):=a_k\text{if}{t}_{k-1}\le t<t_k,\text{for}k=1,2,...,$$ where $\left\{a_k\right\}$ is a given increasing sequence of positive numbers, and $\left\{t_k\right\}$ is chosen at random so that $\{t_k-t_{k-1}\}$ are totally independent random variables uniformly distributed on interval $[0,1]$. We determine the probability of the event that all solutions of the equation tend to zero as $t\to \infty$.

In the paper we study the subject of stability of systems with $h$-differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$-difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$-orders.