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In this paper we establish sufficient conditions for the oscillation of all solutions of equation $$\begin{aligned} \varDelta ^4x(n)+p(n)\varDelta x(n+1)+q(n)x(n-\tau )=0 \end{aligned}$$ Δ 4 x ( n ) + p ( n ) Δ x ( n + 1 ) + q ( n ) x ( n - τ ) = 0 via comparison with some first order delay difference equations whose oscillatory characters are known. The presented criterion is easily verifiable. Examples are also given to illustrate the main result.

We consider discrete Sturm–Liouville-type equations of the form $$\begin{aligned} \varDelta (r_n\varDelta x_n)=a_nf(x_{\sigma (n)})+b_n. \end{aligned}$$ Δ ( r n Δ x n ) = a n f ( x σ ( n ) ) + b n . We present a theory of asymptotic properties of solutions which allows us to control the degree of approximation. Namely, we establish conditions under which for a given sequence y which solves the equation $$\varDelta (r_n\varDelta y_n)=b_n$$ Δ ( r n Δ y n ) = b n , the above equation possesses a solution x with the property $$x_n=y_n+\mathrm {o}(u_n)$$ x n = y n + o ( u n ) , where u is a given positive, nonincreasing sequence. The obtained results are applied to the study of asymptotically periodic solutions. Moreover, these results also allow us to obtain some nonoscillation criteria for the classical Sturm–Liouville equation.

We consider the difference equation of the form ?(rn?(pn?xn)) = anf (x?(n)) + bn. We present sufficient conditions under which, for a given solution y of the equation ?(rn?(pn?yn)) = 0, there exists a solution x of the nonlinear equation with the asymptotic behavior xn = yn + zn, where z is a sequence convergent to zero. Our approach allows us to control the degree of approximation, i.e., the rate of convergence of the sequence We examine two types of approximation: harmonic approximation when zn = o(ns), s ? 0, and geometric approximation when zn = o(?n), ? ? (0, 1).

We consider equations of the form Δ 2 ( r n Δ 2 x n ) = a n f ( x σ ( n ) ) + b n , where a , b are sequences of real numbers, r is a sequence of positive real numbers, σ a sequence of integers. Let Y denote the space of all solutions of the equation Δ 2 ( r n Δ 2 y n ) = 0 , and let s be a fixed nonpositive real number. We present sufficient conditions under which for a given sequence y ∈ Y there exists a solution x with the asymptotic behavior x n = y n + o ( n s ) . Moreover, we establish conditions under which for a given solution x there exists a sequence y ∈ Y such that x has the above asymptotic behavior. The obtained results are applied to the study of solutions of discrete Euler–Bernoulli beam equation.

In this paper we investigate the Volterra difference equation of the form $ \D(r_n\D x_n)=b_n+\sum_{k=1}^{n}K(n,k)f(x_k). $ We establish sufficient conditions for the existence of a solution $x$ of the above equation with the property $ x_n=y_n+\o(n^s), $ where $y$ is a given solution of the equation $\D(r_n\D y_n)=b_n$ and $s$ is nonpositive real number. We also obtain sufficient conditions for the existence of asymptotically periodic solutions.

In this paper, we obtain sufficient conditions for the existence of asymptotically periodic solutions of the second order difference equations of the form Δ ( r n Δ x n ) = a n f ( x n − τ ) + b n . Moreover, we give sufficient conditions under which every bounded solution of the above equation is asymptotically periodic. Our approach allows us to control the degree of approximation.

We study the asymptotic properties of solutions to nonautonomous difference equations of the form Δ m x n = a n f ( n , x σ ( n ) ) + b n , f : 𝕅 × 𝕉 → 𝕉 , σ : 𝕅 → 𝕅 $${\Delta ^m}{x_n} = {a_n}f(n,{x_{\sigma (n)}}) + {b_n},\,\,f:N \times {\Bbb R} \to {\Bbb R},\,\,\sigma :{\Bbb N} \to {\Bbb N}$$ Using the iterated remainder operator and asymptotic difference pairs we establish some results concerning approximative solutions and approximations of solutions. Our approach allows us to control the degree of approximation.

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.

In the paper the general higher order difference equation \begin{document}$(-1)^z Δ^m x(n)=f(n, x(σ_1(n)),x(σ_2(n)),..., x(σ_k(n)))$ \end{document} with several deviating arguments is considered. According to the kind of the deviations \begin{document} $σ_i$ \end{document} sufficient conditions for the equation to have property A and B are established.

The 2-dimensional system of neutral type nonlinear difference equations with delays in the following form \begin{document}$\left\{ \begin{align}&Δ≤(x_1(n)-p_1(n)\,x_1(n-τ_1))=a_1(n)\,f_1(x_1(n-σ_1),x_2(n-σ_2))\\&Δ≤(x_2(n)-p_2(n)\,x_2(n-τ_2))=a_2(n)\,f_2(x_1(n-σ_3),x_2(n-σ_4)),\end{align} \right.$ \end{document} is considered. In this paper we use Schauder's fixed point theorem to study the existence of periodic solutions of the above system.

A higher order nonlinear neutral difference equation with quasi-differences is studied. Sufficient conditions are established under which for every real constant there exists a solution of considered equation convergent to this constant.

We study a third-order delay trinomial difference equation. We transform this equation to a binomial third-order difference equation with quasidifferences. Using comparison theorems with a certain first order delay difference equation we establish results on asymptotic properties of nonoscillatory solutions of the studied equation. We give an easily verifiable criterium which ensures that all nonoscillatory solutions tend to zero.

The $k$-dimensional system of neutral type nonlinear difference equations with delays in the following form \begin{equation*} \begin{cases} \Delta \Big(x_i(n)+p_i(n)\,x_i(n-\tau_i)\Big)=a_i(n)\,f_i(x_{i+1}(n-\sigma_i))+g_i(n),\\ \Delta \Big(x_k(n)+p_k(n)\,x_k(n-\tau_k)\Big)=a_k(n)\,f_k(x_1(n-\sigma_k))+g_k(n), \end{cases} \end{equation*} where $i=1,\dots,k-1$, is considered. The aim of this paper is to present sufficient conditions for the existence of nonoscillatory bounded solutions of the above system with various $(p_i(n))$, $i=1,\dots,k$, $k\geq 2$.

The system of neutral type difference equations with delays {?(x(n) + p(n) x(n - ?))= a(n) f(y(n - l)) ?y(n) = b(n) g(z(n - m)) ?z(n) = c(n) h(x(n - k)) is considered. The aim of this paper is to present sufficient conditions for the existence of nonoscillatory bounded positive solutions of the considered system with various (p(n)).

We study a scalar linear difference equation with several delays by transforming it to a system of Volterra equations without delays. The results obtained for this system are then used to establish oscillation criteria and asymptotic properties of solutions of the considered equation.

We give sufficient conditions for the boundedness of all solutions of some classes of systems of difference equations with delays, by comparison of their norms with the solution of certain auxiliary scalar difference equations.

We study asymptotic behavior of nonoscillatory solutions to second-order neutral difference equation of the form: &Delta, ( r n &Delta, ( x n + p n x n &minus, &tau, ) ) = a n f ( n , x n ) + b n . The obtained results are based on the discrete Bihari type lemma and a Stolz type lemma.

We study the neutral difference equation of the form Explicit sufficient conditions which guarantee that all solutions of the above equation are oscillatory are obtained.

We give sufficient conditions, under which for every real constant, there exists a solution of the nonlinear discrete Volterra equationconvergent to this constant. We give also conditions under which all solutions are asymptotically constant. Sufficient conditions for the existence of asymptotically periodic solutions of the above equation are also derived.

In this paper the nonlinear second-order neutral difference equation of the following form: $\Delta ( a_{n}\Delta(x_{n}-p_{n}x_{n-1}) ) + q_{n}f(x_{n-\tau})=0$ is considered. By suitable substitution the above equation is transformed into a new one, which is a third-order non-neutral difference equation. Using results obtained for the new equation, the asymptotic properties of the neutral difference equation are studied. Some classification of nonoscillatory solutions is presented, as well as an estimation of the solutions. Finally, we present necessary and sufficient conditions for the existence of solutions to both considered equations being asymptotically equivalent to the given sequences.

The goal of this paper is to give sufficient conditions for the existence of bounded solutions of a system of discrete equations u ( k + 1 ) = F ( k , u ( k ) ) , k = a , a + 1 , a + 2 , … with a ∈ N , N = { 0 , 1 , … } and F : N ( a ) × R n → R n . A principle giving sufficient conditions for the existence of at least one initial problem generating solution with its graph staying in a previously prescribed domain for every k ∈ { a , a + 1 , a + 2 , … } is formulated. Illustrative examples are considered as well.

In this paper, we study asymptotic behavior of solutions of second-order neutral difference equation Δ 2 ( x n + px n - k ) + f ( n , x n ) = 0 . We present conditions under which all nonoscillatory solutions of the above equation have the property x n = cn + o ( n ) for some c ∈ R as well as sufficient conditions under which all nonoscillatory solutions are asymptotically linear.

In this note we consider the third order linear difference equations of neutral type \[ \Delta ^{3}[x(n)-p(n)x(\sigma (n))]+\delta q(n)x(\tau (n))=0, \quad n \in N(n_0), \qquad \mathrm{({\mathrm E})}\] where $\delta =\pm 1$, $p,q\: N(n_0)\rightarrow \mathbb R_+;$ $\sigma ,\tau \: N(n_0)\rightarrow \mathbb N$, $\lim _{n \rightarrow \infty }\sigma (n)= \lim \limits _{n \rightarrow \infty }\tau (n)= \infty .$ We examine the following two cases: \[ \BOF\align \lbrace 0 1, \ \sigma (n)=n-k,\ \tau (n)=n-l\rbrace , \BOF\endalign \] where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.

The first-order nonlinear difference equations of neutral type @D(x(n)-p(n)x(@t(n)))+@d@?j=1mq"j(n)f"j(x(@s"j(n)))=0,(E@d) where @d = +/-1, N(n"0) = {n"0, n"0 + 1, }, p, q"j : N(n"0) -> R"+, @t,@s"j : N(n"0) -> N, and lim"n"->"~ @t(n) = lim"n"->"~ @s"j"("n") = ~, @?"j : R -> R, j = 1,2,, m, are investigated. All nonoscillatory solutions of these equations are classified according to their asymptotic behavior. Criteria for oscillation of all solutions of these equations are established.

We consider a class of fourth-order nonlinear difference equations of the form @D(a"n@D(b"n@D(c"n@Dy"n)))+f(n,y"n)=0, where a, b, c : N->@?"+ and (a"n) is bounded away from zero, a"n >= c"n, the series @S"i"="1^~ (1/a"i) is convergent and @S"j"="1^~ (1/c"j) @S"i"="j^~ (1/b"i) is divergent. The classification of nonoscillatory solutions is given. Sufficient conditions for the above equation are obtained to admit the existence of nonoscillatory solutions with special asymptotic properties.

A class of monotone sequences is introduced. A classification of all such monotone solutions of odd order neutral difference equation is established. Necessary and/or sufficient conditions are then derived for the existence of these solutions.

Asymptotic behavior of solutions of the mth-order difference equation of the form (E1) Δ m x n + ƒ(n, x n ,…, Δ m−1 x n ) = h n and some special case (E2) of these equation are investigated.

In this paper, using the fixed point theorem, we investigate the boundedness and asymptotic stability of the zero solution of the discrete Volterra equation