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| ≠ 1" or "μ...

The classical system of functional equations      $1/n{\sum }_{\nu =0}^{n-1}F((x+\nu )/n)=n^{-s}F\left(x\right)$ (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to      $1/n{\sum }_{\nu =0}^{n-1}F((x+\nu )/n)={\sum }_{d=1}^{\infty }{\lambda }_n\left(d\right)F\left(dx\right)$ (n ∈ ℕ) with complex valued sequences ${\lambda }_n$. This leads to new results on the periodic integrable and the aperiodic continuous solutions F:ℝ₊ → ℂ interrelating the theory of functional equations and the theory of arithmetic functions.

The nonlinear difference equation $$x_{n+1}-x_n=a_n{\varphi }_n\left(x_{\sigma \left(n\right)}\right)+b_n,\left(\text{E}\right)$$ where $\left(a_n\right),\left(b_n\right)$ are real sequences, ${\varphi }_nℝ⟶ℝ$, $\left(\sigma \right(n\left)\right)$ is a sequence of integers and ${lim}_{n⟶\infty }\sigma \left(n\right)=\infty$, is investigated. Sufficient conditions for the existence of solutions of this equation asymptotically equivalent to the solutions of the equation $y_{n+1}-y_n=b_n$ are given. Sufficient conditions under which for every real constant there exists a solution of equation () convergent to this constant are also obtained.

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 of noncompactness...

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...

The purpose of this paper is to give some results on the asymptotic relationship between the solutions of a linear difference equation and its perturbed nonlinear equation.