### A class of generalized uniform asymptotic expansions.

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We consider a discrete Schrödinger operator 𝒥 with Wigner-von Neumann potential not belonging to l². We find the asymptotics of orthonormal polynomials associated to 𝒥. We prove a Weyl-Titchmarsh type formula, which relates the spectral density of 𝒥 to a coefficient in the asymptotics of the orthonormal polynomials.

We provide an elementary proof of the asymptotic behavior of solutions of second order differential equations without successive approximation argument.

Asymptotic forms of solutions of half-linear ordinary differential equation $\left(\right|{u}^{\text{'}}{|}^{\alpha -1}{u}^{\text{'}}{)}^{\text{'}}=\alpha (1+b\left(t\right)){\left|u\right|}^{\alpha -1}u$ are investigated under a smallness condition and some signum conditions on $b\left(t\right)$. When $\alpha =1$, our results reduce to well-known ones for linear ordinary differential equations.

In this paper new generalized notions are defined: $\Psi $-boundedness and $\Psi $-asymptotic equivalence, where $\Psi $ is a complex continuous nonsingular $n\times n$ matrix. The $\Psi $-asymptotic equivalence of linear differential systems ${y}^{\text{'}}=A\left(t\right)y$ and ${x}^{\text{'}}=A\left(t\right)x+B\left(t\right)x$ is proved when the fundamental matrix of ${y}^{\text{'}}=A\left(t\right)y$ is $\Psi $-bounded.

Asymptotic representations of some classes of solutions of nonautonomous ordinary differential $n$-th order equations which somewhat are close to linear equations are established.

In this paper we investigate the problem of existence and asymptotic behavior of solutions for the nonlinear boundary value problem $$\u03f5{y}^{\text{'}\text{'}}+ky=f(t,y),\phantom{\rule{1.0em}{0ex}}t\in \langle a,b\rangle ,\phantom{\rule{4pt}{0ex}}k<0,\phantom{\rule{4pt}{0ex}}0<\u03f5\ll 1$$ satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions and delicate estimations.

It is established convergence to a particular equilibrium for weak solutions of abstract linear equations of the second order in time associated with monotone operators with nontrivial kernel. Concerning nonlinear hyperbolic equations with monotone and conservative potentials, it is proved a general asymptotic convergence result in terms of weak and strong topologies of appropriate Hilbert spaces. It is also considered the stabilization of a particular equilibrium via the introduction of an asymptotically...