ESTIMATES OF EIGENVALUES FOR UNIFORMLY ELLIPTIC OPERATOR OF SECOND ORDER

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On stability of the second order neutral differential equation

There exist a well-developed stability theory for neutral differential equations of the first order and only a few results on functional differential equations of the second order. One of the aims of this paper is to fill this gap. Explicit tests for stability of linear neutral delay differential equations of the second order are obtained.     

Positive solutions for quasilinear second order differential equation

It is well known that the Krasnoselskii's fixed point theorem is very very important. It was extensively used for studying the boundary value problems. In this article, the Krasnoselskii's fixed point theorem is extended. The new fixed point theorem is obtained. The second order quasilinear differential equation (Φ (y′))′+a(t)f(t,y,y′)=0,, 0<t<1 subject to mixed boundary condition is studied, where f is a nonnegative continuous function, Φ (v)= |v|^p-2 v, p>1. We show the existence of at least one positive solution by using the new fixed point theorem in cone.     

Nonreal eigenvalues for second order differential operators on networks with circuits

It is shown that on every finite network with at least one circuit there exist second order differential operators having an infinite number of nonreal eigenvalues. The presence of nonreal eigenvalues implies that these operators cannot be selfadjoint with respect to any metric. These eigenvalues reveal also the existence of oscillatory solutions for the corresponding time-dependent partial differential equations.     

On oscillation of nonlinear second order differential equation with damping term

In this paper we investigate the oscillation of nonlinear differdntial equation with damping term. Some new oscillation criteria for the equation are obtained.     

Global monotonicity and oscillation for second order differential equation

Oscillatory properties of the second order nonlinear equation

Oscillation theorems for a second order delay differential equation

A generalized version is proved of the following inequality, arising in a study of invertible measure preserving transformations: $\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}{}^{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}{}^{\mathrm{m}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>m</mtext>}}\text{? (∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}^{\mathrm{mn}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>mn</mtext>}}\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}\text{),}\text{where}\text{x}{}_{\mathrm{i}}\text{? 0, i = 1, 2,…, N,}\text{and}\text{(m ? 1)(n ? 1) > 0}$.     

Measure algebras associated with a second order differential operator

Let $\text{L =}\text{d}{}^2\text{dx}{}^2\text{? q(x)}$ be a Sturm-Liouville operator acting on functions defined on R. The authors have recently shown how to construct commutative associative algebras of distributions of compact support for which L is a centralizer (in the sense that $\text{L(f 1 g) = (Lf) 1 g}$ for distributions f, g of compact support) when q is locally bounded. Here, it is assumed either that q is bounded and $\text{x → (1 + ¦ x ¦) q(x)}$ is integrable, or that q is of bounded variation. A function ψ is then found such that $\text{M}$ψ={μ : μ is a measure on R and | μ |(ψ) < & infin;} becomes a Banach algebra containing the algebra of measures of compact support. The representation theory of $\text{M}$_ψ is discussed and conditions for its semisimplicity are obtained.     

Linear differential operators and operator matrices of the second order

Linear differential operators (equations) of the second order in Banach spaces of vector functions defined on the entire real axis are studied. Conditions of their invertibility are given. The main results are based on putting a differential operator in correspondence with a second-order operator matrix and further use of the theory of first-order differential operators that are defined by the operator matrix. A general scheme is presented for studying the solvability conditions for different classes of second-order equations using second-order operator matrices. The scheme includes the studied problem as a special case.     

A boundary value problem for a differential equation of second order

We find the spectrum and prove a theorem on the expansion of an arbitrary function satisfying certain smoothness conditions in terms of the root functions of a boundary value problem of the type ?y″+q(x)+a/x²y=λy, y(0)=0, M(λ) y(a)+N(λ) y(b)=0, where 0