Yuri V. Rogovchenko

Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations

where the nonlinearity f satis9es certain monotonicity conditions speci9ed in the sequel. We always assume without mentioning that the function f(t; u; v) is continuous in the domain D= {(t; u; v): t ∈ [t0;+∞); u; v∈R}, where t0 ? 1. It is supposed also that all equations and inequalities involving t hold for t? t0 unless otherwise speci9ed, the symbols “o” and “O” have their standard meanings as t → ∞; and we use the notation R+ def =(0;+∞). Eq. (1) is often used for mathematical modelling of various physical, chemical and biological systems and attracts constant interest of researchers. A great deal of papers published during the last three decades are concerned with local and global existence of solutions of Eq. (1) and its numerous particular cases, uniqueness of solutions,

A Contribution to the Study of Functional Differential Equations with Impulses

A periodic boundary value problem for a special type of functional differential equa- tions with impulses at fixed moments is studied. A comparison result is presented that allows to construct a sequence of approximate solutions and to give an existence result. Several particular cases are considered.

On oscillation of nonlinear second order differential equation with damping term

We present new oscillation criteria for the second order nonlinear differential equation with damping. Our theorems are stated in general form; they complement and extend related results known in the literature. The relevance of our results is illustrated with a number of examples. To facilitate computations in these examples, we use Mathematica^(R), a symbolic computer language.

Asymptotic integration of a class of nonlinear differential equations

This work is concerned with the behavior of solutions of a class of second-order nonlinear differential equations locally near infinity. Using methods of the fixed point theory, the existence of solutions with different asymptotic representations at infinity is established. A novel technique unifies different approaches to asymptotic integration and addresses a new type of asymptotic behavior.

Existence and asymptotic behavior of solutions of a boundary value problem on an infinite interval

In this paper, we shall study a boundary value problem on an infinite interval involving a semilinear second-order differential equation. Existence result extending recent researches is obtained by using a fixed-point theorem due to Furi and Pera. Asymptotic behavior of solutions and their first-order derivatives at infinity is discussed. Comparison with relevant known results in literature is also made.

Oscillation theorems for differential equations with a nonlinear damping term

In this paper, we are concerned with a class of nonlinear second-order differential equations with a nonlinear damping term. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity, and, as a consequence, our results apply to wider classes of nonlinear differential equations. Two illustrative examples are considered.

Yan’s oscillation theorem revisited☆

Abstract Yan’s contribution [J. Yan, Oscillation theorems for second order linear differential equations with damping, Proc. Amer. Math. Soc. 98 (1986) 276–282] was an important breakthrough in the development of the Theory of Oscillation. This frequently cited paper has stimulated extensive investigations in the field. During the last decade, an integral oscillation technique has been developed to such an extent as to allow us to revisit Yan’s fundamental oscillation theorem and remove one of the conditions, leaving the other assumptions and the conclusion intact, thus enhancing this keystone result.

Estimates for domains of local invertibility of diffeomorphisms

Using a novel Wintner-type formulation of the classical Peanos existence theorem (Math. Ann. 37 (1890), 182-228), we enhance Wazewskis result on invertibility of maps defined on closed balls (Ann. Soc. Pol. Math. 20 (1947), 81-125) securing the size of the domain of invertibility that agrees with the bounds derived by John (Comm. Pure Appl. Math. 21 (1968), 77-110) and Sotomayor (Z. Angew. Math. Phys. 41 (1990), 306-310).

Global existence of solutions for a class of nonlinear differential equations

By making use of a special Lyapunov-type function and applying the comparison method due to Conti, we prove global existence of solutions for a general class of nonlinear second-order differential equations that includes, in particular, van der Pol, Rayleigh, and Lienard equations, widely encountered in applications. Relevant examples are discussed.

An oscillation criterion for a class of nonlinear functional differential equations

We present a new oscillation criterion for a class of second-order nonlinear functional differential equations obtained by using the integral averaging technique.