Jelena V. Manojlović

Precise asymptotic behavior of solutions of the sublinear Emden–Fowler differential equation

Abstract The aim of this paper is to show that if the sublinear Emden–Fowler differential equation (A) x ″ ( t ) + q ( t ) | x ( t ) | γ sgn x ( t ) = 0 , 0 γ 1 , with regularly varying coefficient q(t) is studied in the framework of regular variation, not only necessary and sufficient conditions for the existence of nontrivial regularly varying solutions of (A) , (A) can be established, but also precise information can be acquired about the asymptotic behavior at infinity of these solutions.

Positive solutions of fourth order Emden–Fowler type differential equations in the framework of regular variation

Abstract The fourth order nonlinear differential equations (A) x ( 4 ) + q ( t ) | x | γ sgn x = 0 , 0 γ 1 , with regularly varying coefficient q ( t ) are studied in the framework of regular variation. It is shown that thorough and complete information can be acquired about the existence of all possible regularly varying solutions of (A) and their accurate asymptotic behavior at infinity.

Oscillation criteria for second-order sublinear differential equation

Abstract The basic purpose of this paper is to present a new oscillation criteria for second-order sublinear differential equation [a(t)γ(x(t))x′(t)]′ + q(t)ƒ(t)) = 0, t ⩾ t 0 > 0 , where a a ϵ C 1 ([ t 0 , ∞)) is positive function, q ϵ C ([ t 0 , ∞)) has no restriction on its sign, γ, ƒ ϵ C 1 (R) are such that γ(x) > 0, xƒ(x) > 0 for x ≠0.

Oscillation and nonoscillation theorems for a class of even-order quasilinear functional differential equations

We are concerned with the oscillatory and nonoscillatory behavior of solutions of even-order quasilinear functional differential equations of the type, where and are positive constants, and are positive continuous functions on, and is a continuously differentiable function such that,. We first give criteria for the existence of nonoscillatory solutions with specific asymptotic behavior, and then derive conditions (sufficient as well as necessary and sufficient) for all solutions to be oscillatory by comparing the above equation with the related differential equation without deviating argument.

Intermediate solutions of fourth order quasilinear differential equations in the framework of regular variation

Intermediate solutions of fourth-order quasilinear differential equation p ( t ) | x ? ( t ) | α - 1 x ? ( t ) ? + q ( t ) | x ( t ) | β - 1 x ( t ) = 0 , α β 0 are studied in the framework of regular variation. Under the assumptions that p ( t ) , q ( t ) are regularly varying functions satisfying conditions ? a ∞ t p ( t ) 1 α dt = ∞ , ? a ∞ t p ( t ) 1 α dt = ∞ and ? a ∞ dt p ( t ) 1 α < ∞ necessary and sufficient conditions are established for the existence of regularly varying intermediate solutions and it is shown that the asymptotic behavior of all such solutions is governed by a unique explicit law.

Regularly varying sequences and Emden–Fowler type second-order difference equations

Abstract We study asymptotic behavior of positive solutions of nonlinear difference equation where are regularly varying sequences. It is shown that with the help of discrete regular variation, complete information can be acquired about the existence of all possible strongly increasing regularly varying solutions of this equation and their accurate asymptotic behavior at infinity.

Oscillation Theorems for a Nonlinear System of Differential Equations

We shall establish some theorems analogous to Knesers oscillation theorems for the linear differential equation of second order, which give sufficient conditions for oscillation of all solutions of the system of differential equations of the type u′i = |u3-i|λi sgn u3-i + (-1)i-1bi(t)ui (i = 1,2), where the functions bi (i = 1, 2) are nonnegative and summable on each finite segment of the interval [0, + ∞) and λi > 0 (i = 1, 2).

Friction at Nanoscale—Self-assembled Monolayers

In many technical fields a contact between two surfaces is very important and often the subject of research. The numerous physical phenomena that occur at the contact between two materials indicate the complexity of the processes that take place at the macro, micro or nanoscale. Therefore, friction, lubrication and wear are the subjects that have been attracting attention for many years, especially as part of tribological investigations. The research has shown that these three components are of fundamental importance for surfaces in contact. The aim of this chapter is to primarily describe friction as a tribological component and lubrication as a process to control friction, at scales of various lengths, especially at the atomic level. At the atomic and molecular scale there are materials with the property to spontaneously assemble themselves into ordered structures and many surface properties are influenced by the formation of such a film. One of the procedures to make these ultrathin organic films of controlled thickness is to prepare self-assembled monolayers. These monolayers are described as a model system to study boundary lubrication.

Regularly Varying Solutions of Half-Linear Diffferential Equations with Retarded and Advanced Arguments

Abstract Sharp conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the half-linear functional differential equation with both retarded and advanced arguments of the form (|x′(t)|α sgn x′(t))′ ± p(t)|x(g(t))|α sgn x(g(t)) ± q(t)|x(h(t))|α sgn x(h(t)) = 0, where α > 0 is a constant, p, q: [a,∞) → (0,∞), a ≧ 0 are continuous functions, g, h are continuous and increasing with g(t) < t, h(t) > t for t ≥ a and .