Asymptotic Formula for Oscillatory Solutions of Some Singular Nonlinear Differential Equation

Abstract

Singular differential equation(p(t)u′)′=p(t)f(u)is investigated. Herefis Lipschitz continuous on ℝ and has at least two zeros 0 andL>0. The functionpis continuous on [0,∞) and has a positive continuous derivative on (0,∞) andp(0)=0. An asymptotic formula for oscillatory solutions is derived.

Nonautonomous Differential Equations in Banach Space and Nonrectifiable Attractivity in Two-Dimensional Linear Differential Systems

“…In the theory of differential equations, the importance of studying the different kind of asymptotic behaviours of the nonautonomous linear differential systems comes from their application in the study of asymptotic and oscillatory behaviour of the second and higher order ordinary differential equations. For instance, in [5], authors study the asymptotic behaviour near = ∞ of oscillatory solutions of the nonlinear second-order differential equation ( ( ) ) = ( ) ( ) via the asymptotic formula for solutions of an auxiliary linear differential system having elements which are absolutely continuous functions on [ 0 , ∞). In [6], authors derive a precise asymptotic behaviour near = ∞ of solutions ( ) of the third-order nonlinear differential equation + (…”

Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations

“…Such a kind of topics has been considered for the first time in [4] but only in the case of the so-called integrable systems (systems that allow all solutions in explicit forms), where the asymptotic integration near = 0 is not required. About the asymptotic integration near = ∞ of differential equations and systems, we refer reader to [5][6][7][8][9][10][11][12]. On several asymptotic properties of twodimensional differential systems near = ∞, let us see, for instance, [13][14][15][16][17][18].…”

“…where matrix ( ) is defined in (23). For system (28), we will use the next generalized Matell's theorem appearing as Theorem 11 in Coppel's book [5,Chapter 4] and Theorem 6.5 in Kiguradze's monograph [7] (see also [12,Theorem 7] and [8]).…”