Abstract
Modified Riccati technique for half-linear differential equations with delay
Highlights
In this paper we study the half-linear differential equation with delay in the form (r(t)Φ(x (t))) + c(t)Φ(x(τ(t))) = 0, Φ(x) := |x|p−2x, p > 1, (1.1)
One of the techniques in oscillation theory of (1.1) which produces reasonably sharp results is the transformation to the first order Riccati type equation. This method is in the qualitative theory of the linear ordinary differential equation (r(t)x ) + c(t)x = 0 (1.3)
For examples of applications of modified Riccati technique which allowed to obtain half-linear versions of the results proved originally for the linear equation using transformation technique see [6] and [7]
Summary
Marík [8, 11, 13, 15, 16]) and one of the crucial problems is to find conditions which ensure that all nonsingular solutions of this equation have infinitely many zeros This is motivation for the following definitions. The Sturm theorem on interlacing property of zeros fails and oscillatory solutions may coexist with nonoscillatory solutions Despite this fact, many results and methods (including the methods which allow to detect oscillation of all solutions) can be extended from the theory of ordinary differential equations to the theory of delay differential equations. One of the techniques in oscillation theory of (1.1) which produces reasonably sharp results is the transformation to the first order Riccati type equation This method is in the qualitative theory of the linear ordinary differential equation (r(t)x ) + c(t)x = 0. We introduce the modified Riccati equation for equation (1.1) and in the third section we use these results to obtain explicit comparison theorems which compare (1.1) with a certain (linear or half-linear) ordinary differential equation
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