New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations

Mervan Pašić

doi:10.1155/2013/735360

Abstract

We establish some new interval oscillation criteria for a general class of second-order forced quasilinear functional differential equations withϕ-Laplacian operator and mixed nonlinearities. It especially includes the linear, the one-dimensionalp-Laplacian, and the prescribed mean curvature quasilinear differential operators. It continues some recently published results on the oscillations of the second-order functional differential equations including functional arguments of delay, advanced, or delay-advanced types. The nonlinear terms are of superlinear or supersublinear (mixed) types. Consequences and examples are shown to illustrate the novelty and simplicity of our oscillation criteria.

Highlights

We study the oscillation of the following three kinds of second-order forced quasilinear functional differential equations of delay, advanced, and delay-advanced types: (r (t)
Bai and Liu [1] have studied the oscillation of second-order delay differential equation: (r (t) τi)󵄨󵄨󵄨󵄨αi sgn x
In Murugadass et al [2], authors have studied the oscillation of the second-order quasilinear delay differential equation: (r (t)

Summary

Introduction

We study the oscillation of the following three kinds of second-order forced quasilinear functional differential equations of delay, advanced, and delay-advanced types:. Bai and Liu [1] have studied the oscillation of second-order delay differential equation: x󸀠. In Murugadass et al [2], authors have studied the oscillation of the second-order quasilinear delay differential equation:. In contrast to the preceding, we use a combination of the Riccati classic transformation, a blow-up argument, and a comparison pointwise principle recently established in [12, 13] but for differential equations without functional arguments. It seems that our criteria are slightly simpler to be verified, which is discussed on some examples given . About the applications of second-order functional differential equations in the mathematical description of certain phenomena in physics, technics, and biology (oscillation in a vacuum tube; interaction of an oscillator with an energy source; coupled oscillators in electronics, chemistry, and ecology; relativistic motion of a mass in a central field; ship course stabilization; moving of the tip of a growing plant; etc.), we suggest reading Kolmanovskii and Myshkis book [21]

Main Results and Examples

Auxiliary Results

Proof of Main Results