Abstract
The oscillation of differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of this equations. The purpose of this article is to establish new oscillatory properties which describe both the necessary and sufficient conditions for a class of nonlinear second-order differential equations with neutral term and mixed delays of the form p(ι)w′(ι)α′+r(ι)uβ(ν(ι))=0,ι≥ι0 where w(ι)=u(ι)+q(ι)u(ζ(ι)). Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.
Highlights
In this paper we present our work in the study of certain oscillation properties of second-order differential equations containing mixed delays
It is worth pointing out that both oscillation and stability criteria are currently used in the studies of nonlinear mathematical models with delay for single species and several species with interactions, in logistic models, α-delay models, mathematical models with varying capacity, mathematical models for food-limited population dynamics with periodic coefficients, diffusive logistic models
In the last few years, the research activity concerning the oscillation of solutions of neutral differential equations has been received considerable attention
Summary
In this paper we present our work in the study of certain oscillation properties of second-order differential equations containing mixed delays. The analysis of qualitative properties of ordinary differential equations is attracting considerable attention from the scientific community due to numerous applications in several contexts as Biology, Physics, Chemistry, and Dynamical Systems. For some details related the recent studies on oscillation and non-oscillation properties, exponential stability, instability, existence of unbounded solutions of the equations under consideration, we refer the reader to the books [1,2]. In the last few years, the research activity concerning the oscillation of solutions of neutral differential equations has been received considerable attention. Neutral equations contribute to many applications in economics, physics, medicine, engineering and biology, see [3–8]. The literature is full of very interesting results linked with the oscillation properties for second-order differential equations
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