New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

Shyam Sundar Santra,Mihai Postolache,Omar Bazighifan

doi:10.3390/math9111159

Abstract

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

Highlights

It is well known that the differential equations have many applications to the study of population growth, decay, Newton’s law of cooling, glucose absorption by the body, the spread of epidemics, Newton’s second law of motion, and interacting species, to name a few
We stress that the modeling of these phenomena is suitably formulated by evolutive partial differential equations, and moment problem approaches appear as a natural instrument in the control theory of neutral-type systems; see [4,5,6,7], respectively
We highlight some current developments in oscillation theory for second-order differential equations of the neutral type

Summary

Introduction

It is well known that the differential equations have many applications to the study of population growth, decay, Newton’s law of cooling, glucose absorption by the body, the spread of epidemics, Newton’s second law of motion, and interacting species (competition), to name a few. In another paper [9], Santra et al established some new oscillation theorems for the differential equations of the neutral type with mixed delays under the canonical operator with 0 ≤ p < 1. Motivated by the above studies, in this paper, we established some new sufficient conditions for the oscillation of solutions to second-order non-linear differential equations in the form: b(θ)(h )∆1 + q(θ)u∆ β(θ) = 0, θ ≥ θ0,. ≥0 for any function ω ∈ C([θ0, ∞), R+), which is decreasing to zero. There exists θ1 > θ0 such that Lemma 2 holds true and h satisfies (3) for θ ≥ θ1. Limθ→∞ b(θ) h (θ) ∆1 exists, and integrating (2) from θ to l, we obtained: b(l) h (l) ∆1 − b(θ) h (θ) ∆1 = − l q(ς)u∆(β(ς)) dς.

Oscillation Theorems

Conclusions and Open Problem