Oscillation and Asymptotic Properties of Differential Equations of Third-Order

R Elayaraja,Clemente Cesarano,Omar Bazighifan,V Ganesan

doi:10.3390/axioms10030192

R Elayaraja, Clemente Cesarano + Show 2 more

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https://doi.org/10.3390/axioms10030192

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Journal: Axioms	Publication Date: Aug 18, 2021
Citations: 4	License type: CC BY 4.0

Affiliation: UniNettuno University, Hadhramout University

Abstract

The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator (a(ι)[(b(ι)x(ι)+p(ι)x(ι−τ)′)′]β)′+∫cdq(ι,μ)xβ(σ(ι,μ))dμ=0, where ι≥ι0 and w(ι):=x(ι)+p(ι)x(ι−τ). New oscillation results are established by using the generalized Riccati technique under the assumption of ∫ι0ιa−1/β(s)ds<∫ι0ι1b(s)ds=∞asι→∞. Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem.

Highlights

New oscillation results are established by using the
The objective of this paper is to provide oscillation theorems for the third order equation as follows: Academic Editor: Martin Bohner a(ι)h b(ι)[ x (ι) + p(ι) x (ι − τ )]0 0 i β 0Received: 8 June 2021Accepted: 14 August 2021Published: 18 August 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. +Z d c q(ι, μ) x β (σ (ι, μ)) dμ = 0, (1)where a(ι), b(ι), p(ι), q(ι) ∈ C ([ι0, +∞)), a(ι), b(ι) > 0, a0 (ι) ≥ 0, q(ι) ≥ 0, β ≥ 1 and 0 ≤ p(ι) ≤ p0 ≤ 1
The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third

Summary

Introduction

New oscillation results are established by using the The objective of this paper is to provide oscillation theorems for the third order equation as follows: Academic Editor: Martin Bohner a(ι) The main results are obtained under the following assumptions: (A1 ) q(ι, μ) ∈ C ([ι0 , +∞) × [c, d], [0, +∞)) and q(ι, μ) does not vanish identically for any half line [ι ∗ , +∞) × [c, d], ι ∗ ≥ ι; (A2 ) σ (ι, μ) ∈ C ([ι0 , +∞) × [c, d], [0, +∞)), σ (ι, μ) + τ ≤ ι, σ (ι, μ) is nondecreasing with respect to ι and μ respectively, lim inf σ (ι, μ) = ∞ and lim inf τ (ι) = ∞. There is an ongoing interest in obtaining several sufficient conditions for the oscillation or non-oscillation of the solutions of different kinds of differential equations; see [2–24] as examples of instant results on this topic.

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