Oscillation of third order nonlinear damped dynamic equation with mixed arguments on time scales

Ying Sui,Shurong Sun

doi:10.1186/s13662-018-1654-3

Abstract

The objective of this paper is to offer sufficient conditions for the oscillation of all solutions of the third order nonlinear damped dynamic equation with mixed arguments of the form \t\t\t(r2(r1(yΔ)α)Δ)Δ(t)+p(t)ψ(t,yΔ(a(t)))+q(t)f(t,y(g(t)))=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\bigl(r_{2}\\bigl(r_{1}\\bigl(y^{\\Delta}\\bigr)^{\\alpha}\\bigr)^{\\Delta}\\bigr)^{\\Delta}(t)+p(t)\\psi \\bigl(t,y^{\\Delta}\\bigl(a(t)\\bigr)\\bigr)+q(t)f\\bigl(t,y\\bigl(g(t)\\bigr) \\bigr)=0 $$\\end{document} on time scales, where a(t)geq t and g(t)leq t. Using Riccati transformation, integral averaging technique, and comparison theorem, we give some new criteria for the oscillation of the studied equation. Our results essentially improve and complement the earlier ones.

Highlights

IntroductionWhere I = [t0, ∞)T, α ≥ 1 is the ratio of positive odd integers
1 Introduction This paper deals with oscillatory behavior of all solutions of the third order nonlinear damped dynamic equation with mixed arguments of the form r2 r1 y α (t) + p(t)ψ t, y a(t) + q(t)f t, y g(t) = 0, t ∈ I, (1.1)
Assume that the conditions are satisfied: (H1) r1, r2, p, q ∈ Crd(I, R+), a ∈ Crd(I, R), g ∈ Cr1d(I, R), where R+ = (0, ∞)T; (H2) a(t) ≥ σ (t) ≥ t, g(t) ≤ t, g (t) ≥ 0 and g(t) → ∞ as t → ∞; (H3) ψ, f ∈ C(T × R, R) such that ψ(t, x(t)) ≥ k1xα(t), ψ(t, –x(t)) = –ψ(t, x(t)), and f (t, x(t)) ≥ max{k2xβ (t), k2xβ (σ (t))}, f (t, –x(t)) = –f (t, x(t)), and x(t) is defined on T, k1, k2 are constants, β is the ratio of positive odd integers

Summary

Introduction

Where I = [t0, ∞)T, α ≥ 1 is the ratio of positive odd integers. (H1) r1, r2, p, q ∈ Crd(I, R+), a ∈ Crd(I, R), g ∈ Cr1d(I, R), where R+ = (0, ∞)T; (H2) a(t) ≥ σ (t) ≥ t, g(t) ≤ t, g (t) ≥ 0 and g(t) → ∞ as t → ∞; (H3) ψ, f ∈ C(T × R, R) such that ψ(t, x(t)) ≥ k1xα(t), ψ(t, –x(t)) = –ψ(t, x(t)), and f (t, x(t)) ≥ max{k2xβ (t), k2xβ (σ (t))}, f (t, –x(t)) = –f (t, x(t)), and x(t) is defined on T, k1, k2 are constants, β is the ratio of positive odd integers.

Objectives

Conclusion