Abstract
In this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0,\\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}$$ \\bigl(r(t) \\bigl(z'(t)\\bigr)^{\\alpha }\\bigr)'+q(t)x^{\\beta } \\bigl(\\sigma (t)\\bigr)=0,\\quad t\\geq t_{0}, $$\\end{document} where z(t)=x(t)+p_{1}(t)x(tau (t))+p_{2}(t)x(lambda (t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.
Highlights
In the article, we consider the oscillatory and asymptotic behavior of solutions to a second order nonlinear advanced differential equation with mixed neutral terms of the form r(t) z (t) α + q(t)xβ σ (t) = 0, t ≥ t0, (1.1)where z(t) = x(t) + p1(t)x(τ (t)) + p2(t)x(λ(t))
We present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form (r(t)(z (t))α) + q(t)xβ (σ (t)) = 0, t ≥ t0, where z(t) = x(t) + p1(t)x(τ (t)) + p2(t)x(λ(t)) and α, β are ratios of two positive odd integers
In this paper we only consider the nontrivial solution of Eq (1.1) which satisfies sup{|x(t)| : t ≥ T} > 0 for all T ≥ Tx
Summary
We present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form (r(t)(z (t))α) + q(t)xβ (σ (t)) = 0, t ≥ t0, where z(t) = x(t) + p1(t)x(τ (t)) + p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Motivated by the above work, we will consider a generalized nonlinear advanced differential equations with mixed neutral terms and establish new sufficient conditions for oscillation of Eq (1.1) under the condition (1.3). 2 Main results we present some lemmas and our new sufficient conditions for oscillation of Eq (1.1).
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