Second-order Half-linear Delay Research Articles

Oscillation result for half-linear delay difference equations of second-order.

In this paper, we obtain the new single-condition criteria for the oscillation of second-order half-linear delay difference equation. Even in the linear case, the sharp result is new and, to our knowledge, improves all previous results. Furthermore, our method has the advantage of being simple to prove, as it relies just on sequentially improved monotonicities of a positive solution. Examples are provided to illustrate our results.

Second-order half-linear delay differential equations: Oscillation tests

In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form $ \bigl(r(y^{\prime})^\gamma\bigr)^{\prime}(t)+ q(t)y^\alpha(\tau(t))=0\,.$ We study this equation under the assumption $\int^{\infty}\big(r(\eta)\big)^{-1/\gamma} d\eta=\infty$ and consider two cases when $\gamma > \alpha$ and $\gamma < \alpha$. We provide examples, illustrating the results and state an open problem.

Second-Order Differential Equation: Oscillation Theorems and Applications

Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form ς y u ′ y a ′ + p y u c ϑ y = 0 , for y ≥ y 0 , under the assumption ∫ ∞ ς η − 1 / a = ∞ . Two cases are considered for a < c and a > c , where a and c are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.

Second-Order Differential Equation with Multiple Delays: Oscillation Theorems and Applications

Differential equations of second order appear in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations, and quantum mechanics. In this paper, necessary and sufficient conditions are established of the solutions to second-order half-linear delay differential equations of the formςyu′ya′+∑j=1mpjyucjϑjy=0 for y≥y0, under the assumption∫∞ςη−1/adη=∞. We consider two cases whena<cjanda>cj, whereaandcjare the quotient of two positive odd integers. Two examples are given to show effectiveness and applicability of the result.

Kneser-type oscillation criteria for second-order half-linear delay differential equations

In the paper, we establish a variant of Kneser oscillation theorem for the second-order half-linear delay differential equation(r(t)|y′(t)|α−1y′(t))′+q(t)|y(τ(t))|α−1y(τ(t))=0,t≥t0>0,under the assumption∫t0∞r−1/α(s)ds=∞,which improves a plenty of results reported in the literature. In case of α ≥ 1, the obtained criterion is sharp in the sense that the oscillation constant is optimal for the corresponding half-linear Euler type equation with delay, while the result for α < 1 is slightly worse. Our methodology differs from the previous ones, since it is only based on sequentially improved monotonicities of the nonoscillatory solution.

Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms

AbstractThis article concerns the oscillatory behavior of solutions to second-order half-linear delay differential equations with mixed neutral terms. The authors present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated with examples.

Necessary and sufficient conditions for oscillation to second-order half-linear delay differential equations

In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form $$\begin{aligned} (r(x^{\prime })^\gamma )^{\prime }(t)+ q(t)x^\alpha (\tau (t))=0. \end{aligned}$$ Under the assumption $$\int ^{\infty }(r(\eta ))^{-1/\gamma } \mathrm{d}\eta =\infty $$ , we consider the two cases when $$\gamma > \alpha $$ and $$\gamma < \alpha $$ . Further, some illustrative examples showing applicability of the new results are included, and state an open problem.

Oscillation for second-order half-linear delay damped dynamic equations on time scales

We investigate oscillation of second-order half-linear variable delay damped dynamic equations \t\t\t[a(t)|xΔ(t)|λ−1xΔ(t)]Δ+b(t)|xΔ(t)|λ−1xΔ(t)+p(t)|x(δ(t))|λ−1x(δ(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}$$ \\bigl[a(t) \\bigl\\vert x^{\\Delta }(t) \\bigr\\vert ^{\\lambda -1}x^{\\Delta }(t) \\bigr]^{\\Delta }+b(t) \\bigl\\vert x ^{\\Delta }(t) \\bigr\\vert ^{\\lambda -1}x^{\\Delta }(t)+p(t) \\bigl\\vert x \\bigl(\\delta (t) \\bigr) \\bigr\\vert ^{\\lambda -1}x \\bigl(\\delta (t) \\bigr)=0 $$\\end{document} on a time scale mathbb{T}. By using the generalized Riccati transformation and the inequality technique, we establish some new oscillation criteria for the equations under the condition \t\t\t∫t0∞[a−1(s)e−b/a(s,t0)]1/λΔs<∞.\\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}$$ \\int ^{\\infty }_{t_{0}} \\bigl[a^{-1}(s)e_{{-b/a}}(s,t_{0}) \\bigr]^{1/\\lambda } \\Delta s< \\infty. $$\\end{document} These results deal with some cases not covered by existing results in the literature.

Oscillatory results for second-order noncanonical delay differential equations

The main purpose of this paper is to improve recent oscillation results for the second-order half-linear delay differential equation \[\left(r(t)\left(y'(t)\right)^\gamma\right)'+q(t)y^\gamma(\tau(t))= 0, \quad t\geq t_0,\] under the condition \[\int_{t_0}^{\infty}\frac{\text{d} t}{r^{1/\gamma}(t)} \lt \infty.\] Our approach is essentially based on establishing sharper estimates for positive solutions of the studied equation than those used in known works. Two examples illustrating the results are given.

Improved oscillation results for second-order half-linear delay differential equations

In this paper, we study the second-order half-linear delay differential equation of the form \[(r(t)(y'(t))^\alpha)'+q(t)y^\alpha(\tau(t))= 0.\:\:\:(E)\] We establish new oscillation criteria for (E), which improve a number of related ones in the literature. Our approach essentially involves establishing sharper estimates for the positive solutions of (E) than those presented in known works and a comparison principle with first-order delay differential inequalities. We illustrate the improvement over the known results by applying and comparing our method with the other known methods on the particular example of Euler-type equations.

A note on oscillation of second-order delay differential equations

The purpose of this paper is to study the second-order half-linear delay differential equation r ( t ) y ′ ( t ) α ′ + q ( t ) y α ( τ ( t ) ) = 0 under the condition ∫ ∞ r − 1 ∕ α ( t ) d t < ∞ . Contrary to most existing results, oscillation of the studied equation is attained via only one condition. A particular example of Euler type equation is provided in order to illustrate the significance of our main results.

Oscillatory behavior of second-order half-linear damped dynamic equations

By refining the generalized Riccati transformation technique, we obtain new oscillation criteria for a class of second-order half-linear delay dynamic equations with damping on a time scale. Assumptions in our theorems are less restrictive, these results complement and improve related contributions to the subject. In particular, the results obtained amend related results reported in the literature.

On the Oscillation for Second-Order Half-Linear Neutral Delay Dynamic Equations on Time Scales

We discuss oscillation criteria for second-order half-linear neutral delay dynamic equations on time scales by using the generalized Riccati transformation and the inequality technique. Under certain conditions, we establish four new oscillation criteria. Our results in this paper are new even for the cases of𝕋=ℝand𝕋=ℤ.

Oscillation Theorems for Second-Order Half-Linear Neutral Delay Dynamic Equations with Damping on Time Scales

We establish the oscillation criteria of Philos type for second-order half-linear neutral delay dynamic equations with damping on time scales by the generalized Riccati transformation and inequality technique. Our results are new even in the continuous and the discrete cases.

New oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique,we establish four new oscillation criteria for certain second-order half-linear delay dynamic equations with damping on time scales.Our results extend and improve some known results,but also unify the oscillation of the second-order half-linear delay differential equations with damping and the second-order half-linear delay difference equations with damping.

Oscillation theorems for second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish a few new oscillation criteria for certain second-order half-linear delay dynamic equations with damping on a time scale. Our results extend and improve some known results, but also unify the oscillation of the second-order half-linear delay differential equation with damping and the second-order half-linear delay difference equation with damping.

Oscillation criteria for second-order half-linear delay dynamic equations with damping on time scales (Ⅱ)

By using the generalized Riccati transformation and the inequality technique,we establish four new oscillation criteria for certain second-order half-linear delay dynamic equations with damping on time scales.Our results extend and improve some known results,but also unify the oscillation of the second-order half-linear delay differential equations with damping and the second-order half-linear delay difference equations with damping.

Oscillation of second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish a few new oscillation criterions for certain second-order half-linear delay dynamic equations with damping on a time scale. Our results extend and improve some known results, but also unify the oscillation of the second-order half-linear delay differential equation with damping and the second-order half-linear delay difference equation with damping.