Second-order Half-linear Differential Equations Research Articles

Asymptotic Behavior Criteria for Solutions Second Order Half Linear Neutral Differential Equation

In this paper the we study the oscillation and asymptotic behavior of second-order Half linear neutral differential equations. some basic and necessary and sufficient condition are established for every solution of , to insure the convergence of all nonoscillatory solutions to zero or tends to infinity as Some examples are given to illustrate the main results.

Integral Comparison Criteria for Half-Linear Differential Equations Seen as a Perturbation

In this paper, we present further developed results on Hille–Wintner-type integral comparison theorems for second-order half-linear differential equations. Compared equations are seen as perturbations of a given non-oscillatory equation, which allows studying the equations on the borderline of oscillation and non-oscillation. We bring a new comparison theorem and apply it to the so-called generalized Riemann–Weber equation (also referred to as a Euler-type equation).

Oscillation of Second-Order Half-Linear Neutral Advanced Differential Equations

The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form $$\begin{aligned} (r(t)((y(t)+p(t)y(\tau (t)))')^\alpha )'+q(t)y^\alpha (\sigma (t))=0,\ t\geqslant t_0, \end{aligned}$$ when $$\int _{}^\infty r^{-\frac{1}{\alpha }}(s){\text{d}}s<\infty $$ . We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation. An example is provided to illustrate the results.

Oscillatory and Nonoscillatory Conditions for a Second-Order Half-Linear Differential Equation

In this paper, we obtain oscillatory and nonoscillatory conditions for a certain second-order half-linear differential equation.

On the oscillation of second-order half-linear functional differential equations with mixed neutral term

In this article, the authors establish new sufficient conditions for the oscillation of solutions to a class of second-order half-linear functional differential equations with mixed neutral term. The results obtained improve and complement some known results in the relevant literature. Examples illustrating the results are included.

Hille-Nehari type oscillation and nonoscillation criteria for linear and half-linear differential equations

Differential equations attract considerable attention in many applications. In particular, it was found out that half-linear differential equations behave in many aspects very similar to that in linear case. The aim of this contribution is to investigate oscillatory properties of the second-order half-linear differential equation and to give oscillation and nonoscillation criteria for this type of equation. It is also considered the linear Sturm- Liouville equation which is the special case of the half-linear equation. Main ideas used in the proof of these criteria are given and Hille-Nehari type oscillation and nonoscillation criteria for the Sturm-Liouville equation are formulated. In the next part, Hille-Nehari type criteria for the half-linear differential equation are presented. Methods used in this investigation are based on the Riccati technique and the quadratic functional, that are very useful instruments in proving oscillation/nonoscillation both for linear and half-linear equation. Conclude that there are given further criteria which guarantee either oscillation or nonoscillation of linear and half-linear equation, respectively. These criteria can be used in the next research in improving some conditions given in theorems of this paper.

New oscillation criteria for second-order half-linear advanced differential equations

The objective of this paper is to offer sufficient conditions for the oscillation and asymptotic behavior of all solutions of second-order half-linear differential equations with advanced argument of the form(r(y′)α)′(t)+q(t)yα(σ(t))=0,when ∫∞r−1/α(s)ds<∞. Our criteria substantially improve, simplify and complement a number of existing ones. The results obtained are illustrated by an example on the Euler type equations.

Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values

AbstractThe paper belongs to the qualitative theory of half-linear equations which are located between linear and non-linear equations and, at the same time, between ordinary and partial differential equations. We analyse the oscillation and non-oscillation of second-order half-linear differential equations whose coefficients are given by the products of functions having mean values and power functions. We prove that the studied very general equations are conditionally oscillatory. In addition, we find the critical oscillation constant.

On conjugacy of second-order half-linear differential equations on the real axis

On conjugacy of second-order half-linear differential equations on the real axis

Conditional oscillation of Riemann-Weber half-linear differential equations with asymptotically almost periodic coefficients

We analyse the oscillation and non-oscillation of second-order half-linear differential equations with periodic and asymptotically almost periodic coefficients, where the equations have the so-called Riemann-Weber form. For these equations, we find an explicit oscillation constant. Corollaries and examples are mentioned as well.

Oscillation criteria for neutral second-order half-linear differential equations with applications to Euler type equations

We study the second-order neutral delay half-linear differential equation , where , and . We use the method of Riccati type substitution and derive oscillation criteria for this equation. By an example of the neutral Euler type equation we show that the obtained results are sharp and improve the results of previous authors. Among others, we improve the results of Sun et al. (Abstr. Appl. Anal. 2012:819342, 2012) and discuss also the case when . MSC:34K11, 34K40.

Oscillation of second-order half-linear differential equations with several neutral terms

We study oscillatory properties of a class of second-order neutral differential equations. New results essentially simplify the examination of the equations studied. In particular, these results extend those reported in the paper by Baculikova and Džurina (Comput. Math. Appl. 61:94–99, 2011).

Oscillation and nonoscillation of second-order half-linear differential equations

We consider the oscillation and nonoscillation of second-order half-linear differential equations. By using some new technique, we establish new oscillation and nonoscillation criteria that extend and improve some known results for second-order linear differential equations. Published in Neliniini Kolyvannya, Vol. 15, No. 3, pp. 344–353, July–September, 2012.

Oscillation of half-linear differential equations with asymptotically almost periodic coefficients

We investigate second-order half-linear differential equations with asymptotically almost periodic coefficients. For these equations, we explicitly find an oscillation constant. If the coefficients are replaced by constants, our main result (concerning the conditional oscillation) reduces to the classical one. We also mention examples and concluding remarks. MSC:34C10, 34C15.

Gaps between zeros of second-order half-linear differential equations

In this paper, for second order half-linear differential equations, we will establish some new inequalities of Lyapunov’s type. These inequalities give results related to the spacing between consecutive zeros of a solution and the spacing between a zero of a solution and/or a zero of its derivative. The results also yield conditions for disfocality, disconjugacy and lower bounds for an eigenvalue of a boundary value problem. The main results will be proved by making use of some generalizations of Opial and Wirtinger type inequalities. Some examples are considered to illustrate the main results.

On the Zeroes and the Critical Points of a Solution of a Second Order Half‐Linear Differential Equation

This paper presents two methods to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second‐order half‐linear differential equation (p(x)Φ(y′))′ + q(x)Φ(y) = 0, with p(x) and q(x) piecewise continuous and p(x) &gt; 0, Φ(t) = |t|r−2t and r being real such that r &gt; 1. It also compares between them in several examples. Lower bounds (i.e., Lyapunov inequalities) for such a distance are also provided and compared with other methods.

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.

Lyapunov type inequalities for second-order half-linear differential equations

In this paper, we establish some new Lyapunov type inequalities for second-order half-linear differential equations, which almost generalize and extend all related existing results in the literature.

Rectifiable oscillations in second-order half-linear differential equations

Second-order half-linear differential equation (H): \({(\Phi (y'))'+f(x)\Phi (y)=0}\) on the finite interval I = (0,1] will be studied, where \({\Phi (u)=|u|^{p-2}u}\) , p > 1 and the coefficient f(x) > 0 on I, \({f\in C^{2}((0,1])}\) , and \({\lim_{x\rightarrow0}f(x)=\infty }\) . In case when p = 2, the equation (H) reduces to the harmonic oscillator equation (P): y′′ + f(x)y = 0. In this paper, we study the oscillations of solutions of (H) with special attention to some geometric and fractal properties of the graph \({G(y)=\{(x,y(x)):0\leq x\leq 1\}\subseteq {\bf {R}}^{2}}\) . We establish integral criteria necessary and sufficient for oscillatory solutions with graphs having finite and infinite arclength. In case when \({f(x)\sim \lambda x^{-\alpha}}\), λ > 0, α > p, we also determine the fractal dimension of the graph G(y) of the solution y(x). Finally, we study the L p nonintegrability of the derivative of all solutions of the equation (H).

Geometrical conditions for oscillation of second-order half-linear differential equations

This paper deals with the second-order half-linear differential equation $$ (\phi _p (\dot x))^ \cdot + a(t)\phi _p (\dot x) + b(t)\phi _p (x) = 0. $$ Here ϕ p (z):= |z|p−2 z is the so-called one-dimensional p-Laplacian operator. Our main purpose is to establish new criteria for all nontrivial solutions to be oscillatory and for those to be nonoscillatory. In our theorems, the parametric curve given by (a(t), b(t)) plays a critical role in judging whether all solutions are oscillatory or nonoscillatory. This paper takes a di-erent approach from most of the previous research. Our results are new even in the linear case (p = 2). The method used here is mainly phase plane analysis for a system equivalent to the half-linear differential equation. Some suitable examples are included to illustrate the main results. Global phase portraits are also attached.