Half-linear Differential Equations Research Articles

Application of generalised Riccati equations to analysis of asymptotic forms of solutions of perturbed half-linear ordinary differential equations

Application of generalised Riccati equations to analysis of asymptotic forms of solutions of perturbed half-linear ordinary differential equations

Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, II

We consider the half-linear differential equation of the form \[ (p(t)|x^{\prime }|^{\alpha }\mathrm{sgn}\,x^{\prime })^{\prime } + q(t)|x|^{\alpha }\mathrm{sgn}\,x = 0\,, \quad t \ge t_{0} \,, \] under the assumption that $p(t)^{-1/\alpha }$ is integrable on $[t_{0}, \infty )$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t \rightarrow \infty $.

Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I

We consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t \to \infty\).

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.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

Abstract We study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation ( p ⁢ ( t ) ⁢ | x ′ | α ⁢ sgn ⁡ x ′ ) ′ + q ⁢ ( t ) ⁢ | x | α ⁢ sgn ⁡ x = 0 , (p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0, where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions ∫ a ∞ d ⁢ s p ⁢ ( s ) 1 / α = ∞ or ∫ a ∞ d ⁢ s p ⁢ ( s ) 1 / α &lt; ∞ . \int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}&lt;\infty. The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

Even Order Half-Linear Differential Equations with Regularly Varying Coefficients

We establish nonoscillation criterion for the even order half-linear differential equation (−1)nfn(t)Φx(n)(n)+∑l=1n(−1)n−lβn−lfn−l(t)Φx(n−l)(n−l)=0, where β0,β1,…,βn−1 are real numbers, n∈N, Φ(s)=sp−1sgns for s∈R, p∈(1,∞) and fn−l is a regularly varying (at infinity) function of the index α−lp for l=0,1,…,n and α∈R. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms fn(t) and fn−l(t) are replaced by the tα and tα−lp, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.

Lyapunov-type inequalities for higher-order half-linear difference equations

In this paper, we will establish some new Lyapunov-type inequalities for some higher-order superlinear–sublinear difference equations with boundary conditions. Our results not only complement the existing results established in the literature, but also furnish a handy tool for the study of qualitative properties of solutions of some difference equations.

Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments

Abstract Neutral differential equations are one of the most important extensions of classical ordinary differential equations and aim to give a better explanation for modeling phenomena where ordinary differential equations are insufficient. Naturally, all the questions studied in the scope of ordinary differential equations attracted the attention also for neutral differential equations. In this paper we study the oscillatory properties of second order half-linear neutral differential equations. We present oscillation criteria derived using a new approach. This approach allows us to reduce common restrictions on the deviations in arguments which are present in the currently known results of this type.

New Oscillation Results for Third-Order Half-Linear Neutral Differential Equations

The main purpose of this paper is to obtain criteria for the oscillation of all solutions of a third-order half-linear neutral differential equation. The main result in this paper is an oscillation theorem obtained by comparing the equation under investigation to two first order linear delay differential equations. An additional result is obtained by using a Riccati transformation technique. Examples are provided to show the importance of the main results.

An integral oscillation criterion for even order half-linear differential equations

Integral tests for detection of oscillatory behavior of solutions of even order half-linear differential equations of the form Lnx+q(t)|x|βsgnx=0,t≥a>0,(A)where L1x(t)=x′(t),Li+1x(t)=[pi(t)|Lix(t)|αisgnLix(t)]′,i=1,…n−1,αi,1≤i≤n−1, and β being positive constants and pi,1≤i≤n−1, and q continuous functions from [a,∞) to (0,∞), are established in the case where α1⋯αn−1=β.

Oscillation criteria for fourth order half-linear differential equations

Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form \begin{equation*} \big (|y^{\prime \prime }|^\alpha {\rm sgn\ } y^{\prime \prime }\big )^{\prime \prime } + q(t)|y|^\alpha {\rm sgn}\ y = 0, \quad t \ge a > 0, A \end{equation*} where $\alpha > 0$ is a constant and $q(t)$ is positive continuous function on $[a,\infty )$, are given in terms of an increasing continuously differentiable function $\omega (t)$ from $[a,\infty )$ to $(0,\infty )$ which satisfies $\int _a^\infty 1/(t\omega (t))\,dt < \infty $.

Reduction of order in the oscillation theory of half-linear differential equations

Reduction of order in the oscillation theory of half-linear differential equations

New Oscillation Conditions for Second Order Half-Linear Advanced Difference Equations

This paper aims to establish adequate conditions that are intended for the oscillation of every solution of the second order advanced type half-linear difference equations with noncanonical form. Initially, we derive a sufficient condition that ensures all solutions of the studied equation are either oscillatory or tending to zero. Secondly, we obtain a criteria for the oscillation of all solutions of the studied equation. These criteria are obtained by using Riccati transformation and summation averaging method. The results established in this paper in essence complement, extend and enhance the existing outcomes recorded in the literature. The improvement of our main results are illustrated through three examples.

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.

Euler type linear and half-linear differential equations and their non-oscillation in the critical oscillation case

This paper is devoted to the analysis of the oscillatory behavior of Euler type linear and half-linear differential equations. We focus on the so-called conditional oscillation, where there exists a borderline between oscillatory and non-oscillatory equations. The most complicated problem involved in the theory of conditionally oscillatory equations is to decide whether the equations from the given class are oscillatory or non-oscillatory in the threshold case. In this paper, we answer this question via a combination of the Riccati and Prüfer technique. Note that the obtained non-oscillation of the studied equations is important in solving boundary value problems on non-compact intervals and that the obtained results are new even in the linear case.

Modified Prüfer angle and conditional oscillation of perturbed linear and half-linear differential equations

The research and results described in this paper belong to the qualitative theory of differential equations (more precisely, the partial differential equations with the one-dimensional p-Laplacian). Using a method whose core is formed by the Prüfer technique, we identify a borderline case between oscillatory and non-oscillatory equations. Moreover, we are able to decide whether the studied equations are oscillatory or not even in the so-called critical (i.e., the borderline) case. The advantage of our approach is the fact that we obtain new and strong results for linear and half-linear equations (i.e., the equations with the one-dimensional p-Laplacian) at the same time. In addition, we are able to work with equations whose coefficients are non-constant and non-periodic. The novelty of our results is documented by examples and corollaries.

Non-Oscillation of half-linear difference equations with asymptotically periodic coefficients

We study oscillatory properties of half-linear difference equations with asymptotically periodic coefficients, i.e., coefficients which can be expressed as the sums of periodic sequences and sequences vanishing at infinity. Using a special variation of the discrete Riccati technique, we prove that the non-oscillation of the studied equations can be determined directly from their coefficients. Thus, the studied equations can be widely used as testing equations. Our main result is new even for linear equations with periodic coefficients. This fact is illustrated by simple corollaries and examples at the end of this paper.

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.

Ground state solutions to nonlinear equations with p-Laplacian

We investigate the existence of positive radial solutions for a nonlinear elliptic equation with p-Laplace operator and sign-changing weight, both in superlinear and sublinear case. We prove the existence of solutions u, which are globally defined and positive outside a ball of radius R, satisfy fixed initial conditions u(R)=c>0, u′(R)=0 and tend to zero at infinity. Our method is based on a fixed point result for boundary value problems on noncompact intervals and on asymptotic properties of suitable auxiliary half-linear differential equations. The results are new also for the classical Laplace operator and may be used for proving the existence of ground state solutions and decaying solutions with exactly k-zeros which are defined in the whole space. Some examples illustrate our results.

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.