Half-linear Differential Equations Research Articles

A refinement of the Hille–Wintner comparison theorem and new nonoscillation criteria for half-linear differential equations

AbstractA refinement of the Hille–Wintner comparison theorem is obtained for two half-linear differential equations of the second order. As a consequence, some new nonoscillation tests for such equations are derived by means of this improved comparison technique. In most of our results coefficients and their integrals do not need to be nonnegative and are allowed to oscillate in any neighborhood of infinity.

Oscillatory Behavior of Higher-Order Differential Equations with Delay Terms

The aim of this research is to study the oscillatory properties of higher -order delay half linear differential equations with non-canonical operators. Two methods for establishing some new conditions for the oscillation of all solutions of higher-order differential equations will be presented. The first method to use the Riccati transformations which differ from those reported in some literature. The second method employs comparison principles with first-order delay differential equations which can easily deduce oscillation of all solutions of studied equations. Not only do the newly proposed criteria improve, extend, and greatly simplify the previously established criteria, but they also have the potential to act as a reference point for the theory of delay differential equations of higher order, which is still in its early stages of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. There will be some examples provided to illustrate the findings.

Oscillation criteria for a second order half-linear differential equation with delay, with monotone nondecreasing delay function

Oscillation criteria for a second order half-linear differential equation with delay, with monotone nondecreasing delay function

Oscillation of linear and half-linear difference equations via modified Riccati transformation

In this paper, we introduce a new oscillation criterion for Euler type half-linear difference equations with positive coefficients which are asymptotically periodic and bounded, respectively. As the mean of proving the criterion, the generalized adapted Riccati transformation is used. The presented main result is the oscillatory counterpart of a previously obtained non-oscillation criterion for the treated equations. Therefore, the result shows that the analyzed equations are conditionally oscillatory. Then we emphasize the novelty of the result in the linear case by several consequences.

New Criteria on Oscillatory Behavior of Third Order Half-Linear Functional Differential Equations

This paper deals with some new criteria for the oscillation of third order half-linear differential equations. The purpose of the present paper is the linearization of equation 1.1 in the sense that we would deduce oscillation of studied equation from that of the linear form and to provide new oscillation criteria via comparison with first order equations whose oscillatory behavior are known. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature for equation 1.1. The results are illustrated by some examples.

Oscillation Criteria for Advanced Half-Linear Differential Equations of Second Order

In this paper, we find new oscillation criteria for second-order advanced functional half-linear differential equations. Our results extend and improve recent criteria for the same equations established previously by several authors and cover the existing classical criteria for related ordinary differential equations. We give some examples to illustrate the significance of the obtained results.

Modification of adapted Riccati equation and oscillation of linear and half-linear difference equations

Applying a modification of the adapted Riccati technique, we introduce a new type of oscillation criteria for generalizations of the Euler half-linear difference equation. Our main criterion is new in many cases. This fact is documented by corollaries and examples. In particular, we obtain new results for linear equations.

Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

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

Half-linear differential equations: Regular variation, principal solutions, and asymptotic classes

We are interested in the structure of the solution space of second-order half-linear differential equations taking into account various classifications regarding asymptotics of solutions. We focus on an exhaustive analysis of the relations among several types of classes which include the classes constructed with respect to the values of the limits of solutions and their quasiderivatives, the classes of regularly varying solutions, the classes of principal and nonprincipal solutions, and the classes of the solutions that obey certain asymptotic formulae. Many of our observations are new even in the case of linear differential equations, and we provide also the revision of existing results.

Oscillation and nonoscillation criteria for second order half-linear neutral difference equations

In this paper, we present some necessary condition of second order half linear neutral difference equations which make certain that all solution there oscillatory or converge to zero as n → ∞. Some examples are demonstrated to validate the conditions.

Oscillation of modified Euler type half-linear differential equations via averaging technique

Oscillation of modified Euler type half-linear differential equations via averaging technique

Nonoscillation of the Mathieu-type half-linear differential equation and its application to the generalized Whittaker–Hill-type equation

Nonoscillation of the Mathieu-type half-linear differential equation and its application to the generalized Whittaker–Hill-type equation

Oscillation and Nonoscillation Criteria for a Half-Linear Difference Equation of the Second Order and the Extended Discrete Hardy Inequality

Oscillation and Nonoscillation Criteria for a Half-Linear Difference Equation of the Second Order and the Extended Discrete Hardy Inequality

Applications of the novel diamond alpha Hardy–Copson type dynamic inequalities to half linear difference equations

This paper is devoted to novel diamond alpha Hardy–Copson type dynamic inequalities, which are complements of the classical ones obtained for and their applications to difference equations. We obtain two kinds of diamond alpha Hardy–Copson type inequalities for , one of which is mixed type and established by the convex linear combinations of the related delta and nabla inequalities while the other one is new and is obtained by using time scale calculus rather than algebra. In contrast to the works existing in the literature, these complements are derived by preserving the directions of the classical inequalities. Therefore both kinds of our results unify some of the known delta and nabla Hardy–Copson type inequalities obtained for into one diamond alpha Hardy–Copson type inequalities and offer new types of diamond alpha Hardy–Copson type inequalities which have the same directions as the classical ones and can be considered as complementary inequalities. Moreover the application of these inequalities in the oscillation theory of half linear difference equations provides several nonoscillation criteria for such equations.

Non-oscillation criterion for Euler type half-linear difference equations with consequences in linear case

Non-oscillation criterion for Euler type half-linear difference equations with consequences in linear case

Nonoscillation criteria for damped half-linear dynamic equations with mixed derivatives on a time scale

In this study, we investigate the use of two types of damped half-linear dynamic equations on a single time scale to provide sufficient conditions to guarantee nonoscillation for nontrivial solutions of both ordinary differential equations and difference equations. If the time scale is selected from the sets of all real numbers or all natural numbers, then these equations are half-linear differential and half-linear difference equations. As a feature of the nonoscillation theorems provided, if both the differential and difference equations are linear, the conditions ensuring nonoscillation are similar. However, if both are nonlinear equations with a one-dimensional p-Laplacian, differences exist in the nonoscillatory conditions. An Euler-type dynamic equation with damping is presented as an example to emphasize the similarities and differences in the nonoscillatory conditions described.

Half-linear differential equations of fourth order: oscillation criteria of solutions

In this paper, we are concerned with the oscillation of solutions to a class of fourth-order delay differential equations with p-Laplacian like operators ( r ( t ) vert x^{prime prime prime } ( t ) vert ^{p_{1}-2}x^{prime prime prime } ( t ) ) ^{prime }+q ( t ) vert x ( tau ( t ) ) vert ^{p_{2}-2}x ( tau ( t ) ) =0 and ( r ( t ) vert x^{prime prime prime } ( t ) vert ^{p_{1}-2}x^{prime prime prime } ( t ) ) ^{prime }+sigma ( t ) vert x^{prime prime prime } ( t ) vert ^{p_{1}-2}x^{ prime prime prime } ( t ) +q ( t ) vert x ( tau ( t ) ) vert ^{p_{2}-2}x ( tau ( t ) ) =0. New oscillation criteria are presented by the comparison technique and employing the Riccati transformation. Moreover, our results are an extension and complement to previous results in the literature. Two examples are shown to illustrate the conclusions.

"New asymptotic results for half-linear differential equations with deviating argument"

"In the paper, we study the oscillation of the half-linear second-order differential equations with deviating argument of the form \begin{equation*} \left(r(t)(y'(t))^{\alpha}\right)'=p(t)y^{\alpha}(\tau(t)). \tag{$E$} \end{equation*} We introduce new monotonic properties of nonoscillatory solutions and use them to offer new criteria for elimination of certain types of solutions. The presented results essentially improve existing ones even for linear differential equations."

On the existence and asymptotic behavior of solutions of half-linear ordinary differential equations

In this paper the half-linear differential equation with one-dimensional p-Laplacian (p=α+1),(|u′|αsgnu′)′=α(λα+1+b(t))|u|αsgnu,t≥t0, is considered, where α and λ are positive constants. It is proved that if the function b(t) is absolutely integrable on [t0,∞), then the above equation has two nonoscillatory solutions u+(t) and u−(t) such that u±(t)∼ce±λt and u±′(t)∼±cλe±λt (t→∞), and moreover, any nontrivial solution u(t) of the above equation satisfies either u(t)∼ceλt, u′(t)∼cλeλt (t→∞) or u(t)∼ce−λt, u′(t)∼−cλe−λt (t→∞). Here, c is a nonzero constant. A weaker result is also shown under the weaker condition that b(t) is conditionally integrable on [t0,∞).

Non-oscillation of modified Euler type linear and half-linear differential equations

Non-oscillation of modified Euler type linear and half-linear differential equations