Half-linear Dynamic Equations Research Articles

Properties of third-order half-linear dynamic equations with an unbounded neutral coefficient

We study oscillation and asymptotic behavior of a class of third-order half-linear functional dynamic equations with an unbounded neutral coefficient. Several comparison theorems are presented that are essentially new.

Asymptotic properties of third-order half-linear neutral dynamic equations with mixed arguments

This work is concerned with the asymptotic behavior of a class of third-order half-linear neutral dynamic equations with mixed arguments on a time scale. Some new criteria and an example are presented.

Oscillation Theorems for Fourth-Order Half-Linear Delay Dynamic Equations with Damping

This article is concerned with oscillatory behavior of a class of fourth-order half-linear delay dynamic equations with damping on a time scale. Some new oscillation criteria are established.

Peculiarities in power type comparison results for half-linear dynamic equations

We look for conditions guaranteeing that oscillatory properties of a half-linear dynamic equation are preserved when the power in the nonlinearities is changed. In particular, we discuss discrepancies (which indeed occur) between the results on different time scales. We provide an example showing an optimality of one of the assumptions. We also present one generalization of a standard Sturm-Picone type comparison theorem and give an important note on the smoothness condition for the coefficient in the leading term which concerns general theory of second order dynamic equations. Many of our observations are new also in the differential and difference equations cases.

Oscillation of Second-Order Half-Linear Dynamic Equations on Time Scales

Oscillation of Second-Order Half-Linear Dynamic Equations on Time Scales

Some dynamic Wirtinger-type inequalities and their applications

In this paper, we present some new Wirtinger-type inequalities on time scales. As special cases, some new continuous and discrete Wirtinger-type inequalities are given. The obtained inequalities are applied to a certain class of half-linear dynamic equations on time scales in order to establish sufficient conditions for disconjugacy.

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.

Interval criteria for second-order super-half-linear functional dynamic equations with delay and advance arguments

Interval oscillation criteria are established for second-order forced super half-linear dynamic equations on time scales containing both delay and advance arguments, where the potentials and forcing term are allowed to change sign. Four discrete examples are provided to illustrate the relevance of the results. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.

Nonoscillation of half-linear dynamic equations

The half-linear dynamic equation on time scales (r(t)|yΔ|α−1sgnyΔ)Δ+p(t)|yσ|α−1sgnyσ=0,α>1, is here considered, under the condition ∫∞r1/(1−α)(s)Δs<∞. Two new characterizations of nonoscillation to this equation are provided, namely in terms of solvability to a related weighted integral Riccati type inequality and in terms of the convergence of a certain function sequence involving the coefficients r and p. These relations are then applied to obtain new oscillation criteria and a comparison theorem, which extend the results known from the continuous or the linear case. The results are new even in the well-studied difference equation setting. A basic classification of nonoscillatory solutions to the equation is also presented. The paper is concluded by indicating some directions for a future research.

A critical oscillation constant as a variable of time scales for half-linear dynamic equations

Abstract We present criteria of Hille-Nehari type for the half-linear dynamic equation (r(t)Φ(y Δ))Δ+p(t)Φ(y σ) = 0 on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from q-calculus, a Hardy type inequality with weights, and further possibilities for study are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case.

A telescoping principle for oscillation of the second order half-linear dynamic equations on time scales

Abstract . We establish the so-called “telescoping principle” for oscillation of the second order half-linear dynamic equation [r(t)Φ(x&lt;sup&gt;Δ&lt;/sup&gt;)]&lt;sup&gt;Δ&lt;/sup&gt; + c(t)Φ(x&lt;sup&gt;σ&lt;/sup&gt;) = 0 on a time scale. This principle provides a method enabling us to construct many new oscillatory equations. Unlike previous works concerning the telescoping principle, we formulate some oscillation results under the weaker assumption r(t) ≠ 0 (instead r(t) &gt; 0).

On the oscillation of second-order half-linear dynamic equations1

We obtain some oscillation criteria for solutions to the second-order half-linear dynamic equation when or . These criteria unify and extend known criteria for corresponding half-linear differential and difference equations. Some of our results are new even in the continuous and the discrete cases.

New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales

Let T be a time scale (i.e., a closed nonempty subset of R ) with sup T = + ∞ . Consider the second-order half-linear dynamic equation ( r ( t ) ( x Δ ( t ) ) α ) Δ + p ( t ) x α ( σ ( t ) ) = 0 , where r ( t ) > 0 , p ( t ) are continuous, ∫ t 0 ∞ ( r ( t ) ) − 1 α Δ t = ∞ , α is a quotient of odd positive integers. In particular, no explicit sign assumptions are made with respect to the coefficient p ( t ) . We give conditions under which every positive solution of the equations is strictly increasing. For α = 1 , T = R , the result improves the original theorem [see: [Lynn Erbe, Oscillation theorems for second-order linear differential equation, Pacific J. Math. 35 (2) (1970) 337–343]]. As applications, we get two comparison theorems and an oscillation theorem for half-linear dynamic equations which improve and extend earlier results. Some examples are given to illustrate our theorems.

Oscillation criteria for half-linear dynamic equations on time scales

This paper is concerned with oscillation of the second-order half-linear dynamic equation ( r ( t ) ( x Δ ) γ ) Δ + p ( t ) x γ ( t ) = 0 , on a time scale T where γ is the quotient of odd positive integers, r ( t ) and p ( t ) are positive rd-continuous functions on T . Our results solve a problem posed by [R.P. Agarwal, D. O'Regan, S.H. Saker, Philos-type oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085–1104; S.H. Saker, Oscillation criteria of second order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 375–387] and our results in the special cases when T = R and T = Z involve and improve some oscillation results for second-order differential and difference equations; and when T = h Z , T = q N 0 and T = N 0 2 , etc., our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.

Philos-Type Oscillation Criteria for Second Order Half-Linear Dynamic Equations on Time Scales

In this paper we establish some oscillation theorems for the second order half-linear dynamic equation ( r(t)(x(t)) )Δ + p(t)x(t) = 0, t ∈ [a, b], on time scales. Special cases of our results include some well-known oscillation results for second-order differential and half-linear differential equations. Our results are new for difference, generalized difference and q difference half-linear equations.

Hardy inequality on time scales and its application to half-linear dynamic equations

A time-scale version of the Hardy inequality is presented, which unifies and extends well-known Hardy inequalities in the continuous and in the discrete setting. An application in the oscillation theory of half-linear dynamic equations is given.

Oscillation criteria of second-order half-linear dynamic equations on time scales

In this paper, by using the Riccati transformation technique, chain rule and inequality A λ - λ AB λ - 1 + ( λ - 1 ) B λ ⩾ 0 , λ > 1 , where A and B are positive constants, we will establish some oscillation criteria for the second-order half-linear dynamic equation ( p ( t ) ( x Δ ( t ) ) γ ) Δ + q ( t ) x γ ( t ) = 0 , t ∈ [ a , b ] on time scales, where γ > 1 is an odd positive integer. Our results not only unify the oscillation of half-linear differential and half-linear difference equations but can be applied on different types of time scales and improve some well-known results in the difference equation case. Some examples are considered here to illustrate our main results.

Positive solutions of second-order half-linear dynamic equations on time scales

Necessary and sufficient conditions are obtained for the existence of positive solutions of a second-order half-linear dynamic equation on time scales. Relations between this equation and an advanced type dynamic equation are also discussed.