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We investigate oscillatory properties of even-order half-linear differential equations and conditions for negativity of the associated energy functional. First, using the relationship between positivity of the functional and nonoscillation of the investigated equation, we prove Hille-Nehari type nonoscillation criteria which extend criteria known in the linear case. In the second part of the paper, we present conditions which guarantee that the energy functional attains a negative value, i.e., it is unbounded below. [ABSTRACT FROM AUTHOR]

In this work, necessary and sufficient conditions for oscillations of the solutions of a class of nonlinear first-order neutral differential equations with several delays of the form ... are established under various ranges of r(t). Finally, two illustrating examples are presented to show the feasibility and the effectiveness of the main results. MSC 2010. 34C10, 34C15, 34K40. [ABSTRACT FROM AUTHOR]

A variation of parameters formula with Green function type and Gronwall type integral inequality are proved for impulsive differential equations involving piecewise constant delay of generalized type. Some results for piecewise constant linear and nonlinear delay differential equations with impulsive effects are obtained. They include existence and uniqueness theorems, a variation of parameters formula, integral inequalities, the oscillation property and some applications. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic properties of the distinguished solutions of Riccati matrix equations and inequalities for discrete symplectic systems. In particular, we generalize the inequalities known for symmetric solutions of Riccati matrix equations to Riccati matrix inequalities. We also justify the definition and properties of the distinguished solution and the recessive solution at minus infinity by relating them to their counterparts at plus infinity. [ABSTRACT FROM AUTHOR]

The authors give sufficient conditions for all solutions of a higher order nonlinear delay dynamic equation on time scales to be nonoscillatory. The results cover both the superlinear and sublinear cases and, in the differential equations case, resolve a question that has been open for 30 years. In order to prove their results, the authors first prove a new Bihari type inequality for functions on time scales. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new concept of a recessive solution for discrete symplectic systems, which does not require any eventual controllability assumption. We prove that the existence of a recessive solution is equivalent to the nonoscillation of the system and that recessive solutions can have any rank between explicitly given lower and upper bounds. The smallest rank corresponds to the minimal recessive solution, which is unique up to a right nonsingular multiple, while the largest rank yields the traditional maximal recessive solution. We also present a method for constructing some (but not all) recessive solutions having a block diagonal structure from systems in lower dimension. Our results are new even for special discrete symplectic systems, such as for even order Sturm–Liouville difference equations and linear Hamiltonian difference systems. [ABSTRACT FROM AUTHOR]

We establish nonoscillation criteria for even order half-linear differential equations. The principal tool we use is the Wirtinger type inequality combined with various perturbation techniques. Our results extend nonoscillation criteria known for linear higher order differential equations. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions for the nonoscillation of the fundamental solutions to a linear autonomous differential equation with distributed delay. The conditions are proposed in both the analytic and geometric forms. [ABSTRACT FROM AUTHOR]

Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation x''(t) + c(t)x(t) = 0 can be oscillating and asymptotically unstable, the delay equation x''(t) + a(t)x(t - h(t)) - b(t)x(t - g(t)) = 0, where c(t) = a(t) - b(t), can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper refute this delusion. [ABSTRACT FROM AUTHOR]

In this paper, we establish necessary and sufficient conditions for the solutions of a second-order nonlinear neutral delay dynamic equation with positive and negative coefficients to be oscillatory or tend to zero asymptotically. We consider three different ranges of the coefficient associated with the neutral part in one of which it is allowed to be oscillatory. Thus, our results improve and generalize the existing results in the literature to arbitrary time scales. Some examples on nontrivial time scales are also given. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of the solutions of a class of first order nonhomogeneous advanced impulsive differential equations with piecewise constant arguments. We also study the conditions of periodicity, oscillation, nonoscillation and global asymptotic stability for some special cases. First published online: 09 Jun 2011

New oscillation and nonoscillation criteria are established for the two-dimensional system , , where are locally integrable functions, for a.e. , and . [ABSTRACT FROM AUTHOR]

We present a sharp explicit condition sufficient for the nonoscillation of solutions to a scalar linear nonautonomous difference equation with several delays. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to study the asymptotic properties and oscillation of the nth order neutral differential equationsObtained results are based on the new comparison theorems, that permit to reduce the problem of the oscillation of the nth order equation to the oscillation of a set of the first order equation. Obtained comparison principles essentially simplify the examination of studied equations and allow to relax some conditions imposed on the coefficients of (E). [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

In this paper, we are concerned with oscillation of the nonlinear difference equation Δ(cn[Δ(dnΔxn)]γ) +qnf(xg(n))=0, n≥n0, where γ > 0 is the quotient of odd positive integers, cn, dn and qn are positive sequences of real numbers, g(n) is a sequence of nonnegative integers and f ϵ C(R,R) such that u f (u) > 0 for u ≠ 0. We establish some new sufficient conditions for oscillation by employing the Riccati substitution and the analysis of the associated Riccati difference inequality. Our results extend and improve some previously obtained ones. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

Third-order nonlinear difference equations of the form Δ (cnΔ (dnΔ xn)) + pnΔ xn+1 + qnf (xn-σ)=0, n ≥ n0 are considered. Here, {cn}, {dn}, {pn}, and {qn} are sequences of positive real numbers for n0 ∈ N, f is a continuous function such that f (u) /u ≥ K> 0 for u ≠ 0. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results. [ABSTRACT FROM AUTHOR]

Abstract: For the system with a distributed delay sufficient conditions for positivity of the fundamental matrix are obtained. In the case when the fundamental matrix is positive, comparison results and sufficient positivity conditions for solutions are established. [Copyright &y& Elsevier]

Abstract: For the system with a distributed delay and a nonnegative fundamental matrix, solution estimates and stability results are obtained. The results are applied to integro–differential equations and equations of mixed type including both distributed and concentrated delays. [Copyright &y& Elsevier]

The objective of this paper is to offer sufficient conditions for the oscillation of all solutions and other asymptotic properties of the third-order nonlinear functional differential equation with mixed arguments, where both cases ∫ a( s) ds=∞ and ∫ a( s) ds<∞ are dealt with. We deduce properties of the studied equations via new comparison theorems. Our results essentially improve and complement earlier ones. We also repair one interesting result of Grace et al. [ABSTRACT FROM AUTHOR]

Abstract: For the delay differential equationwe demonstrate that the inequality is not sufficient for oscillation. Moreover, for any A >0 the relation , generally, does not imply oscillation. A similar result is obtained for difference equations. In terms of the maximal argument, new sufficient oscillation conditions are presented for differential and difference equations. [Copyright &y& Elsevier]

Abstract: The objective of this work is to study oscillatory and asymptotic properties of the third-order nonlinear delay differential equation Applying suitable comparison theorems we present new criteria for oscillation or certain asymptotic behaviors of nonoscillatory solutions of Eq. . The results obtained essentially improve and complement earlier ones. [Copyright &y& Elsevier]

We present some new necessary and sufficient conditions for the oscillation of second order nonlinear dynamic equation on an arbitrary time scale $\mathbb{T}$ , where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on $\mathbb{T}$ . Comparison results with the inequality are established and application to neutral equations of the form are investigated. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to study the oscillation of the second order neutral differential equations The obtained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. The obtained comparison principles essentially simplify the examination of the studied equations. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study asymptotic properties of the third-order neutral differential equation . We will establish two kinds of sufficient conditions which ensure that either all nonoscillatory solutions of (E) converge to zero or all solutions of (E) are oscillatory. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

A well-known Picone identity is extended and generalized to second-order dynamic matrix equations on arbitrary time scales. A comparison theorem is obtained in the spirit of the classical Sturm-Picone comparison theorem that extends known scalar results to matrix equations that include the linear homogeneous and inhomogeneous cases, and nonlinear unperturbed and perturbed cases. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a systematic development of the oscillatory results for the linear and nonlinear impulsive delay differential equations. We also discuss linearized oscillation theory, and give some applications to mathematical biology. [ABSTRACT FROM AUTHOR]

Abstract: The objective of this paper is to study asymptotic properties of the couple of third-order neutral differential equations where , ,  are positive functions,  is a quotient of odd positive integers and , . We will establish some sufficient conditions which ensure that all nonoscillatory solutions of converge to zero. Some examples are considered to illustrate the main results. [Copyright &y& Elsevier]

We prove the existence and uniqueness of the solutions of a class of first order nonhomogeneous advanced impulsive differential equations with piecewise constant arguments. We also study the conditions of periodicity, oscillation, nonoscillation and global asymptotic stability for some special cases. [ABSTRACT FROM AUTHOR]

Some new criteria for the oscillation of nonlinear dynamic equations of the form on a time scale $\mathbb{T}$ are established. [ABSTRACT FROM AUTHOR]

Some new criteria for the oscillation of third order nonlinear difference equations are established. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, necessary and sufficient conditions for the asymptotic behaviour of all bounded solutions of a class of nonlinear higher-order neutral dynamic equations are stated related to various ranges of the coefficient associated with the neutral part, in one of which the coefficient is allowed to oscillate and not tend to zero asymptotically. The new results improve and generalize the known results stated for difference equations and differential equations, and answer some problems left open for the case for arbitrary time scales. Some certain examples, which the results in the literature fail to deliver an answer to, are given to illustrate the applicability and significance of the new results. [Copyright &y& Elsevier]

Abstract: We obtain Willett–Wong-type oscillation and nonoscillation theorems for second order linear dynamic equations with integrable coefficients on a time scale. The results obtained extend and are motivated by oscillation and nonoscillation results due to Willett and Wong for the second order linear differential equation. As applications of the new results obtained, we give the complete classification of oscillation and nonoscillation for the difference equations andfor . We also improve a nonoscillation result of Mingarelli and extend an oscillation result of Del Medico and Kong . [Copyright &y& Elsevier]

Abstract: Consider the Emden–Fowler sublinear dynamic equation where , is a time scale, , where , , with the quotient of odd positive integers, . When , and , (0.1) is the usual sublinear Emden–Fowler equation which has attracted the attention of many researchers. In this paper, we allow the coefficient function to be negative for arbitrarily large values of . We extend a nonoscillation result of Wong for the second order sublinear Emden–Fowler equation in the continuous case to the dynamic equation (0.1). As applications, we show that the sublinear difference equation has a nonoscillatory solution, for , , and the sublinear q-difference equation has a nonoscillatory solution, for , , , . [Copyright &y& Elsevier]

Abstract: In this article, we first study the asymptotic behavior of delay dynamic equations having the following form: where is a time scale unbounded from above, , , , and are delay functions. Then, we extend our results to equations of the form: where is allowed to oscillate. Our results state necessary and sufficient conditions on all solutions of the equations to be oscillatory or convergent () or divergent () depending on various ranges of the coefficient . We generalize and improve some of the results for differential and difference equations using the time scale theory. Most of our results for the equations are not, thus far, presented in the literature–not even for the discrete or the continuous cases. [Copyright &y& Elsevier]

Abstract: We establish some new oscillation criteria for solutions to the second order nonlinear dynamic equation on an arbitrary time scale , where and are ratios of positive odd integers, and are positive rd-continuous functions on . The results are obtained when or . These criteria unify and extend known results for the corresponding half-linear and nonlinear differential and difference equations. Some of our results are new even in the continuous and discrete cases. [Copyright &y& Elsevier]

Abstract: We obtain the complete classification of oscillation and nonoscillation for the -difference equation where , . In particular we prove that this -difference equation is nonoscillatory, if and is oscillatory, if . In the critical case we show that it is oscillatory, if , and is nonoscillatory, if . [Copyright &y& Elsevier]

A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made. [ABSTRACT FROM AUTHOR]