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In this paper, we study the second-order linear delay differential equation of the form (E) y ′ ′ (t) + p (t) y (τ (t)) = 0. We establish new oscillation criteria for (1.1), which improve a number of related ones in the literature. Our approach essentially involves establishing stronger monotonicity properties for the positive solutions of (1.1) than those presented in known works. Two main approaches for the investigation of (1.1) will be used, namely a comparison principle with first-order delay differential inequalities and generalization of very effective Koplatadze's technique. We illustrate the improvement over the known results by applying and comparing our method with the other known results for (1.1). [ABSTRACT FROM AUTHOR]

We investigate the oscillation of a first order delay differential equation with non-negative coefficient and non-monotone arguments. New oscillation criterion of lim sup type is obtained. An example is given to show the applicability and strength of the obtained condition over known ones. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the distribution of zeros of solutions of first order linear delay differential equations with variable coefficients of the form where , . By introducing a class of polynomial functions, we are able to derive new estimates for the lower and upper bounds of the distance between consecutive zeros of solutions of the above equations. We illustrate the obtained results with several examples. [Copyright &y& Elsevier]

Abstract: In this work, we analyze the new generalized Emden–Fowler equation with neutral type delays:where . By use of averaging technique and specific analytical skills, some easily-accessible oscillation and asymptotic criteria are established, which have extended the results in the cited literature. [Copyright &y& Elsevier]

Abstract: Han et al. [Z. Han, T. Li, S. Sun, C. Zhang, Oscillation behavior of third order neutral Emden–Fowler delay dynamic equations on time-scales, Adv. Differential Equations 2010 (2010). Article ID 586312, 23 pages] gave some criteria for the oscillation and asymptotic behavior of solutions of the third order neutral delay dynamic equation on a time scale , where is the quotient of odd positive integers. In this paper, the authors establish some new criteria for the oscillation of this equation that improve and unify those of Han et al. [Copyright &y& Elsevier]

Abstract: We investigate the behavior of population models in a periodically varying environment. In this paper, we introduce the Fox surplus harvesting model with delay We obtain explicit conditions for existence of global solutions, existence of a positive periodic solution, oscillation of all solutions about positive equilibrium and global attractivity of this equilibrium. Numerical simulations illustrate the effectiveness of the conditions for oscillations and stability of the Fox model. It was also found that oscillations in the intrinsic rate r(t) and the environmental carrying capacity K(t) contributed quantitatively and qualitatively to the system’s dynamics in significant ways, and cannot be neglected in mathematical models of biological systems. [Copyright &y& Elsevier]

Abstract: Consider the neutral delay difference equation with positive and negative coefficients, where and and . Some sufficient conditions for the existence of a nonoscillatory solution of the above equation expressed in terms of are obtained. [Copyright &y& Elsevier]

Abstract: We consider a certain second-order nonlinear delay differential equation and prove that the all solutions oscillate when proper impulse controls are imposed. An example is given. [Copyright &y& Elsevier]

Abstract: In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the formThe results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper. [Copyright &y& Elsevier]

Abstract: The Mackey-Glass equation, , is considered, with variable coefficients and a nonconstant delay. Under rather natural assumptions all solutions are positive and bounded. Persistence and extinction conditions are presented for this equation. In the case when there exists a constant positive equilibrium, local asymptotic stability of the constant solution and oscillation about this equilibrium are analyzed. The results are illustrated by numerical examples. In particular, it is demonstrated that with delay in both terms, a solution with positive initial conditions may become negative. [Copyright &y& Elsevier]

Some new oscillation criteria of first order nonlinear neutral delay differential equations with variable coefficients are given. Our results extend and improve some of the well known previously results in the literature. Some examples are considered to illustrate our main results. [Copyright &y& Elsevier]

In this paper, some basic oscillatory and nonoscillatory properties of the delay differential inclusions are established, which are used to study fuzzy generalized Hutchinson''s biodynamics models. [Copyright &y& Elsevier]

In this paper, we establish some oscillation criteria for second-order nonlinear neutral delay difference equations with impulses, and some sufficient conditions for the oscillatory behavior of solutions of third-order impulsive delay difference equations are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we shall consider the nonlinear delay differential equation N′(t) = −δ(t)N(t) + P(t)N(t − mω)e−aN(t−mω, where m is a positive integer, δ(t) and P(t) are positive periodic functions of period ω. In the nondelay case, we shall show that (*) has a unique positive periodic solution N(t), and provide sufficient conditions for the global attractivity of N(t). In the delay case, we shall present sufficient conditions for the oscillation of all positive solutions of (*) about N(t), and establish sufficient conditions for the global attractivity of N(t). [Copyright &y& Elsevier]

• Oscillation of certain fourth order differential equations. • To study the oscillatory properties of fourth order delay differential equations. • Comparison with first and second order differential equations and Riccati transformations. The aim of this work is to study the oscillatory properties of the solutions of fourth-order delay differential equations with a p -Laplacian like operator of the form (r (ν) | w ″ ′ (ν) | p 1 − 2 w ″ ′ (ν)) ′ + q (ν) | w (η (ν)) | p 2 − 2 w (η (ν)) = 0. The results obtained are based on the comparison with first and second-order differential equations and the use of a Riccati transformation. Some examples are provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new oscillatory criteria for the second order noncanonical differential equation with delay argument (r (t) (y ′ (t)) α) ′ + p (t) y β (τ (t)) = 0. Our oscillatory results essentially extend the earlier ones. [ABSTRACT FROM AUTHOR]

Abstract: By means of Riccati transformation technique, we establish some new oscillation criteria for a second-order delay differential equation on time scales in terms of the coefficients. [Copyright &y& Elsevier]