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We study one class of linear differential equations with varying distributed delay. We obtain an effective (in terms of parameters of the initial problem) criterion for the positivity of the Cauchy function of this class of equations. Based on this result, we establish effective criteria for the exponential stability of equations under consideration. [ABSTRACT FROM AUTHOR]

For the delay differential equations and explicit exponential stability conditions are obtained. [ABSTRACT FROM AUTHOR]

Abstract: For a scalar delay differential equation , we obtain new explicit conditions for the existence of a positive solution. [Copyright &y& Elsevier]

Linear difference equations with variable delay are considered. The most important result of this paper is a new oscillation criterion, which should be looked upon as the discrete analogue of a well-known oscillation criterion for first order linear delay differential equations. This criterion constitutes a substantial improvement of an oscillation result due to the first author (Funkcial. Ekvac. 34 (1991), pp. 157-172). The results obtained extend the ones by the authors and Stavroulakis (J. Differ. Equ. Appl. 10 (2004), pp. 419-435) concerning the special case of linear difference equations with constant delay. [ABSTRACT FROM AUTHOR]

We study existence of nonoscillatory solutions and positiveness of the fundamental function for a scalar linear difference equation with several delays:[image omitted] where al can be either positive or negative. Coefficients of arbitrary signs are also considered. [ABSTRACT FROM AUTHOR]

Abstract: Oscillation and nonoscillation of the second order differential equation with delay depending on the unknown functionin the case when holds are consider. The results obtained in this paper can be conjugated with the theorems given by Bainov et al. [J. Comput. Appl. Math. 91 (1998) 87–96]. [Copyright &y& Elsevier]

The article presents an analysis of the oscillary behavior relative to higher-order nonlinear functional differential equation. It indicates the considerable study on the oscillation of equation and some criteria on oscillation. It mentions some quantity of conditions and comparison on the results relative to linear and semilinear second-order differential for the oscillation of equation. It notes the necessary and sufficient terms for the oscillation of equation as well as the results in comparison of its equality.

Abstract: This paper considers the oscillation problem for the second order half-linear difference equation Δ(∣Δx n−1∣ r−1Δx n−1)+ q n ∣x n ∣ r−1 x n =0, where is a nonnegative sequence and r >0. we establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When r =1, our results are complete extensions of work by Zhang and Zhou [Oscillation and nonoscillation for second-order linear difference equations, Comp. Math. Appl. 39 (2000) 1–7], and reduce to the results in [Y.G. Sun, Oscillation and nonoscillation for second-order linear difference equations, Appl. Math. Comput. 170 (2005) 1095–1103]. When r >0, our results generalize and complement Sun’s results in [Y.G. Sun, Oscillation and nonoscillation for half-linear second order difference equations, Indian J. Pure Appl. Math. 35 (2004) 133–142]. [Copyright &y& Elsevier]

Abstract: In this paper, sufficient conditions are obtained for nonoscillation/oscillation of all solutions of a class of nonlinear third order difference equations of the form involving the generalized difference operator which is defined as , . [Copyright &y& Elsevier]

Abstract: In this paper, we consider a food-limited population model with impulsive effect. Several explicit sufficient conditions are established for oscillation and nonoscillation of solutions of the equations. [Copyright &y& Elsevier]

Abstract: We investigate the oscillatory behaviour of a random Euler-type difference equation, intended to serve as a discrete model of a linear Itô stochastic differential equation with vanishing delay. The oscillatory behaviour of the continuous process satisfying this differential equation was partially described in Appleby and Kelly [Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay, Funct. Differential Equation 11(3–4) (2004) 235–265.] The construction of a discrete model that successfully mimics some of the properties of the continuous process would simplify the analysis, allowing the partial description to be completed. However, care must be taken; a uniform Euler discretisation yields spurious oscillatory behaviour. We present a complete analysis of the uniform scheme. [Copyright &y& Elsevier]

Abstract: We study the oscillation problems for the second order half-linear differential equation , where with , and q are locally integrable on ; , a.e. on , and . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When , our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712–723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180–188] on linear equations to the half-linear case for all . These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39–47] on nonoscillation when and on oscillation when . The approach in this paper can also be used to fully extend Elbert''s criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363–373]. [Copyright &y& Elsevier]

Abstract: Some nonoscillation criteria for quasilinear second order differential equations are obtained. These results generalize the classical results of Hille, Wintner and Opial and recent results of Elbert, Yan, Del Pino as well as Takasi and Yoshida. [Copyright &y& Elsevier]

Nonoscillation criteria for certain second order linear difference equation with delay in impulses are established in this paper. Furthermore, necessary and sufficient condition is obtained for the boundedness of the equation if the equation is nonoscillatory. [ABSTRACT FROM AUTHOR]

Abstract: The paper is devoted to the maximum principles for functional equations in the space of measurable essentially bounded functions. The necessary and sufficient conditions for validity of corresponding maximum principles are obtained in a form of theorems about functional inequalities similar to the classical theorems about differential inequalities of the Vallee Poussin type. Assertions about the strong maximum principle are proposed. All results are also true for difference equations, which can be considered as a particular case of functional equations. The problems of validity of the maximum principles are reduced to nonoscillation properties and disconjugacy of functional equations. Note that zeros and nonoscillation of a solution in a space of discontinuous functions are defined in this paper. It is demonstrated that nonoscillation properties of functional equations are connected with the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Simple sufficient conditions of nonoscillation, disconjugacy and validity of the maximum principles are proposed. The known nonoscillation results for equation in space of functions of one variable follow as a particular cases of these assertions. It should be noted that corresponding coefficient tests obtained on this basis cannot be improved. Various applications to nonoscillation, disconjugacy and the maximum principles for partial differential equations are proposed. [Copyright &y& Elsevier]

Abstract: Some new criteria for the oscillation of certain difference equations with mixed nonlinearities are established. The main tool in the proofs is an inequality due to Hardy, Littlewood, and Pólya. [Copyright &y& Elsevier]

In this paper, oscillation and nonoscillation criteria are established for nonlinear neutral delay differential equations with several positive and negative coefficients [x(t) - R (t) x (t - r)]' + ∑ i=1m Pi (t) Hi(x(t - τi)) - ∑ j=1n Qj(t)Hj(x(t - σj)) = 0. Our criteria improve and extend many results known in the literature. In addition we prove that under appropriate hypotheses, if every solution of the associated linear equation with constant coefficients, y'(t) + ∑ i=1m(pi - ∑ k∈Jiqk)y(t - τi) = 0, oscillates, then every solution of the nonlinear equation also oscillates. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the halflinear delay differential equation for and . Using the method of generalized Riccati equation, we give conditions for the existence of positive solutions. [Copyright &y& Elsevier]

Abstract: Some new criteria for the oscillation of fourth order nonlinear difference equations of the form where are investigated. [Copyright &y& Elsevier]

Abstract: Comparison theorem and explicit sufficient conditions are obtained for oscillation and nonoscillation of solutions of nonlinear impulsive delay differential equations which can be utilized to population dynamic models. Our results in this paper generalize and improve several known results. [Copyright &y& Elsevier]

Abstract: The oscillatory behavior of solutions of the neutral difference equations with nonlinear neutral term,is studied in the case when a ≠ 1. A necessary and sufficient oscillatory conditions for the case 0 < a < 1 and an almost “sharp” oscillatory and nonoscillatory criteria for the case a > 1 are obtained. [Copyright &y& Elsevier]

Abstract: Some new criteria for the oscillation of the fourth order functional differential equation where are established. [Copyright &y& Elsevier]

Necessary and sufficient conditions are found for existence of at least one nonoscillatory solution of a class of impulsive differential equations of third order with retarded argument and fixed moments of impulse effect. Some asymptotic properties of the nonoscillating solutions are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study nonoscillatory solutions of a class of second order nonlinear neutral delay differential equations with positive and negative coefficients. Some sufficient conditions for existence of nonoscillatory solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillatory behavior of first order nonlinear neutral delay differential equations. Several new sufficient conditions which ensure that all solutions are oscillatory are given. The obtained results extend and improve several known results in the literature. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the oscillation and nonoscillation behavior of solution of the nonlinear two-dimensional difference system Some necessary and sufficient conditions are given for the system to admit the existence of oscillatory and nonoscillatory solutions with special asymptotic properties. Some important results in the literatures are generalized. Examples to illustrate the result are included. [Copyright &y& Elsevier]

Abstract: We introduce a natural generalization of the concept of regularly varying functions in the sense of Karamata, and show that the class of generalized Karamata functions is a well-suited framework for the study of the asymptotic behavior of nonoscillatory solutions of the half-linear differential equation of the type [Copyright &y& Elsevier]

In this paper, we consider the oscillation of first order sublinear difference equation with positive neutral term Δ(x(n) + p(n)x(τ(n))) + f(n, x(g1(n)), ... ,x(gm(n))) = 0. We obtain necessary and sufficient conditions for the solutions of this equation to be oscillatory. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study nonoscillatory solutions of a class of second-order nonlinear neutral delay differential equations with positive and negative coefficients. Some sufficient conditions for existence of nonoscillatory solutions are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, some sufficient conditions for oscillation and nonoscillation are obtained for the second-order nonlinear neutral differential equation where , , . [Copyright &y& Elsevier]

Abstract: In this paper we shall consider the discrete nonlinear delay population model with Allee effectwhere a(n), b(n) and c(n) are positive sequences of period ω and p and q are positive integers. By using the coincidence degree theorem as well as some priori estimates we will establish a sufficient condition for the existence of positive periodic solution . [Copyright &y& Elsevier]

Abstract: In this paper, distribution of zeros of solutions to functional equations is studied. It will be demonstrated that oscillation properties of functional equations are determined by the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Distances between zeros of solutions are estimated. On this basis, zones of solutions positivity to the Dirichlet boundary value problem for delay PDEs are estimated. [Copyright &y& Elsevier]

Abstract: New oscillation results are obtained for the second order nonlinear difference equation and its functional form The role played by the argument on the oscillation of the functional equation is explored. In particular, we characterize a class of sequences which have a harmless effect on the oscillation of this type of equations. Some of our results generalize, improve or unify known fundamental oscillation results for several particular cases of the above equations. [Copyright &y& Elsevier]

Abstract: New oscillation and nonoscillation theorems are obtained for the second order quasilinear difference equation where is a constant and is a real sequence with . [Copyright &y& Elsevier]

In this paper, the oscillation and asymptotic stability behavior of a third order linear impulsive equation are investigated. A lemma is presented to deal with the sign relation of the nonoscillatory solutions and their derived functions. By the lemma explicit sufficient conditions are obtained for all solutions either oscillating or asymptotically tending to zero. Two illustrative examples are proposed to demonstrate the effectiveness of the conditions. [ABSTRACT FROM AUTHOR]

Abstract: Some new criteria for the oscillation of functional differential equations of the form, are presented in this paper. [Copyright &y& Elsevier]

The oscillations of nonlinear impulsive delay differential equations can be reduced to the corresponding linear impulsive delay differential equations under some condition. In addition, the conditions for the equivalence of oscillations of nonlinear impulsive delay differential equations and the corresponding linear impulsive delay differential equations were given. The example indicates the results' significance. [ABSTRACT FROM AUTHOR]

The second order nonlinear elliptic differential equation with damping … is considered in an exterior domain Ω(1) ⊂ RN, where the coefficients bi(x), i = 1, … , N, and q(x) are not nonnegative in Ω(1). By using the partial Riccati transformation and the function sequence, some nonoscillation and oscillation theorems are established. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the oscillation and nonoscillation of solutions of second-order linear ordinary differential equations with impulses. Our results show that the oscillatory behavior of all solutions of differential equations without impulses can be inherited by impulsive differential equations under certain impulsive perturbations. It is also seen that the oscillatory properties of all solutions of impulsive differential equations may be caused by the impulsive perturbations though the corresponding differential equations without impulses admits a nonoscillatory solution. [Copyright &y& Elsevier]

In this paper, we are concerned with the oscillation of solutions of the second-order quasilinear differential equation (φp(x′))′+r(t)φp(x)=0, where φp(u)=|u|p-1u,p>0,r∈L1[a,∞), for some a>0. [Copyright &y& Elsevier]

The paper deals with the change of integration for integralswhere m>0 and A,b are continuous nonnegative functions on [0,∞). There are given applications for the second-order differential nonlinear equationswhere 0<β⩽α and a,b are continuous positive functions on [0,∞). The first application brings comparison results for oscillation of two generalized Emden–Fowler equations, the second one shows the similarities and discrepancies for nonoscillatory half-linear and linear differential equations. [Copyright &y& Elsevier]

New oscillation criteria for nth-order functional differential equations of the form fx:1185-1 when g(t) is a retarded argument, i.e., g(t) < t and g(t) is an advanced argument, i.e., g(t) > t are established. [Copyright &y& Elsevier]

In this paper, for second-order superlinear differential equations, we present new oscillation criteria. Our results involve comparison with related linear and half-linear second-order differential equations, so that the known oscillation theorems from literature can be employed directly. [Copyright &y& Elsevier]

We study scalar first order linear autonomous neutral delay differential equations with distributed type delays. This article presents some new results on the asymptotic behavior, the nonoscillation and the stability. These results are obtained via a real root (with an appropriate property) of the characteristic equation. Applications to the special cases such as (non-neutral) delay differential equations are also presented. [ABSTRACT FROM AUTHOR]

Several oscillation criteria are given for the second-order damped nonlinear differential equation (a(t)[y′(t)]σ′i +p(t)[y′(t)]σ +q(t)f(y(t)) = 0, where σ > 0 is any quotient of odd integers, a ε C(R, (0, ∞)), p(t) and q(t) are allowed to change sign on [to, ∞), and f ε Cl (R, R) such that xf (x) > 0 for x≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results. [Copyright &y& Elsevier]

The most important result of this paper is a new oscillation criterion for delay difference equations. This criterion constitutes a substantial improvement of the one by Ladas et al. [ J. Appl. Math. Simulation 2 (1989), 101-111] and should be looked upon as the discrete analogue of a well-known oscillation criterion for delay differential equations. [ABSTRACT FROM AUTHOR]

Oscillation criteria for all solutions of the first order delay difference equation of the form $$ x_{n+1}- x_{n} + p_nx_{n-k} = 0, \quad n = 0, 1, 2,..., $$ where { p n} is a sequence of nonnegative real numbers and k is a positive integer are established especially in the case that the well-known oscillation conditions $$\limsup_{n\rightarrow \infty} \sum^n_{i=n-k} p_i > 1 \quad and \quad \liminf_{n\rightarrow \infty} {1\over k} \sum^{n-1}_{i=n-k} p_i > {k^k\over (k+1)^{k+1}}$$ are not satisfied. [ABSTRACT FROM AUTHOR]

Consider the second order self-adjoint neutral difference equation of form In this paper, we will give the classification of nonoscillatory solutions of the above equation; and by the fixed point theorem, we present some existence results for some kinds of nonoscillatory solutions of the equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the nonoscillatory problems of odd-dimensional systems of linear retarded functional differential equations. Based upon the corresponding characteristic equations, we get some criteria for nonoscillations by utilizing the matrix measures. [Copyright &y& Elsevier]