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In this article, we examine the first-order retarded difference equation ∆v(n) + m ∑i=1pi(n)v (δi(n)) = 0, n ≥ 0, where (δi(n))(i = 1, . . ., m) are sequences of positive real numbers such that δi(n) ≤ n for n ≥ 0, and limn→∞ δi(n) = ∞ and (pi(n))(i = 1, . . ., m) are sequences of nonnegative real numbers, When the delay terms δi(n) are not necessarily monotone, we will give an oscillation criterion obtained for the differential equation, but not yet obtained for its discrete analogous. Finally, we give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

The solution of fractional oscillation equations exhibits highly oscillatory behavior. It is a challenging work to find efficient numerical methods for fractional oscillation equations. The objective of this paper is to develop an accurate numerical technique for fractional oscillation equations with oscillatory solutions. The present method combines the variation‐of‐constants formula with the reproducing kernel function approximation. The merit of our approach is that it can preserve the oscillatory structure of the solution to the considered fractional initial value problems. Illustrative examples clearly show the reliability and efficiency of the approach. [ABSTRACT FROM AUTHOR]

This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP's) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP's). The main dificulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm's theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm's classical theorems on comparison and oscillation. [ABSTRACT FROM AUTHOR]

In this work, we study the oscillatory behavior of even-order neutral delay differential equations υ n (l) + b (l) u (η (l)) = 0 , where l ≥ l 0 , n ≥ 4 is an even integer and υ = u + a (u ∘ μ) . By deducing a new iterative relationship between the solution and the corresponding function, new oscillation criteria are established that improve those reported in (T. Li, Yu.V. Rogovchenko in Appl. Math. Lett. 61:35–41, 2016). [ABSTRACT FROM AUTHOR]

In this work, some new oscillation criteria are established for a third order nonlinear mixed neutral dynamic equation. Our results improve and extend some known results in the literature. Several examples are given to illustrate the importance of the results. [ABSTRACT FROM AUTHOR]

In this paper, we present a new sufficient condition for the oscillation of all solutions of linear delay differential equations. The obtained result improves known conditions in the literature. We also give an example to illustrate the applicability and strength of the obtained condition over known ones. [ABSTRACT FROM AUTHOR]

This work is concerned with the oscillatory behavior of solutions of fourth-order neutral differential equations. By using the Riccati transformation and integral averaging techniques we obtain some new Kamenev-type and Philos-type oscillation criteria. Our results extend and improve some known results in the literature. An example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form... where > 0 is a constant and q(t) is positive continuous function on [a,1), are given in terms of an increasing continuously differentiable function !(t) from [a,1) to (0,1) which satisfies... [ABSTRACT FROM AUTHOR]

In this article, studied the properties of the oscillation of fractional difference equations, and we obtain some results. The results we obtained are an expansion and further development of highly known results. Then we showed them with examples. [ABSTRACT FROM AUTHOR]

An iterative procedure is used to establish the oscillatory sufficient conditions of first-order linear difference equations with several non-monotone deviating arguments and nonnegative coefficients. The conditions obtained by this method, achieve a marked improvement on all known conditions in the literature. [ABSTRACT FROM AUTHOR]

Using Riccati type transformations, the authors establish some new oscillation criteria for the fractional differential equation... whereD0+α is the Riemann-Liouville fractional derivative of order α of y, G(t)=...y(s)ds, and α ε (0,1) Examples are provided to illustrate the relevance of the result. [ABSTRACT FROM AUTHOR]

We study the semi-linear Sturm-Liouville equation ..., where α ≥ 1, p > 1 and λ are real parameters. We prove that all non-trivial solutions are oscillatory and unbounded as x approaches 0. Moreover, there exist γ > 0 and δ > 0 such that any solution u ( x ) resembles x-γη (x-δ) near the origin, where η is a non-trivial periodic solution to the Emden-Fowler equation δ²η" + |η|p-1η = 0 in [ 0, ∞) . [ABSTRACT FROM AUTHOR]

Consider the first-order retarded difference equation Δ x ( n ) + p ( n ) x τ ( n ) = 0 , n ∈ N 0 where ( p ( n ) ) n ⩾ 0 is a sequence of nonnegative real numbers, and ( τ ( n ) ) n ⩾ 0 is a sequence of integers such that τ ( n ) ⩽ n - 1 , n ⩾ 0 , and lim n → ∞ τ ( n ) = ∞ . Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving lim inf , is established. An example illustrates the case when the result of the paper implies oscillation while previously known results fail. [ABSTRACT FROM AUTHOR]

Abstract: We study a scalar linear difference equation with several delays by transforming it to a system of Volterra equations without delays. The results obtained for this system are then used to establish oscillation criteria and asymptotic properties of solutions of the considered equation. [Copyright &y& Elsevier]

In this paper we give some sufficient conditions for the oscillation of all solutions of the following system of neutral difference equations with variable coefficients Δ(ui(n) + cui(n-aλ)) + rΣj=l αij(n) u j (n-K)=0 ; i=1,2,…,r where α ij (n) a are real sequences with i, j =1,2,…, r ; , λ , k ϵ Z+ , a = ± 1 and c ϵ [-1, ∞). [ABSTRACT FROM AUTHOR]

The paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term. First it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question, and vice versa. Advantages and drawbacks of such translation are carefully examined. Then an oscillation criterion for the new class of equations is established. The principal result of the paper is compared to those reported in the literature, and an illustrative example to which known oscillation criteria fail to apply is provided. [Copyright &y& Elsevier]

Some new results are given concerning the behavior of the oscillatory solutions of first or second order delay differential equations. These results establish that all oscillatory solutions x of a first or second order delay differential equation satisfy x(t)=O(v(t)) as t→∞, where v is a nonoscillatory solution of a corresponding first or second order linear delay differential equation. Some applications of the results obtained are also presented. [Copyright &y& Elsevier]

A new procedure based on sixth degree (Hexic) C 2 -Spline for the numerical integration of the second order initial value problems (IVPs) y^{\prime\prime}=f(x,y) , including those possessing oscillatory solutions, is presented. The proposed method is essentially an implicit sixth order one-step method. Stability analysis shows that the method possesses (0, 75.3)\bigcup (130.2, 201.9) as interval of periodicity and/or absolute stability. In addition, the method has phase-lag (dispersion) of order six with actual phase-lag H^{6}/774144 . Convergence results yield error bounds \parallel\! s^{(r)}-y^{(r)}\!\parallel\,=O\left(h^{6}\right),r=0,1 , in the uniform norm, provided y\in C^{8}[0,b] . Furthermore, it turns out that the method is a continuous extension of a sixth order four-stage Runge-Kutta (-Nyström) method. Numerical experiments will also be considered. [ABSTRACT FROM AUTHOR]

Some new oscillatory and asymptotic properties of solutions of neutral differential equations with odd-order are established. Through the new results, we give sufficient conditions for the oscillation of all solutions of the studied equations, and this is an improvement of the relevant results. The efficiency of the obtained criteria is illustrated via example. [ABSTRACT FROM AUTHOR]

This paper relied on the investigation of the properties of the stage-structured model of coupled larvae and adult mosquito populations' evolution when parameterized, in general, by time-varying (or stage-dependent) sequences. In particular, the investigated properties were the non-negativity of the solution under non-negative initial conditions, the boundedness of the sequence solutions under any finite non-negative initial conditions, the equilibrium points, and the convergence conditions to them in the event that the parameterizing sequences converge to finite limits. Some further properties that were investigated relied on deriving the oscillation conditions of the solutions under certain conditions of the parameterizations. The use of feedback controls to decrease the foreseen numbers of alive mosquitoes in future evolution stages is also proposed. The proposed control actions are exerted on the birth rate and/or the maximum progression rate sequences. Some illustrative examples are also given. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillation of second-order neutral differential equations with delayed arguments. Some new oscillatory criteria are obtained by a Riccati transformation. To illustrate the importance of the results, one example is also given. [ABSTRACT FROM AUTHOR]

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the fourth order delay differential equations. Some new oscillatory criteria are obtained by using the generalized Riccati transformations and comparison technique with first order delay differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. The effectiveness of the obtained criteria is illustrated via examples. [ABSTRACT FROM AUTHOR]