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This paper focuses on studying the oscillatory properties of a distinctive class of second-order advanced differential equations with distributed deviating arguments in a noncanonical case. Utilizing the Riccati method and the comparison method with first-order equations, in addition to other analytical methods, we have established criteria to test whether the solutions of the studied equation exhibit oscillatory behavior. To verify the validity of the results we obtained and determine their applicability, we present some examples to confirm the strength and accuracy of our proposed criteria. [ABSTRACT FROM AUTHOR]

In this paper, we address the study of the oscillatory properties of the solutions of a class of third-order delay differential equations. The primary objective of this study is to provide new relationships that can be employed to obtain criteria for excluding increasing positive solutions and decreasing positive solutions so that the resulting criteria are easier to apply than other criteria that have appeared in the literature. We have obtained new oscillation criteria that hold up more robustly upon application. Some examples are presented to illustrate the significance of our main findings. [ABSTRACT FROM AUTHOR]

This paper deals with the oscillation and asymptotic behaviour of even order nonlinear differential equations with mixed nonlinear neutral terms. The findings are obtained via utilising an integral criterion as well as a comparison theorem with the oscillatory properties of a first order advanced and/or delay differential equation. We provide novel oscillation criteria that improve, extend, and simplify previously published ones. The results are illustrated by two examples. [ABSTRACT FROM AUTHOR]

This paper focuses on studying certain oscillatory properties of a new class of half-linear second-order differential equations with an advanced argument in a non-canonical case. By employing some new relations between the solution and its higher derivatives and taking into account the symmetry of positive and negative solutions, we have introduced new criteria to test whether all solutions of the studied equation exhibit oscillatory behavior. Our study aims to expand and enhance previous results, helping to understand these properties in the specified context. The results obtained are confirmed and clarified through an example involving Euler-type equations. [ABSTRACT FROM AUTHOR]

In the current paper, we aim to study the oscillatory behavior of a new class of third-order differential equations. In the present study, we are interested in a better understanding of the relationships between the solutions and their derivatives. The recursive nature of these relationships enables us to obtain new criteria that ensure the oscillation of all solutions of the studied equation. In comparison with previous studies, our results are more general and include models in a wider range of applications. Furthermore, our findings are also significant because no additional restrictive conditions are required. The presented examples illustrate the significance of the results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation and asymptotic behavior of solutions of third-order nonlinear neutral differential equations with a middle term and mixed nonlinear neutral terms in the case of the canonical operator. We establish several oscillation criteria that guarantee that all solutions are oscillatory or converge to zero. The given results are obtained by applying the comparison method, the Riccati transformation and the integral averaging technique. The results improve significantly and extend existing ones in the literature. Finally, illustrative examples are given. [ABSTRACT FROM AUTHOR]

In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form ... are established, where m > 0, α ≥ 0, β ≥ 0 are integers and a(n), b(n), c(n), d(n), p(n) are sequences of real numbers. [ABSTRACT FROM AUTHOR]

In the present article, we set the if and only if conditions for the solutions of the class of neutral impulsive delay second-order differential equations. We consider two cases when it is non-increasing and non-decreasing for quotient of two positive odd integers. Our main tool is the Lebesgue's dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem. [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient conditions are establish of the solutions to second-order delay differential equations of the form (r(t)(x′(t))λ)′ + ∑i=1m qi (t) fi (x(σi(t))) = 0 for t ≥ t0, We consider two cases when fi(u)/uβ is non-increasing for β < γ, and non-decreasing for β > γ where β and γ are the quotient of two positive odd integers. Our main tool is Lebesgue's Dominated Convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem. [ABSTRACT FROM AUTHOR]

In this work, sufficient conditions are established for oscillation of all unbounded solutions of a class of fourth order neutral differential equations of the form (r (t) (y (t) + p (t) y (t − τ)) ″) ″ + q (t) G (y (t − α)) − h (t) H (y (t − σ)) = 0 under the non-canonical type assumption ∫ 0 ∞ t r (t) d t < ∞ for various ranges of p(t) with ∣p(t)∣ < ∞. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are obtained for nonoscillation/oscillation of all solutions of a class of higher-order difference equations involving the generalized difference operator of the form ..., where Δa is generalized difference operator which is defined as Δayn = yn+1−ayn, a ̸= 0. [ABSTRACT FROM AUTHOR]

This article studies the oscillation of solutions of a class of second order nonlinear neutral impulsive difference equations of the form: ... for the various ranges of the neutral coefficient. The technique employed here is due to the linearization method by using the Banach contraction principle and Knaster-Tarski fixed point theorem. In addition, some illustrative examples are given to verify our main results. [ABSTRACT FROM AUTHOR]

In this study we investigate some new oscillation and nonoscillation criteria and generalize and improve some results in the literatures for second order nonlinear difference equation with generalized difference operators of the form Δl,a(pnΔl;axn) + qn(Δl,axn)β = F (n, xn, Δl,bxn), where Δl,σ is generalized difference operator such that defined as Δl,σxn = xn+l - σxn, and F: N × R² → included. [ABSTRACT FROM AUTHOR]

This paper is concerned with the third order functional delay difference equation of the form... We obtain some new oscillation criteria by using Riccati transformation technique. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this study, we have found some sufficient conditions for the oscillation of a class of second order impulsive delay dynamic equations on time scale by using impulsive inequality and Riccati transformation technique. Some examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, by using the Riccati technique which reduces the higher order dynamic equations to a Riccati dynamic inequality, we will establish some new sufficient conditions for oscillation of the second order nonlinear neutral dynamic equation (r(t)((x(t) + p(t)x(τ(t)))Δ)γ)Δ + q(t)xα(δ(t)) + v(t)xβ(η(t)) = 0; on time scales where γ, α, β, are quotient of odd positive integers. [ABSTRACT FROM AUTHOR]

In this paper; by using set-valued analysis, fixed point theory, and the method of upper and lower solutions, we prove some existence, oscillatory and nonoscillatory results for some classes of Caputo-Fabrizio fractional differential equations and inclusions. [ABSTRACT FROM AUTHOR]

This paper deals first with existence of bounded solutions, then followed by some oscillation results, for a coupled fractional q-difference system. For the first results, some applications are made of the fixed point theory, and the diagonalization process. Finally, we give two examples illustrating the applicability of the imposed conditions. [ABSTRACT FROM AUTHOR]

This paper deals with the oscillation of certain class of third order nonlinear delay difference equations with damping. Some new criteria of oscillation of the third order equation in terms of oscillation of a related second order linear difference equation without damping are obtained. Examples are provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this work, the necessary and sufficient conditions for oscillation of a class of second order neutral impulsive systems are established and our impulse satisfies a discrete neutral nonlinear equation of similar type. Further, one illustrative example showing applicability of the new result is included. [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient conditions for asymptotic behavior are established of the solutions to second-order neutral delay differential equations of the form d/dt(r(t)(d/dt[x(t) -- p(t)x(τ(t))])γ) + ... qi(t)fi(x(σi(t))) = 0 for t ≥ to. We consider two cases when fi(u)/uβ is nonincreasing for γ > β, and nondecreasing for β > γ, where β and γ are quotients of two positive odd integers. Our main tool is Lebesgue's dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions for oscillation of unbounded solutions of a class of fourth order neutral delay differential equations of the form (r (t) (y (t) + p (t) y (t − τ)) ″) ″ + q (t) G (y (t − α)) − h (t) H (y (t − σ)) = 0 $$\begin{array}{} \displaystyle (r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\alpha))-h(t)H(y(t-\sigma))=0 \end{array}$$ are discussed under the assumption ∫ 0 ∞ t r (t) d t = ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}\text{d}~~ t=\infty \end{array}$$ [ABSTRACT FROM AUTHOR]

We have established sufficient conditions for oscillation of a class of first order neutral impulsive difference equations with deviating arguments and fixed moments of impulsive effect. [ABSTRACT FROM AUTHOR]

The main objective of this article is to improve and complement some of the oscillation criteria published recently in the literature for third order differential equation of the form (r (t) (z ″ (t)) α) ′ + q (t) f (x (σ (t))) = 0 , t ≥ t 0 > 0 , where z (t) = x (t) + p (t) x (τ (t)) and α is a ratio of odd positive integers in the two cases ∫ t 0 ∞ r − 1 α (s) d s < ∞ and ∫ t 0 ∞ r − 1 α (s) d s = ∞ . Some illustrative examples are presented. [ABSTRACT FROM AUTHOR]

A class of third order differential equations with a sublinear neutral term of the type a (t) b (t) x (t) + p (t) x α (τ (t)) ′ ′ ′ + q (t) x γ (σ (t)) = 0 $$\begin{array}{} \displaystyle \left(a(t)\left(b(t)\left[ x(t)+p(t)x^{\alpha }(\tau (t))\right] ^{\prime }\right) ^{\prime }\right) ^{\prime }+q(t)x^{\gamma }(\sigma (t))=0 \end{array}$$ is considered. Some oscillation criteria are presented to improve and complement those in the literature. Several examples are established to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this article, we prove some existence, oscillatory and nonoscillatory results for some classes of conformable fractional differential equations and inclusions. Our results are based on the set-valued analysis, the fixed point theory, and the method of upper and lower solutions. [ABSTRACT FROM AUTHOR]

In this paper, oscillation of all solutions of fourth order functional differential equations of neutral type of the form (r (t) (y (t) + p (t) y (t − τ)) ″) ″ + q (t) G (y (t − σ)) = 0 $$\begin{array}{} \displaystyle (r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\sigma))=0 \end{array}$$ are studied under the assumption ∫ 0 ∞ t r (t) d t < ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}\text{d}t \lt \infty \end{array}$$ for various ranges of p(t) [ABSTRACT FROM AUTHOR]

In this paper, we give new sufficient conditions for both oscillation and nonoscillation of the delay dynamic equation xΔ(t)+p(t)x(τ(t))=0fort∈[t0,∞)T,where p∈Crd([t0,∞)T,R0+) and τ∈Crd([t0,∞)T,T) satisfy τ(t) ≤ σ(t) for all large t and limt→∞τ(t)=∞. As an important corollary, we obtain the time scale invariant integral condition for nonoscillation: ∫τ(t)σ(t)p(η)Δη≤1e for all large t. Also, with some examples, we show that newly presented results are sharp. [ABSTRACT FROM AUTHOR]

The objective of this work is to study oscillatory and asymptotic properties of the third-order nonlinear delay differential equation (p(t)((r(t)x′(t))′)γ′ + ƒ(t,x(τ(t))) = 0 Applying suitable comparison theorems we present new criteria for oscillation or certain asymptotic behaviors of nonoscillatory solutions. [ABSTRACT FROM AUTHOR]

For a delay differential equation... a generalized Riccati inequality is constructed for nonoscillation and exponential stability of the differential equation. [ABSTRACT FROM AUTHOR]

A nonoscillation problem is dealt with the second-order linear difference equation cnxn+1 + cn-1xn-1 = bnxn, where {bn} and {cn} are positive sequences. For all sufficiently large n ∈ N, the ratios cn/cn-1 and cn-1/bn play an important role in the results obtained. To be precise, our nonoscillation criteria are described in terms of the sequence qn =cn-1/bn cn/bn+1 cn/cn-1 = cn2/bnbn+1. These criteria are compared with those that have been reported in previous researches by using some specific examples. Figures are attached to facilitate understanding of the concrete examples. [ABSTRACT FROM AUTHOR]

In this work, oscillatory and asymptotic behavior of solutions of a class of nonlinear neutral differential equations of second-order with several delays of the form ... are studied, for various ranges of the bounded neutral coefficient p, under the assumptions R1 0 d a = 1 and R1 0 d a < 1. Also, an attempt is made to discuss existence of bounded positive solutions of (E). Further, some illustrative examples, showing the applicability of the new results, are included. [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]

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

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: 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]

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]

Oscillation criteria for two-dimensional difference and differential systems of first-order linear difference equations are generalized and extended to nonlinear dynamic equations on arbitrary time scales. This unifies and extends under one theory previous linear results from discrete and continuous systems. An example is given illustrating that a key theorem is sharp on all time scales. [ABSTRACT FROM AUTHOR]

In this paper, we consider two universal higher order dynamic equations with several delay functions. We will establish two oscillatory criteria of the first equation and a sufficient and necessary condition for the second equation with a nonoscillatory solution by employing fixed point theorem. [ABSTRACT FROM AUTHOR]

Abstract: In this study, a new companion transformation is used for the neutral delay difference equation where , are nonnegative sequences and are positive integers. New criteria, which do not need the conditions and/or for all sufficiently large , are introduced. All the recent results in the literature depend on either the condition (⋆) or the limitation (⋆⋆). We give illustrating examples of which neither oscillatory nor nonoscillatory behaviors are known by the results in the literature, and graphics of these examples are plotted by the mathematical programming language Mathematica 6. Even in the scalar case, our results still improve the literature. Moreover, some mistakes in the literature and their corrections are given. [Copyright &y& Elsevier]

Abstract: Oscillation criteria are presented for second-order differential inclusions We note that the results of this paper are new even in the single valued case. [Copyright &y& Elsevier]

Abstract: We obtain Wong-type comparison theorems for second order linear dynamic equations on a time scale. The results obtained extend and are motivated by Wong''s comparison theorems. As a particular application of our results, we show that the difference equation where , , is oscillatory. [Copyright &y& Elsevier]

Abstract: New oscillation and nonoscillation criteria are established for second-order linear equations with damping and forcing terms. Examples are given to illustrate the results. [Copyright &y& Elsevier]

Abstract: We offer some sufficient conditions for the oscillation of all solutions of third order nonlinear functional differential equations of the formandwhen . The case when is also included. [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: 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]