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This paper mainly investigates the oscillatory behaviors in a class of discontinuous equations with a variable delay. Under the restrictions of boundedness and monotonicity on the time-delay function, the type and distribution of zero points of the solution are firstly determined. And then, the non-existence of oscillatory solutions with infinite frequency is proven for the systems with two classes of discontinuous vector fields. In detail, by applying the inequality techniques on a backward shift operator, an exact lower bound of the time is provided to ensure the vanishing of infinite frequency oscillations after this time. The obtained results improve some previous works in the cases of the constant delay and constant vector fields. [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]

Analytical solution of initial value problem for the fractional oscillation equation with two Caputo fractional derivatives x′′(t)+aDtαx(t)+cx′(t)+bDtβx(t)+kx(t)=q(t)$$ {x}&#x0005E;{\prime \prime }(t)&#x0002B;a{D}_t&#x0005E;{\alpha }x(t)&#x0002B;c{x}&#x0005E;{\prime }(t)&#x0002B;b{D}_t&#x0005E;{\beta }x(t)&#x0002B; kx(t)&#x0003D;q(t) $$, where the coefficients and orders satisfy a,b,c,k>0,1<α≤2$$ a,b,c,k>0,1<\alpha \le 2 $$ and 0<β≤1$$ 0<\beta \le 1 $$, is investigated by using the Laplace transform and complex inverse integral method on the principal Riemann surface. It is proved by using the argument principle that the characteristic equation has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface under the assumption that α$$ \alpha $$ and β$$ \beta $$ are not both integers. Then three fundamental solutions, the unit impulse response, the unit initial displacement response, and the unit initial rate response, are derived analytically. Each of these solutions is expressed into a superposition of a classical damped oscillation decaying exponentially and a real Laplace integration decaying in a negative power law. Finally, the asymptotic behaviors of these analytical solutions for sufficiently large t$$ t $$ are determined as monotonous decays in a power of negative exponent. [ABSTRACT FROM AUTHOR]

In this work, we study the oscillatory features of a class of neutral differential equations with multiple delays. We present novel oscillation conditions for this equation by using a comparison method. We create conditions that exclude the positive solution of the examined equation. Using the feature of symmetry between non-oscillatory solutions (positive and negative), these conditions also exclude negative solutions without adding additional restrictions. In our study, we take into account the even and odd cases of the order of the equation. Furthermore, we study the asymptotic properties in three different cases of functional coefficients. Our results are a fundamental extension and generalization of previous relevant findings, and this fact has been demonstrated through comparisons. [ABSTRACT FROM AUTHOR]

The present paper studies the asymptotic and oscillatory properties of solutions of odd-order differential equations with advanced arguments and in a noncanonical case. By providing new and effective relationships between the corresponding function and the solution, we present strict and new criteria for testing whether the studied equation exhibits oscillatory behavior or converges to zero. Our results contribute uniquely to oscillation theory by presenting some theorems that improve and expand upon the results found in the existing literature. We also provide an example to corroborate the validity of our proposed criteria. [ABSTRACT FROM AUTHOR]

This paper employed the well-known Riccati transformation method to deduce a Kneser-type oscillation criterion for second-order dynamic equations. These results are considered an extension and improvement of the known Kneser results for second-order differential equations and are new for other time scales. We have included examples to highlight the significance of the results we achieved. [ABSTRACT FROM AUTHOR]

In this study, we investigate the oscillation behavior of second-order self-adjoint q-difference equations, focusing on the renowned Leighton oscillation theorem. Through an example, we demonstrate that the q-version of Leighton’s classical oscillation theorem does not hold and requires refinement. To address this, we introduce an oscillation-preserving transformation and establish alternative theorems to the ones existing in the literature. The strength of our work lies in the absence of any sign condition on the potential function. We also provide illustrative examples to support our findings and mention directions for future research. [ABSTRACT FROM AUTHOR]

In this paper, we establish iterative Hille-type criteria for advanced functional half-linear dynamic equations of the second order. These results extend and improve recent criteria established by multiple authors for the same equation and encompass classical criteria. We provide an example to demonstrate the significance of the results obtained. [ABSTRACT FROM AUTHOR]

In the sense of an arbitrary time scale, some new sufficient conditions on oscillation are presented in this paper for a class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term. The established oscillation results include known Kamenev and Philos-type oscillation criteria and are new oscillation results so far in the literature. Some inequalities, the Riccati transformation, the integral technique, and the theory of time scale are used in the establishment of these oscillation criteria. The proposed results unify continuous and discrete analysis, and the process of deduction is further extended to another class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term and a damping term. As applications for the established oscillation criteria, some examples are given. [ABSTRACT FROM AUTHOR]

In this paper, new Sturmian comparison results were obtained for linear and nonlinear hyperbolic equations on a rectangular prism. The results obtained for linear equations extended those given by Kreith [Sturmian theorems on hyperbolic equations, Proc. Amer. Math. Soc., 22 (1969), 277-281] in which the Sturmian comparison theorem for linear equations was obtained on a rectangular region in the plane. For the purpose of verification, an application was described using an eigenvalue problem. [ABSTRACT FROM AUTHOR]

Principal and nonprincipal solutions of differential equations play a critical role in studying the qualitative behavior of solutions in numerous related differential equations. The existence of such solutions and their applications are already documented in the literature for differential equations, difference equations, dynamic equations, and impulsive differential equations. In this paper, we make a contribution to this field by examining impulsive dynamic equations and proving the existence of such solutions for second-order impulsive dynamic equations. As an illustration, we prove the famous Leighton and Wong oscillation theorems for impulsive dynamic equations. Furthermore, we provide supporting examples to demonstrate the relevance and effectiveness of the results. [ABSTRACT FROM AUTHOR]

We present two sufficient conditions for the oscillatory behaviour of a class of noncanonical fourth-order dynamic equations on arbitrary time scales. An illustrative example is included to show that new criteria improve related results reported in the literature. [ABSTRACT FROM AUTHOR]

When ions are crowded, the effect of steric repulsion between ions (which can produce oscillations in charge density profiles) becomes significant and the conventional Poisson--Boltzmann (PB) equation should be modified. Several modified PB equations were developed but the associated total ionic charge density has no oscillation. This motivates us to derive a general model of PB equations called the PB-steric equations with a parameter Λ, which not only include the conventional and modified PB equations but also have oscillatory total ionic charge density under different assumptions of steric effects and chemical potentials. As Λ = 0, the PB-steric equation becomes the conventional PB equation, but as Λ > 0, the concentrations of ions and solvent molecules are determined by the Lambert type functions. To approach the modified PB equations, we study the asymptotic limit of PB-steric equations with the Robin boundary condition as Λ goes to infinity. Our theoretical results show that the PB-steric equations (for 0 ≤ Λ ≤ ∞ ) may include the conventional and modified PB equations. On the other hand, we use the PB-steric equations to find oscillatory total ionic charge density which cannot be obtained in the conventional and modified PB equations. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic behavior of solutions to a class of semicanonical third-order delay differential equations with a superlinear neutral term. First, we transform the semi-canonical equation into canonical type and then by applying a Riccati type transformation and integral conditions, we obtain some new sufficient conditions ensuring that every solution of the studied equation either oscillates or converges to zero asymptotically. The results are illustrated via two examples. [ABSTRACT FROM AUTHOR]

This paper studies oscillatory properties of solutions of a dynamic equation on the set of time scales Tλ provided that the graininess function µλ approaches zero as λ → 0. We derived the conditions under which oscillation of solutions of differential equations implies that of solutions of the corresponding equations defined on time scales with the same initial data, and vice versa. [ABSTRACT FROM AUTHOR]

In this study, we investigate the oscillatory properties of solutions of a class of conformable fractional generalized Lienard equations. By using generalized Riccati technique, we present some new oscillation results for the equation. Illustrative examples are also given. [ABSTRACT FROM AUTHOR]

In this work, we research oscillation for a class of fractional dynamic equations on time scales involving delay term. By use of a generalized Riccati function, inequality technique, and especially a certain technique dealing with the delay term, some new sufficient conditions for oscillation and asymptotic behaviour are proposed. The established results unify continuous and discrete analysis as two special cases of arbitrary time scales, and are further extensions of the corresponding oscillatory and asymptotic results for delay dynamic equations involving derivatives of integer order. We also present some examples for the established results. [ABSTRACT FROM AUTHOR]

In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case T = R or T = Z. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Przegląd Elektrotechniczny is the property of Przeglad Elektrotechniczny and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The purpose of this paper is to establish some new criteria for the oscillation of the second-order nonlinear noncanonical dynamic equation (a(t)xΔ(t)) Δ + q(t)xβ(g(t)) = 0 under the condition ∫t∞ 1/a(s) Δs < ∞. The authors consider both delay and advanced equations. An example of Euler type equations is provided to illustrate the significance of the main results. [ABSTRACT FROM AUTHOR]

In this work, we study the oscillation of a kind 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]

Some new criteria for oscillation results are obtained for the first order superlinear delay dynamic equations with and without maxima. Our results can be applied to some oscillation problems that were not studied before, we also give examples to clear our results. [ABSTRACT FROM AUTHOR]

Discussed in this report, the oscillating outcomes for resolve of third order delay difference equation. In addition, deriving some new conditions which are of considerable importance for the study of the target equation. [ABSTRACT FROM AUTHOR]

The research contained in this paper belongs to the qualitative theory of dynamic equations on time scales. Via the detailed analysis of solutions of the associated Riccati equation and an advanced averaging technique, we provide the description of domain of nonoscillation of very general equations. The results are formulated and proved for half‐linear equations (i.e., equations connected to PDEs with one dimensional p‐Laplacian) on time scales. Nevertheless, we obtain new results also for linear difference equations. Moreover, the combination of the presented results with previous ones shows that many useful equations are conditionally oscillatory. Such equations are ideal as testing and comparison equations in real‐world models which are beyond capabilities of known oscillation and nonoscillation criteria often. [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]

Copyright of Journal of College of Education is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we study the oscillation of a class of third-order Emden–Fowler delay dynamic equations with sublinear neutral terms on time scales. By using Riccati transformation and integral inequality, we establish several new theorems to ensure that each solution of the equation oscillates or asymptotically approaches zero, and the results in the literature are supplemented and extended. Examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this work, we consider the first-order dynamic equations xΔ(t) + p(t)x (τ (t)) = 0, t ∈ [t0,∞)T where p ∈ Crd ([t0,∞)T,R+), τ ∈ Crd ([t0,∞)T, T) and τ (t) ≤ t, limt→∞ τ (t) = ∞. When the delay term τ (t) is not necessarily monotone, we present a new sufficient condition for the oscillation of first-order delay dynamic equations on time scales. [ABSTRACT FROM AUTHOR]

In this work, we establish new oscillation criteria for the second order neutral dynamic equation with maxima and mixed arguments of the form: where New comparison theorems are obtained that allow us to examine the oscillation of the given second order dynamic equation by examining the oscillatory behavior of first order dynamic ones. The obtained results extend and complement some known ones in literature. Examples are given to illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

The aim in this work is to investigate oscillation criteria for a class of nonlinear discrete fractional order equations with damping term of the form Δ a (t) Δ r (t) g Δ α x (t) β + p (t) Δ r (t) g Δ α x (t) β + F (t , G (t)) = 0 , t ∈ N t 0 . $$\begin{array}{} \displaystyle \Delta\left[a(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta\right]+p(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta+F(t,G(t))=0, t\in N_{t_0}. \end{array}$$ In the above equation α (0 < α ≤ 1) is the fractional order, G (t) = ∑ s = t 0 t − 1 + α t − s − 1 (− α) x (s) $\begin{array}{} \displaystyle G(t)=\sum\limits_{s=t_0}^{t-1+\alpha}\left(t-s-1\right)^{(-\alpha)}x(s) \end{array}$ and Δα is the difference operator of the Riemann-Liouville (R-L) derivative of order α. We establish some new sufficient conditions for the oscillation of fractional order difference equations with damping term based on a Riccati transformation technique and some inequalities. We provide numerical examples to illustrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, some new oscillation criteria for a class of fractional dynamic equations with damping term on time scales are established by use of the properties of fractional calculus and generalized Riccati transformation technique, where the fractional derivative is defined in the sense of the conformable fractional derivative. The established oscillation criteria unify continuous and discrete analysis, and are new results so far in the literature. Oscillation criteria for corresponding dynamic equations on time scales involving integer order derivative are special cases of the present results. For illustrating the established results, some examples are also presented. [ABSTRACT FROM AUTHOR]

Consider the first‐order dynamic equations xΔ(t)+p(t)xτ(t)=0,t∈[t0,∞)Twhere p∈Crd[t0,∞)T,R+,τ∈Crd[t0,∞)T,T and τ(t)≤t,limt→∞τ(t)=∞. Under the assumption that the τ(t) is not necessarily monotone, we present new sufficient conditions for the oscillation of first‐order delay dynamic equations on time scales. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give general solutions of second-order linear dynamic equations on time scales, which are related to Riemann-Weber-type differential equations. The general solutions naturally include iterated logarithm functions on time scales. Using their properties, we obtain complete information on oscillation for the equations, which are important for comparison purposes. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new oscillation criteria for the second-order nonlinear impulsive delay dynamic equation (r(t)|xΔ(t)|α-1xΔ(t))Δ + p(t)f(x(τ(t))) = 0; t Δ= θk; Δ(r(t)|xΔ(t)|α-1xΔ(t))|t=θk + bkh(x(τ(θk))) = 0 on a time scale T. Our results generalize and extend some pervious results [11, 18, 19, 21] and can be applied to some oscillation problems that not discussed before. These results extend the known results for the dynamic equations with and without impulses. Finally, we give some examples to show that impulses play a dominant part in the oscillations of dynamic equations on time scales and to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we establish necessary and sufficient conditions for oscillation of second‐order strongly superlinear and strongly sublinear dynamic equations. Our results unify and improve many known results in the literature. [ABSTRACT FROM AUTHOR]

In the present manuscript, we study the oscillation theory for a first‐order nonlinear neutral dynamic equations on timescales with variable exponents of the form (x(σ(t))−R(t)xξ(t−η))Δ+T(t)∏m=1n|fm(x(t−τm))|αm(t)sign(x(t−τm))=0,∀t∈[t∗,∞)T,where ξ is a quotient of odd positive integers; t∗∈T be a fixed number; [t∗,∞)T is a timescale interval; η,τm > 0; fm∈C(R,R) for m = 1,2,...,n such that xfm(x)>0∀x∈R\{0};R,T∈Crd([t∗,∞)T,R), and the variable exponents αm(t) satisfy ∑m=1nαm(t)=1. The principal goal of this paper is to establish some new succinct sufficient conditions for oscillation. Furthermore, we introduce a forcing term Ξ(·,x(·))∈Crd(T×R,R) and then study the oscillation. Afterward, some interesting special cases are also studied to obtain similar sufficient conditions of oscillation under certain conditions. Moreover, the oscillatory behaviour of the solutions of a first‐order neutral dynamic equation on timescale with a nonlocal condition and a forced nonlinear neutral dynamic equation on time scale are studied. But the proofs are based on the prior estimates obtained in this paper. Some enthralling examples are constructed to show the effectiveness of our analytic results. These counterparts are quite different in the literature even when T=R. Finally, the Kamenev‐type and Philos‐type oscillation criterions are established. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation of solutions of the second order nonlinear neutral delay dynamic equation with "Maxima" of the form on a time scale. The obtained results improve and generalize some recent published results. Examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this study, we consider the generalized neutral Emden-Fowler equation with damping and distributed delay... where z(t) = x(t) + p(t)x(τ(t)), α,β > 0. By using averaging technique and some analytical skills, we obtain the sufficient conditions to ensure the oscillation for the above equation. Our results improve and generalize some existing results. Finally, two examples are given to show the feasibility of our results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the some new oscillatory conditions for all solutions of higher order difference equation with general argument (*) Δmx(n) + p(n)x (τ(n)) = 0; n = 0; 1; ... ; where (p(n)) is a sequence of nonnegative real numbers and (τ(n)) is a sequence of integers and non-monotone. [ABSTRACT FROM AUTHOR]

We analyze the oscillatory behavior of solutions to a nonlinear second-order neutral delay dynamic equation with a nonpositive neutral coefficient under the assumptions that allow applications to Emden-Fowler type dynamic equations. New theorems complement and improve related contributions to the subject. An example is included. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the dynamics of the rational difference equation xn+1 =xn-1/A + Bxnxn-1 with arbitrary initial conditions A and B as nonzero real numbers. The solution is obtained and analytical study and asymptotic behavior are investigated. The forbidden set is determined. The existence of periodic and oscillatory solutions are discussed. Our results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, some new sufficient conditions for the oscillation of all solutions of nonlinear third order neutral dynamic equations are established aiming at extending some well known results in the literature. By using a generalized Riccati transformation and an integral averaging technique, we obtain some new results which ensure that every solution of this equation oscillates or converges to zero. Moreover, an example is given to illustrate the applicability of these results. [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]

This paper is mainly to investigate the oscillation condition for the solution of a class of delay partial difference equations. Since the oscillation condition is equivalent to the regions where the corresponding characteristic equation has no positive root, by applying the envelope theory, some necessary and sufficient conditions for the oscillatory property of the solutions are obtained. [ABSTRACT FROM AUTHOR]

For autonomous functional-differential equations with delays, we obtain an oscillation criterion, which allows one to reduce the oscillation problem to the calculation of a unique root of a real-valued function determined by the coefficients of the original equation. The criterion is illustrated by examples of equations with concentrated and distributed aftereffect, for which convenient oscillation tests are obtained. [ABSTRACT FROM AUTHOR]

We investigate oscillatory behavior of solutions to a class of second-order nonlinear neutral delay dynamic equations with nonpositive neutral coefficients. In particular, we study the corresponding noncanonical neutral differential equations. New oscillation criteria are established that complement and improve related contributions to the subject. An example is given to illustrate the main results. [ABSTRACT FROM AUTHOR]