Search

In this paper we introduce new effective technique for investigation of oscillation for the second-order trinomial differential equation with retarded and advanced arguments y ′ ′ (t) = p (t) y (τ (t)) + q (t) y (σ (t)). Our criteria improve the existing ones and the progress is illustrated via several examples. [ABSTRACT FROM AUTHOR]

Abstract It is well-known that for the oscillation of all solutions of the linear delay differential equation x ′ (t) + p (t) x (t − τ) = 0 , t ≥ t 0 , with p ∈ C ([ t 0 , ∞) , R +) and τ > 0 it is necessary that B ≔ lim sup t → ∞ A (t) ≥ 1 e , where A (t) ≔ ∫ t − τ t p (s) d s. Our main result shows that if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions B > 1 e implies the oscillation of all solutions of the above linear delay differential equation. The applicability of the obtained results and the importance of the slowly varying assumption on A are illustrated by examples. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to study the second-order nonlinear noncanonical differential equation r (t) y ′ (t) ′ + p (t) f (y (τ (t))) = 0 under the condition ∫ ∞ 1 r (t) d t < ∞. Contrary to most existing results, oscillation of the studied equation is attained via only one condition. We consider both delay and advanced differential equations. A particular example of Euler type equation is provided in order to illustrate the significance of our main results. [ABSTRACT FROM AUTHOR]

Abstract We show by a series of counterexamples that a commonly used assumption in the study of asymptotic and oscillatory properties of solutions to neutral differential equations is not valid. Appearing consequences and related open problems are briefly discussed. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to complement existing oscillation results for third-order neutral delay differential equations by establishing sufficient conditions for nonexistence of so-called Kneser solutions. Combining newly obtained results with existing ones, we attain oscillation of all solutions of the studied equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide an oscillation theorem that can be applied to half-linear differential equations with time-varying coefficients. A parametric curve by the coefficients is focused in order to obtain our theorem. This parametric curve is a generalization of the curve given by the characteristic equation of the second-order linear differential equation with constant coefficients. The obtained theorem is proved by transforming the half-linear differential equation to a standard polar coordinates system and using phase plane analysis carefully. [ABSTRACT FROM AUTHOR]

Applying a modification of the adapted Riccati technique, we introduce a new type of oscillation criteria for generalizations of the Euler half-linear difference equation. Our main criterion is new in many cases. This fact is documented by corollaries and examples. In particular, we obtain new results for linear equations. [ABSTRACT FROM AUTHOR]

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 establish new comparison theorem for deducing oscillation of noncanonical nonlinear differential equation r ( t ) u ′ ( t ) γ ′ + p ( t ) u γ ( τ ( t ) ) = 0 from that of suitable linear equation y ″ ( t ) + p ˜ ( t ) u ( τ ˜ ( t ) ) = 0 . Our results are based on Trench’s theory of canonical operators and comparison technique. [ABSTRACT FROM AUTHOR]

A new sufficient condition is given for the oscillation of all solutions of a linear scalar delay differential equation with slowly varying and uniformly positive coefficient at infinity. Based on the celebrated Myshkis type necessary condition, we show that this new sufficient condition is sharp within the considered class of delay differential equations. [ABSTRACT FROM AUTHOR]

Establishing new properties of nonoscillatory solutions we introduce new oscillatory criteria for the second order advanced differential equation y ′ ′ ( t ) + p ( t ) y ( σ ( t ) ) = 0 . Our oscillatory results essentially extend the earlier ones. The progress is illustrated via Euler differential equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with oscillation of second order superlinear dynamic equations with oscillating coefficients. By using generalized Riccati transformations, oscillation theorems are obtained on an arbitrary time scale. We illustrate the versatility of our results by means of examples. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the second-order half-linear delay differential equation r ( t ) y ′ ( t ) α ′ + q ( t ) y α ( τ ( t ) ) = 0 under the condition ∫ ∞ r − 1 ∕ α ( t ) d t < ∞ . Contrary to most existing results, oscillation of the studied equation is attained via only one condition. A particular example of Euler type equation is provided in order to illustrate the significance of our main results. [ABSTRACT FROM AUTHOR]

We study asymptotic behavior of solutions to a class of higher-order sublinear Emden–Fowler delay differential equations. Our theorems improve several results reported recently in the literature. Two examples are provided to illustrate the importance and advantages of new criteria. [ABSTRACT FROM AUTHOR]

In the paper we study properties of positive solutions for the third-order neutral differential equations of the form ( r ( t ) z ″ ( t ) ) ′ + q ( t ) y ( t ) = 0 , where z ( t ) = y ( t ) + p ( t ) y ( τ ( t ) ) . The lower and upper estimates of positive decreasing solutions of studied equations are provided that yield the rate of convergence to zero for such solutions. [ABSTRACT FROM AUTHOR]

In this paper, the distance between adjacent zeros of oscillatory solutions for second order nonlinear neutral delay differential equations ( a ( t ) z ′ ( t ) ) ′ + q ( t ) f ( x ( t − σ ) ) = 0 , t ≥ t 0 , where z ( t ) = x ( t ) + p ( t ) x ( t − τ ) is investigated. By means of inequality techniques, specific function sequences and nonincreasing solutions for corresponding first order differential inequality, some new estimates for the distribution of zeros of oscillatory solutions have been presented, which have extended and improved some known results. [ABSTRACT FROM AUTHOR]

The distributed-order model is equivalent to weighting the fractional-order model on a given interval. In this paper, a double time-distributed order Maxwell model is developed and formulated to study the characteristics of a start-up pipe flow. The numerical solution of the distributed-order governing equation is obtained by the finite difference method. In order to test the convergence of numerical solutions, the truncation error of the proposed numerical method be calculated. The characteristics of the flow and the effects of fractional parameters and weight functions on the velocity distribution are discussed in detail. By comparing the velocity distribution diagrams, it can be inferred that the distributed-order model controlled by the weight function can describe the memory and elastic properties of fluid more flexibly. [ABSTRACT FROM AUTHOR]

This paper is concerned with oscillation of second order sublinear dynamic equations with oscillating coefficients. By using generalized Riccati transformations, oscillation theorems are obtained on an arbitrary time scale. Our results provide new oscillation criteria for arbitrary time scales. [ABSTRACT FROM AUTHOR]

We study oscillatory behavior of solutions to a class of even-order neutral differential equations relating oscillation of higher-order equations to that of a pair of associated first-order delay differential equations. As illustrated with two examples in the final part of the paper, our criteria improve a number of related results reported in the literature. [ABSTRACT FROM AUTHOR]

We establish some new criteria for the oscillation and asymptotic behaviour of solutions of the third-order nonlinear neutral differential equation ( 1 p ( t ) ( 1 r ( t ) [ x ( t ) + a ( t ) x ( γ ( t ) ) ] ′ ) ′ ) ′ + q ( t ) f ( x ( δ ( t ) ) ) = 0 . Applications illustrating the results are given. [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]

We establish some new oscillation criteria for a class of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient on a time scale. The results obtained supplement and improve those reported in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we deal with oscillatory and asymptotic properties of solutions of a fourth order sub-linear differential equation with the oscillatory operator. We establish conditions for the nonexistence of positive and bounded solutions and an oscillation criterion. [Copyright &y& Elsevier]

Abstract: Erbe’s and Hassan’s contributions regarding oscillation criteria are interesting in the development of oscillation theory of dynamic equations on time scales. The objective of this paper is to amend these results. [Copyright &y& Elsevier]

Abstract: By making use of asymptotic properties of nonoscillatory solutions, the oscillation behavior of solutions for the integro-dynamic equation and the integral equation on time scales is investigated. Easily verifiable sufficient conditions are established for the oscillation of all solutions. The results are new for both continuous and discrete cases. The paper is concluded by an open problem. [Copyright &y& Elsevier]

Abstract: This paper is concerned with oscillatory behavior of a class of second-order delay dynamic equations on a time scale. Two new oscillation criteria are presented that improve some known results in the literature. The results obtained are sharp even for the second-order ordinary differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, the authors initiate the study of oscillation theory for integro-dynamic equations on time-scales. They present some new sufficient conditions guaranteeing that the oscillatory character of the forcing term is inherited by the solutions. [Copyright &y& Elsevier]

Abstract: In this paper, the authors initiate the study of oscillation theory for integro-dynamic equations on time-scales. They present some new sufficient conditions guaranteeing that the oscillatory character of the forcing term is inherited by the solutions. [Copyright &y& Elsevier]

Abstract: This work is concerned with the oscillation of a certain class of fourth-order nonlinear dynamic equations on time scales. A new oscillation result and an example are included. [Copyright &y& Elsevier]

Abstract: A necessary and sufficient condition is given under which a linear time-invariant system of ordinary differential equations has a solution which is nonoscillatory with respect to a fixed set of cones. In contrast with existing results, ours allows that the given cones do not cover the whole space. The proof of the result is based on its discrete counterpart. [Copyright &y& Elsevier]

Abstract: In this paper, we give a nonoscillation criterion for half-linear equations with periodic coefficients under fixed moments of impulse actions. The method is based on the existence of positive solutions of the related Riccati equation and a recently obtained comparison principle. In the special case when the equation becomes impulsive Hill equation new oscillation criteria are also obtained. [Copyright &y& Elsevier]

Abstract: The third-order nonlinear functional differential equations of the form are considered. We present some new oscillatory and asymptotic behavior of solutions of this equation by modifying a method given for second-order differential equations. Our results are applicable to nonlinear functional differential equations of the above form. Several examples are also given to illustrate the importance of our results. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain some sufficient conditions for the oscillation of all solutions of the second order half-linear difference equation where is the forward difference operator,  is a sequence of positive real numbers, is a sequence of nonnegative real numbers and . One open problem in Remark 3 of Rehak (2000)  is solved. [Copyright &y& Elsevier]

Abstract: In this note, we study the oscillatory behavior of the following higher-order half-linear delay differential equation where we assume . An example is given to illustrate the main results. [Copyright &y& Elsevier]