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Oscillation Theory for Neutral Differential Equations with Delay
... e oscillation theory of differential equations is an active area of research. Many scholars have studied the oscillation of solutions to various classes of differential equations and made some progress, especially for the first-order and second-order equations (see, e.g., the monographs [3,4], the papers [5,6], and the references cited therein). ird-order differential equations originate in many different fields of applied mathematics and physics, for example, the deflection of a buckling beam with a fixed or varying cross section, three-layer beam, electromagnetic waves, and the rising tide caused by gravitational blowing. ...
... Proof. Let x(t) be an eventually positive solution of equation (4). Because x(t) has property (II), we obtain lim t⟶∞ x(t) � l ≥ 0. ...
... In this section, we establish some new oscillation criteria for (4). For the following theorem, we introduce a class of function R. Let ...
In this paper, we study oscillatory properties of solutions to a class of third-order differential equation a t r t x ′ t ′ α ′ + p t r t x ′ t ′ α + q t f x σ t = 0 , where f x / x β ≥ k > 0 and α and β are quotients of odd positive integers. By using the generalized Riccati technique, we obtain some oscillation and asymptotic criteria when α ≥ β and α < β . Finally, some examples are given to show the effectiveness of the criteria obtained.
... ( 1 ( )( ′ ( )) ) ′ = 1 ( ) 1 ( ( 1 ( ))) ( 2 ( )( ′ ( )) ) ′ = 2 ( ) 2 ( ( 2 ( ))) , ≥ 0 > 0. (1) where ( ) = ( ) + 1 ( ) ( 1 ( )), ( ) = ( ) + 2 ( ) ( 2 ( )) and > 0 is the ratio of two odd integers. By a solution to system (1), we mean functions ( ) = [ ( ), ( )] which have the properties 1 ( )( ′ ( )) , 2 ( )( ′ ( )) ∈ 1 and satisfies system (1). ...
... ( 1 ( )( ′ ( )) ) ′ = 1 ( ) 1 ( ( 1 ( ))) ( 2 ( )( ′ ( )) ) ′ = 2 ( ) 2 ( ( 2 ( ))) , ≥ 0 > 0. (1) where ( ) = ( ) + 1 ( ) ( 1 ( )), ( ) = ( ) + 2 ( ) ( 2 ( )) and > 0 is the ratio of two odd integers. By a solution to system (1), we mean functions ( ) = [ ( ), ( )] which have the properties 1 ( )( ′ ( )) , 2 ( )( ′ ( )) ∈ 1 and satisfies system (1). A solution ( ) = [ ( ), ( )] of system (1) is said to oscillate if every component of ( ) has arbitrarily large zeros. ...
... Then there exist 2 < 0 such that 2 ( )( ′ ( )) ≤ 2 , ′ ( ) ≤ ( 2 2 ( ) ) 1 . ...
The oscillation criteria for nonlinear systems of neutral differential equations were studied.
Sufficient conditions were obtained to ensure that all bounded solutions to this system either
oscillate or converge to zero when 𝑡→ ∞. Some examples were given to illustrate the results.
... Neutral/delay differential equations are used in a variety of problems in economics, biology, medicine, engineering and physics, including lossless transmission lines, vibration of bridges, as well as vibrational motion in flight, and as the Euler equation in some variational problems, see [1][2][3]. ...
... The authors in [34] used the comparison technique that differs from the one we used in this article. Our approach is based on using integral averaging method and the Riccati technique to reduce the main equation into a first-order inequality to obtain more effective oscillation conditions for Equation (2). Therefore, in order to highlight the novelty of the results that we obtained in this work, we presented a comparison between the previous results and our main results, represented in the Example 2. ...
... Proof. Suppose that x(t) is a positive solution of Equation (2). Then, we can assume that x(t) > 0, x(β(t)) > 0 and x(w(t)) > 0 for t ≥ t 1 . ...
In this work, new criteria for the oscillatory behavior of even-order delay differential equations with neutral term are established by comparison technique, Riccati transformation and integral averaging method. The presented results essentially extend and simplify known conditions in the literature. To prove the validity of our results, we give some examples.
... ( 1 ( )( ′ ( )) ) ′ = 1 ( ) 1 ( ( 1 ( ))) ( 2 ( )( ′ ( )) ) ′ = 2 ( ) 2 ( ( 2 ( ))) , ≥ 0 > 0. (1) where ( ) = ( ) + 1 ( ) ( 1 ( )), ( ) = ( ) + 2 ( ) ( 2 ( )) and > 0 is the ratio of two odd integers. By a solution to system (1), we mean functions ( ) = [ ( ), ( )] which have the properties 1 ( )( ′ ( )) , 2 ( )( ′ ( )) ∈ 1 and satisfies system (1). ...
... ( 1 ( )( ′ ( )) ) ′ = 1 ( ) 1 ( ( 1 ( ))) ( 2 ( )( ′ ( )) ) ′ = 2 ( ) 2 ( ( 2 ( ))) , ≥ 0 > 0. (1) where ( ) = ( ) + 1 ( ) ( 1 ( )), ( ) = ( ) + 2 ( ) ( 2 ( )) and > 0 is the ratio of two odd integers. By a solution to system (1), we mean functions ( ) = [ ( ), ( )] which have the properties 1 ( )( ′ ( )) , 2 ( )( ′ ( )) ∈ 1 and satisfies system (1). A solution ( ) = [ ( ), ( )] of system (1) is said to oscillate if every component of ( ) has arbitrarily large zeros. ...
... Then there exist 2 < 0 such that 2 ( )( ′ ( )) ≤ 2 , ′ ( ) ≤ ( 2 2 ( ) ) 1 . ...
... Neutral delay differential equations (NDDEs) are encountered in several kinds of phenomena, such as electric transmission line problems, which are utilized for interconnecting switching circuits in high-speed computers, the study of vibrating masses connected to elastic bars, the solution of variational problems involving time delays or in the theory of automatic control, and neuro-mechanical systems where inertia is a significant factor (see [6][7][8][9][10]). The reader is directed to consult the references [11][12][13][14][15] for comprehensive insights into the methodologies, techniques, and findings relating to the investigation of oscillatory behavior in third-order NDDEs. ...
... q i ( )(1 − p(τ i ( )).(7) (a) Using the facts J 2 z( ) > 0 and J 3 z( ) ≤ 0, it is obvious that J 2 z( ) → 0 as → ∞. Assume the contrary that 0 > 0. Hence, it follows that J 2 z( ) ≥ 0 > 0. Therefore,J 1 z( ) ≥ 1 1 κ 2 (ρ) J 2 z(ρ)dρ,(8)and soz( ) ≥ 1 1 κ 1 (ρ) J 1 z(ρ)dρ ≥ 1 1 κ 1 (ρ) ρ 1 1 κ 2 (u) J 2 z(u)du dρ (9) ≥ 0 1 1 κ 1 (ρ) ρ 1 1 κ 2 (u) du dρ = 0 1 M 2 (ρ) κ 1 (ρ) dρ > δ 0 M 12 ( ). ...
This study aims to examine the oscillatory behavior of third-order differential equations involving various delays within the context of functional differential equations of the neutral type. The oscillation criteria for the solutions of our equation have been obtained in this study to extend and supplement existing findings in the literature. In this study, a technique that relies on repeatedly improving monotonic properties was used in order to exclude positive solutions to the studied equation. Negative solutions are excluded based on the symmetry between the positive and negative solutions. Our results are important because they become sharper when applied to a Euler-type equation as compared to previous studies of the same equation. The significance of the findings was illustrated through the application of these findings to specific cases of the investigated equation.
... Yildiz et al. [33,34] have considered neutral type nonlinear higher-order functional differential equations with oscillating coefficients. Basic definitions and results on oscillation for neutral type differential equations are given in [2]. It is a well-known fact that the motions on the earth are not always uniform as various kinds of resistance appear during the motions. ...
... if and only if u(x) satisfies (1) on the interval (x 0 , ∞) . (2). Then we will show that u(x) satisfies ...
In the present article, we considered a class of nth order impulsive neutral differential equations. The study on the oscillatory and asymptotic behavior of solutions for the higher-order neutral differential equation is theoretical and practical. Various techniques appeared for these studies. We reduced this class into a class of non-impulsive neutral differential equations by using suitable substitutions. Through a comparison strategy involving first-order differential equations, we studied the oscillatory and asymptotic behavior of solutions. Sufficient conditions are obtained for asymptotic as well as oscillatory bounded solutions. Several examples have illustrated the effectiveness of the requirements.
... The importance of studying delay differential equations DDEs is not limited to the theoretical side only, but the applications of this type of equations extend to many branches of applied science. In fact, the neutral DDEs arise in the examination of vibrating masses attached to an elastic bar, in the solution of variational problems with time delays, and in problems concerning electric networks containing lossless transmission lines (as in high speed computers), see [1,2]. ...
... Proof Assume the contrary that there is a nonoscillatory solution x of (1.1). Then we can assume x ∈ S + , and so x(t), x(τ (t)), and x(g(t, s)) are positive for t ≥ t 1 ...
In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.
... A solution is said to be oscillatory if it has arbitrary large zeros, otherwise is said to be nonoscillatory that is eventually positive or eventually negative. Ladas (1982) [1], [2] established sufficient conditions under which all proper solutions of higher order linear NDE are oscillatory where ( ) ≡ 0. Bainov (1991) [3] and Gyori (1991) [4] obtained sufficient conditions for higher order NDE with constant delays. Parhi (2004) [5] and [6] obtained sufficient conditions for all solutions of (1.1) to oscillate or tend to zero as → ∞, where the delays are constants and ( ) = . ...
... A solution is said to be oscillatory if it has arbitrary large zeros, otherwise is said to be nonoscillatory that is eventually positive or eventually negative. Ladas (1982) [1], [2] established sufficient conditions under which all proper solutions of higher order linear NDE are oscillatory where ( ) ≡ 0. Bainov (1991) [3] and Gyori (1991) [4] obtained sufficient conditions for higher order NDE with constant delays. Parhi (2004) [5] and [6] obtained sufficient conditions for all solutions of (1.1) to oscillate or tend to zero as → ∞, where the delays are constants and ( ) = . ...
In this paper some necessary and sufficient conditions of n – th order neutral differential equations are obtained to insure the convergence of all nonoscillatory solutions to zero or tends to infinity as t → ∞. Some examples are given to illustrate the main results.
... For the basic background on the oscillation theory of differential equations, we refer to the monographs see [2,9,10,28,29] and the references there in. It seems that there has been no work published on the oscillation of systems of impulsive partial differential equations with continuous distributed deviating arguments. ...
... We prove that the inequality (2) has no eventually positive solution if the conditions of Theorem 2 hold. Suppose that Z(t) is an eventually positive solution of inequality (2). Then there exists a number t * t 0 such that Z(θ j 0 (t)) > 0, j = 1, 2, · · · , d for t t * . ...
In this paper, we consider systems of impulsive nonlinear neutral delay partial differential equations with distributed deviating arguments and sufficient conditions for the oscillation of the system under the Dirichlet boundary condition. The main results are illustrated by one example.
... The problem of the oscillation of solutions of differential equations has been widely studied by many authors using a wide variety of techniques ever since the pioneering work of Sturm [18] on second order linear differential equations. In the past 30 years, oscillation theory for second order neutral delay differential equations and third-order retarded delay differential equations has been well developed; see, for example, the monographs [19,20] and papers [3][4][5][6][7][8][9][10][11] as well as the references contained therein. Compared to second order neutral delay differential equations, it seems that considerably less has been done on the oscillation and asymptotic behaviour of solutions of third order neutral differential equations [4,10]. ...
... With u = h(t) and v = ρ(t), (19) becomes ...
The authors present some new oscillation criteria for third order nonlinear differential equations with a nonlinear nonpositive neutral term. The results obtained are new and improve known oscillation criteria appearing in the literature even for equations not having a neutral term. Using comparison methods and integral conditions, the authors obtain five new theorems on the oscillatory behaviour of the solutions. Suggestions for future research are included.
... Laddas (1982) [1], Jaros and Kusano (1990) [2] established sufficient conditions under which all proper solutions of higher order linear NDE are oscillatory where ( ) . Bainov (1991) [3], ...
... Laddas (1982) [1], Jaros and Kusano (1990) [2] established sufficient conditions under which all proper solutions of higher order linear NDE are oscillatory where ( ) . Bainov (1991) [3], ...
In this paper some necessary and sufficient conditions are obtained to insure the oscillation of all solutions of third order nonlinear neutral differential equations. Some examples are given to illustrate our main results
... and Theorem 3.1.3 of the monograph by Bainov and Mishev [3]. Theorem 3.1: Consider the neutral delay impulsive differential equation (2.1) and assume conditions C2.1-C2.3 ...
... By appropriate imposition of impulse controls, all solutions of a certain class of second order neutral impulsive differential equations are observed to be oscillatory. In this paper, we generalized and proved the results of oscillations of second order neutral differential equations with constant coefficients obtained by Bainov and Mishev [3] for impulsive differential equations. ...
This paper deals with the oscillations of a class of second order linear neutral impulsive ordinary differential equations with variable coefficients and constant retarded arguments. Here, we obtain sufficient conditions ensuring the oscillation of all solutions. Examples are provided to illustrate the abstract results.
... Assume that the function x is said to be a solution of (1) if the function y(t), y (t) and b(t)(y (t)) β are continuous differential function and x satisfies Eq. (1). A solution of (1) (which is non-trivial for all large t) is called oscillatory if it has a sequence of large zeros lending to ∞; otherwise we call non-oscillatory. ...
... Of late, much attention is being paid in the research activities related to oscillation and asymptotic behavior of different kinds of differential equations. We refer the readers to the books [1][2][3], the papers [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references cited therein. The applications of neutral differential equations are manifold. ...
By applying Riccati substitution techniques triply, we establish some new oscillation
and asymptotic nature of solutions to the third-order nonlinear differential equations
with mixed neutral type. We present many theorems and related examples in order to
illustrate and substantiate the main theory.
... Neutral impulsive dierential equations are part of impulsive dierential equations with deviating arguments (IDEDA). Generally speaking, IDEDA are a very interesting mixture between impulsive dierential equations (see [1] and [4]) and dierential equations with deviating argument (see [3] and [12]). We note here that [5] is the rst work where IDEDA were considered and where an oscillation theory of such equation was studied. ...
... The following theorems extend Theorem 3.1.4 and Theorem 3.1.5 of the monograph by Bainov and Mishev [3] by imposing impulsive perturbations as appropriate. ...
... Researchers leverage oscillatory differential equations to grapple with intricate systems involving multiple variables (Kusano & Naito, 1997). This field bears substantial significance for numerical analysts, facilitating the simulation of diverse phenomena across science, engineering, and social sciences (Agarwal, Grace, Li & Zhang, 2003;Bainov & Mishev, 1991;Agarwal, Bohner, Li & Zhang, 2013). Notably, it offers solutions for challenges in transportation, mass-spring systems, simple harmonic motion, and dynamic object systems, among others (Agarwal, Bohner, Li & Zhang, 2013;. ...
This research explores the practical implementation and simulation of oscillatory differential equations concerning objects in motion. The methodology incorporates power series polynomials, ensuring adherence to the fundamental properties of these functions. The novel approach is applied to various oscillatory differential equations, encompassing harmonic motion, spring motion, dynamic mass motion, Betiss and Stiefel equations, and nonlinear differential equations. The results demonstrate computational reliability, showcasing enhanced accuracy and quicker convergence compared to currently examined methods. Dynamic motion, Genesio, Harmonic motion, Mass, Spring of motion
... Hence, investigating oscillation criteria, particularly for third-order NDEs, holds paramount importance in both theoretical and practical contexts. This paper delves into obtaining oscillation criteria for third-order NDEs, aiming to establish more precise conditions governing the occurrence of oscillations in the solutions, see [11][12][13][14]. ...
This paper delves into the investigation of quasi-linear neutral differential equations in the third-order canonical case. In this study, we refine the relationship between the solution and its corresponding function, leading to improved preliminary results. These enhanced results play a crucial role in excluding the existence of positive solutions to the investigated equation. By building upon the improved preliminary results, we introduce novel criteria that shed light on the nature of these solutions. These criteria help to distinguish whether the solutions exhibit oscillatory behavior or tend toward zero. Moreover, we present oscillation criteria for all solutions. To demonstrate the relevance of our results, we present an illustrative example. This example validates the theoretical framework we have developed and offers practical insights into the behavior of solutions for quasi-linear third-order neutral differential equations.
... NDDEs have many interesting applications in various branches of science such as, physics, electrical control and engineering, physical chemistry, and mathematical biology, etc., see [4]. ...
In this paper, we study the existence and uniqueness of a periodic solution for a third-order neutral delay differential equation (NDDE) by applying Mawhin’s continuation theorem of coincidence degree and analysis techniques. An illustrative example is given as an application to support our results. To confirm the accuracy of our results, we also present a plot of the behavior of the periodic solution.
... has been studied by Kulenović and Hadžiomerspahić [2], which were extended higher-order linear neutral delay differential equation of the form Zhou and B. G. Zhang [3]. We refer the reader to the papers [4][5][6] for recent contributions concerning the distributed deviating arguments and books [7][8][9][10][11]. ...
In this work, we consider the existence of nonoscillatory solutions of variable coefficient higher order linear neutral differential equations with distributed deviating arguments. We use the Banach contraction principle to obtain new sufficient conditions, which are weaker than those known, for the existence of nonoscillatory solutions.
... With regard to a large number of applications of first-order functional differential equations in many fields of natural sciences and engineering (see, for example, [6,10,13,14] for more details), the oscillation theory of such equations has been developed extensively over the last few decades. The interest in this subject is evidenced by numerous published monographs [1,3,6,10,11,13]. Most efforts, however, were dedicated toward studying * Correspondence: jadlovska@saske.sk the existence or nonexistence of oscillatory solutions whereas only a few authors were interested in determining the location of zeros of solutions of such equations. ...
... The theory of Neutral differential equations has become an independent trend, and the literature on this subject is extensive. We shall mention the survey on the theory of neutral equations by Akhmerov et al. [2], where a classification is made and a statement of the main problems is given, as well as the books by Chukwu [28], Bainov & Mishev [10] and Hale & Lunel [50]. ...
In this thesis, we study the existence of solutions and controllability for retarded semilinear neutral differential equations with non-instantaneous impulses, non-local conditions, and infinite delay. First, we set the problem in a phase space satisfying the Hale-Kato axiomatic theory for retarded differential equations with infinite delay. Second, we assume that the nonlinear functions are locally Lipschitz, and Karakostas’s fixed point theorem is applied to obtain the existence of solutions. Additionally, under some additional conditions, the uniqueness is proved as well. Next, assuming that the nonlinear terms are globally Lipschitz, we consider a more simplified system that allows us to apply the Banach contraction theorem to prove the existence of solutions. Subsequently, we study the associated control problem. On the one hand, we investigate the approximate controllability by using the technique employed by Bashirov and Ghahramanlou, which avoids the use of fixed point theorems. On the other hand, we prove the exact controllability of the same system. To this end, we transform the controllability problem into a fixed point problem. Then, under some conditions on the nonlinear terms, we use Rothe’s fixed point theorem to obtain the desired result.
... Basic definitions and results on oscillation for neutral type differential equations can be found in [5,15]. Due to the wide applicability of neutral differential equations in various fields of science and engineering, there is a great interest in obtaining new oscillation criteria for a different types of differential equations (see [2-4, 6, 12, 13, 17, 22, 25, 26, 29] for instance). ...
In this paper, we consider a class of third-order neutral impulsive differential equations. An equivalent class of neutral differential equations is obtained by using a suitable substitution. Some new oscillation results are proved. Moreover, we discuss the asymptotic behavior of the solution. The results in the abstract are illustrated via examples.
... Many authors have studied various types of models based on noninteger order derivatives [29,32,36]. In recent years, many researchers have shown keen interest in the study of oscillating and nonoscillating behavior of solutions [3,21,32,38,41]. In papers [6,25,38], authors studied the oscillatory nature of different classes of fractional differential equations without impulses. ...
The present article is concerned with the oscillatory nature of the fractional differential equation of order \alpha \in (2, 3) with impulsive effects. By employing a generalized Riccati transformation, we derive several oscillation criteria of Philos type, which are either new or improve several recent results in the literature. Also, we show the stability of the considered problem. To obtain the results, we transform the fractional differential equation into a second-order ordinary differential equation. In addition, we provide examples to show the effectiveness of the results.
... In papers [2,3,10,16,18,26], authors studied the oscillatory behavior of different classes of fractional differential equations without impulses. Oscillation criteria for different orders of neutral differential equations have been discussed in [1]. ...
The present article deals with the oscillatory nature of nonlinear impulsive fractional
differential equations of order \alpha; where \alpha \in (2; 3): Here, some oscillation results are
established using sufficient parts of the differential inequalities established via differential inequality
methods. Also, abstract results are illustrated by an example
... Thus, we can see that investigating the oscillatory and asymptotic behavior of solutions of neutral differential equations is of great importance. During the past period, many papers appeared on the oscillatory behavior of differential equations of neutral and delay type, see [3][4][5][6][7][8][9][10][11][12][13][14][15][16], and the references mentioned therein. ...
In this paper, we aim to explore the oscillation of solutions for a class of second-order neutral functional differential equations. We propose new criteria to ensure that all obtained solutions are oscillatory. The obtained results can be used to develop and provide theoretical support for and further develop the oscillation study for a class of second-order neutral differential equations. Finally, an illustrated example is given to demonstrate the effectiveness of our new criteria.
... The importance of neutral differential equations is that they contribute to many applicationsin physics, engineering, chemistry and medicine, so third-order equations appear in computer engineering, networks, aeromechanical systems and others; see [1][2][3]. ...
In this paper, we analyze the asymptotic behavior of solutions to a class of third-order neutral differential equations. Using different methods, we obtain some new results concerning the oscillation of this type of equation. Our new results complement related contributions to the subject. The symmetry plays a important and fundamental role in the study of oscillation of solutions to these equations. An example is presented in order to clarify the main results.
... In addition, second order neutral equations appear in the theory of automatic control and in aeromechanical systems in which inertia plays an important role. Moreover, second order delay equations play an important role in the study of vibrating masses attached to an elastic bar, as the Euler equation, see: [1][2][3]. ...
In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results.
... The problem of the oscillation of solutions of differential equations has been widely studied by many authors and by many techniques since the pioneering work of Sturm on second-order linear differential equations. In the past 30 years, the oscillation theory for second-order neutral delay differential equations and third-order retarded delay differential equations have been well developed; see, for example, the monographs [1,4] and papers [3, 10, 11, 13−18] as well as the references cites therein. Compared to second-order neutral delay differential equations, it seems that not much work has been done concerning with the oscillation and asymptotic of third-order neutral QUALITATIVE BEHAVIOUR 5 differential equations [10,17]. ...
This paper deals with the oscillatory and asymptotic behavior of third-order nonlinear differential equations with nonlinear neutral terms. We present new oscillation criteria, which improve, extend and simplify existing ones in the literature. Two examples are provided in order to illustrate the significance of our main results.
... They have wide applications in engineering, see [12], in ecology, see [13], in physics, see [14], in electrical power systems, see [15], and applied mathematics, see [16]. This type of NDDEs also appear in the study of vibrating masses attached to an elastic bar, in problems concerning electric networks containing lossless transmission lines (as in high speed computers), and in the solution of variational problems with time delays, see [17,18]. In this article, we consider the following class of non-linear NDDEs of third-order: ...
In this article, we study a class of non-linear neutral delay differential equations of third order. We first prove criteria for non-existence of non-Kneser solutions, and criteria for non-existence of Kneser solutions. We then use these results to provide criteria for the under study differential equations to ensure that all its solutions are oscillatory. An example is given that illustrates our theory.
... In recent decades, an increasing interest in establishing sufficient criteria for oscillatory and non-oscillatory properties of different classes of differential equations has been observed; see, for instance, the monographs [1][2][3] and the references cited therein. Many authors were concerned with the oscillation and nonoscillation of delay differential equations of second-order [4][5][6][7][8][9][10][11][12][13][14][15][16][17] and higher-order [18][19][20][21][22][23]. ...
The motivation for this paper is to create new criteria for oscillation of solutions of second-order nonlinear neutral differential equations. In more than one respect, our results improve several related ones in the literature. As proof of the effectiveness of the new criteria, we offer more than one practical example.
... In aeromechanical systems, where they have a significant role, in the theory of automatic control, in study of vibrating masses attached to an elastic bar (as the Euler equation), in the networks that have lossless transmission lines (as is the case in high-speed computers), and other applications, delay or neutral differential equations can be seen in the modeling of the mentioned phenomena, see [1,2,5,15]. As a result of these applications, research groups including us still study the differential equations with delay. ...
The purpose of this work is to study the oscillation criteria for generalized Emden–Fowler neutral differential equation. We establish new oscillation criteria using both the technique of comparison with first order delay equations and the technique of Riccati transformation. Our new criteria are interesting as they improve, simplify, and complement some results that have been published recently in the literature. Moreover, we present an illustrating example.
... where y is positive and z is positive and increasing. In addition, based on a result given in ( [13], page 28), if y and z are positive and z is decreasing, they then assume that y is also nonincreasing. This leads to the following relation between y and z: ...
The main purpose of this paper is to obtain criteria for the oscillation of all solutions of a third-order half-linear neutral differential equation. The main result in this paper is an oscillation theorem obtained by comparing the equation under investigation to two first order linear delay differential equations. An additional result is obtained by using a Riccati transformation technique. Examples are provided to show the importance of the main results.
... In addition, second order neutral equations appear in the theory of automatic control and in aeromechanical systems, in which inertia plays an important role. Moreover, second order delay equations play an important role in studying vibrating masses attached to an elastic bar, as the Euler equation, see [1,2,7]. One area of active research in recent times is to study the sufficient criterion for oscillation of delay differential equations, see . ...
Abstract The aim of this work is to offer sufficient conditions for the oscillation of neutral differential equation second order (r(t)[(y(t)+p(t)y(τ(t)))′]γ)′+f(t,y(σ(t)))=0, $$ \bigl( r ( t ) \bigl[ \bigl( y ( t ) +p ( t ) y \bigl( \tau ( t ) \bigr) \bigr) ^{\prime } \bigr] ^{\gamma } \bigr) ^{\prime }+f \bigl( t,y \bigl( \sigma ( t ) \bigr) \bigr) =0, $$ where ∫∞r−1/γ(s)ds=∞ $\int ^{\infty }r^{-1/\gamma } ( s ) \,\mathrm{d}s= \infty $. Based on the comparison with first order delay equations and by employ the Riccati substitution technique, we improve and complement a number of well-known results. Some examples are provided to show the importance of these results.
... Note that the operators T and S in Example 4.10 are related to certain neutral delay differential expressions, see e.g. [4,Chapt. 3]. More precisely, if we define ...
We introduce the concept of essential numerical range $W_{\!e}(T)$ for unbounded Hilbert space operators $T$ and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the properties known for the bounded case do \emph{not} carry over to the unbounded case, and new interesting phenomena arise which we illustrate by some striking examples. A key feature of the essential numerical range $W_{\!e}(T)$ is that it captures spectral pollution in a unified and minimal way when approximating $T$ by projection methods or domain truncation methods for PDEs.
... In [4], Gopasamy, Lalli, and Zhang considered the linear equation (2), we refer the reader to the monographs by Bainow and Mishev [2], Erbe, Kong, and Zhang [3], and Gyǒri and Ladas [8] as well as the papers of Agarwal and Saker [1], Pahri [15], Saker and Elabbasy [17], Tanaka [18], and Zhou [21] And the references contained therein. ...
In this paper, the author established some new integral conditions for the oscillation of all solutions of nonlinear first order neutral delay differential equations. Examples are inserted to illustrate the results.
... Laddas (1982) [1], Jaros and Kusano (1990) [2] established sufficient conditions under which all proper solutions of higher order linear NDE are oscillatory where ( ) . Bainov (1991) [3], ...
... The origins of the oscillation theory of these equations are traced back to Norkin's pioneering paper [17] in 1977, although the first result establishing an efficient oscillatory criterion, essentially different from the classical criteria for non-neutral equations, was published by Zahariev and Bainov [24] in 1980. The subsequent rapid development of the theory has been laid out in detail in the monograph by Bainov and Mishev [5] in 1991. ...
It is shown on a series of counterexamples that the assumption
traditionally used 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.
... Ordinary differential delay equations are well understood, we refer exemplarily to the expositions [7,15,13]. For parabolic partial differential equations, we only mention [5] and the references cited therein, since this book investigates oscillation effects for nonlinear partial differential equations with delay that we observe also for (1.1). Our parabolic delay equation is nonlinear and contains a nonlocal Pyragas type feedback term defined by a measure. ...
We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a non-differentiable term with the measure-norm of the control. Existence, uniqueness and regularity of the solution of the state equation, as well as differentiability properties of the control-to-state operator are obtained. Next, we provide first order optimality conditions for local solutions. Finally, the control space is suitably discretized and we prove convergence of the solutions of the discrete problems to the solutions of the original problem. Several numerical examples are included to illustrate the theoretical results.
... The origins and applications of such equations occur in a variety of different fields, such as fluid dynamics, heat conduction and diffusion, to describe the motion of waves in physics, modeling chemical reactions in chemistry, the population growth of species. We refer the monographs in the literature [12,13,14,9,15,16]. The qualitative theory of partial differential equations has attracted a great deal of attention over the last few decades. ...
In this article, we investigate the oscillatory behavior of nonlinear
partial differential equations (1) with the boundary condition (2). By using
integral averaging method, we will obtain some new oscillation criteria for
given system. The main results are illustrated through suitable example.
In this research, some properties of oscillation and nonoscillation of
a system of second order half linear delay differential equations
were studied, as some conditions were obtained that ensure the
oscillation of all solutions of the system, or the convergence of
nonoscillating ones to zero. Some illustrative examples of the
obtained results are given.
This paper deals with some new criteria for the oscillation of third order half-linear differential equations. The purpose of the present paper is the linearization of equation 1.1 in the sense that we would deduce oscillation of studied equation from that of the linear form and to provide new oscillation criteria via comparison with first order equations whose oscillatory behavior are known. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature for equation 1.1. The results are illustrated by some examples.
This paper deals with the asymptotic and oscillatory behaviour of third-order non-linear differential equations with mixed non-linear neutral terms and a canonical operator. The results are obtained via utilising integral conditions as well as comparison theorems with the oscillatory properties of first-order advanced and/or delay differential equations. The proposed theorems improve, extend, and simplify existing ones in the literature. The results are illustrated by two numerical examples.
In this paper, the authors obtained some new sufficient conditions for the oscillation of all solutions of the fourth order nonlinear difference equation of the form ( ) ( ) 0 1 3 n n n n n n a x p x q f x n = 0,1,2, … ., where an, pn, qn positive sequences. The established results extend, unify and improve some of the results reported in the literature. Examples are provided to illustrate the main result.
In this paper, using the classical Schauder fixed point theorem, we prove that a nonlinear class of integral equations has at least one solution in the space of functions satisfying the Hölder condition. Also, some important examples showing the applicability of the existence theorem presented in this study are given.
The properties of oscillation and non-oscillation of solutions in neutral differential equations (NDEs) paid attention many researchers. This attention produced many publications in this context. The majority of these studies were associated with (NDEs) with constant delays. A fewer number that discussed sufficient conditions signified that solutions of impulsive neutral differential equations (INDEs) oscillate.
In this thesis, some sufficient conditions are obtained to ensure the oscillatory property for all solutions of impulsive neutral differential equations with positive and negative coefficients (INDEWPNCs) and variable delays of first and second order. Additionally, sufficient conditions are obtained for the asymptotic behavior of all non-oscillatory solutions of (INDEWPNCs).
In other words, sufficient conditions are obtained to ensure the convergence to zero for all non-oscillatory solutions. For this purpose, some lemmas are proved to get the results. Furthermore, these results are obtained by introducing modified transformations to deal with oscillation property and asymptotic behavior of this type of equations. Also, some known lemmas in the literatures are improved and generalized.
During this work, different types of impulsive conditions for (INDEWPNCs) are presented to get sufficient conditions for oscillation and asymptotic behavior of all solutions. Illustrative examples are given to clarify the applicability of the obtained results.
In this paper necessary and sufficient conditions were obtained to insure that every solution of neutral integro-differential equations oscillates these results improve and generalized Lemma 2.1, Theorem 2.2, Theorem 2.3 in Olach(2005).
In this paper oscillation criterion is investigated for all solutions of the third-order non linear neutral differential equations with positive and negative coefficients: [í µí±¥(í µí±¡) + í µí±(í µí±¡)í µí±(í µí±¥(í µí¼(í µí±¡)))]′′′ + í µí±(í µí±¡)í µí± (í µí±¥(í µí¼(í µí±¡))) − í µí±(í µí±¡)í µí± (í µí±¥(í µí»¼(í µí±¡))) = 0, í µí±¡ ≥ í µí±¡ 0 (1.1) Some sufficient conditions are established so that every solution of eq.(1.1) oscillate. We improved theorem 2.4 and theorem 2.10 in [5]. Examples are given to illustrated our main results.
In this paper, the problem of asymptotic behavior of solutions for impulsive neutral partial differential equations has been investigated. Using Riccati transform method and impulsive differential inequalities, some new sufficient conditions are derived for a solution of the proposed equation which converges to zero. Finally, the effectiveness of the derived main results has been shown in numerical section.
Based on the properties of Riemann-Liouville difference and sum operators, sufficient conditions are established to guarantee the oscillation of solutions for forced and damped nabla fractional difference equations. Numerical examples are presented to show the applicability of the proposed results. We finish the paper by a concluding remark.
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