In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational c...In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.展开更多
In this paper,the oscillation criteria for the solutions of the nonlinear differential equations of neutral type of the forms:[x(t)+p(t)x(σ(t))]″+q(t)f(x(τ(t)))g(x′(t))=0and[x(t)+p(t)x(σ(t))]″+q(t)f(x(t),x(τ(t)...In this paper,the oscillation criteria for the solutions of the nonlinear differential equations of neutral type of the forms:[x(t)+p(t)x(σ(t))]″+q(t)f(x(τ(t)))g(x′(t))=0and[x(t)+p(t)x(σ(t))]″+q(t)f(x(t),x(τ(t)))g(x′(t))=0are obtained.展开更多
This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained b...This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam’s novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.展开更多
In this paper,some sufficient conditions are obtained for the oscillation for solutions of systems of highd order partial differential equations of neutral type.
This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several differen...This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.展开更多
The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equa...The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.展开更多
Delay differential equations (DDEs), as well as neutral delay differential equations (NDDEs), are often used as a fundamental tool to model problems arising from various areas of sciences and engineering. However, NDD...Delay differential equations (DDEs), as well as neutral delay differential equations (NDDEs), are often used as a fundamental tool to model problems arising from various areas of sciences and engineering. However, NDDEs particularly the systems of these equations are special transcendental in nature;it has therefore, become a challenging task or times almost impossible to obtain a convergent approximate analytical solution of such equation. Therefore, this study introduced an analytical method to obtain solution of linear and nonlinear systems of NDDEs. The proposed technique is a combination of Homotopy analysis method (HAM) and natural transform method, and the He’s polynomial is modified to compute the series of nonlinear terms. The presented technique gives solution in a series form which converges to the exact solution or approximate solution. The convergence analysis and the maximum estimated error of the approach are also given. Some illustrative examples are given, and comparison for the accuracy of the results obtained is made with the existing ones as well as the exact solutions. The results reveal the reliability and efficiency of the method in solving systems of NDDEs and can also be used in various types of linear and nonlinear problems.展开更多
This paper discusses a linear neutral stochastic differential equation with variable delays. By using fixed point theory, the necessary and sufficient conditions are given to ensure that the trivial solution to such a...This paper discusses a linear neutral stochastic differential equation with variable delays. By using fixed point theory, the necessary and sufficient conditions are given to ensure that the trivial solution to such an equation is pth moment asymptotically stable. These conditions do not require the boundedness of delays, nor derivation of delays. An example was also given for illustration.展开更多
Oscillation criteria are established for third-order neutral delay differential equations with deviating arguments. These criteria extend and generalize those results in the literature. Moreover, some illustrating exa...Oscillation criteria are established for third-order neutral delay differential equations with deviating arguments. These criteria extend and generalize those results in the literature. Moreover, some illustrating examples are also provided to show the importance of our results.展开更多
The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equa...The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.展开更多
By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation wit...By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.展开更多
This paper gives the rules of oscillations of two classes of neutraldifferential equations with forced terms, and some oscillation criteria undercertain conditions are presented according to the equations having diffe...This paper gives the rules of oscillations of two classes of neutraldifferential equations with forced terms, and some oscillation criteria undercertain conditions are presented according to the equations having differentcharacters.展开更多
we consider the third-order neutral functional differential equations with deviating arguments. A new theorem is presented that improves a number of results reported in the literature. Examples are included to illustr...we consider the third-order neutral functional differential equations with deviating arguments. A new theorem is presented that improves a number of results reported in the literature. Examples are included to illustrate new results.展开更多
Abstract In this paper, the higher order neutral differential equation with continuous distributed delay is concerned and the oscillatory criteria are given.
The objective of this paper is to develop monotone techniques for obtaining extremal solutions of initial value problem for nonlinear neutral delay differential equations.
This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration(PDA)feedback,which can be used to understand human balance in quiet standing.The closed-loop system is ...This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration(PDA)feedback,which can be used to understand human balance in quiet standing.The closed-loop system is described by a neutral delay differential equation(NDDE).The optimal feedback gains(OFGs)that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4.Such a property is called multiplicity-induced dominancy of time-delay systems,and has been discussed intensively by many authors for retarded delay differential equations(RDDEs).This paper shows that multiplicity-induced dominancy can be achieved in NDDEs.In addition,the OFGs are delay-dependent,and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays.Thus,the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains.The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.展开更多
Neutral Delay Differential Equation(NDDE)is a differential problem that has regularly existed in numerous occurrences and has represented a significant role in dealing with real-life phenomena,especially on their appl...Neutral Delay Differential Equation(NDDE)is a differential problem that has regularly existed in numerous occurrences and has represented a significant role in dealing with real-life phenomena,especially on their application in biological and physiological processes.A fifth-order two-point hybrid implicit multistep block method(2PIH5)has been formulated in this research for the numerical solution of Neutral Delay Differential Equation(NDDE).A Taylor series interpolation polynomial has been implemented in the formulation of the proposed 2PIH5.The order,consistency,and zero-stability for 2PIH5 have been illustrated.The analyses of convergence and stability test have been performed and discussed.The initial value problems for the first-order NDDE with constant or proportional delay have been solved using the proposed block method.Some numerical results for the proposed method have been presented to prove the adaptability and applicability of the proposed method in solving NDDE.The proposed method is proved to be comparable with the other existing methods.It is assumed to be reliable and efficient for solving the first-order NDDE with constant or proportional delay.展开更多
In this paper, we study two types of neutral functional differential equations with finite or unbounded distributed deviating arguments. By Banach contraction princi-ple, we obtain some sufficient conditions for the e...In this paper, we study two types of neutral functional differential equations with finite or unbounded distributed deviating arguments. By Banach contraction princi-ple, we obtain some sufficient conditions for the existence of positive solutions to such equations.展开更多
The aim of this paper is to the discussion of the exponential stability of a class of impulsive neutral stochastic functional differential equations with Markovian switching.Under the influence of impulsive disturbanc...The aim of this paper is to the discussion of the exponential stability of a class of impulsive neutral stochastic functional differential equations with Markovian switching.Under the influence of impulsive disturbance,the solution for the system is discontinuous.By using the Razumikhin technique and stochastic analysis approaches,as well as combining the idea of mathematical induction and classification discussion,some sufficient conditions for the pth moment exponential stability and almost exponential stability of the systems are obtained.The stability conclusion is full time-delay.The results show that impulse,the point distance of impulse and Markovain switching affect the stability for the system.Finally,two examples are provided to illustrate the effectiveness of the results proposed.展开更多
文摘In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.
文摘In this paper,the oscillation criteria for the solutions of the nonlinear differential equations of neutral type of the forms:[x(t)+p(t)x(σ(t))]″+q(t)f(x(τ(t)))g(x′(t))=0and[x(t)+p(t)x(σ(t))]″+q(t)f(x(t),x(τ(t)))g(x′(t))=0are obtained.
基金supported by Ministry of Human Resource and Development(MHR-02-23-200-429/304)
文摘This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam’s novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.
文摘In this paper,some sufficient conditions are obtained for the oscillation for solutions of systems of highd order partial differential equations of neutral type.
基金Supported by NSFC (11001091)Chinese UniversityResearch Foundation (2010MS129)
文摘This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.
基金Sponsored by HUST Foundation(0125011017)the National NSFC under grant(70671047)
文摘The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.
文摘Delay differential equations (DDEs), as well as neutral delay differential equations (NDDEs), are often used as a fundamental tool to model problems arising from various areas of sciences and engineering. However, NDDEs particularly the systems of these equations are special transcendental in nature;it has therefore, become a challenging task or times almost impossible to obtain a convergent approximate analytical solution of such equation. Therefore, this study introduced an analytical method to obtain solution of linear and nonlinear systems of NDDEs. The proposed technique is a combination of Homotopy analysis method (HAM) and natural transform method, and the He’s polynomial is modified to compute the series of nonlinear terms. The presented technique gives solution in a series form which converges to the exact solution or approximate solution. The convergence analysis and the maximum estimated error of the approach are also given. Some illustrative examples are given, and comparison for the accuracy of the results obtained is made with the existing ones as well as the exact solutions. The results reveal the reliability and efficiency of the method in solving systems of NDDEs and can also be used in various types of linear and nonlinear problems.
基金The National Natural Science Foundation of China (No.10671078)
文摘This paper discusses a linear neutral stochastic differential equation with variable delays. By using fixed point theory, the necessary and sufficient conditions are given to ensure that the trivial solution to such an equation is pth moment asymptotically stable. These conditions do not require the boundedness of delays, nor derivation of delays. An example was also given for illustration.
文摘Oscillation criteria are established for third-order neutral delay differential equations with deviating arguments. These criteria extend and generalize those results in the literature. Moreover, some illustrating examples are also provided to show the importance of our results.
基金Sponsored by HUST Foundation(0125011017) the National NSFC under grant(70671047)
文摘The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.
文摘By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.
文摘This paper gives the rules of oscillations of two classes of neutraldifferential equations with forced terms, and some oscillation criteria undercertain conditions are presented according to the equations having differentcharacters.
文摘we consider the third-order neutral functional differential equations with deviating arguments. A new theorem is presented that improves a number of results reported in the literature. Examples are included to illustrate new results.
文摘Abstract In this paper, the higher order neutral differential equation with continuous distributed delay is concerned and the oscillatory criteria are given.
文摘The objective of this paper is to develop monotone techniques for obtaining extremal solutions of initial value problem for nonlinear neutral delay differential equations.
基金supported by the National Natural Science Foundation of China(No.12072370)。
文摘This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration(PDA)feedback,which can be used to understand human balance in quiet standing.The closed-loop system is described by a neutral delay differential equation(NDDE).The optimal feedback gains(OFGs)that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4.Such a property is called multiplicity-induced dominancy of time-delay systems,and has been discussed intensively by many authors for retarded delay differential equations(RDDEs).This paper shows that multiplicity-induced dominancy can be achieved in NDDEs.In addition,the OFGs are delay-dependent,and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays.Thus,the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains.The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.
基金All authors gratefully acknowledge for the financial support by Putra Grant(project code:GP-IPS/2018/9625400)Graduate Research Fellowship(GRF)from Universiti Putra Malaysia.The authors are also thankful to the referees for their useful comments and suggestions.
文摘Neutral Delay Differential Equation(NDDE)is a differential problem that has regularly existed in numerous occurrences and has represented a significant role in dealing with real-life phenomena,especially on their application in biological and physiological processes.A fifth-order two-point hybrid implicit multistep block method(2PIH5)has been formulated in this research for the numerical solution of Neutral Delay Differential Equation(NDDE).A Taylor series interpolation polynomial has been implemented in the formulation of the proposed 2PIH5.The order,consistency,and zero-stability for 2PIH5 have been illustrated.The analyses of convergence and stability test have been performed and discussed.The initial value problems for the first-order NDDE with constant or proportional delay have been solved using the proposed block method.Some numerical results for the proposed method have been presented to prove the adaptability and applicability of the proposed method in solving NDDE.The proposed method is proved to be comparable with the other existing methods.It is assumed to be reliable and efficient for solving the first-order NDDE with constant or proportional delay.
基金sponsored by the National Natural Science Foundation of China (11071001)the NSF of Anhui Province (1208085MA13)+3 种基金the NSF of Education Bureau of Anhui Province(KJ2009A005Z KJ2010ZD02 2010SQRL159)Innovative Research Team Program of Anhui University, College doctoral special research foundation (20093401110001)
文摘In this paper, we study two types of neutral functional differential equations with finite or unbounded distributed deviating arguments. By Banach contraction princi-ple, we obtain some sufficient conditions for the existence of positive solutions to such equations.
基金This research was supported by the National Nature Science Foundation of China under Grant No.11571245Young Crop Project of Sichuan University under Grant No.2020SCUNL111.
文摘The aim of this paper is to the discussion of the exponential stability of a class of impulsive neutral stochastic functional differential equations with Markovian switching.Under the influence of impulsive disturbance,the solution for the system is discontinuous.By using the Razumikhin technique and stochastic analysis approaches,as well as combining the idea of mathematical induction and classification discussion,some sufficient conditions for the pth moment exponential stability and almost exponential stability of the systems are obtained.The stability conclusion is full time-delay.The results show that impulse,the point distance of impulse and Markovain switching affect the stability for the system.Finally,two examples are provided to illustrate the effectiveness of the results proposed.