A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive...A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.展开更多
The finite-time stability and the finite-time contractive stability of solutions for nonlinear fractional differential equations with bounded delay are investigated. The derivative of Lyapunov function along solutions...The finite-time stability and the finite-time contractive stability of solutions for nonlinear fractional differential equations with bounded delay are investigated. The derivative of Lyapunov function along solutions of the considered system is defined in terms of the Caputo fractional Dini derivative. Based on the Lyapunov-Razumikhin method, several sufficient criteria are established to guarantee the finite-time stability and the finite-time contractive stability of solutions for the related systems. An example is provided to illustrate the effectiveness of the obtained results.展开更多
This paper has given necessary and sufficient conditions for oscillation of a classof higher order nonlinear delay differential equations, and given some sufficientconditions or necessary conditions for oscillation of...This paper has given necessary and sufficient conditions for oscillation of a classof higher order nonlinear delay differential equations, and given some sufficientconditions or necessary conditions for oscillation of a forced delay differentialequation.展开更多
In this paper, we construct a composite Milstein method for nonlinear stochastic differential delay equations. Then we analyze the mean square stability for this method and obtain the step size condition under which t...In this paper, we construct a composite Milstein method for nonlinear stochastic differential delay equations. Then we analyze the mean square stability for this method and obtain the step size condition under which the composite Milstein method is mean square stable. Moreover, we get the step size condition under which the composite Milstein method is global mean square stable. A nonlinear test stochastic differential delay equation is given for numerical tests. The results of numerical tests verify the theoretical results proposed.展开更多
The aim of this paper is to apply Adomian decomposition method (ADM) for solving some classes of nonlinear delay differential equations (NDDEs) with accelerated Adomian polynomial called El-kalla polynomial proposed b...The aim of this paper is to apply Adomian decomposition method (ADM) for solving some classes of nonlinear delay differential equations (NDDEs) with accelerated Adomian polynomial called El-kalla polynomial proposed by El-kalla [1]. The main advantages of El-kalla polynomials can be summarized in the following main three points: 1) El-kalla polynomials are recursive and do not have derivative terms so, El-kalla formula is easy in programming and save much time on the same processor compared with the traditional Adomian polynomials formula;2) Solution using El-Kalla polynomials converges faster than the traditional Adomian polynomials;3) El-Kalla polynomials used directly in estimating the maximum absolute truncated error of the series solution. Some convergence remarks are studied and some numerical examples are solved using the Adomian decomposition method using the two polynomials (Adomian polynomial and El-kalla polynomial). In all applied cases, we obtained an excellent performance that may lead to a promising approach for many applications.展开更多
The nonlinear vector differential equation of the sixth order with constant delay is considered in this article. New criteria for instability of the zero solution are established using the Lyapunov-Krasovskii function...The nonlinear vector differential equation of the sixth order with constant delay is considered in this article. New criteria for instability of the zero solution are established using the Lyapunov-Krasovskii functional approach and the differential inequality techniques. The result of this article improves previously known results.展开更多
In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the loca...In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.展开更多
In this paper, oscillatory properties for solutions of certain nonlinear impulsive parabolic equations with several delays are investigated and a series of new sufficient conditions for oscillations of the equation ar...In this paper, oscillatory properties for solutions of certain nonlinear impulsive parabolic equations with several delays are investigated and a series of new sufficient conditions for oscillations of the equation are established.展开更多
In this paper, we give some new criteria for the asymptotic behavior and oscillation of third-order delay differential equation. The oscillation of the studied equation is studied under two conditions, and our results...In this paper, we give some new criteria for the asymptotic behavior and oscillation of third-order delay differential equation. The oscillation of the studied equation is studied under two conditions, and our results improve some ones in D?urina et al.(2018). Some examples are given to illustrate the main results with Euler-type differential equations.展开更多
The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solut...The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solutions, the existence of the periodic solutions, the existence and uniqueness of the stationary oscillation are obtained by means of the Liapunov's second method, The conclusion in the literatures are generalized.展开更多
This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measur...This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing setvalued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.展开更多
In this paper, we present and analyze a single interval Legendre-Gaussspectral collocation method for solving the second order nonlinear delay differentialequations with variable delays. We also propose a novel algori...In this paper, we present and analyze a single interval Legendre-Gaussspectral collocation method for solving the second order nonlinear delay differentialequations with variable delays. We also propose a novel algorithm for the singleinterval scheme and apply it to the multiple interval scheme for more efficient implementation. Numerical examples are provided to illustrate the high accuracy ofthe proposed methods.展开更多
In this paper,we investigate the numerical performance of a family of P-stable two-step Maruyama schemes in mean-square sense for stochastic differential equations with time delay proposed in[8,10]for a certain class ...In this paper,we investigate the numerical performance of a family of P-stable two-step Maruyama schemes in mean-square sense for stochastic differential equations with time delay proposed in[8,10]for a certain class of nonlinear stochastic delay differential equations with multiplicative white noises.We also test the convergence of one of the schemes for a time-delayed Burgers’equation with an additive white noise.Numerical results show that this family of two-step Maruyama methods exhibit similar stability for nonlinear equations as that for linear equations.展开更多
Presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDE). Discussion on the numerical analogous results of the natural Runge-Kutta (NRK) methods for...Presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDE). Discussion on the numerical analogous results of the natural Runge-Kutta (NRK) methods for the same class of nonlinear NDDE; Review of the related concepts and results on RK methods; Information on the asymptotic stability and global stability of the induced NRK method.展开更多
By the averaging technique,we establish some oscillation theorems for a class of second order delay differential equations with nonlinear neutral term,the results obtained extend some known results in the previous lit...By the averaging technique,we establish some oscillation theorems for a class of second order delay differential equations with nonlinear neutral term,the results obtained extend some known results in the previous literatures.展开更多
In this paper, we investigate stochastic asymptotic stability of the zero solution for certain third-order nonlinear stochastic delay differential equations by constructing Lyapunov functionals.
Sufficient conditions are established for the asymptotic behavior of solutions of nonlinear delay differential equationswhere equipped with the sup norm for some r > 0. A new result is established, some known resul...Sufficient conditions are established for the asymptotic behavior of solutions of nonlinear delay differential equationswhere equipped with the sup norm for some r > 0. A new result is established, some known results are improved.展开更多
文摘A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.
基金Natural Science Foundation of Shanghai,China (No.19ZR1400500)。
文摘The finite-time stability and the finite-time contractive stability of solutions for nonlinear fractional differential equations with bounded delay are investigated. The derivative of Lyapunov function along solutions of the considered system is defined in terms of the Caputo fractional Dini derivative. Based on the Lyapunov-Razumikhin method, several sufficient criteria are established to guarantee the finite-time stability and the finite-time contractive stability of solutions for the related systems. An example is provided to illustrate the effectiveness of the obtained results.
文摘This paper has given necessary and sufficient conditions for oscillation of a classof higher order nonlinear delay differential equations, and given some sufficientconditions or necessary conditions for oscillation of a forced delay differentialequation.
基金Supported by National Natural Science Foundation of China(No.61272024)Anhui Provincial Natural Science Foundation(No.11040606M06)
文摘In this paper, we construct a composite Milstein method for nonlinear stochastic differential delay equations. Then we analyze the mean square stability for this method and obtain the step size condition under which the composite Milstein method is mean square stable. Moreover, we get the step size condition under which the composite Milstein method is global mean square stable. A nonlinear test stochastic differential delay equation is given for numerical tests. The results of numerical tests verify the theoretical results proposed.
基金supported by the National Natural Science Foundation of China(61370136)the Hainan Province Science and Technology Cooperation Fund Project(KJHZ2015-36)the Hainan Province Introduced and Integrated Demonstration Projects(YJJC20130009)
文摘The aim of this paper is to apply Adomian decomposition method (ADM) for solving some classes of nonlinear delay differential equations (NDDEs) with accelerated Adomian polynomial called El-kalla polynomial proposed by El-kalla [1]. The main advantages of El-kalla polynomials can be summarized in the following main three points: 1) El-kalla polynomials are recursive and do not have derivative terms so, El-kalla formula is easy in programming and save much time on the same processor compared with the traditional Adomian polynomials formula;2) Solution using El-Kalla polynomials converges faster than the traditional Adomian polynomials;3) El-Kalla polynomials used directly in estimating the maximum absolute truncated error of the series solution. Some convergence remarks are studied and some numerical examples are solved using the Adomian decomposition method using the two polynomials (Adomian polynomial and El-kalla polynomial). In all applied cases, we obtained an excellent performance that may lead to a promising approach for many applications.
文摘The nonlinear vector differential equation of the sixth order with constant delay is considered in this article. New criteria for instability of the zero solution are established using the Lyapunov-Krasovskii functional approach and the differential inequality techniques. The result of this article improves previously known results.
基金supported by National Natural Science Foundation of China (Grant No. 11571128)
文摘In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.
基金This work is supported by National Natural Science Foundation of China (40373003 and 40372121).
文摘In this paper, oscillatory properties for solutions of certain nonlinear impulsive parabolic equations with several delays are investigated and a series of new sufficient conditions for oscillations of the equation are established.
基金This work was supported by Youth Program of National Natural Science Foundation of China under Grant 61304008Youth Program of Natural Science Foundation of Shandong Province under Grant ZR2013FQ033.
文摘In this paper, we give some new criteria for the asymptotic behavior and oscillation of third-order delay differential equation. The oscillation of the studied equation is studied under two conditions, and our results improve some ones in D?urina et al.(2018). Some examples are given to illustrate the main results with Euler-type differential equations.
文摘The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solutions, the existence of the periodic solutions, the existence and uniqueness of the stationary oscillation are obtained by means of the Liapunov's second method, The conclusion in the literatures are generalized.
基金supported by the National Natural Science Foundation of China(11471230,11671282)。
文摘This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing setvalued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.
基金The first author is supported in part by the National Science Foundation of China(Nos.11226330 and 11301343)the Research Fund for the Doctoral Program of Higher Education of China(No.20113127120002)+5 种基金the Research Fund for Young Teachers Program in Shanghai(No.shsf018)and the Fund for E-institute of Shanghai Universities(No.E03004)The second author is supported in part by the National Science Foundation of China(No.11171225)the Research Fund for the Doctoral Program of Higher Education of China(No.20133127110006)the Innovation Program of Shanghai Municipal Education Commission(No.12ZZ131)the Fund for E-institute of Shanghai Universities(No.E03004).
文摘In this paper, we present and analyze a single interval Legendre-Gaussspectral collocation method for solving the second order nonlinear delay differentialequations with variable delays. We also propose a novel algorithm for the singleinterval scheme and apply it to the multiple interval scheme for more efficient implementation. Numerical examples are provided to illustrate the high accuracy ofthe proposed methods.
基金This work was supported by the NSF of China(No.10901036)and AIRFORCE MURI.The authors thank the referees for their helpful suggestions for improving the paper.The first author also would like to thank Professor George Em Karniadakis for his hospitality when she was visiting Division of Applied Mathematics at Brown University.
文摘In this paper,we investigate the numerical performance of a family of P-stable two-step Maruyama schemes in mean-square sense for stochastic differential equations with time delay proposed in[8,10]for a certain class of nonlinear stochastic delay differential equations with multiplicative white noises.We also test the convergence of one of the schemes for a time-delayed Burgers’equation with an additive white noise.Numerical results show that this family of two-step Maruyama methods exhibit similar stability for nonlinear equations as that for linear equations.
文摘Presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDE). Discussion on the numerical analogous results of the natural Runge-Kutta (NRK) methods for the same class of nonlinear NDDE; Review of the related concepts and results on RK methods; Information on the asymptotic stability and global stability of the induced NRK method.
文摘By the averaging technique,we establish some oscillation theorems for a class of second order delay differential equations with nonlinear neutral term,the results obtained extend some known results in the previous literatures.
文摘In this paper, we investigate stochastic asymptotic stability of the zero solution for certain third-order nonlinear stochastic delay differential equations by constructing Lyapunov functionals.
文摘Sufficient conditions are established for the asymptotic behavior of solutions of nonlinear delay differential equationswhere equipped with the sup norm for some r > 0. A new result is established, some known results are improved.
文摘By a Riccati transformation, we establish some new oscillation criteria which im-prove and generalize some known results in the previous literatures.