New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces

This article deals with the use of Krasnoselskii’s fixed point theorem and Leray–Schauder alternative theorem combined with resolvent operator and some analytical methods to investigate the exact controllability and continuous dependence of a class of neutral fractional integro-differential systems with state-dependent delay in Banach spaces. An application to exemplify the concept is provided at the end.     

Bound of solutions to third-order nonlinear differential equations with bounded delay

A theorem is presented that contains some sufficient conditions to ensure bound of solutions to a third-order nonlinear delay differential equation. It is shown that this theorem improves the result of Ponzo [On the stability of certain nonlinear differential equations, IEEE Trans. Automatic Control AC-10 (1965) 470-472] to the bound of solutions for the equation considered.     

状态依赖时滞型微分方程正周期解的存在性

本文利用Krasnoselskii不动点定理,考虑了状态依赖时滞性微分方程x′(t) = &;#8722;A(t, x(t))x(t) + B(x(t))F(x(t &;#8722;  (t, x(t))))正周期解的存在性, 得到了该方程存在与不存在正周期解的充分条件.     

Controllability of mixed Volterra–Fredholm-type integro-differential inclusions in Banach spaces

The paper establishes a sufficient condition for the controllability of semilinear mixed Volterra–Fredholm-type integro-differential inclusions in Banach spaces. We use Bohnenblust–Karlin's fixed point theorem combined with a strongly continuous operator semigroup. Our main condition (A5) only depends upon the local properties of multivalued map on a bounded set. An example is also given to illustrate our main results.     

Analytical and numerical stability of nonlinear neutral delay integro-differential equations

In this paper, we are concerned with the analytical and numerical stability of nonlinear neutral delay integro-differential equations (NDIDEs). First, sufficient conditions for the analytical stability of nonlinear NDIDEs with a variable delay are derived. Then, we show that any A-stable linear multistep method can preserve the asymptotic stability of the analytical solution for nonlinear NDIDEs with a constant delay. At last, we validate our conclusions by numerical experiments.     

On the asymptotic behavior of solutions of certain second-order differential equations

In this paper, the second order non-linear differential equation

     

Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay

In this paper, a sufficient condition is established for the controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay in Banach spaces. The approach used is analytic semigroups and fractional powers of closed operators and nonlinear alternative of Leray–Schauder type for multivalued maps due to D. O'Regan.     

On the qualitative behaviors of solutions of some differential equations of higher order with multiple deviating arguments

We study the boundedness of the solutions to a non-autonomous and non-linear differential equation of second order with two constant deviating arguments. We give two examples to illustrate the main results. By this work, we extend some boundedness results obtained for a differential equation with a constant deviating argument in the literature to the boundedness of the solutions of a differential equation with two constant deviating arguments.     

Solutions of some partial differential equations (with tables)

A simple method has been demonstrated for obtaining some solutions of the principal equations of mathematical physics. Tables of solutions have been included in the Appendix so that problems in field theory can be solved with a minimum of labor.     

Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects

In this paper, we study stability of a class of stochastic differential delay equations with nonlinear impulsive effects. First, we establish the equivalent relation between the stability of this class of stochastic differential delay equations with impulsive effects and that of a corresponding stochastic differential delay equations without impulses. Then, some sufficient conditions ensuring various stabilities of the stochastic differential delay equations with impulsive effects are obtained. Finally, two examples are also discussed to illustrate the efficiency of the obtained results.     

Optimal control of stochastic functional neutral differential equations with time lag in control

In this work, we consider an optimal control problem of a class of stochastic differential equations driven by additive noise with aftereffect appearing in control. We develop a semigroup theory of the driving deterministic neutral system and identify explicitly the adjoint operator of the corresponding infinitesimal generator. We formulate the time delay equation under consideration into an infinite dimensional stochastic control system without time lag by means of the adjoint theory established. Consequently, we can deal with the associated optimal control problem through the study of a Hamilton–Jacob–Bellman (HJB) equation. Last, we present an example whose optimal control can be explicitly determined to illustrate our theory.     

The pth moment boundedness of stochastic functional differential equations with Markovian switching

This paper gives some Razumikhin-type theorems on pth moment boundedness of stochastic functional differential equations with Markovian switching (SFDEwMS) by using Razumikhin technique and comparison principle. Some improved conditions on pth moment stability are also proposed. The main results of this paper allow the estimated upper bound of the diffusion operator associated with the underlying SFDEwMS of the Lyapunov function to have time-varying coefficients (the coefficients may even be sign-changing functions). Examples are provided to illustrate the effectiveness of the proposed results.     

Delay-dependent stability analysis of trapezium rule for second order delay differential equations with three parameters

This paper is concerned with the study of the stability properties of trapezium rule for second order delay differential equations with three parameters. We start with introducing the analytical stability of a model equation. Then by using the boundary locus method, the delay-dependent stability region of the trapezium rule is analyzed and its boundary is found. Finally, a comparison between analytical and numerical stability regions is made and it is proved that the trapezium rule can completely preserve the delay-dependent stability of the underlining equations.     

On the stability and boundedness of a class of higher order delay differential equations

We establish some sufficient conditions which guarantee asymptotic stability of the null solution and boundedness of all the solutions of the following nonlinear differential equation of third order with the variable delay, r(t)

     

Existence of nonoscillatory solutions of first and second order neutral differential equations with distributed deviating arguments

In this paper, we consider the existence of nonoscillatory solutions of variable coefficient first and second-order linear neutral differential equations with distributed deviating arguments. We use the Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solution.     

pth Moment exponential stability of impulsive stochastic functional differential equations with Markovian switching

In this paper, the pth moment exponential stability for a class of impulsive stochastic functional differential equations with Markovian switching is investigated. Based on the Lyapunov function, Dynkin formula and Razumikhin technique with stochastic version as well as stochastic analysis theory, many new sufficient conditions are derived to ensure the pth moment exponential stability of the trivial solution. The obtained results show that stochastic functional differential equations with/without Markovian switching may be pth moment exponentially stabilized by impulses. Moreover, our results generalize and improve some results obtained in the literature. Finally, a numerical example and its simulations are given to illustrate the theoretical results.     

Razumikhin-type theorems on pth moment boundedness of neutral stochastic functional differential equations with Makovian switching

This paper investigates pth moment boundedness of neutral stochastic functional differential equations with Markovian switching (NSFDEsMS) based on Razumikhin technique and comparison principle. And pth moment stability is examined as a special case. Since the stochastic disturbances and neutral delays are incorporated, the considered system becomes more complex. Besides, the coefficients of the estimated upper bound for the diffusion operation associated with the underlying NSFDEsMS also may be chosen to be sign-changing functions instead of constant functions or negative definite functions, as a result, our results can work in general non-autonomous neutral stochastic systems. Finally, two examples are provided to show the effects of the proposed methods.     

Representation and realization of operational differential equations with time-varying coefficients

An algebraic treatment of operational differential equations with time-varying coefficients is presented in terms of skew rings of differential polynomials defined over a Noetherian ring. Included in this framework are delay differential equations with time- varying coefficients. The operator equations are characterized by transfer matrices which are utilized to construct realizations given by first-order vector differential equations with operator coefficients. It is shown that the realization of matrix equations can be reduced to the realization of scalar equations. Finally, a simple procedure is derived for realizing scalar equations.     

On group and phase velocities manifested through exact solutions to the equations of some propagation phenomena

A system of non-Hermite, non-orthogonal functions that are new for concrete applications is introduced to generate explicit and previously unknown, physically meaningful solutions for bar and plate equations. The approach allows in some cases to establish completeness of these sets of solutions. Various other separate results technically related to these basic development are presented. In the available literature there are no registered efforts to extract the two velocities from exact solutions of the equations of the processes in which the two parameters are explicitly present.