Functional Differential Systems Research Articles

On Approximate Controllability of Functional Impulsive Evolution Inclusions in a Hilbert Space

In this paper, we study approximate controllability of semilinear impulsive functional differential systems with finite delay. Simple sufficient conditions for such controllability are given. An example is provided to illustrate the application of the obtained results.

Positive solutions of Riemann-Stieltjes integral boundary problems for the nonlinear coupling system involving fractional-order differential

In this article, we study Riemann-Stieltjes integral boundary value problems of nonlinear fractional functional differential coupling system involving higher-order Caputo fractional derivatives. Some sufficient criteria are obtained for the existence, multiplicity, and nonexistence of positive solutions by applying fixed-point theorems on a convex cone. As applications, some examples are provided to illustrate our main results.

Homoclinic solutions for a second-order p-Laplacian functional differential system with local condition

By means of critical point theory and some analysis methods, the existence of homoclinic solutions for the p-Laplacian system with delay, d d t [ | u ′ ( t ) | p − 2 u ′ (t)]= ∇ x G(t,u(t),u(t+τ))+ ∇ y G(t−τ,u(t−τ),u(t))+e(t), is investigated. Some new results are obtained. The interesting thing is that the function G(t,x,y) is only required to satisfy a local condition. Furthermore, the results are all explicitly related to the value of delay τ.MSC:34C37, 58E05, 70H05.

Periodic solutions for a kind of neutral functional differential systems

In this paper, we analyze some properties of the linear difference operator A: C T → C T , [Ax](t)=x(t)−V(t)x(t−τ), and then, by using the coincidence degree theory of Mawhin, a kind of neutral differential systems with non-constant matrix is studied. Some new results on the existence of periodicity are obtained. It is worth noting that V(t) is no longer a constant matrix, which is different from the corresponding ones of past work.

IMPULSIVE CONTROLLABILITY OF MULTI-VALUED FUNCTIONAL DIFFERENTIAL SYSTEMS

This paper is concerned with the study of controllability for a class of multi-valued functional differential systems with impulses and delayed control. By making use of fixed point theorem for multi-valued maps, we prove that our system is completely controllable. The result is obtained in the sense of Caratheodory, and without the requirement that the linear part

Linear overdetermined boundary value problems in Hilbert space

The general linear boundary value problem for an abstract functional differential equation is considered in the case that the number of boundary conditions is greater than the dimension of the null-space to the corresponding homogeneous equation. Sufficient conditions of the solvability of the problem are obtained. A case of a functional differential system with aftereffect is considered separately.

Asymptotic Stability Criteria for a Class of Impulsive Functional Differential Systems

In this paper, a class of impulsive functional differential systems is inve stigated. It is proved that for the asymptotic stability of the zero solution of the system considered, it is sufficient that only som e components of the right-hand side of the system are bounded for unbounded values of time. For functional differential equations with out impulses, similar results were proved by Burton and Makay using Lyapunov-Krasovskii functionals. The goal of this paper is to prove these criteria for a class of impulsive functional differential systems with variable impulsive perturbations applying the Lyapunov-Razumikhin technique.

BOUNDEDNESS IN NONLINEAR PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS

In this paper, we investigate bounds for solutions of nonlinear perturbed functional differential systems.

On some stability problems of impulsive stochastic Cohen–Grossberg neural networks with mixed time delays

This paper covers the topic of both the pth moment (p⩾2) and almost sure stability of impulsive stochastic Cohen–Grossberg neural networks with mixed time delays. We partially use a known result on exponential stability of impulsive stochastic functional differential systems, based on the Razumikhin type technique, and extend it to the case of stochastic neural networks using the Lyapunov function method and a Gronwall type inequality. Additionally, we consider the stability with respect to a general decay function which includes exponential, but also more general lower rate decay functions as the polynomial and the logarithmic ones. This fact gives us the opportunity to study general decay almost sure stability, even when the exponential one cannot be discussed. Suitable examples which support the theory are also presented.

Integral manifolds of impulsive fractional functional differential systems

In this paper, a class of impulsive Caputo fractional functional differential systems with variable impulsive perturbations is investigated. Sufficient conditions for the existence of integral manifolds are obtained. The main results are proved by means of piecewise continuous Lyapunov functions and the fractional comparison principle. The demonstrated techniques can be applied in studying properties of many applied problems of diverse interest.

BOUNDEDNESS IN NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS

In this paper, we investigate bounds for solutions of nonlinear perturbed differential systems. The behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. There are three useful methods for showing the qualitative behavior of the solutions of perturbed non- linear system : Lyapunov's second method, the use of integral inequalities , and the method of variation of constants formula. The method incorporating in- tegral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of differential equations. In the presence the method of integral inequalities is as efficient as the direct Lyapunov's method. The notion of h-stability (hS) was introduced by Pinto (15,16) with the in- tention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential asymptotic stability) under some pertur- bations. That is, Pinto extended the study of exponential asymptotic stability to a variety of reasonable systems called h-systems. Using this notion, Choi and Ryu (3,5) investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional differential systems. Also, Goo et al. (8) studied the boundedness of solutions for nonlinear perturbed systems. In this paper, we obtain some results on boundedness of solutions of nonlinear perturbed differential systems under suitable conditions on perturbed term. To do this we need some integral inequalities.

Control problems for semilinear neutral differential equations in Hilbert spaces.

We construct some results on the regularity of solutions and the approximate controllability for neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the controllability of the neutral equations, we first consider the existence and regularity of solutions of the neutral control system by using fractional power of operators and the local Lipschitz continuity of nonlinear term. Our purpose is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature. Finally we give a simple example to which our main result can be applied.

Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. I

In this paper, we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for solutions of the considered systems in the critical case. Using the ideas of the center manifold method, we show the existence of the so-called critical manifold that is positively invariant for trajectories of the initial system. We establish that the asymptotics for solutions of the system on this manifold defines the asymptotics for all solutions of the initial system. In the first part of this work, we propose an algorithm for an approximate construction of the critical manifold. Moreover, we establish the unique solvability for auxiliary algebraic problems that occur within the algorithm implementation.

Practical stability of impulsive functional differential systems via analysis techniques

Practical stability of impulsive functional differential systems via analysis techniques

Existence and controllability results for damped second order impulsive functional differential systems with state-dependent delay

In this paper, we investigate the existence and controllability of mild solutions for a damped second order impulsive functional differential equation with state-dependent delay in Banach spaces. The results are obtained by using Sadovskii's fixed point theorem combined with the theories of a strongly continuous cosine family of bounded linear operators. Finally, an example is provided to illustrate the main results.

Generalized Cauchy problems for hyperbolic functional differential systems

Generalized Cauchy problems for hyperbolic functional differential systems

Center Manifold Method in the Asymptotic Integration Problem for Functional Differential Equations with Oscillatory Decreasing Coefficients. II

In this paper we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for the solutions of the considered systems in a critical case. In the second part of the work we establish the existence of a critical manifold for the considered class of systems and study its main properties. We also investigate the asymptotic integration problem for a reduced system. We illustrate the proposed method with an example of constructing the asymptotics for the solutions of a certain scalar delay differential equation.

Existence of Positive Periodic Solutions forn-Dimensional Nonautonomous System

In this paper we consider the existence, multiplicity, and nonexistence of positive periodic solutions forn-dimensional nonautonomous functional differential systemx'(t)=H(t,x(t))-λB(t)F(x(t-τ(t))), wherehiareω-periodic intand there existω-periodic functionsαi,βi∈C(R,R+)such thatαi(t)≤(hi(t,x)/xi)≤βi(t),∫0ω‍αi(t)dt>0,forx∈R+nall withxi>0,andt∈R,limxi→0+(hi(t,x)/xi)exist fort∈R;bi∈C(R,R+)areω-periodic functions and∫0ω‍bi(t)dt>0;fi∈C(R+n,R+),fi(x)>0forx >0;τ∈(R,R)is anω-periodic function. We show that the system has multiple or no positiveω-periodic solutions for sufficiently large or smallλ>0, respectively.

Existence and regularity of solutions for hyperbolic functional differential problems

A generalized Cauchy problem for quasilinear hyperbolic functional differential systems is considered. A theorem on the local existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions for this system is proved by using a method of successive approximations. We show a theorem on the differentiability of solutions with respect to initial functions which is the main result of the paper.

Euclidean and Function Space Null-Controllability of Nonlinear Delay Systems with Restrained Controls

The basic tool we use in our investigation is the nonlinear variation of constants formula for delay systems whilst we determine the controllability of nonlinear functional differential systems. This paper also touches on capture in nonlinear functional differential games of pursuit.