Optimal control of impulsive Volterra equations with variable impulse times

We obtain necessary conditions of optimality for impulsive Volterra integral equations with switching and impulsive controls, with variable impulse time-instants. The present work continues and complements our previous work on impulsive Volterra control with fixed impulse times.     

A new algebraic procedure to construct exact solutions of nonlinear differential-difference equations

This paper presents a new algebraic procedure to construct exact solutions of selected nonlinear differential-difference equations. The discrete sine-Gordon equation and differential-difference asymmetric Nizhnik-Novikov-Veselov equations are chosen as examples to illustrate the efficiency and effectiveness of the new procedure, where various types of exact travelling wave solutions for these nonlinear differential-difference equations have been constructed. It is anticipated that the new procedure can also be used to produce solutions for other nonlinear differential-difference equations.     

Uniform eventual Lipschitz stability of impulsive systems on time scales

In this paper, we discuss the uniform eventual Lipschitz stability of impulsive system on time scales. By using comparison method, Lyapunov function and analysis technology, some criteria of such stability for system with impulses on time scales are obtained. An example is presented to illustrate the efficiency of proposed results.     

Existence theory for perturbed impulsive hyperbolic differential inclusions with variable times

In this paper, the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is proved under the mixed generalized Lipschitz and Carathéodory's conditions.     

Comparison principles for impulsive hyperbolic equations of first order

Strong and weak comparison principles for impulsive hyperbolic equations of first order are proved. Uniqueness criterion is obtained.     

Singular perturbations of abstract wave equations

Given, on the Hilbert space H₀, the self-adjoint operator B and the skew-adjoint operators C₁ and C₂, we consider, on the Hilbert space H?D(B)⊕H₀, the skew-adjoint operator

     

A computational method for singularly perturbed nonlinear differential-difference equations with small shift

This paper is devoted to the numerical study of the boundary value problems for nonlinear singularly perturbed differential-difference equations with small delay. Quasilinearization process is used to linearize the nonlinear differential equation. After applying the quasilinearization process to the nonlinear problem, a sequence of linearized problems is obtained. To obtain parameter-uniform convergence, a piecewise-uniform mesh is used, which is dense in the boundary layer region and coarse in the outer region. The parameter-uniform convergence analysis of the method has been discussed. The method has shown to have almost second-order parameter-uniform convergence. The effect of small shift on the boundary layer(s) has also been discussed. To demonstrate the performance of the proposed scheme two examples have been carried out. The maximum absolute errors and uniform rates of convergence have been presented in the form of the tables.     

Uniform attractors for impulsive reaction-diffusion equations

The goal of this paper is to consider the long time behavior of solutions of reaction-diffusion equations with impulsive effects at fixed moment of time. Under a new class of impulse function, we prove the existence of uniform attractors in the spaces and L^2p-2(Ω), respectively.     

Global stability results for impulsive differential equations

Global results concerning stability properties of impulsive differential equations are established, employing piecewise continuous Lyapunov functions which are then applied for proving stability and boundedness properties     

The numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.     

Exponential stability of nonlinear impulsive neutral integro-differential equations

In this paper, a nonlinear impulsive neutral integro-differential equation with time-varying delays is considered. By establishing a singular impulsive delay integro-differential inequality and transforming the n$n$-dimensional impulsive neutral integro-differential equation to a 2n$2n$-dimensional singular impulsive delay integro-differential equation, some sufficient conditions ensuring the global exponential stability in PC¹$P{C}^1$ of the zero solution of an impulsive neutral integro-differential equation are obtained. The results extend and improve the earlier publications. An example is also discussed to illustrate the efficiency of the obtained results.     

Monotone iterative methods for impulsive hyperbolic differential-functional equations

Theorems on impulsive hyperbolic differential-functional inequalities are considered. Comparison results and a uniqueness criterion are obtained. A method of approximation of the solutions of impulsive hyperbolic differential-functional equations by means of solutions of the associated linear problems is established. The difference between the exact and the approximate solutions is estimated.     

Lipschitz regularity for viscous Hamilton-Jacobi equations with Lp terms

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable.     

Almost periodic solutions for forced perturbed impulsive differential equations

Sufficirnt condition for the existence of almost periodic solutions of forced perturbed systems of impulsive differential equations with impulsive effect at fixed Moments are considered.     

Exponential boundedness of solutions for impulsive delay differential equations

For an impulsive delay differential equation

exponent estimates of solutions have been obtained.     

Oscillation criteria of neutral type impulsive hyperbolic equations

In this paper, oscillatory properties of all solutions for neutral type impulsive hyperbolic equations with several delays under the Robin boundary condition are investigated and several new sufficient conditions for oscillation are presented.     

The existence of mild solutions for impulsive fractional partial differential equations

This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010) [2] are corrected, and some previous results are generalized.     

First-order impulsive ordinary differential equations with advanced arguments

This paper deals with impulsive advanced ordinary differential equations with boundary conditions. We investigate the existence of solutions and quasisolutions for advanced impulsive differential equations. To obtain such results we apply Schauder's fixed point theorem. Corresponding results are also formulated for differential inequalities.     

On convergence of solutions to difference equations with additive perturbations

Various types of stabilizing controls lead to a deterministic difference equation with the following property: once the initial value is positive, the solution tends to the unique positive equilibrium. Introducing additive perturbations can change this picture: we give examples of difference equations experiencing additive perturbations which have solutions staying around zero rather than tending to the unique positive equilibrium. When perturbations are stochastic with a bounded support, we give an upper estimate for the probability that the solution can stay around zero. Applying extra conditions on the behaviour of the map function f at zero or on the amplitudes of stochastic perturbations, we prove that the solution tends to the unique positive equilibrium almost surely. In particular, this holds either for all amplitudes when the right derivative of the map f at zero exceeds one or, independently of the behaviour of f at zero, when the amplitudes are not square summable.