Variation Of Parameters Formula Research Articles

Nonautonomous impulsive differential equations of alternately advanced and retarded type

A variation of parameters formula and Gronwall type integral inequality are proved for an impulsive differential equation involving piecewise alternately advanced and retarded argument. At the same time, we study its oscillation and asymptotic stability properties. Our results are new, extend and improve earlier ones. Several numerical examples and simulations are also given to show the feasibility of our results.

Approximate controllability for finite delay nonlocal neutral integro-differential equations using resolvent operator theory

In this paper, our purpose is to study the approximate controllability of abstract nonlocal neutral integro-differential equations with finite delay in a Hilbert space using the resolvent operator theory. We derive a variation of parameters formula for representing a solution of the given neutral integro-differential system in the form of resolvent operators and then define a mild solution of the system. We also study the existence of a mild solution of the system with the help of resolvent operator theory. The fractional power theory, $$\alpha $$ -norm, resolvent operator theory, semigroup theory and Krasnoselskii’s fixed point theorem are used to prove the approximate controllability of the system. Finally, we illustrate our results with the help of an example.

Stability And Boundedness In Nonlinear And Neutral Difference Equations using New Variation of Parameters Formula And Fixed Point Theory

In the case of nonlinear problems, whether in differential or difference equations, it is difficult and in some cases impossible to invert the problem and obtain a suitable mapping that can be effectively used in fixed point theory to qualitatively analyze its solutions. In this paper we consider the existence of a positive sequence and utilize it in the capacity of integrating factor to obtain a new variation of parameters formula. Then, we will use the obtained new variation of parameters formula and revert to the contraction principle to arrive at results concerning, boundedness, periodicity and stability. The author is working on parallel results for the continuous case.

Weyl-almost periodic and asymptotically Weyl-almost periodic properties of solutions to linear and semilinear abstract Volterra integro-differential equations

The main purpose of paper is to consider Weyl-almost periodic and asymptotically Weyl-almost periodic solutions of linear and semilinear abstract Volterra integro-differential equations. We focus our attention to the investigations of Weyl-almost periodic and asymptotically Weyl-almost periodic properties of both, finite and infinite convolution product, working in the setting of complex Banach spaces. We introduce the class of asymptotically (equi)-Weyl-p-almost periodic functions depending on two parametres and prove a composition principle for the class of asymptotically equi-Weylp-almost periodic functions. Basically, our results are applicable in any situations where the variation of parameters formula takes a role. We provide several new contributions to abstract linear and semilinear Cauchy problems, including equations with the Weyl Liouville fractional derivatives and the Caputo fractional derivatives. We provide some applications of our abstract theoretical results at the end of paper, considering primarily abstract degenerate differential equations, including the famous Poisson heat equation and its fractional analogues.

Synchronization of stochastic discrete-time complex networks with partial mixed impulsive effects

In this paper, the synchronization problem is studied for a class of stochastic discrete-time complex networks with partial mixed impulsive effects. The involving impulsive effects, called partial mixed impulses, can be regarded as local and time-varying impulses, which means that impulses are not only injected into a fraction of nodes in networks but also contain synchronizing and desynchronizing impulses at the same time. In order to handle this case, several mathematical techniques are proposed to tackle mixed impulsive effects in discrete-time dynamical systems. Based on the variation of parameters formula, several sufficient criteria are derived to ensure that synchronization of the addressed networks can be achieved in mean square. The obtained criteria not only rely on the strengths of mixed impulses and the impulsive intervals, but also can reduce conservativeness. Finally, a numerical example is presented to show the effectiveness of our results for neural networks.

On generalized impulsive piecewise constant delay differential equations

A variation of parameters formula with Green function type and Gronwall type integral inequality are proved for impulsive differential equations involving piecewise constant delay of generalized type. Some results for piecewise constant linear and nonlinear delay differential equations with impulsive effects are obtained. They include existence and uniqueness theorems, a variation of parameters formula, integral inequalities, the oscillation property and some applications.

Interval Analytic Method in Existence Result for Hyperbolic Partial Differential Equation

Without the usual assumption of monotonicity, we establish some results on the theory of hyperbolic differential inequalities which enable us to produce a majorising interval function for the solution of the hyperbolic initial value problem. Using this function, a variation of parameters formula and interval iterative technique, the existence of solution to the problem is established.

Nonlinear Variation of Parameters Formula for Impulsive Differential Equations with Initial Time Difference and Application

This paper establishes variation of parameters formula for impulsive differential equations with initial time difference. As an application, one of the results is used to investigate stability properties of solutions.

Necessary and sufficient conditions for uniform stability of Volterra integro-dynamic equations using new resolvent equation

AbstractWe consider the system of Volterra integro-dynamic equations and obtain necessary and sufficient conditions for the uniform stability of the zero solution employing the resolvent equation coupled with the variation of parameters formula. The resolvent equation that we use for the study of stability will have to be developed since it is unknown for time scales. At the end of the paper, we furnish an example in which we deploy an appropriate Lyapunov functional. In addition to generalization, the results of this paper provides improvements for its counterparts in integro-differential and integro-difference equations which are the most important particular cases of our equation.

CONTINUOUS LYAPUNOV DYNAMICAL SYSTEMS - ARTIFICIAL NEURAL NETWORK APPROACH

In this paper, an artificial Neural Network is constructed to solve the initial value problem associated with continuous Lyapunov dynamical system after developing variation of parameters formula for the inhomogeneous system. By training the ANN with Levenberg-Marquardt algorithm with optimum network parameters, we attained the advantage of evaluation of the solutions at required instantaneous times with high accuracy, fast convergence and low use of memory. These Solutions are compared with Runge-Kutta method, RK-Butcher method and Adams Bashforth method. Numerical examples are presented for the developed methodology.

Variation of parameters formula and Gronwall inequality for differential equations with a general piecewise constant argument

A variation of parameters formula and Gronwall type integral inequality are proved for a differential equation involving general piecewise alternately advanced and retarded argument.

PRINCIPAL MATRIX SOLUTIONS AND VARIATION OF PARAMETERS FOR VOLTERRA INTEGRO-DYNAMIC EQUATIONS ON TIME SCALES

AbstractWe introduce the principal matrix solution Z(t, s) of the linear Volterra-type vector integro-dynamic equation and prove that it is the unique matrix solution of We also show that the solution of is unique and given by the variation of parameters formula

Smoothed integral equations

For a linear integral equation x(t)=a(t)−∫0tB(t,s)x(s)ds there is a resolvent equation R(t,s)=B(t,s)−∫stB(t,u)R(u,s)du and a variation of parameters formula x(t)=a(t)−∫0tR(t,s)a(s)ds. It is assumed that B is a perturbed convex function and that a(t) may be badly behaved in several ways. When the first two equations are treated separately by means of a Liapunov functional, restrictive conditions are required separately on a(t) and B(t,s). Here, we treat them as a single equation f(t)=S(t)−∫0tB(t,u)f(u)du where S is an integral combination of a(t) and B(t,s). There are two distinct advantages. First, possibly bad behavior of a(t) is smoothed. Next, properties of S needed in the Liapunov functional can be obtained from an array of properties of a(t) and B(t,s) yielding considerable flexibility not seen in standard treatment. The results are used to treat nonlinear perturbation problems. Moreover, the function y(t)=a(t)−∫0tB(t,s)a(s)ds is shown to converge pointwise and in L2[0,∞) to x(t).

Exponential asymptotic stability for nonlinear neutral systems with multiple delays

In this paper, we first consider exponential properties of solutions to a class of nonlinear delay differential inequalities. Then, based on the delay differential inequalities and the variation of parameters formula, we derive the new sufficient conditions for exponential asymptotic stability of the nonlinear neutral differential systems with delay.

Sense from sensitivity and variation of parameters

It is known that Alekseev's variation of parameters formula for ordinary differential equations can be generalized to other types of causal equations (including delay differential equations and Volterra integral equations), and corresponding discrete forms. Such variation of parameters formulae can be employed, together with appropriate inequalities, in discussing the behaviour of solutions of continuous and discretized problems, the significance of parameters in mathematical models, sensitivity and stability issues, bifurcation, and (in classical numerical analysis) error control, convergence and super-convergence of densely defined approximations and error analysis in general. However, attempts to extend Alekseev's formula to nonlinear Volterra integral equations are not straightforward, and difficulties can recur in attempts to analyze the sensitivity of functionally-dependent or structurally-dependent solutions. In analyzing sensitivity we discuss behaviour for infinitesimally small perturbations. In discussions of stability we need to establish the existence of bounds on changes to solutions (or their decay in the limit as t → ∞ ) that ensue from perturbations in the problem. Yet the two topics are related, not least through variation of parameters formulae, and (motivated by some of our recent results) we discuss this and related issues within a general framework.

Variational Lyapunov Method for Difference Equations

In this article we consider the difference equationwith a smoothness condition on f with respect to y. We study the dependence of solutions of (A) on initial data and then obtain a discrete analog of Alekseev's variation of parameters formula. In addition, we develop a Variational Lyapunov Method (VLM) and use it to study the stability properties of difference equations.

Stability Properties of Linear Volterra Discrete Systems with Nonlinear Perturbation

We consider a Volterra discrete system with nonlinear perturbation x ( n +1)= A ( n ) x ( n )+ ~ s =0 n B ( n , s ) x ( s )+ g ( n , x ( n ) and obtain necessary and sufficient conditions for stability properties of the zero solution employing the resolvent equation coupled with the variation of parameters formula.

Qualitative properties of nonlinear difference systems

This paper gives criteria for the existence and uniqueness of solutions to two and multipoint boundary value problems associated with a typical system of kth order nonlinear Lyapunov system. A variation of parameters formula for nonlinear kth order system is established. Using this formula, we also study various concepts of stability and establish sufficient conditions for the kth order difference system to be Lipschitz stable and uniform Lipschitz stable.

Two (multi) point nonlinear Lyapunov systems associated with an nth order nonlinear system of differential equations – existence and uniqueness

This paper presents a criterion for the existence and uniqueness of solutions to two and multipoint boundary value problems associated with an nth order nonlinear Lyapunov system. A variation of parameters formula is developed and used as a tool to obtain existence and uniqueness. We discuss solution of the second order problem by the ADI method and develop a fixed point method to find the general solution of the nth order Lyapunov system. The results of Barnett (SIAM J. Appl. Anal. 24(1), 1973) are a particular case.

Two-point boundary value problems associated with first order non-linear difference system — Existence and uniqueness

The general solution of the non homogeneous matrix difference system is established in terms of two fundamental matrix solutions by using the variation of parameters formula. Uniqueness of solution to two point boundary value problem is established by using modified QR algorithm and Barterls—Stewart algorithm.