Retarded Differential Equations Research Articles

Boundedness of solutions of nonlinear retarded differential equations by using new integral inequalities

Boundedness of solutions of nonlinear retarded differential equations by using new integral inequalities

Periodic perturbations of reducible scalar second order functional differential equations

Using a topological approach we investigate the structure of the set of forced oscillations of a class of reducible second order functional retarded differential equations subject to periodic forcing. More precisely, we consider a delay-type functional dependence involving a gamma probability distribution and, using a linear chain trick, we formulate a first order system of ODEs whose T -periodic solutions correspond to those of the functional equation.

METHOD OF TIME-DISCRETIZATION TO A MULTITERM NONLINEAR RETARDED DIFFERENTIAL EQUATION

We study the existence of a strong solution for a class of second-order nonlinear retarded differential equations involving a lower-order fractional term. We prove the existence result by using Rothe’s time-discretization method. We also provide some examples to examine the existence of solutions for a class of second-order fractional differential equations.

A note on regularity property of stochastic convolutions for a class of functional differential equations

This is a continuation of [5] which is concerned about the regularity property of stochastic convolutions for abstract linear stochastic retarded differential equations with unbounded operators on delay terms. In this work, we improve and generalize the main results in [5] by considering those delay operators which may have the same order as the infinitesimal generator of the system under consideration. To this end, we need restrict the weight function of distributed delay term to be Hölder continuous type in this system.

Mild Solution and Approximate Controllability of Retarded Semilinear Systems with Control Delays and Nonlocal Conditions

This work studies the approximate controllability of a class of first-order retarded semilinear differential equations with delays in control and nonlocal conditions. First we deduce the existence of mild solutions using fixed point approach. For this, the nonlinear function is supposed to be locally Lipschitz, which is a weaker condition than Lipschitz continuity. Controllability results of the system are shown by using the approximate and iterative technique. The results are illustrated by providing an example.

Mild Solution and Approximate Controllability of Second-Order Retarded Systems with Control Delays and Nonlocal Conditions

This work studies the approximate controllability of a class of second-order retarded semilinear differential equations with nonlocal conditions and with delays in control. First, we deduce the existence of mild solutions using cosine family and fixed point approach. For this, the nonlinear function is supposed to be locally Lipschitz. Controllability of the system is shown using an approximate and iterative technique. The results are illustrated using an example.

Large spectral gap induced by small delay and its application to reduction

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.

On qualitative properties of single-delay linear retarded differential equations: Characteristic roots of maximal multiplicity are necessarily dominant

This paper presents necessary and sufficient conditions for the existence of a real root of maximal multiplicity in the spectrum of a linear time-invariant single-delay equation of retarded type. We also prove that this root is always strictly dominant, and hence determines the asymptotic behavior of the system. These results are based on improved a priori bounds on the imaginary part of roots on the complex right half-plane.

Further Nonlinear Retarded Integral Inequalities for Gronwall–Bellman Type and Their Applications

In this article, we will derive several new nonlinear retarded integral inequalities for Gronwall–Bellman type and then apply them as handy tools to examine explicit bounds for solutions of initial value problems, and will also use to evaluate the qualitative as well as the quantitative behavior of solutions of nonlinear retarded differential and integral equations. These inequalities extend some present inequalities to the current literature. An application is also given to demonstrate the strength of our results.

Fractional retarded differential equations and their numerical solution via a multistep collocation method

In this paper, we consider the nonlinear fractional retarded differential equations (FRDE). We extend the results of the existence and uniqueness of the solution, the propagation of derivative discontinuities and the dependence of the solution on the parameters of the equation. Next, we develop an efficient multistep collocation method for solving this type of equations. The proposed scheme is especially suited for FRDEs with piecewise smooth solutions, due to its essential feature of local approximations on subintervals. The stability of the scheme is accessed, and the convergence analysis is studied for functions in appropriate Sobolev spaces. Numerical results confirm the spectral accuracy and the stability of the proposed method for large domain calculations.

Lipschitz stability for generalized ordinary differential equations and impulsive retarded differential equations

Lipschitz stability for generalized ordinary differential equations and impulsive retarded differential equations

PREDICTION OF MULTI-DIMENSIONAL MILLING BEHAVIOR

Milling process is a multi-dimensional cutting process accompanied by chatter vibrations because of material removal discontinuity. Also, chatter phenomenon has a great impact on final milled products quality. In this paper, an adequate criterion derived from mathematical literature is used to predict stability of multi-dimensional chatter milling. For that, firstly, the chatter milling system is represented by a system of three Degrees Of Freedom (DOF) and its dynamics is modeled by a system of Retarded Differential Equations (RDEs). Afterward, stability in Lyapunov sense is computed on the basis of the quasi-polynomial characteristic function in frequency domain. Finally, this method is used to predict stability of chatter milling process under different machining conditions and the output related to each case is verified by resolving the system of RDEs and visualizing the displacements in time domain using Matlab software.

Harnack Inequality and Applications for Stochastic Retarded Differential Equations Driven by Fractional Brownian Motion

Harnack Inequality and Applications for Stochastic Retarded Differential Equations Driven by Fractional Brownian Motion

Impulsive retarded fractional differential equations with variable moments

Integer order impulsive functional differential equations with fi xed and variable times and the fractional order ones with fi xed times have been studied in the related literature several times, so far. To the extent we know, the fractional order ones with variable times have not yet been considered. In this work, we deal with establishing some existence results for solutions of impulsive retarded functional differential equations of fractional order with variable moments.

Converse Lyapunov–Krasovskii theorems for uncertain retarded differential equations

In this article we give a collection of converse Lyapunov–Krasovskii theorems for uncertain retarded differential equations. We show that the existence of a weakly-degenerate Lyapunov–Krasovskii functional is a necessary and sufficient condition for the global exponential stability of linear retarded functional differential equations. This is carried out using a switched system representation approach.

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.

On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation

Abstract In this paper, we investigate some new nonlinear retarded integral inequalities of Gronwall–Bellman–Pachpatte type. These inequalities generalize some former famous inequalities and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of some nonlinear retarded differential and integral equations. An application is also presented to illustrate the usefulness of some of our results in estimation of solution of certain retarded nonlinear differential equations with the initial conditions.

Impulsive stabilization of high-order nonlinear retarded differential equations

In this paper, impulsive stabilization of high-order nonlinear retarded differential equations is investigated by using Lyapunov functions and some analysis methods. Our results show that several non-impulsive unstable systems can be stabilized by imposition of impulsive controls. Some recent results are extended and improved. An example is given to demonstrate the effectiveness of the proposed control and stabilization methods.

Massera problem for non-autonomous retarded differential equations

In this paper, under Acquistapace–Terreni conditions, we study the existence of periodic solutions for some non-autonomous semilinear partial functional differential equation with delay. We assume that the linear part is non-densely defined. The delayed part is assumed to be ω-periodic with respect to the first argument. Using a fixed point theorem for multivalued mapping, some sufficient conditions are given to prove the existence of periodic solutions. An example is shown to illustrate our results.

On the solutions of a system of linear retarded and advanced differential equations by the Bessel collocation approximation

In this study, we present a numerical approximation for the solutions of the system of high-order linear retarded and advanced differential equations with variable coefficients under the mixed conditions. This method is based on taking the truncated Bessel expansion of the functions in the retarded and advanced differential equation system. By the aid of the matrix operations and the collocation points, the problem is transformed into a matrix equation with the unknown Bessel coefficients. By solving this matrix equation, the unknown coefficients of the approximate solutions are computed. In addition, examples that illustrate the pertinent features of the method are presented, and the results of this investigation are discussed.