Asymptotic constancy for linear neutral Volterra integrodifferential equations

Two-dimensional wavelets scheme for numerical solutions of linear and nonlinear Volterra integro-differential equations

In this note we establish a criterion to obtain classical solutions for a quasi-linear abstract retarded functional differential equation with infinite delay and we apply this result to characterize the infinitesimal generators of several strongly continuous semigroup of linear operators that arise in the theory of linear abstract retarded functional differential equations with infinite delay on a phase space defined axiomatically.

In this paper we consider periodic functional differential and integral equations. Massera proved that the existence of a bounded solution implies the existence of a periodic solution for linear (inhomogeneous) ordinary differential equations. Chow proved a similar result for linear functional differential equations with finite delay. We give a new proof for Chow's result that works for linear functional differential equations with infinite delay and also for integral equations. We also give examples that do not satisfy uniform boundedness or uniform ultimate boundedness, the conditions frequently used to prove that there is a periodic solution.

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.

Ważewski's Principle for retarded functional differential equations

The idea of this research is to find approximate solution of Linear Volterra Integro-Differential Equation (LVIDE) of the first order and second kind by using different degrees of the Touchard polynomials is presented. Because the simplicity definition applying this method will give high resolution results in low cost and short time, also can be easily differentiated, integrated and converge to an exact solution.The algorithm and example given are to illustrate the solution in this way and compare it with the exact solution.

By applying the Laplace transform to a certain Volterra convolution equation complex function theory can be made to bear on a linear autonomous retarded functional differential equation (RFDE) with finite delay.

In this paper, we apply an extension of the abstract Hopf bifurcation theorem stated in Jacimovic (Non Anal TMA 73(8):2426---2432, 2010) to abstract integral equations (AIE) and retarded functional differential equations (RFDE). This yields sufficient conditions of what we refer to as extended Hopf bifurcation for AIE and RFDE, in which we have a relaxation of the non-resonance condition on the eigenvalues of the generator of corresponding semigroup. We illustrate our results with an explicit example of a system of two delay differential equations (DDE), undergoing extended Hopf bifurcation at the resonant eigenvalue.

In this paper, by using the collocation method the operational matrix of Caputo fractional derivative was constructed for the cubic B-spline scaling function and wavelets. We apply the operational matrix and the Collocation method to solve linear fractional Volterra integro-differential equations. By using the principal characteristic of this technique, we have the operational matrix, which is applied to solve any linear fractional Volterra integro-differential equation by reducing the time, computer memory occupation and convert to a system of linear equations. Finally, the validity and applicability of the new technique will be shown by some numerical examples and convergence analysis.

Different results regarding different integro-differentials are usually not properly generalized, as they often do not satisfy some of the constraints. The field of fuzzy integro-differentials is very rich these days because of their different applications and functions in different physical phenomena. Solutions of linear fuzzy Volterra integro-differential equations (FVIDEs) are more generalized and have better applications. In this report, the Sumudu decomposition method (SDM) was used to find the solution to some linear and nonlinear fuzzy integro-differential equations (FIDEs). Some examples are given to show the validity of the presented method.

This paper deals with the approximation of the eigenvalues of evolution operators for linear retarded functional differential equations through the reduction to finite dimensional operators by a pseudospectral collocation. Fundamental applications such as determination of asymptotic stability of equilibria and periodic solutions of nonlinear autonomous retarded functional differential equations follow at once. Numerical tests are provided.

Introduction to the Series. Preface. Overview. Progress in Stability Theory by the First Approximation. Invariant Foliations for Caratheodory Type Differential Equations. On Exponential Asymptotic Stability for Functional Differential Equations with Causal Operators. Lyapunov Problems on Stability. Contemporary Development of Lyapunov's Ideas of Direct Method. Vector Lyapunov Function: Nonlinear, Time-Varying, Ordinary and Functional Differential Equations. Some Results on Total Stability Properties for Singular Systems. Stability Theory of Voltera Difference Equations. Consistent Lyapunov Methodology for Exponential Stability: PCUP Approach. Advances in Stability Theory of Lyapunov: Old and New. Matrix Lyapunov Functions and Stability Analysis of Dynamical Systems. Stability Theorems in Impulsive Functional Differential Equations with Infinite Delay. The Asymptotic Behavior of Solutions of Stochastic Functional Differential Equations with Finite Delays by Lyapunov-Razumikhin Method. A Non-standard Approach to the Study of the Dynamic System Stability. Stability of Solutions. A Survey of Starzhinskii's Works on Stability of Periodic Motions and Nonlinear Oscillations. Implications of the Stability of an Orbit for Its Omega Limit Set. Some Concepts of Periodic Motions and Stability Originated by Analysis of Homogenous Systems. Stability Criteria for Periodic Solutions of Autonomous Hamiltonian Systems. Selected Applications. Stability in Models of Agriculture-Industry-Environment. Bifurcations of Periodic Solutions of the Three Body Problem. Complex Mechanical Systems: Steady State Motions, Oscillations, Stability. Progress in Stability of Impulsive Systems with Applications to Population Growth Models. Contemporary Development of Lyapunov's Ideas of Direct Method. Stability of Solutions to Periodic Differential Systems. Selected Applications.

This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained.

In this article, we investigate the existence and uniqueness of solution of controlled hybrid neutral stochastic differential equations with infinite delay (HNSFDEswID). It is known that the time lag generated by the controller in each discrete observation must be different. The controlled HNSFDEswID are affected by the variable delay induced by the control function, the infinite time delay, and the highly nonlinear coefficients of the systems itself, which makes our problem more sophisticated. Different from the classical Khasminskii‐type conditions in the literature, we provide some sufficient conditions for the mean square and almost sure exponential stability for controlled HNSFDEswID by using the ‐matrix technique and a suitable Lyapunov functional. In this sense, the main contributions of our results, compared with those in the literature, are the infinite time delay, the neutral term, and the new Lyapunov functions. Finally, we illustrate our results by a numerical example.

In this paper, we use the variational iteration method (VIM) to construct approximate solutions for the general [Formula: see text]th-order integro-differential equations. We show that his method can be effectively and easily used to solve some classes of linear and nonlinear Volterra integro-differential equations. Finally, some numerical examples with exact solutions are given.