Exact null controllability of semilinear integrodifferential systems in Hilbert spaces

Sufficient conditions for exact null controllability of the semilinear integrodifferential systems in Hilbert spaces are obtained. It is shown that under some natural conditions exact null controllability of the semilinear integrodifferential system is implied by the exact null controllability of the corresponding linear system with additive term. An application to partial integrodifferential equations is given.     

On some new integrodifferential inequalities of the Wendroff type

This paper presents some new integrodifferential inequalities in two independent variables of the Wendroff type which can be used in the analysis of a class of hyperbolic partial integrodifferential equations as handy tools.     

Regularity of solutions of evolution integrodifferential equations with deviating argument

In this paper we prove the existence of mild and classical solutions of delay integrodifferential equation and delay integrodifferential evolution equations with nonlocal condition in Banach spaces. The regularity solutions of integrodifferential evolution equations interconnected with viscoelastic material is derived to guarantee the stabilization. The results are established by using the resolvent operators and the fixed point principles. Finally, an example is given to show the potential of the proposed techniques.     

Study of Volterra integro-differential equations arising in viscoelasticity theory

Volterra integrodifferential equations with unbounded operator coefficients in a Hilbert space that are operator models of integrodifferential equations arising in viscoelasticity theory are studied. These equations are shown to be well-posed in Sobolev spaces of vector functions, and spectral analysis is applied to the operator functions that are the symbols of the given equations.     

Existence and regularity of solutions for neutral partial functional integrodifferential equations

In this paper, we study a class of neutral partial functional integrodifferential equations with finite delay by using the theory of resolvent operators. We give some sufficient conditions ensuring the existence, uniqueness and regularity of solutions. As an application, we also consider a diffusive neutral partial functional integrodifferential equation.     

Existence of solutions to some retarded integrodifferential equations via the topological transversality theorem

We prove the existence of solutions to some retarded integrodifferential equations under some suitable conditions on the involved functions. Some applications of our results are also provided.     

Existence of asymptotically almost periodic solutions for some second-order hyperbolic integrodifferential equations

Semigroup Forum - In this paper we study and obtain the existence of asymptotically almost periodic solutions to some classes of second-order hyperbolic integrodifferential equations of...     

Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations

The authors study integrodifferential equations in Hilbert space. The coefficients of the equations are unbounded and the principal part is an abstract hyperbolic equation perturbed by terms with Volterra integral operators. Such equations can be regarded as an abstract generalization of the Gurtin–Pipkin integrodifferential equation that describes heat transfer in materials with memory and has a number of other applications. Well-defined solvability of initial boundary value problems for such equations is established in weighted Sobolev spaces on the positive semi-axis. The authors examine spectral problems for operator-valued functions representing the symbols of the said equations and study the spectrum of the abstract Gurtin–Pipkin integrodifferential equation.     

非线性积分微分方程的稳定性

本文研究一类非线性积分微分方程的稳定性问题．给出了简洁、实用的稳定性新准则．文末例子说明了本文结果的优越性。     

The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces

In this paper we prove the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces. Further, Cauchy problems with nonlocal initial conditions are discussed for the aforementioned fractional integrodifferential equations. At the end, an example is presented.     

Almost periodic solutions for some integrodifferential equations with infinite delay

By using a Liapunov functional, the conditions of existence and uniqueness of almost periodic solutions for some integrodifferential equations are obtained.     

Pseudo almost automorphic solution to stochastic differential equation driven by Lévy process

Almost automorphic is a particular case of the recurrent motion, which has been studied in differential equations for a long time. We introduce square-mean pseudo almost automorphic and some of its properties, and then study the pseudo almost automorphic solution in the distribution sense to stochastic differential equation driven by Lévy process.     

Regularity for solutions of nonlinear second order evolution equations

This paper deals with the existence of solutions for the class of nonlinear second order evolution equations. The regularity and a variation of solutions of the given equations are also given. As particular cases of our general formulation, some results for Volterra integrodifferential equations of the hyperbolic type are given.     

Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law). The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.     

On a solution to fractional order integrodifferential equations with analytic semigroups

In this paper we study a class of fractional order integrodifferential equations considered in an arbitrary Banach space. Using the theory of analytic semigroups we establish the existence, uniqueness and regularity of a mild solution to these fractional order integrodifferential equations.     

Existence results for fractional impulsive integrodifferential equations in Banach spaces

This paper is mainly concerned with the existence of solutions of first order nonlinear impulsive fractional integrodifferential equations in Banach spaces. The results are obtained by using fixed point principles. Further, some interesting examples are presented to illustrate the theory.     

Instability in the critical case of pair of pure imaginary roots for a class of systems with aftereffect

The stability of a system described by Volterra integrodifferential equations is investigated in the critical case when the characteristic equation has a pair of pure imaginery roots. Conditions for instability, analogous to the well-known conditions from the theory of differential equations [1], are derived. (A similar result was established previously in [2] for integrodifferential equations of simpler structure with integral kernels of exponential-polynomial type). For the proof, several manipulations are used to simplify the original equation and, in particular, to reduce the linearized equation to the form of a differential equation with constant diagonal matrix. (An analogous approach was used to analyse instability for Volterra integrodifferential equations in the critical case of zero root in [3, 4]). As an example, the sign of the Lyapunov constant in the problem of the rotational motion of a rigid body with viscoelastic supports is calculated.     

Existence and Regularity of Solutions for Some Partial Integrodifferential Equations Involving the Nonlocal Conditions

The aim of this work is to prove some results about the existence and regularity of solutions for some partial integrodifferential equations with nonlocal conditions. We suppose that the linear part has a resolvent operator in the sens given by Grimmer. The non linear part is assumed to be continuous and Lipschitzian with respect to the second argument.     

Oscillation and Nonoscillation in Delay or Advanced Differential Equations and in Integrodifferential Equations

Some new oscillation and nonoscillation criteria are given for linear delay or advanced differential equations with variable coefficients and not (necessarily) constant delays or advanced arguments. Moreover, some new results on the existence and the nonexistence of positive solutions for linear integrodifferential equations are obtained.     

Positive almost automorphic solutions for a class of non-linear delay integral equations

This article is concerned with a class of non-linear delay integral equations, which unify some extensively studied delay integral equations. We establish a new existence and uniqueness theorem about positive almost automorphic solutions of the delay integral equations. Our theorem can deal with some cases to which many known results are not applicable. Two examples are given to illustrate our results.