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 fundamental integrodifferential inequalities and their discrete analogues

Some fundamental integrodifferential inequalities and their discrete analogues which can be used in the analysis of a class of integrodifferential and summary difference equations as handy tools are discussed.     

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.     

Construction of periodic solutions of integrodifferential equations of viscoelasticity

A theorem on the estimation of the periodic solutions of a linear integrodifferential equation and a theorem of existence and uniqueness of the periodic solution of a nonlinear integrodifferential equation are presented without proof.     

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.     

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.     

Variation of parameters formulae with initial time difference for linear integrodifferential equations

This article investigates the relationship between an unperturbed integrodifferential system and a perturbed integrodifferential system that have different initial positions and an initial time difference (ITD). Variation of parameter techniques are employed to obtain integral formulae and to define Lyapunov-like functions.     

Null controllability of semilinear integrodifferential systems in Banach space

Sufficient conditions for null controllability of semilinear integrodifferential systems with unbounded linear operators in Banach space are established. The results are obtained using semigroup of linear operators, fractional powers of operators, and the Schauder fixed point theorem. An application to partial integrodifferential equations is given.     

The Cauchy Problem for a Class of Coupled Systems Containing a Convolution Operator

Some results on the invariant regions ,existence and uniqueness of solutions to a class of integrodifferential systems are established. Applying these results to integrodifferential systems with a small parameter ε > 0, we obtain,in particular, some estimates of solutions uniform in ε > 0.     

Controllability Results for Second Order Neutral Impulsive Integrodifferential Systems

In this paper, we address the problem of controllability for abstract second order neutral integrodifferential systems with impulsive conditions in Banach spaces. Furthermore, we discuss the controllability of second-order neutral integrodifferential systems in finite dimensional spaces. The results are obtained by using strongly continuous cosine families of operators and the Banach fixed point theorem.     

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.     

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.     

Reduction method for studying localized solutions of neural field equations on the Poincaré disk

We present a reduction method to study localized solutions of an integrodifferential equation defined on the Poincaré disk. This equation arises in a problem of texture perception modeling in the visual cortex. We first derive a partial differential equation which is equivalent to the initial integrodifferential equation and then deduce that localized solutions which are radially symmetric satisfy a fourth order ordinary differential equation.     

Properties of finite-difference schemes for singular integrodifferential equations of index 1

Systems of integrodifferential equations with a singular matrix multiplying the highest derivative of the unknown vector function are considered. An existence theorem is formulated, and a numerical solution method is proposed. The solutions to singular systems of integrodifferential equations are unstable with respect to small perturbations in the initial data. The influence of initial perturbations on the behavior of numerical processes is analyzed. It is shown that the finite-difference schemes proposed for the systems under study are self-regularizing.     

Controllability of McKean–Vlasov Stochastic Integrodifferential Evolution Equation in Hilbert Spaces

Sufficient conditions for controllability of McKean–Vlasov stochastic integrodifferential evolution equation in Hilbert spaces are obtained by using Banach fixed point analysis approach. This paper is motivated primarily by the work of Ahmed and Ding (Stochastic Processes Appl. 1995; 60:65–85), which is extended to a more general class of McKean–Vlasov stochastic integrodifferential equation in Hilbert space for establishing controllability result.     

An Identification Problem for an Ultraparabolic Integrodifferential Equation

We prove the well posedness of an inverse problem related to an abstract integrodifferential problem in two time variables with applications to ultraparabolic partial integrodifferential equations of Volterra type.     

The method of integrodifferential relations for analysing the natural oscillations of membranes

An approach to solving of problems of the free oscillations of membranes, based on the method of integrodifferential relations, is proposed. The variational properties of the integrodifferential problems are investigated and their relation to classical variational principles is demonstrated. Integral and local criteria for the quality of the approximate solutions are proposed. A numerical-analytical algorithm is developed for finding the characteristic frequencies and modes of oscillations. Problems involving the free transverse motions of circular and elliptic membranes are considered as an example.     

Local Controllability of Functional Integrodifferential Equations in Banach Space

In this paper, we establish a set of sufficient conditions for the local controllability of functional integrodifferential equations in Banach space. The results are obtained by using the methods of analytic semigroups, fractional powers of operators, and a fixed-point argument. These results generalize previous results on bounded linear operators to unbounded linear operators in which the equation involves a nonlinear delay term. An application to a partial integrodifferential equation is given.     

Asymptotic methods in the nonlinear theory of viscoelasticity

Dynamic and quasistatic problems of the nonlinear theory of viscoelasticity are described by nonlinear integrodifferential and integral equations. Methods of averaging various classes of nonlinear integrodifferential and integral equations are described and asymptotic expansions of the solutions of these equations are given.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 221–229, March–April, 1974.     

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.