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.     

Analytic solutions of Volterra equations via semigroups

We show that solutions of Volterra integrodifferential equations are analytic provided the leading operator generates an analytic semigroup and the convolution kernel is in a space of analytic functions. Similar results have been obtained via Laplace transform, so far. We show that it is possible to obtain such results using a semigroup approach.     

Well-posed solvability of volterra integro-differential equations in Hilbert space

We study the well-posed solvability of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are an abstract form of linear partial integro-differential equations that arise in the theory of viscoelasticity and have a series of other important applications. We obtain results on the wellposed solvability of the considered integro-differential equations in weighted Sobolev spaces of vector functions defined on the positive half-line and ranging in a Hilbert space.     

Analysis of Operator Models Arising in Problems of Hereditary Mechanics

The correct solvability of initial-value problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces is determined. Numerous problems of hereditary mechanics and thermal physics have motivated the study of such equations. Models taking into account a Kelvin–Voight friction are considered. A spectral analysis of the operator functions, which are the symbols of integro-differential equations considered in this paper, is made.     

Solvability of some classes of nonlinear integro-differential equations with noncompact operator

For a class of nonlinear integrodifferential equations with a noncompact Urysohn-type operator we prove the existence of nonnegative bounded solutions. We study the asymptotic behavior of solutions at infinity. We give some examples that are of practical interest.     

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.     

Inhomogeneous degenerate Sobolev type equations with delay

Under consideration is the first order linear inhomogeneous differential equation in an abstract Banach space with a degenerate operator at the derivative, a relatively p-radial operator at the unknown function, and a continuous delay operator. We obtain conditions of unique solvability of the Cauchy problem and the Showalter problem by means of degenerate semigroup theory methods. These general results are applied to the initial boundary value problems for systems of integrodifferential equations of the type of phase field equations.     

Integrodifferential equations in viscoelasticity theory

We prove the correct solvability of the initial problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces. Such equations occur in many problems of the theory of viscoelasticity with memory and the heat transfer theory.     

Correct Solvability and Representation of Solutions of Volterra Integrodifferential Equations with Fractional Exponential Kernels

Doklady Mathematics - For abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space, the correct solvability of initial value problems is studied and the...     

Existence of solutions for some functional integrodifferential equations with nonlocal conditions

In this paper, we discuss the existence of mild solution of functional integrodifferential equation with nonlocal conditions. To establish this results by using the resolvent operator theory and Sadovskii-Krasnosel'skii type of fixed point theorem and to show the usefulness and the applicability of our results to a broad class of functional integrodifferential equations, an example is given to illustrate the theory.     

Existence results for non-local operators of elliptic type

In this paper, we investigate the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. We make use of homological linking and Morse theory.     

Controllability for some integrodifferential equations driven by vector measures

In this work, we consider the question of controllability of a class of integrodifferential equations on Hilbert space with measures as controls. We assume that the linear part has a resolvent operator in the sense given by R. Grimmer. We generalize the original work of N. Ahmed on vector measures, and we use it to develop necessary and sufficient conditions for weak and the exact controllability of the integrodifferential equation. Using the latter, we prove that exact controllability of the integrodifferential equation implies exact controllability of a perturbed integrodifferential equation. Controllability problem for the perturbed system is formulated fixed point problem in the space of vector measures. Our results cover impulsive controls as well as regular controls. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

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.     

Eigenfunctions of Linearized Integrable Equations Expanded Around an Arbitrary Solution

Eigenfunctions of linearized integrable equations expanded around an arbitrary solution are obtained for the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy and the Korteweg–de Vries (KdV) hierarchy. It is shown that the linearization operators and the integrodifferential operator that generates the hierarchy are commutable. Consequently, eigenfunctions of the linearization operators are precisely squared eigenfunctions of the associated eigenvalue problem. Similar results are obtained for the adjoint linearization operators as well. These results make a simple connection between the direct soliton/multisoliton perturbation theory and the inverse-scattering based perturbation theory for these hierarchy equations.     

Estimates of the solutions of two-point boundary-value problems for systems of first-order integrodifferential equations and the method of lines

One proves theorems on the estimates of the solutions of the systems of first-order integrodifferential equations with the boundary conditions On the basis of these theorems, one suggests a method for estimating the norms of integrodifferential equations by the method of the lines for the solutions of the periodic boundary-value problems for second-order integrodifferential equations of parabolic type. On the basis of the established theorem, on the solvability and on the estimate of the solution of the nonlinear equation $$Tx + F\left( x \right) = 0$$ in a Banach space X, where T is a linear unbounded operator, one investigates the convergence of the method of lines for solving the periodic boundary-value problem for a second-order nonlinear integrodifferential equation of parabolic type.     

Generating operators for integrable nonlinear evolution equations

For linear problems which are associated with known, exactly integrable nonlinear evolution equations, one gives the corresponding integrodifferential Λ-operators. Relative to the expansions with respect to the elgenfunctions of Λ-operators, the method of the inverse scattering problem can be considered as the analog of the Fourier transform of linear problems, while the Λ-operators are the analogues of the differentiation operator. One considers the equations: Koteweg-de Vries, the nonlinear Schrödinger equations, the nonlinear Schrödinger equations with a derivative, the system of three waves, the matricial analog of the KdV equation, the Toda chain equation.     

Approximate controllability of a parabolic integrodifferential equation

In this paper, we obtain the approximate controllability of a parabolic integrodifferential equation with interior controls. The proof relies on the linear operator semigroup theory and the controllability theory for abstract equations. Copyright © 2013 John Wiley & Sons, Ltd.