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.     

Controllability of nonlinear neutral evolution integrodifferential systems

Sufficient conditions for controllability of nonlinear neutral evolution integrodifferential systems in a Banach space are established. The results are obtained by using the resolvent operators and the Schaefer fixed-point theorem. An application to partial integrodifferential equation is given.     

Approximate Controllability for a Class of Abstract Second-Order Functional Evolution Equations

Results providing sufficient conditions for the approximate controllability of a class of second-order abstract functional evolution equations governed by the generator of a strongly continuous cosine family of linear operators, together with nonlocal initial conditions, are developed. This work extends results included in recent work by Naito, Park, Zhou, and others. The abstract theory is then applied to a parabolic partial integrodifferential equation.     

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.     

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.     

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.     

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.     

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.     

Some properties and applications of the integrodifferential operators of hadamard-marchaud type in the class of harmonic functions

In the class of harmonic functions, we study the properties of some integrodifferential operators generalizing the operators of fractional derivation in the sense of Hadamard and Hadamard-Marchaud. By way of application of the so-obtained properties, we consider some boundary value problems for the Laplace equation in the ball.     

Existence of weak and strong solutions of an integrodifferential equation in Banach spaces

A class of evolution integrodifferential equation has been studied over the analytic semigroup of operators in a Banach space. Further the existence of weak and strong solutions in Banach space have been proved and extended to a maximum interval of existence.     

A PERTURBATION THEOREM ANDITS APPLICATION

1IntroductionInpractice,manydifferentialequationsandintegrodifferentialeqllationscanbeformulatedasabstractCauchyproblemswhicharedependentonmulti-parameters[l,2,3]whereA(E)areclosedoperatorsonaBanachspaceandEisamulti-parameter.InordertostudydifferentiabilitywithrespecttotheparameterEoftheso-lutionof(l.1),severalstudieshavebeendevotedtothedifferentiabilitywithrespecttoparametersofCO--semigroupwhichisgeneratedbytheoperatorA(e)[l,5,6,7].AninterestingquestionisraisedwhenwestudVthefollowingabstr…     

Integral equations as evolution equations

We study a class of time-dependent linear integrodifferential equations (VE) with the evolution equation approach. We determine the generators of a time-dependent evolution equation (DE) which is equivalent to the given integrodifferential equation. Under very general assumptions we prove the well-posedness and continuity of (VE) from the stability of (DE). The related question of convergence of a family of approximate solutions is examined. As an application, we include an example of hyperbolic integro-partial-differential equation to illustrate the theory.     

On stabilization of solutions to incomplete second-order integrodifferential equations

We consider abstract incomplete linear second-order integrodifferential equations in a Hilbert space. Operator coefficients of the equations are unbounded selfadjoint nonnegative operators. These equations arise naturally in viscoelasticity and hydroelasticity. We prove a theorem on asymptotic stability of strong solutions of the equations.     

Numerical-analytical algorithm to solve the equation of particle transport in plane geometry

We describe a numerical-analytical algorithm to solve the boundary-value problem for the integrodifferential equation of particle transport in a plane homogeneous medium. The general scheme of approximate solution of this problem is based on its reduction to the solution of some integral equation by summator operators of function approximation theory. Solvability conditions are established for the approximate equations and the algorithm errors are estimated. Working formulas are presented for the algorithm implemented in the form of a computer program. The summator operators in this algorithm are the algebraic interpolation operators with nodes at the extremal points of Chebyshev polynomials of first kind.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 69–76, 1987.     

Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution

A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ?₊, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.     

Hyperbolic integro-differential equations of convolution type

Making use of the theory of Wiener-Hopf operators in the scale of abstract Krein spaces, we prove existence and uniqueness of unbounded solutions for the linear hyperbolic integrodifferential equation (P_o). We extend herewith results obtained in [8] for hyperbolic evolution equations, where the convolution integral was absent. The method utilizes Dunford's functional calculus and permits thus a constructive existence proof for solutions exhibiting an exponential growth rate when time increases. Our approach bases upon the fundamental hypothesis that the spectrum of the time-independent mapping -A shows a parabolic condensation along the negative real axis. This condition completely determines the admissible geometry of the spectral set of the convolution integral operator, and a fortiori the magnitude of the exponential growth rate. The theory works in arbitrary reflexive Banach spaces.     

Stability and growth estimates for electric fields in non-conducting material dielectrics

Employing results derived by the author for solutions of an abstract integrodifferential equation in Hilbert space, we obtain stability and growth estimates for electric fields in nonconducting material dielectrics. It is assumed that a linear constitutive equation of Maxwell-Hopkinson type relates the electric field and the electric displacement field in the dielectric; specific results for a simple memory function of exponential type are given.     

Existence results of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces

This paper investigates a class of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces. The existence of mild solutions of these inclusions is determined by using the theory of cosine families of bounded linear operators and a recent fixed point theorem for multivalued maps due to Dhage.     

Estimates for the solutions of systems of linear algebraic equations and their application to the investigation of the convergence of the method of finite differences

One proves theorems regarding the estimates of the solutions of systems of linear algebraic equations and inequalities. On the basis of these theorems one suggests a method for estimating the norm of the inverse matrix of a system of difference equations which approximates a boundary-value problem for an integrodifferential equation. The method allows us to eliminate the restrictions which are usually imposed on the coefficients of the integrodifferential equation in order to ensure the diagonal dominance in the system of the difference equations. One considers an application to nonlinear problems.