On Volterra Integro-Differential Equations with Kernels Representable by Stieltjes Integrals

Differential Equations - We consider abstract linear inhomogeneous second-order integro-differential equations in a Hilbert space that are defined on the positive half-line and have unbounded...     

Exponential Stability of Semigroups Generated by Volterra Integro-Differential Equations with Singular Kernels

Differential Equations - In a separable Hilbert space, we study abstract linear inhomogeneous second-order Volterra integro-differential equations on the positive half-line with operator...     

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...     

Square-mean almost periodic solutions to some stochastic evolution equations

This paper concerns the square-mean almost periodic mild solutions to a class of abstract nonautonomous functional integro-differential stochastic evolution equations in a real separable Hilbert space. By using the so-called "Acquistapace–Terreni" conditions and the Banach fixed point theorem, we establish the existence, uniqueness and the asymptotical stability of square-mean almost periodic solutions to such nonautonomous stochastic differential equations. As an application, almost periodic solution to a concrete nonautonomous stochastic integro-differential equation is considered to illustrate the applicability of our abstract results.     

非自伴非紧的抽象边值问题

在Ｈｉｌｂｅｒｔ空间研究了一类非自伴非紧的抽象边值问题,利用Ｗｉｅｎｅｒ－Ｈｏｐｆ方法构造了空间Ｘ的分裂投影,进而得出了方程解的适定性．     

Degenerate integro-differential operators in Banach spaces and their applications

In this paper we consider linear integro-differential equations in Banach spaces with Fredholm operators at the highest-order derivatives and convolution-type Volterra integral parts. We obtain sufficient conditions for the unique solvability (in the classical sense) of the Cauchy problem for the mentioned equations and illustrate the abstract results with pithy examples. The studies are carried out in classes of distributions in Banach spaces with the help of the theory of fundamental operator functions of degenerate integro-differential operators. We propose a universal technique for proving theorems on the form of fundamental operator functions.     

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.     

Variational inequalities in magneto-hydrodynamics

We study subdifferential initial boundary-value problems for the magneto-hydrodynamics (MHD) equations of a viscous incompressible liquid. We construct a solvability theory for an abstract evolution inequality in Hilbert space for operators with quadratic nonlinearity. The results obtained are applied to the study of MHD flows. For three-dimensional flows, we prove the existence of weak solutions of variational inequalities “globally” with respect to time, while, for two-dimensional flows, we establish the existence and uniqueness of strong solutions.     

Well-posed solvability of functional-differential equations with unbounded operator coefficients

We study functional-differential equations with unbounded variable operator coefficients and variable delays in a Hilbert space. We prove the well-posed solvability of initial-boundary value problems for the above-mentioned equations in Sobolev spaces of vector functions on the positive half-line.     

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.     

Solvability and Optimal Controls of Fractional Impulsive Stochastic Evolution Equations with Nonlocal Conditions

This paper deals with the solvability and optimal controls of a class of impulsive fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. Firstly, the existence and uniqueness of mild solutions for the considered system are investigated. Then, we derive the existence conditions of optimal pairs to the control systems. In the end, an example is presented to illustrate the effectiveness of our abstract results.     

Long-time behavior of nonlinear integro-differential evolution equations

We study the long-time behavior as time tends to infinity of globally bounded strong solutions to certain integro-differential equations in Hilbert spaces. Based on an appropriate new Lyapunov function and the Łojasiewicz–Simon inequality, we prove that any globally bounded strong solution converges to a steady state in a real Hilbert space.     

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space. II

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.     

Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect

We study functional differential equations with unbounded operator coefficients in Hilbert spaces such that the principal part of the equation is an abstract hyperbolic equation perturbed by terms with delay and terms containing Volterra integral operators. The well-posed solvability of initial boundary-value problems for the specified problems in weighted Sobolev spaces on the positive semi-axis is established.     

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. II. Application

This paper is a continuation of the author’s paper in 2009,where the abstract theory of fold completeness in Banach spaces has been presented.Using obtained there abstract results,we consider now very general boundary value problems for ODEs and PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions.Moreover,equations and boundary conditions may contain abstract operators as well.So,we deal,generally,with integro-differential equations,functional-differential equations,nonlocal boundary conditions,multipoint boundary conditions,integro-differential boundary conditions.We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach Lq-framework,in contrast to previously known results in the Hilbert L 2-framework.Some concrete mechanical problems are also presented.     

A Reproducing Kernel Hilbert Space Method for Solving Integro-Differential Equations of Fractional Order

In this article, we implement a relatively new analytical technique, the reproducing kernel Hilbert space method (RKHSM), for solving integro-differential equations of fractional order. The solution obtained by using the method takes the form of a convergent series with easily computable components. Two numerical examples are studied to demonstrate the accuracy of the present method. The present work shows the validity and great potential of the reproducing kernel Hilbert space method for solving linear and nonlinear integro-differential equations of fractional order.     

Spectral decomposition of self-adjoint cyclically compact operators and partial integral equations

We prove a spectral decomposition theorem for self-adjoint cyclically compact operators on Hilbert–Kaplansky module over a ring of bounded measurable functions. We apply this result to partial integral equations on the space with mixed norm of measurable functions. We give a condition of solvability of partial integral equations with self-adjoint kernel.     

Study of Volterra Integro-Differential Equations with Kernels Depending on a Parameter

We carry out spectral analysis of operator functions that are the symbols of integro-differential equations with unbounded operator coefficients in a separable Hilbert space. The structure and localization of the spectrum of operator functions which are symbols of these equations play an important role in studies of the asymptotic behavior of their solutions.     

Dynamics of an ideal liquid with a free surface in conformal variables

Problems of mathematical hydrodynamics with a free surface in conformal variables are studied. Analytical solvability in Hilbert space scale and numerical techniques of finding approximate solutions are considered. The lifetime for solutions, a constructive evaluation, and application of mathematical statistics to the solvability of nonlinear equations are studied. Multiple numerical experiments of the methods considered are shown. A lot of these methods can be applied not only to problems of mathematical hydrodynamics with a free surface but to abstract Cauchy–Kovalevskaya problems in Banach spaces scale as well. Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 28, Hydrodynamics, 2008.