Unbounded perturbation of the controllability for evolution equations

In this paper we prove that the controllability for evolution equations in Banach spaces is not destroyed, if we perturb the equation by “small” unbounded linear operator. This is done by employing a perturbation principle from linear operator theory and a characterization of surjective operators in Banach spaces. Finally, we apply these to a control system governed by partial integro-differential equations.     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

Study of functional-differential equations with unbounded operator coefficients

Functional-differential and integro-differential equations with the principal part being an abstract hyperbolic equation perturbed by terms with unbounded variable operator coefficients multiplying variable delays are studied. Additionally, Volterra integral operators are considered. For the equations under study, the well-posedness of initial value problems in Sobolev spaces of vector functions is proved. In the autonomous case, spectral analysis of the operator functions that are the symbols of the indicated equations is performed.     

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.     

一类非线性算子方程解的存在唯一性及其应用

该文在更广泛的条件下,利用锥理论和Banach压缩映象原理证明了序Banach空间中一类非线性算子方程解的存在唯一性定理,并应用到Banach空间中二阶非线性混合型微分积分方程初值问题,改进并推广了已有的一些结果.     

Nonlinear random equations involving completely closed operators

We introduce the concept of completely closed operators in Banach spaces and then obtain the existence of random solutions of operator equations involving such operators. As simple corollaries we obtain the existence theorems for random operator equations involving monotone operators as well as operators of type (M).     

一个增算子不动点定理及其在Banach 空间非线性方程的应用

给出了Banach空间的一个增算子不动点定理,将这一定理应用到Banach空间的积分-微分方程,给出了一类积分-微分方程的连续可微最大解和连续可微最小解的存在性定理.     

On an abstract integro-differential problem

In an arbitrary Banach space we consider the Cauchy problem for an integro-differential equation with unbounded operator coefficients. We establish the solvability of the problem in the weight spaces of integrable functions under certain conditions on the data. To prove this, the solution of the problem is written in explicit form with the help of analytic semigroups generated by fractional powers of the operator coefficients. Here an important role is played by the conditions on the coefficients ensuring the coercive estimates of the corresponding integral operators. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 887–896, December, 1999.     

Evolution Systems for Differential Equations with Variable Time Lags

We consider nonautonomous retarded functional differential equations under hypotheses which are designed for the application to equations with variable time lags, which may be unbounded, and construct an evolution system of solution operators which are continuously differentiable. These operators are defined on manifolds of continuously differentiable functions. The results apply to pantograph equations and to nonlinear Volterra integro-differential equations, for example. For linear equations we also provide a simpler evolution system with solution operators on a Banach space of continuous functions.     

序Banach空间非混合单调二元算子不动点的存在唯一性及应用

利用锥理论和Banach压缩映象原理在更一般的条件下建立了序Banach空间中一类非混合单调二元算子不动点的存在唯一性定理,并应用到Banach空间中二阶非线性Volterra型微分-积分方程初值问题,改进并推广了已有的一些结果.     

Linear differential operators and operator matrices of the second order

Linear differential operators (equations) of the second order in Banach spaces of vector functions defined on the entire real axis are studied. Conditions of their invertibility are given. The main results are based on putting a differential operator in correspondence with a second-order operator matrix and further use of the theory of first-order differential operators that are defined by the operator matrix. A general scheme is presented for studying the solvability conditions for different classes of second-order equations using second-order operator matrices. The scheme includes the studied problem as a special case.     

Representation of solutions of one integro-differential operator equation

We describe solutions of an integro-differential operator equation in the class of linear continuous operators acting in spaces of functions analytic in domains.     

On ℒ Characteristic of Some Potential Type Operators with Oscillating Symbols and Singularities of the Kernels on a Sphere

We obtain some new existence theorems for the maximal and minimal fixed points of discontinuous increasing operators. As applications, we consider the existence of the maximal and minimal solutions of nonlinear integro-differential equations with discontinuous terms in Banach spaces. Our results generalize and improve many well-known conclusions. This revised version was published online in June 2006 with corrections to the Cover Date.     

一个非连续增算子不动点定理及其应用

本文在无限区间上给出了一个新的非连续增算子不动点定理。作为应用,我们研究了无限区间上Banach空间中非线性积－微分方程初值问题解的存在性。     

Banach空间一类H-增生算子的混合拟变分包含的邻近算子方程

本文在Banach空间中引入一类H-增生算子的混合拟变分包含,并提出求该变分包含问题解的邻近点法.通过H-增生算子的预解算子技术,建立了混合拟变分包含问题与邻近算子方程的等价关系,由这个等价关系得到求解邻近算子方程的迭代算法,该算法收敛于上述混合拟变分包含问题的解.     

Fundamental Operator-Functions of Degenerate Differential and Difference-Differential Operators in Banach Spaces Which Have a Noether Operator in the Principal Part

Continuing the study by the first author, we construct fundamental operator-functions for some classes of degenerate differential operators (both ordinary and partial) and difference-differential operators in Banach spaces which have a Noether operator at the leading (time) derivative. To this end, we use the method of Jordan chains of Noether operators and the methods of distribution theory in Banach spaces.     

Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces

This paper investigates the existence of solutions for a class of second-order boundary-value problems with integral boundary conditions of nonlinear impulsive integro-differential equations in Banach spaces. The arguments are based upon the fixed point theorem of strict set contraction operators. Meanwhile, an example is worked out to demonstrate the main results.     

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.     

Systems of degenerate differential equations in Banach spaces

We consider systems of degenerate differential equations in Banach spaces of a special form. The main instrument of research is the technique of distributions in Banach spaces; namely, the construction of a fundamental operator function introduced by the first author. We translate the results obtained previously for a single equation to the systems of various types and illustrate them with examples.     

Mappings of higher order and nonlinear equations in some spaces of almost periodic functions

In the first part of the paper we examine mappings of higher order from a general point of view, that is, in normed spaces of bounded real-valued functions defined on R$\mathbb{R}$. Particular attention is paid to the relation of such mappings with the so-called autonomous superposition operators. Next we investigate mappings of higher order in Banach spaces of almost periodic functions and their perturbations. We also give necessary and sufficient conditions guaranteeing that a nonautonomous superposition operator acts in the space of almost periodic functions in the sense of Levitan and is uniformly continuous. In the Banach space of bounded almost periodic functions in the sense of Levitan we discuss mappings of higher order and a convolution operator. Some applications to nonlinear differential and integral equations are given.