On Bounded Solutions of Some Classes of Two-Parameter Difference Equations in a Banach Space

We obtain criteria for the existence of bounded solutions of some classes of linear two-parameter difference equations with operator coefficients in a Banach space.     

Bounded Solutions for Some Classes of Difference Equations with Operator Coefficients

We obtain necessary and sufficient conditions for the existence and uniqueness of bounded solutions for some classes of linear one- and two-parameter difference equations with operator coefficients in a Banach space.     

On the approximation of a bounded solution of a linear differential equation in a Banach space

We investigate the problem of approximation of a bounded solution of a linear differential equation by solutions of the corresponding difference equations in a Banach space. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1268–1271, September, 1998.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

On BV-Solutions of Some Nonlinear Integral Equations

In this paper we prove uniqueness theorems for bounded variation (shortly: BV) solutions and continuous BV-solutions of the Hammerstein and the Volterra-Hammerstein integral equations. We investigate real-valued functions and functions with values in a Banach space. Submitted: August 16, 2001?Revised: September 13, 2002     

Solvability criterion and representation of solutions of <Emphasis Type="Italic">n</Emphasis>-normal and <Emphasis Type="Italic">d</Emphasis>-normal linear operator equations in a Banach space

On the basis of a generalization of the well-known Schmidt lemma to the case of n-normal and d-normal linear bounded operators in a Banach space, we propose constructions of generalized inverse operators. We obtain criteria for the solvability of linear equations with these operators and formulas for the representation of solutions of these equations.     

解一类抽象积—微分方程的积分半群方法

在这篇文章内，研究一类含参数线性一算子族Ａ（ｔ）和线性算子Ｂ的积－微分方程，应用积分半群方法证明了该方程存在指数有界解，和含Ａ（ｔ），Ｂ的二阶微分方程存在指数有界和分解。这里Ａ（ｔ），Ｂ不必满足Ｈｉｌｌｅ－Ｙｏｓｉｄａ条件。     

具有可变时滞的偶数阶非线性中立型差分方程的正解

本文考虑具有可变时滞的偶数阶非线性中立型差分方程，利用Ｂａｎａｃｈ空间的压缩映象原理，得到了这类方程存在最终正解的判别准则．     

MULTIPLE POSITIVE PERIODIC SOLUTIONS OF A CLASS OF DELAY DIFFERENCE EQUATIONS

Based on the Leggett-Williams fixed point theorem for a Banach space, we establish the existence of three positive periodic solutions for a class of delay difference equations.     

MULTIPLICITY OF NONNEGATIVE SOLUTIONS TO SECOND-ORDER SINGULAR DIFFERENTIAL EQUATION WITH INTEGRAL BOUNDARY CONDITIONS IN BANACH SPACES

This paper investigates the existence and multiplicity of nonnegative solutions to a singular nonlinear boundary value problem of second order differential equations with integral boundary conditions in a Banach space. The arguments are based on the construction of a nonempty bounded open convex set and fixed point index theory. Our nonlinearity possesses singularity and first derivative which makes it different with that in [10].     

Existence and Approximate Controllability of Solutions for an Impulsive Evolution Equation with Nonlocal Conditions in Banach Space

In this article, we study the existence of mild solutions and approximate controllability for non-autonomous impulsive evolution equations with nonlocal conditions in Banach space. The existence of mild solutions and some conditions for approximate controllability of these non-autonomous impulsive evolution equations are given by using the Krasnoselskii''s fixed point theorem, the theory of evolution family and the resolvent operator. In particular,the impulsive functions are supposed to be continuous and the nonlocal item is divided into Lipschitz continuous and completely bounded. An example is given as an application of the results.     

On the properties of ambiguity and irreversibility of dynamical maps of the initial data space of Cauchy problem

We investigate the Cauchy problems for evolutionary differential equations which possess the following properties: the solutions of considered problems admit the arising on the bounded time interval of singularities such that destroying of existence or uniqueness of solution and unbounded growth of norm of solution in Cauchy problem Banach space. The opportunity of continuation (probably of many-valued continuation) of the dynamical maps of the space of initial data by the procedure of passage to the limit for the sequences of approximating Cauchy problems is studied.     

BVφ-solutions of nonlinear integral equations

In this paper we investigate solutions of nonlinear Hammerstein and Volterra-Hammerstein integral equations in the space of functions of bounded φ-variation in the sense of Young. We prove the existence and in some cases the existence and uniqueness of local and global solutions in this class. Real-valued as well as vector-valued functions are under our consideration. The method of our proofs is based on an application of the Banach contraction principle as well as the Leray-Schauder alternative for contractions.     

Existence and global attractivity of solutions of a nonlinear functional integral equation

In this paper we prove a result on the existence and global attractivity of solutions of a nonlinear functional integral equations with deviating arguments. The investigations are placed in the Banach space of real functions defined, continuous and bounded on the real half-axis. The main tool used in consideration is a fixed point theorem of Krasnosel’skii type. A few examples illustrating the obtained results are also included.     

The existence of positive solutions of the nonlinear neutral differential equations in a Banach space

1.IntroductionTheexistenceanduniquenessofsolutionsofdelaydifferentialequationsinaBanachspacehavebeenstudiedbymanyauthors(see[1--4]).Inthispaper,weshallapplythefixedpoillttheoremstoinvestigatetheexistenceofpositivesolutionsofnonlinearneutraldifferentialdifferenceequationsinaBanachspace.LetRbethefieldofrealnumbers,andletR ={tERIt20},to>0,J=[to, co)'ConsiderthenonlinearneutraldifferentialdifferenceequationinaBanachspaceE.mI(x(t)~Zci(t)x(t~ri))' ZPj(t)fj(x(t~D))=0,(1)i=1j=1wherefiEC[P,PI,f…     

Approximation of a bounded solution of a linear difference equation on Z2 by solutions of corresponding boundary-value problems in a Banach space

We study the problem of existence and uniqueness of solutions of boundary-value difference problems in a Banach space that correspond to a certain difference equation on Z². We prove a theorem on approximation of the unique bounded solution of the considered equation by solutions of the corresponding boundary-value problems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 535–541, April, 1995.     

增算子不动点定理及其对含间断项非线性脉冲方程的应用

设E是Banach空间,本文在比C[I,E]更一般的Banach空间中得到了几个新的非连续增算子不动点定理.作为应用,我们研究了Banach空间中含间断项的非线性脉冲积分方程和微分方程的最大解和最小解的存在性.     

Existence of mild solutions for semilinear differential equations with nonlocal and impulsive conditions

This paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure of noncompactness, and the linear part generates only a strongly continuous evolution system.     

On a linear nth-order differential equation with special right-hand side in a Banach space

We show that the classical algorithm for finding a particular solution of a scalar linear differential equation can be applied to linear differential equations with bounded operator coefficients in a Banach space.