On the stability of abstract monotone impulsive differential equations in terms of two measures

We consider differential equations in a Banach space subjected to an impulsive influence at fixed times. It is assumed that a partial ordering is introduced in the Banach space by using a normal cone and that the differential equations are monotone with respect to the initial data. We propose a new approach to the construction of comparison systems in finite-dimensional spaces without using auxiliary Lyapunov-type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures. In this case, a Birkhoff measure is chosen as the measure of initial displacements, and the norm in the given Banach space is used as the measure of current displacements. We present some examples of investigations of the impulsive systems of differential equations in the critical cases and linear impulsive systems of partial differential equations.     

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 solutions and periodic solutions for nonlinear evolution inclusions

In this paper we consider nonlinear-dependent systems with multivalued perturbations in the framework of an evolution triple of spaces. First we prove a surjectivity result for generalized pseudomonotone operators and then we establish two existence theorems: the first for a periodic problem and the second for a Cauchy problem. As applications we work out in detail a periodic nonlinear parabolic partial differential equation and an optimal control problem for a system driven by a nonlinear parabolic equation.     

Banach空间中脉冲积分-微分方程的初值问题

给出了 Banach空间中一阶线性脉冲积分 -微分方程初值问题解的存在唯一性的一个新证法 ,改进了已有结果 .利用它讨论了一阶非线性脉冲积分 -微分方程初值问题的解 ,所得结果大大推广了已有的相关结果 .     

Adjoint for operators in Banach spaces

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.

     

Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator

The paper deals with second order nonlinear evolution inclusions and their applications. We study evolution inclusions involving a Volterra-type integral operator, which are considered within the framework of an evolution triple of spaces. First, we deliver a result on the unique solvability of the Cauchy problem for the inclusion by combining a surjectivity result for multivalued pseudomonotone operators and the Banach contraction principle. Next, we provide a theorem on the continuous dependence of the solution to the inclusion with respect to the operators involved in the problem. Finally, we consider a dynamic frictional contact problem of viscoelasticity for materials with long memory and indicate how the result on evolution inclusion is applicable to the model of the contact problem.     

Product formulas,nonlinear semigroups,and accretive operators

A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency is equivalent to convergence for nonlinear contractive algorithms. (The linear case is due to Chernoff.) The proof uses ideas of Crandall, Liggett, and Baillon. Other applications of our theorem include results concerning the generation of nonlinear semigroups (e.g., a nonlinear Hille-Yosida theorem for “nice” Banach spaces that includes the familiar Hilbert space result), the geometry of Banach spaces, extensions of accretive operators, invariance criteria, and the asymptotic behavior of nonlinear semigroups and resolvents. The equivalence between resolvent consistency and convergence for nonlinear contractive algorithms seems to be new even in Hilbert space. Our nonlinear Hille-Yosida theorem is the first of its kind outside Hilbert space. It establishes a biunique correspondence between m-accretive operators and semigroups on nonexpansive retracts of “nice” Banach spaces and provides affirmative answers to two questions of Kato.     

Controllability Results for Second Order Neutral Impulsive Integrodifferential Systems

In this paper, we address the problem of controllability for abstract second order neutral integrodifferential systems with impulsive conditions in Banach spaces. Furthermore, we discuss the controllability of second-order neutral integrodifferential systems in finite dimensional spaces. The results are obtained by using strongly continuous cosine families of operators and the Banach fixed point theorem.     

Impulsive nonlocal differential equations through differential equations on time scales

We propose a non-standard approach to impulsive differential equations in Banach spaces by embedding this type of problems into differential (dynamic) problems on time scales. We give an existence result for dynamic equations and, as a consequence, we obtain an existence result for impulsive differential equations.     

Banach空间中非线性脉冲Volterra型积分方程组的可解性

在较弱的条件下，利用锥理论和单调迭代方法首先建立了Banach空间中一类非线性算子方程组最小最大解的存在性定理；然后作为应用，利用一个新的比较结果，得到了Banach空间中非线性脉冲Volterra型积分方程组的整体解，改进了最近的许多结果．     

Existence of nonzero positive solutions of systems of second order

Existence of nonzero positive solutions  of systems of second order elliptic boundary value problems under sublinear conditions is obtained. The methodology is to establish a new result on existence of nonzero positive solutions of a fixed point equation in real Banach spaces by using the well-known theory of fixed point index for compact maps defined on cones, where the fixed point equation involves composition of a compact linear operator and a continuous nonlinear map. The conditions imposed on the nonlinear maps involve the spectral radii of the compact linear operators. Moreover, the nonlinear maps are not required to be increasing in  ordered Banach spaces.     

Banach空间非线性脉冲Volterra型积分方程整体解的存在性定理及应用

利用一个新的比较结果和Ｍｏｎｅｃｈ不动点定理证明了Ｂａｎａｃｈ空间中非线性脉冲Ｖｏｌｔｅｒｒａ型积分方程整体解的存在性定理,并给出了对Ｂａｎａｃｈ空间中一阶脉冲微分方程初值问题的应用,改进了文（１－３）中的主要结果。     

非线性中立发展积微分系统解的存在性和可控性(英文)

In this paper, we establish sufficient conditions for existence and controllability of nonlinear neutral evolution integroditferential systems in Banach spaces. The result is obtained by using the resolvent operators and fixed point analysis approach.     

Controllability of Damped Second-Order Impulsive Neutral Functional Differential Systems with Infinite Delay

In this paper, the controllability problem is discussed for the damped second-order impulsive neutral functional differential systems with infinite delay in Banach spaces. Sufficient conditions for controllability results are derived by means of the Sadovskii fixed point theorem combined with a noncompact condition on the cosine family of operators. An example is provided to illustrate the theory.     

On interpolation of bilinear operators

In this paper we study interpolation of bilinear operators between products of Banach spaces generated by abstract methods of interpolation in the sense of Aronszajn and Gagliardo. A variant of bilinear interpolation theorem is proved for bilinear operators from corresponding weighted c₀ spaces into Banach spaces of non-trivial the periodic Fourier cotype. This result is then extended to the spaces generated by the well-known minimal and maximal methods of interpolation determined by quasi-concave functions. In the case when a maximal construction is generated by Hilbert spaces, we obtain a general variant of bilinear interpolation theorem. Combining this result with the abstract Grothendieck theorem of Pisier yields further results. The results are applied in deriving a bilinear interpolation theorem for Calderón-Lozanovsky, for Orlicz spaces and an embedding interpolation formula for (E,p)-summing operators.     

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.     

Banach空间非线性脉冲Volterra积分方程组的整体解

研究Banach空间中定义在无穷区间R+上具有无穷多个脉冲点的非线性脉冲Volterra积分方程组解的存在性。给出了若干极值解的存在定理,改进了定义在有限区间上具有有限个脉冲点情形时该类方程的相应结果,并利用该结果讨论了一个无穷维积分方程组。     

Banach空间一阶脉冲微分积分方程组初值问题的解

研究了Banach空间非线性一阶脉冲微分积分方程组初值问题解的存在性，改进并推广了近期的一些结果．     

Spaces of integrable functions with respect to a vector measure and factorizations through and Hilbert spaces

We use the integration structure of the spaces of scalar integrable functions with respect to a vector measure to provide factorization theorems for operators between Banach function spaces through Hilbert spaces. A broad class of Banach function spaces can be represented as spaces of scalar integrable functions with respect to a vector measure, but this representation (the vector measure) is not unique. Since our factorization depends on the vector measure that is used for the representation we also give a characterization of those vector measures whose corresponding spaces of integrable functions coincide.