§1 Introduction and preliminariesA set T Rn×Rnis called a monotone operator on Rn,if T has the property(x,y) ,(x′,y′)∈T 〈x -x′,y -y′〉≥0 ,where〈·,·〉denotes the inner product on Rn.T is maximal if(considered as a graph) itis not strictly contained in any other monotone operator on Rn.It is well known that thetheory of maximal monotone operators plays an important role in the study of convexprogramming and variational inequalities since itcan provide a powerful general framework… 相似文献
It will be shown that a normed partially ordered vector space is linearly, norm, and order isomorphic to a subspace of a normed Riesz space if and only if its positive cone is closed and its norm p satisfies p(x)p(y) for all x and y with -yxy. A similar characterization of the subspaces of M-normed Riesz spaces is given. With aid of the first characterization, Krein's lemma on directedness of norm dual spaces can be directly derived from the result for normed Riesz spaces. Further properties of the norms ensuing from the characterization theorem are investigated. Also a generalization of the notion of Riesz norm is studied as an analogue of the r-norm from the theory of spaces of operators. Both classes of norms are used to extend results on spaces of operators between normed Riesz spaces to a setting with partially ordered vector spaces. Finally, a partial characterization of the subspaces of Riesz spaces with Riesz seminorms is given. 相似文献
We consider continuous-time hysteresis operators, defined to be causal and rate independent operators mapping input signals to output signals . We show how a hysteresis operator defined on the set of continuous piecewise monotone functions can be naturally extended to piecewise continuous piecewise monotone functions. We prove that the extension is also a hysteresis operator and that a number of important properties of the original operator are inherited by the extension. Moreover, we define the concept of a discrete-time hysteresis operator and we show that discretizing continuous-time hysteresis operators using standard sampling and hold operations leads to discrete-time hysteresis operators. We apply the concepts and results described above in the context of sampled-data feedback control of linear systems with input hysteresis. 相似文献
Recently, the authors studied the connection between each maximal monotone operator and a family of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities.
The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator.
In this paper we study the question ``When does a perfect generalized ordered space have a -closed-discrete dense subset?' and we characterize such spaces in terms of their subspace structure, -mappings to metric spaces, and special open covers. We also give a metrization theorem for generalized ordered spaces that have a -closed-discrete dense set and a weak monotone ortho-base. That metrization theorem cannot be proved in ZFC for perfect GO-spaces because if there is a Souslin line, then there is a non-metrizable, perfect, linearly ordered topological space that has a weak monotone ortho-base.
We consider Dirichlet–type problems for weakly connected systems of elliptic equations of second order with a strong degeneracy at an inner point of the domain, when, in a neighborhood of this point, the asymptotics of a solution is additionally given. The form of the given asymptotics essentially depends on the properties of the coefficients at the first–order derivatives. We prove the existence and uniqueness of solutions of the problems considered in Hölder function classes. 相似文献
We consider a Dirichlettype problem for a system of elliptic equations of second order with a strong degeneracy at an inner point of the domain, when, in a neighborhood of this point, the principal term of the asymptotics of a solution is additionally given. We prove the existence and uniqueness of a solution of the problem considered in a weighted class of Hölder vector functions. 相似文献
Let be a real Banach space partially ordered by a closed convex cone with nonempty interior . We study the continuation method for the monotone operator which satisfies
for all , , where . Thompson's metric is among the main tools we are using. 相似文献
The method of quasilinearization is a procedure for obtaining approximate solutions of differential equations. In this paper, this technique is applied to a differential-algebraic problem. Under some natural assumptions, monotone sequences converge quadratically to a unique solution of our problem. 相似文献