LITTLEWOOD-PALEY THEOREM FOR SCHR(O)DINGER OPERATORS

Let H be a Schr(o)dinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

Extremal Solutions for Quasilinear Elliptic Inclusions in All of ℝN with State-Dependent Subdifferentials

This paper deals with quasilinear elliptic differential inclusions defined in all of ^N and governed in general by a nonpotential quasilinear elliptic operator of the Leray–Lions type and a multivalued term in form of a (nonmonotone) state-dependent subdifferential. We prove the existence of entire extremal solutions within a sector of an ordered pair of appropriately defined upper and lower solutions without imposing any condition at infinity. Therefore, standard variational methods cannot be applied here. Furthermore, due to the unboundedness of the domain and due to lack of monotonicity of the operators involved, no comparison results are available such that the problem under consideration becomes even more difficult.     

Estimation of systems with statistically-constrained inputs

This paper discusses the estimation of a class of discrete-time linear stochastic systems with statistically-constrained unknown inputs (UI), which can represent an arbitrary combination of a class of un-modeled dynamics, random UI with unknown covariance matrix and deterministic UI. In filter design, an upper bound filter is explored to compute, recursively and adaptively, the upper bounds of covariance matrices of the state prediction error, innovation and state estimate error. Furthermore, the minimum upper bound filter (MUBF) is obtained via online scalar parameter convex optimization in pursuit of the minimum upper bounds. Two examples, a system with multiple piecewise UIs and a continuous stirred tank reactor (CSTR), are used to illustrate the proposed MUBF scheme and verify its performance.     

Block monotone iterative algorithms for variational inequalities with nonlinear operators

Some block iterative methods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established. Compared with the research work in given by Pao in 1995 for nonlinear equations and research work in given by Zeng and Zhou in 2002 for elliptic variational inequalities, the algorithms proposed in this paper are independent of the boundedness of the derivatives of the nonlinear operator.     

Existence, upper and lower solutions and quasilinearization for singular differential equations

In this paper, we discuss existence theorems in the presenceof upper and lower solutions as well as the method of quasilinearization(QSL) for general non-linear second-order singular ordinarydifferential equations. We show the existence of solutions underthe assumption of weak continuity of the non-linear part. Ifthe non-linear part is monotone decreasing, a solution may beobtained by the QSL method as the strong limit of a quadraticallyconvergent sequence of approximate solutions. Under strongerassumptions on the linear and the non-linear parts, a solutionis quadratically bracketed between two monotone sequences ofapproximate solutions of certain related linear equations.     

关于积分上限函数的几个定理

变上限积分是积分学的一个重要理论，其运算结果仍以函数的形式体现．研究这类函数，得出几个颇有理论意义的定理。     

Cubic and Quartic Convergence for First-Order Periodic Boundary-Value Problems

In this paper, the results of Lakshmikantham et al. (Ref. 1) for first-order periodic boundary-value problems are extended, by using the extended method of quaislinearization and rapid convergence for initial-value problems of Mohapatra et al. (Ref. 2). Also, it is shown that monotone sequences converge cubically to the unique solution when the forcing function in the differential equation is 2–hyperconvex and converge quartically when the forcing function is 3–hyperconvex. Several other generalizations of the problem are also presented.     

关于对数平均的上界和下界

利用广义对数平均，给出了对数平均的上界和下界控制     

非空交定理及其应用

本文得到了广义区间空间中几个参数型非空交定理，并用此结果证明了一些新型极大极小定理，截口定理，重合定理，以及变分不等式解的存在性定理。本文中的结果包含（１，３，５－１２，１４－１６）中相应结果。     

Convergence of splitting and Newton methods for complementarity problems: An application of some sensitivity results

This paper is concerned with two well-known families of iterative methods for solving the linear and nonlinear complementarity problems. For the linear complementarity problem, we consider the class of matrix splitting methods and establish, under a finiteness assumption on the number of solutions, a necessary and sufficient condition for the convergence of the sequence of iterates produced. A rate of convergence result for this class of methods is also derived under a stability assumption on the limit solution. For the nonlinear complementarity problem, we establish the convergence of the Newton method under the assumption of a pseudo-regular solution which generalizes Robinson's concept of a strongly regular solution. In both instances, the convergence proofs rely on a common sensitivity result of the linear complementarity problem under perturbation.This work was based on research supported by the National Science Foundation under grant ECS-8717968.