Error bounds for systems of lower semicontinuous functions in Asplund spaces

In this paper, using the Fréchet subdifferential, we derive several sufficient conditions ensuring an error bound for inequality systems in Asplund spaces. As an application we obtain in the context of Banach spaces a global error bound for quadratic nonconvex inequalities and we derive necessary optimality conditions for optimization problems.     

Conditions for the solvability of boundary value problems for quasi-elliptic systems in a half-space

We consider boundary value problems in a half-space for a class of quasi-elliptic systems with constant coefficients. We assume that the boundary value problems satisfy the Lopatinskii condition. We obtain necessary and sufficient conditions for their unique solvability in Sobolev spaces.     

Positive Solutions in the Space of Lipschitz Functions for Fractional Boundary Value Problems with Integral Boundary Conditions

In this paper, we study the existence of positive solutions for a class of nonlinear fractional boundary value problems with integral boundary conditions in Hölder spaces. Our analysis relies on a sufficient condition for the relative compactness in Hölder spaces and the classical Schauder fixed point theorem.     

The Fermat rule for multifunctions for super efficiency

Using variational analysis, we study the vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In terms of the coderivatives and normal cones, we present Fermat’s rules as necessary or sufficient conditions for a super efficient solution of the above problems.     

非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结...     

Extremum principles for optimal control in normed linear spaces

Extremum principles intended for use in optimal control are derived in the form of necessary conditions and sufficient conditions, formulated in general normed linear spaces. The method of application is illustrated by several examples involving optimal control problems, mathematical programming problems, lumped-parameter systems, and distributed-parameter systems. The basic theorems provide a unified approach which is applicable to a wide variety of problems in open-loop optimal control.     

Conditions for the embedding of classes of sequences and their consequences

We obtain necessary and sufficient conditions for the embedding of classes, which consist of sequences and are generated by certain functions. We use these conditions to study a number of problems in the theory of the coordinate spaces that are generalizations of Orlicz spaces, and in the theory of the superposition operator acting in these spaces.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 681–692, November, 1976.     

The antipodal mapping theorem and difference equations in Banach spaces†

We employ the Borsuk–Krasnoselskii antipodal theorem to prove a new fixed point theorem in ordered Banach spaces. Then, the applicability of the result is shown by presenting sufficient conditions for the existence of solutions to initial value problems for first-order difference equations in Banach spaces. To prove that result we shall employ set valued analysis techniques.     

General Regularizing Functionals for Solving Ill-Posed Problems in Lebesgue Spaces

We study sufficient conditions for general integral functionals in Lebesgue spaces to possess regularizing properties required for solving nonlinear ill-posed problems. We select special classes of such functionals: uniformly convex and quasiuniformly convex (in the extended sense). We give a series of examples and, in particular, a functional that can be used in a generalized version of the maximum entropy method in Lebesgue spaces.     

Solvability of Elliptic Differential Equations,Set in Three Habitats with Skewness Boundary Conditions at the Interfaces

In this work, we study an elliptic differential equation set in three habitats with skewness boundary conditions at the interfaces. It represents the linear stationary case of dispersal problems of population dynamics which incorporate responses at interfaces between the habitats. Existence, uniqueness and regularity of the solution of these problems are obtained in Hölder spaces under necessary and sufficient conditions on the data. Our techniques are based on the semigroup theory, the fractional powers of linear operators, the \(H^{\infty }\) functional calculus for sectorial operators in Banach spaces and some properties of real interpolation spaces.     

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON TRIEBEL SPACES

1IntroductionIn a series of recent publications operator-valued Fourier multipliers on vector-valued func-tion spaces were studied(see e.g.[1,2,3,5,6,7,14,16]).They are needed to establish existence anduniqueness as well as regularity of di?erential equat…     

Completeness of the Root Function System of a Nonlocal Problem in Lp

We obtain sufficient conditions for the completeness of the root function system for nonlocal elliptic problems in the Sobolev spaces W_p ^l(Q).     

Nonlinear optimization and approximation

This paper is concerned with nonlinear optimization problems in normed linear spaces. Necessary and sufficient conditions for optimal points are given and the range of applicability of these conditions is studied.The results are applied to nonlinear approximation problems.     

非凸分式优化问题的最优性充分条件和对偶

通过引入广义弧连通概念,在Rn空间中,研究极大极小非凸分式规划问题的最优性充分条件及其对偶问题.首先获得了极大极小非凸分式规划问题的最优性充分条件;然后建立分式规划问题的一个对偶模型并得到了弱对偶定理,强对偶定理和逆对偶定理.     

Nonlinear stability behaviour of linear multistep methods

For linear multistep methods sufficient conditions are derived such that the numerical solutions of stable nonlinear initial value problems in Banach spaces are also stable.     

Banach-Saks property in Marcinkiewicz spaces

We catalogue all Marcinkiewicz function and sequence spaces with the Banach-Saks property and present necessary and sufficient conditions for a wide subclass of spaces to possess the p-Banach-Saks property, 1<p<∞. We apply our results to several open problems.     

The Lagrange multiplier rule for super efficiency in vector optimization

In this paper, we study constrained multiobjective optimization problems with objectives being closed-graph multifunctions in Banach spaces. In terms of the coderivatives and Clarke's normal cones, we establish Lagrange multiplier rules for super efficiency as necessary or sufficient optimality conditions of the above problems.     

Boundary value problems for quasielliptic systems

We consider the boundary value problems in the half-space for a class of quasielliptic systems with variable coefficients. Assuming that the boundary value problems satisfy the Lopatinski? condition, we establish sufficient conditions for unique solvability in Sobolev spaces.     

关于一类向量极值问题弱有效解的充分必要条件

本文研究带集合约束的向量极值问题。运用局部凸Hausdorff拓扑向量空间中广义次似凸映射的择一定理和其他一些结论，得到了关于集合约束向量极值问题弱有效解的几个充分必要条件．     

Optimality conditions for vector equilibrium problems with constraint in Banach spaces

In this paper, vector equilibrium problems with constraint in Banach spaces are investigated. Kuhn–Tucker-like conditions for weakly efficient solutions are given by using the Gerstewitz’s function and nonsmooth analysis. Moreover, the sufficient conditions of weakly efficient solutions are established under the assumption of generalized invexity. As applications, necessary conditions of weakly efficient solutions for vector variational inequalities with constraint and vector optimization problems with constraint are obtained.