Variational Analysis of Directional Minimal Time Functions and Applications to Location Problems

This paper is devoted to the study of directional minimal time functions that specify the minimal time for a vector to reach an object following its given direction. We provide a careful analysis of general and generalized differentiation properties of this class of functions. The analysis allows us to study a new model of facility location that involves sets.     

Time-dependent Variational Inequalities and Applications to Equilibrium Problems

We present two different applications of time-dependent variational inequalities. First we propose a new model of time-dependent distributed markets networks which includes delay effects. Afterwards, we deal with time-dependent and elastic models of transportation networks.     

Signed Fuzzy-Valued Measures and Radon-Nikodym Theorem of Fuzzy-Valued Measurable Functions

In this paper, we introduce and study the signed fuzzy-valued measure and using the representation theorem of fuzzy numbers, we establish the Radon-Nikodym theorem for fuzzy-valued measurable functions with respect to fuzzy-valued measures.AMS Subject Classification (2000): 28E10, 04A72The research of this work is supposed by a grant of the education committee of Liaoning Province, China #20161049     

Hessian Measures of Convex Functions and Applications to Area Measures

The Hessian measures of a (semi-)convex function can be introducedas coefficients of a local Steiner formula. The investigationof Hessian measures is continued by the provision of a geometriccharacterization of the support of these measures. Then theRadon–Nikodym derivative and the absolute continuity ofHessian measures with respect to Lebesgue measure are explored.As special cases of the results, known results for surface areameasures of convex bodies are recovered.     

Topics on Variational Analysis and Applications to Equilibrium Problems

We observe how many equilibrium problems obey a generalized complementarity condition, which in general leads to a variational inequality. We illustrate this fact, by studying the elastic–plastic torsion problem and finding the related Lagrange multipliers.     

拟凸函数的近似次微分及其在多目标优化问题中的应用

针对拟凸函数提出一类新的近似次微分,研究其性质,并将近似次微分应用到拟凸多目标优化问题近似解的刻画中.首先,对已有的近似次微分进行改进,得到拟凸函数新的近似次微分,并给出其与已有次微分之间的关系及一系列性质.随后,利用新的近似次微分给出拟凸多目标优化问题近似有效解、近似真有效解的最优性条件.     

Variational Analysis of Marginal Functions with Applications to Bilevel Programming

This paper pursues a twofold goal. First goal is to derive new results on generalized differentiation in variational analysis focusing mainly on a broad class of intrinsically nondifferentiable marginal/value functions. Then the results established in this direction are applied to deriving necessary optimality conditions for the optimistic version of bilevel programs, which occupy a remarkable place in optimization theory and its various applications. We obtain new sets of optimality conditions in both smooth and nonsmooth settings of finite-dimensional and infinite-dimensional spaces.     

广义有界变差函数的发展与应用

本文讨论下列问题：１．１９７２年至１９９２年间广义有界变美函数的定义，性质包括函数类东省ΛＢＶ，ΦＢＶ，ΛＢＭＶ，ΦΛＢＭＶ，φΛＢＶ，φΛＢＭＶ等等．２．广义有界变差函数在Ｆｏｕｒｉｃｒ分析及逼近论中的应用．     

KKM Theorems on FC-spaces with Their Applications to Variational Inequalities

A new concept of finitely continuous topological spaces （in short, FC-space） without convexity structure and linear structure was introduced. Some KKM type theorems in noncompact FC-spaces were obtained, and from which, some section theorems and wriational inequality theorems were proved under much weak assumptions. Our results improve and generalize the corresponding conclusions in recent literature.     

Global $$L^q$$-Gevrey Functions and Their Applications

The goal of this paper is to introduce a class of \(C^\infty \) functions derivatives of which satisfy quantitative size estimates. The estimates, called global \(L^q\) Gevrey estimates, first arose in the work of Boggess and Raich ( J. Fourier Anal. Appl. 19:180–224, 2013) when they investigated how to capture a particular type of exponential decay through estimates on the Fourier transform. In the present work, we refine the notion of global \(L^q\)-Gevrey functions and include a discussion of the function theory as well as the relationship to Gevrey classes and known function spaces. In addition, we present explicit examples of global \(L^q\)-Gevrey functions and ways to generate new global \(L^q\)-Gevrey functions from old ones. We conclude with three applications: The first is solving a Carleman-type problem for constructing functions derivatives of which are a given sequence of global \(L^q\)-Gevrey functions. The other two applications concern extensions of a given global \(L^q\)-Gevrey function: the first is constructing an almost analytic extension, and the second is building an approximate solution to a first-order complex vector field coefficients of which are global \(L^q\)-Gevrey functions.     

Shifted Hermite-Biehler Functions and Their Applications

We investigate a particular subclass of so-called symmetric indefinite Hermite-Biehler functions and give a characterization of functions of this class in terms of the location of their zeros. For the proof we employ the theory of de Branges Pontryagin spaces of entire functions. We apply our results to obtain information on the eigenvalues of some boundary value problems.     

Further on Set-Valued Equilibrium Problems and Applications to Browder Variational Inclusions

In this paper, we introduce some concepts of convexity and semicontinuity for real set-valued mappings similar to those of real single-valued mappings. Then, we obtain different results on the existence of solutions of set-valued equilibrium problems generalizing in a common way several old ones for both single-valued and set-valued equilibrium problems. Applications to Browder variational inclusions, with weakened conditions on the involved set-valued operator, are given.     

Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities

This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle.     

An Extension of Pseudolinear Functions and Variational Inequality Problems

In optimization, objective functions which are both pseudoconvex and pseudoconcave have been studied extensively. Generalizing these results, we characterize pseudomonotone maps F where -F is also pseudomonotone and explore their properties in variational inequality problems. In particular, we extend recent results by Jeyakumar and Yang which were derived for optimization problems.     

D-Gap Functions for Nonsmooth Variational Inequality Problems

We study the Clarke generalized gradient of the D-gap functions for the variational inequality problem (VIP) defined by a locally Lipschitz, but not necessarily differentiable, function in an Euclidean space. Using these results, we study the relationship between minimizing sequences and stationary sequences of the D-gap function, regardless of the existence of solutions of (VIP).     

F值函数的微分及其在FNN中的应用

本文引进了集值函数的ｓ－可微和模糊值（Ｆ值）函数的Ｆｓ－可微概念。给出了这两种可微性的几个判别条件。最后研究并得到了一类模糊神经网络（ＦＮＮ）的Ｆｓ－可微性和连续性。     

三步投影算法的收敛性及其在变分不等式组中的应用

First a general model for a three-step projection method is introduced, and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let H be a real Hilbert space and K be a nonempty closed convex subset of H. For arbitrarily chosen initial points x0, y0, z0 ∈ K,compute sequences xn, yxn, zxn such that { xn+1=(1-αn-rn)xn+αxPk[yn-ρTyn]+rnun,yn=(1-β-δn)xn+βnPk[zn-ηTxn]+δnun,zn=(1-an-λn)xn+akPk[xn-γTxn]+λnwn.For η, ρ,γ>0 are constants,{αn}, {βn}, {an}, {rn}, {δn}, {λn} C [0,1], {un}, {vn}, {wn} are sequences in K, and 0≤n + rn ≤ 1,0 ≤βn + δn ≤ 1,0 ≤ an + λn ≤ 1,(A)n ≥ 0, where T : K → H is a nonlinear mapping onto K. At last three-step models are applied to some variational inequality problems.