VECTORIAL EKELAND'S VARIATIONAL PRINCIPLE WITH A W-DISTANCE AND ITS EQUIVALENT THEOREMS

By using the properties of w-distances and Gerstewitz＇s functions, we first give a vectorial Takahashi＇s nonconvex minimization theorem with a w-distance. From this, we deduce a general vectorial Ekeland＇s variational principle, where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value. From the general vectorial variational principle, we deduce a vectorial Caristfs fixed point theorem with a w-distance. Finally we show that the above three theorems are equivalent to each other. The related known results are generalized and improved. In particular, some conditions in the theorems of [Y. Araya, Ekeland＇s variational principle and its equivalent theorems in vector optimization, J. Math. Anal. Appl. 346（2008）, 9-16] are weakened or even completely relieved.

EKELAND'S PRINCIPLE FOR SET-VALUED VECTOR EQUILIBRIUM PROBLEMS

In this paper, we introduce a concept of quasi C-lower semicontinuity for setvalued mapping and provide a vector version of Ekeland＇s theorem related to set-valued vector equilibrium problems. As applications, we derive an existence theorem of weakly efficient solution for set-valued vector equilibrium problems without the assumption of convexity of the constraint set and the assumptions of convexity and monotonicity of the set-valued mapping. We also obtain an existence theorem of ε-approximate solution for set-valued vector equilibrium problems without the assumptions of compactness and convexity of the constraint set.

非对称度量空间中的Takahashi极小化定理

利用分析方法将Takahashi极小化定理推广到上完备的非对称度量空间这一情形;进一步,将所得结果应用到Caristi不动点定理和Ekelandε变分原理上.

Ekeland变分原理在正则性证明中的一个应用

应用Ekeland变分原理证明了一类具有临界Sobolev增长的积分泛函ω 极小的局部H lder连续性.

集值映射的Ekeland变分原理

引进了集值映射向量优化问题的εｅ－有效解，给出了这一解的存在性的一个充分条件．作为主要结果，我们得到了一个关于集值映射的Ｅｋｅｌａｎｄ变分原理．

集值映射型广义Ekeland变分原理

在序拓扑向量空间中定义了集合的非线性标量化函数,并证明了它的一些性质.利用集合的非线性标量化函数,给出了关于集值映射的广义Ekeland变分原理,此变分原理是向量形式广义Ekland变分原理的推广.

R^3上的一类非齐次Kirchhoff-Poisson系统的多重解

运用临界点理论中的Ekeland变分原理和山路定理以及一些分析技巧证明了一类非齐次Kirchhoff-Poisson系统在R3上的多重解是存在的.

Ekeland变分原理的一个注记及应用

给出Ｅｋｅｌａｎｄ变分原理的一个推论，并据此简单地证明几个熟知定理。

具有年龄结构的SIR传染病模型的最优接种和治疗策略

研究了一类染病者具有年龄结构的SIR传染病模型的最优接种和治疗策略问题.利用Banach压缩映射原理和Gronwall引理,证明了该传染病模型非负解的唯一性以及解对控制变量的连续依赖性.借助切锥法锥技巧给出最优接种和治疗策略的必要条件.根据Ekeland变分原理确立了最优接种和治疗策略的存在性和唯一性.

基于改进集的集值Ekeland变分原理

Ekeland变分原理在最优化理论及应用研究中具有十分重要的作用.利用非线性标量化函数及相应的非凸分离定理建立了基于改进集的集值Ekeland变分原理.新的Ekeland变分原理包含了一些经典的Ekeland变分原理作为其特例.

矩形b-度量空间的Ekeland变分原理

Ekeland变分原理在不动点理论中占据着重要地位,它在优化理论,控制理论、临界点理论等数学的许多领域都有应用,也是研究偏微分方程的有力工具。从二十世纪七十年代Ekeland提出了著名的Ekeland变分原理至今,国内外许多学者致力于Ekeland变分原理的研究,取得了许多重要的研究成果。本文主要结合矩形b-度量空间的特点和性质,给出矩形b-度量空间的Ekeland变分原理以及Caristi不动点定理。

P-Distances, Q-Distances and a Generalized Ekeland's Variational Principle in Uniform Spaces

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland＇s variational principle. In the framework of uniiorm spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance （particularly, a p-distance）, which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi＇s nonconvex minimization theorem, a generalized Ekeland＇s variational principle and a generalized Caristi＇s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland＇s variational principle, we deduce a number of particular versions of Ekeland＇s principle, which include many known versions of the principle and their improvements.

A General Vectorial Ekeland's Variational Principle with a P-distance

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle （in short EVP）, where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space （which is a uniform space） has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi＇s fixed point theorem and a general vectorial Takahashi＇s nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.

On Ha's Version of Set-valued Ekeland's Variational Principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187-206 （2005）] established a new version of Ekeland＇s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha＇s version of set-valued Ekeland＇s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha＇s version, we deduce a Caristi-Kirk＇s fixed point theorem and a Takahashi＇s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.

EKELAND'S VARIATIONAL PRINCIPLE AND CARISTI'S FIXED POINT THEOREM IN PROBABILISTIC METRIC SPACE

The main purpose of this paper is to establish the Ekeland’s variational principle andCaristi’s fixed point theorem in probabilistic metric spaces and to give a direct simple proofof the equivalence between these two theorems in the probabilistic metric space. The resultspresented in this paper generalize the corresponding results of [9--12].

Sequentially Lower Complete Spaces and Ekeland's Variational Principle

By using sequentially lower complete spaces（see [Zhu, J., Wei, L., Zhu, C. C.： Caristi type coincidence point theorem in topological spaces. J. Applied Math., 2013, ID 902692（2013）]）, we give a new version of vectorial Ekeland＇s variational principle. In the new version, the objective function is defined on a sequentially lower complete space and taking values in a quasi-ordered locally convex space, and the perturbation consists of a weakly countably compact set and a non-negative function p which only needs to satisfy p（x, y） = 0 iff x = y. Here, the function p need not satisfy the subadditivity.From the new Ekeland＇s principle, we deduce a vectorial Caristi＇s fixed point theorem and a vectorial Takahashi＇s non-convex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. By considering some particular cases, we obtain a number of corollaries,which include some interesting versions of fixed point theorem.

Ekeland’s Variational Principle and the Mountain Pass Lemma

Ekeland’s variational principle is a fundamental theorem in nonconves analysis. Its general statement is as the following:Ekeland’s Variational Principle"’a:. Let V be a complete metric space, and F: F—*-RU{ + °°} a lower semicontinuous function, not identically +00 and bounded from, below. Let s>0 be given, and a point u^V such thatF(u)<infF+e.vThen there exists some point v ￡ V such that

F-型拓扑空间中集值Caristi型定理

给出了F 型拓扑空间中集值Caristi型重合定理及加强形式的集值Caristi型不动点定理 .证明了在F 空间中Ekeland变分原理与加强形式的集值Caristi型不动点定理等价 .统一和推广了相关的结果 .

POSITIVE SOLUTIONS FOR CRITICAL QUASILINEAR ELLIPTIC EQUATIONS WITH MIXED DIRICHLET-NEUMANN BOUNDARY CONDITIONS

The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.

Study of optimal control problems for hybrid dynamical systems

From the viewpoint of continuous systems, optimal control problem is proposed for a class of controlled Hybrid dynamical systems. Then a mathematical method- HDS minimum principle is put forward, which can solve the above problem. The HDS minimum principle is proved by means of Ekeland＇ s variational principle.