The classical bipolar theorem plays an important role in functional analysis. This paper generalizes this theorem to fuzzy quasi-normed spaces, which include asymmetric normed space and fuzzy normed space as special cases. First, the concept of the asymmetric polar of a subset is introduced in the fuzzy quasi-normed space, its basic properties, such as closedness and compactness, are investigated. After the notion of asymmetric bipolar being proposed, the bipolar theorem is established. Additionally, some conclusions are presented based on the bipolar theorem. For example, a necessary and sufficient condition for the linear hull of a subset to be dense is given, a representation of the gauge of a subset is presented, and a characteristic of the family of equicontinuous linear functional is proved. These results generalize the existing results.

Motivated by some deep problems in optimization and control theory, convexity theory has been extended to the various infinite dimensional functional spaces. The separation theorems play key roles in developing convexity theory. In this paper, we focus on the convexity problem in the framework of fuzzy quasi-normed spaces. First, we give some separation results of convex sets, and show a characterization of a closed convex subset with the aid of the dual of a fuzzy quasi-normed space. Second, we introduce the concept of an extreme point and generalize the famous Krein–Milman theorem to a fuzzy quasi-normed space. The obtained results of this paper will play key roles in convex programming and optimization problems of fuzzy quasi-normed spaces.

Motivated by the fact that the fuzzy quasi-normed space provides a suitable framework for complexity analysis and has important roles in discussing some questions in theoretical computer science, this paper aims to study the nearest point problems in fuzzy quasi-normed spaces. First, by using the theory of dual space and the separation theorem of convex sets, the properties of the fuzzy distance from a point to a set in a fuzzy quasi-normed space are studied comprehensively. Second, more properties of the nearest point are given, and the existence, uniqueness, characterizations, and qualitative properties of the nearest points are obtained. The results obtained in this paper are of great significance for expanding the application fields of optimization theory.

The hyperplane theory plays a very important role in the research of optimization, it can help us better understand and solve various optimization problems. Therefore, the development of hyperplane theory has always been concerned by the scholars. This paper mainly studies the quasi-support hyperplane, which is the generalization of a support hyperplane, on the asymmetric normed space. First, the properties of the distance from a point to a non-empty set are studied more comprehensively, the isometric property of the distance from a point to a half space and to the corresponding hyperplane is proved. Second, the problem of supremum (infimum, resp.) of a linear function on a set is reduced to the problem of upper quasi-support (lower quasi-supports, resp.) hyperplane of the set. Additionally, some equivalent conditions for the quasi-support hyperplane are given by using the theory of dual space of asymmetric normed spaces. The obtained results enrich the functional analysis in asymmetric normed spaces and have great significance to applications in optimization theory, approximation problems, complexity analysis and the other fields.

In the study of the classical approximation and optimization problems, the dual method is a powerful tool. The present paper initiates the problems of the duality for best approximation in fuzzy quasi-normed spaces. It gives the characterization of the nearest points, and extends the well-known Arzela formula for the distance from a point to a hyperplane in a normed space to the case of a fuzzy quasi-normed space. The obtained notions and results show the validity of dual method in the study of best approximation for fuzzy quasi-normed spaces, and will play an important role in the research of this domain.

Since the notion of a fuzzy quasi-normed space has important applications in constructing suitable mathematical models in theoretical computer science, the study of fuzzy quasi-normed spaces has received a lot of attention in the last years. In this paper, the dual space of a fuzzy quasi-normed space with the general continuous t -norm is studied; some fundamental properties are presented, such as the bicompleteness. After introducing the concept of weak ^* topology, the Alaoglu–Bourbaki theorem is generalized to the fuzzy quasi-normed spaces.

大多数普通人是无法像数学家一样思考的.德国克里斯蒂安·黑塞在《像数学家一样思考》一书中认为普通人能像"数学家一样思考",但是从数学教育的角度来讲,数学是冰冷的美丽;从数学家的角度来说,数学家是赌徒;从思维方式的角度来说,数学家的思维方式是无定式.另外,数学家的品质比其思考方式更为重要.大多数大学生或数学工作者选择数学作为自己的专业也未必情愿."像数学家一样思考"反映的是一种精英的教育主义观念.

In this paper, firstly, we introduce the concepts of continuity and boundedness of linear operators between two fuzzy quasi-normed spaces with general continuous t-norms, prove the equivalence of them, and point out that the set of all continuous linear operators forms a convex cone. Secondly, we establish the family of star quasi-seminorms on the cone of continuous linear operators, and construct a fuzzy quasi-norm of a continuous linear operator.

In this paper, two cubic functional equations are shown to be equivalent, Hyers-Ulam-Rassias stability of them is proved under some suitable conditions by the fixed point method in fuzzy normed spaces. Moreover, the fuzzy continuity of the solution of the functional equation is discussed.

Fuzzy quasi-normed space provides an ideal mathematical framework for studying asymmetric phenomena. In this paper, we prove a version of the Ekeland variational principle in fuzzy quasi-normed spaces and apply it to Caristi???s fixed point theorem and Takahashi minimization theorem. Moreover, we prove the equivalence relations among these theorems.

LFSTM为中学数学解题等价辅助题目链的产生提供了方向,但是当等价辅助题目链应用LFSTM无法产生或是产生多个分支时,就会给解题带来困惑.通过对思维主体、思维客体和思维的社会历史性分析,确定了一种运用LFSTM将等价辅助题目链由原题目等价变换成"已知或是明显可以解答的题目"所采取的行动,即基于LFSTM的中学数学解题靶向.

In this article, we study some topological properties of fuzzy quasi-normed spaces, and prove the open mapping and the closed graph theorems in the framework of fuzzy quasi-normed spaces. The obtained results are of important roles in developing fuzzy functional analysis and its applications.

In this paper, the Hyers–Ulam–Rassias stabilities of two functional equations, fax+by=rfx+sfy and fx+y+z=2fx+y/2+fz, are investigated in the framework of fuzzy normed spaces.

In this paper, the fuzzy soft net is explored to study the properties of a fuzzy soft topological space. Some important results about the closure, separation, and compactness are obtained. It is demonstrated that the net is a powerful tool for studying fuzzy soft topological spaces.

在复杂性理论中,复杂性分析主要是针对算法的效率进行分析.通常地,在复杂性分析中更多的是研究算法的渐近效率.近年来,拟度量在复杂性分析中的应用受到学者们的广泛研究,但是它也存在着一定的局限性,例如拟度量并不适合刻画算法渐近效率的高低.为了解决这个问题,本文引入了复杂性函数集上的模糊拟度量,并以此刻画了算法渐近效率的高低.同时,通过研究它的基本性质,建立了一个不动点定理,并应用该不动点定理研究了与分治算法相关的递归方程的解的存在性和唯一性,以及与快速排序算法相关的递归方程的解的存在性和唯一性.以上结果构建起了模糊拟度量和算法的渐近效率之间的联系,为模糊拟度量在算法应用方面的进一步研究提供了一种新的有效途径.

In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasi-normed spaces.

由于模糊度量在彩色图像滤波等方面的成功应用,近年来该领域的研究引起了人们的重视.在将经典度量空间中的重要结论推广到模糊度量空间中的同时,研究方法上的创新显得特别重要.其中,将模糊度量分解为一族经典度量,建立模糊度量的分解定理无疑是十分有意义的.已有的分解定理主要是针对取小算子的模糊度量展开的,在应用上具有很大的局限性.本文引入了星伪度量族的概念作为对伪度量族概念的推广,利用这一概念,建立了针对取一般连续t-模的模糊度量的分解定理.同时,本文给出了模糊度量空间与伪度量族空间等距同构的充分条件和必要条件,由此建构起模糊度量与伪度量族之间的联系,为一般意义下的模糊度量的研究提供了一种新的有效途径.

In this paper, we introduce the convergence of a fuzzy soft filter with the help of the Q-neighborhoods and study the relations between fuzzy soft nets and fuzzy soft filters. In addition, we use fuzzy soft filters to characterize some basic concepts of a fuzzy soft topological space, such as open sets, closure, T2 separation and continuity.

In this paper, a partial order and its basic properties are exploited in a fuzzy quasi-metric space in the sense of George and Veeramani (1994) [17]. Based on this partial order, Caristi's fixed point theorem, Ekeland's variational principle and Takahashi's maximization theorem are extended to fuzzy quasi-metric spaces by using the Brézis and Browder principle on ordered sets. Moreover, an equivalence chain among these theorems is provided.

This paper deals with fuzzy quasinormed spaces in the sense of Alegre and Romaguera. After introducing the concept of the family of star quasiseminorms, we prove the decomposition theorem for a fuzzy quasinorm with general t-norm, characterize fuzzy quasinorms in terms of families of star quasiseminorms, and establish the connection between the fuzzy quasinorm and the family of quasinorms.