Stability of solutions for variational relation problems with applications

In this paper, we prove that most of problems in variational relations (in the sense of Baire category) are essential and that, for any problem in variational relations, there exists at least one essential component of its solution set. As applications, we deduce the existence of essential components of the set of Ky Fan’s points based on Ky Fan’s minimax inequality theorem, the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games, the existence of essential component of the set of solutions for vector Ky Fan’s minimax inequality, the existence of essential components of the set of KKM points and the existence of essential components of the set of solutions for Ky Fan’s section theorem.     

Existence of vector mixed variational inequalities in Banach spaces

The purpose of this paper is to study the solvability for vector mixed variational inequalities (for short, VMVI) in Banach spaces. Utilizing Ky Fan’s Lemma and Nadler’s theorem, we derive the solvability for VMVIs with compositely monotone vector multifunctions. On the other hand, we first introduce the concepts of compositely complete semicontinuity and compositely strong semicontinuity for vector multifunctions. Then we prove the solvability for VMVIs without monotonicity assumption by using these concepts and by applying Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.     

Weighted quasi-equilibrium problems with lower and upper bounds

This paper is devoted to studying the solution existence of weighted quasi-equilibrium problems with lower and upper bounds by using maximal element theorems, a fixed point theorem of set-valued mappings and Fan–KKM theorem, respectively. Some new results are obtained.     

Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G$G$-convex spaces properly. In this work, we show that a large number of results in KKM theory on G$G$-convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.     

Generalized Vector Variational-Like Inequalities

Using a generalized Fan’s KKM theorem, some existence results for generalized vector variational-like inequalities in noncompact settings are established. Some applications to vector optimization problems are given. The results presented in this paper extend and unify corresponding results of other authors.     

On extremum properties of orthogonal quotients matrices

In this paper we explore the extremum properties of orthogonal quotients matrices. The orthogonal quotients equality that we prove expresses the Frobenius norm of a difference between two matrices as a difference between the norms of two matrices. This turns the Eckart-Young minimum norm problem into an equivalent maximum norm problem. The symmetric version of this equality involves traces of matrices, and adds new insight into Ky Fan’s extremum problems. A comparison of the two cases reveals a remarkable similarity between the Eckart-Young theorem and Ky Fan’s maximum principle. Returning to orthogonal quotients matrices we derive “rectangular” extensions of Ky Fan’s extremum principles, which consider maximizing (or minimizing) sums of powers of singular values.     

The KKM principle in abstract convex spaces: Equivalent formulations and applications

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its “open” version. In this paper, we clearly derive a sequence of a dozen statements which characterize the KKM spaces and equivalent formulations of the partial KKM principle. As their applications, we add more than a dozen statements including generalized formulations of von Neumann minimax theorem, von Neumann intersection lemma, the Nash equilibrium theorem, and the Fan type minimax inequalities for any KKM spaces. Consequently, this paper unifies and enlarges previously known several proper examples of such statements for particular types of KKM spaces.     

Generalized 2-KKM theorems and their applications in hyperconvex metric spaces

In this work, we first define the 2-KKM mapping and the generalized 2-KKM mapping on a metric space, and then we apply the property of the hyperconvex metric space to get a KKM theorem and a fixed point theorem without a compactness assumption. Next, by using this KKM theorem, we get some variational inequality theorems and minimax inequality theorems.     

Remarks on Caristi’s fixed point theorem and Kirk’s problem

Without assumptions on the continuity and the subadditivity of η, by means of Caristi’s fixed point theorem, we investigated the existence of fixed points for a Caristi type mapping which partially answered Kirk’s problem and improved Caristi’s fixed point theorem, Jachymski’s fixed point theorem and Khamsi’s fixed point theorem since φ is not necessarily assumed to be bounded below on X.     

On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces

We prove a strong convergence theorem for multivalued nonexpansive mappings which includes Kirk’s convergence theorem on CAT(0) spaces. The theorem properly contains a result of Jung for Hilbert spaces. We then apply the result to approximate a common fixed point of a countable family of single-valued nonexpansive mappings and a compact valued nonexpansive mapping.     

Remarks on some basic concepts in the KKM theory

In the KKM theory, some authors adopt the concepts of the compact closure (ccl), compact interior (cint), transfer compactly closed-valued multimap, transfer compactly l.s.c. multimap, and transfer compactly local intersection property, respectively, instead of the closure, interior, closed-valued multimap, l.s.c. multimap, and possession of a finite open cover property. In this paper, we show that such adoption is inappropriate and artificial. In fact, any theorem with a term with “transfer” attached is equivalent to the corresponding one without “transfer”. Moreover, we can invalidate terms with “compactly” attached by giving a finer topology on the underlying space. In such ways, we obtain simpler formulations of KKM type theorems, Fan-Browder type fixed point theorems, and other results in the KKM theory on abstract convex spaces.     

Some new results and generalizations in metric fixed point theory

The main purpose of this paper is the study of the generalization of some results given in [M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl. 326 (2007) 772-782] and references therein. Some generalizations of the Mizoguchi-Takahashi fixed point theorem, Kannan’s fixed point theorems and Chatterjea’s fixed point theorems are established by using our new fixed point theorems.     

On general best proximity pairs and equilibrium pairs in free abstract economies

In this paper, using Lassonde’s fixed point theorem for Kakutani factorizable multifunctions and Park’s fixed point theorem for acyclic factorizable multifunctions, we will prove new existence theorems for general best proximity pairs and equilibrium pairs for free abstract economies, which generalize the previous best proximity theorems and equilibrium existence theorems due to Srinivasan and Veeramani [P.S. Srinivasan, P. Veeramani, On best approximation pair theorems and fixed point theorems, Abstr. Appl. Anal. 2003 (1) (2003) 33–47; P.S. Srinivasan, P. Veeramani, On existence of equilibrium pair for constrained generalized games, Fixed Point Theory Appl. 2004 (1) (2004) 21–29], and Kim and Lee [W.K. Kim, K.H. Lee, Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl. 316 (2006) 433–446] in several aspects.     

Some generalizations of Caristi’s fixed point theorem with applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem

In this paper, we first prove some generalizations of Caristi’s fixed point theorem. Then we give some applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem.     

A fixed point theorem for weakly Zamfirescu mappings

In [5], Zamfirescu (1972) gave a fixed point theorem that generalizes the classical fixed point theorems by Banach, Kannan, and Chatterjea. In this paper, we follow the ideas of Dugundji and Granas to extend Zamfirescu’s fixed point theorem to the class of weakly Zamfirescu maps. A continuation method for this class of maps is also given.     

Well-posed Ky Fan’s point,quasi-variational inequality and Nash equilibrium problems

We give a unified approach to Hadamard well-posedness for some nonlinear problems such as those of Ky Fan’s point and quasi-variational inequality. As applications, we obtain some well-posed theorems for Nash equilibrium points.     

Strong convergence theorems for 2-generalized hybrid mappings in Hilbert spaces

In this paper, we prove a strong convergence theorem of Halpern’s type for 2-generalized hybrid mappings in a Hilbert space. We also deal with strong convergence theorems by hybrid methods for these nonlinear mappings in a Hilbert space.     

Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces

Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in Kurokawa and Takahashi [Y. Kurokawa, W. Takahashi, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. 73 (2010) 1562-1568]. We explored an auxiliary mapping in our theorems and proofs and this also yielded a strong convergence theorem of Halpern’s type for our class of mappings and hence resolved in the affirmative an open problem posed by Kurokawa and Takahashi in their final remark for the case where the mapping T is averaged.     

A generalization of Browder’s fixed point theorem with applications

In this paper, the concept of c$c$-compact mapping is introduced. A generalization of Browder’s fixed point theorem and some equivalence forms are given. As applications, the existence of solutions for some variational inequalities and monotone operator equations is discussed.     

Integrable solutions of a mixed type operator equation

In this paper we prove the existence of integrable solutions for a generalized mixed type operator equation, which contains many key integral and functional equations appearing frequently in Mathematical literature. Our main tool is a Krasnosel’skii type fixed point theorem recently proved by Latrach and Taoudi, the first author. An existence theory for a class of nonlinear transport equations is also developed.