On the metrical properties of the configuration space

We study some topological and metrical properties of configuration spaces. In particular, we introduce a family of metrics on the configuration space Γ, which makes it a Polish space. Compact functions on Γ are also considered. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Limit sets for maps of the circle

A trick which reduces some questions concerning the dynamics of self-maps of the circle to similar questions about self-maps of the interval is suggested and applied to answer two questions of Block and Coppel.     

Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings

In this paper, the famous Banach contraction principle and Caristi's fixed point theorem are generalized to the case of multi-valued mappings. Our results are extensions of the well-known Nadler's fixed point theorem [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-487], as well as of some Caristi type theorems for multi-valued operators, see [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188; J.P. Aubin, Optima and Equilibria. An Introduction to Nonlinear Analysis, Grad. Texts in Math., Springer-Verlag, Berlin, 1998, p. 17; S.S. Zhang, Q. Luo, Set-valued Caristi fixed point theorem and Ekeland's variational principle, Appl. Math. Mech. 10 (2) (1989) 111-113 (in Chinese), English translation: Appl. Math. Mech. (English Ed.) 10 (2) (1989) 119-121], etc.     

Scalar Flat Metrics of Eguchi-Hanson Type

We use a new method to construct a class of asymptotically locally flat, scalar flat metrics. These metrics were constructed via algebraic geometry method by LeBrun before and provide counterexamples to the generalized positive action conjecture of Hawking and Pope.     

Fixed point theorems for set-valued contractions in complete metric spaces

The fixed point theory of set-valued contractions which was initiated by Nadler [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-488] was developed in different directions by many authors, in particular, by [S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972) 26-42; N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188; Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112]. In the present paper, the concept of contraction for set-valued maps in metric spaces is introduced and the conditions guaranteeing the existence of a fixed point for such a contraction are established. One of our results essentially generalizes the Nadler and Feng-Liu theorems and is different from the Mizoguchi-Takahashi result. The second result is different from the Reich and Mizoguchi-Takahashi results. The method used in the proofs of our results is inspired by Mizoguchi-Takahashi and Feng-Liu's ideas. Comparisons and examples are given.     

The Connection Between the Metric and Generalized Projection Operators in Banach Spaces

In this paper we study the connection between the metric projection operator PK ： B →K, where B is a reflexive Banach space with dual space B^＊ and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K ： B → K and πK ： B^＊ → K. We also present some results in non-reflexive Banach spaces.     

Regularity in approach theory

The setting of approach frames is an exquisite environment to compare the various notions of regularity in topology, frames and approach spaces and in this paper we make this investigation. This brings to light some interesting similarities and differences and proves that there are at least two different unifying notions in the setting of approach frames each with specific advantages. The third author is an Aspirant of the Foundation for Scientific Research Flanders.     

Plasticity in metric spaces

In this paper we examine the properties of EC-plastic metric spaces, spaces which have the property that any noncontractive bijection from the space onto itself must be an isometry.     

On Embeddings for Classes of Functions with Generalized Smoothness on Metric Spaces

Given a metric space with a Borel measure, we consider the classes of functions whose increment is controlled by the measure of a ball containing the corresponding points and a nonnegative function summable with some power. We prove embedding theorems for these spaces defined by two different measures satisfying the doubling condition.     

Generic well-posedness of variational problems without convexity assumptions

The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In our previous work a generic well-posedness result (with respect to variations of the integrand of the integral functional) without the convexity condition was established for a class of optimal control problems satisfying the Cesari growth condition. In this paper we extend this generic well-posedness result to two classes of variational problems in which the values at the end points are also subject to variations. The main results of the paper are obtained as realizations of a general variational principle.