Startpoints and -Contractions in Quasi-Pseudometric Spaces

We introduce the concept of startpoint and endpoint for multivalued maps defined on a quasi-pseudometric space. We investigate the relation between these new concepts and the existence of fixed points for these set valued maps. Dedicated to my beloved Clémence on the occasion of her 25th birthday 1. Introduction In the last few years there has been a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces (see, e.g., [1]), because such a theory provides an important tool and a convenient framework in the study of several problems in theoretical computer science, applied physics, approximation theory, and convex analysis. Many works on general topology have been done in order to extend the well-known results of the classical theory. In particular, various types of completeness are studied in [2], showing, for instance, that the classical concept of Cauchy sequences can be accordingly modified. In the same reference, which uses an approach based on uniformities, the bicompletion of a -quasi-pseudometric has been explored. It is worth mentioning that, in the fixed point theory, completeness is a key element, since most of the constructed sequences will be assumed to have a Cauchy type property. It is the aim of this paper to continue the study of quasi-pseudometric spaces by proving some fixed point results and investigating a bit more the behaviour of set-valued mappings. Thus, in Section 3 a suitable notion of -contractive mapping is given for self-mappings defined on quasi-pseudometric spaces and some fixed point results are discussed. In Sections 4 and 5, the notions of startpoint and endpoint for set-valued mappings are introduced and different variants of such concepts, as well as their connections with the fixed point of a multivalued map, are characterized. For recent results in the theory of asymmetric spaces, the reader is referred to [3–8]. 2. Preliminaries Definition 1. Let be a nonempty set. A function is called a quasi-pseudometric on if (i) for all ,(ii) for all .Moreover, if , then is said to be a -quasi-pseudometric. The latter condition is referred to as the -condition. Remark 2. (i) Let be a quasi-pseudometric on ; then the map defined by whenever is also a quasi-pseudometric on , called the conjugate of . In the literature, is also denoted by or . (ii) It is easy to verify that the function defined by , that is, , defines a metric on whenever is a -quasi-pseudometric on . Let be a quasi-pseudometric space. For and , denotes the open -ball at . The