New Results in the Startpoint Theory for Quasipseudometric Spaces

We give two generalizations of Theorem 35 proved by Gaba (2014). More precisely, we change the structure of the contractive condition; namely, we introduce a function instead of a simple constant . Dedicated to Professor Guy A. Degla for his mentorship 1. Introduction and Preliminaries In [1], we introduced the concept of startpoint and endpoint for set-valued mappings defined on quasipseudometric spaces. As mentioned there, the purpose of this theory is to study fixed point like related properties. In the present, we give more results from the theory. More precisely, we generalize Theorem 35 of [1] by changing the structure of the contractive condition; namely, we introduce a function instead of a simple constant (as it appears in the original statement). This new condition is interesting in the sense that it allows us to have a condition involving a functional of the variables and not just the variables themselves. For the convenience of the reader, we will recall some necessary definitions but for a detailed exposé of the definition and examples, the interested reader is referred to [1]. Definition 1. Let be a nonempty set. A function is called quasipseudometric on if (i)？？;(ii)？？. Moreover, if , then is said to be a -quasipseudometric. The latter condition is referred to as the -condition. Remark 2. (i) Let be quasipseudometric on ; then the map defined by whenever is also a quasipseudometric on , called the conjugate of . In the literature, is also denoted as or . (ii) It is easy to verify that the function defined by , that is, , defines a metric on whenever is a -quasipseudometric on . The quasipseudemetric induces a topology on . Definition 3. Let be a quasipseudometric space. The -convergence of a sequence to a point , also called left-convergence and denoted by , is defined in the following way: Similarly, we define the -convergence of a sequence to a point or right convergence and denote it by , in the following way: Finally, in a quasipseudometric space , we will say that a sequence ()？？-converges to if it is both left and right convergent to , and we denote it as or ？？when there is no confusion. Hence Definition 4. A sequence in quasipseudometric is called (a)left -Cauchy if, for every , there exist and such that (b)left -Cauchy if, for every , there exists such that (c)-Cauchy if, for every , there exists such that Dually, we define right -Cauchy and right -Cauchy sequences. Definition 5. A quasipseudometric space is called(i)left -complete provided that any left -Cauchy sequence is -convergent,(ii)left Smyth sequentially complete if any