On Quasi-Pseudometric Type Spaces

We introduce the concept of a quasi-pseudometric type space and prove some fixed point theorems. Moreover, we connect this concept to the existing notion of quasi-cone metric space. 1. Introduction Cone metric spaces were introduced in [1] and many fixed point results concerning mappings in such spaces have been established. In [2], Khamsi connected this concept with a generalised form of metric that he named metric type. Recently in [3], Shadda and Md Noorani discussed the newly introduced notion of quasi-cone metric spaces and proved some fixed point results of mappings on such spaces. Basically, cone metric spaces are defined by substituting, in the definition of a metric, the real line by a real Banach space that we endowed with a partial order. The fact that the introduced order is not linear does not allow us to always compare any two elements and then gives rise to a kind of duality in the definition of the induced topology, hence the convergence in such space. We introduce a quasi-pseudometric type structure and show that some proofs follow closely the classical proofs in the quasi-pseudometric case but generalize them. 2. Preliminaries In this section, we recall some elementary definitions from the asymmetric topology which are necessary for a good understanding of the work below. Definition 1. Let be a nonempty set. A function is called a quasi-pseudometric on if(i) ,(ii) .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. Let be a quasi-pseudometric space. Then for each and , the set denotes the open -ball at with respect to . It should be noted that the collection yields a base for the topology induced by on . In a similar manner, for each and , we define known as the closed -ball at with respect to . Also the collection yields a base for the topology induced by on . The set is -closed but not -closed in general. The balls with respect to are often called forward balls and the topology is called forward topology, while the balls with respect to are often called backward balls and the topology is called backward topology. Definition 3. Let be a quasi-pseudometric space. The convergence of a sequence to with respect to , called -convergence or left-convergence and denoted by , is