Fixed Point Theorems and Error Bounds in Partial Metric Spaces

This paper is introduced as a survey of result on some generalization of Banach’s fixed point and their approximations to the fixed point and error bounds, and it contains some new fixed point theorems and applications on dualistic partial metric spaces. 1. Introduction The partial metric spaces were introduced in [1] as a part of the study of denotational semantics of dataflow networks. He established the precise relationship between partial metric spaces and the weightable quasi-metric spaces, and proved a partial metric generalization of Banach contraction mapping theorem. A partial metric [1] on a set is a function such that for all (1) ; (2) ; (3) ; (4) . A partial metric space is a pair , where is a partial metric on . If is a partial metric on , then the function given by is a (usual) metric on . Each partial metric on induces a topology on which has as a basis the family of open -balls , where for all and . Similarly, closed -ball is defined as . A sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) [1]. Note that is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space [1]. A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to to a point such that [1]. A mapping is said to be continuous at , if for , there exists such that [2]. Definition 1 (see [1]). An open ball for a partial metric is a set of the form for each and . In [3], O’Neill proposed one significant change to Matthews definition of the partial metrics, and that was to extend their range from to . In the following, partial metrics in the O’Neill sense will be called dualistic partial metrics and a pair such that is a nonempty set and is a dualistic partial metric on will be called a dualistic partial metric space. A dualistic partial metric on a set is a partial metric . A dualistic partial metric space is a pair , where is a dualistic partial metric on . A quasi-metric on a set is a nonnegative real-valued function on such that for all , , (i) ,(ii) . Lemma 2 (see [1]). If is a dualistic partial metric space, then the function defined by , is a quasi-metric on such that . Lemma 3 (see [1]). A dualistic partial metric space is complete if and only if the metric space is complete. Furthermore if and only if . Before stating our main results, we establish some (essentially known) correspondences between dualistic partial metrics and quasi-metric spaces. Also refer to definition of -Fixed point and the existence of -Fixed point for . Our basic references for quasi-metric