A Fixed Point Theorem for Multifunctions in Partial Metric spaces

Abstract

Fixed Point theorems on partial metric spaces have been the subject of recent work, with the interest generated in partial metric spaces (as a suitable structure for studies in theoretical computer science). Several approaches to fixed point theory for point-valued functions on complete metric spaces have been generalized to partial metric spaces (see, for instance, Alghamdi [1]). On the other hand, it appears that substantial work may still be done to generalize the theory (in the partial metric space context) to set-valued functions. Recently, Damjanovic et al [3] looked into pairs of multi-valued and single-valued maps in complete metric spaces, and used coincidence and common fixed points, to establish a theorem on fixed points for pairs of multivalued functions. In this paper we take off from Damjanovic and proceed to establish the same result in the setting of partial metric spaces. As a consequence of our generalization, we are able to include as special cases the theorem of Aydi et al [2] and our [9] generalization of [4]. Further, Reich's result is also generalized to multivalued functions in partial metric spaces. Special cases include the partial metric space version of Kannan's theorem, as well as that due to Hardy and Rogers.

A generalized Nadler's theorem in dislocated quasi-metric spaces

“…Of note is the generalization of Nadler [14] to multifunctions on metric spaces satisfying a contraction condition. Some recent treatments and extensions of fixed point theorems for multivalued functions are given in [2], [3], and [4] .…”

“…In [4] we used the approach of Damjanovic et al [5] who looked into pairs of multi-valued and single-valued maps in complete metric spaces, and used coincidence and common fixed points, to establish a theorem on fixed points for pairs of multivalued functions. In that paper, we proved a similar result, in the setting of partial metric spaces.…”