Nadler's Fixed Point Theorem for Multi-valued functions (see [6]) was generalized by H. Aydi et al in their recent paper [1]. We use the approach of Gordji et. al. [3] to further generalize this result for a larger class of multifunctions (with a more general contraction condition) on complete partial metric spaces. The result yields the main theorem in [1] as a special case.

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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.

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Let G be a graph and S be a closed orientable surface. Litjens defined G to have a nowhere-identity dihedral 4-flow if for each S in which G can be cellularly embedded, there is a rotation system π of G which gives a nowhere-identity dihedral 4-flow [4]. Furthermore, he proved the existence of nowhere-identity dihedral 4flows for bridgeless cubic graphs with 16 or less vertices. We give a method for exhibiting the nowhere-identity dihedral n-flows in generalized Petersen graphs and illustrate the results by exhibiting the nowhere-identity dihedral 4-flows of G(3, 1).