In this article, we define the new concept of a fixed set for a set valued map with set valued domain in the setting of metric space endowed with a directed graph. This notion of fixed set is analogous to the notion of a fixed point for a multivalued map and not for a classical single-valued map. We also introduce the new concept of the start set of a graph whose vertices are closed and bounded subsets of a metric space. Characterizations for such a graph to have a start set are given. Further, the notion of a self-path set valued map is defined and its relation with the start set is established. Finally, the existence of fixed sets is established in this context. MSC: 47H10; 54H25; 54E50

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Abstract:In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and establish its relation with start point in the setting of a metric space endowed with a directed graph. Further, some fixed point theorems for set valued maps have been proven in this context. A version of the Knaster-Tarski theorem has also been established using our results.

We provide some new fixed point results which are inspired by the works of Suzuki and Kannan. The results are proved using the properties of sequentially convergent mappings and A-contractions. Existence and uniqueness of fixed points of self maps satisfying certain conditions are investigated in a complete metric space.