| Summary: | In this paper we introduce the notion of tvs-cone normed spaces, discuss related topological concepts and characterize the tvs-cone norm in various directions. We construct generalize locally convex tvs generated by a family of tvs-cone seminorms. The class of weak contractions properly includes large classes of highly applicable contractions like Banach, Kannan, Chatterjea and quasi etc. We prove fixed point results in tvs-cone Banach spaces for nonexpansive self mappings and self/non-self weak contractive mappings. We discuss the necessary conditions for T -stability of Picard iteration. To ensure the novelty of our work we establish an application in homotopy theory without the assumption of normality on cone and many non-trivial examples.
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