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Rectangular TVS-cone metric spaces are introduced and Kannan's fixed pointtheorem is proved in these spaces. Two approaches are followed for the proof.At first we prove the theorem by a direct method using the structure of thespace itself. Secondly, we use the nonlinear scalarization used recently byWei-Shih Du in [A note on cone metric fixed point theory and its equivalence,{Nonlinear Analysis},72(5),2259-2261 (2010).] to prove the equivalence of theBanach contraction principle in cone metric spaces and usual metric spaces. Theproof is done without any normality assumption on the cone of the locallyconvex topological vector space, and hence generalizing several previouslyobtained results.

Locally Convex Valued Rectangular Metric Spaces and The Kannan's Fixed Point Theorem