Derivation of Fixed-Point Theorem Using Expansive Mapping Approach
Koech Vincent
Department of Physical Sciences, Chuka University, P.O. Box 109-60400, Chuka, Kenya.
Musundi W. Sammy *
Department of Physical Sciences, Chuka University, P.O. Box 109-60400, Chuka, Kenya.
Kinyanjui Jeremiah
Department of Pure and Applied Sciences, Kirinyaga University, P.O BOX 143-10300, Kerugoya, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Application of Fixed-Point Theorem has tremendously increased in different areas of interest and research. Fixed Point Theorem presents that if T:X→X is a contraction mapping on a complete metric space (X, d) then there exists a unique fixed point in X. A lot has been done on application of contraction mapping in Fixed Point Theorem on metric spaces such as Cantor set with the contraction constant of 1/3 , the Sierpinski triangle also with contraction constant of 1/2 . On the other hand, a mapping T:X → X on (X, d) such that ∀x, y ∈ X: d(Tx, Ty) ≥ d (x, y) is called an expansive mapping. There are various types of expansive mappings such as; isometry expansive mapping, proper/strict expansive mapping and anti-contraction expansive mapping. From the available literature, Fixed Point Theorem has been derived using contraction mapping approach. In this paper, we establish that it is also possible to derive Fixed Point Theorem using expansive mapping approach.
Keywords: Fixed point theorem, expansive mapping, contractive mapping
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References
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