Fixed Point Theorem on Uncomplete G-Metric Spaces

Mustafa, Zead; Shatanawi, Wasfi; Bataineh, Malik
October 2008
Journal of Mathematics & Statistics;2008, Vol. 4 Issue 4, p196
Academic Journal
Problem statement: Over the past two decades the development of fixed point theory in metric spaces has attracted considerable attention due to numerous applications in areas such as variational and linear inequalities, optimization and approximation theory. Therefore, different Authors proved many fixed points results for self mappings defined on complete G-metric space. The objectives of this study are to: (i) prove fixed point results for mapping satisfying certain conditions and (ii) finding the suitable conditions on these mapping to avoid using the complete property of Gmetric space. Approach: A G-metric space (X,G) had been defined and several contractive self mapping on (X,G) has been define, to avoid using the complete property each one of these mapping assumed to satisfy various extra conditions such as, a mapping T is G-continuous at a point u where the sequence of iterates {Tn (x)} is G-convergent to u and a G-continuous map satisfy contractive condition defined only on an every where dense subset M of X with the existence of sequence of iterates which G-converge to appoint x in X. Results: The existence and uniqueness of fixed point theory for contractive mapping defined on un complete G-metric space (X,G) can be proved but with assumed certain conditions on these contractive mapping. Conclusion: The complete property of Gmetric space (X,G) can be replaced with some conditions, but these conditions do not guarantee the complete property of G-metric spaces.


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