Fixed Point Theorems in b-Metric Spaces
In this dissertation, we study the concept of fuzzy b-metric space which is the generalization of fuzzy metric spaces and b-metric spaces. The Banach contraction principle is extended in the setting of fuzzy b-metric spaces and this result has been illustrated by an example. The notion of g-orbitally upper semi continuous function is also introduced in fuzzy metric space and the xed point result of Hicks and Rhoads is generalized in the setting of fuzzy b-metric space. Some xed point results are also proved by introducing a novel and rational contraction and using a control function in fuzzy b-metric spaces. Some applications are also highlighted as consequences of our results. This idea is further used to prove some new xed point results and some common xed point results for Geraghty-type contraction in G-complete fuzzy b-metric spaces. Further, the notion of generalized fuzzy metric space is introduced. Many topological spaces like fuzzy metric spaces, fuzzy b- metric spaces and dislocated fuzzy metric spaces have been generalized by this new generalized fuzzy metric space. It is also proved that the class of generalized fuzzy metric spaces contains the classes of fuzzy metric spaces, fuzzy b-metric spaces and dislocated fuzzy metric spaces as proper sub-classes. The Banach contraction principle and Ciric’s quasi-contraction theorem are demonstrated in the setting of generalized fuzzy metric space. As consequences of our results, we obtain Jleli and Samet’s and many other author’s recent results as corollaries. We also present an application related to our main result for nonlinear integral equation.