Abstract
In this paper, we first introduce a new notion of the property (M_{C}) to improve and generalize the property (G_{C}). After that, we present two new concepts, proximal b-cyclic contraction of first type and second type, on b-metric spaces. Then, we obtain two best proximity point results for such mappings in the frameworks of best proximally complete b-metric spaces by using the property (M_{C}). Hence, we generalize some results existing in the literature. Finally, we present some illustrative and interesting examples.
Highlights
Introduction and PreliminariesFixed point theory has great importance in dealing with various problems in di¤erential equations, approximation theory, control systems, nonlinear analysis and game theory
We ...rst introduce a new notion of the property (MC ) to improve and generalize the property (GC ): After that, we present two new concepts, proximal b-cyclic contraction of the ...rst type and second type, on b-metric spaces
Many authors have studied to develop ...xed point theory. It was proved the Banach contraction principle [5] which is considered the start of the ...xed point theory on metric spaces
Summary
Introduction and PreliminariesFixed point theory has great importance in dealing with various problems in di¤erential equations, approximation theory, control systems, nonlinear analysis and game theory. We obtain two best proximity point theorems for such mappings in the framework of best proximally complete b-metric spaces by using the property (MC ). The following de...nitions and theorems about best proximity point theory are important for our results.
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