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In this article, we introduce the notion of weak <inline-formula> <tex-math notation="LaTeX">$P_\gamma $ </tex-math></inline-formula>-property and <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>-controlled proximal contraction in the setting of <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula>-metric spaces and prove best proximity results for such mappings. By restricting these results, we get some new results to study the existence of best proximity points and fixed points of mappings.