University of Isfahan Mathematics and Society 2345-6493 11 2 2025 11 03 Some results of the $ \phi $-best proximity point in the complete metric space Some results of the $ \phi $-best proximity point in the complete metric space 41 64 29658 10.22108/msci.2025.143009.1706 FA Maryam Shams Department of pure Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran. Farah Rashidi Department of pure Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran Journal Article 2024 10 19 In this paper, we introduce two types of proximal contractions. First, we define the $ (\phi, \varphi, \psi, H )$-proximal contraction and subsequently the weak $(\phi, \varphi, \psi, H)$-proximal contraction. Then, we investigate the existence and uniqueness of the $ \varphi $-best proximity point for these contractions in a complete metric space, considering specific conditions. One of these specific conditions, required in both theorems, is that the function $ \psi $ is non-decreasing and the function $ \varphi$ is lower semi-continuous. The main theorems obtained are generalizations and extensions of existing $ \varphi $-best proximity point theorems for proximal contractions related to the control function $ H $. If, in the main theorems, the two subsets $ A $ and $ B $ are equal, then the existence and uniqueness of a fixed point for the corresponding self-mappings are obtained. Subsequently, we illustrate the importance and applicability of the main theorems with the help of examples. In this paper, we introduce two types of proximal contractions. First, we define the $ (\phi, \varphi, \psi, H )$-proximal contraction and subsequently the weak $(\phi, \varphi, \psi, H)$-proximal contraction. Then, we investigate the existence and uniqueness of the $ \varphi $-best proximity point for these contractions in a complete metric space, considering specific conditions. One of these specific conditions, required in both theorems, is that the function $ \psi $ is non-decreasing and the function $ \varphi$ is lower semi-continuous. The main theorems obtained are generalizations and extensions of existing $ \varphi $-best proximity point theorems for proximal contractions related to the control function $ H $. If, in the main theorems, the two subsets $ A $ and $ B $ are equal, then the existence and uniqueness of a fixed point for the corresponding self-mappings are obtained. Subsequently, we illustrate the importance and applicability of the main theorems with the help of examples. https://math-sci.ui.ac.ir/article_29658_f6c89f79b0c8da92777aff0abea52cb4.pdf