It is the purpose of the present paper to obtain certain fixed point outcomes in the sense of $ C^* $-algebra valued metric spaces. Here, we present the definitions of the gauge function, the Bianchini-Grandolfi gauge function, $ \alpha $-admissibility, and $ (\alpha, \beta) $-admissible Geraghty contractive mapping in the sense of $ C^* $-algebra. Using these definitions, we define $ (\alpha, \beta) $-Bianchini-Grandolfi gauge contraction of type I and type II. Next, we prove our primary results that the function satisfying our contraction condition has to have a unique fixed point. We also explain our results using examples. Additionally, we discuss some consequent results that can be easily obtained from our primary outcomes. Finally, there is a useful application to integral calculus.
Citation: Moirangthem Pradeep Singh, Yumnam Rohen, Khairul Habib Alam, Junaid Ahmad, Walid Emam. On fixed point and an application of $ C^* $-algebra valued $ (\alpha, \beta) $-Bianchini-Grandolfi gauge contractions[J]. AIMS Mathematics, 2024, 9(6): 15172-15189. doi: 10.3934/math.2024736
It is the purpose of the present paper to obtain certain fixed point outcomes in the sense of $ C^* $-algebra valued metric spaces. Here, we present the definitions of the gauge function, the Bianchini-Grandolfi gauge function, $ \alpha $-admissibility, and $ (\alpha, \beta) $-admissible Geraghty contractive mapping in the sense of $ C^* $-algebra. Using these definitions, we define $ (\alpha, \beta) $-Bianchini-Grandolfi gauge contraction of type I and type II. Next, we prove our primary results that the function satisfying our contraction condition has to have a unique fixed point. We also explain our results using examples. Additionally, we discuss some consequent results that can be easily obtained from our primary outcomes. Finally, there is a useful application to integral calculus.
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Moirangthem Pradeep Singh, Yumnam Rohen, Khairul Habib Alam, Junaid Ahmad, Walid Emam. On fixed point and an application of $ C^* $-algebra valued $ (\alpha, \beta) $-Bianchini-Grandolfi gauge contractions[J]. AIMS Mathematics, 2024, 9(6): 15172-15189. doi: 10.3934/math.2024736
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