On fixed point and an application of $C^{\ast }$-algebra valued $(\alpha ,\beta )$-Bianchini-Grandolfi gauge contractions

It is the purpose of the present paper to obtain certain fixed point outcomes in the sense of $C^{\ast }$-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 $C^{\ast }$-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.