Some Fixed Point Theorems in $(\alpha ,\beta )$- Metric Spaces with applications to Fredholm integral and non-linear differential equations

In this paper, we presented a new type of metric space called $(\alpha ,\beta )$-metric space along with some novel contraction mappings named $(\alpha ,\beta )$-contraction and weak $(\alpha ,\beta )$-contraction mapping. We established some fixed point theorem for these newly introduced contractive mappings. Our results extended some fixed point results in the existing literature. We also provide an example which holds for the weak $(\alpha ,\beta )$-contraction. Furthermore, we proved Kannan’s fixed point theorem and Reich’s fixed point theorem in the setting of $(\alpha ,\beta )$-metric spaces. At the end, as applications the Fredholm integral and non-linear differential equations are solved in order to validate the theoretically obtained conclusions