Abstract
In this manuscript, we present and develop different F-contraction methods using new kinds of contractions, namely F-g-contraction and extended F-g-contraction in the context of controlled b-Branciari metric type space. We then suggest an easy and effective solution for Fredholm integral equations using the fixed point method in the framework of controlled b-Branciari metric type space. We also provide an illustrative example for the existence of solution to second order boundary value problem to demonstrate the efficiency of the work that has been developed.