New Fixed-Point Results in Controlled Metric Type Spaces with Applications

In this manuscript, we present several novel results in fixed-point theory for a complete controlled metric space. The first presented result is inspired from the Caristi contraction where we explore the existence and uniqueness of fixed points under specific conditions. Furthermore, we propose a graphical form of it by endowing the considered space with a graph and develop a new fixed-point theorem, which is illustrated by two examples. Also, we establish a theorem for the alpha-admissible mapping. To demonstrate its effectiveness, the last theorem proposes an approach to solve a second-order differential equation.