Comparison of F-contraction and φ-contractions

During the last two decades the concept of F-contraction, its modifications, extensions and applications, is in the spotlight of scientific research in the area of Metric Fixed Point Theory. It is natural to ask what is the relation between the newly introduced F-contraction and some well-known classes of contraction mappings. We will discuss the relation between F-contraction and some of its modification on one side and Boyd-Wong contraction or Matkowski contraction on the other. The main aim is to compare the conditions under which those classes of contractions obtain the unique fixed point on a complete metric space and to prove under which assumptions we assert that F-contraction is a Boyd-Wong or Matkowski contraction.