Two fixed point theorems in complete metric spaces

Two new classes of self-mappings defined on a complete metric space $(M,d)$ are introduced. The first one, called the class of $p$-contractions with respect to a family of mappings, includes mappings $F:M\to M$ satisfying a contraction involving a finite number of mappings $S_i:M\times M\to M$. The second one, called the class of $(\psi ,\mathrm{\Gamma },\alpha )$-contractions, includes mappings $F:M\to M$ satisfying a contraction involving the famous ratio $\psi \left(\frac{\mathrm{\Gamma }(t+1)}{\mathrm{\Gamma }(t+\alpha )}\right)$, where $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a function, $\mathrm{\Gamma }$ is the Euler Gamma function, and $\alpha \in (0,1)$ is a given constant. For both classes, under suitable conditions, we establish the existence and uniqueness of fixed points of $F$. Our results are supported by some examples in which the Banach fixed point theorem is inapplicable. Moreover, the paper includes some interesting questions related to our work for further studies in the future. These questions will push forward the development of fixed point theory and its applications.