Fixed point theorem via measure of non-compactness for a new kind of contractions

In this paper, we will use the notion of $\alpha$-admissible mappings in Banach spaces, to introduce the concept of $T_\beta$-contractive mappings and establish a fixed point theorem for this type of contractions. Our theorems generalize and improve many results in the literature. Moreover, we apply the main result to prove the existence of a solution for Volterra-integral equation, under more general assumptions than previously made.