In the article, we introduce a new concept of contraction and prove a fixed point theorem which generalizes Banach contraction principle in a different way than in the known results from the literature. The article includes an example which shows the validity of our results, additionally there is delivered numerical data which illustrates the provided example.

In this paper, we introduce the notion of an F-weak contraction and prove a fixed point theorem for F-weak contractions. Examples are given to show that our result is a proper extension of some results known in the literature

The main results of the paper concern the existence of fixed points of nonlinear -contraction and the sum of this type of mapping with a compact operator. The results of Krasnosel'skii type are obtained with a usage of the Hausdorff measure of noncompactness and condensing mappings. The presented new tools give the possibility to verify the existence problems of the solutions for some classes of integral equations.