Locally Weak Version of the Contraction Mapping Principle

The Banach Contraction Mapping Principle has many generalizations and extensions in different directions. Here we define $(\varepsilon,\psi)$-Uniformly Local Weak Contractions and show that these mappings admit unique fixed points. We obtain a generalization of two existing results. We construct an example which illustrates of our main result in which the mapping is neither a $\psi$-weak contraction nor a local contraction.