Around averaged mappings

This paper is intended for a general mathematical audience. The examples show how the study of existence of fixed points of averaged mappings $$T_{\lambda }= (1-\lambda )I+ \lambda T$$ T λ = ( 1 - λ ) I + λ T , where $$0<\lambda <1$$ 0 < λ < 1 and I is the identity operator, can help in the study of existence of fixed points of mappings T.