on a bounded, closed and convex subset $C$ of $H$ has fixed points in $C$ and that the set of all its fixed points is convex. In order to approximate a fixed point of an enriched nonexpansive mapping, we used the Krasnoselskij iteration for which we have proven a strong convergence result (Theorem \[th1\]) as well as two weak convergence theorems (Theorems \[th2\] and \[th3\]).

We illustrated the richness of the new class of mappings by means of Examples \[ex1\] and Corollary \[cor2\]. We conclude that all nonexpansive mappings are included in the class of enriched nonexpansive mappings, which is also independent of that of quasinonexpansive mappings (and which includes all nonexpansive mappings possessing fixed points). Moreover, see the proof of Corollary \[cor2\], all Lipschitz and generalized pseudocontractive mappings are also included in the class of [*enriched nonexpansive mappings*]{}. Note also that, similarly to the case of nonexpansive mappings, any enriched nonexpansive mapping is continuous.

Our results extend some convergence theorems in [@BroP67] from nonexpansive mappings to enriched nonexpansive mappings and thus include many other important related results from literature as particular cases, see [@Ber02], [@Ber02a], [@Ber07], [@BroP66], [@Chi], [@Kra55], [@Pet], [@Sch] etc.

For some other old and recent related 