researchers for the study of nonexpansive mappings, the main aim of the present paper is to introduce a larger class of mappings of nonexpansive type, which will be called [*enriched nonexpansive mappings*]{}, and to answer in the affirmative the two problems a) and b) mentioned above in that context.

It is our personal belief that the concept of [*enriched nonexpansive mapping*]{} introduced here for the first time as well as the technique of enrichment of contractive mappings itself, by means of which we obtained the results presented in this paper and in [@BerP19]-[@Pac19b], will open new and productive research avenues in the field of nonlinear analysis.

Enriched nonexpansive mappings in Hilbert spaces
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\[def0\] Let $(X,\|\cdot\|)$ be a linear normed space. A mapping $T:X\rightarrow X$ is said to be an [*enriched nonexpansive mapping*]{} if there exists $b\in[0,ü\infty)$ such that $$\label{eq3}
\|b(x-y)+Tx-Ty\|\leq (b+1) \|x-y\|,\forall x,y \in X.$$ To indicate the constant involved in we shall also call $T$ as a $b$-[*enriched nonexpansive mapping*]{}.

It is easy to see that any nonexpansive mapping $T$ is a $0$-enriched mapping, i.e., it satisfies with $b=0$.

We note that, according to Theorem 12.1 in [@Goe], in a Hilbert space any enriched nonexpansive mapping which is 