developments related to nonexpansive type mappings we refer to [@Ag]-[@BerP09], [@Bro65]-[@Xu16] and references therein.

As mentioned in the Introduction, the study of nonexpansive mappings attracted a large number of researchers who contributed significantly to the development of this area of research. As a consequence, various single-valued and multi-valued self and nonself mappings related to nonexpansive mappings were introduced and studied independently of in connection to nonexpansive mappings (see MathScinet or zbMATH): quasi-nonexpansive mappings (see [@Dot]), strictly quasi-nonexpansive mappings (see [@Tri]), firmly nonexpansive mappings (see [@Bru73]), asymptotically nonexpansive mappings (see [@Goe72]), asymptotically quasi-nonexpansive mappings, generalized asymptotically nonexpansive mappings, generalized nonexpansive mappings (see [@Goe73]), $\alpha$-nonexpansive mappings (see [@Ao11]), Suzuki nonexpansive mappings ([@Suz]), Suzuki generalized nonexpansive mappings ([@Fal]), pseudocontractive mappings and strictly $k$-pseudocontractive mappings (see [@Bro65c]), $k$-demicontractive mappings ([@Mar73], [@Mar77], [@Hic]), Bregman nonexpansive mappings, Bregman strongly nonexpansive mappings (see [@Reich10]), Bregman relatively nonexpansive mappings, affine nonexpansive mappings, weakly nonexpansive mappings ([@BerK]), Berinde nonexpansive mappings ([@Bun18]-[@Bus18]), Preši' c nonexpansive mappings ([@BerP09]), Preši' c-Kannan nonexpansive mappings ([@Fuk]), nearly nonexpansive mappings (see [@Sahu]), $G$-nonexpansive mappings, iterated nonexpansive mappings (see [@Ben]), $I$-nonexpansive mappings, $Q$-nonexpansive mappings, $\Phi$-nonexpansive mappings, ($L$)-type mappings (see [@Llo]), non-spreading mappings (see [@Koh]), hybrid mappings (see [@Tak10]), $\lambda$-hybrid mappings (see [@Ao]) etc.

So, a challenging 