Moduli R(a,X) and M(X) of direct sums of Banach spaces

The moduli R(a,X) and M(X), introduced by Domínguez Benavides, play an important role in the fixed point theory for nonexpansive mappings. In the paper we show that if infi∈I⁡M(Xi)>1, then M((⨁i∈IXi)Z)>1, where (⨁i∈IXi)Z is the direct sum of Banach spaces Xi with respect to a Banach lattice Z, under some conditions for Z and I. Similar results are obtained for the modulus R(a,X).