If, in Theorem 10, we assume that f(x) = u [member of] C, a

constant mapping, then we get the following corollary.

(Lemma 3.1) Let K : D [right arrow] comp A be a continuous multifunction and v : D [right arrow] [R.sup.n] be a piecewise constant mapping such that d (v (x, y, z), K (x, y, z)) < [rho] for every (x, y, z) [member of] D.

We shall construct, for every n [greater than or equal to] 1, a continuous mapping [g.sup.n] : K [right arrow] [L.sup.1] (D; [R.sup.n]) such that, for every u [member of] K, [g.sup.n] (u) is a piecewise constant mapping of D into A which satisfies, at every (x, y, z) [member of] D,

there exists a piecewise constant mapping [v.sup.n.sub.m] : D [right arrow] A and a point [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that, at every (x, y, z) [member of] D,

a piecewise constant mapping [v.sup.p+1.sub.(m,s)] : D [right arrow] A which satisfies, at every (x, y, z) [member of] D,

(P2): For [for all] (a, i) [member of] [S.sub.[alpha]], (b, j) [member of] [S.sub.[beta], we consider the following situation separately: ([[gamma].sup.(a,i).sub.[alpha],[alpha]) ([[gamma].sup.(a,i).sub.[beta],[beta]) is a constant mapping on [[LAMBDA].sub.[alpha]] and we denote the constant value by

(b) If [alpha],[beta],[delta], [member of] Y with [alpha],[beta][greater than or equal to] [delta] ([[gamma].sup.(a,i).sub.[alpha],[alpha]) ([[gamma].sup.(a,i).sub.[beta],[beta]) is a constant mapping on [[LAMBDA].sub.[alpha]] = k, then, ([[gamma].sup.(a,i).sub.[alpha],[alpha]) ([[gamma].sup.(a,i).sub.[beta],[beta]) = k, then ([[gamma].sup.(ab,k).sub.[alpha],[beta][delta])

Fisher, "Fixed point and

constant mappings on metric spaces," Atti della Accademia Nazionale dei Lincei.