UPPER ESTIMATES FOR SUPERPOSITIOI\ OPERATORS AND SOME APPLICATIONS

The application of many principles of nonlinear analysis to the study of dif-ferent types of equations requires upper estimates for the operators which are gen-erated by given functions. Such estimates are usually obtained (by elementary means) from analogous estimates that are true for these functions. In this connection it is tacitly assumed that the estimates for the operators and those for the corresponding functions are equivalent (at least in a weak sense), and this is in fact true for many types of operators in the space C of continuous functions. However, for operators between the spaces Lo and In such an equivalence does not even hold for the simplest nonlinear opeiator, the superposition operator (Nemytskij operator). It is the aim of this paper to describe and discuss this situation in Z, spaces. As appli-cation, certain necessary and sufficient conditions for the differentiability and asym-ptotic differentiability of the superposition operator are given. ' l. The ca;se P:q-L To begin with, let us recall the following well known theorem of M. A. Kras-nosel'skii ([3], see also [2], [8]): Given an open subset OqRN and a Carath6odory function /: OXR*R, the superposition operator Fx(s)::/(s, x(s)) maps the space lr(o) into itself if and only if there is a pait (a,b) of positive numbers such that (1) lf(t, u)l a(s)+blal (s€O, z€R) forsomefunction a(L{Q) with llallr=a. (Inthiscase F isautomatically bounded and continuous.) Denote the set of such pairs (a, å) associated with / by T(f) and let (2) vr(o):- inf {o*bQ: @, bW(f)l (8 = 0). 150 J. Appnn Moreover, we associate with the operator F a function p " which is defined by(3) pr(Q):: sup {llrxll,:llxl [ = e] (e = 0) and describes the growth of F on balls centered at the origin. Theorem l. The functions p " and t1 are equiualent in the sense tha