The purpose of the paper is to introduce a class of “j-nonexpansive” mappings and to prove fixed point theorems for such mappings. They naturally arise in the existence theory of functional differential equations. These mappings act in spaces without specific geometric properties as, for instance, uniform convexity. Critical examples are given.

Let {φν: ν ∈ N (non-negative integers)} ⊆ C[0, 1] be a complete orthonormal sequence of complex-valued functions in L2[0, 1], {λν: ν ∈ N} and {λνμ: ν, μ ∈ N} be sequences of complex numbers. In this paper, the necessary and sufficient conditions are developed for the series to converge and also to exist, in C[0, 1] for each f ∈ L1[0, 1] where .

Proposition 1.6 states that the category Alg(P*) is hypercomplete but non-compact. This is true, but the argument for non-compactness must be corrected as follows.

Let A be the category of algebras (X, x′, x″) where x′, x″: P*X → X are operations with x′(φ) = x″(φ), and homomorphisms are mappings which are P*-homomorphisms with respect to both operations. The embedding E: Alg(P*) → A with E(X, x) = (X, x, x) preserves colimits, although it is not a left adjoint. In fact, the preservation of colimits C = colim D is obvious in case C is finite, and for the infinite case the original argument presented in the paper is correct (namely, one of the colimit maps is onto). E is not a left adjoint because given A = (X, x′, x″) in A with X infinite and x′(M) ≠ x″(M) for any M ≠ φ, then there is no universal arrow into A with respect to E.