B.S.T.J. Brief: An Observation Concerning the Application of the Contraction-Mapping Fixed-Point Theorem, and a Result Concerning the Norm-Boundedness of Solutions of Nonlinear Functional Equations

An Observation Concerning the Application of the Contraction-Mapping Fixed-Point T h e o r e m , and a Result Concerning the Norm-Boiindedness of Solutions of Nonlinear Functional E q u a t i o n s By I. W. SANDBEHG (Manuscript received July 20, 1965) PART I Let (B denote a Banach space over the real or complex field Let 9((B) denote the set of (not necessarily linear) operators that map (B into itself, with / the identity operator, and let || T || denote the "Lipshitz norm" of T for all T e 9((B) (i.e., II r || 4 sup x,Ue(B || X ~ V || Observation: Let A and B belong to 9 ((B), and let g e (B. Suppose that there exists c £ 5 such that (i) (/ + cA ) _ 1 exists on (B, (ii) || A (/ + cA)_1|| and || B - cl || are finite, and (iii) || A (/ + cA)~l|-|| B - cl || 1. Then (B contains exactly one element / such that g = / + ABf. (It can be verified that under our assumptions, / e (B satisfies g = f + ABf if and only if / satisfies g = f + A (/ + cA )~{B cl)f + eg].) For the special case in which A is a linear operator, this result is well known* and has been applied often in the engineering literature [see, for example, Ref. 2]. The fact that it can be generalized as indicated suggests that the scope of its range of applicability to engineering problems can be extended significantly. * The linearity of A plays an essential role in all of the previous proofs known to this writer. See, for example, Ref. 1. 1809 1810 PART II THE B E L L SYSTEM TECHNICAL JOURNAL, OCTOBER 1905 Let X denote ail abstract linear space, over the real or complex field JF, that contains a normcd linear space £ with norm || · ||.