PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 86, Number 3, November 1982 CHARACTERISTIC EXPONENTSAND SOMEAPPLICATIONS TO DIFFERENTIALEQUATIONS F. S. DE BLASI AND M. A. BOUDOURIDES Abstract. A generalization of the Lozinskiï logarithmic norm is introduced and some applications to stability of differential equations are given. 1. Introduction. Let A be a continuous linear function from a real Banach space E into itself. Following Lozinskiï [8] (see also [4]) the logarithmic norm p[A] of A is defined by p[A] = lim/l_0+(ll^ + ^11 ~ 1)M> where / denotes the identity and ||7-|- hA\] = supi^j \x + hA(x)\. This notion has been used to bound solutions of differential equations and to obtain asymptotic stability [1, 3, 9]. A definition of logarithmic norm for functions A, which are continuous but not necessarily linear, is given by Martin [10], who presents also some applications to differential equations. In this note we consider continuous, possibly nonlinear, functions A from E into itself such that A(0) = 0. For such functions we define upper and lower characteristic exponents of order a > 1, denoted by i/° [A] and vf[A] respectively. When A is linear and E is a real Hubert space, we have u\[A\ = p[A] and v\[A\ = —p[—A]. The upper and lower characteristic exponents are related to stability properties of the differential equation (1.1) x'=A(x). For example (Theorem 3.1), v^[A\ < 0(^f[A] > 0) implies asymptotic stability (instability) of the zero solution of (1.1). A characterization of these numbers in terms of some properties of solutions of (1.1) is also given (Theorem 3.2). 2. Characteristic exponents. Let £bea real Banach space with norm | • |. Set Sd = {x GE] ]x] <d), d> 0. Let U C E be a nonempty open convex set containing the origin. Denote by ? = 7(U) the set of continuous functions from U into E such that A(0) = 0. For any a>l,Qa = Q.a(U) is the subset of 7 of all A G 7 such that |A|cjQ = limsupI_>0|A(:r)]/|2;|0' < +co; Ba = Ba(U) is the subset of Qa of all A G Qa which satisfy \A(x)\ < LaW* (La > 0), for each xEU. Clearly Qa and Ba are linear spaces; \-\ç>a is a seminorm on Qa. Notice that for any a > 1, the set Qa is contained in Q.\. DEFINITION 2.1. For each A G 7 and x G U, set N[A,x] = lim (]x + hA(x)\ - \x\)/h. Received by the editors May 22, 1981. This paper has been presented by the second author to the special session on "Qualitative theory of differential equations" at the 87th annual meeting of the A.M.S. in San Francisco (January 7-11, 1981). 1980 Mathematics Subject Classification. Primary 34D05, 34D20. Key words and phrases. Characteristic exponents, logarithmic norm, differential equations, stability. © 1982 American Mathematical Society 0002-9939/81/0000-0458/S02.25 464 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use