Volterra functional equations via projective techniques

Abstract

One of the most interesting and specific instruments of investigating a large class of problems pertaining especially to functional equations theory is, undoubtedly, that represented by the so-called Hilbert’s and Thompson’s projective metrics. The first of them, introduced by Hilbert in his 1903 paper [ 151 and subsequently developed (under a “modern” form) by Birkhoff [7] and Bushel1 [lo], appears to be a very adequate tool for solving some delicate questions about linear functional equations (see, in this direction, the well-known Jentzsch’s theorem on integral equations with positive kernels (discussed, especially, in the above quoted Birkhoff paper), the Perron-Frobenius theorem on positive matrices (Samuelson [27]) as well as the multiplicative processes theory (Birkhoff (8,9])), but a somehow inadequate one in the treatment of the nonlinear versions of these problems (as typical examples of this kind being Theorem 1.2 of Bushel1 [ 111 and Theorem 4.1 of Turinici [38]). On the contrary, the second of these metrics-introduced by Thompson in his 1963 paper [33]-may be considered as a complementary instrument with respect to the preceding one since it was demonstrated that a large number of nonlinear functional equations (such as, e.g., those generated by (uniformly) “concave” operators in Bakhtin’s sense [2] (cf. also Krasnoselskii and Stetsenko [20])) may be treated by this procedure. Under these lines, it is, the main aim of the present paper to introduce a class of projective metrics on ordered linear spaces comprising Thompson’s metric as a particular case and, subsequently, to investigate-under this perspective-some specific examples of (nonlinear) Volterra functional equations-not reductible, in general, to other “nonprojective” techniques-including, among others, a “vectorial” andogue of an interesting mathematical model for metabolic growth processes due to Bertalanffy [5, Chapter VII]. It’s not without importance to specify at this moment that the basic instrument employed in our construction is the monotone semigroups theory on ordered linear spaces; some 211 0022-247X/84 $3.00