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Abstract Gronwall-Bellman inequalities on ordered metrizable uniform spaces. (English) Zbl 0555.47025

From the author’s Introduction: ”...the main intent of the present paper is to formulate a number of results concerning the relationships between the operator inequality \((OI)x\leq Tx,\) and its corresponding operator equation \((OE)x=Tx,\) where T is a monotone, order-compact self-mapping of the ordered metrizable uniform space X - thus extending to ordered metrizable uniform structures...” results of J. Chandra and B. A. Fleischman [J. Math. Anal. Appl. 31, 668-681 (1970; Zbl 0179.203)] proved for partially ordered Banach spaces. The basic tool in proving such an extension is a maximality principle on ordered metrizable uniform spaces (Theorem 1) comparable to earlier results of H. Brezis and F. E. Browder [Adv. Math. 21, 355-364 (1976; Zbl 0339.47030]. Variants of Theorem 1 yield extensions or abstract versions of the maximality principle of I. Ekeland [J. Math. Anal. Appl. 47, 324- 353 (1974; Zbl 0286.49015)] and an abstract version of a fixed point theorem of J. Caristi [Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)]. Applications are given (Theorems 2, 3) to (OI), (OE), and in Theorems 4,5 to systems of Volterra functional inequalities and the existence of maximal solutions of the associated Volterra functional equations. It is of interest that this existence theorem does not follow from the classical Schauder-Tikhonov fixed point principle.
Reviewer: P.R.Beesack

MSC:

47B60 Linear operators on ordered spaces
45D05 Volterra integral equations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
26D10 Inequalities involving derivatives and differential and integral operators
47H10 Fixed-point theorems
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