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Iterated operator inequalities on ordered linear spaces. (English) Zbl 1263.47023

The Gronwall-Bellman inequality asserts that, if the function \(u : \mathbb{R}_+\to \mathbb{B}_+\) is continuous and \[ u(t)\leq b(t)+\int_0^t k(s)u(s)\, ds,\quad t \in\mathbb{R}_+, \] where \(b, k : \mathbb{R}_+\to \mathbb{B}_+\) are continuous functions, then \[ u(t) \leq b(t)+\int_0^t \exp[\int_0^s k(r)dr]b(s)\, ds,\quad t\in \mathbb{R}_+. \] Some generalizations of this relation have appeared in the literature. In this paper, the author finds an operator version of this inequality for a real linear space \(X\) with a convex cone \(X_+\). He finds an upper bound for \(u\in X_+\) when \(u\leq S(u)\) for some increasing positive map \(S\) on \(X\) with some special conditions, using fixed point theory.

MSC:

47A63 Linear operator inequalities
26D10 Inequalities involving derivatives and differential and integral operators
46A40 Ordered topological linear spaces, vector lattices
54E40 Special maps on metric spaces

Citations:

Zbl 0587.26013
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Full Text: Euclid