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Abstract comparison principles and multivariable Gronwall-Bellman inequalities. (English) Zbl 0613.47037

The paper is a continuation of the previous papers by the author [Atti Accad. Naz. Lincei 8, 66, 189-193 (1979; Zbl 0452.47062), Bull. Acad. Polon. Sci. Ser. Sci. Math. 30, 161-166 (1982; Zbl 0497.47038), Demonst. Math. 15, 145-153 (1982; Zbl 0509.47045)]. The paper divided into four chapters contains seven theorems, four lemmas, many remarks, and examples.

In the first chapter the author studies compactness of increasing sequences of elements in an ordered metrizable uniform space X with denumerable sufficient family of semi-metrics. In the second chapter the author studies comparison theorems between the solutions in Y of an operator inequality (O.I.) $x\le Tx$ and associated equation (O.E.) $x=Tx$ where T:Y$\to Y$ is increasing mapping on some ordered closed subset Y of X. The statement is e.g.: to any solution u of (O.I.) there corresponds solution v of (O.E.) such that $u\le v$ and if there exists other solution w of (O.I.) such that $v\le w$ then $v=w·$

In the next two chapters applying the above described theorems the author studies multivariable (and scalars) Gronwall type inequalities

$x\left(t\right)\le p\left(t\right)+{\int }_{0}^{t}k\left(x\right)\left(s\right)ds,\phantom{\rule{1.em}{0ex}}t\in {R}_{+}^{n}$

and many of its particular cases. Here $x,p\in {X}_{n}^{m}$, $k:{X}_{n}^{m}\to {X}_{n}^{m}$ is an increasing mapping and ${X}_{n}^{m}$ denotes the class of all continuous functions from ${R}_{+}^{n}$ to ${R}^{m}$.

Reviewer: J.Popenda

##### MSC:
 47B60 Operators on ordered spaces 26D10 Inequalities involving derivatives, differential and integral operators