On Some Vector-Valued Inequalities of Gronwall Type

Dah-Chin Luor

doi:10.1155/2014/973586

Abstract

In this paper we established some vector-valued inequalities of Gronwall type in ordered Banach spaces. Our results can be applied to investigate systems of real-valued Gronwall-type inequalities. We also show that the classical Gronwall-Bellman-Bihari integral inequality can be generalized from composition operators to a variety of operators, which include integral operators, maximal operators, geometric mean operators, and geometric maximal operators.

Highlights

It is well known that the Gronwall-type inequalities play an important role in the study of qualitative properties of solutions to differential equations and integral equations
The Gronwall inequality was established in 1919 by Gronwall [1] and it was generalized by Bellman [2]
Where η ≥ 0 and u, and g are nonnegative continuous functions on [0, b], t u (t) ≤ η exp (∫ g (s) ds), 0 ≤ t ≤ b. This result plays a key role in studying stability and asymptotic behavior of solutions to differential equations and integral equations

Summary

Introduction

It is well known that the Gronwall-type inequalities play an important role in the study of qualitative properties of solutions to differential equations and integral equations. Where η ≥ 0 and u, and g are nonnegative continuous functions on [0, b], t u (t) ≤ η exp (∫ g (s) ds) , 0 ≤ t ≤ b This result plays a key role in studying stability and asymptotic behavior of solutions to differential equations and integral equations. Many results on the various generalizations of real-valued Gronwall-BellmanBihari type inequalities are established. See [4–12], [13, CH.XII], [14–16], and the references given in this literature Another direction of generalizations is the development of the abstract Gronwall lemma. We extend the Gronwall-Bellman-Bihari inequality (3) and (4) to the form (5) and the operator A in (5) is generalized from a composition operator to the class F of operators. By restricting our results to Euclidean spaces we study systems of real-valued Gronwall-type inequalities. We denote by C(X1, X2) and C1(X1, X2) the space of continuous operators and the space of continuously Frechet differentiable operators, respectively, from X1 into X2

Preliminaries

Some Vector-Valued Gronwall-Type Inequalities

Systems of Real-Valued Gronwall-Type Inequalities

A Real-Valued Gronwall-Type Inequality