DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

CONTINUOUS AND DISCRETE NONLINEAR INEQUALITIES
AND APPLICATIONS TO BOUNDARY VALUE PROBLEMS
WING-SUM CHEUNG
We give some new nonlinear integral and discrete inequalities of the Gronwall-Bellman-
Ou-Iang type in two variables. These on the one hand generalize and on the other hand
furnish a handy tool for the study of qualitative as well as quantitative properties of solutions
of differential and difference equations. We illustrate this by applying our new
results to certain boundary value problems.
Copyright © 2006 Wing-Sum Cheung. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In studying the boundedness behavior of the solutions of certain second-order differential
equations, Ou-Iang established the following Gronwall-Bellman-type integral inequality
which is now known as Ou-Iang’s inequality in the literature.
Theorem 1.1 (Ou-Iang [17]). If u and f are nonnegative functions on [0,∞) satisfying
∫ x
u 2 (x) ≤ k 2 +2 f (s)u(s)ds (1.1)
0
for all x ∈ [0,∞),wherek ≥ 0 is a constant, then
∫ x
u(x) ≤ k + f (s)ds, ∀x ∈ [0,∞). (1.2)
0
An important feature of Ou-Iang-type inequalities or more generally Gronwall-
Bellman-Ou-Iang-type inequalities [2, 10] is that they provide explicit bounds on the
unknown function in terms of known functions. This makes such inequalities especially
important in many practical situations. In fact, over the years, such inequalities and their
generalizations to various settings have proven to be very effective in the study of many
qualitative as well as quantitative properties of solutions of differential equations. These
include, among others, the global existence, boundedness, uniqueness, stability, and continuous
dependence on initial data (see, e.g., [3–5, 7, 8, 11, 13–16, 18–22]). For example,
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 299–313