2
Gronwall’s Inequality
and
Systems of Integral Equations
Bradford Garcia
Undergraduate Thesis Project
Honors College
Texas A&M University – Commerce
3
Abstract
Differential Equations is a field of research which deals in the modeling of dynamical systems, be it a population of rabbits or the transmission of the Zika virus. Expanding our discussion to include integral and integro-differential equations, we similarly expand our applications to include more complex systems, such as those undergoing oscillation. Hence, the number of disciplines to which the study of differential equations is valuable is substantial, with biology, physics, engineering, and economics being just a few of them. However, in spite of the great advances achieved in the past century, a lot remains to be discovered, and even more remains to be refined. Therefore, this thesis project aims to make improvements upon the results of some of the predecessors in the field, specifically with regard to further generalizing Gronwall’s Inequality and applying such generalizations to systems of integral equations in order to estimate solutions.
Mathematical study and research are very suggestive of mountaineering. Whymper made several efforts before he climbed the Matterhorn in the 1860’s and even then it cost the life of four of his party. Now, however, any tourist can be hauled up for a small cost, and perhaps does not appreciate the difficulty of the original ascent. So in mathematics, it may be found hard to realize the great initial difficulty of making a little step which now seems so natural and obvious, and it may not be surprising if such a step has been found and lost again.
-Louis Joel Mordell (1888−1972) 4
Historical Introduction
Named after Thomas Hakon Grönwall, Gronwall’s inequality was a huge breakthrough in
the field of differential equations. Originally solved by Grönwall in 1919 and further generalized
by Richard Bellman in 1943, only a finite number of improvements have been made since then.
Gronwall's Inequality concerns itself with finding a bound for a function that is known to satisfy
a certain differential or integral inequality. Development of Gronwall's Inequality usually takes
the form of altering the conditions in order to widen the breadth of its applications, or else
tightening the bound(s) in order to achieve a more accurate estimate of the solution(s).
Grönwall’s original result was discovered while studying a system of differential
equations with a parameter. The following lemma was given as thus in that 1919 paper:
Theorem 1.1 (Gronwall’s Original Inequality): Let α, a, b and h be nonnegative constants, and
[ ] [ be continuous. If
then
As previously mentioned, Bellman worked to make generalizations of Gronwall’s
inequality some 24 years after its first conception. One such example (shown below) allowed for
b to be a function f and represented integration of a over α ≤ t ≤ α + h as a constant c:
0 5
Theorem 1.2 (Gronwall’s Inequality): Let α, β and c be nonnegative constants, and
[ ] [ be continuous. If
then
Later, in 1958, Bellman took these generalizations one step further by allowing that c be
a nonnegative and non-decreasing function, which is stated as following:
Theorem 1.3 (Gronwall-Bellman’s Inequality): If u(t) and α(t) are real valued continuous
functions on [a,b], α(t) is non-decreasing, and β(t) ≥ 0 is integrable on [a,b] with
then
( u 6
B. G. Pachpatte assembled a thorough compilation of the innovations made to Gronwall's
Inequality up to 1998, the year in which his book was published. While this mathematical
anthology of sorts does much to show just how much research of Gronwall-type inequalities has
exploded in recent years, the generalization of particular interest to this thesis project comes
from a 2012 publication by Bohner, Hristova and Stefanova. Therein, they generalize Gronwall-
Bellman’s Inequality to the following nonlinear inequality:
(1)
Then, in 2014, Wang generalized and simplified Inequality (1) so that becomes a
more general composite function, . He also unified the two summations and allowed
for a product of more terms as follows:
(2)
In progressing past the historical background of Gronwall’s Inequality, we will now
generalize Inequality (2) further so that K will be a nonnegative and non-decreasing function,
and will become , forming what is known as a nonlinear Volterra-type integral
inequality, as seen below:
Nonlinear Volterra-type Integral Inequality
(3)
( u 7
With each subsequent generalization of Theorem 1.1, the list of notations and conditions
will necessarily grow larger; so it is with Inequality (3):
Theorem 2.1: Define [ . Let and T be constants such that .
(A1) [ [ is continuous and non-decreasing;
(A2) [ are non-decreasing and [ ;
(A3) [ [ are continuous and non-decreasing in terms of t
for ;
(A4) are non-decreasing, and if ;
(A5) is increasing, , and ;
(A6) [ .
If satisfies Inequality (3) with conditions (A1) through (A6), then
Note that
8
Proof: Let η be an arbitrary constant such that . Then by (A1) and (A3),
Inequality (3) becomes, for [ ],
Define, for ,
It follows directly that is non-decreasing, and ( ) [ ]. By (A5),
exists and has the same properties as those of w. Therefore,
In addition,
Thus, for [ ],
( w ( Z u(max 9
Let
Clearly, , and . Then
That is,
Integrate the above inequality from to [ ], to get
( Z ( R ( d 10
Since W is continuous and increasing, exists. Hence,
Letting in the above inequality results in
Since is arbitrary and , we finally obtain that
Let us consider a very simple case of Inequality (3) for applications:
W 11
Corollary 2.2: Given
or
where g, f, H, w and u satisfy (A1), (A3), (A4), (A5), and (A6), respectively. Then
where
System of Integral Equations
(4)
Moving on toward what it actually looks like applying Gronwall’s Inequality in order to
estimate solutions of an equation or inequality, above is what would be classified as a system of
integral equations. Before continuing, note promptly that, by the magnitude of a square
matrix A, we mean | | √Σ Σ
.
( w 12
In 2009, Wang examined equations of the form (4) through the lens of Lyapunov’s
second method, a useful tool for estimating solutions of differential and integral equations. The
resulting theorem follows:
Theorem 3.1: Let be continuous on and be an real matrix of
continuous functions on . Assume that
| |
,
| | for , where is the derivative from the right
with respect to t.
If is a solution of (1), then
where
In order to distinguish ourselves from Wang’s result, let’s apply the following variant of
Gronwall’s Inequality in place of Lyapunov’s second method:
13
Theorem 3.2: Let and f be nonnegative continuous functions defined on [ ].
Let and its partial derivative
be nonnegative continuous functions for
and
Then
where
We can now proceed with the result by applying Gronwall’s Inequality in the form of
Theorem 3.2 to estimate the solution to systems of integral equations in the form of (4):
( u 14
Theorem 3.3: Consider the system of integral equations
Let be continuous on and be an matrix of continuous functions with
| | and
| | exists. Further, let and . Then
where
Proof: Because Theorem 3.2 is applicable only to integral equations whose lower bound of
integration is constant, we must solve the problem of our variable lower bound of integration.
Hence, use b to split the integral in two according to the algebra of integration, resulting in
X 15
Taking the magnitude of both sides and applying the well-known triangle inequality gives
Keeping in mind that , it follows that
We can now apply Gronwall’s Inequality in the form of Theorem 3.2 to this inequality by setting
| | | | and | |. Doing so
will bring us to the conclusion of our proof.
Conclusion
While we were able to solve the problem of the variable lower bound of integration, the
simple patch of designating an arbitrary constant b sacrifices significant accuracy and tightness
of the estimate bound in most cases. Nevertheless, the knowledge of and ability to apply so
useful a tool as Gronwall’s Inequality is invaluable in its own right. If mathematics truly is like
mountaineering, then it should be just as much about the journey as the destination.
X X 16
Bibliography
1. R. Bellman, The stability of solutions of linear differential equations, Duke Math. J., 10(1943), 643-647.
2. R. Bellman, Asymptotic Series for the Solutions of Linear Differential-Difference Equations, Rendiconti del Circolo Matematico di Palermo, 7(1958), no. 3, 261-269.
3. M. Bohner, S. Hristova, K. Stefanova, Nonlinear Integral Inequalities Involving Maxima of the Unknown Scalar Functions, Math. Inequal. Appl. 15(2012), no. 4, 811-825.
4. T. A. Burton, Volterra Integral and Differential Equations, second edition, Elsevier, New York, (2005).
5. T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of differential equations, Ann. of Math., 20(2)(1919), 292-296.
6. B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, London, (1998).
7. T. Wang, Generalization of Gronwall’s Inequality and Its Applications in Functional Differential Equations, Communications in Applied Analysis, (2014).
8. T. Wang, Inequalities of solutions of Volterra integral and differential equations, E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, (2009), No. 28., pp. 1-10.
9. T. Wang, Stability in Abstract Functional Differential Equations, Part I: General Theorems, J. Math. Anal. Appl., No.2, pp.534-558, Vol.186 (1994).