On a Generalized Retarded Integral Inequality with Two Variables

Abstract

This paper improves Pachpatte's results on linear integral inequalities with two variables, and gives an estimation for a general form of nonlinear integral inequality with two variables. This paper does not require monotonicity of known functions. The result of this paper can be applied to discuss on boundedness and uniqueness for a integrodifferential equation.

1. Introduction

Gronwall-Bellman inequality [1, 2] is an important tool in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations and integral equations. There can be found a lot of its generalizations in various cases from literature (see, e.g., [1–12]). In [11], Pachpatte obtained an estimation for the integral inequality

(1.1)

His results were applied to a partial integrodifferential equation:

(1.2)

for boundedness and uniqueness of solutions.

In this paper, we discuss a more general form of integral inequality:

(1.3)

for all . Obviously, appears linearly in (1.1), but in our (1.3) it is generalized to nonlinear terms: and . Our strategy is to monotonize functions s with other two nondecreasing ones such that one has stronger monotonicity than the other. We apply our estimation to an integrodifferential equation, which looks similar to (1.2) but includes delays, and give boundedness and uniqueness of solutions.

2. Main Result

Throughout this paper, are given numbers. Let and . Consider inequality (1.3), where we suppose that is strictly increasing such that , and are nondecreasing, such that and , , and are given, and are functions satisfying and for all .

Define functions

(2.1)

Obviously, , and in (2.1) are all nondecreasing and nonnegative functions and satisfy . Let

(2.2)

(2.3)

(2.4)

Obviously, , and are strictly increasing in , and therefore the inverses , and are well defined, continuous, and increasing. We note that

(2.5)

Furthermore, let which is also nondecreasing in for each fixed , and and satisfies .

Theorem 2.1.

If inequality (1.3) holds for the nonnegative function , then

(2.6)

for all , where

(2.7)

and is arbitrarily given on the boundary of the planar region

(2.8)

Here denotes the domain of a function.

Proof.

By the definition of functions and , from (1.3) we get

(2.9)

for all .

Firstly, we discuss the case that for all . It means that for all . In such a circumstance, is positive and nondecreasing on and

(2.10)

Regarding (1.3), we consider the auxiliary inequality

(2.11)

for all , where is chosen arbitrarily. We claim that

(2.12)

for all , where is defined by (2.8).

Let denote the right-hand side of (2.11), which is a nonnegative and nondecreasing function on . Then, (2.11) is equivalent to

(2.13)

By the fact that for and the monotonicity of , and , we have

(2.14)

for all . Integrating the above from to , we get

(2.15)

for all . Let

(2.16)

From (2.15), (2.16), we obtain

(2.17)

for all . Let denote the right-hand side of (2.17), which is a nonnegative and nondecreasing function on . Then, (2.17) is equivalent to

(2.18)

From (2.13), (2.16), and (2.18), we have

(2.19)

for all , where is defined by (2.8). By the definitions of , and , is continuous and nondecreasing on and satisfies for . Let . Since and for , from (2.19) we have

(2.20)

for all . Integrating the above from to , by (2.4) we get

(2.21)

for all . By (2.19) and the above inequality, we obtain

(2.22)

for all , where is defined by (2.8). It follows from (2.5) that

(2.23)

which proves the claimed (2.12).

We start from the original inequality (1.3) and see that

(2.24)

for all ; namely, the auxiliary inequality (2.11) holds for . By (2.12), we get

(2.25)

for all . This proves (2.6).

The remainder case is that for some . Let

(2.26)

where is an arbitrary small number. Obviously, for all . Using the same arguments as above, where is replaced with , we get

(2.27)

for all . Letting , we obtain (2.6) because of continuity of in and continuity of , and . This completes the proof.

3. Applications

In [11], the partial integrodifferential equation (1.2) was discussed for boundedness and uniqueness of the solutions under the assumptions that

(3.1)

respectively. In this section, we further consider the nonlinear delay partial integrodifferential equation

(3.2)

for all , where , and are supposed to be as in Theorem 2.1; , , , and are all continuous functions such that . Obviously, the estimation obtained in [11] cannot be applied to (3.2).

We first give an estimation for solutions of (3.2) under the condition

(3.3)

Corollary 3.1.

If is nondecreasing in and and (3.3) holds, then every solution of (3.2) satisfies

(3.4)

where

(3.5)

and , and are defined as in Theorem 2.1.

Corollary 3.1 actually gives a condition of boundedness for solutions. Concretely, if there is a positive constant such that

(3.6)

on , then every solution of (3.2) is bounded on .

Next, we give the condition of the uniqueness of solutions for (3.2).

Corollary 3.2.

Suppose

(3.7)

where are defined as in Theorem 2.1. There is a positive number such that

(3.8)

on . Then, (3.2) has at most one solution on , where are defined as in Theorem 2.1.