A Note on the Integral Inequalities with Two Dependent Limits

Abstract

The theorem on the Gronwall's type integral inequalities with two dependent limits is established. In application, the boundedness of the solutions of nonlinear differential equations is presented.

1. Introduction

Integral inequalities play a significant role in the study of qualitative properties of solutions of integral, differential and integro-differential equations (see, e.g., [1–4] and the references given therein). One of the most useful inequalities in the development of the theory of differential equations is given in the following lemma (see [5]).

Lemma 1.1.

Let and be real-valued nonnegative continuous functions for all . If

(1.1)

for all , where is a real constant, then

(1.2)

for all .

Note that the generalization of this integral inequality and its discrete analogies are given in papers [5–8]. In paper [9] the following useful inequality with two dependent limits was established.

Lemma 1.2.

Let be a real-valued nonnegative continuous function defined on and let and be nonnegative constants. Then the inequality

(1.3)

implies that

(1.4)

The theory of integral inequalities with several dependent limits and its applications to differential equations has been investigated in [10–14].

The present study involves some Gronwall's type integral inequalities with two dependent limits. Section 2 includes some new integral inequalities with two dependent limits and relevant proofs. Subsequently, Section 3 includes an application on the boundedness of the solutions of nonlinear differential equations.

2. A Main Statement

Our main statement is given by the following theorem.

Theorem 2.1.

Let , , , , , and be real-valued nonnegative continuous functions defined on .

• (i) Let be a nonnegative constant. If

  

  (2.1)

for then

(2.2)

for all

• (ii) Let be a real constant. If

  

  (2.3)

for then

(2.4)

for all

• (iii) Let be a real-valued positive continuous and nondecreasing function defined on and be a real constant. If

  

  (2.5)

for then

(2.6)

for all

• (iv) Let and its partial derivative be real-valued nonnegative continuous functions on and let be even function in If

  

  (2.7)

for then

(2.8)

for all . Here

(2.9)

where

(2.10)

(2.11)

Proof.

• (i) Define a function by

  

  (2.12)

Note that is a nonnegative function and Then (2.1) can be rewritten as

(2.13)

It is easy to see that is an even function.

First, let ; then (2.12) can be rewritten as

(2.14)

Differentiating (2.14) and using (2.13), we get

(2.15)

Dividing both sides of (2.15) by , we get

(2.16)

Integrating the last inequality from to , we get

(2.17)

Second, let . Then, (2.12) can be written as

(2.18)

Differentiating (2.18) and using (2.13), we get

(2.19)

Dividing both sides of (2.19) by , we get

(2.20)

Integrating (2.20) from to 0, we get

(2.21)

Finally, using (2.17) and (2.21), we obtain

(2.22)

The inequality (2.2) follows from (2.13) and (2.22).

• (ii) Define a function by

  

  (2.23)

It is evident that is an even and nonnegative function. We have that

(2.24)

Using Young's inequality (see, e.g., [2]), we obtain that

(2.25)

Let . Then

(2.26)

Differentiating (2.26), we get

(2.27)

Using (2.24) and (2.25), we get

(2.28)

Denoting

(2.29)

we get

(2.30)

From that it follows that

(2.31)

for any Integrating the last inequality from to and using , we get

(2.32)

It is easy to see that

(2.33)

Then

(2.34)

Since , we have that

(2.35)

Applying (2.24), we obtain

(2.36)

From (2.36), and (2.29) it follows (2.4) for Let ; then

(2.37)

Using (2.24) and (2.25), we get

(2.38)

From that it follows that

(2.39)

for any Integrating the last inequality from to and using , we get

(2.40)

It is easy to see that

(2.41)

Then

(2.42)

Since , we have that

(2.43)

Applying (2.43) and (2.24), we obtain (2.36) for Then from (2.36) and (2.29), (2.4) follows for

• (iii) Since is a positive, continuous, and nondecreasing function for , we have that

  

  (2.44)

Now the application of the inequality proven in (ii) yields the desired result in (2.6).

• (iv) We define a function by

  

  (2.45)

Evidently, the function is a nonnegative, monotonic, and nondecreasing in and We have that

(2.46)

Let . Then

(2.47)

Differentiating (2.47), we get

(2.48)

Using (2.46) and Young's inequality, we obtain that

(2.49)

Using (2.29), we get

(2.50)

Applying the differential inequality, we get

(2.51)

Since , we have that

(2.52)

Using (2.33), we get

(2.53)

Since , we have that

(2.54)

Using (2.9) and (2.11), we get

(2.55)

Let . Then

(2.56)

Differentiating (2.56), we get

(2.57)

Using (2.46) and Young's inequality, we obtain that

(2.58)

Using (2.29), we get

(2.59)

Applying the differential inequality, we get

(2.60)

Since , we have that

(2.61)

Using (2.41), we get

(2.62)

Since , we have that

(2.63)

Using (2.9) and (2.10), we get

(2.64)

The inequality (2.8) follows from (2.29), (2.55), and (2.64). Theorem 2.1 is proved.

3. An Application

In this section, we indicate an application of Theorem 2.1 (part (ii)) to obtain the explicit bound on the solution of the following boundary value problem for one dimensional partial differential equations:

(3.1)

where is a fixed real number and . Let , , , , , be smooth functions and problem (3.1) has a unique smooth solution Assume that

(3.2)

for all Here and are real-valued nonnegative continuous functions defined on .

This allows us to reduce the nonlocal boundary-value (3.1) to the initial-value problem

(3.3)

in a Hilbert space with a self-adjoint positive definite operator defined by the formula with the domain (see, e.g., [15, 16]).

Let us give a corollary of Theorem 2.1.

Theorem 3.1.

The solution of problem (3.1) satisfies the estimates

(3.4)

for all Here

Proof.

It is known thatthe formula (see, e.g., [15, 16])

(3.5)

gives a solution of problem ( 3.3 ). Here

(3.6)

Applying the triangle inequality, condition (3.2), formula (3.5), and estimates (see, e.g., [17])

(3.7)

we get

(3.8)

Since

(3.9)

we have that

(3.10)

Denote that Then

(3.11)

for Applying the integral inequality (2.4), we get

(3.12)

We have that

(3.13)

Therefore, the inequality (3.4) follows from the last inequality. Theorem 3.1 is proved.