Abstract and Applied Analysis

Volume 2014 (2014), Article ID 973586, 8 pages

http://dx.doi.org/10.1155/2014/973586

Research Article

On Some Vector-Valued Inequalities of Gronwall Type

Dah-Chin Luor

Department of Applied Mathematics, I-Shou University, No. 1, Sec. 1, Syuecheng Road, Dashu District, Kaohsiung City 84001, Taiwan

Received 25 October 2013; Accepted 12 December 2013; Published 22 January 2014

Academic Editor: Weinian Zhang

Copyright © 2014 Dah-Chin Luor. 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.

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.

1. 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. The Gronwall inequality was established in 1919 by Gronwall [1] and then it was generalized by Bellman [2]. In fact, if

where and , and are nonnegative continuous functions on , then

This result plays a key role in studying stability and asymptotic behavior of solutions to differential equations and integral equations. One of the important nonlinear generalizations of (1) and (2) was established by Bihari [3]. Assume that , and are nonnegative continuous functions on , and is a positive increasing continuous function on . Bihari showed that if

and , then

where , . By choosing , inequality (4) can be reduced to the form (2). Many results on the various generalizations of real-valued Gronwall-Bellman-Bihari 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. These results are closely related to the fixed points of operators. See [17, 18], [13, CH.XIV], [19], [20, Proposition 7.15], and the references given in this literature.

Inequality (3) can be written in a general form

where is a positive operator on continuous functions. If is a composition operator defined by , then (5) is reduced to (3). We show that if belongs to the class of operators which is defined in Section 5, then we have an upper estimate of which is similar to the form (4). It is worth pointing out that the class includes integral operators

maximal operators

geometric mean operators

and geometric maximal operators

We discuss these operators and the class in Section 5. We extend the Gronwall-Bellman-Bihari inequality (3) and (4) to the form (5) and the operator in (5) is generalized from a composition operator to the class of operators.

The aim of this paper is to investigate vector-valued inequalities in ordered Banach spaces. Under suitable conditions we have estimates for which are similar to (4). By restricting our results to Euclidean spaces we study systems of real-valued Gronwall-type inequalities. For the one-dimensional case, real-valued inequalities of the form (5) with are discussed.

Throughout this paper, let , , where , let be Banach spaces, ; and be an ordered Banach space with an order cone (see [20, Definition 7.1]). We denote by and the space of continuous operators and the space of continuously Fréchet differentiable operators, respectively, from into .

2. Preliminaries

For , the integral of on and the derivative of at , which are denoted by and , respectively, can be defined as generalizations of the usual definitions of integral and derivative for real-valued functions. One may see [20, Sections 3.1 and 3.2] for the definitions and properties of the integral and derivative. In particular, if exists for all , where and are defined by one-sided limits, and if is continuous on , then . If we define for all , then exists for all and . See [20, Propositions 3.5 and 3.7]. By defining

we see that is an ordered Banach space with sup-norm and the order cone . If in , then .

By [21, Theorem 7.1.9] we see that, for , the Fréchet derivative exists if and only if exists and

Here we denote by the value of at .

For , , let and we define . Then is a norm and , with coordinatewise linear operations, is a Banach space. If are ordered Banach spaces with ordered cones , respectively, then is an ordered Banach space with an order cone . It is easy to see that a sequence of points in converges to a point if and only if converges to in for each . For , , we define . Then . If all exist for , then the derivative of at exists and .

Let , , and be Banach spaces, and , are nonempty open subsets of , , respectively, and let and . Suppose that is continuous and Fréchet differentiable at a point and that is continuous and Fréchet differentiable at the point . Then is continuous and Fréchet differentiable at , and

Let , , and be Banach spaces, and is a nonempty open subset of , and let be given by . For , let and let for all . It is clear that is open and . If has a Fréchet derivative at , then we define the partial Fréchet derivative of at with respect to the variable to be ; it is a linear operator of into . The derivative is defined similarly. If is Fréchet differentiable at , then is Fréchet differentiable with respect to both variables at and

for all . Moreover, is continuously Fréchet differentiable in a neighborhood of if and only if all partial Fréchet derivatives are continuous in a neighborhood of . Similar results hold for maps of the form .

Notation. Here we give notations used in this paper for reader’s convenience. , , , , , and is defined by

3. Some Vector-Valued Gronwall-Type Inequalities

In this section we consider vector-valued inequalities of the form

Theorem 1 gives an estimate of which is similar to the form (4).

Theorem 1. Let . Suppose that and is bijective and is of monotone type, and suppose that there exist monotone increasing operators , , such that for all ,

We also suppose that there exist and , , such that is bijective and monotone increasing and is of monotone type, and for ,

If satisfies (15), then

where is defined by

Moreover, if there exists such that for all , then

where in (21) is the constant function in with value .

Proof of Theorem 1. Let . By (15) we have for all . By the chain rule we see that, for and ,

Since , and we see that

This implies

Since is bijective, we write to be the solution of the equation . By in and [20, Proposition 7.37] we see that . This shows that for and hence

Since is monotone increasing, we see that is positive and hence monotone increasing for each . Therefore for ,

For we obtain

By letting and the condition ,

for all . Since and are continuous on , inequality (28) holds for all . This implies for all . Therefore and in .

Since and are monotone increasing, it follows that if for all then in and we obtain (21). This completes the proof.

In the following we consider two particular cases as examples of Theorem 1. We see that under these cases, conditions in Theorem 1 can be reduced to more simpler forms.

In the case and , we see that is given by

Moreover, is the zero operator of into and is the identity operator of into . We have the following corollary.

Corollary 2. Let and let , , , , , , , and be given as in Theorem 1 with . Suppose that , and

for each and for all . If satisfies

then we have (19)–(21).

Remark 3. If and for each , then (32) is reduced to the following integral inequality:

By [20, Proposition 3.7] the item in (17) and (30) can be replaced by . Moreover, if is monotone increasing for each , then we can choose , , in Corollary 2 and condition (30) is redundant.

Consider the case that are ordered Banach spaces, , and . We see that is given by

Moreover, is the zero operator of into and is the operator of into such that for , the th element of is and the other elements are zero.

Remark 4. Let , , and is defined by for . Suppose that each is injective and is of monotone type. Suppose that for any , where , there exist such that for each . Then is bijective and is of monotone type.

The following corollary can be obtained by Theorem 1.

Corollary 5. Let and let , , , , and be given as in Theorem 1 with . Let , be given as in Remark 4. Suppose that , , and there exist and monotone increasing operators , , such that

for and for all , where and the th element of is and the other elements are zero. If satisfies the systems

for all , then we have (19)–(21).

Remark 6. If and for each , then (37) is reduced to the system of integral inequalities

The item in (35) can be replaced by . Moreover, if is monotone increasing for each , then we can choose , , in Corollary 5 and condition (35) is redundant.

4. Systems of Real-Valued Gronwall-Type Inequalities

In this section we apply results in Section 3 to obtain systems of real-valued Gronwall-type inequalities. Consider the case , , , , and . Let , , , , , and , , , . Let , .

Define by

Then (15) can be reduced to the system

Remark 7. Suppose that is injective and is of monotone type for each . Suppose that for any , where , there exist such that for each . Then is bijective and is of monotone type.

Since

and, for , the linear transform can be represented by the matrix , conditions (16)-(17) are reduced to

where , , are monotone increasing operators. Here the vector in (43) is written as a column matrix. In particular, if , then and (43) is reduced to

Remark 8. Define by . Suppose that is injective and monotone increasing and is of monotone type for each , and, for any , there exist such that for each . Then is bijective and monotone increasing and is of monotone type. Moreover, the linear transform can be represented by the Jacobian matrix , , . Hence (18) is reduced to

where . Here the vectors and in (45) are written as column matrices.

Theorem 9. Let and the conditions of , be given in Remark 7. Suppose that , and there exist monotone increasing operators , , such that (42)-(43) are satisfied. We also suppose that conditions for , in Remark 8 hold and (45) is satisfied. If satisfies the system (40), then one has (19)–(21) with .

Let , . If we define by

then (15) can be reduced to the system

Remark 10. Suppose that is bijective and is of monotone type for each . Then defined by (46) is bijective and is of monotone type, and

where . In particular, the function is given by

Example 11. Define , where . If is strictly increasing from onto , then is bijective and is of monotone type, and (48) can be reduced to

where .

Remark 12. Let , . Define by

Suppose that is strictly increasing from onto for ; then it is easy to see that is bijective and monotone increasing and is of monotone type, and

Moreover, the Jacobian matrix is a diagonal matrix with diagonal entries , . Hence in (18) can be represented by the diagonal matrix with diagonal entries , .

The following theorem can be obtained by Theorem 1.

Theorem 13. Let and the conditions of , be given as in Remark 10. Suppose that , , and there exist monotone increasing operators , , such that (42)-(43) are satisfied. We also suppose that there exist , , and , , , such that conditions for and in Remark 12 hold, and (18) with are satisfied. If satisfies (47), then we have

where

Moreover, if there exists such that for all , , then

where in (55) is the constant function in with value .

Consider the particular case and , of Theorem 13. In this case and (47) is reduced to

Assume that

We also assume that there exist monotone increasing operators , , such that

We also suppose that there exist and , , such that conditions for in Remark 12 hold and satisfy the following condition:

where

Then Theorem 13 implies that if satisfies (56) then we have (53) with

Remark 14. Condition (59) can be satisfied if there exist , , such that

and we choose

where is some constant. Moreover, defined by (61) satisfies , where

By a little modification of Theorem 13 we have Corollary 15.

Corollary 15. Let and the conditions of be given as in Remark 10. Suppose that (57) holds, and there exist monotone increasing operators , , such that (58) is satisfied. For , we suppose that there exist and such that (62) holds. Define by (63). Assume that for all , where is defined by (61). If satisfies (56), then we have (53) and (55) with (64).

As an example we consider the system of integral inequalities

Here is strictly increasing from onto , , , and . Assume that for each , is monotone increasing and there exist and such that

for all . Moreover, we suppose that for all , where is given in (63). By Example 11 and Corollary 15 with , , , and , we have

5. A Real-Valued Gronwall-Type Inequality

In this section we establish a real-valued Gronwall-type inequality which extends (3) from composition operators to . Consider the case of (65):

Here , is strictly increasing and , , , and . Let be the class of operators such that is positive and monotone increasing, and there exist and such that for all increasing , we have

By some modification of results in Section 4 we have Theorem 16.

Theorem 16. Let and , , , and be given as above. Suppose . Let , where and . Assume that . If satisfies (68), then for all we have

It is easy to see that if , then for . If , then . Moreover, we also have , where . Hence the class of operators is closed under multiplication and linear combination with nonnegative coefficients. This and the following examples show that Theorem 16 can be applied to a variety of operators in (68).

Example 17. Let

where is increasing. Then with in (69). In this case and , Theorem 16 is reduced to (3)-(4).

Example 18. Let

Then with and in (69).

Example 19. Let

where and is increasing. Then with in (69).

Example 20. Let

where and is increasing. Then with in (69).

Example 21. Let

where and are given as in Example 19. Then with in (69).

Example 22. Let

where and are given as in Example 20. Then with in (69).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Science Council, Taipei [NSC 100-2115-M-214-004].

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