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).