Abstract and Applied Analysis

Volume 2013 (2013), Article ID 582161, 8 pages

http://dx.doi.org/10.1155/2013/582161

Research Article

Measure Functional Differential Equations in the Space of Functions of Bounded Variation

S. M. Afonso¹ and A. Rontó²

¹Departamento de Matemática, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista (UNESP), Campus de Rio Claro, CP 178, 13506-900 Rio Claro, SP, Brazil

²Institute of Mathematics, Academy of Sciences of the Czech Republic, Branch in Brno, 22 Žižkova Street, 616 62 Brno, Czech Republic

Received 21 May 2013; Accepted 8 August 2013

Academic Editor: Rodrigo Lopez Pouso

Copyright © 2013 S. M. Afonso and A. Rontó. 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.

We establish general conditions for the unique solvability of nonlinear measure functional differential equations in terms of properties of suitable linear majorants.

1. Introduction, Motivation, and Problem Setting

Let , be the norm in , and let be the Banach space of functions of bounded variation with the standard norm , where .

Our aim is to examine the solvability of the equation

is a, generally speaking, nonlinear operator and is a nonlinear vector functional. The integral on the right-hand side of (1) is the Kurzweil-Stieltjes integral with respect to a nondecreasing function . We refer to [1–5] for the definition and properties of this kind of an integral, recalling only that (1) is a particular case of a generalised differential equation [2, 6]. It is important to note that, for any , the Kurzweil-Stieltjes integral in (1) exists (see, e.g., [4, 7]) and, therefore, the equation itself makes sense.

By a solution of (1), we mean a vector function which has bounded variation and satisfies (1) on the interval .

Equation (1) is an extension of a measure differential equation studied systematically, for example, in [2, 8–10]. It is a fairly general object which includes many other types of equations such as differential equations with impulses [11] or functional dynamic equations on time scales [12] (see, e.g., [13, 14]). In particular, if , , (1) takes the form

and, thus, in the absolutely continuous case, reduces to the nonlocal boundary value problem for a functional differential equation

whose various particular types are the object of investigation of many authors (see, e.g., [15–19]). A more general choice of in (1) allows one to cover further cases where solutions lose their absolute continuity at some points. For example, consider the impulsive functional differential equation [16, page 191]

where for any function from (in fact, if, as is customary [11] in that context, a solution is assumed to be left continuous). Here, , the jumps may occur at the preassigned times , and their action is described by the operators , . By the usual integration argument [11], one can represent (4) alternatively in the form

It follows, in particular, from [14, Lemma  2.4] that (5) is equivalent to the measure functional differential equation

with , , and defined by the relation

Thus, system (4) can be considered as a particular case of (1). Likewise, an appropriate construction [13, 20] allows one to regard differential equations on time scales [12] as measure differential equations. The same is true for equations involving functional components; in the case of a differential equation on a time scale with retarded argument, by choosing suitably [13], one arrives at the equation

In (8), is a functional in the first variable, is from the space of continuous functions on , and the Krasovsky notation , , is used [21, Chapter VI]. Finally, eliminating the initial function from (8) in a standard way by transforming it to a forcing term (see [15]), we conclude that the resulting equation falls into the class of equations of form (1).

Note that, by measure functional differential equations, the Volterra type equations of form (8) are usually meant in the existing bibliography on the subject (see, e.g., [8, 13, 22]), whereas equations with more general types of argument deviation are rather scarce (we can cite, perhaps, only [4, page 217]). Comparing (8) with (1), we find that the latter includes non-Volterra cases as well.

This list of examples can be continued. It is interesting to observe that solutions of problems of type (3) studied in the literature up to now are always assumed, at least locally, to be absolutely continuous [16], or even continuously differentiable [23]. In contrast to this, the gauge integral involved in (1) allows one to deal with a considerably wider class of solutions, which are, in fact, assumed to be of bounded variation only. A possible noteworthy consequence for systems with impulses may be that the unpleasant effect of the so-called pulsation phenomenon [11, page 5] might be more natural to be dealt with in the framework of the space . Our interest in (1), originally motivated by a relation to problems of type (3), has strengthened still further due to the last observation.

The general character of the object represented by (1) suggests a natural idea to examine its solvability by comparing it to simpler linear equations with suitable properties. Here, we show that such statements can indeed be obtained rather easily by analogy to [24–26]. The key assumption is that certain linear operators associated with the nonlinear operator appearing in (1) possess the following property.

Definition 1. Let be a linear mapping. One says that a linear operator belongs to the set if the equation

has a unique solution for any from , and, moreover, the solution is nonnegative for any nonnegative .

The property described by Definition 1, in fact, means that the linear operator associated with (9) is positively invertible on , and thus it corresponds to the existence and positivity of Green’s operator for a boundary value problem [15].

Remark 2. The inclusion , generally speaking, does not imply that for !

The question on the unique solvability of (1) is thus reduced to estimating the nonlinearities suitably, so that the appropriate majorants generate linear equations with a unique solution depending monotonously on forcing terms. The problem of finding such majorants is a separate topic not discussed here. We only note that, in a number of cases, the existing results on differential inequalities can be applied (see, e.g., [17–19]).

Note that, due to the nature of the techniques used, statements of this kind available in the literature on problems of type (3), as a rule, are established separately in every concrete case (see, e.g., [27–29]). Here, we provide a simple unified proof, which is, in a sense, independent on the character of the equation and also allows one to gain a considerable degree of generality. The results may be useful in studies of the solvability of various measure functional differential equations and, in particular, of problem (3) and its generalisations (note that, e.g., rather complicated neutral-type functional differential equations [23] can be formulated in form (1); see also [4, 30]).

2. Unique Solvability Conditions

We are going to show that the knowledge of the property for certain linear operators and associated with (1) allows one to guarantee its unique solvability.

2.1. Nonlinear Equations

The following statement is true.

Theorem 3. Assume that there exist certain linear operators , , such that

for arbitrary functions , with the property

Furthermore, let the inclusions

be fulfilled with some linear functionals , . Then (1) has a unique solution for an arbitrary such that

whenever (11) holds.

The inequality sign and modulus for vectors in (10), (11), (13), and similar relations below are understood componentwise. The theorem as well as the other statements formulated below will be proved later.

Theorem 4. Let there exist certain linear operators , , and linear functionals , satisfying the inclusions

and such that (13) and the inequality

is true for arbitrary functions and of bounded variation with property (11). Then (1) is uniquely solvable.

Theorem 4 is, in fact, an alternative form of Theorem 3, where the estimate of a “linear part” is more visible.

In other statements, we need the following natural notion of positivity of a linear operator in the space .

Definition 5. A linear operator will be called positive if is a nonnegative function for an arbitrary nonnegative from .

Note that no monotonicity assumptions are imposed on in Theorem 4. In the cases where the positivity of certain linear majorants is known, the following statement may be of use.

Corollary 6. Assume that there exist some positive linear operators , , such that the inequalities

hold on for any and from with property (11). Moreover, let one can specify linear functionals , , satisfying (13), and such that the inclusions

hold for a certain . Then (1) has a unique solution.

Corollary 6 allows one to obtain, in particular, the following statements.

Corollary 7. Assume that, for arbitrary and from with property (11), and satisfy estimates (13) and (16) with some linear functionals , and positive linear operators , . Then the inclusions

guarantee that (1) is uniquely solvable.

Corollary 8. The assertion of Corollary 7 is true with (18) replaced by the condition

The statements formulated above express fairly general properties of (1) and extend, in particular, the corresponding results of [25, 27, 29, 31].

2.2. Linear Equations

Let us now assume that in (1) is an affine mapping, and, therefore, (1) has the form

where and are linear, and is a given function.

Corollary 9. Assume that there exist certain linear operators , , and a linear mapping such that the inclusions

hold, and the estimate

is satisfied for any nonnegative . Then (20) has a unique solution.

We also have the following.

Corollary 10. Let there exist positive linear operators , , satisfying the inclusions

and such that the inequalities

are true for an arbitrary nonnegative function of bounded variation. Then (20) has a unique solution for any .

We conclude this note by considering the case where in (20) is a linear mapping admitting a decomposition into the sum of its positive and negative parts; that is,

where , , are linear and positive. In that case, for the equation of the form

where is linear and , the following result is obtained.

Corollary 11. Let the linear vector functional and the linear positive operators ,   , be such that the inclusions

are satisfied. Then (26) has a unique solution for any .

It is interesting to observe the second condition in (27); it thus turns out that property for one half of the operator under the integral sign in (26) ensures the unique solvability of the original equation (26).

3. Proofs

Let be real Banach space, let be a given vector, and let be a mapping. Let , , be closed cones inducing the corresponding partial orderings , so that if and only if . The following statement [32, 33] on the abstract equation

will be used below.

Theorem 12 (see [33], Theorem  49.4). Let the cone be normal and generating. Furthermore, let , , be linear operators such that and exist and possess the properties

and, furthermore, let the order relation

be satisfied for any pair such that . Then (28) has a unique solution for an arbitrary element .

Recall that is normal if all the sets order bounded with respect to are also norm bounded and that is generating if and only if (see, e.g., [33, 34]).

Let (resp., ) be the set of all the nonnegative (resp., nonnegative and nondecreasing) functions from .

Lemma 13. (1) The set is a cone in the space .

(2) The set is a normal and generating cone in .

Proof. The first assertion of the lemma being obvious, only the second one should be verified.

It follows directly from the definition of the set that it is a cone in , which is also generating due to the Jordan decomposition of a function of bounded variation (see, e.g., [3]). In order to verify its normality, it will be sufficient to show [32, Theorem  4.1] that the set

is bounded for any . Indeed, if , then the functions and are both nonnegative and nondecreasing. Therefore,

and, hence,

The last estimate shows that the norms of all such are uniformly bounded.

Let be a linear operator and a linear functional. Let us put

for any from . It follows immediately from Definition 1 that the linear operator defined by (34) has the following property.

Lemma 14. If is a linear operator such that

then is invertible and, moreover, its inverse satisfies the inclusion

We will also use the obvious identity

which is valid for any linear , .

3.1. Proof of Theorem 3

Let us set and put

for any from . Then (1) takes the form of (28) with . Since and are both from , it follows (see, e.g., [30]) that the function

also belongs to . Therefore, given by (38) is an operator acting in .

Note that relation (10) is equivalent to inequalities

for any and from with properties (11). Integrating (40) with respect to , we obtain

and, therefore, according to (38),

for all . Taking assumption (13) into account and using notation (34), we get

for all and and from with properties (11). Furthermore, it follows immediately from (34) and (38) that, for any ,

Therefore, by virtue of inequality (43) and assumption (10), the function is nonnegative and nondecreasing and, hence,

In the same manner, one shows that

Considering (45) and (46), we conclude that satisfies condition (30) with

, and

By virtue of Lemma 13, is a normal and generating cone in .

Since, by assumption (12), , it follows that is invertible and the inclusion

holds. Furthermore, by (12) and Lemma 14, the operator () exists and coincides with the inverse operator to . It is moreover positive in the sense that

Combining (49) and (50), we see that the inverse operators and exist and possess properties (29) with respect to cones (48). Applying now Theorem 12, we prove the unique solvability of (28) and, hence, that of (1).

3.2. Proof of Theorem 4

Rewriting relations (15) in the form

and putting

we find that admits estimate (10) with and defined by (52). Therefore, it remains only to note that assumption (14) ensures the validity of inclusions (12), and to apply Theorem 3.

3.3. Proof of Corollary 6

It turns out that, under assumptions (16) and (17), the operators , , defined by the formulae

with , satisfy conditions (14) and (15) of Theorem 4. Indeed, estimate (16) and the positivity of the operator imply that, for any and with properties (11) and all , the relations

are true. This means that admits estimate (15) with the operators and of form (53). It is easy to verify that assumption (17) ensures the validity of inclusions (14) for operators (53), and, therefore, Theorem 4 can be applied.

3.4. Proof of Corollaries 7 and 8

The results follow directly from Corollary 6 if one puts and , respectively.

3.5. Proof of Corollary 9

If , one should apply Theorem 4 with , , , and , . For a nonzero , one can modify the theorem slightly by incorporating the forcing term directly into (1) similarly to (20). Then we find that the argument of Section 3.1 remains almost unchanged.

3.6. Proof of Corollary 10

Corollary 7 with , , and is applied.

3.7. Proof of Corollary 11

It is sufficient to note that, under these assumptions, the linear operators ,   , defined by the formulae

satisfy conditions (21) and (22) of Corollary 9.

4. Comments

The following can be pointed out in relation to the above said.

4.1. Remark on Constants

The conditions presented in Sections 2.1 and 2.2 are, in a sense, optimal and cannot be improved. For example, it follows from [26] that assumption (14) of Corollary 7 can be replaced neither by the condition

nor by the condition

no matter how small may be. Likewise, counterexamples show that the assertion of Corollary 11 is not true any more if condition (27) is replaced by either of its weaker versions

and

with a positive . The same holds for the other inequalities and constants.

4.2. Equations with Matrix-Valued Functions

It is clear from the proofs given above that similar statements can also be obtained in the case where the integrals of matrix-valued functions are considered in (1), as described, for example, in [3, 4].

4.3. The Case of a Nonmonotone Measure

Results similar to those stated above can also be formulated in the case where the function involved in (1) is of bounded variation only and not necessarily nondecreasing. For this purpose, one should use the representation

where , , are nondecreasing functions, and modify the definition of the set in the following way.

Definition 15. A pair of operators is said to belong to if the equation

has a unique solution for any from and, moreover, the solution is nonnegative for nonnegative .

In that case, an analogue of the assertion of Theorem 3 is obtained if assumption (12) is replaced by the pair of conditions

The proof of this fact is pretty similar to the argument given in Section 3.1 and uses Theorem 12 with the operators , ,

instead of those defined by (47).

Acknowledgment

This research is supported in part by RVO: 67985840 (A. Rontó).

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