Abstract
We consider the initial value problem for a functional differential inclusion with a Volterra multivalued mapping that is not necessarily decomposable in . The concept of the decomposable hull of a set is introduced. Using this concept, we define a generalized solution of such a problem and study its properties. We have proven that standard results on local existence and continuation of a generalized solution remain true. The question on the estimation of a generalized solution with respect to a given absolutely continuous function is studied. The density principle is proven for the generalized solutions. Asymptotic properties of the set of generalized approximate solutions are studied.
1. Introduction
During the last years, mathematicians have been intensively studying (see [1, 2]) perturbed inclusions that are generated by the algebraic sum of the values of two multivalued mappings, one of which is decomposable. Many types of differential inclusions can be represented in this form (ordinary differential, functional differential, etc.). In the above-mentioned papers, the authors investigated the solvability problem for such inclusions. Estimates for the solutions were obtained similar to the estimates, which had been obtained by Filippov for ordinary differential inclusions (see [3, 4]). The concept of quasisolutions is introduced and studied. The density principle and the βbang-bangβ principle are proven. In papers [5β8], the perturbed inclusions with internal and external perturbations are considered, and the conjecture that βsmallβ internal and external perturbations can significantly change the solution set of the perturbed inclusion is proven. Let us remark that, in the cited papers, the proofs of the obtained results essentially depend on the assumption that the multivalued mapping, which generates the algebraic sum of the values, is decomposable. Therefore, these studies once again confirm V. M. Tikhomirov's conjecture that decomposability is a specific feature of the space and plays the same role as the concept of convexity in Banach spaces. The decomposability is implicitly used in many fields of mathematics: optimization theory, differential inclusions theory, and so forth. If a multivalued mapping is not necessarily decomposable, then the methods known for multivalued mappings cannot even be applied to the solvability problem of the perturbed inclusion. Furthermore, in this case, the equality between the set of quasisolutions of the perturbed inclusion and the solution set of the perturbed inclusion with the decomposable hull of the right-hand side fails. This equality for the ordinary differential inclusions was proven by WaΕΌewski (see [9]). The point is that, in this case, the closure (in the weak topology of ) of the set of the values of this multivalued mapping does not coincide with the closed convex hull of this set. As a result, we have that fundamental properties of the solution sets (the density principle and βbang-bangβ principle) do not hold any more (see [3, 10β13]). The situation cannot be improved even if the mapping in question is continuous.
In this paper, we consider the initial value problem for a functional differential inclusion with a multivalued mapping. We assume that this mapping is not necessarily decomposable. Some mathematical models can naturally be described by such an inclusion. For instance, so do certain mathematical models of sophisticated multicomponent systems of automatic control (see [14]), where, due to the failure of some devices, objects are controlled by different control laws (different right-hand sides) with the diverse sets of the control admissible values. This means that the object's control law consists of a set of the controlling subsystems. These subsystems may be linear as well as nonlinear. For example, this occurs in the control theory of the hybrid systems (see [15β20]). Due to the failure of a device, the control object switches from one control law to another. The control of an object must be guaranteed in spite of the fact that failures (switchings) may take place any time. Therefore, the mathematical model should treat all available trajectories (states) corresponding to all switchings. The generalized solutions of the inclusion make up the set of all such trajectories. The concept of a generalized solution should be then introduced and its properties should be studied.
We consider a functional differential inclusion with a Volterra-Tikhonov type (in the sequel simply Volterra type) multivalued mapping and we prove that for such an inclusion, the theorem on existence and continuation of a local generalized solution holds true. This justifies one of the requirements, which were formulated in the monograph of Filippov [4] for generalized solutions of differential equations with discontinuous right-hand sides. In the present paper, it is also proven that in the regular case, that is, when a multivalued mapping is decomposable, a generalized solution coincides with an ordinary solution. At the same time, the concept of a generalized solution discussed in the present paper does not satisfy all the requirements that are usually put on generalized (in the sense of the monograph [4]) solutions of differential equations with discontinuous right-hand sides. For instance, the limit of generalized (in the sense of the present paper) solutions is not necessarily a generalized solution itself. The reason for that is that a multivalued mapping that determines a generalized solution (the definition is given below) may not be closed in the weak topology of as this mapping is not necessarily convex-valued.
2. Preliminaries
We start with the notation and some definitions. Let be a normed space with the norm . Let be the closed ball in the space with the center at and of radius ; if then . Let . Then is the closure of , is the convex hull of ; , is the set of all extreme points of ; . Let . Let if and .
Let be the distance from the point to the set in the space ; let be the Hausdorff semideviation of the set from the set ; let be the Hausdorff distance between the subsets and of .
We denote by (resp., ) the set of all nonempty compact subsets of (resp., the set of all nonempty, bounded, closed in the space , and relatively compact in the weak topology on the space subsets of ). Let be the set of all nonempty bounded subsets of
Let be a system of subsets of (a subset of ). We denote by the set of all nonempty convex subsets of belonging to the system (the set of all nonempty convex subsets of belonging to ).
Let be the space of all -dimensional column vectors with the norm . We denote by (resp., ) the space of continuous (resp., absolutely continuous) functions with norm (resp., . Let be a measurable set (βthe Lebesgue measure). We denote by the space of all functions such that is integrable (if ) and the space of all measurable, essentially bounded (if ) functions with the norms respectively.
Let . The set is called integrally bounded if there exists a function such that for each and almost all The set is said to be decomposable if for each and every measurable set the inclusion holds, where is the characteristic function of the set . We denote by (resp., ) the set of all nonempty, closed, and integrally bounded (resp., nonempty, bounded, closed, and decomposable) subsets of the space .
Let be a measurable mapping. Then by definition, . By (resp., ), denote the cone of all nonnegative functions of the space (resp., ).
Let be a mapping between two partially ordered sets and (the partial order of both sets is denoted by ). The mapping is isotonic if whenever
In this paper, the expression βmeasurability of a single-valued functionβ is always used in the sense of Lebesgue measurability and βmeasurability of a multivalued functionβ in the sense of [21]. Let be a space with finite positive measure and let be a multivalued mapping from to A set () of measurable mappings from to is said to approximate the multivalued mapping if the set is measurable for any , and the set belongs to the closure of its intersection with the set for almost all A multivalued mapping from to is called measurable if there exists a countable set of measurable mappings from to that approximates the mapping
Further, let us introduce the main characteristic properties of a set that is decomposable.
Lemma 2.1. Let Then there exists a function such that for each function and almost all
Proof. Let , be a sequence
of functions such that
Let us show that there exists a sequence of functions , such that the
equality (2.2) holds and for almost all
Indeed, let and , where Since we see that the
sequence , has the
following properties: for almost all , the inequalities (2.3) hold and for each Hence, from
this property and equality (2.2), it follows that the sequence , satisfies
(2.2). Further, we consider a measurable function defined by Since the set is bounded, we
see, using Fatou's lemma (see [22]), that Moreover, by
the definition of the function and due to
(2.2), for every
measurable set Now, let us
show that the function defined by
(2.4) satisfies the assumptions of the lemma. Indeed, if the contrary is true,
then there exist a function and a
measurable set () such that for each This implies
that which
contradicts (2.5). This completes the proof.
Lemma 2.2. Let and , be a sequence that is dense in Further, let a measurable set be defined by Then
Proof. Since and the sequence , is dense in we have, due to the closedness of the set the relation Let us prove that Let For each , , put which are measurable sets. For , let , and for , let Then if By the definition of the mapping for each , we have Let , be a sequence of measurable functions such that Since the set is decomposable, we see that for each . Moreover, from Lemma 2.1 and the definition of the set , it follows that for the functions , we have the estimates where satisfies the assertions of Lemma 2.1. From (2.8) and (2.10), it follows that in as Since the set is closed, we have that . Hence Thus
Lemma 2.3. Let measurable sets , be integrally bounded, then if and only if for almost all
Proof. First
of all, it is evident that if for almost all , then .
Let and let , be a countable
set, which is dense in and which
approximates (see [21]).
Thus for each and by the
definition of the set we have that for almost all Since the
sequence , approximates
the map it follows from
the previous inclusion that for almost all
Corollary 2.4. Let and let , be measurable sets such that Then for almost all
Remark 2.5. If then a measurable set , that satisfies uniquely determines the set
3. Decomposable Hull of a Set in the Space of Integrable Functions
We introduce the concept of the decomposable hull of a set in the space We consider a multivalued mapping that is not necessarily decomposable. For such a mapping, we construct its decomposable hull and investigate topological properties of this hull.
Definition 3.1. Let be a nonempty subset of By , we denote the set of all finite combinations of elements , where the disjoint measurable subsets , of the segment are such that
Lemma 3.2. The set is decomposable for any nonempty set
Proof. Let Let also be a measurable set. Without loss of generality, it can be assumed that where , , and the measurable disjoint sets , , are such that , (if the number of summands in (3.2) is not the same, we may use arbitrary functions multiplied by the characteristic functions of the empty sets). Further, from the equality it follows that Hence, the set is decomposable.
Remark 3.3. Note that even if a set is bounded, the set is not necessarily bounded. For example, let us check that Indeed, let and , be measurable sets with the following properties: if , , for each , the inequality holds. Then , and Therefore, and consequently, the equality (3.4) holds.
Remark 3.4. From (3.4), it follows that if a set is relatively compact in the weak topology of then the set does not necessarily possess this property.
Remark 3.5. Note that if a set is convex in then this set is not necessarily decomposable. The ball is an example of such a set.
Remark 3.6. If a set is integrally bounded, then by Lemma 2.2, for the set , there exists a measurable and integrally bounded mapping such that
Lemma 3.7. If a set is decomposable, then
Proof. Evidently, . We claim that . The proof is made by induction over . By the
definition of the switching convexity, any expression (3.1) including two
elements and two
measurable sets belongs to .
Suppose now that for , the combination of the form (3.1) belongs to . Let and let be disjoint
measurable sets such that . Let By the inductive assumption, and therefore . Since we have that Hence . This concludes the proof.
Corollary 3.8. If then the set is the minimal set which is decomposable and which contains
Proof. Consider any set which is decomposable and which satisfies Then, by Lemma 3.7, we have
Lemma 3.9. If a set is convex, then so is the set
Proof. Let be given by the formula (3.2). It follows from the convexity of the set and the equality that for any Thus, the set is convex.
Similar to the definition of the convex hull in a normed space, the set will, in the sequel, be called the decomposable hull of the set in the space of integrable functions, or simply the decomposable hull of the set . Likewise, is addressed as the closed decomposable hull of the set .
Remark 3.10. If then the closed decomposable hull of the set (the set ) can be constructed as described in Remark 3.6. To do it, one needs a measurable and integrally bounded (see Remark 3.6) mapping that satisfies (3.7). Note that finding this mapping is easier than constructing the set . At the same time, when one studies the metrical relations between the sets and their decomposable hulls (see Lemma 3.12), it is more convenient to use Definition 3.1.
Lemma 3.11. Let and let a set be decomposable. Then for any disjoint measurable sets such that , one has
Proof. Indeed,
let and satisfy It follows from
this estimate that This yields
Further, let us show that the opposite inequality is
valid. Let , where is the set of
of all mappings from restricted to , and suppose
that the functions , satisfy Since the set is
decomposable, it follows that the map defined by belongs to the set By (3.15), we
have This implies that Comparing (3.14)
and (3.18), we obtain (3.12).
Lemma 3.12. If and there exists a function such that for any measurable set , then for any measurable set
Proof. Let be a measurable
set, . Let and , Suppose also
that the functions and disjoint
measurable sets , such that satisfy the
equality
Further, by , we denote the
restrictions of these functions to and put ,
From (3.21) and Lemma 3.11, it follows that
From (3.19), we obtain that for each
Therefore, (3.22) and (3.23) imply Since (3.24)
holds for any it follows from
(3.24) that (3.20) holds as well.
Remark 3.13. Note that the function (see (3.19)) provides a uniform with respect to measurable sets estimate for the Hausdorff semideviation of the set from the set
Remark 3.14. The inequality (3.20) holds true even if the set is replaced with its closure ,
We say that a multivalued mapping is integrally bounded on a set if the image is integrally bounded.
Let We introduce an operator by the formula
Note that even if a mapping is continuous, the mapping given by (3.25) may be discontinuous. To illustrate this, let us consider an example.
Example 3.15. We define an
integrable function by
We also define a multivalued mapping by the formula
Note that for any , but at the same time, for any
Using Lemma 3.12, we obtain the following continuity conditions for the operator given by (3.25).
Definition 3.16. Let One says that a mapping is symmetric on the set if for any One says that a mapping is continuous in the second variable at a point belonging to the diagonal of if for any sequence such that as it holds that One says that a mapping is continuous in the second variable on the diagonal of if is continuous in the second variable at each point of this diagonal. Continuity in the fist variable is defined similarly.
Definition 3.17. Let Suppose also that for any One says that a mapping has property on the set if it is continuous in the second variable on the diagonal of it has property on the set if it is continuous in the first variable on the diagonal of it has property on the set if it is continuous on the diagonal of and symmetric on the set
Theorem 3.18. Let Suppose also that for a mapping there exists a mapping such that for any and any measurable set Then for the mapping given by (3.25), the inequality (3.30), where is satisfied as well as for any and any measurable set
Corollary 3.19. If the mapping in Theorem 3.18 has property (resp., , ) on the set then the operator given by (3.25) is Hausdorff lower semicontinuous (resp., Hausdorff upper semicontinuous, Hausdorff continuous) on the set
We say that the mapping satisfying the inequality (3.30) for any measurable set is a majorant mapping for on the set
Let a mapping , be measurable as a composite function for every Let also be integrally bounded for every bounded set Consider a mapping given by where the mapping , is the Nemytskii operator generated by the mapping , For the operator given by (3.31), the majorant mapping can be defined as
It follows from Theorem 3.18 that the operator given by (3.32) is also a majorant mapping for the mapping given by (3.25), where If the mapping , is Hausdorff lower semicontinuous (resp., Hausdorff upper semicontinuous and Hausdorff continuous) in the second variable, then by Corollary 3.19, the mapping given by (3.25) is Hausdorff lower semicontinuous (resp., Hausdorff upper semicontinuous and Hausdorff continuous).
Definition 3.20. One says that a multivalued mapping has Property (resp., and ) if for this mapping there exists a majorant mapping satisfying Property (resp., and ).
4. Basic Properties of Generalized Solutions of Functional Differential Inclusions
Using decomposable hulls, we introduce in this section the concept of a generalized solution of a functional differential inclusion with a right-hand side which is not necessarily decomposable. Using, as mentioned in Section 3, basic topological properties of a mapping given by (3.25), we study the properties of a generalized solution of the initial value problem.
Consider the initial value problem for the functional differential inclusion where the mapping satisfies the following condition: for every bounded set , the image is integrally bounded. Note that the right-hand side of the inclusion (4.1) is not necessarily decomposable. Note also that in (4.1) is not treated as a derivative at a point but as an element of (see [10, 23β25]). When we study such a problem, there may appear some difficulties described in the introduction. In this connection, we will introduce the concept of a generalized solution of the problem (4.1) and study the properties of this solution. Using the Nemytskii operator, which is decomposable, the initial value problem for a classical differential inclusion, that is, one without delay (see [10, 23β25]), can be reduced to (4.1).
Definition 4.1. An absolutely continuous function is called a generalized solution of the problem (4.1) if
Note that from Lemma 3.7, it follows that if the set (see(4.1)) is decomposable, then a generalized solution of the problem (4.1) coincides with a classical solution.
Example 4.2. Consider an ordinary differential equation, Its solution is the function
We assume that the parameter may take two
values: 1 or 2. Then the trajectories of such a system are described by the
differential inclusion where is a
multivalued function with the values from the set Note that that is, the
set in the right-hand side of the inclusion is decomposable. In this case, a
generalized solution of the inclusion coincides with a classical solution.
The latter differential inclusion describes the model
that is controlled by the differential equation either with the parameter value or with the
parameter value In this model,
switchings from one law (equation) to another may take place any time.
In the limit case, all possible solutions fill
entirely the set of all points between the graphs of the functions and
Example 4.3. Consider a
simple pendulum. It consists of a mass hanging from a
string of length and fixed at a
pivot point . When displaced to an initial angle and released, the
pendulum will swing back and forth with periodic motion. The equation of motion
for the pendulum is given by where is the
angular displacement at the moment , , is the
acceleration of gravity, and is the length
of the string.
If the amplitude of angular displacement is small
enough that the small angle approximation holds true, then the equation of
motion reduces to the equation of simple harmonic motion Let us now assume that the length of the string may change,
that is, it may take an value from a finite set In this case,
the equation of simple harmonic motion transforms to the differential inclusion
with a multivalued mapping where
We assume that switching from one length (equation) to
another may take place any time. Then the generalized solutions of the
inclusion treat all available trajectories (states) corresponding to all
switchings.
Definition 4.4. An operator is called a Volterra-Tikhonov (or simply a Volterra) operator (see [26]) if the equality on , implies where is the set of all functions from restricted to
In what follows, we assume that the operator (the right-hand side of the inclusion (4.1)) is a Volterra operator. This implies that the operator given by (3.25) is also a Volterra operator.
Let Let us determine the continuous mapping by
Definition 4.5. One says that an absolutely continuous function is a generalized solution of the problem (4.1) on the interval , if satisfies and where the continuous mapping is given by (4.8).
A function which is absolutely continuous on any interval , is called a generalized solution of the problem (4.1) on the interval if for each the restriction of to is a generalized solution of the problem (4.1) on the interval
A generalized solution of the problem (4.1) on the interval is said to be nonextendable if there is no generalized solution of the problem (4.1) on any larger interval (here, if and if ) such that for each
In Theorems 4.6β4.12 below, we assume that the mapping has Property Due to Corollary 3.19, the mapping given by (3.25) is lower semicontinuous. Due to [27, 28], the mapping admits a continuous selection. Therefore, the following propositions on local solutions of the problem (4.1) are straightforward.
Theorem 4.6. There exists such that a generalized solution of the problem (4.1) is defined on the interval .
Theorem 4.7. A generalized solution of the problem (4.1) admits a continuation if and only ifββ
Theorem 4.8. If is a generalized solution of the problem (4.1) on the interval , then there exists a nonextendable solution of the problem (4.1) defined on the interval , or on the entire interval such that for each
Let be the set of all generalized solutions of the problem (4.1) on the interval
We say that generalized solutions of the problem (4.1) admit a uniform a priori estimate if there exists a number such that for every , there is no generalized solution satisfying
Theorems 4.6β4.8 yield the following result.
Theorem 4.9. Let the generalized solutions of the problem (4.1) admit a uniform a priori estimate. Then for any and there exists a number such that for any ,
Definition 4.10. One says that a mapping has Property if there exists an isotonic continuous operator satisfying the following conditions:(i)for any function and any measurable set , one has where the continuous mapping is given by (ii)the local solutions of the problem admit a uniform a priori estimate.
Lemma 4.11. Suppose that a multivalued mapping has Property Then so does the mapping given by (3.25).
Proof. It suffices to show that for any function and any measurable set Indeed, let a function be as in (3.1). By (4.9), for each Hence, we have that for the function , the estimate is satisfied as well. This gives the inequality (4.12). The proof is complete.
Let a continuous operator be given by
Theorem 4.12. Suppose that a mapping has Property Then the set is nonempty for any and there exists a number such that for any ,
Proof. Indeed, let (). From Lemma 4.11, it follows that for any , where the function is given by (4.15). Due to the theorem on integral inequalities for an isotonic operator (see [29]), this implies that we actually have where is the upper solution of the problem (4.11). Thus, there is no satisfying the inequality From this, it follows that the set of all local generalized solutions of the problem (4.1) admits a uniform a priori estimate. Applying Theorem 4.9 completes the proof.
Let a linear continuous operator be given by We say that is the operator of integration.
Theorem 4.13.
Let
the set of all local generalized solutions of the problem (4.1) admit a uniform
a priori estimate. Suppose also that has Property Then for any
function and any , there exists a generalized solution of the problem
(4.1) such that for any
measurable set
If then the
theorem is also valid for
Proof. Let have Property Then by Corollary 3.19, the mapping given by (3.25) is continuous. Therefore (see [30β32]), given a number and a function , there exists a continuous mapping satisfying and for any and any measurable set It follows from Theorem 4.9 that for any , and that there exists a number such that for each , Now, we show that there exists satisfying (4.18). Consider the problem where the continuous mapping is given by We denote by the set of all solutions of the problem (4.20). Let us show that It follows from the definition of the mapping (see (4.21)) that Let us prove that Assume the converse. Then there exists such that Since we have This implies that there exists a number such that ( is the restriction of the function to ). By (4.21), we have This contradicts to the definition of the number Hence, Consider a continuous operator given by where the operator is the operator of integration defined by (4.17), and is a continuous selection of the mapping given by (3.25). The function ia also assumed to satisfy (4.19). Since the operator is bounded, we obtain that the image is a relatively compact subset of Hence, the set is a convex compact set. Since the operator given by (4.22) takes the set into itself, we have, by Schauder theorem, that the mapping has a fixed point. This fixed point is the solution of the problem (4.20). It follows from the above equality that this solution is a generalized solution of the problem (4.1). Since we see that (4.19) implies (4.18).
Let us prove the second statement of the theorem. Let Suppose also that has Property Then by Lemma 3.9, Hence for each , there exists a generalized solution of the problem (4.1) such that for any measurable set , the inequality (4.18) is valid for and Since the set is bounded, we see that the sequence is weakly compact in Without loss of generality, it can be assumed that weakly in and in as Let us show that is a generalized solution of the problem (4.1). In other words, we have to prove that Assume that the functions , satisfy (as these functions do exist). It follows from (4.23) that Since the mapping given by (3.25) is continuous, we obtain, by (4.24), that in as Since weakly in as we have that weakly in as Therefore, the convexity of the set implies that (see [21]). Thus, is a generalized solution of the problem (4.1).
Further, let us show that (4.19) holds for the solution and for Since weakly in as we have, by [21], that for each , there exist numbers , , satisfying the following conditions: the sequence tends to in Since for each it follows, due to the choice of the sequence that for each .
Since it follows that letting in the previous inequality, we obtain Finally, note that by the decomposability of the set this equality holds for any measurable set This completes the proof.
Theorems 4.12 and 4.13 yield the following result.
Corollary 4.14. Suppose that a mapping has
Properties and Then for any function and any , there exists a generalized solution of the problem
(4.1) such that (4.18) holds for any measurable set
If then the
corollary is also valid for
Remark 4.15. Consider the convex compact set where the mapping is given by Here, the operators and are determined by (3.25) and (4.21), respectively. If a number is such that for any , then due to the the coincidence of the sets and (see the proof of Theorem 4.13),
Definition 4.16. Given , , , one says that a mapping has Property if there exists an isotonic and continuous Volterra operator satisfying the following conditions:(i)(ii)for any functions and any measurable set , one has where the continuous mapping is determined by (4.10);(iii)the set of all local solutions of the problem admits a uniform a priori estimate.
Given and , the following estimate will be used in the sequel: for each measurable set
Theorem 4.17. Let functions and satisfy the inequality (4.32) for each measurable set Suppose that a mapping has Property where , , and is the initial condition of the problem (4.1). Then for any generalized solution of the problem (4.1) satisfying for any measurable set , the following conditions are satisfied: (1) for each where the function is the upper solution of the problem (4.31) for and and the mapping is given by (4.15);(2) for almost all
Proof. First,
note that since the mapping has Property it follows from
Theorem 3.18 that so does the mapping determined by
(3.25). Further, the inequality (4.33) yields that for any
measurable set
Remark 4.15 and relations (4.36), (4.32), and (4.30)
imply that for any measurable set , we obtain the inequality where the mapping is given by
(4.10). It follows from this inequality that for almost all Since for all (see (4.10),
(4.15)) and the operator (see (4.38)) is
isotonic, we have that for almost all Therefore,
(4.39) and the theorem on differential inequalities with an isotonic operator
(see [29]) imply (4.34) for any The inequality
(4.35) follows from (4.34) and (4.39). The proof is complete.
Theorems 4.13 and 4.17 yield the following result.
Theorem 4.18.
Let
functions and satisfy (4.32)
for each measurable set
Suppose that a
mapping has Property where , , is the initial
condition in the problem (4.1). Let the set of all local generalized solutions
of the problem (4.1) admit a uniform a priori estimate. Then for , there exists a generalized solution
of the problem
(4.1) which satisfies (4.34) and (4.35) for all
and for almost
all respectively.
If then the theorem
is also valid for
Corollary 4.19.
Let
functions and satisfy (4.32)
for each measurable set
Suppose that a
mapping has properties and where , ,
is the initial
condition in the problem (4.1). Then for , there exists a generalized solution
of the problem
(4.1) which satisfies (4.34) and (4.35) for all
and for almost
all respectively.
If then the
corollary is also valid for
Remark 4.20. It follows from the proof of Theorem 4.17 that Theorems 4.17, 4.18, and Corollary 4.19 are also valid if the functions and satisfy for each measurable set
Definition 4.21. An absolutely continuous function is called a generalized quasisolution of the problem (4.1) if there exists a sequence of functions , such that the following conditions hold:(i) in as (ii) and for each .
Note that by Lemma 3.7, if the set mentioned in Definition 4.21 is decomposable, then a generalized quasisolution coincides with a quasisolution defined in [9, 33], where is the Nemytskii operator. Note also that this definition of a generalized quasisolution differs from the definition of a quasitrajectory given in [9, 33, 34] due to the condition Using Definition 4.21, we can obtain more general results on the properties of quasisolutions (see Remark 4.23). Moreover, this definition is more suitable for applications.
Let be the set of all generalized quasisolutions of the problem (4.1).
We define a mapping by the formula
We call the convex decomposable hull.
Consider the problem (4.1) with the convex decomposable hull given by (4.41) leading to
Let be the set of all solutions of the problem (4.42) on the interval .
Theorem 4.22.
Proof. First,
we will show that Let By [35], for , there exists a sequence , such that weakly in as This implies
that in as where is the operator
of integration (see (4.17)). Hence,
Let us now prove that Let Then there exists
a sequence , satisfying the
following conditions: (1) (see (4.41))
and for each ; (2) in as Since the
sequence , is weakly
compact, we can assume without loss of generality that weakly in as Since (see (4.41)),
it follows that (see [21]).
Hence and therefore
Remark 4.23. Theorem 4.22 may still remain valid even if the mapping is discontinuous and its image is not integrally bounded for every bounded set The proof of Theorem 4.22 is only based on the fact that every value of this mapping is integrally bounded, rather than on the assumption that is a Volterra operator.
Definition 4.24. One says that a compact convex set has Property if , and for any , there exists a sequence of absolutely continuous functions , such that(i) in as (ii), and for each .
Lemma 4.25. Suppose that the set of all local solutions of the problem (4.41) admits a uniform a priori estimate. Then, there exists a set satisfying Property
Proof. It follows from Theorem 4.22 and Remark 4.15 that the set has Property Here, the mapping is determined by (4.29), where
Lemma 4.26. Let sets , satisfy , where are measurable mappings. Then for any measurable set , one has
Proof. Let be a measurable set. Put The set is measurable. Since we have This implies (4.43), and the proof is completed.
Let be a measurable mapping. Let a mapping be defined by
Corollary 4.27. Let sets , satisfy , where are measurable mappings. Then for any measurable set , one has
Proof. By [35], we have that , Therefore, for any measurable set Since, for any measurable set , we obtain, due to (4.43), the inequality (4.48).
Definition 4.28. One says that a mapping has Property if Property is satisfied and the following conditions hold:(i)(ii)on every interval (), there exists a unique zero solution of the problem (4.31), where , ,
Theorem 4.29. Suppose that the set of all local generalized solutions of the problem (4.1) admits a uniform a priory estimate. Suppose also that a mapping satisfies Property Then, and where is the closure of the set in
Proof. Let us first prove that the set is closed in Indeed, suppose
that a sequence , tends to in as Since the
sequence is integrally
bounded, it follows that and weakly in as For each , let the function satisfy where is the convex
decomposable hull given by (4.41). Since the mapping given by (3.25)
is Hausdorff continuous, it follows from (4.48) that so is the mapping Therefore,
(4.52) implies that in as Hence, weakly in as Since the set is convex, we
have (see [21]) that Therefore, the
set is closed in
Now, let us prove the equality (4.51). The closedness
of the set yields that Further, let us
show that Suppose Then from
Theorem 4.22, it follows that there exists a sequence , such that , , ( is the initial
condition in the problem (4.1)) and in as Since the
mapping has Property we see that,
due to (4.30), for each and any
measurable set Here, the
operator is given by
(4.10). Since the mapping is continuous
and we have that in as Since the
problem (4.31) with , and only has the
zero solution on each interval (), we see that
the set of all local solutions of the problem (4.31) with , , and admits a
uniform a priori estimate starting from some (see [36]).
Renumerating, we may assume without loss of generality that this holds true for
all . This implies (see [29]) that for each , there exists the upper solution of the problem
(4.31) with , , and Hence, it
follows from Theorem 4.18 that for each , there exists a generalized solution of the problem
(4.1) satisfying where the
continuous operator is given by
(4.15). Since in as we have that as Since in as we see that in as Therefore, and
consequently This yields
(4.51). The proof is complete.
Corollary 4.30. Suppose that a mapping has Properties and Then and the equality (4.51) is satisfied.
Remark 4.31. If the solution set of a differential inclusion with nonconvex multivalued mapping is dense in the solution set of the convexified inclusion, then such a property is called the density principle. The density principle is a fundamental property in the theory of differential inclusions (see [13]). Many papers (e.g., [3, 4, 6, 10β12, 23, 24, 25, 29, 30, 31, 32, 37, 38, 39]) deal with the justification of the density principle. Theorem 4.29 and Corollary 4.30 justify the density principle for the generalized solutions of the problem (4.1).
5. Generalized Approximate Solutions of the Functional Differential Equation
Approximate solutions are of great importance in the study of differential equations and inclusions (see [4, 40β43]). They are used in the theorems on existence (e.g., Euler curves) as well as in the study of the dependence of a solution on initial conditions and the right-hand side of the equation. In [40, 41], the definition of an approximate solution of a differential equation with piecewise continuous right-hand side was given, using so-called internal and external perturbations. This definition not only deals with small changes of the right-hand side within its domain of continuity, but also with the small changes in the boundaries of these domains. A more general definition of an approximate solution, which can be used not only for the study of functional differential equations with discontinuous right-hand sides but also for differential inclusions with upper semicontinuous convex right-hand sides, was given in [4]. In this paper, the following important property was justified for such an inclusion: the limit of approximate solutions is again a solution of functional differential inclusion. In the present paper, we introduce various definitions of generalized approximate solutions of a functional differential inclusion. The main difference of our definitions from the one given in [4] is that the values of a multivalued mapping are not convexified. Due to this, the topological properties of the sets of generalized approximate solutions are studied and the density principle is proven.
Since a generalized solution of the problem (4.1) is determined by the closed decomposable hull of a set, it is natural to raise the following question: how robust is the set of the generalized solutions of (4.1) with respect to small perturbations of It follows from Remark 3.10 that constructing for each fixed is equivalent to finding a measurable, integrally bounded mapping satisfying The mapping is, in the sequel, written as and called a mapping generating the mapping given by (3.25).
Denote by the set of all continuous functions satisfying the following conditions:(1)for each , (2)for each , there exists a function such that for almost all and all (3) for almost all
Since the mappings and are related by the equality (5.1), we have that the robustness of the set of the generalized solutions of (4.1) with respect to small perturbations of can be studied via the robustness properties of Assume that the perturbation (e.g., an error in measurements of ) is given by where (here, is an -neighborhood of the set see Preliminaries).
Note that (5.2) yields for all Thus, (5.3) implies that for each function almost all and all Therefore, all mappings defined by (5.2) and depending on are close (in the sense of (5.4)) to the mapping The mapping is called the approximating operator.
We define a mapping by the formula where the operator is given by (5.2). The equalities (5.3) and (5.5) imply that for any
It follows from (5.6) and the Lebesgue theorem that
Thus, all mappings defined by (5.2) and (5.5) and depending on are close (in the sense of (5.7)) to the mapping given by (3.25).
Lemma 5.1 (see [6]). Let be a normed space and let be a convex set. Then for all and all
Denote by the set of all continuous functions such that for any and for any
Let be a closed convex set and let We define a multivalued mapping by
The inequality (5.8) yields the following result.
Lemma 5.2. Let be a closed convex set and let Then, a multivalued mapping given by (5.9) is Hausdorff continuous.
We define a mapping by the formula where the mapping is given by (5.9) and the mapping generates the mapping given by (3.25).
It is natural to address the value of the function at the point as the modulus of continuity of the mapping at the point with respect to the variable We call the function the radius of continuity, while the function itself is called the modulus of continuity of the mapping with respect to the radius of continuity
Definition 5.3. One says that a mapping has Property if the mapping generating the mapping given by (3.25) is Hausdorff continuous in the second variable for almost all
Lemma 5.4. Suppose that
for a mapping , there exists an isotonic continuous operator
satisfying the
following conditions: (i)(ii)the inequality (4.30), where is satisfied
for any and any measurable
set
Then the mapping has Property
Proof. Let in as Let us show
that for almost all
For each , put Due to Theorem
3.18, (4.43), and the isotonity of the operator for each and almost all , we have Since the sequence , decreases, we
obtain, due to the continuity of the mapping and the
equality the equality
(5.11). This completes the proof.
Lemma 5.5. Let be a nonempty, convex, compact set in the space and let Suppose also that a mapping has Property Then the mapping given by (5.10) has the following properties: (i) is measurable for any (ii) is continuous on for almost all (iii)for any and for almost all , (iv)there exists an integrable function such that for almost all any and all
Definition 5.6. Let One says that the function provides on a uniform with respect to the radius of continuity estimate from above for the modulus of continuity of the mapping ; if for any there exists such that for almost all all , and , one has where is given by (5.10).
Let and One defines a function by
Lemma 5.1 yields the following result.
Corollary 5.7. Let be a nonempty, convex, compact set in the space and let Suppose also that a mapping has Property Then the mapping given by (5.15) belongs to the set and provides a uniform (in the sense of Definition 5.6) estimate from above for the modulus of continuity of the mapping
Remark 5.8. Corollary 5.7 yields that if is a nonempty, convex, compact set in the space and a mapping has Property then for a given , there exists at least one function that provides a uniform (in the sense of Definition 5.6) estimate from above for the modulus of continuity of the mapping
Let For each , consider the initial value problem where the mapping is given by (5.1) and (5.5).
Since the operator given by (3.25) is a Volterra operator, we see that the mapping has the following property: if on (), then for almost all This property, (5.1), and (5.4) imply that the operator is a Volterra operator for each
Any solution of the problem (5.16) with a given is said to be a generalized -solution (a generalized approximate solution with external perturbations) of the problem (4.1). We denote by the set of all generalized -solutions of (4.1) belonging to
Theorem 5.9. Suppose that a set has Property Then for any function that provides a uniform (in the sense of Definition 5.6) estimate from above for the modulus of continuity of the mapping , one has where is the closure of in
Proof. First,
let us prove that Let Let us show
that is a limit
point of the set for any By Theorem 4.22, is a
generalized quasisolution of the problem (4.1). Moreover, Since the set has Property we see that
there exists a sequence of absolutely continuous functions , such that the
following conditions hold: in as ; , and for each Suppose that provides a
uniform (in the sense of Definition 5.6) estimate from above for the modulus of
continuity of the mapping Then there
exists such that for each . This implies that for each . Therefore, for each By Definition
5.6, there exists a number such that for any and almost all
The inequality (5.19) yields that for each and almost all By (5.20), for each This implies
that is a limit
point of the set Therefore, , and consequently (5.18), is satisfied.
Let us prove the opposite inclusion Let This implies
that for each , there exists satisfying Suppose that
functions satisfy for each and almost all Let us show
that Since by the Lebesgue
theorem, we have that By (5.22), the
estimates are satisfied for each and almost all Therefore, for each . By (5.24) and due to the continuity of the mapping given by
(3.25), we have (5.23).
Since weakly in as we have that weakly in as Therefore, by
[21], and hence Thus, (5.21) is valid. Hence, (5.17) holds and the
proof is complete.
Theorem 5.10. Suppose that a set has Property Then, for any if and only if the equality (4.51) is satisfied.
Proof. Let us
prove the sufficiency. Assume
that (4.51) holds. Let us show that the equality (5.27) is satisfied
for any function By the
definition of the problem (5.16), the inclusion is satisfied for any Therefore, for
any , we have the inclusion and consequently the inclusion holds. Now, let us check that the opposite relation which is, by (4.51), equivalent to the inclusion holds as well. The latter relation can be proven
similarly to Theorem 5.9.
The necessity follows readily from Theorem 5.9. The proof is
complete.
Remark 5.11. Note that the equality (5.27) describes the robustness property of the set with respect to external perturbations These external perturbations (e.g., ) characterize an error in measurements of the values of the mapping given by (3.25).
On the other hand, each generalized solution of the problem (4.1) may also be measured with a certain error. This error may be described by a function belonging to the set and it may be characterized by so-called internal perturbations, which are defined below. Let us show further that internal perturbations influence essentially the properties of generalized solutions of the problem (4.1).
We define a mapping by where is the mapping generating the operator given by (3.25); see the definition of in Section 2. Let us remark that the mapping is measurable (see [25]) and integrally bounded for each
Consider the operator given by where the mapping is given by (5.33).
Remark 5.12. Note that for each , the set has the following property Also, is the minimal set among all nonempty closed in decomposable subsets of satisfying (5.35).
Consider the problem We call any solution (resp., quasisolution) of (5.36) a generalized extreme (in the sense of the definition in Section 2) solution (resp., generalized extreme quasisolution) of the problem (4.1).
Let be the set of all generalized extreme quasisolutions of the problem (4.1). Theorem 4.22, Remark 4.23, and equality (5.35) imply the following result.
Corollary 5.13.
Let , Let also be a convex closed set in We define mappings , , , by the formulas where the mappings , , are given by the equalities (5.5), (5.9), and (5.33), respectively.
For each , consider the following problems on : where the mappings , are given by (5.37).
We call any solution of the problem (5.38) with a fixed a generalized -solution of the problem (4.1), or a generalized approximate solution of (4.1) with external and internal perturbations. For each , we denote by () the set of all solutions of the problem (5.38) ((5.39)) on Since for any and any we see that for any .
Theorem 5.14. Let the set have Property Then for any , , one has where and are the closures of the sets and respectively, in the space
Proof. First
of all, let us check that Let We show that is a limit
point of the set for any By Corollary
5.13, is a
generalized extreme quasisolution of the problem (4.1). Since the set has Property we see that and there
exists a sequence of absolutely continuous functions , with the
following properties: in as ; , and for each . Here, the operator is given by
(5.33) and (5.34).
Let us prove also that there exists a number such that for
each , Since we see that
there exists a number such that for
each , This implies
that for each , we have that (see (5.9)).
Therefore, for each , the inclusion holds. Hence,
for each , we have (5.42). This means that is a limit
point of the set Therefore, and (5.41) is
satisfied.
The relation can be proven similarly to (5.21) (see the proof of
Theorem 5.9). The second equality of (5.40) can be proven in the same way. This
completes the proof of the theorem.
Remark 5.15. Theorem 5.14 says that no measurement accuracy of the values of the mapping could guarantee the βreconstructionβ of the set by means of That is only possible if the density principle holds for the generalized solutions.
6. Conclusion
The main results of the paper can be summarized as follows. For the decomposable hull of a mapping, we have obtained the conditions for the property of the Hausdorff lower semicontinuity (resp., upper semicontinuity and continuity). We considered a functional differential inclusion with a Volterra multivalued mapping which is not necessarily decomposable. The concept of a generalized solution of the initial value problem for such an inclusion was introduced and its properties were studied. Conditions for the local existence and continuation of a generalized solution to the initial value problem were obtained. We have offered some estimates, which characterize the closeness of generalized solutions and a given absolutely continuous function. These estimates were derived from the conditions for the existence of a generalized solution satisfying the inequality (4.18) (see Theorem 4.13 and Corollary 4.14).
The concept of a generalized quasisolution of the initial value problem was introduced. We proved that the set of all generalized quasisolutions of the initial value problem coincides with the solution set of the functional differential inclusion with the convex decomposable hull of the right-hand side. Using this fact as well as the estimates characterizing the closeness of generalized solutions and a given absolutely continuous function, we obtained the density principle for the generalized solutions.
Asymptotic properties of the set of generalized approximate solutions (generalized -solutions) were studied. It was proven that the limit of the closures of the sets of generalized approximate solutions coincides with the closure of the set of the generalized solutions if and only if the density principle holds for the generalized solutions.
Acknowledgments
The first author is supported by CIGENE (Center for Integrative Genetics) and by LΓ₯nekassen (Norwegian State Educational Loan Fund). The second author is supported by CIGENE and partially supported by the Russian FBR Grant no. 07-01-00305. The third author is partially supported by the Russian FBR Grant no. 07-01-00305.