# Generalized Solutions of Functional Differential Inclusions

Authors: Anna Machina; Aleksander Bulgakov; Anna Grigorenko
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

We consider the initial value problem for a functional differential inclusion with a Volterra multivalued mapping that is not necessarily decomposable inL1n[a,b]. 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.


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## Body

## 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 “bang-bang”
principle are proven. In papers [5–8], the perturbed inclusions with
internal and external perturbations are considered, and the conjecture that
“small” 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 L1n[a,b] 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  Ważewski (see [9]). The
point is that, in this case, the closure (in the weak topology of L1n[a,b]) 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 “bang-bang” principle) do
not hold any more (see [3, 10–13]). 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 [15–20]). 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 L1n[a,b], as this mapping
is not necessarily  convex-valued.


## 2. Preliminaries

We start with the notation and some definitions. LetX be a normed
space with the norm ∥⋅∥X. Let BX[x,ε] be the closed
ball in the space X with the
center at x∈X and of radius ε>0; if ε=0, then BX[x,0]≡x. Let U⊂X. Then U¯ is the closure
of U, coU is the convex
hull of U; co¯U≡coU¯, extU is the set of
all extreme points of U; ext¯U=extU¯. Let ∥U∥X=sup⁡u∈U∥u∥X. Let Uε≡⋃u∈UB[u,ε]¯ if ε>0 and U0≡U¯.LetρX[x;U] be the distance
from the point x∈X to the set U in the space X; let hX+[U1;U]≡sup⁡x∈U1ρX[x,U] be the
Hausdorff semideviation of the set U1 from the set U; let hX[U1;U]=max⁡{hX+[U1;U];hX+[U;U1]} be the
Hausdorff distance between the subsets U1 and U of X.We denote bycomp[X] (resp., comp[X*]) the set of
all nonempty compact subsets of X (resp.,
the set of all nonempty, bounded, closed in the space X, and relatively compact in the weak topology on the
space X subsets of X). Let 2X be the set of
all nonempty bounded subsets of X.Let𝒫 be a system of subsets of X (a subset of X). We denote by Ω(𝒫) the set of all nonempty convex
subsets of X, belonging to
the system 𝒫 (the set of all nonempty
convex subsets of X, belonging to 𝒫).Letℝn be the space of
all n-dimensional
column vectors with the norm |⋅|. We denote by Cn[a,b] (resp., Dn[a,b]) the space of
continuous (resp., absolutely continuous) functions x:[a,b]→ℝn with norm ∥x∥Cn[a,b]=max⁡{|x(t)|:t∈[a,b]} (resp., ∥x∥Dn[a,b]=|x(a)|+∫ab|x˙(s)|ds). Let 𝒰⊂[a,b] be a measurable
set μ(𝒰)>0 (μ—the Lebesgue
measure). We denote by Lpn(𝒰) the space of
all functions x:𝒰→ℝn such that (x(s))p is integrable
(if p<∞) and the space
of all measurable, essentially bounded (if p=∞) functions x:𝒰→ℝn with the norms (2.1)∥x∥Lpn(𝒰)=(∫𝒰|x(s)|pds)1/p,∥x∥L∞n(𝒰)=vraisups∈𝒰|x(s)|, respectively.LetΦ⊂L1n[a,b]. The set Φ is called
integrally bounded if there exists a function φΦ∈L11[a,b] such that |x(t)|≤φΦ(t) for each x∈Φ and almost all t∈[a,b]. The set Φ is said to be decomposable if for each x,y∈Φ and every
measurable set 𝒰⊂[a,b] the inclusion χ(𝒰)x+χ([a,b]∖𝒰)y∈Φ holds, where χ(V) is the
characteristic function of the set V. We denote by Q[L1n[a,b]] (resp., Π[L1n[a,b]]) the set of
all nonempty, closed, and integrally bounded (resp., nonempty, bounded,
closed, and decomposable) subsets of the space L1n[a,b].LetF:[a,b]→comp[ℝn] be a measurable
mapping. Then by definition, S(F)={y∈L1n[a,b]:y(t)∈F(t)for almost allt∈[a,b]}. By C+1[a,b] (resp., L+1[a,b]), denote the
cone of all nonnegative functions of the space C11[a,b] (resp., L11[a,b]).Letf:P→Q be a mapping
between two partially ordered sets P and Q (the partial
order of both sets is denoted by ≤). The mapping f is isotonic if f(x)≤f(y), whenever x≤y.In this paper, the expression “measurability of a
single-valued function” is always used in the sense of Lebesgue
measurability and “measurability of a multivalued function” in the
sense of [21]. Let (T,Σ,μ) be a space with
finite positive measure and let F be a
multivalued mapping from T to ℝn. A set {xν(⋅)}  (ν∈ℕ) of measurable
mappings from T to ℝn is said to
approximate the multivalued mapping F if the set {t∈T∣xν(t)∈F(t)} is measurable
for any ν∈ℕ, and the set F(t) belongs to the
closure of its intersection with the set ⋃ν∈ℕ{xν(t)} for almost all t∈T. A multivalued
mapping F from T to ℝn is called
measurable if there exists a countable set of measurable mappings from T to ℝn that
approximates the mapping F.Further, let us introduce the main characteristic
properties of a set that is decomposable.Lemma 2.1.
LetΦ∈Π[L1n[a,b]].  Then there
exists a function  u∈L11[a,b]  such that |φ(t)|≤u(t)  for each function  φ∈Φ  and almost all  t∈[a,b].Proof.
Letφi∈Φ, i=1,2,…, be a sequence
of functions such that (2.2)lim⁡i→∞∥φi∥L1n[a,b]=∥Φ∥L1n[a,b].
Let us show that there exists a sequence of functionsφ˜i∈Φ, i=1,2,…, such that the
equality (2.2) holds and (2.3)|φ˜1(t)|≤|φ˜2(t)|≤|φ˜3(t)|≤⋯≤|φ˜i(t)|≤|φ˜i+1(t)|≤⋯  for almost all t∈[a,b].
Indeed, letφ˜1=φ1 and φ˜i+1=χ(𝒰i)φ˜i+χ([a,b∖𝒰i)φi+1,  i=1,2,…, where 𝒰i={t∈[a,b]:|φ˜i(t)|≥|φi+1(t)|}. Since Φ∈Π[L1n[a,b]], we see that the
sequence φ˜i∈Φ,  i=1,2,…, has the
following properties: for almost all t∈[a,b], the inequalities (2.3) hold and ∥φ˜i∥L1n[a,b]≥∥φi∥L1n[a,b] for each i=1,2,…. Hence, from
this property and equality (2.2), it follows that the sequence φ˜i, i=1,2,…, satisfies
(2.2). Further, we consider a measurable function u:[a,b]→[0,∞) defined by (2.4)u(t)=lim⁡i→∞|φ˜i(t)|.  Since the set Φ is bounded, we
see, using Fatou's lemma (see [22]), that u∈L11[a,b]. Moreover, by
the definition of the function u and due to
(2.2), (2.5)∫𝒰u(t)dt=∥Φ∥L1n(𝒰)  for every
measurable set 𝒰⊂[a,b]. Now, let us
show that the function u defined by
(2.4) satisfies the assumptions of the lemma. Indeed, if the contrary is true,
then there exist a function φ∈Φ and a
measurable set 𝒰1⊂[a,b]  (μ(𝒰1)>0) such that |φ(t)|>u(t) for each t∈𝒰1. This implies
that ∫𝒰1|φ1(t)|dt>∫𝒰1u(t)dt, which
contradicts (2.5). This completes the proof.Lemma 2.2.
LetΦ∈Π[L1n[a,b]]  and  φi∈Φ,  i=1,2,…,  be a sequence
that is dense in  Φ.
Further, let a
measurable set  F:[a,b]→comp[ℝn]  be defined by  (2.6)F(t)={φi(t),i=1,2,…}¯. Then S(F)=Φ.Proof.
Sinceφi∈S(F) and the
sequence φi, i=1,2,…, is dense in Φ, we have, due to
the closedness of the set Φ, the relation Φ⊂S(F). Let us prove
that S(F)⊂Φ. Let x∈S(F). For each k, i=1,2,…, put (2.7)Eik={t∈[a,b]:|x(t)−φi(t)|≤1k}, which are measurable sets. For i=1, let E˜1k=E1k, and for i=2,3,…, let E˜ik=Eik∖⋃j=1i−1Ejk. Then E˜ik⋂E˜jk=∅ if i≠j. By the
definition of the mapping F:[a,b]→comp[ℝn], for each k=1,2,…, we have (2.8)μ(⋃i=1∞E˜ik)=b−a.  Let xk:[a,b]→ℝn, k=1,2,…, be a sequence
of measurable functions such that (2.9)xk(t)={φi(t)ift∈E˜ik,i=1,2,…,k,φ1(t)ift∈[a,b]∖⋃i=1kE˜ik. Since the set Φ is
decomposable, we see that xk∈Φ for each k=1,2,…. Moreover, from Lemma 2.1 and the definition of the
set E˜ik, it follows that for the functions xk,  k=1,2,…, we have the
estimates (2.10)∥x−xk∥L1n[a,b]≤b−ak+2∫[a,b]∖∪i=1kE˜iku(t)dt,  where u satisfies the
assertions of Lemma 2.1. From (2.8) and (2.10), it follows that xk→x in Ln[a,b] as k→∞. Since the set Φ is closed, we
have that x∈Φ. Hence S(F)⊂Φ. Thus S(F)=Φ.Lemma 2.3.
Let
measurable setsFi:[a,b]→comp[ℝn],  i=1,2,…,
be integrally
bounded, then  S(F1(⋅))⊂S(F2(⋅))  if and only if  F1(t)⊂F2(t)  for almost all  t∈[a,b].Proof.
First
of all, it is evident that if for almost allt∈[a,b], F1(t)⊂F2(t), then S(F1(⋅))⊂S(F2(⋅)).
LetS(F1(⋅))⊂S(F2(⋅)) and let φi∈L1n[a,b], i=1,2,…, be a countable
set, which is dense in S(F1) and which
approximates F1:[a,b]→comp[ℝn] (see [21]).
Thus φi∈S(F2(⋅)) for each i=1,2,… and by the
definition of the set S(F2(⋅)), we have that {φi(t):i=1,2,…}⊂F2(t) for almost all t∈[a,b]. Since the
sequence φi, i=1,2,…, approximates
the map F1:[a,b]→comp[ℝn], it follows from
the previous inclusion that F1(t)⊂F2(t) for almost all t∈[a,b].Corollary 2.4.
LetΦ∈Π[L1n[a,b]]  and let  Fi:[a,b]→comp[ℝn],   i=1,2,  be measurable
sets such that  Φ=S(F1)=S(F2).  Then  F1(t)=F2(t)  for almost all  t∈[a,b].Remark 2.5.
IfΦ∈Π[L1n[a,b]], then a
measurable set F:[a,b]→comp[ℝn], that satisfies S(F)=Φ, uniquely
determines the set Φ.


## 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 spaceL1n[a,b]. 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.
LetΦ be a nonempty
subset of L1n[a,b]. By decΦ, we denote the set of all finite combinations (3.1)y=χ(𝒰1)x1+χ(𝒰2)x2+⋯+χ(𝒰m)xm  of elements xi∈Φ, i=1,2,…,m, where the
disjoint measurable subsets 𝒰i, i=1,2,…,m, of the segment [a,b] are such that ⋃i=1m𝒰i=[a,b].Lemma 3.2.
The setdecΦ
is decomposable
for any nonempty set  Φ⊂L1n[a,b].Proof.
Lety1,y2∈decΦ. Let also 𝒰⊂[a,b] be a measurable
set. Without loss of generality, it can be assumed that (3.2)yi=χ(𝒰1i)x1i+χ(𝒰2i)x2i+⋯+χ(𝒰mi)xmi,  where xji∈Φ, j=1,2,…,m, i=1,2, and the
measurable disjoint sets 𝒰ji⊂[a,b], j=1,2,…,m, i=1,2, are such that [a,b]=⋃j=1m𝒰ji, i=1,2, (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 (3.3)χ(𝒰)y1+χ([a,b]∖𝒰)y2=∑i=1mχ(𝒰∩𝒰i1)xi1+∑i=1mχ(([a,b]∖𝒰)∩𝒰i2)xi2, it follows that χ(𝒰)y1+χ([a,b]∖𝒰)y2∈decΦ. Hence, the set decΦ is
decomposable.Remark 3.3.
Note that even if a setΦ⊂L1n[a,b] is bounded, the
set decΦ is not
necessarily bounded. For example, let us check that (3.4)dec[BLpn[a,b][0,1]]=Lpn[a,b](p∈[1,∞)).  Indeed, let z∈Lpn[a,b] and ei, i=1,2,…,m, be measurable
sets with the following properties: ei∩ej=∅ if i≠j, i,j=1,2,…,m, ⋃i=1mei=[a,b]; for each i=1,2,…,m, the inequality (3.5)∫ei|z(s)|pds<1 holds.   Then zi=χ(ei)z∈BLpn[a,b][0,1], i=1,2,…,m, and (3.6)z=χ(e1)z1+χ(e2)z2+⋯+χ(em)zm. Therefore, z∈dec[BLpn[a,b][0,1]] and
consequently, the equality (3.4) holds.Remark 3.4.
From (3.4), it follows that if a set Φ⊂L1n[a,b] is relatively
compact in the weak topology of L1n[a,b], then the set decΦ does not
necessarily possess this property.Remark 3.5.
Note
that if a set is convex inL1n[a,b], then this set
is not necessarily decomposable. The ball BL1n[a,b][0,1] is an example
of such a set.Remark 3.6.
If a setΦ⊂L1n[a,b] is integrally
bounded, then by Lemma 2.2, for the set dec¯Φ∈Π[L1n[a,b]], there exists a measurable and integrally bounded
mapping Fdec¯Φ:[a,b]→comp[Rn] such that (3.7)dec¯Φ=S(Fdec¯Φ(⋅)).Lemma 3.7.
If a
setΦ⊂L1n[a,b]  is
decomposable, then  decΦ=Φ.Proof.
Evidently,Φ⊂decΦ. We claim that decΦ⊂Φ. The proof is made by induction over m. By the
definition of the switching convexity, any expression (3.1) including two
elements x1,x2∈Φ and two
measurable sets 𝒰1,𝒰2⊂[a,b] belongs to Φ.
Suppose now that form=k, the combination of the form (3.1) belongs to Φ. Let x1,x2,…,xm+1∈Φ and let 𝒰1,𝒰2,…,𝒰m+1⊂[a,b] be disjoint
measurable sets such that [a,b]=⋃i=1m+1𝒰i. Let (3.8)z=χ(𝒰2∪𝒰1)x2+χ(𝒰3)x3+⋯+χ(𝒰m+1)xm+1. By the inductive assumption, z∈Φ and therefore χ(𝒰1)x1+χ([a,b]∖𝒰1)z∈Φ. Since (3.9)χ([a,b]∖𝒰1)z=χ(𝒰2)x2+χ(𝒰3)x3+⋯+χ(𝒰m+1)xm+1, we have that (3.10)χ(𝒰1)x1+χ(𝒰2)x2+⋯+χ(𝒰m+1)xm+1∈Φ. Hence decΦ⊂Φ. This concludes the proof.Corollary 3.8.
IfΦ⊂L1n[a,b],  then the set  decΦ  is the minimal
set which is decomposable and which contains  Φ.Proof.
Consider any setU⊂L1n[a,b] which is
decomposable and which satisfies Φ⊂U. Then, by Lemma
3.7, we have Φ⊂decΦ⊂decU=U.Lemma 3.9.
If a
setΦ⊂L1n[a,b]  is convex, then
so is the set  decΦ∈L1n[a,b].Proof.
Lety1,y2∈decΦ be given by the
formula (3.2). It follows from the convexity of the set Φ⊂L1n[a,b] and the
equality (3.11)λy1+(1−λ)y2=∑i,j=1mχ(𝒰i1∩𝒰j2)(λxi1+(1−λ)xj2) that λy1+(1−λ)y2∈decΦ for any λ∈[0,1]. Thus, the set decΦ is convex.Similar to the definition of the convex hull in a
normed space, the setdecΦ will, in the
sequel, be called the decomposable hull of the setΦin the space of
integrable functions, or simply the decomposable
hull of the setΦ. Likewise, dec¯Φ is addressed as
the closed decomposable hull of the setΦ.Remark 3.10.
IfΦ∈Q[L1n[a,b]], then  the closed
decomposable hull of the set Φ (the set dec¯Φ) can be
constructed as described in Remark 3.6. To do it, one needs a measurable and
integrally bounded (see Remark  3.6)
mapping Fdec¯Φ:[a,b]→comp[ℝn] that satisfies
(3.7). Note that finding  this mapping Fdec¯Φ is easier than
constructing  the set dec¯Φ. At the same time, when one studies the metrical
relations between the sets Φ1,Φ2⊂L1n[a,b] and their
decomposable hulls (see Lemma 3.12), it is more convenient to use Definition
3.1.Lemma 3.11.
Letv∈L1n(𝒰)(𝒰⊂[a,b])       and let a set  Φ⊂L1n[a,b]
be
decomposable. Then for any disjoint measurable sets  𝒰1,𝒰2⊂𝒰  such that  𝒰1∪𝒰2=𝒰, one has(3.12)ρL1n(𝒰)[v;Φ]=ρL1n(𝒰1)[v;Φ]+ρL1n(𝒰2)[v;Φ].Proof.
Indeed,
letε>0 and y∈Φ satisfy ∥v−y∥L1n(𝒰)<ρL1n(𝒰)[v;Φ]+ε. It follows from
this estimate that (3.13)ρL1n(𝒰1)[v;Φ]+ρL1n(𝒰2)[v;Φ]≤∥v−y∥L1n(𝒰1)+∥v−y∥L1n(𝒰2)<ρL1n(𝒰)[v;Φ]+ε. This yields (3.14)ρL1n(𝒰1)[v;Φ]+ρL1n(𝒰2)[v;Φ]≤ρL1n(𝒰)[v;Φ].
Further, let us show that the opposite inequality is
valid. Letyi∈Φ|𝒰i, i=1,2, where Φ|𝒰i is the set of
of all mappings from Φ, restricted to 𝒰i, i=1,2, and suppose
that the functions yi, i=1,2, satisfy (3.15)∥v−yi∥L1n(𝒰i)<ρL1n(𝒰i)[v;Φ]+ε2,i=1,2.  Since the set Φ is
decomposable, it follows that the map y:𝒰→ℝ defined by (3.16)y(t)={y1(t)ift∈𝒰1,y2(t)ift∈𝒰2 belongs to the set Φ|𝒰. By (3.15), we
have (3.17)ρL1n(𝒰)[v;Φ]≤∥v−y∥L1n(𝒰)<ρL1n(𝒰1)[v;Φ]+ρL1n(𝒰2)[v;Φ]+ε. This implies that (3.18)ρL1n(𝒰)[v;Φ]≤ρL1n(𝒰1)[v;Φ]+ρL1n(𝒰2)[v;Φ].  Comparing (3.14)
and (3.18), we obtain (3.12).Lemma 3.12.
IfΦ1,Φ2∈Q[L1n[a,b]]  and there
exists a function  ω∈L+1[a,b]  such that  (3.19)hL1n(𝒰)+[Φ1;Φ2]≤∫𝒰ω(s)ds  for any
measurable set 𝒰⊂[a,b], then (3.20)hL1n(𝒰)+[decΦ1;decΦ2]≤∫𝒰ω(s)ds  for any
measurable set 𝒰⊂[a,b].Proof.
Let𝒰⊂[a,b] be a measurable
set, μ(𝒰)>0. Let z∈decΦ1 and zi∈Φ1, i=1,2,…,m. Suppose also
that the functions zi and disjoint
measurable sets e˜i⊂[a,b], i=1,2,…,m, such that [a,b]=⋃i=1me˜i, satisfy the
equality (3.21)z=χ(e˜1)z1+χ(e˜2)z2+⋯+χ(e˜m)zm.
Further, byz,zi, i=1,2,…,m, we denote the
restrictions of these functions to 𝒰 and put ei=e˜i∩𝒰, i=1,2,…,m.
From (3.21) and Lemma 3.11, it follows that (3.22)ρL1n(𝒰)[z;decΦ2]=∑i=1mρL1n(ei)[zi;decΦ2]≤∑i=1mρL1n(ei)[zi;Φ2].
From (3.19), we obtain that (3.23)ρL1n(ei)[zi;Φ2]≤∫eiω(s)ds  for each i=1,2,…,m.
Therefore, (3.22) and (3.23) imply (3.24)ρL1n(𝒰)[z;decΦ2]≤∫𝒰ω(s)ds.  Since (3.24)
holds for any z∈decΦ1, it follows from
(3.24) that (3.20) holds as well.Remark 3.13.
Note that the functionω∈L+1[a,b] (see (3.19))
provides a uniform with respect to measurable sets 𝒰⊂[a,b] estimate for
the Hausdorff semideviation of the set Φ1 from the set Φ2.Remark 3.14.
The
inequality (3.20) holds true even if the set decΦi is replaced
with its closure dec¯Φi, i=1,2.We say that a multivalued mappingΦ:Cn[a,b]→Q[L1n[a,b]] is integrally bounded on a setK⊂Cn[a,b] if the image Φ(K) is integrally
bounded.LetΦ:Cn[a,b]→Q[L1n[a,b]]. We introduce an
operator Φ˜:Cn[a,b]→Π[L1n[a,b]] by the formula (3.25)Φ˜(x)=dec¯Φ(x).Note that even if a mappingΦ:Cn[a,b]→Q[L1n[a,b]] is continuous,
the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by (3.25)
may be discontinuous. To illustrate this, let us consider an example.Example 3.15.
We  define an
integrable functionφ:[0,2]×[0,1]×[0,2]→ℝ1 by (3.26)φ(x,r)(t)={1ift∈[x,x+r]∩[0,2],r≠0,0ift∈¯[x,x+r]∩[0,2],r≠0,0ifr=0.
We also define a multivalued mappingΦ:[0,1]→Q[L11[0,2]] by the formula (3.27)Φ(r)={⋃x∈[0,2]φ(x,r)ifr≠0,0ifr=0.
Note that(3.28)hL11[0,2][Φ(r1);Φ(r2)]=|r1−r2| for any r1,r2∈[0,1], but at the same time, (3.29)hL11[0,2][Φ˜(0);Φ˜(r)]=2 for any r∈(0,1].Using Lemma3.12, we obtain the following continuity
conditions for the operator Φ˜:Cn[a,b]→Π[L1n[a,b]] given by
(3.25).Definition 3.16.
LetU⊂Cn[a,b]. One says that a
mapping P:U×U→L+1[a,b] is symmetric on
the set U if P(x,y)=P(y,x) for any x,y∈U. One says that a
mapping P:U×U→L+1[a,b] is continuous
in the second variable at a point (x,x) belonging to
the diagonal of U×U if for any
sequence yi∈U such that yi→x as i→∞ it holds that P(x,x)=lim⁡i→∞P(x,yi). One says that a
mapping P:U×U→L+1[a,b] is continuous
in the second variable on the diagonal of U×U if P is continuous
in the second variable at each point of this diagonal. Continuity in the fist
variable is defined similarly.Definition 3.17.
LetU⊂Cn[a,b]. Suppose also
that P(x,x)=0 for any x∈U. One says that a
mapping P:U×U→L+1[a,b] has property 𝒜 on the set U if it is
continuous in the second variable on the diagonal of U×U; it   has property ℬ on the set U if it is
continuous in the first variable on the diagonal of U×U; it has property 𝒞 on the set U if it is
continuous on the diagonal of U×U and symmetric
on the set U.Theorem 3.18.
LetU⊂Cn[a,b].
Suppose also
that for a mapping  Φ:Cn[a,b]→Q[L1n[a,b]]  there exists a
mapping  P:U×U→L+1[a,b]  such that  (3.30)hL1n(𝒰)+[Φ(x),Φ(y)]≤∥P(x,y)∥L11(𝒰)  for any x,y∈U and any
measurable set 𝒰⊂[a,b]. Then for the
mapping Φ˜:Cn[a,b]→Π[L1n[a,b]]  given by
(3.25), the inequality (3.30), where Φ(⋅)≡Φ˜(⋅), is satisfied as
well as for any x,y∈U and any
measurable set 𝒰⊂[a,b].Corollary 3.19.
If
the mappingP:U×U→L+1[a,b]  in Theorem 3.18
has property  𝒜  (resp.,  ℬ,  𝒞) on the set  U⊂Cn[a,b],
then the
operator  Φ˜:Cn[a,b]→Π[L1n[a,b]]  given by (3.25)
is Hausdorff lower semicontinuous (resp., Hausdorff upper
semicontinuous, Hausdorff continuous) on the set  U⊂Cn[a,b].We say that the mappingP:U×U→L+1[a,b] satisfying the
inequality (3.30) for any measurable set 𝒰⊂[a,b] is a majorant mapping for Φ:Cn[a,b]→Q[L1n[a,b]] on the set U.Let a mappingFi:[a,b]×ℝn→comp[ℝn], i=1,2, be measurable
as a composite function for every x∈Cn[a,b]. Let also Fi be integrally
bounded for every bounded set K⊂ℝn. Consider a
mapping ℳ:Cn[a,b]→Q[L1n[a,b]] given by (3.31)ℳ(x)=𝒩1(x)∪𝒩2(x),  where the
mapping 𝒩i:Cn[a,b]→Π[L1n[a,b]], i=1,2, is the
Nemytskii operator generated by the mapping Fi:[a,b]×ℝn→comp[ℝn], i=1,2. For the
operator ℳ:Cn[a,b]→Q[L1n[a,b]] given by
(3.31), the majorant mapping P˜:Cn[a,b]×Cn[a,b]→L+1[a,b] can be defined
as (3.32)P˜(x,y)(t)=max⁡{h+[F1(t,x(t));F1(t,y(t))];h+[F2(t,x(t));F2(t,y(t))]}.It follows from Theorem3.18 that the operator P˜(⋅,⋅) given by (3.32)
is also a majorant mapping for the mapping ℳ˜:Cn[a,b]→Π[L1n[a,b]] given by
(3.25), where Φ(⋅)≡ℳ(⋅). If the mapping Fi:[a,b]×ℝn→comp[ℝn], i=1,2, is Hausdorff
lower semicontinuous (resp., Hausdorff upper semicontinuous and
Hausdorff continuous) in the second variable, then by Corollary 3.19, the
mapping ℳ˜:Cn[a,b]→Π[L1n[a,b]] given by (3.25)
is Hausdorff lower semicontinuous (resp., Hausdorff upper semicontinuous
and Hausdorff continuous).Definition 3.20.
One says that a
multivalued mappingΦ:Cn[a,b]→Q[L1n[a,b]]has Property𝒜 (resp., ℬ and 𝒞) if for this
mapping there exists a majorant mapping P:Cn[a,b]×Cn[a,b]→L+1[a,b] satisfying
Property 𝒜 (resp., ℬ and 𝒞).


## 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 Section3, 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(4.1)x˙∈Φ(x),x(a)=x0(x0∈ℝn),  where the
mapping Φ:Cn[a,b]→Q[L1n[a,b]] satisfies the
following condition: for every bounded set U⊂Cn[a,b], the image Φ(U) is integrally
bounded. Note that the right-hand side of the inclusion (4.1) is not
necessarily decomposable. Note also that x˙ in (4.1) is not
treated as a derivative at a point but as an element of L1n[a,b] (see
[10, 23–25]). 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, 23–25]), can be reduced to (4.1).Definition 4.1.
An absolutely
continuous functionx:[a,b]→ℝn is called a
generalized solution of the problem (4.1) if (4.2)x˙∈dec¯Φ(x),x(a)=x0(x0∈ℝn).Note that from Lemma3.7, it follows that if the set Φ(x) (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,x∈[0,1],(4.3)x˙=kx,x(0)=1. Its solution is the function x=ekt.
We assume that the parameterk may take two
values: 1 or 2. Then the trajectories of such a system are described by the
differential inclusion (4.4)x˙∈Φ(t)x(t),x(0)=1, where Φ(t) is a
multivalued function with the values from the set {1,2}. Note that  dec¯Φ(t)=Φ(t), 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 valuek=1 or with the
parameter value k=2. 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 functionset and e2t.Example 4.3.
Consider a
simple pendulum. It consists of a massm hanging from a
string of length l and fixed at a
pivot point P. 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 (4.5)x¨=−a⁢⁢sin⁢⁡x, where x(t) is the
angular  displacement at the moment t, a=g/l, g is the
acceleration of gravity, and l 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(4.6)x¨=−ax. Let us now assume that the length of the string l may change,
that is, it may take an value from a finite set {l1,…,lm}. In this case,
the equation of simple harmonic motion transforms to the differential inclusion
with a multivalued mapping (4.7)x¨∈Φ(x), where Φ(x)=⋃i=1m−(g/li)x.
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 operatorΦ:Cn[a,b]→Q[L1n[a,b]] is called a
Volterra-Tikhonov (or simply a Volterra) operator (see [26]) if the equality x=y on [a,τ], τ∈(a,b], implies (Φ(x))|τ=(Φ(y))|τ, where (Φ(z))|τ is the set of
all functions from Φ(z) restricted to [a,τ].In what follows, we assume that the operatorΦ:Cn[a,b]→Q[L1n[a,b]] (the right-hand
side of the inclusion (4.1)) is a Volterra operator. This implies that the
operator Φ˜:Cn[a,b]→Q[L1n[a,b]] given by (3.25)
is also a Volterra operator.Letτ∈(a,b]. Let us
determine the continuous mapping Vτ:Cn[a,τ]→Cn[a,b] by (4.8)(Vτx)={x(t)ift∈[a,τ],x(τ)ift∈(τ,b].Definition 4.5.
One says that an
absolutely continuous functionx:[a,τ]→ℝn is a
generalized solution of the problem (4.1) on the interval [a,τ], τ∈(a,b], if x satisfies x˙∈(dec¯Φ(Vτ(x)))|τ and x(a)=x0, where the
continuous mapping Vτ:Cn[a,τ]→Cn[a,b] is given by (4.8).A functionx:[a,c)→ℝn, which is
absolutely continuous on any interval [a,τ]⊂[a,c), c∈(a,b], is called a
generalized solution of the problem (4.1) on the interval [a,c) if for each τ∈(a,c) the restriction
of x to [a,τ] is a
generalized solution of the problem (4.1) on the interval [a,τ].A generalized solutionx:[a,c)→ℝn of the problem
(4.1) on the interval [a,c) is said to be
nonextendable if there is no generalized solution y of the problem
(4.1) on any larger interval [a,τ] (here, τ∈(c,b] if c<b and τ=b if c=b) such that x(t)=y(t) for each t∈[a,c).In Theorems4.6–4.12 below, we assume that the mapping Φ:Cn[a,b]→Q[L1n[a,b]] has Property 𝒜. Due to
Corollary 3.19, the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by (3.25)
is lower semicontinuous. Due to [27, 28], the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] admits a
continuous selection. Therefore, the following propositions on local solutions
of the problem (4.1) are straightforward.Theorem 4.6.
There existsτ∈(a,b]
such that a
generalized solution of the problem (4.1) is defined on the interval  [a,τ].Theorem 4.7.
A
generalized solutionx:[a,c)→ℝn  of the problem
(4.1) admits a continuation if and only if  lim⁡¯t→c−0|x(t)|<∞.Theorem 4.8.
Ify
is a
generalized solution of the problem (4.1) on the interval  [a,τ],  τ∈(a,b),
then there
exists a nonextendable solution  x
of the problem
(4.1) defined on the interval  [a,c)(c∈(τ,b]),
or on the
entire interval  [a,b],  such that  x(t)=y(t)  for each  t∈[a,τ].LetH(x0,τ) be the set of
all generalized solutions of the problem (4.1) on the interval [a,τ](τ∈(a,b]).We say that generalized solutions of the problem (4.1) admit a uniform a priori estimate if there exists a number r>0 such that for
every τ∈(a,b], there is no generalized solution y∈H(x0,τ) satisfying ∥y∥Cn[a,τ]>r.Theorems4.6–4.8 yield the following result.Theorem 4.9.
Let
the generalized solutions of the problem (4.1) admit a uniform a priori
estimate. Then  H(x0,τ)≠∅  for any  τ∈(a,b]
and there
exists a number  r>0  such that  ∥y∥Cn[a,τ]≤r  for any  τ∈(a,b],  y∈H(x0,τ).Definition 4.10.
One says that a
mappingΦ:Cn[a,b]→Q[L1n[a,b]] has Property Γ1 if there exists
an isotonic continuous operator Γ1:C+1[a,b]→L+1[a,b] satisfying the
following conditions:(i)
for any functionx∈Cn[a,b] and any
measurable set 𝒰⊂[a,b], one has(4.9)∥Φ(x)∥L1n(𝒰)≤∥Γ1(Zx)∥L11(𝒰),  where the
continuous mapping Z:Cn[a,b]→C+1[a,b] is given by (4.10)(Zx)(t)=|x(t)|;(ii)
the local solutions of the problem(4.11)y˙=Γ1(y),y(a)=|x0|  admit a uniform
a priori estimate.Lemma 4.11.
Suppose that a multivalued mappingΦ:Cn[a,b]→Q[L1n[a,b]]  has Property  Γ1.  Then so does
the mapping  Φ˜:Cn[a,b]→Π[L1n[a,b]]  given by
(3.25).Proof.
It
suffices to show that(4.12)∥decΦ(x)∥L1n(𝒰)≤∥Γ1(Zx)∥L11(𝒰)  for any
function x∈Cn[a,b] and any
measurable set 𝒰⊂[a,b]. Indeed, let a
function y∈decΦ(x) be as in (3.1).
By (4.9), (4.13)∫𝒰i∩𝒰|xi(s)|ds≤∥Γ1(Zx)∥L11(𝒰i∩𝒰) for each i=1,2,…,m. Hence, we have
that for the function y∈decΦ(x), the estimate (4.14)∫𝒰|y(s)|ds≤∥Γ1(Zx)∥L11(𝒰) is satisfied as well. This gives the inequality (4.12).
The proof is complete.Let a continuous operatorΘ:Dn[a,b]→C+1[a,b] be given by (4.15)(Θz)(t)=|z(a)|+∫at|z˙(s)|ds.Theorem 4.12.
Suppose that a mappingΦ:Cn[a,b]→Q[L1n[a,b]]  has Property  Γ1.  Then the set  H(x0,τ)  is nonempty for
any  τ∈(a,b]
and there
exists a number  r>0  such that  ∥y∥Cn[a,τ]≤r  for any  y∈H(x0,τ),  τ∈(a,b].Proof.
Indeed,
letx∈H(x0,τ) (τ∈(a,b]). From Lemma 4.11,
it follows that for any t∈[a,τ], (4.16)(Θx)(t)≤|x0|+∫at(Γ1(Zx))(s)ds≤|x0|+∫at(Γ1(Θx))(s)ds, where the function Θx is given by
(4.15). Due to the theorem on integral inequalities for an isotonic operator
(see [29]), this implies that we actually have Θx≤ξ0, where ξ0 is the upper
solution of the problem (4.11). Thus, there is no x∈H(x0,τ) satisfying the
inequality ∥x∥Cn[a,τ]>∥ξ0∥C1[a,b]. 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 operatorΛ:L1n[a,b]→Cn[a,b] be given by (4.17)(Λz)(t)=∫atz(s)ds,t∈[a,b].  We say that Λ:L1n[a,b]→Cn[a,b] 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  Φ:Cn[a,b]→Q[L1n[a,b]]  has Property  𝒞.  Then for any
function  v∈L1n[a,b]  and any  ε>0, there exists a generalized solution  x∈Dn[a,b]  of the problem
(4.1) such that  (4.18)∥x˙−v∥L1n(𝒰)≤ρL1n(𝒰)[v,dec¯Φ(x)]+εμ(𝒰)  for any
measurable set 𝒰⊂[a,b].
IfΦ:Cn[a,b]→Ω(Q[L1n[a,b]]), then the
theorem is also valid for ε=0.Proof.
LetΦ:Cn[a,b]→Q[L1n[a,b]] have Property 𝒞. Then by
Corollary 3.19, the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by (3.25)
is continuous. Therefore (see [30–32]), given  a number ε>0 and a function v∈L1n[a,b], there exists a continuous mapping φ:Cn[a,b]→L1n[a,b] satisfying φ(y)∈Φ˜(y) and (4.19)∥φ(y)−v∥L1n(𝒰)≤ρL1n(𝒰)[v,dec¯Φ(y)]+εμ(𝒰)  for any y∈Cn[a,b] and any
measurable set 𝒰⊂[a,b]. It follows from
Theorem 4.9 that H(x0,τ)≠∅ for any τ∈(a,b], and that there exists a number r>0 such that ∥y∥Cn[a,τ]≤r for each τ∈(a,b], y∈H(x0,τ). Now, we show
that there exists x∈H(x0,b) satisfying
(4.18). Consider the problem (4.20)x˙∈dec¯Φ(Wr(x)),x(a)=x0(x0∈ℝn),  where the
continuous mapping Wr:Cn[a,b]→Cn[a,b] is given by (4.21)(Wrx)(t)={x(t)if|x(t)|≤r+2,r+2|x(t)|x(t)if|x(t)|>r+2.  We denote by H(W) the set of all
solutions of the problem (4.20). Let us show that H(W)=H(x0,b). It follows from
the definition of the mapping Wr:Cn[a,b]→Cn[a,b] (see (4.21))
that H(x0,b)⊂H(W). Let us prove
that H(W)⊂H(x0,b). Assume the
converse. Then there exists y∈H(W) such that ∥y∥Cn[a,b]>r+2. Since y(a)=x0, we have |y(a)|<r+2. This implies
that there exists a number τ∈(a,b] such that ∥y|τ∥Cn[a,τ]=r+1 (y|τ is the
restriction of the function y to [a,τ]). By (4.21),
we have y|τ∈H(x0,τ). This contradicts
to the definition of the number r. Hence, H(x0,b)=H(W). Consider a
continuous operator Ψ:Cn[a,b]→Cn[a,b] given by (4.22)Ψ(x)=x0+Λφ(Wr(x)),  where the
operator Λ:L1n[a,b]→Cn[a,b] is the operator
of integration defined by (4.17), and φ:Cn[a,b]→L1n[a,b] is a continuous
selection of the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by
(3.25). The function φ ia also assumed
to satisfy (4.19). Since the operator Wr:Cn[a,b]→Cn[a,b] is bounded, we
obtain that the image Ψ(Cn[a,b]) is a relatively
compact subset of Cn[a,b]. Hence, the set U=co¯Ψ(Cn[a,b]) is a convex
compact set. Since the operator Ψ:Cn[a,b]→Cn[a,b] given by (4.22)
takes the set U into itself, we
have, by Schauder theorem, that the mapping Ψ(⋅) has a fixed
point. This fixed point x is the solution
of the problem (4.20). It follows from the above equality H(W)=H(x0,b) that this
solution x∈H(W) is a
generalized solution of the problem (4.1). Since x˙=φ(x), we see that
(4.19) implies (4.18).Let us prove the second statement of the theorem. LetΦ:Cn[a,b]→Ω(Q[L1n[a,b]]). Suppose also
that Φ has Property 𝒞. Then by Lemma
3.9, Φ˜:Cn[a,b]→Ω(Π[L1n[a,b]]). Hence for each i=1,2,…, there exists a generalized solution xi∈Dn[a,b] of the problem
(4.1) such that for any measurable set 𝒰⊂[a,b], the inequality (4.18) is valid for x˙=x˙i and ε=1/i. Since the set H(x0,b) is bounded, we
see that the sequence {x˙i} is weakly
compact in L1n[a,b]. Without loss of
generality, it can be assumed that x˙i→x˙ weakly in L1n[a,b] and xi→x in Cn[a,b] as i→∞. Let us show
that x is a
generalized solution of the problem (4.1). In other words, we have to prove
that x˙∈dec¯Φ(x). Assume that the
functions yi∈dec¯Φ(x),  i=1,2,…, satisfy (4.23)∥yi−x˙i∥L1n[a,b]=ρL1n[a,b][x˙i;dec¯Φ(x)]  (as dec¯Φ(x)∈Π[L1n[a,b]], these functions
do exist). It follows from (4.23) that (4.24)∥yi−x˙i∥L1n[a,b]≤hL1n[a,b][dec¯Φ(xi);dec¯Φ(x)].
Since the mapping Φ˜:Cn[a,b]→Ω(Π[L1n[a,b]]) given by (3.25)
is continuous, we obtain, by (4.24), that yi−x˙i→0 in L1n[a,b] as i→∞. Since x˙i→x˙ weakly in L1n[a,b] as i→∞, we have that yi→x˙ weakly in L1n[a,b] as i→∞. Therefore, the
convexity of the set dec¯Φ(x) implies that x˙∈dec¯Φ(x) (see [21]).
Thus, x is a generalized
solution of the problem (4.1).Further, let us show that (4.19) holds for the
solution x and for ε=0. Since x˙i→x˙ weakly in L1n[a,b] as i→∞, we have, by
[21], that for each m=1,2,…, there exist numbers i(m), λjm≥0, j=1,2,…,i(m), satisfying the
following conditions: ∑j=1i(m)λjm=1; the sequence {βm=∑j=1i(m)λjmx˙j+m} tends to x˙ in L1n[a,b]. Since (4.25)∥x˙−v∥L1n[a,b]≤∥x˙−βm∥L1n[a,b]+∑j=1i(m)λjm∥x˙j+m−v∥L1n[a,b] for each m=1,2,…, it follows, due
to the choice of the sequence {x˙i}, that (4.26)∥x˙−v∥L1n[a,b]≤∥x˙−βm∥L1n[a,b]+∑j=1i(m)λjmρL1n[a,b][v;dec¯Φ(xj+m)]+(b−a)∑j=1i(m)λjm1j+m for each m=1,2,….Since(4.27)lim⁡i→∞ρL1n[a,b][v;dec¯Φ(xi)]=ρL1n[a,b][v;dec¯Φ(x)], it follows that letting m→∞ in the previous
inequality, we obtain (4.28)∥x˙−v∥L1n[a,b]=ρL1n[a,b][v;dec¯Φ(x)]. Finally, note that by the decomposability of the set dec¯Φ(x), this equality
holds for any measurable set 𝒰⊂[a,b]. This completes
the proof.Theorems4.12 and 4.13 yield the following result.Corollary 4.14.
Suppose that a mappingΦ:Cn[a,b]→Q[L1n[a,b]]  has
Properties  Γ1  and  𝒞.  Then for any function  v∈L1n[a,b]  and any  ε>0, there exists a generalized solution  x∈Dn[a,b]  of the problem
(4.1) such that (4.18) holds for any measurable set  𝒰⊂[a,b].
IfΦ:Cn[a,b]→Ω(Q[L1n[a,b]]), then the
corollary is also valid for ε=0.Remark 4.15.
Consider the convex compact setU=co¯Ψ˜(Cn[a,b])⊂Cn[a,b], where the
mapping Ψ˜:Cn[a,b]→2Cn[a,b] is given by (4.29)Ψ˜(x)=x0+ΛΦ˜(Wr(x)).  Here, the
operators Φ˜:Cn[a,b]→Π[L1n[a,b]] and Wr:Cn[a,b]→Cn[a,b] are determined
by (3.25) and (4.21), respectively. If a number r>0 is such that ∥y∥Cn[a,τ]≤r for any τ∈(a,b], y∈H(x0,τ), then due to the
the coincidence of the sets H(W) and H(x0,b) (see the proof
of Theorem 4.13), H(x0,b)⊂U.Definition 4.16.
Givenε≥0, p≥0, u∈L+1[a,b], one says that a mapping Φ:Cn[a,b]→Q[L1n[a,b]] has Property Γ2u,ε,p if there exists
an isotonic and continuous Volterra operator Γ2:C+1[a,b]→L+1[a,b] satisfying the
following conditions:(i)
Γ2(0)=0;(ii)
for any functionsx,y∈Cn[a,b] and any
measurable set 𝒰⊂[a,b], one has (4.30)hL1n(𝒰)[Φ(x);Φ(y)]≤∥Γ2(Z(x−y))∥L11(𝒰),  where the
continuous mapping Z:Cn[a,b]→C+1[a,b] is determined
by (4.10);(iii)
the set of all local solutions of the problem(4.31)y˙=u+ε+Γ2(y),y(a)=p,  admits a
uniform a priori estimate.Giveny∈Dn[a,b] and ϰ∈L+1[a,b], the following estimate will be used in the sequel: (4.32)ρL1n(𝒰)[y˙;Φ(y)]≤∫𝒰ϰ(s)ds  for each
measurable set 𝒰⊂[a,b].Theorem 4.17.
Let
functionsy∈Dn[a,b]  and  ϰ∈L+1[a,b]  satisfy the
inequality (4.32) for each measurable set  𝒰⊂[a,b].
Suppose that a
mapping  Φ:Cn[a,b]→Q[L1n[a,b]]  has Property  Γ2ϰ,ε,p,  where  ε≥0,  p=|x0−y(a)|, and  x0  is the initial
condition of the problem (4.1). Then for any generalized solution of the
problem (4.1) satisfying  (4.33)∥x˙−y˙∥L1n(𝒰)≤ρL1n(𝒰)[y˙;dec¯Φ(x)]+εμ(𝒰)  for any
measurable set 𝒰⊂[a,b], the following conditions are satisfied:
(1)
(4.34)Θ(x−y)(t)≤ξ(ϰ,ε,p)(t)  for each t∈[a,b], where the
function ξ(ϰ,ε,p)∈D1[a,b] is the upper
solution of the problem (4.31) for u=ϰ and p=|x0−y(a)|, and the mapping Θ:Dn[a,b]→C+1[a,b] is given by
(4.15);(2)
(4.35)|x˙(t)−y˙(t)|≤ϰ(t)+ε+(Γ2(ξ(ϰ,ε,p)))(t)  for almost all t∈[a,b].Proof.
First,
note that since the mappingΦ:Cn[a,b]→Q[L1n[a,b]] has Property Γ2ϰ,ε,p, it follows from
Theorem 3.18 that so does the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] determined by
(3.25). Further, the inequality (4.33) yields that (4.36)∥x˙−y˙∥L1n(𝒰)≤ρL1n(𝒰)[y˙;dec¯Φ(y)]+hL1n(𝒰)[dec¯Φ(y);dec¯Φ(x)]+εμ(𝒰)  for any
measurable set 𝒰⊂[a,b].
Remark4.15 and relations (4.36), (4.32), and (4.30)
imply that for any measurable set 𝒰⊂[a,b], we obtain the inequality (4.37)∥x˙−y˙∥L1n(𝒰)≤∫𝒰(ϰ(s)+ε+Γ2(Z(x−y))(s))ds, where the mapping Z:Cn[a,b]→C+1[a,b] is given by
(4.10). It follows from this inequality that (4.38)|x˙(t)−y˙(t)|≤ϰ(t)+ε+Γ2(Z(x−y))(t)  for  almost all t∈[a,b]. Since Z(x−y)(t)≤Θ(x−y)(t) for all t∈[a,b] (see (4.10),
(4.15)) and the operator Γ2:C+1[a,b]→L+1[a,b]] (see (4.38)) is
isotonic, we have that (4.39)|x˙(t)−y˙(t)|=Θ˙(x−y)(t)≤ϰ(t)+ε+Γ2(Θ(x−y))(t)  for almost all t∈[a,b]. Therefore,
(4.39) and the theorem on differential inequalities with an isotonic operator
(see [29]) imply (4.34) for any t∈[a,b]. The inequality
(4.35) follows from (4.34) and (4.39). The proof is complete.Theorems4.13 and 4.17 yield the following result.Theorem 4.18.
Let
functionsy∈Dn[a,b]  and  ϰ∈L+1[a,b]  satisfy (4.32)
for each measurable set  𝒰⊂[a,b].
Suppose that a
mapping  Φ:Cn[a,b]→Q[L1n[a,b]]  has Property  Γ2ϰ,ε,p,  where  ε≥0,  p=|x0−y(a)|,  x0  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  ε>0, there exists a generalized solution  x∈Dn[a,b]
of the problem
(4.1) which satisfies (4.34) and (4.35) for all  t∈[a,b]
and for almost
all  t∈[a,b],  respectively.
IfΦ:Cn[a,b]→Ω(Q[L1n[a,b]]), then the theorem
is also valid for ε=0.Corollary 4.19.
Let
functionsy∈Dn[a,b]  and  ϰ∈L+1[a,b]  satisfy (4.32)
for each measurable set  𝒰⊂[a,b].
Suppose that a
mapping  Φ:Cn[a,b]→Q[L1n[a,b]]  has properties  Γ1  and  Γ2ϰ,ε,p,  where  ε≥0,  p=|x0−y(a)|,  x0
is the initial
condition in the problem (4.1). Then for  ε>0, there exists a generalized solution  x∈Dn[a,b]
of the problem
(4.1) which satisfies (4.34) and (4.35) for all  t∈[a,b]
and for almost
all  t∈[a,b],  respectively.
IfΦ:Cn[a,b]→Ω(Q[L1n[a,b]]), then the
corollary is also valid for ε=0.Remark 4.20.
It
follows from the proof of Theorem4.17 that Theorems 4.17, 4.18, and Corollary 4.19
are also valid if the functions y∈Dn[a,b] and ϰ∈L+1[a,b] satisfy (4.40)ρL1n(𝒰)[y˙;dec¯Φ(y)]≤∫𝒰ϰ(s)ds for each measurable set 𝒰⊂[a,b].Definition 4.21.
An absolutely
continuous functionx∈Dn[a,b] is called a
generalized quasisolution of the problem (4.1) if there exists a sequence of
functions xi∈Dn[a,b], i=1,2,…, such that the
following conditions hold:(i)
xi→x in Cn[a,b] as i→∞;(ii)
x˙i∈dec¯Φ(x) and xi(a)=x0 for each i=1,2,….Note that by Lemma3.7, if the set Φ(x) mentioned in
Definition 4.21 is decomposable, then a generalized quasisolution coincides with
a quasisolution defined in [9, 33], where Φ(⋅) 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 x˙i∈dec¯Φ(x). 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.Letℋ(x0) be the set of
all generalized quasisolutions of the problem (4.1).We define a mappingΦ˜co:Cn[a,b]→Ω(Π[L1n[a,b]]) by the formula (4.41)Φ˜co(x)=co¯(dec¯Φ(x)).We callΦ˜co:Cn[a,b]→Ω(Π[L1n[a,b]]) the convex
decomposable hull.Consider the problem (4.1) with the convex
decomposable hull Φ˜co:Cn[a,b]→Ω(Π[L1n[a,b]])  given by (4.41)
leading to (4.42)x˙∈Φ˜co(x),x(a)=x0(x0∈ℝn).LetHco(x0,τ) be the set of
all solutions of the problem (4.42) on the interval [a,τ](τ∈(a,b]).Theorem 4.22.
ℋ(x0)=Hco(x0,b).Proof.
First,
we will show thatHco(x0,b)⊂ℋ(x0). Let x∈Hco(x0,b). By [35], for x˙∈L1n[a,b], there exists a sequence yi∈dec¯Φ(x), i=1,2,…, such that yi→x˙ weakly in  L1n[a,b] as i→∞. This implies
that xi=x0+Λyi→x in Cn[a,b] as i→∞, where Λ:L1n[a,b]→Cn[a,b] is the operator
of integration (see (4.17)). Hence, Hco(x0,b)⊂ℋ(x0).
Let us now prove thatℋ(x0)⊂Hco(x0,b). Let x∈ℋ(x0). Then there exists
a sequence xi∈Dn[a,b], i=1,2,…, satisfying the
following conditions: (1) x˙i∈dec¯Φ(x) (see (4.41))
and xi(a)=x0 for each i=1,2,…; (2) xi→x in Cn[a,b] as i→∞. Since the
sequence x˙i, i=1,2,…, is weakly
compact, we can assume without loss of generality that x˙i→x˙ weakly in L1n[a,b] as i→∞. Since x˙i∈Φ˜co(x) (see (4.41)),
it follows that x˙∈Φ˜co(x) (see [21]).
Hence x∈Hco(x0,b) and therefore ℋ(x0)⊂Hco(x0,b).Remark 4.23.
Theorem4.22 may still remain valid
even if the mapping Φ:Cn[a,b]→Q[L1n[a,b]] is
discontinuous and its image Φ(ℬ) is not
integrally bounded for every bounded set ℬ⊂Cn[a,b]. 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 Φ is a Volterra
operator.Definition 4.24.
One says that a
compact convex setU⊂Cn[a,b] has Property 𝒟 if ℋ(x0)⊂U, and for any x∈ℋ(x0), there exists a sequence of absolutely continuous
functions xi:[a,b]→ℝn, i=1,2,…, such that(i)
xi→x in Cn[a,b] as i→∞;(ii)
xi∈U, x˙i∈dec¯Φ(x) and xi(a)=x0 for each i=1,2,….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  U⊂Cn[a,b]
satisfying
Property  𝒟.Proof.
It
follows from Theorem4.22 and Remark 4.15 that the set U=co¯Ψ˜(Cn[a,b]) has Property 𝒟. Here, the
mapping Ψ˜:Cn[a,b]→2Cn[a,b] is determined
by (4.29), where Φ˜(⋅)≡Φ˜co(⋅).Lemma 4.26.
Let setsΦi∈Π[L1n[a,b]],  i=1,2,  satisfy  Φi=S(Fi(⋅)),  i=1,2,  where  Fi:[a,b]→comp[ℝn]
are measurable
mappings. Then for any measurable set  𝒰⊂[a,b], one has  (4.43)hL1n(𝒰)[Φ1;Φ2]≤∫𝒰h[F1(t);F2(t)]dt≤2hL1n(𝒰)[Φ1;Φ2].Proof.
Let𝒰⊂[a,b] be a measurable
set. Put (4.44)𝒰˜={t∈𝒰:h+[F1(t);F2(t)]≥h+[F2(t);F1(t)]}. The set 𝒰˜⊂𝒰 is measurable.
Since (4.45)∫𝒰h[F1(t);F2(t)]dt=∫𝒰˜h+[F1(t);F2(t)]dt+∫𝒰∖𝒰˜h+[F2(t);F1(t)]dt, we have (4.46)∫𝒰h[F1(t);F2(t)]dt=hL1n(𝒰˜)+[Φ1;Φ2]+hL1n(𝒰∖𝒰˜)+[Φ2;Φ1]. This implies (4.43), and the proof is completed.LetF:[a,b]→comp[ℝn] be a measurable
mapping. Let a mapping coF:[a,b]→comp[ℝn] be defined by (4.47)(coF)(t)=co(F(t)).Corollary 4.27.
Let
setsΦi∈Π[L1n[a,b]],  i=1,2,  satisfy  Φi=S(Fi(⋅)),  i=1,2,  where  Fi:[a,b]→comp[ℝn]  are measurable
mappings. Then for any measurable set  𝒰⊂[a,b], one has  (4.48)hL1n(𝒰)[co¯(Φ1);co¯(Φ2)]≤2hL1n(𝒰)[Φ1;Φ2].Proof.
By
[35], we have that co¯(Φi)=S(coFi(⋅)), i=1,2. Therefore, (4.49)hL1n(𝒰)[co¯(Φ1);co¯(Φ2)]≤∫𝒰h[(coF1)(t);(coF2)(t)]dt for any measurable set 𝒰⊂[a,b]. Since, for any
measurable set 𝒰⊂[a,b], (4.50)∫𝒰h[(coF1)(t);(coF2)(t)]dt≤∫𝒰h[(F1)(t);(F2)(t)]dt, we obtain, due to (4.43), the inequality (4.48).Definition 4.28.
One says that a
mappingΦ:Cn[a,b]→Q[L1n[a,b]] has Property Γ3 if Property Γ20,0,0 is satisfied
and the following conditions hold:(i)
Γ2(0)=0;(ii)
on every interval[a,τ] (τ∈(a,b]), there exists a unique zero solution of the problem
(4.31), where u=0, ε=0, p=0.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  Φ:Cn[a,b]→Q[L1n[a,b]]  satisfies
Property  Γ3.  Then,  H(x0,b)≠∅  and  (4.51)H(x0,b)¯=Hco(x0,b),  where H(x0,b)¯ is the closure
of the set H(x0,b) in Cn[a,b].Proof.
Let us first prove that the setHco(x0) is closed in Cn[a,b]. Indeed, suppose
that a sequence xi∈Hco(x0),  i=1,2,…, tends to x in Cn[a,b] as i→∞. Since the
sequence {x˙i} is integrally
bounded, it follows that xi∈Dn[a,b] and x˙i→x˙ weakly in L1n[a,b] as i→∞. For each i=1,2,…, let the function zi∈Φ˜co(x) satisfy (4.52)∥x˙i−zi∥L1n[a,b]=ρL1n[a,b][x˙i;Φ˜co(x)],  where Φ˜co:Cn[a,b]→Ω(Π[L1n[a,b]]) is the convex
decomposable hull given by (4.41). Since the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by (3.25)
is Hausdorff continuous, it follows from (4.48) that so is the mapping Φ˜co(x):Cn[a,b]→Ω(Π[L1n[a,b]]). Therefore,
(4.52) implies that x˙i−zi→0 in L1n[a,b] as i→∞. Hence, zi→x˙ weakly in L1n[a,b] as i→∞. Since the set Φ˜co(x) is convex, we
have (see [21]) that x˙∈Φ˜co(x). Therefore, the
set Hco(x0) is closed in Cn[a,b].
Now, let us prove the equality (4.51). The closedness
of the set Hco(x0) yields that H(x0,b)¯⊂Hco(x0). Further, let us
show that Hco(x0)⊂H(x0,b)¯. Suppose x∈Hco(x0). Then from
Theorem 4.22, it follows that there exists a sequence yi∈Dn[a,b], i=1,2,…, such that yi∈Φ˜(x), yi(a)=x0, i=1,2,… (x0 is the initial
condition in the problem (4.1)) and yi→x in Cn[a,b] as i→∞. Since the
mapping Φ:Cn[a,b]→Q[L1n[a,b]] has Property Γ3, we see that,
due to (4.30), (4.53)ρL1n(𝒰)[y˙i;Φ(yi)]≤hL1n(𝒰)[Φ(x);Φ(yi)]≤∫𝒰(Γ2(Z(x−yi)))(s)ds for each i=1,2,… and any
measurable set 𝒰⊂[a,b]. Here, the
operator Z:Cn[a,b]→C+1[a,b] is given by
(4.10). Since the mapping Γ2:C+1[a,b]→L+1[a,b] is continuous
and Γ2(0)=0, we have that ϰi=Γ2(Z(x−yi))→0 in L11[a,b] as i→∞. Since the
problem (4.31) with u=0, ε=0, and p=0 only has the
zero solution on each interval [a,τ] (τ∈(a,b]), we see that
the set of all local solutions of the problem (4.31) with u=ϰi, ε=1/i, and p=0 admits a
uniform a priori estimate starting from some i=1,2,… (see [36]).
Renumerating, we may assume without loss of generality that this holds true for
all i=1,2,…. This implies (see [29]) that for each i=1,2,…, there exists the upper solution ξ(ϰi,1/i,0) of the problem
(4.31) with u=ϰi, ε=1/i, and p=0. Hence, it
follows from Theorem 4.18 that for each i=1,2,…, there exists a generalized solution xi∈Dn[a,b] of the problem
(4.1) satisfying Θ(xi−yi)≤ξ(ϰi,1/i,0), where the
continuous operator Θ:Dn[a,b]→C+1[a,b] is given by
(4.15). Since ξ(ϰi,1/i,0)→0 in C1[a,b] as i→∞, we have that Θ(xi−yi)→0 as i→∞. Since yi→x in Cn[a,b] as i→∞, we see that xi→x in Cn[a,b] as i→∞. Therefore, x∈H(x0,b)¯ and
consequently Hco(x0)⊂H(x0,b)¯. This yields
(4.51). The proof is complete.Corollary 4.30.
Suppose that a mappingΦ:Cn[a,b]→Q[L1n[a,b]]  has
Properties   Γ1  and  Γ3.  Then  H(x0,b)≠∅  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, 10–12, 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, 40–43]). 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 dec¯Φ(x)? It follows from
Remark 3.10 that constructing dec¯Φ(x) for each fixed x∈Cn[a,b] is equivalent
to finding a measurable, integrally bounded mapping Δx:[a,b]→comp[ℝn] satisfying (5.1)dec¯Φ(x)=S(Δx(⋅)).  The mapping Δx:[a,b]→comp[ℝn] is, in the
sequel, written as Δ:[a,b]×Cn[a,b]→comp[ℝn] and called a
mapping  generating the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by
(3.25).Denote byK([a,b]×[0,∞)) the set of all
continuous functions η:[a,b]×[0,∞)→[0,∞) satisfying the following conditions:(1)
for eachδ≥0, η(⋅,δ)∈L11[a,b];(2)
for eachδ≥0, there exists a function βδ(⋅)∈L11[a,b] such that η(t,τ)≤βδ(t) for almost all t∈[a,b] and all τ∈[0,δ];(3)
lim⁡δ→0+0η(t,δ)=η(t,0)=0 for almost all t∈[a,b].Since the mappingsΔ(⋅,⋅) and Φ(⋅) 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 dec¯Φ(x) can be studied
via the robustness properties of Δ. Assume that the
perturbation Δη(t,x,δ) (e.g., an error
in measurements of Δ(t,x)) is given by (5.2)Δη(t,x,δ)=(Δ(t,x))η(t,δ),  where η(⋅,⋅)∈K([a,b]×[0,∞)) (here, (Δ(t,x))η(t,δ) is an η-neighborhood
of the set Δ(t,x), see
Preliminaries).Note that (5.2) yields (5.3)h[Δ(t,x);Δη(t,x,δ)]=η(t,δ)  for all (t,x)∈[a,b]×Cn[a,b]. Thus, (5.3)
implies that (5.4)lim⁡δ→+0h[Δ(t,x);Δη(t,x,δ)]=0  for each
function η(⋅,⋅)∈K([a,b]×[0,∞)), almost all t∈[a,b], and all x∈Cn[a,b]. Therefore, all
mappings Δη:[a,b]×Cn[a,b]×[0,∞)→comp[ℝn] defined by
(5.2) and depending on η(⋅,⋅)∈K([a,b]×[0,∞)) are close (in
the sense of (5.4)) to the mapping Δ:[a,b]×Cn[a,b]→comp[ℝn]. The mapping Δη:[a,b]×Cn[a,b]×[0,∞)→comp[ℝn] is called the  approximating operator.We define a mappingΦ˜η:Cn[a,b]×[0,∞)→Π[L1n[a,b]] by the formula (5.5)Φ˜η(x,δ)=S(Δη(⋅,x,δ)),  where the
operator Δη:[a,b]×Cn[a,b]×[0,∞)→comp[ℝn] is given by
(5.2). The equalities (5.3) and (5.5) imply that (5.6)hL1n[a,b][Φ˜η(x,δ);Φ˜(x)]=∫abη(t,δ)dt  for any x∈Cn[a,b].It follows from (5.6) and the Lebesgue theorem that (5.7)lim⁡δ→0+0hL1n[a,b][Φ˜η(x,δ);Φ˜(x)]=0.Thus, all mappingsΦ˜η:Cn[a,b]×[0,∞)→Π[L1n[a,b]] defined by
(5.2) and (5.5) and depending on η(⋅,⋅)∈K([a,b]×[0,∞)) are close (in
the sense of (5.7)) to the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by
(3.25).Lemma 5.1 (see [6]).
LetX  be a normed
space and let  U⊂X
be a convex
set. Then  (5.8)hX[BX[x1,r1]∩U;BX[x2,r2]∩U]≤∥x1−x2∥X+|r2−r1|  for all x1,x2∈U and all r1,r2>0.Denote byP(Cn[a,b]×[0,∞)) the set of all
continuous functions ω:Cn[a,b]×[0,∞)→[0,∞) such that ω(x,0)=0 for any x∈Cn[a,b] and ω(x,δ)>0 for any (x,δ)∈Cn[a,b]×(0,∞).LetU⊂Cn[a,b] be a closed
convex set and let ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)). We define a
multivalued mapping MU(ω):U×[0,∞)→Ω(U) by (5.9)MU(ω)(x,δ)=BCn[a,b][x,ω(x,δ)]¯∩U.The inequality (5.8) yields the following result.Lemma 5.2.
LetU⊂Cn[a,b]  be a closed
convex set and let  ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)).
Then, a
multivalued mapping  MU(ω):U×[0,∞)→Ω(U)
given by (5.9)
is Hausdorff continuous.We define a mappingφU(ω):[a,b]×U×[0,∞)→[0,∞) by the formula (5.10)φU(ω)(t,x,δ)=sup⁡y∈MU(ω)(x,δ)h[Δ(t,x);Δ(t,y)],  where the
mapping MU(ω):U×[0,∞)→Ω(U) is given by
(5.9) and the mapping Δ:[a,b]×Cn[a,b]→comp[ℝn] generates the
mapping Φ˜ given by
(3.25).It is natural to address the value of the functionφU(ω)(⋅,⋅,⋅) at the point (t,x,δ)∈[a,b]×U×[0,∞) as the  modulus of continuity of the mappingΔ:[a,b]×Cn[a,b]→comp[ℝn] at the point (t,x) with respect to
the variable x∈U. We call the
function ω(⋅,⋅) the  radius of continuity, while the function φU(⋅,⋅,⋅) itself is
called the modulus of continuity of
the mappingΔ:[a,b]×Cn[a,b]→comp[ℝn]with respect to
the radius of continuityω(⋅,⋅).Definition 5.3.
One says that a
mappingΦ:Cn[a,b]→Q[Ln[a,b]] has Property C˜ if the mapping Δ:[a,b]×Cn[a,b]→comp[ℝn] generating the
mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by (3.25)
is Hausdorff continuous in the second variable for almost all t∈[a,b].Lemma 5.4.
Suppose that
for a mappingΦ:Cn[a,b]→Q[Ln[a,b]], there exists an isotonic continuous operator  Γ:C+1[a,b]→L+1[a,b]
satisfying the
following conditions: (i)
Γ(0)=0;(ii)
the inequality (4.30), where Γ2≡Γ, is satisfied
for any x,y∈Cn[a,b] and any measurable
set 𝒰⊂[a,b].
Then the mappingΦ(⋅) has Property 𝒞˜.Proof.
Letxi→x in Cn[a,b] as i→∞. Let us show
that (5.11)lim⁡i→∞h[Δ(t,xi);Δ(t,x)]=0  for almost all t∈[a,b].
For eachi=1,2,…, put yi=sup⁡j≥i∥xj−x∥Cn[a,b]. Due to Theorem
3.18, (4.43), and the isotonity of the operator Γ:C+1[a,b]→L+1[a,b], for each i=1,2,… and almost all t∈[a,b], we have (5.12)h[Δ(t,xi);Δ(t,x)]≤2Γ(Z(xi−x))(t)≤2Γ(yi)(t). Since the sequence Γ(yi),  i=1,2,…, decreases, we
obtain, due to the continuity of the mapping Γ(⋅) and the
equality Γ(0)=0, the equality
(5.11). This completes the proof.Lemma 5.5.
LetU
be a nonempty,
convex, compact set in the space  Cn[a,b]  and let  ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)).
Suppose also
that a mapping  Φ:Cn[a,b]→Q[L1n[a,b]]  has Property  C˜.  Then the
mapping  φU(ω):[a,b]×U×[0,∞)
given by (5.10)
has the following properties: (i)
φU(ω)(⋅,x,δ) is measurable
for any (x,δ)∈U×[0,∞);(ii)
φU(t,⋅,⋅) is continuous
on U×[0,∞) for almost all t∈[a,b];(iii)
for anyx∈U and for almost
all t∈[a,b], (5.13)lim⁡z→x,δ→0+0φU(ω)(t,x,δ)=0;(iv)
there exists an integrable functionpU:[a,b]→[0,∞) such that φU(ω)(t,x,δ)≤pU(t) for almost all t∈[a,b], any x∈U, and all δ∈[0,∞).Definition 5.6.
LetU⊂Cn[a,b]. One says that the
function η(⋅,⋅)∈K([a,b]×[0,∞)) provides on U a uniform with
respect to the radius of continuity ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)) estimate from
above for the modulus of continuity of the mapping Δ:[a,b]×Cn[a,b]→comp[ℝn]; if for any ε>0 there exists δ(ε)>0 such that for
almost all t∈[a,b], all x∈U, and δ∈(0,δ(ε)], one has (5.14)φU(ω)(t,x,δ)≤η(t,ε),  where φU:[a,b]×U×[0,∞)→[0,∞) is given by
(5.10).LetU⊂Cn[a,b] and ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)). One defines a
function λU(ω):[a,b]×[0,∞)→[0,∞) by (5.15)λU(ω)(t,δ)=sup⁡x∈UφU(ω)(t,x,δ).Lemma5.1 yields the following result.Corollary 5.7.
LetU
be a nonempty,
convex, compact set in the space  Cn[a,b]  and let  ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)).
Suppose also
that a mapping  Φ:Cn[a,b]→Q[L1n[a,b]]  has Property  C˜.  Then the
mapping  λU(ω):[a,b]×[0,∞)→[0,∞)
given by (5.15)
belongs to the set  K([a,b]×[0,∞))
and provides a
uniform (in the sense of Definition 5.6) estimate from above for the modulus of
continuity of the mapping  Δ:[a,b]×Cn[a,b]→comp[ℝn].Remark 5.8.
Corollary5.7 yields that if U is a nonempty,
convex, compact set in the space Cn[a,b] and a mapping Φ:Cn[a,b]→Q[L1n[a,b]] has Property C˜, then for a
given ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)), there exists at least one function η(⋅,⋅)∈K([a,b]×[0,∞)) that provides a
uniform (in the sense of Definition 5.6) estimate from above for the modulus of
continuity of the mapping Δ:[a,b]×Cn[a,b]→comp[ℝn].Letη(⋅,⋅)∈K([a,b]×[0,∞)). For each δ∈[0,∞), consider the initial value problem (5.16)x˙∈Φ˜η(x,δ),x(a)=x0(x0∈ℝn),  where the
mapping Φ˜η:Cn[a,b]×[0,∞)→Π[L1n[a,b]] is given by
(5.1) and (5.5).Since the operatorΦ˜:Cn[a,b]→Π[L1n[a,b]] given by (3.25)
is a Volterra operator, we see that the mapping Δ:[a,b]×Cn[a,b]→comp[ℝn] has the
following property: if x=y on [a,τ]   (τ∈(a,b]), then Δ(t,x)=Δ(t,y) for almost all t∈[a,τ]. This property,
(5.1), and (5.4)  imply that the
operator Φ˜η:Cn[a,b]×[0,∞)→Π[L1n[a,b]] is a Volterra
operator for each δ∈[0,∞).Any solution of the problem (5.16) with a given δ>0 is said to be a
generalized δ-solution (a
generalized approximate solution with external perturbations) of the problem
(4.1). We denote by Hη(δ)(U) the set of all
generalized δ-solutions of
(4.1) belonging to U⊂Cn[a,b].Theorem 5.9.
Suppose that a setU⊂Cn[a,b]  has Property  𝒟.
Then for any
function  η(⋅,⋅)∈K([a,b]×[0,∞))
that provides a
uniform (in the sense of Definition 5.6) estimate from above for the modulus of
continuity of the mapping  Δ:[a,b]×Cn[a,b]→comp[ℝn], one has  (5.17)Hco(x0,b)=⋂δ>0Hη(δ)(U)¯,  where Hη(δ)(U)¯ is the closure
of Hη(δ)(U) in Cn[a,b].Proof.
First,
let us prove that(5.18)Hco(x0,b)⊂⋂δ>0Hη(δ)(U)¯.  Let x∈Hco(x0,b). Let us show
that x is a limit
point of the set Hη(δ)(U) for any δ>0. By Theorem 4.22, x is a
generalized quasisolution of the problem (4.1). Moreover, x∈U. Since the set U has Property 𝒟, we see that
there exists a sequence of absolutely continuous functions xi:[a,b]→ℝn, i=1,2,…, such that the
following conditions hold: xi→x in Cn[a,b] as i→∞;   xi∈U, x˙i∈dec¯Φ(x), and xi(a)=x0 for each i=1,2,…. Suppose that η(⋅,⋅)∈K([a,b]×[0,∞)) provides a
uniform (in the sense of Definition 5.6) estimate from above for the modulus of
continuity of the mapping Δ:[a,b]×Cn[a,b]→comp[ℝn]. Then there
exists i1 such that ∥x−xi∥Cn[a,b]<ω(x,δ) for each i≥i1. This implies that xi∈BCn[a,b][x,ω(x,δ)]¯ for each i≥i1. Therefore, xi∈MU(ω)(x,δ) for each i≥i1. By Definition
5.6, there exists a number i2≥i1 such that (5.19)φU(ω)(t,x,∥x−xi∥Cn[a,b])≤η(t,δ)  for any i≥i2 and almost all t∈[a,b].
The inequality (5.19) yields that (5.20)ρ[x˙i(t);Δ(t,xi)]≤h[Δ(t,x);Δ(t,xi)]≤φU(ω)(t,x,∥x−xi∥Cn[a,b])≤η(t,δ)  for each i≥i2 and almost all t∈[a,b]. By (5.20), xi∈Hη(δ)(U) for each i≥i2. This implies
that x is a limit
point of the set Hη(δ)(U). Therefore, x∈Hη(δ)(U)¯, and consequently (5.18), is satisfied.
Let us prove the opposite inclusion(5.21)⋂δ>0Hη(δ)(U)¯⊂Hco(x0,b).  Let x∈⋂δ>0Hη(δ)(U)¯. This implies
that for each i=1,2,…, there exists xi∈Hη(1/i)(U) satisfying ∥x−xi∥Cn[a,b]<1/i. Suppose that
functions zi∈dec¯Φ(x) satisfy (5.22)|x˙i(t)−zi(t)|=ρ[x˙i(t);Δ(t,x)]  for each i=1,2,… and almost all t∈[a,b]. Let us show
that (5.23)lim⁡i→∞∥x˙i−zi∥L1n[a,b]=0.  Since η(⋅,⋅)∈K([a,b]×[0,∞)), by the Lebesgue
theorem, we have that (5.24)lim⁡i→∞∫abη(t,1i)dt=0.  By (5.22), the
estimates (5.25)|x˙i(t)−zi(t)|≤h[Δ(t,xi)η(t,1/i);Δ(t,x)]≤η(t,1i)+h[Δ(t,x);Δ(t,xi)] are satisfied for each i=1,2,… and almost all t∈[a,b]. Therefore, (5.26)∫ab|x˙i(t)−zi(t)|dt≤∫abη(t,1i)dt+2hL1n[a,b][dec¯Φ(x);dec¯Φ(xi)] for each i=1,2,…. By (5.24) and due to the continuity of the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by
(3.25), we have (5.23).
Sincex˙i→x˙ weakly in L1n[a,b] as i→∞, we have that zi→x˙ weakly in L1n[a,b] as i→∞. Therefore, by
[21], x˙∈Φ˜co(x) and hence x∈H(x0,b). Thus, (5.21) is valid. Hence, (5.17) holds and the
proof is complete.Theorem 5.10.
Suppose that a setU⊂Cn[a,b]  has Property  𝒟.  Then,  (5.27)H(x0,b)¯=⋂δ>0Hη(δ)(U)¯  for any η(⋅,⋅)∈K([a,b]×[0,∞)) if and only if
the equality (4.51) is satisfied.Proof.
Let us
prove thesufficiency. Assume
that (4.51) holds. Let us show that the equality (5.27) is satisfied
for any function η(⋅,⋅)∈K([a,b]×[0,∞)). By the
definition of the problem (5.16), the inclusion (5.28)H(x0,b)⊂Hη(δ)(U) is satisfied for any δ>0. Therefore, for
any δ>0, we have the inclusion (5.29)H(x0,b)¯⊂Hη(δ)(U)¯ and consequently the inclusion (5.30)H(x0,b)¯⊂⋂δ>0Hη(δ)(U)¯ holds. Now, let us check that the opposite relation (5.31)⋂δ>0Hη(δ)(U)¯⊂H(x0,b)¯, which is, by (4.51), equivalent to the inclusion (5.32)⋂δ>0Hη(δ)(U)¯⊂Hco(x0,b), holds as well. The latter relation can be proven
similarly to Theorem 5.9.
Thenecessity 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 H(x0,b) with respect to
external perturbations η(⋅,⋅)∈K([a,b]×[0,∞)). These external
perturbations (e.g., η(⋅,⋅)∈K([a,b]×[0,∞))) characterize
an error in measurements of the values of the mapping Φ˜:Cn[a,b]→Π[L1n[a,b]] given by
(3.25).On the other hand, each generalized solutionx:[a,b]→ℝn 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 P(Cn[a,b]×[0,∞)) 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 mappingΔext:[a,b]×Cn[a,b]→comp[ℝn] by (5.33)Δext(t,x)=ext¯(coΔ(t,x)),  where Δ:[a,b]×Cn[a,b]→comp[ℝn] is the mapping
generating the operator Φ˜:Cn[a,b]→Π[L1n[a,b]] given by
(3.25); see the definition of ext¯(coΔ(t,x)) in
Section 2. Let us remark that the mapping Δext(⋅,x) is measurable
(see [25]) and integrally bounded for each x∈Cn[a,b].Consider the operatorΦ˜ext:Cn[a,b]→Π[L1n[a,b]] given by (5.34)Φ˜ext(x)=S(Δext(⋅,x)),  where the
mapping Δext:[a,b]×Cn[a,b]→comp[ℝn] is given by
(5.33).Remark 5.12.
Note that for eachx∈Cn[a,b], the set Φ˜ext(x) has the
following property (5.35)co¯(Φ˜ext(x))=co¯(dec¯Φ(x)).  Also, Φ˜ext(x) is the minimal
set among all nonempty closed in L1n[a,b] decomposable
subsets of dec¯Φ(x) satisfying
(5.35).Consider the problem(5.36)x˙∈Φ˜ext(x),x(a)=x0(x0∈ℝn).  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).Letℋext(x0) 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.
ℋext(x0)=Hco(x0,b).Letη(⋅,⋅)∈K([a,b]×[0,∞)),  ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)). Let also U be a convex
closed set in Cn[a,b]. We define
mappings Φ˜η,ω:U×[0,∞)→comp[L1n[a,b]*],    Δext,η:[a,b]×Cn[a,b]×[0,∞)→comp[ℝn],   Φ˜ext,η:Cn[a,b]×[0,∞)→Π[L1n[a,b]],   Φ˜ext,η,ω:Cn[a,b]×[0,∞)→comp[L1n[a,b]*] by the formulas (5.37)Φ˜η,ω(x,δ)=Φ˜η((MU(ω)(x,δ)),δ),(Δext,η)(t,x,δ)=(Δext(t,x))η(t,δ),(Φ˜ext,η)(x,δ)=S(Δext,η(⋅,x,δ)),Φ˜ext,η,ω(x,δ)=Φ˜ext,η((MU(ω)(x,δ)),δ),  where the mappings
Φ˜η:Cn[a,b]×[0,∞)→Π[L1n[a,b]], MU(ω):U×[0,∞)→Ω(U), Δext:[a,b]×Cn[a,b]×[0,∞)→comp[ℝn] are given by
the equalities (5.5),  (5.9), and (5.33), respectively.For eachδ>0, consider the following problems on U⊂Cn[a,b]: (5.38)(5.39)x˙∈Φ˜ext,η,ω(x,δ),x(a)=x0(x0∈ℝn),  where the
mappings Φ˜η,ω:U×[0,∞)→comp[L1n[a,b]*],   Φ˜ext,η,ω:U×[0,∞)→comp[L1n[a,b]*] are given by (5.37).We call any solution of the problem (5.38) with a fixed δ>0 a  generalizedδ-solution of the problem (4.1), or a generalized approximate solution of (4.1) with external
and internal perturbations. For each δ>0, we denote by Hη(δ),ω(δ)(U) (Hext,η(δ),ω(δ)(U)) the set of
all solutions of the problem (5.38)  ((5.39)) on U⊂Cn[a,b]. Since Φ˜ext,η,ω(x,δ)⊂Φ˜η,ω(x,δ) for any δ>0 and any x∈U, we see that Hext,η(δ),ω(δ)(U)⊂Hη(δ),ω(δ)(U) for any δ>0.Theorem 5.14.
Let
the setU⊂Cn[a,b]  have Property  𝒟.  Then for any  η(⋅,⋅)∈K([a,b]×[0,∞)),  ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)), one has  (5.40)Hco(x0,b)=⋂δ>0Hext,η(δ),ω(δ)(U)¯=⋂δ>0Hη(δ),ω(δ)(U)¯,  where Hext,η(δ),ω(δ)(U)¯ and Hη(δ),ω(δ)(U)¯ are the
closures of the sets Hext,η(δ),ω(δ)(U) and Hη(δ),ω(δ)(U), respectively,
in the space Cn[a,b].Proof.
First
of all, let us check that(5.41)Hco(x0,b)⊂⋂δ>0Hext,η(δ),ω(δ)(U)¯.  Let x∈H(x0,b). We show that x is a limit
point of the set Hext,η(δ),ω(δ)(U) for any δ>0. By Corollary
5.13, x is a
generalized extreme quasisolution of the problem (4.1). Since the set U has Property 𝒟, we see that x∈U and there
exists a sequence of absolutely continuous functions xi:[a,b]→ℝn, i=1,2,…, with the
following properties: xi→x in Cn[a,b] as i→∞; xi∈U, x˙i∈Φ˜ext(x), and xi(a)=x0 for each i=1,2,…. Here, the operator Φ˜ext:Cn[a,b]→Π[L1n[a,b]] is given by
(5.33) and (5.34).
Let us prove also that there exists a numberi0 such that for
each i≥i0, (5.42)xi∈Hext,η(δ),ω(δ)(U).  Since ω(⋅,⋅)∈P(Cn[a,b]×[0,∞)), we see that
there exists a number i0 such that for
each i≥i0,   x∈BCn[a,b][xi;ω(xi,δ)]. This implies
that for each i≥i0, we have that x∈MU(xi,δ) (see (5.9)).
Therefore, for each i≥i0, the inclusion Φ˜ext(x)⊂Φ˜ext,η,ω(xi,δ) holds. Hence,
for each i≥i0, we have (5.42). This means that x is a limit
point of the set Hext,η(δ),ω(δ)(U). Therefore, x∈Hext,η(δ),ω(δ)(U)¯ and (5.41) is
satisfied.
The relation(5.43)⋂δ>0Hext,η(δ),ω(δ)(U)¯⊂Hco(x0,b) 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.
Theorem5.14 says that no measurement accuracy of the values of the mapping Δ:[a,b]×Cn[a,b]→comp[ℝn] could guarantee
the “reconstruction” of the set H(x0,b)¯ by means of Hη(δ),ω(δ)(U)¯. 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 (generalizedδ-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.



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*Source: 829701-2008-02-17.xml*