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Abstract and Applied Analysis
Volume 2008, Article ID 829701, 35 pages
http://dx.doi.org/10.1155/2008/829701
Research Article

Generalized Solutions of Functional Differential Inclusions

1Center for Integrative Genetics (CIGENE), Norwegian University of Life Sciences, 1432 Aas, , Norway
2Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Aas 1432, Norway
3Department of Algebra and Geometry, Tambov State University, Tambov 392000, Russia
4Department of Higher Mathematics, Faculty of Electronics and Computer Sciences, Moscow State Forest University, Moscow 141005, Russia

Received 12 March 2007; Revised 4 July 2007; Accepted 12 September 2007

Academic Editor: Yong Zhou

Copyright © 2008 Anna Machina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the initial value problem for a functional differential inclusion with a Volterra multivalued mapping that is not necessarily decomposable in 𝐿𝑛1[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.

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 [58], 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 𝐿𝑛1[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 𝐿𝑛1[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, 1013]). 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 [1520]). 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 𝐋𝑛1[𝑎,𝑏], as this mapping is not necessarily convex-valued.

2. Preliminaries

We start with the notation and some definitions. Let 𝑋 be a normed space with the norm 𝑋. Let 𝐵𝑋[𝑥,𝜀] be the closed ball in the space 𝑋 with the center at 𝑥𝑋 and of radius 𝜀>0; if 𝜀=0, then 𝐵𝑋[𝑥,0]𝑥. Let 𝑈𝑋. Then 𝑈 is the closure of 𝑈, co𝑈 is the convex hull of 𝑈; co𝑈co𝑈, ext𝑈 is the set of all extreme points of 𝑈; ext𝑈=ext𝑈. Let 𝑈𝑋=sup𝑢𝑈𝑢𝑋. Let 𝑈𝜀𝑢𝑈𝐵[𝑢,𝜀] if 𝜀>0 and 𝑈0𝑈.

Let 𝜌𝑋[𝑥;𝑈] be the distance from the point 𝑥𝑋 to the set 𝑈 in the space 𝑋; let +𝑋[𝑈1;𝑈]sup𝑥𝑈1𝜌𝑋[𝑥,𝑈] be the Hausdorff semideviation of the set 𝑈1 from the set 𝑈; let 𝑋[𝑈1;𝑈]=max{+𝑋[𝑈1;𝑈];+𝑋[𝑈;𝑈1]} be the Hausdorff distance between the subsets 𝑈1 and 𝑈 of 𝑋.

We denote by comp[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.

Let 𝒫 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).

Let 𝑛 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]𝑛 with norm 𝑥𝐶𝑛[𝑎,𝑏]=max{|𝑥(𝑡)|𝑡[𝑎,𝑏]} (resp., 𝑥𝐷𝑛[𝑎,𝑏]=|𝑥(𝑎)|+𝑏𝑎|̇𝑥(𝑠)|𝑑𝑠). Let 𝒰[𝑎,𝑏] be a measurable set 𝜇(𝒰)>0 (𝜇—the Lebesgue measure). We denote by 𝐿𝑛𝑝(𝒰) the space of all functions 𝑥𝒰𝑛 such that (𝑥(𝑠))𝑝 is integrable (if 𝑝<) and the space of all measurable, essentially bounded (if 𝑝=) functions 𝑥𝒰𝑛 with the norms 𝑥𝐿𝑛𝑝(𝒰)=𝒰||||||𝑥(𝑠)𝑝𝑑𝑠1/𝑝,𝑥𝐿𝑛(𝒰)=vraisup𝑠𝒰||||||𝑥(𝑠),(2.1) respectively.

Let Φ𝐿𝑛1[𝑎,𝑏]. The set Φ is called integrally bounded if there exists a function 𝜑Φ𝐋11[𝑎,𝑏] such that |𝑥(𝑡)|𝜑Φ(𝑡) for each 𝑥Φ and almost all 𝑡[𝑎,𝑏]. The set Φ is said to be decomposable if for each 𝑥,𝑦Φ and every measurable set 𝒰[𝑎,𝑏] the inclusion 𝜒(𝒰)𝑥+𝜒([𝑎,𝑏]𝒰)𝑦Φ holds, where 𝜒(𝑉) is the characteristic function of the set 𝑉. We denote by 𝑄[𝐋𝑛1[𝑎,𝑏]] (resp., Π[𝐿𝑛1[𝑎,𝑏]]) the set of all nonempty, closed, and integrally bounded (resp., nonempty, bounded, closed, and decomposable) subsets of the space 𝐿𝑛1[𝑎,𝑏].

Let 𝐹[𝑎,𝑏]comp[𝑛] be a measurable mapping. Then by definition, 𝑆(𝐹)={𝑦𝐿𝑛1[𝑎,𝑏]𝑦(𝑡)𝐹(𝑡)foralmostall𝑡[𝑎,𝑏]}. By 𝐶1+[𝑎,𝑏] (resp., 𝐿1+[𝑎,𝑏]), denote the cone of all nonnegative functions of the space 𝐶11[𝑎,𝑏] (resp., 𝐿11[𝑎,𝑏]).

Let 𝑓𝑃𝑄 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 𝑥𝑦.

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 𝑛. A set {𝑥𝜈()} (𝜈) of measurable mappings from 𝑇 to 𝑛 is said to approximate the multivalued mapping 𝐹 if the set {𝑡𝑇𝑥𝜈(𝑡)𝐹(𝑡)} is measurable for any 𝜈, and the set 𝐹(𝑡) belongs to the closure of its intersection with the set 𝜈{𝑥𝜈(𝑡)} for almost all 𝑡𝑇. A multivalued mapping 𝐹 from 𝑇 to 𝑛 is called measurable if there exists a countable set of measurable mappings from 𝑇 to 𝑛 that approximates the mapping 𝐹.

Further, let us introduce the main characteristic properties of a set that is decomposable.

Lemma 2.1. Let ΦΠ[𝐿𝑛1[𝑎,𝑏]]. Then there exists a function 𝑢𝐿11[𝑎,𝑏] such that |𝜑(𝑡)|𝑢(𝑡) for each function 𝜑Φ and almost all 𝑡[𝑎,𝑏].

Proof. Let 𝜑𝑖Φ, 𝑖=1,2,, be a sequence of functions such that lim𝑖𝜑𝑖𝐿𝑛1[𝑎,𝑏]=Φ𝐿𝑛1[𝑎,𝑏].(2.2)
Let us show that there exists a sequence of functions 𝜑𝑖Φ, 𝑖=1,2,, such that the equality (2.2) holds and |||𝜑1||||||(𝑡)𝜑2||||||(𝑡)𝜑3||||||(𝑡)𝜑𝑖||||||(𝑡)𝜑𝑖+1|||(𝑡)(2.3) for almost all 𝑡[𝑎,𝑏].
Indeed, let 𝜑1=𝜑1 and 𝜑𝑖+1=𝜒(𝒰𝑖)𝜑𝑖+𝜒([𝑎,𝑏𝒰𝑖)𝜑𝑖+1, 𝑖=1,2,, where 𝒰𝑖={𝑡[𝑎,𝑏]|𝜑𝑖(𝑡)||𝜑𝑖+1(𝑡)|}. Since ΦΠ[𝐿𝑛1[𝑎,𝑏]], we see that the sequence 𝜑𝑖Φ, 𝑖=1,2,, has the following properties: for almost all 𝑡[𝑎,𝑏], the inequalities (2.3) hold and 𝜑𝑖𝐿𝑛1[𝑎,𝑏]𝜑𝑖𝐿𝑛1[𝑎,𝑏] for each 𝑖=1,2,. Hence, from this property and equality (2.2), it follows that the sequence 𝜑𝑖, 𝑖=1,2,, satisfies (2.2). Further, we consider a measurable function 𝑢[𝑎,𝑏][0,) defined by 𝑢(𝑡)=lim𝑖|||𝜑𝑖|||(𝑡).(2.4) Since the set Φ is bounded, we see, using Fatou's lemma (see [22]), that 𝑢𝐿11[𝑎,𝑏]. Moreover, by the definition of the function 𝑢 and due to (2.2), 𝒰Φ𝑢(𝑡)𝑑𝑡=𝐿𝑛1(𝒰)(2.5) 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 𝜑Φ and a measurable set 𝒰1[𝑎,𝑏] (𝜇(𝒰1)>0) such that |𝜑(𝑡)|>𝑢(𝑡) for each 𝑡𝒰1. This implies that 𝒰1|𝜑1(𝑡)|𝑑𝑡>𝒰1𝑢(𝑡)𝑑𝑡, which contradicts (2.5). This completes the proof.

Lemma 2.2. Let ΦΠ[𝐿𝑛1[𝑎,𝑏]] and 𝜑𝑖Φ, 𝑖=1,2,, be a sequence that is dense in Φ. Further, let a measurable set 𝐹[𝑎,𝑏]comp[𝑛] be defined by 𝐹(𝑡)=𝜑𝑖(𝑡),𝑖=1,2,.(2.6) Then 𝑆(𝐹)=Φ.

Proof. Since 𝜑𝑖𝑆(𝐹) and the sequence 𝜑𝑖, 𝑖=1,2,, is dense in Φ, we have, due to the closedness of the set Φ, the relation Φ𝑆(𝐹). Let us prove that 𝑆(𝐹)Φ. Let 𝑥𝑆(𝐹). For each 𝑘, 𝑖=1,2,, put 𝐸𝑘𝑖=|||𝑡[𝑎,𝑏]𝑥(𝑡)𝜑𝑖|||1(𝑡)𝑘,(2.7) which are measurable sets. For 𝑖=1, let 𝐸𝑘1=𝐸𝑘1, and for 𝑖=2,3,, let 𝐸𝑘𝑖=𝐸𝑘𝑖𝑖1𝑗=1𝐸𝑘𝑗. Then 𝐸𝑘𝑖𝐸𝑘𝑗= if 𝑖𝑗. By the definition of the mapping 𝐹[𝑎,𝑏]comp[𝑛], for each 𝑘=1,2,, we have 𝜇(𝑖=1𝐸𝑘𝑖)=𝑏𝑎.(2.8) Let 𝑥𝑘[𝑎,𝑏]𝑛, 𝑘=1,2,, be a sequence of measurable functions such that 𝑥𝑘𝜑(𝑡)=𝑖𝐸(𝑡)if𝑡𝑘𝑖𝜑,𝑖=1,2,,𝑘,1(𝑡)if𝑡[𝑎,𝑏]𝑘𝑖=1𝐸𝑘𝑖.(2.9) Since the set Φ is decomposable, we see that 𝑥𝑘Φ for each 𝑘=1,2,. Moreover, from Lemma 2.1 and the definition of the set 𝐸𝑘𝑖, it follows that for the functions 𝑥𝑘, 𝑘=1,2,, we have the estimates 𝑥𝑥𝑘𝐿𝑛1[𝑎,𝑏]𝑏𝑎𝑘+2[𝑎,𝑏]𝑘𝑖=1𝐸𝑘𝑖𝑢(𝑡)𝑑𝑡,(2.10) where 𝑢 satisfies the assertions of Lemma 2.1. From (2.8) and (2.10), it follows that 𝑥𝑘𝑥 in 𝐿𝑛[𝑎,𝑏] as 𝑘. Since the set Φ is closed, we have that 𝑥Φ. Hence 𝑆(𝐹)Φ. Thus 𝑆(𝐹)=Φ.

Lemma 2.3. Let measurable sets 𝐹𝑖[𝑎,𝑏]comp[𝑛], 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 all 𝑡[𝑎,𝑏], 𝐹1(𝑡)𝐹2(𝑡), then 𝑆(𝐹1())𝑆(𝐹2()).
Let 𝑆(𝐹1())𝑆(𝐹2()) and let 𝜑𝑖𝐿𝑛1[𝑎,𝑏], 𝑖=1,2,, be a countable set, which is dense in 𝑆(𝐹1) and which approximates 𝐹1[𝑎,𝑏]comp[𝑛] (see [21]). Thus 𝜑𝑖𝑆(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[𝑛], it follows from the previous inclusion that 𝐹1(𝑡)𝐹2(𝑡) for almost all 𝑡[𝑎,𝑏].

Corollary 2.4. Let ΦΠ[𝐿𝑛1[𝑎,𝑏]] and let 𝐹𝑖[𝑎,𝑏]comp[𝑛], 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. If ΦΠ[𝐿𝑛1[𝑎,𝑏]], then a measurable set 𝐹[𝑎,𝑏]comp[𝑛], 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 space 𝐋𝑛1[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. Let Φ be a nonempty subset of 𝐋𝑛1[𝑎,𝑏]. By decΦ, we denote the set of all finite combinations 𝒰𝑦=𝜒1𝑥1𝒰+𝜒2𝑥2𝒰++𝜒𝑚𝑥𝑚(3.1) of elements 𝑥𝑖Φ, 𝑖=1,2,,𝑚, where the disjoint measurable subsets 𝒰𝑖, 𝑖=1,2,,𝑚, of the segment [𝑎,𝑏] are such that 𝑚𝑖=1𝒰𝑖=[𝑎,𝑏].

Lemma 3.2. The set decΦ is decomposable for any nonempty set Φ𝐋𝑛1[𝑎,𝑏].

Proof. Let 𝑦1,𝑦2decΦ. Let also 𝒰[𝑎,𝑏] be a measurable set. Without loss of generality, it can be assumed that 𝑦𝑖𝒰=𝜒𝑖1𝑥𝑖1𝒰+𝜒𝑖2𝑥𝑖2𝒰++𝜒𝑖𝑚𝑥𝑖𝑚,(3.2) where 𝑥𝑖𝑗Φ, 𝑗=1,2,,𝑚, 𝑖=1,2, and the measurable disjoint sets 𝒰𝑖𝑗[𝑎,𝑏], 𝑗=1,2,,𝑚, 𝑖=1,2, are such that [𝑎,𝑏]=𝑚𝑗=1𝒰𝑖𝑗, 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 𝜒(𝒰)𝑦1𝑦+𝜒[𝑎,𝑏]𝒰2=𝑚𝑖=1𝜒𝒰𝒰1𝑖𝑥1𝑖+𝑚𝑖=1𝜒[𝑎,𝑏]𝒰𝒰2𝑖𝑥2𝑖,(3.3) it follows that 𝜒(𝒰)𝑦1+𝜒([𝑎,𝑏]𝒰)𝑦2decΦ. Hence, the set decΦ is decomposable.

Remark 3.3. Note that even if a set Φ𝐋𝑛1[𝑎,𝑏] is bounded, the set decΦ is not necessarily bounded. For example, let us check that 𝐵dec𝐋𝑛𝑝[𝑎,𝑏][0,1]=𝐋𝑛𝑝[𝑎,𝑏]𝑝[1,).(3.4) Indeed, let 𝑧𝐋𝑛𝑝[𝑎,𝑏] and 𝑒𝑖, 𝑖=1,2,,𝑚, be measurable sets with the following properties: 𝑒𝑖𝑒𝑗= if 𝑖𝑗, 𝑖,𝑗=1,2,,𝑚, 𝑚𝑖=1𝑒𝑖=[𝑎,𝑏]; for each 𝑖=1,2,,𝑚, the inequality 𝑒𝑖||||||𝑧(𝑠)𝑝𝑑𝑠<1(3.5) holds. Then 𝑧𝑖=𝜒(𝑒𝑖)𝑧𝐵𝐋𝑛𝑝[𝑎,𝑏][0,1], 𝑖=1,2,,𝑚, and 𝑒𝑧=𝜒1𝑧1𝑒+𝜒2𝑧2𝑒++𝜒𝑚𝑧𝑚.(3.6) Therefore, 𝑧dec[𝐵𝐋𝑛𝑝[𝑎,𝑏][0,1]] and consequently, the equality (3.4) holds.

Remark 3.4. From (3.4), it follows that if a set Φ𝐋𝑛1[𝑎,𝑏] is relatively compact in the weak topology of 𝐋𝑛1[𝑎,𝑏], then the set decΦ does not necessarily possess this property.

Remark 3.5. Note that if a set is convex in 𝐋𝑛1[𝑎,𝑏], then this set is not necessarily decomposable. The ball 𝐵𝐋𝑛1[𝑎,𝑏][0,1] is an example of such a set.

Remark 3.6. If a set Φ𝐋𝑛1[𝑎,𝑏] is integrally bounded, then by Lemma 2.2, for the set decΦΠ[𝐋𝑛1[𝑎,𝑏]], there exists a measurable and integrally bounded mapping 𝐹decΦ[𝑎,𝑏]comp[𝑅𝑛] such that 𝐹decΦ=𝑆decΦ().(3.7)

Lemma 3.7. If a set Φ𝐋𝑛1[𝑎,𝑏] is decomposable, then decΦ=Φ.

Proof. Evidently, ΦdecΦ. We claim that decΦΦ. The proof is made by induction over 𝑚. By the definition of the switching convexity, any expression (3.1) including two elements 𝑥1,𝑥2Φ and two measurable sets 𝒰1,𝒰2[𝑎,𝑏] belongs to Φ.
Suppose now that for 𝑚=𝑘, the combination of the form (3.1) belongs to Φ. Let 𝑥1,𝑥2,,𝑥𝑚+1Φ and let 𝒰1,𝒰2,,𝒰𝑚+1[𝑎,𝑏] be disjoint measurable sets such that [𝑎,𝑏]=𝑚+1𝑖=1𝒰𝑖. Let 𝒰𝑧=𝜒2𝒰1𝑥2𝒰+𝜒3𝑥3𝒰++𝜒𝑚+1𝑥𝑚+1.(3.8) By the inductive assumption, 𝑧Φ and therefore 𝜒(𝒰1)𝑥1+𝜒([𝑎,𝑏]𝒰1)𝑧Φ. Since 𝜒[𝑎,𝑏]𝒰1𝒰𝑧=𝜒2𝑥2𝒰+𝜒3𝑥3𝒰++𝜒𝑚+1𝑥𝑚+1,(3.9) we have that 𝜒𝒰1𝑥1𝒰+𝜒2𝑥2𝒰++𝜒𝑚+1𝑥𝑚+1Φ.(3.10) Hence decΦΦ. This concludes the proof.

Corollary 3.8. If Φ𝐋𝑛1[𝑎,𝑏], then the set decΦ is the minimal set which is decomposable and which contains Φ.

Proof. Consider any set 𝑈𝐋𝑛1[𝑎,𝑏] which is decomposable and which satisfies Φ𝑈. Then, by Lemma 3.7, we have ΦdecΦdec𝑈=𝑈.

Lemma 3.9. If a set Φ𝐋𝑛1[𝑎,𝑏] is convex, then so is the set decΦ𝐋𝑛1[𝑎,𝑏].

Proof. Let 𝑦1,𝑦2decΦ be given by the formula (3.2). It follows from the convexity of the set Φ𝐋𝑛1[𝑎,𝑏] and the equality 𝜆𝑦1+(1𝜆)𝑦2=𝑚𝑖,𝑗=1𝜒𝒰1𝑖𝒰2𝑗𝜆𝑥1𝑖+(1𝜆)𝑥2𝑗(3.11) that 𝜆𝑦1+(1𝜆)𝑦2decΦ for any 𝜆[0,1]. Thus, the set decΦ is convex.

Similar to the definition of the convex hull in a normed space, the set decΦ will, in the sequel, be called the decomposable hull of the set Φin the space of integrable functions, or simply the decomposable hull of the set Φ. Likewise, decΦ is addressed as the closed decomposable hull of the set Φ.

Remark 3.10. If Φ𝑄[𝐋𝑛1[𝑎,𝑏]], then the closed decomposable hull of the set Φ (the set decΦ) can be constructed as described in Remark 3.6. To do it, one needs a measurable and integrally bounded (see Remark 3.6) mapping 𝐹decΦ[𝑎,𝑏]comp[𝑛] that satisfies (3.7). Note that finding this mapping 𝐹decΦ is easier than constructing the set decΦ. At the same time, when one studies the metrical relations between the sets Φ1,Φ2𝐋𝑛1[𝑎,𝑏] and their decomposable hulls (see Lemma 3.12), it is more convenient to use Definition 3.1.

Lemma 3.11. Let 𝑣𝐋𝑛1(𝒰)(𝒰[𝑎,𝑏]) and let a set Φ𝐋𝑛1[𝑎,𝑏] be decomposable. Then for any disjoint measurable sets 𝒰1,𝒰2𝒰 such that 𝒰1𝒰2=𝒰, one has 𝜌𝐋𝑛1(𝒰)[𝑣;Φ]=𝜌𝐋𝑛1(𝒰1)[𝑣;Φ]+𝜌𝐋𝑛1(𝒰2)[𝑣;Φ].(3.12)

Proof. Indeed, let 𝜀>0 and 𝑦Φ satisfy 𝑣𝑦𝐋𝑛1(𝒰)<𝜌𝐋𝑛1(𝒰)[𝑣;Φ]+𝜀. It follows from this estimate that 𝜌𝐋𝑛1(𝒰1)[𝑣;Φ]+𝜌𝐋𝑛1(𝒰2)[𝑣;Φ]𝑣𝑦𝐋𝑛1(𝒰1)+𝑣𝑦𝐋𝑛1(𝒰2)<𝜌𝐋𝑛1(𝒰)[𝑣;Φ]+𝜀.(3.13) This yields 𝜌𝐋𝑛1(𝒰1)[𝑣;Φ]+𝜌𝐋𝑛1(𝒰2)[𝑣;Φ]𝜌𝐋𝑛1(𝒰)[𝑣;Φ].(3.14)
Further, let us show that the opposite inequality is valid. Let 𝑦𝑖Φ|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 𝑣𝑦𝑖𝐋𝑛1(𝒰𝑖)<𝜌𝐋𝑛1(𝒰𝑖)𝜀[𝑣;Φ]+2,𝑖=1,2.(3.15) Since the set Φ is decomposable, it follows that the map 𝑦𝒰 defined by 𝑦𝑦(𝑡)=1(𝑡)if𝑡𝒰1,𝑦2(𝑡)if𝑡𝒰2(3.16) belongs to the set Φ|𝒰. By (3.15), we have 𝜌𝐋𝑛1(𝒰)[𝑣;Φ]𝑣𝑦𝐋𝑛1(𝒰)<𝜌𝐋𝑛1(𝒰1)[𝑣;Φ]+𝜌𝐋𝑛1(𝒰2)[𝑣;Φ]+𝜀.(3.17) This implies that 𝜌𝐋𝑛1(𝒰)[𝑣;Φ]𝜌𝐋𝑛1(𝒰1)[𝑣;Φ]+𝜌𝐋𝑛1(𝒰2)[𝑣;Φ].(3.18) Comparing (3.14) and (3.18), we obtain (3.12).

Lemma 3.12. If Φ1,Φ2𝑄[𝐋𝑛1[𝑎,𝑏]] and there exists a function 𝜔𝐋1+[𝑎,𝑏] such that +𝐋𝑛1(𝒰)Φ1;Φ2𝒰𝜔(𝑠)𝑑𝑠(3.19) for any measurable set 𝒰[𝑎,𝑏], then +𝐋𝑛1(𝒰)decΦ1;decΦ2𝒰𝜔(𝑠)𝑑𝑠(3.20) for any measurable set 𝒰[𝑎,𝑏].

Proof. Let 𝒰[𝑎,𝑏] be a measurable set, 𝜇(𝒰)>0. Let 𝑧decΦ1 and 𝑧𝑖Φ1, 𝑖=1,2,,𝑚. Suppose also that the functions 𝑧𝑖 and disjoint measurable sets 𝑒𝑖[𝑎,𝑏], 𝑖=1,2,,𝑚, such that [𝑎,𝑏]=𝑚𝑖=1𝑒𝑖, satisfy the equality 𝑧=𝜒𝑒1𝑧1+𝜒𝑒2𝑧2++𝜒𝑒𝑚𝑧𝑚.(3.21)
Further, by 𝑧,𝑧𝑖, 𝑖=1,2,,𝑚, we denote the restrictions of these functions to 𝒰 and put 𝑒𝑖=𝑒𝑖𝒰, 𝑖=1,2,,𝑚.
From (3.21) and Lemma 3.11, it follows that 𝜌𝐋𝑛1(𝒰)𝑧;decΦ2=𝑚𝑖=1𝜌𝐋𝑛1(𝑒𝑖)𝑧𝑖;decΦ2𝑚𝑖=1𝜌𝐋𝑛1(𝑒𝑖)𝑧𝑖;Φ2.(3.22)
From (3.19), we obtain that 𝜌𝐋𝑛1(𝑒𝑖)𝑧𝑖;Φ2𝑒𝑖𝜔(𝑠)𝑑𝑠(3.23) for each 𝑖=1,2,,𝑚.
Therefore, (3.22) and (3.23) imply 𝜌𝐋𝑛1(𝒰)𝑧;decΦ2𝒰𝜔(𝑠)𝑑𝑠.(3.24) Since (3.24) holds for any 𝑧decΦ1, it follows from (3.24) that (3.20) holds as well.

Remark 3.13. Note that the function 𝜔𝐋1+[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 decΦ𝑖, 𝑖=1,2.

We say that a multivalued mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] is integrally bounded on a set 𝐾𝐂𝑛[𝑎,𝑏] if the image Φ(𝐾) is integrally bounded.

Let Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]]. We introduce an operator Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] by the formula Φ(𝑥)=decΦ(𝑥).(3.25)

Note that even if a mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] is continuous, the mapping Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] given by (3.25) may be discontinuous. To illustrate this, let us consider an example.

Example 3.15. We define an integrable function 𝜑[0,2]×[0,1]×[0,2]1 by 𝜑(𝑥,𝑟)(𝑡)=1if𝑡[𝑥,𝑥+𝑟][0,2],𝑟0,0if𝑡[𝑥,𝑥+𝑟][0,2],𝑟0,0if𝑟=0.(3.26)
We also define a multivalued mapping Φ[0,1]𝑄[𝐿11[0,2]] by the formula Φ(𝑟)=𝑥[0,2]𝜑(𝑥,𝑟)if𝑟0,0if𝑟=0.(3.27)
Note that 𝐿11[0,2]Φ𝑟1𝑟;Φ2=|||𝑟1𝑟2|||(3.28) for any 𝑟1,𝑟2[0,1], but at the same time, 𝐿110,2]Φ(0);Φ(𝑟)=2(3.29) for any 𝑟(0,1].

Using Lemma 3.12, we obtain the following continuity conditions for the operator Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] given by (3.25).

Definition 3.16. Let 𝑈𝐶𝑛[𝑎,𝑏]. One says that a mapping 𝑃𝑈×𝑈𝐿1+[𝑎,𝑏] is symmetric on the set 𝑈 if 𝑃(𝑥,𝑦)=𝑃(𝑦,𝑥) for any 𝑥,𝑦𝑈. One says that a mapping 𝑃𝑈×𝑈𝐿1+[𝑎,𝑏] is continuous in the second variable at a point (𝑥,𝑥) belonging to the diagonal of 𝑈×𝑈 if for any sequence 𝑦𝑖𝑈 such that 𝑦𝑖𝑥 as 𝑖 it holds that 𝑃(𝑥,𝑥)=lim𝑖𝑃(𝑥,𝑦𝑖). One says that a mapping 𝑃𝑈×𝑈𝐿1+[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. Let 𝑈𝐶𝑛[𝑎,𝑏]. Suppose also that 𝑃(𝑥,𝑥)=0 for any 𝑥𝑈. One says that a mapping 𝑃𝑈×𝑈𝐿1+[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 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. Let 𝑈𝐶𝑛[𝑎,𝑏]. Suppose also that for a mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] there exists a mapping 𝑃𝑈×𝑈𝐿1+[𝑎,𝑏] such that +𝐿𝑛1(𝒰)Φ(𝑥),Φ(𝑦)𝑃(𝑥,𝑦)𝐿11(𝒰)(3.30) for any 𝑥,𝑦𝑈 and any measurable set 𝒰[𝑎,𝑏]. Then for the mapping Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] given by (3.25), the inequality (3.30), where Φ()Φ(), is satisfied as well as for any 𝑥,𝑦𝑈 and any measurable set 𝒰[𝑎,𝑏].

Corollary 3.19. If the mapping 𝑃𝑈×𝑈𝐿1+[𝑎,𝑏] in Theorem 3.18 has property 𝒜 (resp., , 𝒞) on the set 𝑈𝐶𝑛[𝑎,𝑏], then the operator Φ𝐶𝑛[𝑎,𝑏]Π[𝐿𝑛1[𝑎,𝑏]] given by (3.25) is Hausdorff lower semicontinuous (resp., Hausdorff upper semicontinuous, Hausdorff continuous) on the set 𝑈𝐶𝑛[𝑎,𝑏].

We say that the mapping 𝑃𝑈×𝑈𝐿1+[𝑎,𝑏] satisfying the inequality (3.30) for any measurable set 𝒰[𝑎,𝑏] is a majorant mapping for Φ𝐶𝑛[𝑎,𝑏]𝑄[𝐿𝑛1[𝑎,𝑏]] on the set 𝑈.

Let a mapping 𝐹𝑖[𝑎,𝑏]×𝑛comp[𝑛], 𝑖=1,2, be measurable as a composite function for every 𝑥𝐶𝑛[𝑎,𝑏]. Let also 𝐹𝑖 be integrally bounded for every bounded set 𝐾𝑛. Consider a mapping 𝐶𝑛[𝑎,𝑏]𝑄[𝐿𝑛1[𝑎,𝑏]] given by (𝑥)=𝒩1(𝑥)𝒩2(𝑥),(3.31) where the mapping 𝒩𝑖𝐶𝑛[𝑎,𝑏]Π[𝐿𝑛1[𝑎,𝑏]], 𝑖=1,2, is the Nemytskii operator generated by the mapping 𝐹𝑖[𝑎,𝑏]×𝑛comp[𝑛], 𝑖=1,2. For the operator 𝐶𝑛[𝑎,𝑏]𝑄[𝐿𝑛1[𝑎,𝑏]] given by (3.31), the majorant mapping 𝑃𝐶𝑛[𝑎,𝑏]×𝐶𝑛[𝑎,𝑏]𝐿1+[𝑎,𝑏] can be defined as 𝑃(𝑥,𝑦)(𝑡)=max+𝐹1𝑡,𝑥(𝑡);𝐹1𝑡,𝑦(𝑡);+𝐹2𝑡,𝑥(𝑡);𝐹2𝑡,𝑦(𝑡).(3.32)

It follows from Theorem 3.18 that the operator 𝑃(,) given by (3.32) is also a majorant mapping for the mapping 𝐶𝑛[𝑎,𝑏]Π[𝐿𝑛1[𝑎,𝑏]] given by (3.25), where Φ()(). If the mapping 𝐹𝑖[𝑎,𝑏]×𝑛comp[𝑛], 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 𝐶𝑛[𝑎,𝑏]Π[𝐿𝑛1[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 mapping Φ𝐶𝑛[𝑎,𝑏]𝑄[𝐿𝑛1[𝑎,𝑏]]has Property 𝒜 (resp., and 𝒞) if for this mapping there exists a majorant mapping 𝑃𝐶𝑛[𝑎,𝑏]×𝐶𝑛[𝑎,𝑏]𝐿1+[𝑎,𝑏] satisfying Property 𝒜 (resp., 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 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 ̇𝑥Φ(𝑥),𝑥(𝑎)=𝑥0𝑥0𝑛,(4.1) where the mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] satisfies the following condition: for every bounded set 𝑈𝐂𝑛[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 𝐋𝑛1[𝑎,𝑏] (see [10, 2325]). 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, 2325]), can be reduced to (4.1).

Definition 4.1. An absolutely continuous function 𝑥[𝑎,𝑏]𝑛 is called a generalized solution of the problem (4.1) if ̇𝑥decΦ(𝑥),𝑥(𝑎)=𝑥0𝑥0𝑛.(4.2)

Note that from Lemma 3.7, 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],̇𝑥=𝑘𝑥,𝑥(0)=1.(4.3) Its solution is the function 𝑥=𝑒𝑘𝑡.
We assume that the parameter 𝑘 may take two values: 1 or 2. Then the trajectories of such a system are described by the differential inclusion ̇𝑥Φ(𝑡)𝑥(𝑡),𝑥(0)=1,(4.4) where Φ(𝑡) is a multivalued function with the values from the set {1,2}. Note that decΦ(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 value 𝑘=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 functions 𝑒𝑡 and 𝑒2𝑡.

Example 4.3. Consider a simple pendulum. It consists of a mass 𝑚 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 ̈𝑥=𝑎sin𝑥,(4.5) 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 ̈𝑥=𝑎𝑥.(4.6) 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 ̈𝑥Φ(𝑥),(4.7) where Φ(𝑥)=𝑚𝑖=1(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 operator Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[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 operator Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] (the right-hand side of the inclusion (4.1)) is a Volterra operator. This implies that the operator Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] given by (3.25) is also a Volterra operator.

Let 𝜏(𝑎,𝑏]. Let us determine the continuous mapping 𝑉𝜏𝐂𝑛[𝑎,𝜏]𝐂𝑛[𝑎,𝑏] by 𝑉𝜏𝑥=𝑥(𝑡)if𝑡[𝑎,𝜏],𝑥(𝜏)if𝑡(𝜏,𝑏].(4.8)

Definition 4.5. One says that an absolutely continuous function 𝑥[𝑎,𝜏]𝑛 is a generalized solution of the problem (4.1) on the interval [𝑎,𝜏], 𝜏(𝑎,𝑏], if 𝑥 satisfies ̇𝑥(decΦ(𝑉𝜏(𝑥)))|𝜏 and 𝑥(𝑎)=𝑥0, where the continuous mapping 𝑉𝜏𝐂𝑛[𝑎,𝜏]𝐂𝑛[𝑎,𝑏] is given by (4.8).

A function 𝑥[𝑎,𝑐)𝑛, 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 solution 𝑥[𝑎,𝑐)𝑛 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 Theorems 4.64.12 below, we assume that the mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] has Property 𝒜. Due to Corollary 3.19, the mapping Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] given by (3.25) is lower semicontinuous. Due to [27, 28], the mapping Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] admits a continuous selection. Therefore, the following propositions on local solutions of the problem (4.1) are straightforward.

Theorem 4.6. There exists 𝜏(𝑎,𝑏] such that a generalized solution of the problem (4.1) is defined on the interval [𝑎,𝜏].

Theorem 4.7. A generalized solution 𝑥[𝑎,𝑐)𝑛 of the problem (4.1) admits a continuation if and only if  lim𝑡𝑐0|𝑥(𝑡)|<.

Theorem 4.8. If 𝑦 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].

Let 𝐻(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 𝑦𝐻(𝑥0,𝜏) satisfying 𝑦𝐂𝑛[𝑎,𝜏]>𝑟.

Theorems 4.64.8 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 𝑦𝐂𝑛[𝑎,𝜏]𝑟 for any 𝜏(𝑎,𝑏], 𝑦𝐻(𝑥0,𝜏).

Definition 4.10. One says that a mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] has Property Γ1 if there exists an isotonic continuous operator Γ1𝐂1+[𝑎,𝑏]𝐋1+[𝑎,𝑏] satisfying the following conditions:(i)for any function 𝑥𝐂𝑛[𝑎,𝑏] and any measurable set 𝒰[𝑎,𝑏], one hasΦ(𝑥)𝐋𝑛1(𝒰)Γ1(𝑍𝑥)𝐋11(𝒰),(4.9) where the continuous mapping 𝑍𝐂𝑛[𝑎,𝑏]𝐂1+[𝑎,𝑏] is given by ||||||(𝑍𝑥)(𝑡)=𝑥(𝑡);(4.10)(ii)the local solutions of the problem ̇𝑦=Γ1|||𝑥(𝑦),𝑦(𝑎)=0|||(4.11) admit a uniform a priori estimate.

Lemma 4.11. Suppose that a multivalued mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] has Property Γ1. Then so does the mapping Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] given by (3.25).

Proof. It suffices to show that decΦ(𝑥)𝐋𝑛1(𝒰)Γ1(𝑍𝑥)𝐋11(𝒰)(4.12) for any function 𝑥𝐂𝑛[𝑎,𝑏] and any measurable set 𝒰[𝑎,𝑏]. Indeed, let a function 𝑦decΦ(𝑥) be as in (3.1). By (4.9), 𝒰𝑖𝒰|||𝑥𝑖|||Γ(𝑠)𝑑𝑠1(𝑍𝑥)𝐋11(𝒰𝑖𝒰)(4.13) for each 𝑖=1,2,,𝑚. Hence, we have that for the function 𝑦decΦ(𝑥), the estimate 𝒰||||||Γ𝑦(𝑠)𝑑𝑠1(𝑍𝑥)𝐋11(𝒰)(4.14) is satisfied as well. This gives the inequality (4.12). The proof is complete.

Let a continuous operator Θ𝐃𝑛[𝑎,𝑏]𝐂1+[𝑎,𝑏] be given by ||||||+(Θ𝑧)(𝑡)=𝑧(𝑎)𝑡𝑎||||||̇𝑧(𝑠)𝑑𝑠.(4.15)

Theorem 4.12. Suppose that a mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[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 𝑦𝐂𝑛[𝑎,𝜏]𝑟 for any 𝑦𝐻(𝑥0,𝜏), 𝜏(𝑎,𝑏].

Proof. Indeed, let 𝑥𝐻(𝑥0,𝜏) (𝜏(𝑎,𝑏]). From Lemma 4.11, it follows that for any 𝑡[𝑎,𝜏], |||𝑥(Θ𝑥)(𝑡)0|||+𝑡𝑎Γ1|||𝑥(𝑍𝑥)(𝑠)𝑑𝑠0|||+𝑡𝑎Γ1(Θ𝑥)(𝑠)𝑑𝑠,(4.16) 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 Θ𝑥𝜉0, where 𝜉0 is the upper solution of the problem (4.11). Thus, there is no 𝑥𝐻(𝑥0,𝜏) satisfying the inequality 𝑥𝐂𝑛[𝑎,𝜏]>𝜉0𝐂1[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 operator Λ𝐋𝑛1[𝑎,𝑏]𝐂𝑛[𝑎,𝑏] be given by (Λ𝑧)(𝑡)=𝑡𝑎𝑧(𝑠)𝑑𝑠,𝑡[𝑎,𝑏].(4.17) We say that Λ𝐋𝑛1[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 Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] has Property 𝒞. Then for any function 𝑣𝐋𝑛1[𝑎,𝑏] and any 𝜀>0, there exists a generalized solution 𝑥𝐃𝑛[𝑎,𝑏] of the problem (4.1) such that ̇𝑥𝑣𝐋𝑛1(𝒰)𝜌𝐋𝑛1(𝒰)𝑣,decΦ(𝑥)+𝜀𝜇(𝒰)(4.18) for any measurable set 𝒰[𝑎,𝑏].
If Φ𝐂𝑛[𝑎,𝑏]Ω(𝑄[𝐋𝑛1[𝑎,𝑏]]), then the theorem is also valid for 𝜀=0.

Proof. Let Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] have Property 𝒞. Then by Corollary 3.19, the mapping Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] given by (3.25) is continuous. Therefore (see [3032]), given a number 𝜀>0 and a function 𝑣𝐋𝑛1[𝑎,𝑏], there exists a continuous mapping 𝜑𝐂𝑛[𝑎,𝑏]𝐋𝑛1[𝑎,𝑏] satisfying 𝜑(𝑦)Φ(𝑦) and 𝜑(𝑦)𝑣𝐋𝑛1(𝒰)𝜌𝐋𝑛1(𝒰)𝑣,decΦ(𝑦)+𝜀𝜇(𝒰)(4.19) for any 𝑦𝐂𝑛[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 𝑦𝐂𝑛[𝑎,𝜏]𝑟 for each 𝜏(𝑎,𝑏], 𝑦𝐻(𝑥0,𝜏). Now, we show that there exists 𝑥𝐻(𝑥0,𝑏) satisfying (4.18). Consider the problem ̇𝑥𝑊decΦ𝑟(𝑥),𝑥(𝑎)=𝑥0𝑥0𝑛,(4.20) where the continuous mapping 𝑊𝑟𝐂𝑛[𝑎,𝑏]𝐂𝑛[𝑎,𝑏] is given by 𝑊𝑟𝑥||||||(𝑡)=𝑥(𝑡)if𝑥(𝑡)𝑟+2,𝑟+2||||||||||||𝑥(𝑡)𝑥(𝑡)if𝑥(𝑡)>𝑟+2.(4.21) 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 𝑊𝑟𝐂𝑛[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 𝑦𝐻(𝑊) such that 𝑦𝐂𝑛[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|𝜏𝐂𝑛[𝑎,𝜏]=𝑟+1 (𝑦|𝜏 is the restriction of the function 𝑦 to [𝑎,𝜏]). By (4.21), we have 𝑦|𝜏𝐻(𝑥0,𝜏). This contradicts to the definition of the number 𝑟. Hence, 𝐻(𝑥0,𝑏)=𝐻(𝑊). Consider a continuous operator Ψ𝐂𝑛[𝑎,𝑏]𝐂𝑛[𝑎,𝑏] given by Ψ(𝑥)=𝑥0𝑊+Λ𝜑𝑟(𝑥),(4.22) where the operator Λ𝐋𝑛1[𝑎,𝑏]𝐂𝑛[𝑎,𝑏] is the operator of integration defined by (4.17), and 𝜑𝐂𝑛[𝑎,𝑏]𝐋𝑛1[𝑎,𝑏] is a continuous selection of the mapping Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] given by (3.25). The function 𝜑 ia also assumed to satisfy (4.19). Since the operator 𝑊𝑟𝐂𝑛[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=coΨ(𝐂𝑛[𝑎,𝑏]) is a convex compact set. Since the operator Ψ𝐂𝑛[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 𝑥𝐻(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. Let Φ𝐂𝑛[𝑎,𝑏]Ω(𝑄[𝐋𝑛1[𝑎,𝑏]]). Suppose also that Φ has Property 𝒞. Then by Lemma 3.9, Φ𝐂𝑛[𝑎,𝑏]Ω(Π[𝐋𝑛1[𝑎,𝑏]]). Hence for each 𝑖=1,2,, there exists a generalized solution 𝑥𝑖𝐃𝑛[𝑎,𝑏] of the problem (4.1) such that for any measurable set 𝒰[𝑎,𝑏], the inequality (4.18) is valid for ̇𝑥=̇𝑥𝑖 and 𝜀=1/𝑖. Since the set 𝐻(𝑥0,𝑏) is bounded, we see that the sequence {̇𝑥𝑖} is weakly compact in 𝐋𝑛1[𝑎,𝑏]. Without loss of generality, it can be assumed that ̇𝑥𝑖̇𝑥 weakly in 𝐋𝑛1[𝑎,𝑏] and 𝑥𝑖𝑥 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 ̇𝑥decΦ(𝑥). Assume that the functions 𝑦𝑖decΦ(𝑥), 𝑖=1,2,, satisfy 𝑦𝑖̇𝑥𝑖𝐋𝑛1[𝑎,𝑏]=𝜌𝐋𝑛1[𝑎,𝑏]̇𝑥𝑖;decΦ(𝑥)(4.23) (as decΦ(𝑥)Π[𝐋𝑛1[𝑎,𝑏]], these functions do exist). It follows from (4.23) that 𝑦𝑖̇𝑥𝑖𝐋𝑛1[𝑎,𝑏]𝐋𝑛1[𝑎,𝑏]𝑥decΦ𝑖;decΦ(𝑥).(4.24) Since the mapping Φ𝐂𝑛[𝑎,𝑏]Ω(Π[𝐋𝑛1[𝑎,𝑏]]) given by (3.25) is continuous, we obtain, by (4.24), that 𝑦𝑖̇𝑥𝑖0 in 𝐋𝑛1[𝑎,𝑏] as 𝑖. Since ̇𝑥𝑖̇𝑥 weakly in 𝐋𝑛1[𝑎,𝑏] as 𝑖, we have that 𝑦𝑖̇𝑥 weakly in 𝐋𝑛1[𝑎,𝑏] as 𝑖. Therefore, the convexity of the set decΦ(𝑥) implies that ̇𝑥decΦ(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 ̇𝑥𝑖̇𝑥 weakly in 𝐋𝑛1[𝑎,𝑏] as 𝑖, we have, by [21], that for each 𝑚=1,2,, there exist numbers 𝑖(𝑚), 𝜆𝑚𝑗0, 𝑗=1,2,,𝑖(𝑚), satisfying the following conditions: 𝑖(𝑚)𝑗=1𝜆𝑚𝑗=1; the sequence {𝛽𝑚=𝑖(𝑚)𝑗=1𝜆𝑚𝑗̇𝑥𝑗+𝑚} tends to ̇𝑥 in 𝐋𝑛1[𝑎,𝑏]. Since ̇𝑥𝑣𝐋𝑛1[𝑎,𝑏]̇𝑥𝛽𝑚𝐋𝑛1[𝑎,𝑏]+𝑖(𝑚)𝑗=1𝜆𝑚𝑗̇𝑥𝑗+𝑚𝑣𝐋𝑛1[𝑎,𝑏](4.25) for each 𝑚=1,2,, it follows, due to the choice of the sequence {̇𝑥𝑖}, that ̇𝑥𝑣𝐋𝑛1[𝑎,𝑏]̇𝑥𝛽𝑚𝐋𝑛1[𝑎,𝑏]+𝑖(𝑚)𝑗=1𝜆𝑚𝑗𝜌𝐋𝑛1[𝑎,𝑏]𝑣;𝑥decΦ𝑗+𝑚+(𝑏𝑎)𝑖(𝑚)𝑗=1𝜆𝑚𝑗1𝑗+𝑚(4.26) for each 𝑚=1,2,.

Since lim𝑖𝜌𝐋𝑛1[𝑎,𝑏]𝑣;𝑥decΦ𝑖=𝜌𝐋𝑛1[𝑎,𝑏]𝑣;decΦ(𝑥),(4.27) it follows that letting 𝑚 in the previous inequality, we obtain ̇𝑥𝑣𝐋𝑛1[𝑎,𝑏]=𝜌𝐋𝑛1[𝑎,𝑏]𝑣;decΦ(𝑥).(4.28) Finally, note that by the decomposability of the set decΦ(𝑥), this equality holds for any measurable set 𝒰[𝑎,𝑏]. This completes the proof.

Theorems 4.12 and 4.13 yield the following result.

Corollary 4.14. Suppose that a mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] has Properties Γ1 and 𝒞. Then for any function 𝑣𝐋𝑛1[𝑎,𝑏] and any 𝜀>0, there exists a generalized solution 𝑥𝐃𝑛[𝑎,𝑏] of the problem (4.1) such that (4.18) holds for any measurable set 𝒰[𝑎,𝑏].
If Φ𝐂𝑛[𝑎,𝑏]Ω(𝑄[𝐋𝑛1[𝑎,𝑏]]), then the corollary is also valid for 𝜀=0.

Remark 4.15. Consider the convex compact set 𝑈=coΨ(𝐂𝑛[𝑎,𝑏])𝐂𝑛[𝑎,𝑏], where the mapping Ψ𝐂𝑛[𝑎,𝑏]2𝐂𝑛[𝑎,𝑏] is given by Ψ(𝑥)=𝑥0Φ𝑊+Λ𝑟(𝑥).(4.29) Here, the operators Φ𝐂𝑛[𝑎,𝑏]Π[𝐋𝑛1[𝑎,𝑏]] and 𝑊𝑟𝐂𝑛[𝑎,𝑏]𝐂𝑛[𝑎,𝑏] are determined by (3.25) and (4.21), respectively. If a number 𝑟>0 is such that 𝑦𝐂𝑛[𝑎,𝜏]𝑟 for any 𝜏(𝑎,𝑏], 𝑦𝐻(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. Given 𝜀0, 𝑝0, 𝑢𝐋1+[𝑎,𝑏], one says that a mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] has Property Γ2𝑢,𝜀,𝑝 if there exists an isotonic and continuous Volterra operator Γ2𝐂1+[𝑎,𝑏]𝐋1+[𝑎,𝑏] satisfying the following conditions:(i)Γ2(0)=0;(ii)for any functions 𝑥,𝑦𝐂𝑛[𝑎,𝑏] and any measurable set 𝒰[𝑎,𝑏], one has 𝐋𝑛1(𝒰)ΓΦ(𝑥);Φ(𝑦)2𝑍(𝑥𝑦)𝐋11(𝒰),(4.30) where the continuous mapping 𝑍𝐂𝑛[𝑎,𝑏]𝐂1+[𝑎,𝑏] is determined by (4.10);(iii)the set of all local solutions of the problem ̇𝑦=𝑢+𝜀+Γ2(𝑦),𝑦(𝑎)=𝑝,(4.31) admits a uniform a priori estimate.

Given 𝑦𝐃𝑛[𝑎,𝑏] and 𝜘𝐋1+[𝑎,𝑏], the following estimate will be used in the sequel: 𝜌𝐋𝑛1(𝒰)̇𝑦;Φ(𝑦)𝒰𝜘(𝑠)𝑑𝑠(4.32) for each measurable set 𝒰[𝑎,𝑏].

Theorem 4.17. Let functions 𝑦𝐃𝑛[𝑎,𝑏] and 𝜘𝐋1+[𝑎,𝑏] satisfy the inequality (4.32) for each measurable set 𝒰[𝑎,𝑏]. Suppose that a mapping Φ𝐂𝑛[𝑎,𝑏]𝑄[𝐋𝑛1[𝑎,𝑏]] has Property Γ2𝜘,𝜀,𝑝, where 𝜀0, 𝑝=|𝑥0𝑦(𝑎)|, and 𝑥0 is the initial condition of the problem (4.1). Then for any generalized solution of the problem (4.1) satisfying ̇𝑥̇𝑦𝐋𝑛1(𝒰)𝜌𝐋𝑛1(𝒰)̇𝑦;decΦ(𝑥)+𝜀𝜇(𝒰)(4.33) for any measurable set 𝒰[𝑎,𝑏], the following conditions are satisfied: (1)Θ(𝑥𝑦)(𝑡)𝜉(𝜘,𝜀,𝑝)(𝑡)(4.34) 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=|𝑥0𝑦(𝑎)|, and the mapping Θ𝐃𝑛[𝑎,𝑏]𝐂1+[𝑎,𝑏] is given by (4.15);(2)||||||Γ̇𝑥(𝑡)̇𝑦(𝑡)𝜘(𝑡)+𝜀+2𝜉(𝜘,𝜀,𝑝)(𝑡)(4.35) for almost all