Generalized Solutions of Functional Differential Inclusions

Abstract

We consider the initial value problem for a functional differential inclusion with a Volterra multivalued mapping that is not necessarily decomposable in 𝐿^𝑛1[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 [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 𝐿^𝑛1[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 𝐿^𝑛1[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 𝐋^𝑛1[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 𝐵_𝑋[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 𝐵_𝑋[𝑥,0]≡𝑥. Let 𝑈⊂𝑋. Then 𝑈 is the closure of 𝑈, co𝑈 is the convex hull of 𝑈; co𝑈≡co𝑈, ext𝑈 is the set of all extreme points of 𝑈; ext𝑈=ext𝑈. Let ‖𝑈‖_𝑋=sup_𝑢∈𝑈‖𝑢‖_𝑋. Let 𝑈^𝜀≡⋃_𝑢∈𝑈𝐵[𝑢,𝜀] if 𝜀>0 and 𝑈⁰≡𝑈.

Let 𝜌_𝑋[𝑥;𝑈] be the distance from the point 𝑥∈𝑋 to the set 𝑈 in the space 𝑋; let ℎ^+𝑋[𝑈₁;𝑈]≡sup_𝑥∈𝑈1𝜌_𝑋[𝑥,𝑈] be the Hausdorff semideviation of the set 𝑈₁ from the set 𝑈; let ℎ_𝑋[𝑈₁;𝑈]=max{ℎ^+𝑋[𝑈₁;𝑈];ℎ^+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 2^𝑋 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 ℝ^𝑛 be the space of all 𝑛-dimensional column vectors with the norm |⋅|. We denote by 𝐶^𝑛[𝑎,𝑏] (resp., 𝐷^𝑛[𝑎,𝑏]) the space of continuous (resp., absolutely continuous) functions 𝑥∶[𝑎,𝑏]→ℝ^𝑛 with norm ‖𝑥‖_𝐶^𝑛_[𝑎,𝑏]=max{|𝑥(𝑡)|∶𝑡∈[𝑎,𝑏]} (resp., ‖𝑥‖_𝐷^𝑛_[𝑎,𝑏]=|𝑥(𝑎)|+∫^𝑏𝑎|𝑥̇(𝑠)|𝑑𝑠). Let 𝒰⊂[𝑎,𝑏] be a measurable set 𝜇(𝒰)>0 (𝜇—the Lebesgue measure). We denote by 𝐿^𝑛𝑝(𝒰) the space of all functions 𝑥∶𝒰→ℝ^𝑛 such that (𝑥(𝑠))^𝑝 is integrable (if 𝑝<∞) and the space of all measurable, essentially bounded (if 𝑝=∞) functions 𝑥∶𝒰→ℝ^𝑛 with the norms ‖‖𝑥‖‖_{𝐿𝑛𝑝(𝒰)}=_𝒰|||𝑥(𝑠)|||^𝑝𝑑𝑠^1/𝑝,‖‖𝑥‖‖_{𝐿𝑛∞(𝒰)}=vraisup_𝑠∈𝒰|||𝑥(𝑠)|||,(2.1) respectively.

Let Φ⊂𝐿^𝑛1[𝑎,𝑏]. The set Φ is called integrally bounded if there exists a function 𝜑_Φ∈𝐋¹¹[𝑎,𝑏] such that |𝑥(𝑡)|≤𝜑_Φ(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[𝐋^𝑛1[𝑎,𝑏]] (resp., Π[𝐿^𝑛1[𝑎,𝑏]]) the set of all nonempty, closed, and integrally bounded (resp., nonempty, bounded, closed, and decomposable) subsets of the space 𝐿^𝑛1[𝑎,𝑏].

Let 𝐹∶[𝑎,𝑏]→comp[ℝ^𝑛] be a measurable mapping. Then by definition, 𝑆(𝐹)={𝑦∈𝐿^𝑛1[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 𝑓∶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 ℝ^𝑛. A set {𝑥_𝜈(⋅)} (𝜈∈ℕ) of measurable mappings from 𝑇 to ℝ^𝑛 is said to approximate the multivalued mapping 𝐹 if the set {𝑡∈𝑇∣𝑥_𝜈(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{𝑥_𝜈(𝑡)} for almost all 𝑡∈𝑇. A multivalued mapping 𝐹 from 𝑇 to ℝ^𝑛 is called measurable if there exists a countable set of measurable mappings from 𝑇 to ℝ^𝑛 that approximates the mapping 𝐹.

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

Lemma 2.1. Let Φ∈Π[𝐿^𝑛1[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. Let 𝜑_𝑖∈Φ, 𝑖=1,2,…, be a sequence of functions such that lim_𝑖→∞‖‖𝜑_𝑖‖‖_{𝐿𝑛1[𝑎,𝑏]}=‖‖Φ‖‖_{𝐿𝑛1[𝑎,𝑏]}.(2.2)
Let us show that there exists a sequence of functions 𝜑_𝑖∈Φ, 𝑖=1,2,…, such that the equality (2.2) holds and |||𝜑₁(𝑡)|||≤|||𝜑₂(𝑡)|||≤|||𝜑₃(𝑡)|||≤⋯≤|||𝜑_𝑖(𝑡)|||≤|||𝜑_𝑖+1(𝑡)|||≤⋯(2.3) for almost all 𝑡∈[𝑎,𝑏].
Indeed, let 𝜑₁=𝜑₁ and 𝜑_𝑖+1=𝜒(𝒰_𝑖)𝜑_𝑖+𝜒([𝑎,𝑏⧵𝒰_𝑖)𝜑_𝑖+1, 𝑖=1,2,…, where 𝒰_𝑖={𝑡∈[𝑎,𝑏]∶|𝜑_𝑖(𝑡)|≥|𝜑_𝑖+1(𝑡)|}. Since Φ∈Π[𝐿^𝑛1[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‖_𝐿^𝑛1_[𝑎,𝑏]≥‖𝜑_𝑖‖_𝐿^𝑛1_[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 𝑢(𝑡)=lim_𝑖→∞|||𝜑_𝑖(𝑡)|||.(2.4) 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), _𝒰𝑢(𝑡)𝑑𝑡=‖‖Φ‖‖_{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 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. Let Φ∈Π[𝐿^𝑛1[𝑎,𝑏]] and 𝜑_𝑖∈Φ, 𝑖=1,2,…, be a sequence that is dense in Φ. Further, let a measurable set 𝐹∶[𝑎,𝑏]→comp[ℝ^𝑛] be defined by 𝐹(𝑡)=𝜑^𝑖(𝑡),𝑖=1,2,….(2.6) Then 𝑆(𝐹)=Φ.

Proof. Since 𝜑_𝑖∈𝑆(𝐹) and the sequence 𝜑_𝑖, 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 𝐸^𝑘𝑖=𝑡∈[𝑎,𝑏]∶|||𝑥(𝑡)−𝜑_𝑖(𝑡)|||≤1𝑘,(2.7) which are measurable sets. For 𝑖=1, let 𝐸^𝑘1=𝐸^𝑘1, and for 𝑖=2,3,…, let 𝐸^𝑘𝑖=𝐸^𝑘𝑖⧵⋃^{𝑖−1𝑗=1}𝐸^𝑘𝑗. Then 𝐸^𝑘𝑖⋂𝐸^𝑘𝑗=∅ if 𝑖≠𝑗. By the definition of the mapping 𝐹∶[𝑎,𝑏]→comp[ℝ^𝑛], for each 𝑘=1,2,…, we have 𝜇(^∞_𝑖=1𝐸^𝑘𝑖)=𝑏−𝑎.(2.8) Let 𝑥_𝑘∶[𝑎,𝑏]→ℝ^𝑛, 𝑘=1,2,…, be a sequence of measurable functions such that 𝑥_𝑘(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝜑^𝑖𝜑(𝑡)if𝑡∈𝐸^𝑘𝑖,𝑖=1,2,…,𝑘,₁(𝑡)if𝑡∈[𝑎,𝑏]⧵^𝑘_𝑖=1𝐸^𝑘𝑖.(2.9) Since the set Φ is decomposable, we see that 𝑥_𝑘∈Φ for each 𝑘=1,2,…. Moreover, from Lemma 2.1 and the definition of the set 𝐸^𝑘𝑖, it follows that for the functions 𝑥_𝑘, 𝑘=1,2,…, we have the estimates ‖‖𝑥−𝑥_𝑘‖‖_{𝐿𝑛1[𝑎,𝑏]}≤𝑏−𝑎𝑘+2_{[𝑎,𝑏]⧵∪}^𝑘𝑖=1_𝐸^𝑘𝑖𝑢(𝑡)𝑑𝑡,(2.10) where 𝑢 satisfies the assertions of Lemma 2.1. From (2.8) and (2.10), it follows that 𝑥_𝑘→𝑥 in 𝐿^𝑛[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 sets 𝐹_𝑖∶[𝑎,𝑏]→comp[ℝ^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 𝜑_𝑖∈𝐿^𝑛1[𝑎,𝑏], 𝑖=1,2,…, be a countable set, which is dense in 𝑆(𝐹₁) and which approximates 𝐹₁∶[𝑎,𝑏]→comp[ℝ^𝑛] (see [21]). Thus 𝜑_𝑖∈𝑆(𝐹₂(⋅)) for each 𝑖=1,2,… and by the definition of the set 𝑆(𝐹₂(⋅)), we have that {𝜑_𝑖(𝑡)∶𝑖=1,2,…}⊂𝐹₂(𝑡) for almost all 𝑡∈[𝑎,𝑏]. Since the sequence 𝜑_𝑖, 𝑖=1,2,…, approximates the map 𝐹₁∶[𝑎,𝑏]→comp[ℝ^𝑛], it follows from the previous inclusion that 𝐹₁(𝑡)⊂𝐹₂(𝑡) for almost all 𝑡∈[𝑎,𝑏].

Corollary 2.4. Let Φ∈Π[𝐿^𝑛1[𝑎,𝑏]] and let 𝐹_𝑖∶[𝑎,𝑏]→comp[ℝ^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 Φ∈Π[𝐿^𝑛1[𝑎,𝑏]], then a measurable set 𝐹∶[𝑎,𝑏]→comp[ℝ^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 𝐋^𝑛1[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 𝐋^𝑛1[𝑎,𝑏]. By decΦ, we denote the set of all finite combinations 𝑦=𝜒𝒰₁𝑥¹+𝜒𝒰₂𝑥²+⋯+𝜒𝒰_𝑚𝑥^𝑚(3.1) of elements 𝑥_𝑖∈Φ, 𝑖=1,2,…,𝑚, where the disjoint measurable subsets 𝒰_𝑖, 𝑖=1,2,…,𝑚, of the segment [𝑎,𝑏] are such that ⋃^𝑚𝑖=1𝒰_𝑖=[𝑎,𝑏].

Lemma 3.2. The set decΦ is decomposable for any nonempty set Φ⊂𝐋^𝑛1[𝑎,𝑏].

Proof. Let 𝑦₁,𝑦₂∈decΦ. Let also 𝒰⊂[𝑎,𝑏] be a measurable set. Without loss of generality, it can be assumed that 𝑦_𝑖=𝜒𝒰^𝑖1𝑥^𝑖1+𝜒𝒰^𝑖2𝑥^𝑖2+⋯+𝜒𝒰^𝑖𝑚𝑥^𝑖𝑚,(3.2) where 𝑥^𝑖𝑗∈Φ, 𝑗=1,2,…,𝑚, 𝑖=1,2, and the measurable disjoint sets 𝒰^𝑖𝑗⊂[𝑎,𝑏], 𝑗=1,2,…,𝑚, 𝑖=1,2, are such that [𝑎,𝑏]=⋃^𝑚𝑗=1𝒰^𝑖𝑗, 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)𝑦₁+𝜒[𝑎,𝑏]⧵𝒰𝑦₂=^𝑚_𝑖=1𝜒𝒰∩𝒰^1𝑖𝑥^1𝑖+^𝑚_𝑖=1𝜒[𝑎,𝑏]⧵𝒰∩𝒰^2𝑖𝑥^2𝑖,(3.3) 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 set Φ⊂𝐋^𝑛1[𝑎,𝑏] is bounded, the set decΦ is not necessarily bounded. For example, let us check that dec𝐵_𝐋^𝑛𝑝_[𝑎,𝑏][0,1]=𝐋^𝑛𝑝[𝑎,𝑏]𝑝∈[1,∞).(3.4) Indeed, let 𝑧∈𝐋^𝑛𝑝[𝑎,𝑏] and 𝑒_𝑖, 𝑖=1,2,…,𝑚, be measurable sets with the following properties: 𝑒_𝑖∩𝑒_𝑗=∅ if 𝑖≠𝑗, 𝑖,𝑗=1,2,…,𝑚, ⋃^𝑚𝑖=1𝑒_𝑖=[𝑎,𝑏]; for each 𝑖=1,2,…,𝑚, the inequality _𝑒𝑖|||𝑧(𝑠)|||^𝑝𝑑𝑠<1(3.5) holds. Then 𝑧_𝑖=𝜒(𝑒_𝑖)𝑧∈𝐵_𝐋^𝑛𝑝_[𝑎,𝑏][0,1], 𝑖=1,2,…,𝑚, and 𝑧=𝜒𝑒₁𝑧¹+𝜒𝑒₂𝑧²+⋯+𝜒𝑒_𝑚𝑧^𝑚.(3.6) Therefore, 𝑧∈dec[𝐵_𝐋^𝑛𝑝_[𝑎,𝑏][0,1]] and consequently, the equality (3.4) holds.

Remark 3.4. From (3.4), it follows that if a set Φ⊂𝐋^𝑛1[𝑎,𝑏] is relatively compact in the weak topology of 𝐋^𝑛1[𝑎,𝑏], then the set decΦ does not necessarily possess this property.

Remark 3.5. Note that if a set is convex in 𝐋^𝑛1[𝑎,𝑏], then this set is not necessarily decomposable. The ball 𝐵_𝐋^𝑛1_[𝑎,𝑏][0,1] is an example of such a set.

Remark 3.6. If a set Φ⊂𝐋^𝑛1[𝑎,𝑏] is integrally bounded, then by Lemma 2.2, for the set decΦ∈Π[𝐋^𝑛1[𝑎,𝑏]], there exists a measurable and integrally bounded mapping 𝐹_decΦ∶[𝑎,𝑏]→comp[𝑅^𝑛] such that decΦ=𝑆𝐹_decΦ(⋅).(3.7)

Lemma 3.7. If a set Φ⊂𝐋^𝑛1[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 for 𝑚=𝑘, the combination of the form (3.1) belongs to Φ. Let 𝑥₁,𝑥₂,…,𝑥_𝑚+1∈Φ and let 𝒰₁,𝒰₂,…,𝒰_𝑚+1⊂[𝑎,𝑏] be disjoint measurable sets such that [𝑎,𝑏]=⋃^𝑚+1𝑖=1𝒰_𝑖. Let 𝑧=𝜒𝒰₂∪𝒰₁𝑥²+𝜒𝒰₃𝑥³+⋯+𝜒𝒰_𝑚+1𝑥^𝑚+1.(3.8) By the inductive assumption, 𝑧∈Φ and therefore 𝜒(𝒰₁)𝑥₁+𝜒([𝑎,𝑏]⧵𝒰₁)𝑧∈Φ. Since 𝜒[𝑎,𝑏]⧵𝒰₁𝑧=𝜒𝒰²𝑥²+𝜒𝒰₃𝑥³+⋯+𝜒𝒰_𝑚+1𝑥^𝑚+1,(3.9) we have that 𝜒𝒰₁𝑥¹+𝜒𝒰₂𝑥²+⋯+𝜒𝒰_𝑚+1𝑥^𝑚+1∈Φ.(3.10) Hence decΦ⊂Φ. This concludes the proof.

Corollary 3.8. If Φ⊂𝐋^𝑛1[𝑎,𝑏], then the set decΦ is the minimal set which is decomposable and which contains Φ.

Proof. Consider any set 𝑈⊂𝐋^𝑛1[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 set Φ⊂𝐋^𝑛1[𝑎,𝑏] is convex, then so is the set decΦ∈𝐋^𝑛1[𝑎,𝑏].

Proof. Let 𝑦₁,𝑦₂∈decΦ be given by the formula (3.2). It follows from the convexity of the set Φ⊂𝐋^𝑛1[𝑎,𝑏] and the equality 𝜆𝑦₁+(1−𝜆)𝑦₂=^𝑚_𝑖_,𝑗=1𝜒𝒰^1𝑖∩𝒰^2𝑗𝜆𝑥^1𝑖+(1−𝜆)𝑥^2𝑗(3.11) 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 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 Φ∈𝑄[𝐋^𝑛1[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 𝐹_decΦ∶[𝑎,𝑏]→comp[ℝ^𝑛] that satisfies (3.7). Note that finding this mapping 𝐹_decΦ is easier than constructing the set decΦ. At the same time, when one studies the metrical relations between the sets Φ₁,Φ₂⊂𝐋^𝑛1[𝑎,𝑏] and their decomposable hulls (see Lemma 3.12), it is more convenient to use Definition 3.1.

Lemma 3.11. Let 𝑣∈𝐋^𝑛1(𝒰)(𝒰⊂[𝑎,𝑏]) and let a set Φ⊂𝐋^𝑛1[𝑎,𝑏] be decomposable. Then for any disjoint measurable sets 𝒰₁,𝒰₂⊂𝒰 such that 𝒰₁∪𝒰₂=𝒰, one has 𝜌_𝐋^𝑛1_(𝒰)[𝑣;Φ]=𝜌_𝐋^𝑛1_(𝒰1)[𝑣;Φ]+𝜌_𝐋^𝑛1_(𝒰2)[𝑣;Φ].(3.12)

Proof. Indeed, let 𝜀>0 and 𝑦∈Φ satisfy ‖𝑣−𝑦‖_𝐋^𝑛1_(𝒰)<𝜌_𝐋^𝑛1_(𝒰)[𝑣;Φ]+𝜀. It follows from this estimate that 𝜌_𝐋^𝑛1_(𝒰1)[𝑣;Φ]+𝜌_𝐋^𝑛1_(𝒰2)[𝑣;Φ]≤‖‖𝑣−𝑦‖‖_{𝐋𝑛1(𝒰1)}+‖‖𝑣−𝑦‖‖_{𝐋𝑛1(𝒰2)}<𝜌_𝐋^𝑛1_(𝒰)[𝑣;Φ]+𝜀.(3.13) This yields 𝜌_𝐋^𝑛1_(𝒰1)[𝑣;Φ]+𝜌_𝐋^𝑛1_(𝒰2)[𝑣;Φ]≤𝜌_𝐋^𝑛1_(𝒰)[𝑣;Φ].(3.14)
Further, let us show that the opposite inequality is valid. Let 𝑦_𝑖∈Φ|_𝒰𝑖, 𝑖=1,2, where Φ|_𝒰𝑖 is the set of of all mappings from Φ, restricted to 𝒰_𝑖, 𝑖=1,2, and suppose that the functions 𝑦_𝑖, 𝑖=1,2, satisfy ‖‖𝑣−𝑦_𝑖‖‖_{𝐋𝑛1(𝒰𝑖)}<𝜌_𝐋^𝑛1_(𝒰𝑖)[𝑣;Φ]+𝜀2,𝑖=1,2.(3.15) Since the set Φ is decomposable, it follows that the map 𝑦∶𝒰→ℝ defined by 𝑦(𝑡)=𝑦¹𝑦₂(𝑡)if𝑡∈𝒰₁,(𝑡)if𝑡∈𝒰₂(3.16) belongs to the set Φ|_𝒰. By (3.15), we have 𝜌_𝐋^𝑛1_(𝒰)[𝑣;Φ]≤‖‖𝑣−𝑦‖‖_{𝐋𝑛1(𝒰)}<𝜌_𝐋^𝑛1_(𝒰1)[𝑣;Φ]+𝜌_𝐋^𝑛1_(𝒰2)[𝑣;Φ]+𝜀.(3.17) This implies that 𝜌_𝐋^𝑛1_(𝒰)[𝑣;Φ]≤𝜌_𝐋^𝑛1_(𝒰1)[𝑣;Φ]+𝜌_𝐋^𝑛1_(𝒰2)[𝑣;Φ].(3.18) Comparing (3.14) and (3.18), we obtain (3.12).

Lemma 3.12. If Φ₁,Φ₂∈𝑄[𝐋^𝑛1[𝑎,𝑏]] and there exists a function 𝜔∈𝐋¹⁺[𝑎,𝑏] such that ℎ^+𝐋_𝑛1(𝒰)Φ¹;Φ₂≤_𝒰𝜔(𝑠)𝑑𝑠(3.19) for any measurable set 𝒰⊂[𝑎,𝑏], then ℎ^+𝐋_𝑛1(𝒰)decΦ¹;decΦ₂≤_𝒰𝜔(𝑠)𝑑𝑠(3.20) for any measurable set 𝒰⊂[𝑎,𝑏].

Proof. Let 𝒰⊂[𝑎,𝑏] be a measurable set, 𝜇(𝒰)>0. Let 𝑧∈decΦ₁ and 𝑧_𝑖∈Φ₁, 𝑖=1,2,…,𝑚. Suppose also that the functions 𝑧_𝑖 and disjoint measurable sets 𝑒_𝑖⊂[𝑎,𝑏], 𝑖=1,2,…,𝑚, such that [𝑎,𝑏]=⋃^𝑚𝑖=1𝑒_𝑖, satisfy the equality 𝑧=𝜒𝑒₁𝑧¹+𝜒𝑒₂𝑧²+⋯+𝜒𝑒_𝑚𝑧^𝑚.(3.21)
Further, by 𝑧,𝑧_𝑖, 𝑖=1,2,…,𝑚, we denote the restrictions of these functions to 𝒰 and put 𝑒_𝑖=𝑒_𝑖∩𝒰, 𝑖=1,2,…,𝑚.
From (3.21) and Lemma 3.11, it follows that 𝜌_𝐋^𝑛1_(𝒰)𝑧;decΦ²=^𝑚_𝑖=1𝜌_𝐋^𝑛1_(𝑒𝑖)𝑧^𝑖;decΦ₂≤^𝑚_𝑖=1𝜌_𝐋^𝑛1_(𝑒𝑖)𝑧^𝑖;Φ₂.(3.22)
From (3.19), we obtain that 𝜌_𝐋^𝑛1_(𝑒𝑖)𝑧^𝑖;Φ₂≤_𝑒𝑖𝜔(𝑠)𝑑𝑠(3.23) for each 𝑖=1,2,…,𝑚.
Therefore, (3.22) and (3.23) imply 𝜌_𝐋^𝑛1_(𝒰)𝑧;decΦ²≤_𝒰𝜔(𝑠)𝑑𝑠.(3.24) 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 function 𝜔∈𝐋¹⁺[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 decΦ_𝑖 is replaced with its closure decΦ_𝑖, 𝑖=1,2.

We say that a multivalued mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] is integrally bounded on a set 𝐾⊂𝐂^𝑛[𝑎,𝑏] if the image Φ(𝐾) is integrally bounded.

Let Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]]. We introduce an operator Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] by the formula Φ(𝑥)=decΦ(𝑥).(3.25)

Note that even if a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] is continuous, the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[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]→ℝ¹ by 𝜑(𝑥,𝑟)(𝑡)=⎧⎪⎪⎨⎪⎪⎩100iiifff𝑡𝑡𝑟∈∈=[[𝑥0𝑥.,,𝑥𝑥++𝑟𝑟]]∩∩[[00,,22]],,𝑟𝑟≠≠00,,(3.26)
We also define a multivalued mapping Φ∶[0,1]→𝑄[𝐿¹¹[0,2]] by the formula Φ(𝑟)=⎧⎪⎨⎪⎩_𝑥∈[0,2]0𝜑(𝑥,𝑟)iiff𝑟𝑟≠=00,.(3.27)
Note that ℎ_𝐿¹¹_[0,2]Φ𝑟¹;Φ𝑟²=|||𝑟¹−𝑟₂|||(3.28) for any 𝑟₁,𝑟₂∈[0,1], but at the same time, ℎ_𝐿11^0,2]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 Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25).

Definition 3.16. Let 𝑈⊂𝐶^𝑛[𝑎,𝑏]. One says that a mapping 𝑃∶𝑈×𝑈→𝐿¹⁺[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→𝐿¹⁺[𝑎,𝑏] is continuous in the second variable at a point (𝑥,𝑥) belonging to the diagonal of 𝑈×𝑈 if for any sequence 𝑦_𝑖∈𝑈 such that 𝑦_𝑖→𝑥 as 𝑖→∞ it holds that 𝑃(𝑥,𝑥)=lim_𝑖→∞𝑃(𝑥,𝑦_𝑖). One says that a mapping 𝑃∶𝑈×𝑈→𝐿¹⁺[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 𝑈⊂𝐶^𝑛[𝑎,𝑏]. Suppose also that 𝑃(𝑥,𝑥)=0 for any 𝑥∈𝑈. One says that a mapping 𝑃∶𝑈×𝑈→𝐿¹⁺[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. Let 𝑈⊂𝐶^𝑛[𝑎,𝑏]. Suppose also that for a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] there exists a mapping 𝑃∶𝑈×𝑈→𝐿¹⁺[𝑎,𝑏] such that ℎ^+𝐿_𝑛1(𝒰)Φ(𝑥),Φ(𝑦)≤‖‖𝑃(𝑥,𝑦)‖‖_𝐿¹¹_(𝒰)(3.30) for any 𝑥,𝑦∈𝑈 and any measurable set 𝒰⊂[𝑎,𝑏]. Then for the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[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 mapping 𝑃∶𝑈×𝑈→𝐿¹⁺[𝑎,𝑏] in Theorem 3.18 has property 𝒜 (resp., ℬ, 𝒞) on the set 𝑈⊂𝐶^𝑛[𝑎,𝑏], then the operator Φ∶𝐶^𝑛[𝑎,𝑏]→Π[𝐿^𝑛1[𝑎,𝑏]] given by (3.25) is Hausdorff lower semicontinuous (resp., Hausdorff upper semicontinuous, Hausdorff continuous) on the set 𝑈⊂𝐶^𝑛[𝑎,𝑏].

We say that the mapping 𝑃∶𝑈×𝑈→𝐿¹⁺[𝑎,𝑏] satisfying the inequality (3.30) for any measurable set 𝒰⊂[𝑎,𝑏] is a majorant mapping for Φ∶𝐶^𝑛[𝑎,𝑏]→𝑄[𝐿^𝑛1[𝑎,𝑏]] on the set 𝑈.

Let a mapping 𝐹_𝑖∶[𝑎,𝑏]×ℝ^𝑛→comp[ℝ^𝑛], 𝑖=1,2, be measurable as a composite function for every 𝑥∈𝐶^𝑛[𝑎,𝑏]. Let also 𝐹_𝑖 be integrally bounded for every bounded set 𝐾⊂ℝ^𝑛. Consider a mapping ℳ∶𝐶^𝑛[𝑎,𝑏]→𝑄[𝐿^𝑛1[𝑎,𝑏]] given by ℳ(𝑥)=𝒩₁(𝑥)∪𝒩₂(𝑥),(3.31) where the mapping 𝒩_𝑖∶𝐶^𝑛[𝑎,𝑏]→Π[𝐿^𝑛1[𝑎,𝑏]], 𝑖=1,2, is the Nemytskii operator generated by the mapping 𝐹_𝑖∶[𝑎,𝑏]×ℝ^𝑛→comp[ℝ^𝑛], 𝑖=1,2. For the operator ℳ∶𝐶^𝑛[𝑎,𝑏]→𝑄[𝐿^𝑛1[𝑎,𝑏]] given by (3.31), the majorant mapping 𝑃∶𝐶^𝑛[𝑎,𝑏]×𝐶^𝑛[𝑎,𝑏]→𝐿¹⁺[𝑎,𝑏] can be defined as 𝑃(𝑥,𝑦)(𝑡)=maxℎ⁺𝐹¹𝑡,𝑥(𝑡);𝐹¹𝑡,𝑦(𝑡);ℎ⁺𝐹²𝑡,𝑥(𝑡);𝐹²𝑡,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 ℳ∶𝐶^𝑛[𝑎,𝑏]→Π[𝐿^𝑛1[𝑎,𝑏]] given by (3.25), where Φ(⋅)≡ℳ(⋅). If the mapping 𝐹_𝑖∶[𝑎,𝑏]×ℝ^𝑛→comp[ℝ^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 ℳ∶𝐶^𝑛[𝑎,𝑏]→Π[𝐿^𝑛1[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 Φ∶𝐶^𝑛[𝑎,𝑏]→𝑄[𝐿^𝑛1[𝑎,𝑏]]has Property 𝒜 (resp., ℬ and 𝒞) if for this mapping there exists a majorant mapping 𝑃∶𝐶^𝑛[𝑎,𝑏]×𝐶^𝑛[𝑎,𝑏]→𝐿¹⁺[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 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 𝑥̇∈Φ(𝑥),𝑥(𝑎)=𝑥₀𝑥⁰∈ℝ^𝑛,(4.1) where the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] satisfies the following condition: for every bounded set 𝑈⊂𝐂^𝑛[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 𝑥̇ in (4.1) is not treated as a derivative at a point but as an element of 𝐋^𝑛1[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 function 𝑥∶[𝑎,𝑏]→ℝ^𝑛 is called a generalized solution of the problem (4.1) if 𝑥̇∈decΦ(𝑥),𝑥(𝑎)=𝑥₀𝑥⁰∈ℝ^𝑛.(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],𝑥̇=𝑘𝑥,𝑥(0)=1.(4.3) Its solution is the function 𝑥=𝑒^𝑘𝑡.
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 𝑥̇∈Φ(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 𝑒^𝑡 and 𝑒^2𝑡.

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 𝑥̈=−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 𝑥̈=−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,…,𝑙_𝑚}. In this case, the equation of simple harmonic motion transforms to the differential inclusion with a multivalued mapping 𝑥̈∈Φ(𝑥),(4.7) where Φ(𝑥)=⋃^𝑚𝑖=1−(𝑔/𝑙_𝑖)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 Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[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 Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] (the right-hand side of the inclusion (4.1)) is a Volterra operator. This implies that the operator Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] given by (3.25) is also a Volterra operator.

Let 𝜏∈(𝑎,𝑏]. Let us determine the continuous mapping 𝑉_𝜏∶𝐂^𝑛[𝑎,𝜏]→𝐂^𝑛[𝑎,𝑏] by 𝑉^𝜏𝑥=𝑥𝑥((𝑡𝜏))iiff𝑡𝑡∈∈[(𝑎𝜏,,𝜏𝑏]].,(4.8)

Definition 4.5. One says that an absolutely continuous function 𝑥∶[𝑎,𝜏]→ℝ^𝑛 is a generalized solution of the problem (4.1) on the interval [𝑎,𝜏], 𝜏∈(𝑎,𝑏], if 𝑥 satisfies 𝑥̇∈(decΦ(𝑉_𝜏(𝑥)))|_𝜏 and 𝑥(𝑎)=𝑥₀, where the continuous mapping 𝑉_𝜏∶𝐂^𝑛[𝑎,𝜏]→𝐂^𝑛[𝑎,𝑏] is given by (4.8).

A function 𝑥∶[𝑎,𝑐)→ℝ^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)→ℝ^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 4.6–4.12 below, we assume that the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property 𝒜. Due to Corollary 3.19, the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25) is lower semicontinuous. Due to [27, 28], the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[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)→ℝ^𝑛 of the problem (4.1) admits a continuation if and only if  lim_{𝑡→𝑐−0}|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 𝑦∈𝐻(𝑥₀,𝜏) satisfying ‖𝑦‖_𝐂^𝑛_[𝑎,𝜏]>𝑟.

Theorems 4.6–4.8 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‖_𝐂^𝑛_[𝑎,𝜏]≤𝑟 for any 𝜏∈(𝑎,𝑏], 𝑦∈𝐻(𝑥₀,𝜏).

Definition 4.10. One says that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property Γ₁ if there exists an isotonic continuous operator Γ₁∶𝐂¹⁺[𝑎,𝑏]→𝐋¹⁺[𝑎,𝑏] satisfying the following conditions:(i)for any function 𝑥∈𝐂^𝑛[𝑎,𝑏] and any measurable set 𝒰⊂[𝑎,𝑏], one has‖‖Φ(𝑥)‖‖_{𝐋𝑛1(𝒰)}≤‖‖Γ₁(𝑍𝑥)‖‖_𝐋11(𝒰),(4.9) where the continuous mapping 𝑍∶𝐂^𝑛[𝑎,𝑏]→𝐂¹⁺[𝑎,𝑏] is given by (𝑍𝑥)(𝑡)=|||𝑥(𝑡)|||;(4.10)(ii)the local solutions of the problem 𝑦̇=Γ₁(𝑦),𝑦(𝑎)=|||𝑥₀|||(4.11) admit a uniform a priori estimate.

Lemma 4.11. Suppose that a multivalued mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property Γ₁. Then so does the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[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 𝑥∈𝐂^𝑛[𝑎,𝑏] and any measurable set 𝒰⊂[𝑎,𝑏]. Indeed, let a function 𝑦∈decΦ(𝑥) be as in (3.1). By (4.9), _{𝒰𝑖∩𝒰}|||𝑥_𝑖(𝑠)|||𝑑𝑠≤‖‖Γ₁(𝑍𝑥)‖‖_{𝐋11(𝒰𝑖∩𝒰)}(4.13) for each 𝑖=1,2,…,𝑚. Hence, we have that for the function 𝑦∈decΦ(𝑥), the estimate _𝒰|||𝑦(𝑠)|||𝑑𝑠≤‖‖Γ₁(𝑍𝑥)‖‖_𝐋11(𝒰)(4.14) is satisfied as well. This gives the inequality (4.12). The proof is complete.

Let a continuous operator Θ∶𝐃^𝑛[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 Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[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‖_𝐂^𝑛_[𝑎,𝜏]≤𝑟 for any 𝑦∈𝐻(𝑥₀,𝜏), 𝜏∈(𝑎,𝑏].

Proof. Indeed, let 𝑥∈𝐻(𝑥₀,𝜏) (𝜏∈(𝑎,𝑏]). From Lemma 4.11, it follows that for any 𝑡∈[𝑎,𝜏], (Θ𝑥)(𝑡)≤|||𝑥₀|||+^𝑡𝑎Γ¹(𝑍𝑥)(𝑠)𝑑𝑠≤|||𝑥₀|||+^𝑡𝑎Γ¹(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 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‖_𝐂^𝑛_[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 operator Λ∶𝐋^𝑛1[𝑎,𝑏]→𝐂^𝑛[𝑎,𝑏] be given by (Λ𝑧)(𝑡)=^𝑡𝑎𝑧(𝑠)𝑑𝑠,𝑡∈[𝑎,𝑏].(4.17) We say that Λ∶𝐋^𝑛1[𝑎,𝑏]→𝐂^𝑛[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∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property 𝒞. Then for any function 𝑣∈𝐋^𝑛1[𝑎,𝑏] and any 𝜀>0, there exists a generalized solution 𝑥∈𝐃^𝑛[𝑎,𝑏] of the problem (4.1) such that ‖‖𝑥̇−𝑣‖‖_{𝐋𝑛1(𝒰)}≤𝜌_𝐋^𝑛1_(𝒰)𝑣,decΦ(𝑥)+𝜀𝜇(𝒰)(4.18) for any measurable set 𝒰⊂[𝑎,𝑏].
If Φ∶𝐂^𝑛[𝑎,𝑏]→Ω(𝑄[𝐋^𝑛1[𝑎,𝑏]]), then the theorem is also valid for 𝜀=0.

Proof. Let Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] have Property 𝒞. Then by Corollary 3.19, the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25) is continuous. Therefore (see [30–32]), given a number 𝜀>0 and a function 𝑣∈𝐋^𝑛1[𝑎,𝑏], there exists a continuous mapping 𝜑∶𝐂^𝑛[𝑎,𝑏]→𝐋^𝑛1[𝑎,𝑏] satisfying 𝜑(𝑦)∈Φ(𝑦) and ‖‖𝜑(𝑦)−𝑣‖‖_{𝐋𝑛1(𝒰)}≤𝜌_𝐋^𝑛1_(𝒰)𝑣,decΦ(𝑦)+𝜀𝜇(𝒰)(4.19) for any 𝑦∈𝐂^𝑛[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‖_𝐂^𝑛_[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 𝑥̇∈decΦ𝑊_𝑟(𝑥),𝑥(𝑎)=𝑥₀𝑥⁰∈ℝ^𝑛,(4.20) where the continuous mapping 𝑊_𝑟∶𝐂^𝑛[𝑎,𝑏]→𝐂^𝑛[𝑎,𝑏] is given by 𝑊^𝑟𝑥(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝑥𝑟(+𝑡)2if|||𝑥(𝑡)|||≤𝑟+2,|||𝑥(𝑡)|||𝑥(𝑡)if|||𝑥(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 𝑊_𝑟∶𝐂^𝑛[𝑎,𝑏]→𝐂^𝑛[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‖_𝐂^𝑛_[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‖_𝐂^𝑛_[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∶𝐂^𝑛[𝑎,𝑏]→𝐂^𝑛[𝑎,𝑏] given by Ψ(𝑥)=𝑥₀+Λ𝜑𝑊_𝑟(𝑥),(4.22) where the operator Λ∶𝐋^𝑛1[𝑎,𝑏]→𝐂^𝑛[𝑎,𝑏] is the operator of integration defined by (4.17), and 𝜑∶𝐂^𝑛[𝑎,𝑏]→𝐋^𝑛1[𝑎,𝑏] is a continuous selection of the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25). The function 𝜑 ia also assumed to satisfy (4.19). Since the operator 𝑊_𝑟∶𝐂^𝑛[𝑎,𝑏]→𝐂^𝑛[𝑎,𝑏] is bounded, we obtain that the image Ψ(𝐂^𝑛[𝑎,𝑏]) is a relatively compact subset of 𝐂^𝑛[𝑎,𝑏]. Hence, the set 𝑈=coΨ(𝐂^𝑛[𝑎,𝑏]) is a convex compact set. Since the operator Ψ∶𝐂^𝑛[𝑎,𝑏]→𝐂^𝑛[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 𝑥̇=𝜑(𝑥), we see that (4.19) implies (4.18).

Let us prove the second statement of the theorem. Let Φ∶𝐂^𝑛[𝑎,𝑏]→Ω(𝑄[𝐋^𝑛1[𝑎,𝑏]]). Suppose also that Φ has Property 𝒞. Then by Lemma 3.9, Φ∶𝐂^𝑛[𝑎,𝑏]→Ω(Π[𝐋^𝑛1[𝑎,𝑏]]). Hence for each 𝑖=1,2,…, there exists a generalized solution 𝑥_𝑖∈𝐃^𝑛[𝑎,𝑏] of the problem (4.1) such that for any measurable set 𝒰⊂[𝑎,𝑏], the inequality (4.18) is valid for 𝑥̇=𝑥̇_𝑖 and 𝜀=1/𝑖. Since the set 𝐻(𝑥₀,𝑏) is bounded, we see that the sequence {𝑥̇_𝑖} is weakly compact in 𝐋^𝑛1[𝑎,𝑏]. Without loss of generality, it can be assumed that 𝑥̇_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] and 𝑥_𝑖→𝑥 in 𝐂^𝑛[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 𝑥̇∈decΦ(𝑥). Assume that the functions 𝑦_𝑖∈decΦ(𝑥), 𝑖=1,2,…, satisfy ‖‖𝑦_𝑖−𝑥̇_𝑖‖‖_{𝐋𝑛1[𝑎,𝑏]}=𝜌_𝐋^𝑛1_[𝑎,𝑏]𝑥̇^𝑖;decΦ(𝑥)(4.23) (as decΦ(𝑥)∈Π[𝐋^𝑛1[𝑎,𝑏]], these functions do exist). It follows from (4.23) that ‖‖𝑦_𝑖−𝑥̇_𝑖‖‖_{𝐋𝑛1[𝑎,𝑏]}≤ℎ_𝐋^𝑛1_[𝑎,𝑏]decΦ𝑥_𝑖;decΦ(𝑥).(4.24) Since the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Ω(Π[𝐋^𝑛1[𝑎,𝑏]]) given by (3.25) is continuous, we obtain, by (4.24), that 𝑦_𝑖−𝑥̇_𝑖→0 in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞. Since 𝑥̇_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞, we have that 𝑦_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞. Therefore, the convexity of the set decΦ(𝑥) implies that 𝑥̇∈decΦ(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 𝑥̇_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞, we have, by [21], that for each 𝑚=1,2,…, there exist numbers 𝑖(𝑚), 𝜆^𝑚𝑗≥0, 𝑗=1,2,…,𝑖(𝑚), satisfying the following conditions: ∑^{𝑖(𝑚)𝑗=1}𝜆^𝑚𝑗=1; the sequence {𝛽_𝑚=∑^{𝑖(𝑚)𝑗=1}𝜆^𝑚𝑗𝑥̇_𝑗+𝑚} tends to 𝑥̇ in 𝐋^𝑛1[𝑎,𝑏]. Since ‖‖𝑥̇−𝑣‖‖_{𝐋𝑛1[𝑎,𝑏]}≤‖‖𝑥̇−𝛽_𝑚‖‖_{𝐋𝑛1[𝑎,𝑏]}+^𝑖(𝑚)_𝑗=1𝜆^𝑚𝑗‖‖𝑥̇_𝑗+𝑚−𝑣‖‖_{𝐋𝑛1[𝑎,𝑏]}(4.25) for each 𝑚=1,2,…, it follows, due to the choice of the sequence {𝑥̇_𝑖}, that ‖‖𝑥̇−𝑣‖‖_{𝐋𝑛1[𝑎,𝑏]}≤‖‖𝑥̇−𝛽_𝑚‖‖_{𝐋𝑛1[𝑎,𝑏]}+^𝑖(𝑚)_𝑗=1𝜆^𝑚𝑗𝜌_𝐋^𝑛1_[𝑎,𝑏]𝑣;decΦ𝑥_𝑗+𝑚+(𝑏−𝑎)^𝑖(𝑚)_𝑗=1𝜆^𝑚𝑗1𝑗+𝑚(4.26) for each 𝑚=1,2,….

Since lim_𝑖→∞𝜌_𝐋^𝑛1_[𝑎,𝑏]𝑣;decΦ𝑥_𝑖=𝜌^{𝐋𝑛1[𝑎,𝑏]}𝑣;decΦ(𝑥),(4.27) it follows that letting 𝑚→∞ in the previous inequality, we obtain ‖‖𝑥̇−𝑣‖‖_{𝐋𝑛1[𝑎,𝑏]}=𝜌_𝐋^𝑛1_[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 Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Properties Γ₁ and 𝒞. Then for any function 𝑣∈𝐋^𝑛1[𝑎,𝑏] and any 𝜀>0, there exists a generalized solution 𝑥∈𝐃^𝑛[𝑎,𝑏] of the problem (4.1) such that (4.18) holds for any measurable set 𝒰⊂[𝑎,𝑏].
If Φ∶𝐂^𝑛[𝑎,𝑏]→Ω(𝑄[𝐋^𝑛1[𝑎,𝑏]]), then the corollary is also valid for 𝜀=0.

Remark 4.15. Consider the convex compact set 𝑈=coΨ(𝐂^𝑛[𝑎,𝑏])⊂𝐂^𝑛[𝑎,𝑏], where the mapping Ψ∶𝐂^𝑛[𝑎,𝑏]→2^{𝐂𝑛[𝑎,𝑏]} is given by Ψ(𝑥)=𝑥₀+ΛΦ𝑊_𝑟(𝑥).(4.29) Here, the operators Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] and 𝑊_𝑟∶𝐂^𝑛[𝑎,𝑏]→𝐂^𝑛[𝑎,𝑏] are determined by (3.25) and (4.21), respectively. If a number 𝑟>0 is such that ‖𝑦‖_𝐂^𝑛_[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. Given 𝜀≥0, 𝑝≥0, 𝑢∈𝐋¹⁺[𝑎,𝑏], one says that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property Γ^{2𝑢,𝜀,𝑝} if there exists an isotonic and continuous Volterra operator Γ₂∶𝐂¹⁺[𝑎,𝑏]→𝐋¹⁺[𝑎,𝑏] satisfying the following conditions:(i)Γ₂(0)=0;(ii)for any functions 𝑥,𝑦∈𝐂^𝑛[𝑎,𝑏] and any measurable set 𝒰⊂[𝑎,𝑏], one has ℎ_𝐋^𝑛1_(𝒰)Φ(𝑥);Φ(𝑦)≤‖‖Γ²𝑍(𝑥−𝑦)‖‖_𝐋¹¹_(𝒰),(4.30) where the continuous mapping 𝑍∶𝐂^𝑛[𝑎,𝑏]→𝐂¹⁺[𝑎,𝑏] is determined by (4.10);(iii)the set of all local solutions of the problem 𝑦̇=𝑢+𝜀+Γ₂(𝑦),𝑦(𝑎)=𝑝,(4.31) admits a uniform a priori estimate.

Given 𝑦∈𝐃^𝑛[𝑎,𝑏] and 𝜘∈𝐋¹⁺[𝑎,𝑏], the following estimate will be used in the sequel: 𝜌_𝐋^𝑛1_(𝒰)𝑦;̇Φ(𝑦)≤_𝒰𝜘(𝑠)𝑑𝑠(4.32) for each measurable set 𝒰⊂[𝑎,𝑏].

Theorem 4.17. Let functions 𝑦∈𝐃^𝑛[𝑎,𝑏] and 𝜘∈𝐋¹⁺[𝑎,𝑏] satisfy the inequality (4.32) for each measurable set 𝒰⊂[𝑎,𝑏]. Suppose that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property Γ^{2𝜘,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 ‖‖𝑥̇−𝑦‖‖̇_{𝐋𝑛1(𝒰)}≤𝜌_𝐋^𝑛1_(𝒰)𝑦;̇decΦ(𝑥)+𝜀𝜇(𝒰)(4.33) for any measurable set 𝒰⊂[𝑎,𝑏], the following conditions are satisfied: (1)Θ(𝑥−𝑦)(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=|𝑥₀−𝑦(𝑎)|, and the mapping Θ∶𝐃^𝑛[𝑎,𝑏]→𝐂¹⁺[𝑎,𝑏] is given by (4.15);(2)|||𝑥̇(𝑡)−𝑦(̇𝑡)|||≤𝜘(𝑡)+𝜀+Γ₂𝜉(𝜘,𝜀,𝑝)(𝑡)(4.35) for almost all 𝑡∈[𝑎,𝑏].

Proof. First, note that since the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property Γ^{2𝜘,𝜀,𝑝}, it follows from Theorem 3.18 that so does the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] determined by (3.25). Further, the inequality (4.33) yields that ‖‖𝑥̇−𝑦‖‖̇_{𝐋𝑛1(𝒰)}≤𝜌_𝐋^𝑛1_(𝒰)𝑦;̇decΦ(𝑦)+ℎ_𝐋^𝑛1_(𝒰)decΦ(𝑦);decΦ(𝑥)+𝜀𝜇(𝒰)(4.36) for any measurable set 𝒰⊂[𝑎,𝑏].
Remark 4.15 and relations (4.36), (4.32), and (4.30) imply that for any measurable set 𝒰⊂[𝑎,𝑏], we obtain the inequality ‖‖𝑥̇−𝑦‖‖̇_{𝐋𝑛1(𝒰)}≤_𝒰𝜘(𝑠)+𝜀+Γ²𝑍(𝑥−𝑦)(𝑠)𝑑𝑠,(4.37) where the mapping 𝑍∶𝐂^𝑛[𝑎,𝑏]→𝐂¹⁺[𝑎,𝑏] is given by (4.10). It follows from this inequality that |||𝑥̇(𝑡)−𝑦(̇𝑡)|||≤𝜘(𝑡)+𝜀+Γ₂𝑍(𝑥−𝑦)(𝑡)(4.38) 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∶𝐂¹⁺[𝑎,𝑏]→𝐋¹⁺[𝑎,𝑏]] (see (4.38)) is isotonic, we have that |||𝑥̇(𝑡)−𝑦(̇𝑡)|||=Θ̇(𝑥−𝑦)(𝑡)≤𝜘(𝑡)+𝜀+Γ₂Θ(𝑥−𝑦)(𝑡)(4.39) 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.

Theorems 4.13 and 4.17 yield the following result.

Theorem 4.18. Let functions 𝑦∈𝐃^𝑛[𝑎,𝑏] and 𝜘∈𝐋¹⁺[𝑎,𝑏] satisfy (4.32) for each measurable set 𝒰⊂[𝑎,𝑏]. Suppose that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property Γ^{2𝜘,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∈𝐃^𝑛[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.
If Φ∶𝐂^𝑛[𝑎,𝑏]→Ω(𝑄[𝐋^𝑛1[𝑎,𝑏]]), then the theorem is also valid for 𝜀=0.

Corollary 4.19. Let functions 𝑦∈𝐃^𝑛[𝑎,𝑏] and 𝜘∈𝐋¹⁺[𝑎,𝑏] satisfy (4.32) for each measurable set 𝒰⊂[𝑎,𝑏]. Suppose that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has properties Γ₁ and Γ^{2𝜘,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∈𝐃^𝑛[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.
If Φ∶𝐂^𝑛[𝑎,𝑏]→Ω(𝑄[𝐋^𝑛1[𝑎,𝑏]]), then the corollary is also valid for 𝜀=0.

Remark 4.20. It follows from the proof of Theorem 4.17 that Theorems 4.17, 4.18, and Corollary 4.19 are also valid if the functions 𝑦∈𝐃^𝑛[𝑎,𝑏] and 𝜘∈𝐋¹⁺[𝑎,𝑏] satisfy 𝜌_𝐋^𝑛1_(𝒰)𝑦;̇decΦ(𝑦)≤_𝒰𝜘(𝑠)𝑑𝑠(4.40) for each measurable set 𝒰⊂[𝑎,𝑏].

Definition 4.21. An absolutely continuous function 𝑥∈𝐃^𝑛[𝑎,𝑏] is called a generalized quasisolution of the problem (4.1) if there exists a sequence of functions 𝑥_𝑖∈𝐃^𝑛[𝑎,𝑏], 𝑖=1,2,…, such that the following conditions hold:(i)𝑥_𝑖→𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞;(ii)𝑥̇_𝑖∈decΦ(𝑥) and 𝑥_𝑖(𝑎)=𝑥₀ for each 𝑖=1,2,….

Note that by Lemma 3.7, 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 𝑥̇_𝑖∈decΦ(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.

Let ℋ(𝑥₀) be the set of all generalized quasisolutions of the problem (4.1).

We define a mapping Φ_co∶𝐂^𝑛[𝑎,𝑏]→Ω(Π[𝐋^𝑛1[𝑎,𝑏]]) by the formula Φ_co(𝑥)=codecΦ(𝑥).(4.41)

We call Φ_co∶𝐂^𝑛[𝑎,𝑏]→Ω(Π[𝐋^𝑛1[𝑎,𝑏]]) the convex decomposable hull.

Consider the problem (4.1) with the convex decomposable hull Φ_co∶𝐂^𝑛[𝑎,𝑏]→Ω(Π[𝐋^𝑛1[𝑎,𝑏]]) given by (4.41) leading to 𝑥̇∈Φ_co(𝑥),𝑥(𝑎)=𝑥₀𝑥⁰∈ℝ^𝑛.(4.42)

Let 𝐻_co(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)=𝐻_co(𝑥₀,𝑏).

Proof. First, we will show that 𝐻_co(𝑥₀,𝑏)⊂ℋ(𝑥₀). Let 𝑥∈𝐻_co(𝑥₀,𝑏). By [35], for 𝑥̇∈𝐋^𝑛1[𝑎,𝑏], there exists a sequence 𝑦_𝑖∈decΦ(𝑥), 𝑖=1,2,…, such that 𝑦_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞. This implies that 𝑥_𝑖=𝑥₀+Λ𝑦_𝑖→𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞, where Λ∶𝐋^𝑛1[𝑎,𝑏]→𝐂^𝑛[𝑎,𝑏] is the operator of integration (see (4.17)). Hence, 𝐻_co(𝑥₀,𝑏)⊂ℋ(𝑥₀).
Let us now prove that ℋ(𝑥₀)⊂𝐻_co(𝑥₀,𝑏). Let 𝑥∈ℋ(𝑥₀). Then there exists a sequence 𝑥_𝑖∈𝐃^𝑛[𝑎,𝑏], 𝑖=1,2,…, satisfying the following conditions: (1) 𝑥̇_𝑖∈decΦ(𝑥) (see (4.41)) and 𝑥_𝑖(𝑎)=𝑥₀ for each 𝑖=1,2,…; (2) 𝑥_𝑖→𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞. Since the sequence 𝑥̇_𝑖, 𝑖=1,2,…, is weakly compact, we can assume without loss of generality that 𝑥̇_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞. Since 𝑥̇_𝑖∈Φ_co(𝑥) (see (4.41)), it follows that 𝑥̇∈Φ_co(𝑥) (see [21]). Hence 𝑥∈𝐻_co(𝑥₀,𝑏) and therefore ℋ(𝑥₀)⊂𝐻_co(𝑥₀,𝑏).

Remark 4.23. Theorem 4.22 may still remain valid even if the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] is discontinuous and its image Φ(ℬ) is not integrally bounded for every bounded set ℬ⊂𝐂^𝑛[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 set 𝑈⊂𝐂^𝑛[𝑎,𝑏] has Property 𝒟 if ℋ(𝑥₀)⊂𝑈, and for any 𝑥∈ℋ(𝑥₀), there exists a sequence of absolutely continuous functions 𝑥_𝑖∶[𝑎,𝑏]→ℝ^𝑛, 𝑖=1,2,…, such that(i)𝑥_𝑖→𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞;(ii)𝑥_𝑖∈𝑈, 𝑥̇_𝑖∈decΦ(𝑥) and 𝑥_𝑖(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⊂𝐂^𝑛[𝑎,𝑏] satisfying Property 𝒟.

Proof. It follows from Theorem 4.22 and Remark 4.15 that the set 𝑈=coΨ(𝐂^𝑛[𝑎,𝑏]) has Property 𝒟. Here, the mapping Ψ∶𝐂^𝑛[𝑎,𝑏]→2^{𝐂𝑛[𝑎,𝑏]} is determined by (4.29), where Φ(⋅)≡Φ_co(⋅).

Lemma 4.26. Let sets Φ_𝑖∈Π[𝐋^𝑛1[𝑎,𝑏]], 𝑖=1,2, satisfy Φ_𝑖=𝑆(𝐹_𝑖(⋅)), 𝑖=1,2, where 𝐹_𝑖∶[𝑎,𝑏]→comp[ℝ^𝑛] are measurable mappings. Then for any measurable set 𝒰⊂[𝑎,𝑏], one has ℎ_𝐋^𝑛1_(𝒰)Φ¹;Φ₂≤_𝒰ℎ𝐹₁(𝑡);𝐹₂(𝑡)𝑑𝑡≤2ℎ_𝐋^𝑛1_(𝒰)Φ¹;Φ₂.(4.43)

Proof. Let 𝒰⊂[𝑎,𝑏] be a measurable set. Put 𝒰=𝑡∈𝒰∶ℎ⁺𝐹¹(𝑡);𝐹₂(𝑡)≥ℎ⁺𝐹²(𝑡);𝐹₁(𝑡).(4.44) The set 𝒰⊂𝒰 is measurable. Since _𝒰ℎ𝐹₁(𝑡);𝐹₂(𝑡)𝑑𝑡=_𝒰ℎ⁺𝐹¹(𝑡);𝐹₂(𝑡)𝑑𝑡+_{𝒰𝒰⧵}ℎ⁺𝐹²(𝑡);𝐹₁(𝑡)𝑑𝑡,(4.45) we have _𝒰ℎ𝐹₁(𝑡);𝐹₂(𝑡)𝑑𝑡=ℎ^+𝐋_{𝑛1(𝒰)}Φ¹;Φ₂+ℎ^+𝐋𝑛1_{(𝒰⧵𝒰)}Φ²;Φ₁.(4.46) This implies (4.43), and the proof is completed.

Let 𝐹∶[𝑎,𝑏]→comp[ℝ^𝑛] be a measurable mapping. Let a mapping co𝐹∶[𝑎,𝑏]→comp[ℝ^𝑛] be defined by (co𝐹)(𝑡)=co𝐹(𝑡).(4.47)

Corollary 4.27. Let sets Φ_𝑖∈Π[𝐋^𝑛1[𝑎,𝑏]], 𝑖=1,2, satisfy Φ_𝑖=𝑆(𝐹_𝑖(⋅)), 𝑖=1,2, where 𝐹_𝑖∶[𝑎,𝑏]→comp[ℝ^𝑛] are measurable mappings. Then for any measurable set 𝒰⊂[𝑎,𝑏], one has ℎ_𝐋^𝑛1_(𝒰)coΦ₁;coΦ₂≤2ℎ^{𝐋𝑛1(𝒰)}Φ¹;Φ₂.(4.48)

Proof. By [35], we have that co(Φ_𝑖)=𝑆(co𝐹_𝑖(⋅)), 𝑖=1,2. Therefore, ℎ_𝐋^𝑛1_(𝒰)coΦ₁;coΦ₂≤_𝒰ℎco𝐹₁(𝑡);co𝐹²(𝑡)𝑑𝑡(4.49) for any measurable set 𝒰⊂[𝑎,𝑏]. Since, for any measurable set 𝒰⊂[𝑎,𝑏], _𝒰ℎco𝐹₁(𝑡);co𝐹²(𝑡)𝑑𝑡≤_𝒰ℎ𝐹₁(𝑡);𝐹²(𝑡)𝑑𝑡,(4.50) we obtain, due to (4.43), the inequality (4.48).

Definition 4.28. One says that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property Γ₃ if Property Γ^20,0,0 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∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] satisfies Property Γ₃. Then, 𝐻(𝑥₀,𝑏)≠∅ and 𝐻𝑥₀,𝑏=𝐻_co𝑥⁰,𝑏,(4.51) where 𝐻(𝑥₀,𝑏) is the closure of the set 𝐻(𝑥₀,𝑏) in 𝐂^𝑛[𝑎,𝑏].

Proof. Let us first prove that the set 𝐻_co(𝑥₀) is closed in 𝐂^𝑛[𝑎,𝑏]. Indeed, suppose that a sequence 𝑥_𝑖∈𝐻_co(𝑥₀), 𝑖=1,2,…, tends to 𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞. Since the sequence {𝑥̇_𝑖} is integrally bounded, it follows that 𝑥_𝑖∈𝐃^𝑛[𝑎,𝑏] and 𝑥̇_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞. For each 𝑖=1,2,…, let the function 𝑧_𝑖∈Φ_co(𝑥) satisfy ‖‖𝑥̇_𝑖−𝑧_𝑖‖‖_{𝐋𝑛1[𝑎,𝑏]}=𝜌_𝐋^𝑛1_[𝑎,𝑏]𝑥̇^𝑖;Φ_co(𝑥),(4.52) where Φ_co∶𝐂^𝑛[𝑎,𝑏]→Ω(Π[𝐋^𝑛1[𝑎,𝑏]]) is the convex decomposable hull given by (4.41). Since the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25) is Hausdorff continuous, it follows from (4.48) that so is the mapping Φ_co(𝑥)∶𝐂^𝑛[𝑎,𝑏]→Ω(Π[𝐋^𝑛1[𝑎,𝑏]]). Therefore, (4.52) implies that 𝑥̇_𝑖−𝑧_𝑖→0 in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞. Hence, 𝑧_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞. Since the set Φ_co(𝑥) is convex, we have (see [21]) that 𝑥̇∈Φ_co(𝑥). Therefore, the set 𝐻_co(𝑥₀) is closed in 𝐂^𝑛[𝑎,𝑏].
Now, let us prove the equality (4.51). The closedness of the set 𝐻_co(𝑥₀) yields that 𝐻(𝑥₀,𝑏)⊂𝐻_co(𝑥₀). Further, let us show that 𝐻_co(𝑥₀)⊂𝐻(𝑥₀,𝑏). Suppose 𝑥∈𝐻_co(𝑥₀). Then from Theorem 4.22, it follows that there exists a sequence 𝑦_𝑖∈𝐃^𝑛[𝑎,𝑏], 𝑖=1,2,…, such that 𝑦_𝑖∈Φ(𝑥), 𝑦_𝑖(𝑎)=𝑥₀, 𝑖=1,2,… (𝑥₀ is the initial condition in the problem (4.1)) and 𝑦_𝑖→𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞. Since the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property Γ₃, we see that, due to (4.30), 𝜌_𝐋^𝑛1_(𝒰)𝑦̇^𝑖;Φ𝑦_𝑖≤ℎ^{𝐋𝑛1(𝒰)}Φ(𝑥);Φ𝑦^𝑖≤_𝒰Γ²𝑍𝑥−𝑦^𝑖(𝑠)𝑑𝑠(4.53) for each 𝑖=1,2,… and any measurable set 𝒰⊂[𝑎,𝑏]. Here, the operator 𝑍∶𝐂^𝑛[𝑎,𝑏]→𝐂¹⁺[𝑎,𝑏] is given by (4.10). Since the mapping Γ₂∶𝐂¹⁺[𝑎,𝑏]→𝐋¹⁺[𝑎,𝑏] is continuous and Γ₂(0)=0, we have that 𝜘_𝑖=Γ₂(𝑍(𝑥−𝑦_𝑖))→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=𝜘_𝑖, 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(𝜘_𝑖,1/𝑖,0) of the problem (4.31) with 𝑢=𝜘_𝑖, 𝜀=1/𝑖, and 𝑝=0. Hence, it follows from Theorem 4.18 that for each 𝑖=1,2,…, there exists a generalized solution 𝑥_𝑖∈𝐃^𝑛[𝑎,𝑏] of the problem (4.1) satisfying Θ(𝑥_𝑖−𝑦_𝑖)≤𝜉(𝜘_𝑖,1/𝑖,0), where the continuous operator Θ∶𝐃^𝑛[𝑎,𝑏]→𝐂¹⁺[𝑎,𝑏] is given by (4.15). Since 𝜉(𝜘_𝑖,1/𝑖,0)→0 in 𝐂¹[𝑎,𝑏] as 𝑖→∞, we have that Θ(𝑥_𝑖−𝑦_𝑖)→0 as 𝑖→∞. Since 𝑦_𝑖→𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞, we see that 𝑥_𝑖→𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞. Therefore, 𝑥∈𝐻(𝑥₀,𝑏) and consequently 𝐻_co(𝑥₀)⊂𝐻(𝑥₀,𝑏). This yields (4.51). The proof is complete.

Corollary 4.30. Suppose that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[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 decΦ(𝑥)? It follows from Remark 3.10 that constructing decΦ(𝑥) for each fixed 𝑥∈𝐂^𝑛[𝑎,𝑏] is equivalent to finding a measurable, integrally bounded mapping Δ_𝑥∶[𝑎,𝑏]→comp[ℝ^𝑛] satisfying decΦ(𝑥)=𝑆Δ_𝑥(⋅).(5.1) The mapping Δ_𝑥∶[𝑎,𝑏]→comp[ℝ^𝑛] is, in the sequel, written as Δ∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛] and called a mapping generating the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25).

Denote by 𝐾([𝑎,𝑏]×[0,∞)) the set of all continuous functions 𝜂∶[𝑎,𝑏]×[0,∞)→[0,∞) satisfying the following conditions:(1)for each 𝛿≥0, 𝜂(⋅,𝛿)∈𝐋¹¹[𝑎,𝑏];(2)for each 𝛿≥0, there exists a function 𝛽_𝛿(⋅)∈𝐋¹¹[𝑎,𝑏] such that 𝜂(𝑡,𝜏)≤𝛽_𝛿(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 mappings Δ(⋅,⋅) 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 decΦ(𝑥) can be studied via the robustness properties of Δ. Assume that the perturbation Δ_𝜂(𝑡,𝑥,𝛿) (e.g., an error in measurements of Δ(𝑡,𝑥)) is given by Δ_𝜂(𝑡,𝑥,𝛿)=Δ(𝑡,𝑥)^{𝜂(𝑡,𝛿)},(5.2) 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 ℎΔ(𝑡,𝑥);Δ_𝜂(𝑡,𝑥,𝛿)=𝜂(𝑡,𝛿)(5.3) for all (𝑡,𝑥)∈[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]. Thus, (5.3) implies that lim_𝛿→+0ℎΔ(𝑡,𝑥);Δ_𝜂(𝑡,𝑥,𝛿)=0(5.4) for each function 𝜂(⋅,⋅)∈𝐾([𝑎,𝑏]×[0,∞)), almost all 𝑡∈[𝑎,𝑏], and all 𝑥∈𝐂^𝑛[𝑎,𝑏]. Therefore, all mappings Δ_𝜂∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]×[0,∞)→comp[ℝ^𝑛] 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]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛]. The mapping Δ_𝜂∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]×[0,∞)→comp[ℝ^𝑛] is called the approximating operator.

We define a mapping Φ_𝜂∶𝐂^𝑛[𝑎,𝑏]×[0,∞)→Π[𝐋^𝑛1[𝑎,𝑏]] by the formula Φ_𝜂(𝑥,𝛿)=𝑆Δ_𝜂(⋅,𝑥,𝛿),(5.5) where the operator Δ_𝜂∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]×[0,∞)→comp[ℝ^𝑛] is given by (5.2). The equalities (5.3) and (5.5) imply that ℎ_𝐋^𝑛1_[𝑎,𝑏]Φ^𝜂(𝑥,𝛿);Φ(𝑥)=^𝑏𝑎𝜂(𝑡,𝛿)𝑑𝑡(5.6) for any 𝑥∈𝐂^𝑛[𝑎,𝑏].

It follows from (5.6) and the Lebesgue theorem that lim_𝛿→0+0ℎ_𝐋^𝑛1_[𝑎,𝑏]Φ^𝜂(𝑥,𝛿);Φ(𝑥)=0.(5.7)

Thus, all mappings Φ_𝜂∶𝐂^𝑛[𝑎,𝑏]×[0,∞)→Π[𝐋^𝑛1[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∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25).

Lemma 5.1 (see [6]). Let 𝑋 be a normed space and let 𝑈⊂𝑋 be a convex set. Then ℎ_𝑋𝐵^𝑋𝑥¹,𝑟₁∩𝑈;𝐵^𝑋𝑥²,𝑟₂∩𝑈≤‖‖𝑥¹−𝑥₂‖‖_𝑋+|||𝑟₂−𝑟₁|||(5.8) for all 𝑥₁,𝑥₂∈𝑈 and all 𝑟₁,𝑟₂>0.

Denote by 𝑃(𝐂^𝑛[𝑎,𝑏]×[0,∞)) the set of all continuous functions 𝜔∶𝐂^𝑛[𝑎,𝑏]×[0,∞)→[0,∞) such that 𝜔(𝑥,0)=0 for any 𝑥∈𝐂^𝑛[𝑎,𝑏] and 𝜔(𝑥,𝛿)>0 for any (𝑥,𝛿)∈𝐂^𝑛[𝑎,𝑏]×(0,∞).

Let 𝑈⊂𝐂^𝑛[𝑎,𝑏] be a closed convex set and let 𝜔(⋅,⋅)∈𝑃(𝐂^𝑛[𝑎,𝑏]×[0,∞)). We define a multivalued mapping 𝑀_𝑈(𝜔)∶𝑈×[0,∞)→Ω(𝑈) by 𝑀_𝑈(𝜔)(𝑥,𝛿)=𝐵_𝐂^𝑛_[𝑎,𝑏]𝑥,𝜔(𝑥,𝛿)∩𝑈.(5.9)

The inequality (5.8) yields the following result.

Lemma 5.2. Let 𝑈⊂𝐂^𝑛[𝑎,𝑏] be a closed convex set and let 𝜔(⋅,⋅)∈𝑃(𝐂^𝑛[𝑎,𝑏]×[0,∞)). Then, a multivalued mapping 𝑀_𝑈(𝜔)∶𝑈×[0,∞)→Ω(𝑈) given by (5.9) is Hausdorff continuous.

We define a mapping 𝜑_𝑈(𝜔)∶[𝑎,𝑏]×𝑈×[0,∞)→[0,∞) by the formula 𝜑_𝑈(𝜔)(𝑡,𝑥,𝛿)=sup_{𝑦∈𝑀𝑈(𝜔)(𝑥,𝛿)}ℎΔ(𝑡,𝑥);Δ(𝑡,𝑦),(5.10) where the mapping 𝑀_𝑈(𝜔)∶𝑈×[0,∞)→Ω(𝑈) is given by (5.9) and the mapping Δ∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛] generates the mapping Φ given by (3.25).

It is natural to address the value of the function 𝜑_𝑈(𝜔)(⋅,⋅,⋅) at the point (𝑡,𝑥,𝛿)∈[𝑎,𝑏]×𝑈×[0,∞) as the modulus of continuity of the mapping Δ∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛] at the point (𝑡,𝑥) with respect to the variable 𝑥∈𝑈. We call the function 𝜔(⋅,⋅) the radius of continuity, while the function 𝜑_𝑈(⋅,⋅,⋅) itself is called the modulus of continuity of the mapping Δ∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛]with respect to the radius of continuity 𝜔(⋅,⋅).

Definition 5.3. One says that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛[𝑎,𝑏]] has Property 𝐶 if the mapping Δ∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛] generating the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[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 mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛[𝑎,𝑏]], there exists an isotonic continuous operator Γ∶𝐂¹⁺[𝑎,𝑏]→𝐋¹⁺[𝑎,𝑏] satisfying the following conditions: (i)Γ(0)=0;(ii)the inequality (4.30), where Γ₂≡Γ, is satisfied for any 𝑥,𝑦∈𝐂^𝑛[𝑎,𝑏] and any measurable set 𝒰⊂[𝑎,𝑏].
Then the mapping Φ(⋅) has Property 𝒞.

Proof. Let 𝑥_𝑖→𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞. Let us show that lim_𝑖→∞ℎΔ𝑡,𝑥_𝑖;Δ(𝑡,𝑥)=0(5.11) for almost all 𝑡∈[𝑎,𝑏].
For each 𝑖=1,2,…, put 𝑦_𝑖=sup_𝑗≥𝑖‖𝑥_𝑗−𝑥‖_𝐂^𝑛_[𝑎,𝑏]. Due to Theorem 3.18, (4.43), and the isotonity of the operator Γ∶𝐂¹⁺[𝑎,𝑏]→𝐋¹⁺[𝑎,𝑏], for each 𝑖=1,2,… and almost all 𝑡∈[𝑎,𝑏], we have ℎΔ𝑡,𝑥_𝑖;Δ(𝑡,𝑥)≤2Γ𝑍𝑥^𝑖−𝑥(𝑡)≤2Γ𝑦_𝑖(𝑡).(5.12) Since the sequence Γ(𝑦_𝑖), 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. Let 𝑈 be a nonempty, convex, compact set in the space 𝐂^𝑛[𝑎,𝑏] and let 𝜔(⋅,⋅)∈𝑃(𝐂^𝑛[𝑎,𝑏]×[0,∞)). Suppose also that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property 𝐶. Then the mapping 𝜑_𝑈(𝜔)∶[𝑎,𝑏]×𝑈×[0,∞) given by (5.10) has the following properties: (i)𝜑_𝑈(𝜔)(⋅,𝑥,𝛿) is measurable for any (𝑥,𝛿)∈𝑈×[0,∞);(ii)𝜑_𝑈(𝑡,⋅,⋅) is continuous on 𝑈×[0,∞) for almost all 𝑡∈[𝑎,𝑏];(iii)for any 𝑥∈𝑈 and for almost all 𝑡∈[𝑎,𝑏], lim_{𝑧→𝑥,𝛿→0+0}𝜑_𝑈(𝜔)(𝑡,𝑥,𝛿)=0;(5.13)(iv)there exists an integrable function 𝑝_𝑈∶[𝑎,𝑏]→[0,∞) such that 𝜑_𝑈(𝜔)(𝑡,𝑥,𝛿)≤𝑝_𝑈(𝑡) for almost all 𝑡∈[𝑎,𝑏], any 𝑥∈𝑈, and all 𝛿∈[0,∞).

Definition 5.6. Let 𝑈⊂𝐂^𝑛[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(𝐂^𝑛[𝑎,𝑏]×[0,∞)) estimate from above for the modulus of continuity of the mapping Δ∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛]; 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 𝜑_𝑈(𝜔)(𝑡,𝑥,𝛿)≤𝜂(𝑡,𝜀),(5.14) where 𝜑_𝑈∶[𝑎,𝑏]×𝑈×[0,∞)→[0,∞) is given by (5.10).

Let 𝑈⊂𝐂^𝑛[𝑎,𝑏] and 𝜔(⋅,⋅)∈𝑃(𝐂^𝑛[𝑎,𝑏]×[0,∞)). One defines a function 𝜆_𝑈(𝜔)∶[𝑎,𝑏]×[0,∞)→[0,∞) by 𝜆_𝑈(𝜔)(𝑡,𝛿)=sup_𝑥∈𝑈𝜑_𝑈(𝜔)(𝑡,𝑥,𝛿).(5.15)

Lemma 5.1 yields the following result.

Corollary 5.7. Let 𝑈 be a nonempty, convex, compact set in the space 𝐂^𝑛[𝑎,𝑏] and let 𝜔(⋅,⋅)∈𝑃(𝐂^𝑛[𝑎,𝑏]×[0,∞)). Suppose also that a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property 𝐶. Then the mapping 𝜆_𝑈(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]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛].

Remark 5.8. Corollary 5.7 yields that if 𝑈 is a nonempty, convex, compact set in the space 𝐂^𝑛[𝑎,𝑏] and a mapping Φ∶𝐂^𝑛[𝑎,𝑏]→𝑄[𝐋^𝑛1[𝑎,𝑏]] has Property 𝐶, then for a given 𝜔(⋅,⋅)∈𝑃(𝐂^𝑛[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]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛].

Let 𝜂(⋅,⋅)∈𝐾([𝑎,𝑏]×[0,∞)). For each 𝛿∈[0,∞), consider the initial value problem 𝑥̇∈Φ_𝜂(𝑥,𝛿),𝑥(𝑎)=𝑥₀𝑥⁰∈ℝ^𝑛,(5.16) where the mapping Φ_𝜂∶𝐂^𝑛[𝑎,𝑏]×[0,∞)→Π[𝐋^𝑛1[𝑎,𝑏]] is given by (5.1) and (5.5).

Since the operator Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25) is a Volterra operator, we see that the mapping Δ∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛] 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∶𝐂^𝑛[𝑎,𝑏]×[0,∞)→Π[𝐋^𝑛1[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 𝐻_𝜂(𝛿)(𝑈) the set of all generalized 𝛿-solutions of (4.1) belonging to 𝑈⊂𝐂^𝑛[𝑎,𝑏].

Theorem 5.9. Suppose that a set 𝑈⊂𝐂^𝑛[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]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛], one has 𝐻_co𝑥⁰,𝑏=_𝛿>0𝐻_𝜂(𝛿)(𝑈),(5.17) where 𝐻_𝜂(𝛿)(𝑈) is the closure of 𝐻_𝜂(𝛿)(𝑈) in 𝐂^𝑛[𝑎,𝑏].

Proof. First, let us prove that 𝐻_co𝑥⁰,𝑏⊂_𝛿>0𝐻_𝜂(𝛿)(𝑈).(5.18) Let 𝑥∈𝐻_co(𝑥₀,𝑏). Let us show that 𝑥 is a limit point of the set 𝐻_𝜂(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 𝑥_𝑖∶[𝑎,𝑏]→ℝ^𝑛, 𝑖=1,2,…, such that the following conditions hold: 𝑥_𝑖→𝑥 in 𝐂^𝑛[𝑎,𝑏] as 𝑖→∞; 𝑥_𝑖∈𝑈, 𝑥̇_𝑖∈decΦ(𝑥), and 𝑥_𝑖(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]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛]. Then there exists 𝑖₁ such that ‖𝑥−𝑥_𝑖‖_𝐂^𝑛_[𝑎,𝑏]<𝜔(𝑥,𝛿) for each 𝑖≥𝑖₁. This implies that 𝑥_𝑖∈𝐵_𝐂^𝑛_[𝑎,𝑏][𝑥,𝜔(𝑥,𝛿)] for each 𝑖≥𝑖₁. Therefore, 𝑥_𝑖∈𝑀_𝑈(𝜔)(𝑥,𝛿) for each 𝑖≥𝑖₁. By Definition 5.6, there exists a number 𝑖₂≥𝑖₁ such that 𝜑_𝑈(𝜔)𝑡,𝑥,‖‖𝑥−𝑥_𝑖‖‖_{𝐂𝑛[𝑎,𝑏]}≤𝜂(𝑡,𝛿)(5.19) for any 𝑖≥𝑖₂ and almost all 𝑡∈[𝑎,𝑏].
The inequality (5.19) yields that 𝜌𝑥̇_𝑖𝑡);Δ𝑡,𝑥_𝑖≤ℎΔ(𝑡,𝑥);Δ𝑡,𝑥^𝑖≤𝜑^𝑈(𝜔)𝑡,𝑥,‖‖𝑥−𝑥_𝑖‖‖_{𝐂𝑛[𝑎,𝑏]}≤𝜂(𝑡,𝛿(5.20) for each 𝑖≥𝑖₂ and almost all 𝑡∈[𝑎,𝑏]. By (5.20), 𝑥_𝑖∈𝐻_𝜂(𝛿)(𝑈) for each 𝑖≥𝑖₂. This implies that 𝑥 is a limit point of the set 𝐻_𝜂(𝛿)(𝑈). Therefore, 𝑥∈𝐻_𝜂(𝛿)(𝑈), and consequently (5.18), is satisfied.
Let us prove the opposite inclusion _𝛿>0𝐻_𝜂(𝛿)(𝑈)⊂𝐻_co𝑥⁰,𝑏.(5.21) Let 𝑥∈⋂_𝛿>0𝐻_𝜂(𝛿)(𝑈). This implies that for each 𝑖=1,2,…, there exists 𝑥_𝑖∈𝐻_𝜂(1/𝑖)(𝑈) satisfying ‖𝑥−𝑥_𝑖‖_𝐂^𝑛_[𝑎,𝑏]<1/𝑖. Suppose that functions 𝑧_𝑖∈decΦ(𝑥) satisfy |||𝑥̇_𝑖(𝑡)−𝑧_𝑖(𝑡)|||=𝜌𝑥̇_𝑖(𝑡);Δ(𝑡,𝑥)(5.22) for each 𝑖=1,2,… and almost all 𝑡∈[𝑎,𝑏]. Let us show that lim_𝑖→∞‖‖𝑥̇_𝑖−𝑧_𝑖‖‖_{𝐋𝑛1[𝑎,𝑏]}=0.(5.23) Since 𝜂(⋅,⋅)∈𝐾([𝑎,𝑏]×[0,∞)), by the Lebesgue theorem, we have that lim_𝑖→∞^𝑏𝑎𝜂𝑡,1𝑖𝑑𝑡=0.(5.24) By (5.22), the estimates |||𝑥̇_𝑖(𝑡)−𝑧_𝑖(𝑡)|||≤ℎ[Δ𝑡,𝑥_𝑖^{𝜂(𝑡,1/𝑖)};Δ(𝑡,𝑥)]≤𝜂𝑡,1𝑖+ℎΔ(𝑡,𝑥);Δ𝑡,𝑥^𝑖(5.25) are satisfied for each 𝑖=1,2,… and almost all 𝑡∈[𝑎,𝑏]. Therefore, ^𝑏𝑎|||𝑥̇_𝑖(𝑡)−𝑧_𝑖(𝑡)|||𝑑𝑡≤^𝑏𝑎𝜂𝑡,1𝑖𝑑𝑡+2ℎ^{𝐋𝑛1[𝑎,𝑏]}decΦ(𝑥);decΦ𝑥_𝑖(5.26) for each 𝑖=1,2,…. By (5.24) and due to the continuity of the mapping Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25), we have (5.23).
Since 𝑥̇_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞, we have that 𝑧_𝑖→𝑥̇ weakly in 𝐋^𝑛1[𝑎,𝑏] as 𝑖→∞. Therefore, by [21], 𝑥̇∈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 set 𝑈⊂𝐂^𝑛[𝑎,𝑏] has Property 𝒟. Then, 𝐻𝑥₀,𝑏=_𝛿>0𝐻_𝜂(𝛿)(𝑈)(5.27) for any 𝜂(⋅,⋅)∈𝐾([𝑎,𝑏]×[0,∞)) if and only if the equality (4.51) is satisfied.

Proof. Let us prove the sufficiency. Assume that (4.51) holds. Let us show that the equality (5.27) is satisfied for any function 𝜂(⋅,⋅)∈𝐾([𝑎,𝑏]×[0,∞)). By the definition of the problem (5.16), the inclusion 𝐻𝑥₀,𝑏⊂𝐻_𝜂(𝛿)(𝑈)(5.28) is satisfied for any 𝛿>0. Therefore, for any 𝛿>0, we have the inclusion 𝐻𝑥₀,𝑏⊂𝐻_𝜂(𝛿)(𝑈)(5.29) and consequently the inclusion 𝐻𝑥₀,𝑏⊂_𝛿>0𝐻_𝜂(𝛿)(𝑈)(5.30) holds. Now, let us check that the opposite relation _𝛿>0𝐻_𝜂(𝛿)(𝑈)⊂𝐻𝑥₀,𝑏,(5.31) which is, by (4.51), equivalent to the inclusion _𝛿>0𝐻_𝜂(𝛿)(𝑈)⊂𝐻_co𝑥⁰,𝑏,(5.32) holds as well. The latter relation can be proven similarly to Theorem 5.9.
The necessity follows readily from Theorem 5.9. The proof is complete.

Remark 5.11. Note that the equality (5.27) describes the robustness property of the set 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∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25).

On the other hand, each generalized solution 𝑥∶[𝑎,𝑏]→ℝ^𝑛 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(𝐂^𝑛[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 mapping Δ_ext∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛] by Δ_ext(𝑡,𝑥)=extcoΔ(𝑡,𝑥),(5.33) where Δ∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛] is the mapping generating the operator Φ∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by (3.25); see the definition of ext(coΔ(𝑡,𝑥)) in Section 2. Let us remark that the mapping Δ_ext(⋅,𝑥) is measurable (see [25]) and integrally bounded for each 𝑥∈𝐂^𝑛[𝑎,𝑏].

Consider the operator Φ_ext∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]] given by Φ_ext(𝑥)=𝑆Δ_ext(⋅,𝑥),(5.34) where the mapping Δ_ext∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛] is given by (5.33).

Remark 5.12. Note that for each 𝑥∈𝐂^𝑛[𝑎,𝑏], the set Φ_ext(𝑥) has the following property coΦ_ext(𝑥)=codecΦ(𝑥).(5.35) Also, Φ_ext(𝑥) is the minimal set among all nonempty closed in 𝐋^𝑛1[𝑎,𝑏] decomposable subsets of decΦ(𝑥) satisfying (5.35).

Consider the problem 𝑥̇∈Φ_ext(𝑥),𝑥(𝑎)=𝑥₀𝑥⁰∈ℝ^𝑛.(5.36) We call any solution (resp., quasisolution) of (5.36) a generalized extreme (in the sense of the definition in Section 2) solution (resp., generalized extreme quasisolution) of the problem (4.1).

Let ℋ_ext(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. ℋ_ext(𝑥₀)=𝐻_co(𝑥₀,𝑏).

Let 𝜂(⋅,⋅)∈𝐾([𝑎,𝑏]×[0,∞)), 𝜔(⋅,⋅)∈𝑃(𝐂^𝑛[𝑎,𝑏]×[0,∞)). Let also 𝑈 be a convex closed set in 𝐂^𝑛[𝑎,𝑏]. We define mappings Φ_𝜂,𝜔∶𝑈×[0,∞)→comp[𝐋^𝑛1[𝑎,𝑏]^∗], Δ_ext,𝜂∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]×[0,∞)→comp[ℝ^𝑛], Φ_ext,𝜂∶𝐂^𝑛[𝑎,𝑏]×[0,∞)→Π[𝐋^𝑛1[𝑎,𝑏]], Φ_{ext,𝜂,𝜔}∶𝐂^𝑛[𝑎,𝑏]×[0,∞)→comp[𝐋^𝑛1[𝑎,𝑏]^∗] by the formulas Φ_𝜂,𝜔(𝑥,𝛿)=Φ_𝜂Δ𝑀^𝑈(𝜔)(𝑥,𝛿),𝛿_ext,𝜂,Φ_ext,𝜂(𝑡,𝑥,𝛿)=Δ^ext(𝑡,𝑥)^{𝜂(𝑡,𝛿)},Φ(𝑥,𝛿)=𝑆Δ^ext,𝜂(⋅,𝑥,𝛿)_{ext,𝜂,𝜔},(𝑥,𝛿)=Φ_ext,𝜂𝑀^𝑈(𝜔)(𝑥,𝛿),𝛿,(5.37) where the mappings Φ_𝜂∶𝐂^𝑛[𝑎,𝑏]×[0,∞)→Π[𝐋^𝑛1[𝑎,𝑏]], 𝑀_𝑈(𝜔)∶𝑈×[0,∞)→Ω(𝑈), Δ_ext∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]×[0,∞)→comp[ℝ^𝑛] are given by the equalities (5.5), (5.9), and (5.33), respectively.

For each 𝛿>0, consider the following problems on 𝑈⊂𝐂^𝑛[𝑎,𝑏]: 𝑥𝑥̇∈̇∈ΦΦ_{𝜂,𝜔ext,𝜂,𝜔}(𝑥,𝛿),𝑥(𝑎)=𝑥₀𝑥⁰∈ℝ^𝑛,(5.38)(𝑥,𝛿),𝑥(𝑎)=𝑥₀𝑥⁰∈ℝ^𝑛,(5.39)Φ_𝜂,𝜔∶𝑈×[0,∞)→comp[𝐋^𝑛1[𝑎,𝑏]^∗] where the mappings Φ_{ext,𝜂,𝜔}∶𝑈×[0,∞)→, comp[𝐋^𝑛1[𝑎,𝑏]^∗]𝛿>0 are given by (5.37).

We call any solution of the problem (5.38) with a fixed 𝛿 a generalized 𝛿>0-solution of the problem (4.1), or a generalized approximate solution of (4.1) with external and internal perturbations. For each 𝐻_{𝜂(𝛿),𝜔(𝛿)}(𝑈), we denote by 𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈) (𝑈⊂𝐂^𝑛[𝑎,𝑏].) the set of all solutions of the problem (5.38) ((5.39)) on Φ_{ext,𝜂,𝜔}(𝑥,𝛿)⊂Φ_𝜂,𝜔(𝑥,𝛿) Since 𝛿>0 for any 𝑥∈𝑈, and any 𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈)⊂𝐻_{𝜂(𝛿),𝜔(𝛿)}(𝑈) we see that 𝛿>0 for any 𝑈⊂𝐂^𝑛[𝑎,𝑏].

Theorem 5.14. Let the set 𝒟. have Property 𝜂(⋅,⋅)∈𝐾([𝑎,𝑏]×[0,∞)) Then for any 𝜔(⋅,⋅)∈𝑃(𝐂^𝑛[𝑎,𝑏]×[0,∞)), 𝐻_co𝑥⁰,𝑏=_𝛿>0𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈)=_𝛿>0𝐻_{𝜂(𝛿),𝜔(𝛿)}(𝑈),(5.40), one has 𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈) where 𝐻_{𝜂(𝛿),𝜔(𝛿)}(𝑈) and 𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈) are the closures of the sets 𝐻_{𝜂(𝛿),𝜔(𝛿)}(𝑈), and 𝐂^𝑛[𝑎,𝑏]. respectively, in the space 𝐻_co𝑥⁰,𝑏⊂_𝛿>0𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈).(5.41)

Proof. First of all, let us check that 𝑥∈𝐻(𝑥₀,𝑏). Let 𝑥 We show that 𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈) is a limit point of the set 𝛿>0. for any 𝑥 By Corollary 5.13, 𝑈 is a generalized extreme quasisolution of the problem (4.1). Since the set 𝒟, has Property 𝑥∈𝑈 we see that 𝑥_𝑖∶[𝑎,𝑏]→ℝ^𝑛 and there exists a sequence of absolutely continuous functions 𝑖=1,2,…,, 𝑥_𝑖→𝑥 with the following properties: 𝐂^𝑛[𝑎,𝑏] in 𝑖→∞ as 𝑥_𝑖∈𝑈; 𝑥̇_𝑖∈Φ_ext(𝑥),, 𝑥_𝑖(𝑎)=𝑥₀ and 𝑖=1,2,… for each Φ_ext∶𝐂^𝑛[𝑎,𝑏]→Π[𝐋^𝑛1[𝑎,𝑏]]. Here, the operator 𝑖₀ is given by (5.33) and (5.34).
Let us prove also that there exists a number 𝑖≥𝑖₀ such that for each 𝑥_𝑖∈𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈).(5.42), 𝜔(⋅,⋅)∈𝑃(𝐂^𝑛[𝑎,𝑏]×[0,∞)), Since 𝑖₀ we see that there exists a number 𝑖≥𝑖₀ such that for each 𝑥∈𝐵_𝐂^𝑛_[𝑎,𝑏][𝑥_𝑖;𝜔(𝑥_𝑖,𝛿)]., 𝑖≥𝑖₀ This implies that for each 𝑥∈𝑀_𝑈(𝑥_𝑖,𝛿), we have that 𝑖≥𝑖₀ (see (5.9)). Therefore, for each Φ_ext(𝑥)⊂Φ_{ext,𝜂,𝜔}(𝑥_𝑖,𝛿), the inclusion 𝑖≥𝑖₀ holds. Hence, for each 𝑥, we have (5.42). This means that 𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈). is a limit point of the set 𝑥∈𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈) Therefore, _𝛿>0𝐻_{ext,𝜂(𝛿),𝜔(𝛿)}(𝑈)⊂𝐻_co𝑥⁰,𝑏(5.43) and (5.41) is satisfied.
The relation Δ∶[𝑎,𝑏]×𝐂^𝑛[𝑎,𝑏]→comp[ℝ^𝑛] can be proven similarly to (5.21) (see the proof of Theorem 5.9). The second equality of (5.40) can be proven in the same way. This completes the proof of the theorem.

Remark 5.15. Theorem 5.14 says that no measurement accuracy of the values of the mapping 𝐻(𝑥₀,𝑏) could guarantee the “reconstruction” of the set 𝐻_{𝜂(𝛿),𝜔(𝛿)}(𝑈). by means of 𝛿 That is only possible if the density principle holds for the generalized solutions.

6. Conclusion

The main results of the paper can be summarized as follows. For the decomposable hull of a mapping, we have obtained the conditions for the property of the Hausdorff lower semicontinuity (resp., upper semicontinuity and continuity). We considered a functional differential inclusion with a Volterra multivalued mapping which is not necessarily decomposable. The concept of a generalized solution of the initial value problem for such an inclusion was introduced and its properties were studied. Conditions for the local existence and continuation of a generalized solution to the initial value problem were obtained. We have offered some estimates, which characterize the closeness of generalized solutions and a given absolutely continuous function. These estimates were derived from the conditions for the existence of a generalized solution satisfying the inequality (4.18) (see Theorem 4.13 and Corollary 4.14).

The concept of a generalized quasisolution of the initial value problem was introduced. We proved that the set of all generalized quasisolutions of the initial value problem coincides with the solution set of the functional differential inclusion with the convex decomposable hull of the right-hand side. Using this fact as well as the estimates characterizing the closeness of generalized solutions and a given absolutely continuous function, we obtained the density principle for the generalized solutions.

Asymptotic properties of the set of generalized approximate solutions (generalized 𝛿-solutions) were studied. It was proven that the limit of the closures of the sets of generalized approximate solutions coincides with the closure of the set of the generalized solutions if and only if the density principle holds for the generalized solutions.

Acknowledgments

The first author is supported by CIGENE (Center for Integrative Genetics) and by Lånekassen (Norwegian State Educational Loan Fund). The second author is supported by CIGENE and partially supported by the Russian FBR Grant no. 07-01-00305. The third author is partially supported by the Russian FBR Grant no. 07-01-00305.