On Maximal and Minimal Solutions for Set-Valued Differential Equations with Feedback Control

Abstract

In this paper, we present the existence of extremal solutions of set-valued differential equations with feedback control on semilinear Hausdorff space under Hukuhara derivative which is developed under the form 𝐷_𝐻𝑋(𝑡)=𝐹(𝑡,𝑋(𝑡),𝐻(𝑡,𝑋(𝑡))), 𝑋(0)=𝑋₀,  for all 𝑡∈[0,𝑇] with the monotone iterative technique and we will verify that monotone sequence of approximate solutions converging uniformly to the solution of the problem, that is useful for optimization problems.

1. Introduction

Recently, the study of set differential equations was initiated in a metric space and some basic results of interest were obtained. Some interesting results in this direction have been obtained in a series of works of Professor V. Lakshmikantham and other authors (see [1–5]). Professor V. Lakshmikantham and the other authors considered set differential equations (SDEs) and had some important results on existence, comparison, and stability criteria for SDEs:𝐷_𝐻𝑋(𝑡)=𝐹(𝑡,𝑋(𝑡)), (1.1) where 𝑋(𝑡₀)=𝑋₀, 𝑋(𝑡)∈𝐾_𝐶𝐶(𝑅^𝑛), and𝑡∈[𝑡₀,𝑇]=𝐼⊂𝑅₊.

Based on these results, the authors gave the concept of set-valued control differential equation and studied existence and comparison of its solutions (see [6]). In this paper, we investigate an existence result of Peano’s type and then consider the existence of extremal solution of set-valued control differential equations. For this purpose, one needs to introduce a partial order in (𝐾_𝑐(𝑅^𝑛),𝐷), prove the required comparison result for strict inequalities, and then, utilizing it, discuss the existence of extremal solutions.

This paper is organized as follows: in Section 2, we recall some basic concepts and notations which are useful in next sections. In Section 3, we present on the existence of extremal solutions for SSDEs on semilinear Hausdorff space with the monotone iterative technique and we will verify that monotone sequence of approximate solutions converging uniformly to the solution of the problem.

2. Preliminaries

We recall some notations and concepts presented in detail in recent series works of Professor V. Lakshmikantham et al. (see [1]). Let 𝐾_𝐶𝐶(𝑅^𝑛) denote the collection of all nonempty compact convex subsets of 𝑅^𝑛. Given 𝐴,𝐵∈𝐾_𝐶𝐶(𝑅^𝑛), the Hausdorff distance between 𝐴 and 𝐵 is defined by𝐷[𝐴,𝐵]=maxsup_𝑎∈𝐴inf_𝑏∈𝐵‖𝑎−𝑏‖_𝑅𝑛,sup_𝑏∈𝐵inf_𝑎∈𝐴‖𝑏−𝑎‖_𝑅𝑛, (2.1) where ‖⋅‖_𝑅𝑛 denotes the Euclidean norm in 𝑅^𝑛 and {𝜃^𝑛}—the zero points set in 𝐾_𝐶𝐶(𝑅^𝑛). It is known that (𝐾_𝐶C(𝑅^𝑛),𝐷) is a complete metric space and 𝐾_𝐶𝐶(𝑅^𝑛) is a complete and separable with respect to metric 𝐷.

We define the magnitude of a nonempty subset of 𝐴:𝐷𝐴,𝜃^𝑛=‖𝐴‖=sup‖𝑎‖_𝑅𝑛,𝑎∈𝐴, (2.2) where 𝜃^𝑛 is the zero element of 𝑅^𝑛 which is regarded as a one point set. ‖𝐴‖=𝐷[𝐴,𝜃^𝑛]-norm in 𝐾_𝐶𝐶(𝑅^𝑛) is finite when the supremum in (2.2) is attained with 𝐴∈𝐾_𝐶𝐶(𝑅^𝑛).

The Hausdorff metric (1.1) satisfies the properties below:𝐷[][][][],𝐷[][],𝐷[][][],𝐷𝐴+𝐶,𝐵+𝐶=𝐷𝐴,𝐵,𝐷𝐴,𝐵=𝐷𝐵,𝐴𝜆𝐴,𝜆𝐵=𝜆𝐷𝐴,𝐵𝐴,𝐵≤𝐷𝐴,𝐶+𝐷𝐶,𝐵𝐴+𝐴^′,𝐵+𝐵^′≤𝐷[𝐴,𝐵]+𝐷𝐴^′,𝐵^′ (2.3)

for all 𝐴,𝐵,𝐶,𝐴^′,𝐵^′∈𝐾_𝐶𝐶(𝑅^𝑛) and 𝜆∈𝑅₊. If 𝛼,𝛽∈𝑅, and 𝐴,𝐵∈𝐾_𝐶𝐶(𝑅^𝑛), then𝛼(𝐴+𝐵)=𝛼𝐴+𝛼𝐵,𝛼𝛽𝐴=𝛼𝛽𝐴,1⋅𝐴=𝐴. (2.4) It is known that (𝐾_𝐶𝐶(𝑅^𝑛),𝐷) is a complete metric space and if the space 𝐾_𝐶𝐶(𝑅^𝑛) is equipped with the natural algebraic operations of addition and nonnegative scalar multiplication, then 𝐾_𝐶𝐶(𝑅^𝑛) becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space.

Let 𝐴,𝐵∈𝐾_𝐶𝐶(𝑅^𝑛). The set 𝐶∈𝐾_𝐶𝐶(𝑅^𝑛) satisfying 𝐴=𝐵+𝐶 is called the Hausdorff difference (the geometric difference) of the sets 𝐴 and 𝐵 and is denoted by the symbol 𝐴−𝐵. Given an interval 𝐼 in 𝑅₊. We say that the set mapping 𝐹∶𝐼→𝐾_𝐶𝐶(𝑅^𝑛) has a Hukuhara derivative 𝐷_𝐻𝐹(𝑡₀) at a point 𝑡₀∈𝐼, iflim_ℎ→0+ℎ⁻¹𝐹𝑡₀+ℎ−𝐹𝑡₀,lim_ℎ→0+ℎ⁻¹𝐹𝑡₀−𝐹𝑡₀−ℎ (2.5) exist in the topology of 𝐾_𝐶𝐶(𝑅^𝑛) and are equal to 𝐷_𝐻𝐹(𝑡₀).

By embedding 𝐾_𝐶𝐶(𝑅^𝑛) as a complete cone in a corresponding Banach space and taking into account result on the differentiation of Bochner integral, we find that if𝐹(𝑡)=𝑋₀+^𝑡_𝑡0Φ(𝑠)𝑑𝑠,𝑋₀∈𝐾_𝐶𝐶𝑅^𝑛, (2.6)where Φ∶𝐼→𝐾_𝐶𝐶(𝑅^𝑛) is integrable in the sense of Bochner, then 𝐷_𝐻𝐹(𝑡) exists and the equality 𝐷_𝐻𝐹(𝑡)=Φ(𝑡) a.e on 𝐼 holds.

The Hukuhara integral of 𝐹 is given by_𝐼𝐹(𝑠)𝑑𝑠=cl_𝐼𝑓(𝑠)𝑑𝑠∶𝑓isacontinuousselectorof𝐹 (2.7) for any compact set 𝐼⊂𝑅₊.

Some properties of the Hukuhara integral are in [1]. If 𝐹∶𝐼→𝐾_𝐶𝐶(𝑅^𝑛) is integrable, one has^𝑡2_𝑡0𝐹(𝑠)𝑑𝑠=^𝑡1_𝑡0𝐹(𝑠)𝑑𝑠+^𝑡2_𝑡1𝐹(𝑠)𝑑𝑠,𝑡₀≥𝑡₁≥𝑡₂,_𝑡𝑡0𝜆𝐹(𝑠)𝑑𝑠=𝜆^𝑡_𝑡0𝐹(𝑠)𝑑𝑠,𝜆∈𝑅. (2.8) If 𝐹,𝐺∶𝐼→𝐾_𝐶𝐶(𝑅^𝑛) are integrable, then 𝐷[𝐹(⋅),𝐺(⋅)]∶𝐼→𝑅 is integrable and𝐷^𝑡_𝑡0𝐹(𝑠)𝑑𝑠,^𝑡_𝑡0𝐺(𝑠)𝑑𝑠≤^𝑡_𝑡0𝐷[𝐹(𝑠),𝐺(𝑠)]𝑑𝑠. (2.9)

3. Main Results

We consider the set-valued differential equations (SSDEs) with feedback control under the form𝐷_𝐻𝑋(𝑡)=𝐹(𝑡,𝑋(𝑡),𝐻(𝑡,𝑋(𝑡))),𝑋(0)=𝑋₀,∀𝑡∈[0,𝑇], (3.1) where 𝐹∶[0,𝑇]×𝐾_𝐶𝐶(𝑅^𝑛)×𝐾_𝐶𝐶(𝑅^𝑑)→𝐾_𝐶𝐶(𝑅^𝑛) and 𝐻∶[0,𝑇]×K_𝐶𝐶(𝑅^𝑛)→𝐾_𝐶𝐶(𝑅^𝑑) is a feedback control, state set 𝑋₀,𝑋(𝑡)∈𝐾_𝐶𝐶(𝑅^𝑛).

Definition 3.1. The mapping set 𝑋∶[0,𝑇]→𝐾_𝐶𝐶(𝑅^𝑛) is called to be a solution of (3.1) on [0,𝑇] if and only if the following conditions are satisfied:
(i)𝑋(𝑡)∈𝐶¹[[0,𝑇],𝐾_𝐶𝐶(𝑅^𝑛)] with Hukuhara derivative 𝐷_𝐻 by 𝑡;(ii)𝑋(0)=𝑋₀;(iii)𝐹(𝑡,𝑋(𝑡),𝐻(𝑡,𝑋(𝑡)) is integrable on [0,𝑇];(iv)for all 𝑡∈[0,𝑇],𝑋(𝑡)=𝑋₀+^𝑡₀𝐹(𝑆,𝑋(𝑠),𝐻(𝑠,𝑋(𝑠)))𝑑𝑠 (3.2) the integral in (3.2) is Hukuhara integral.
In this section, we will use the monotone iterative technique to solve the minimal and maximal solutions of (3.1). To construct the set monotone sequence, we first introduce the following definition.

Definition 3.2. We denote(i)by 𝐾⁺(𝐾^0,+) the subfamily of 𝐾_𝐶𝐶(𝑅^𝑛) consisting of sets 𝑋∈𝐾_𝐶𝐶(𝑅^𝑛) such that any 𝑥∈𝑋 is a nonnegative (positive) vector of 𝑛 components satisfying 𝑥_𝑖≥0(𝑥_𝑖>0) for 𝑖=1,2,…,𝑛,(ii)by 𝐾⁻(𝐾^0,−) the subfamily of 𝐾_𝐶𝐶(𝑅^𝑛) consisting of sets 𝑋∈𝐾_𝐶𝐶(𝑅^𝑛) such that any 𝑥∈𝑋 is a nonpostive (negative) vector of 𝑛 components satisfying 𝑥_𝑖≤0(𝑥_𝑖<0) for 𝑖=1,2,…,𝑛.By Definition 3.2, we notice that 𝐾⁺ is a positive cone in 𝐾_𝐶𝐶(𝑅^𝑛) and 𝐾^0,+ is the nonempty interior of 𝐾⁺. 𝐾⁻ is a negative cone in 𝐾_𝐶𝐶(𝑅^𝑛) and 𝐾^0,− is the nonempty interior of 𝐾⁻. We can therefore induce a partial odering in 𝐾_𝐶𝐶(𝑅^𝑛). Thus, if 𝑋∈𝐾⁺(𝐾^0,+) is 𝑋≥𝜃^𝑛(𝑋>𝜃^𝑛), that is, with any 𝑥∈𝑋 is satisfying 𝑥_𝑖≥0(𝑥_𝑖>0) for 𝑖=1,2,…,𝑛 and 𝑋∈𝐾⁻(𝐾^0,−) is 𝑋≤𝜃^𝑛(𝑋<𝜃^𝑛), that is, with any 𝑥∈𝑋 is satisfying 𝑥_𝑖≤0(𝑥_𝑖<0) for 𝑖=1,2,…,𝑛. Now we define the ordering in 𝐾_𝐶𝐶(𝑅^𝑛).

Definition 3.3. For any 𝑋,𝑌∈𝐾_𝐶𝐶(𝑅^𝑛), if there exists a 𝑍∈𝐾_𝐶𝐶(𝑅^𝑛) such that 𝑍∈𝐾⁺(𝐾^0,+) and 𝑋=𝑌+𝑍, then we write 𝑋≥𝑌(𝑋>𝑌). Similarly, if there exists a 𝑍∈𝐾_𝐶𝐶(𝑅^𝑛) such that 𝑍∈𝐾⁻(𝐾^0,−) and 𝑋=𝑌+𝑍, then we write 𝑋≤𝑌(𝑋<𝑌).

Theorem 3.4. Assume the following:(H1)𝐹(⋅,𝑋(⋅),𝑈(⋅))∈𝐶[[0,𝑇],𝐾_𝐶𝐶(𝑅^𝑛)] is monotone nondecreasing in (𝑋,𝑈)∈𝐾_𝐶𝐶(𝑅^𝑛)×𝐾_𝐶𝐶(𝑅^𝑑) for every 𝑡∈[0,𝑇], that is, for fixed 𝑡∈[0,𝑇], 𝐹(𝑡,𝑋₁,𝑈₁))≥𝐹(𝑡,𝑋₂,𝑈₂) wherever 𝑋₁≥𝑋₂, 𝑈₁≥𝑈₂ and 𝐻(⋅,𝑋(⋅))∈𝐶[[0,𝑇],𝐾_𝐶𝐶(𝑅^𝑛)] is monotone nondecreasing in 𝑋∈𝐾_𝐶𝐶(𝑅^𝑛) for every 𝑡∈[0,𝑇], that is, for fixed 𝑡∈[0,𝑇], 𝐻(𝑡,𝑋₁)≥𝐻(𝑡,𝑋₂) wherever 𝑋₁≥𝑋₂; (H2) there exist 𝑉,𝑊∈𝐶¹[𝐼,𝐾_𝐶𝐶(𝑅^𝑛)] such that 𝐷_𝐻𝑉(𝑡)≤𝐹(𝑡,𝑉(𝑡),𝐻(𝑡,𝑉(𝑡))) and𝐷_𝐻𝑊(𝑡)≥𝐹(𝑡,𝑊(𝑡),𝐻(𝑡,𝑊(𝑡))); (3.3)(H3) for any 𝑋₁,𝑋₂∈𝐾_𝐶𝐶(𝑅^𝑛) with 𝑋₁≥𝑋₂ and some positive number real 𝐿>0 such that𝐹𝑡,𝑋₁,𝐻𝑡,𝑋₁≤𝐹𝑡,𝑋₂,𝐻𝑡,𝑋₂+𝐿𝑋₁−𝑋₂; (3.4) then 𝑉(𝑡)≤𝑊(𝑡) for 𝑡∈𝐼 provided 𝑉(0)≤𝑊(0).

Proof. For any 𝜂=(𝜂₁,𝜂₂,…,𝜂_𝑛)>𝜃^𝑛, we define 𝑊_𝜂(𝑡)=𝑊(𝑡)+𝜂𝑒^2𝐿𝑡 and we note that 𝑉(0)≤𝑊(0)≤𝑊_𝜂(0). By using (2.5), we infer 𝐷_𝐻𝑊_𝜂(𝑡)=𝐷_𝐻𝑊(𝑡)+2𝐿𝜂𝑒^2𝐿𝑡. Let 𝑡_𝛿 be the supremum of all positive number 𝛿 such that 𝑉(0)<𝑊_𝜂(0) implies 𝑉(𝑡)<𝑊_𝜂(𝑡) on [0,𝛿]. Thus 𝑡_𝛿>0 and 𝑉(𝑡_𝛿)<𝑊_𝜂(𝑡_𝛿). Using (H1)–(H3), we get 𝐷_𝐻𝑉𝑡_𝛿≤𝐹𝑡_𝛿,𝑉𝑡_𝛿,𝐻𝑡_𝛿,𝑉𝑡_𝛿≤𝐹𝑡_𝛿,𝑊_𝜂𝑡_𝛿,𝐻𝑡_𝛿,𝑊_𝜂𝑡_𝛿≤𝐹𝑡_𝛿,𝑊𝑡_𝛿,𝐻𝑡_𝛿,𝑊𝑡_𝛿+𝐿⋅𝑊_𝜂𝑡_𝛿−𝑊𝑡_𝛿≤𝐹𝑡_𝛿,𝑊𝑡_𝛿,𝐻𝑡_𝛿,𝑊𝑡_𝛿+𝐿𝜂𝑒^{2𝐿𝑡𝛿}≤𝐷_𝐻𝑊𝑡_𝛿+2𝐿𝜂𝑒^{2𝐿𝑡𝛿}=𝐷_𝐻𝑊_𝜂𝑡_𝛿. (3.5) Equation (2.5), together with (3.5), implies that there exists an ℎ>0 such that 𝑊_𝜂𝑡_𝛿−𝑉𝑡_𝛿<𝑊_𝜂(𝑡)−𝑉(𝑡),𝑡_𝛿<𝑡<𝑡_𝛿+ℎ. (3.6) This contradicts that 𝑡_𝛿>0 is the supremum in view of the continuity of function involved and consequently that the inequality 𝑉(𝑡)<𝑊_𝜂(𝑡) holds for 𝑡∈𝐼. Taking the limit 𝜂→0 yields the desired result. This proof is complete.

Corollary 3.5. Let 𝑉,𝑊∈𝐶¹[𝐼,𝐾_𝐶𝐶(𝑅^𝑛)] such that 𝐷_𝐻𝑉(𝑡)≤𝐷_𝐻𝑊(𝑡) for all 𝑡∈𝐼, then 𝑉(0)≤𝑊(0) implies that 𝑉(𝑡)≤𝑊(𝑡) for all 𝑡∈𝐼.

Proof. It is clear from the proof of Theorem 3.4.

Definition 3.6. 𝑉,𝑊∈𝐶¹[𝐼,𝐾_𝐶𝐶(𝑅^𝑛)] are said to be the lower solution and upper solution of (3.1) respectively if 𝐷_𝐻𝑉(𝑡)((𝑡)((𝑡(0)(0),𝐷≤𝐹𝑡,𝑉,𝐻𝑡,𝑉))),𝑉≤𝑋_𝐻𝑊(𝑡)≥𝐹(𝑡,𝑊(𝑡),𝐻(𝑡,𝑊(𝑡))),𝑊(0)≥𝑋(0). (3.7)

Theorem 3.7 (existence of solution). Assume 𝑉,𝑊∈𝐶¹[𝐼,𝐾_𝐶𝐶(𝑅^𝑛)] are lower solution and upper solution of (3.1), respectively, and assumptions (H1), (H3) are satisfied, then there exists solution 𝑋(𝑡) of (3.1).

Proof. For any 𝜂=(𝜂₁,𝜂₂,…,𝜂_𝑛)>𝜃^𝑛, we define 𝑊_𝜂(𝑡)=𝑊(𝑡)+𝜂𝑒^2𝐿𝑡 and 𝑉_𝜂(𝑡)+𝜂𝑒^2𝐿𝑡=𝑉(𝑡). Let 𝑡_𝛿 be the supremum of all positive number 𝛿 such that 𝑉(0)<𝑊_𝜂(0) implies 𝑉(𝑡)<𝑊_𝜂(𝑡) on [0,𝜂]. Thus 𝑡_𝛿>0 and 𝑉(𝑡_𝛿)<𝑊_𝜂(𝑡_𝛿), by putting the above we infer that 𝑊_𝜂𝑡_𝛿=𝑊𝑡_𝛿+𝜂𝑒^{2𝐿(𝑡𝛿)}>𝑊𝑡_𝛿,𝑊_𝜂(0)=𝑊(0)+𝜂>𝑊(0). (3.8) Similarly, 𝑉_𝜂(𝑡_𝛿)<𝑉(𝑡_𝛿) and 𝑊_𝜂(0)<𝑉(0). By using Theorem 3.4, we have 𝑉_𝜂(𝑡_𝛿)<𝑉(𝑡_𝛿)<𝑊(𝑡_𝛿)<𝑊_𝜂(𝑡_𝛿). Since 𝑉,𝑊 are lower and upper solutions of (3.1), we have that 𝑉_𝜂(0)<𝑉(0)<𝑋(0)<𝑊(0)<𝑊_𝜂(0),𝑉_𝜂𝑡_𝛿<𝑋𝑡_𝛿<𝑊_𝜂𝑡_𝛿, (3.9) where 𝑋(𝑡_𝛿) is solution of (3.1). Now, we wish to show that 𝑉_𝜂(𝑡)<𝑋(𝑡)<𝑊_𝜂(𝑡) on 𝐼. If it is not true, then there exists a 𝑡₁∈𝐼 such that 𝑋(𝑡₁)=𝑊_𝜂(𝑡₁) and 𝑉_𝜂(𝑡)<𝑋(𝑡)<𝑊_𝜂(𝑡) on 𝐼⧵{𝑡₁}. This implies that 𝑋(𝑡₁)=𝑊_𝜂(𝑡₁)>𝑊(𝑡₁) and 𝐷_𝐻𝑋(𝑡₁)>𝐷_𝐻𝑊(𝑡₁). Equation (2.5), together with 𝐷_𝐻𝑋(𝑡₁)>𝐷_𝐻𝑊(𝑡₁), implies that there exists an ℎ>0 such that 𝑋𝑡₁−𝑊𝑡₁<𝑋(𝑡)−𝑊(𝑡),𝑡_𝛿<𝑡₁<𝑡_𝛿+ℎ. (3.10) This contradicts that 𝑋(𝑡₁)>𝑊(𝑡₁), hence we have that 𝑋(𝑡)<𝑊_𝜂(𝑡). Similarly, we can show that 𝑉_𝜂(𝑡)<𝑋(𝑡) and hence relation 𝑉_𝜂(𝑡)<𝑋(𝑡)<𝑊_𝜂(𝑡) holds for all 𝑡∈𝐼. Now as 𝜂→0, we conclude that 𝑉(𝑡)≤𝑋(𝑡)≤𝑊(𝑡). The proof is complete.

Definition 3.8. Let 𝑀(𝑡), 𝑀(𝑡) are said to be minimal and maximal solutions of (3.1), respectively, if they both are solution of (3.1) and satisfy 𝑀(𝑡)≤𝑋(𝑡)≤𝑀(𝑡) for every solution 𝑋(𝑡) of (3.1) with 𝑉(𝑡)≤𝑋(𝑡)≤𝑊(𝑡) for all 𝑡∈𝐼, where 𝑉,𝑊 are the lower and upper solutions of (3.1) respectively with 𝑉(𝑡)≤𝑊(𝑡) for all 𝑡∈𝐼.

Theorem 3.9. Assume that(M1) equation (3.1) has the lower solution 𝑉 and upper solution 𝑊 with 𝑉,𝑊∈𝐶¹[𝐼,𝐾_𝐶𝐶(𝑅^𝑛)] and 𝑉(𝑡)≤𝑊(𝑡) for all 𝑡∈𝐼;(M2) hypotheses (H1), (H3) satisfy;(M3)𝐹 is map bounded sets into bounded sets in 𝐾_𝐶𝐶(𝑅^𝑛).Then there exists monotone sequence {𝑊^𝑛(𝑡)} and {𝑉^𝑛(𝑡)} in 𝐾_𝐶𝐶(𝑅^𝑛) such that 𝑉^𝑛(𝑡)→𝑀(𝑡), 𝑊^𝑛(𝑡)→𝑀(𝑡) as 𝑛→∞ in 𝐾_𝐶𝐶(𝑅^𝑛), where 𝑀(𝑡), 𝑀(𝑡) are the minimal and maximal solutions of (3.1), respectively.

Proof. Let us construct the set of integrodifferential sequences by 𝑉^𝑛+1(𝑡)=𝑋₀+^𝑡₀𝐹𝑠,𝑉^𝑛(𝑠),𝐻𝑠,𝑉^𝑛(𝑠)(),𝑊0.1_𝑛+1(𝑡)=𝑋₀+^𝑡₀𝐹𝑠,𝑊^𝑛(𝑠),𝐻𝑠,𝑊^𝑛(𝑠)𝑑𝑠(0.2) (3.11) for 𝑛=0,1,2,…, we prescribe 𝑉⁰(𝑡)=𝑉(𝑡) and 𝑊⁰(𝑡)=𝑊(𝑡), for all 𝑡∈𝐼. From (2.5), (3.1) and using Definition 3.1 we get 𝐷_𝐻𝑉^𝑛+1(𝑡)=𝐹𝑡,𝑉^𝑛(𝑡),𝐻𝑡,𝑉^𝑛(𝑡),𝑉^𝑛+1(0)=𝑋₀(),𝐷0.4_𝐻𝑊^𝑛+1(𝑡)=𝐹𝑡,𝑊^𝑛(𝑡),𝐻𝑡,𝑊^𝑛(𝑡),𝑊^𝑛+1(0)=𝑋₀(0.5). (3.12) First, we claim that the iterations are such that 𝑉⁰≤𝑉¹≤⋯≤𝑉^𝑛≤𝑊^𝑛≤⋯≤𝑊¹≤𝑊⁰. (3.13) Now we show that 𝑉⁰(𝑡)≤𝑉¹(𝑡)≤𝑊¹(𝑡)≤𝑊⁰(𝑡). Consequently, we have to show that (i) 𝑉⁰(𝑡)≤𝑉¹(𝑡), (ii) 𝑊¹(𝑡)≤𝑊⁰(𝑡) and (iii) 𝑉¹(𝑡)≤𝑊¹(𝑡). By using Definition 3.6 and (3.11), (3.12), then (i) is proved. Indeed, by 𝑉⁰(𝑡)=𝑉(𝑡) is a lower solution of (3.1) and following Definition 3.6 we get 𝑉⁰(0)≤𝑋₀=𝑉¹(0), addition 𝐷_𝐻𝑉¹(𝑡)=𝐹𝑡,𝑉⁰(𝑡),𝐻𝑡,𝑉⁰(𝑡),𝐷_𝐻𝑉⁰(𝑡)≤𝐹𝑡,𝑉⁰(𝑡),𝐻𝑡,𝑉⁰(𝑡). (3.14) Hence 𝐷_𝐻𝑉¹(𝑡)≥𝐷_𝐻𝑉⁰(𝑡) and using Corollary 3.5 we infer 𝑉⁰(𝑡)≤𝑉¹(𝑡) for all 𝑡∈𝐼. Similarly, we use Definition 3.6 and (3.11), (3.12), then (ii) is proved. Using (M1), we get 𝐷_𝐻𝑉¹(𝑡)=𝐹𝑡,𝑉⁰(𝑡),𝐻𝑡,𝑉⁰(𝑡)≤𝐹𝑡,𝑊⁰(𝑡),𝐻𝑡,𝑊⁰(𝑡)=𝐷_𝐻𝑊¹(𝑡), (3.15) addition 𝑉¹(0)=𝑋₀=𝑊¹(0) and Corollary 3.5, then (iii) is proved.
By using inductive method, we assume 𝑉⁰≤𝑉^𝑛−1≤𝑉^𝑛≤𝑊^𝑛≤𝑊^𝑛−1≤𝑊⁰ on 𝐼, then we have to claim that 𝑉^𝑛≤𝑉^𝑛+1≤𝑊^𝑛+1≤𝑊^𝑛, by means of the monotone property of 𝐹,𝐺 we obtain 𝐷_𝐻𝑉^𝑛(𝑡)=𝐹𝑡,𝑉^𝑛−1(𝑡),𝐻𝑡,𝑉^𝑛−1(𝑡)≤𝐹𝑡,𝑉^𝑛(𝑡),𝐻𝑡,𝑉^𝑛(𝑡)=𝐷_𝐻𝑉^𝑛+1(𝑡),𝐷_𝐻𝑊^𝑛(𝑡)=𝐹𝑡,𝑊^𝑛−1(𝑡),𝐻𝑡,𝑊^𝑛−1(𝑡)≥𝐹𝑡,𝑊^𝑛(𝑡),𝐻𝑡,𝑊^𝑛(𝑡)=𝐷_𝐻𝑊^𝑛+1(𝑡). (3.16) From 𝑉^𝑛(0)=𝑉^𝑛+1(0)=𝑊^𝑛+1(0)=𝑊^𝑛(0)=𝑋₀, and by virtue of Corollary 3.5 we get 𝑉^𝑛≤𝑉^𝑛+1 and 𝑊^𝑛+1≤𝑊^𝑛 for all 𝑡∈𝐼. Again, by means of the monotone property of 𝐹,𝐺 and our assumption, we have 𝐷_𝐻𝑉^𝑛+1(𝑡)=𝐹𝑡,𝑉^𝑛(𝑡),𝐻𝑡,𝑉^𝑛(𝑡)≤𝐹𝑡,𝑊^𝑛(𝑡),𝐻𝑡,𝑊^𝑛(𝑡)=𝐷_𝐻𝑊^𝑛+1(𝑡), (3.17) for all 𝑡∈𝐼. Using again Corollary 3.5, we get 𝑉^𝑛+1≤𝑊^𝑛+1. Consequently, 𝑉⁰≤𝑉¹≤⋯≤𝑉^𝑛≤𝑊^𝑛≤⋯≤𝑊¹≤𝑊⁰. (3.18) Combining (3.11) and 𝐹 is continuous multiplication, it follows that 𝑉^𝑛, 𝑊^𝑛 are continuous for 𝑛=0,1,2,….
Now using the corresponding of (3.11) and the properties of the Hausdorff metric and the Hukuhara integral, together with the assumption (M3), we prove the equicontinuity of the sequences {𝑉^𝑛} and {𝑊^𝑛} below. Consider for any 𝑡<𝑡_ℎ,𝑡,𝑡_ℎ∈𝐼, we have 𝐷𝑉^𝑛𝑡_ℎ,𝑉^𝑛(𝑡)≤𝐷^𝑡ℎ_𝑡𝐹𝑠,𝑉^𝑛(𝑠),𝐻𝑡,𝑉^𝑛(𝑡)𝑑𝑠,𝜃^𝑛≤𝑀𝑡_ℎ−𝑡. (3.19) Hence {𝑉^𝑛} and {𝑊^𝑛} are uniformly bounded and equicontinuity on 𝐼. On using Ascoli-Arzela theorem (see [1]) in this setup, we obtain a subsequence {𝑉^𝑛𝑘}which converges uniformly to 𝑀(𝑡)∈𝐾_𝐶𝐶(𝑅^𝑛) on 𝐼. Arguing in a similarly to the {𝑉^𝑛}, we conclude that {𝑊^𝑛} converges uniformly to 𝑀(𝑡) on 𝐼. Next, we again consider (3.12), (3.18), respectively, and by using the convergence properties we infer that 𝐷_𝐻𝑀(𝑡)=𝐹𝑡,𝑀(𝑡),𝐻𝑡,𝑀(𝑡),𝑀(0)=𝑋₀,𝐷_𝐻𝑀(𝑡)=𝐹𝑡,𝑀(𝑡),𝐻𝑡,𝑀(𝑡),𝑀(0)=𝑋₀. (3.20) Moreover, by means of (3.18) we easily get 𝑉⁰(𝑡)≤𝑀(𝑡)≤𝑀(𝑡)≤𝑊⁰(𝑡) on 𝐼.
Finally, we show that 𝑀(𝑡) and 𝑀(𝑡) are the minimal and maximal solutions of (3.1), respectively. Let 𝑋(𝑡) be any solution of (3.1) such that 𝑉⁰(𝑡)≤𝑋(𝑡)≤𝑊⁰(𝑡) for all 𝑡∈𝐼 and 𝑉⁰(0)≤𝑋₀≤𝑊⁰(0) and we need to prove that 𝑉⁰(𝑡)≤𝑀(𝑡)≤𝑋(𝑡)≤𝑀(𝑡)≤𝑊⁰(𝑡) on 𝐼. Suppose that for some 𝑛, 𝑉^𝑛≤𝑋(𝑡)≤𝑊^𝑛 on 𝐼. By using monotone nondecreasing of 𝐹, 𝐺, we get 𝐷_𝐻𝑋(𝑡)=𝐹(𝑡,𝑋(𝑡),𝐻(𝑡,𝑋(𝑡)))≥𝐹𝑡,𝑉^𝑛(𝑡),𝐻𝑡,𝑉^𝑛(𝑡)=𝐷_𝐻𝑉^𝑛+1(𝑡), (3.21) where 𝑋₀=𝑉^𝑛+1(0). Applying Corollary 3.5, then we get 𝑋(𝑡)≤𝑉^𝑛+1(𝑡) on 𝐼 for all 𝑛. Similarly, we get 𝑉^𝑛+1≤𝑋(𝑡) for all 𝑛. By using assumption 𝑉⁰(𝑡)≤𝑋(𝑡)≤𝑊⁰(𝑡) from the principle of mathematical induction, we infer that 𝑉^𝑛≤𝑋(𝑡)≤𝑊^𝑛for all 𝑛. Taking limit as 𝑛→∞, then we obtain 𝑀(𝑡)≤𝑋(𝑡)≤𝑀(𝑡). The proof is complete.

Corollary 3.10. If addition to the assumptions of Theorem 3.7 assume that 𝐹 satisfies 𝐹𝑡,𝑋₁,𝐻𝑡,𝑋₁+𝐿₁𝑋₁−𝑋₂≥𝐹𝑡,𝑋₂,𝐻𝑡,𝑋₂ (3.22) for 𝑋₁≥𝑋₂∈𝐾_𝐶𝐶(𝑅^𝑛) and 𝐿₁>0, then 𝑀(𝑡)=𝑋(𝑡)=𝑀(𝑡) is the unique solution of (3.1).

Example 3.11. We consider set-valued differential equation with feedback control in 𝐾_𝐶𝐶(ℝ): 𝐷_𝐻𝑋(𝑡)(𝑡)(𝑡)𝑋(𝑡)(0)=[]𝜋=3𝑋+sin,𝑋1,2,𝑡∈0,2, (3.23) where 𝐹(𝑡,𝑋(𝑡),𝑈(𝑡))=3𝑋(𝑡)+sin(𝑡)𝑋(𝑡)with 𝐻(𝑡,𝑋(𝑡))=sin(𝑡)𝑋(𝑡)is a contraction feedback.
We see that 𝐹(𝑡,𝑋(𝑡),𝐻(𝑡,𝑋(𝑡)))satisfies (M1)–(M3). Now, we show that (3.23) exits as extremal solutions on [0,𝜋/2]. We prescribe 𝑉(𝑡)=𝑋₀𝑒^3𝑡−𝜀, 𝑊(𝑡)=𝑋₀𝑒^4𝑡+𝜀 are lower and upper solutions of (3.23) for all 𝜀>0. We note that 𝐷_𝐻𝑉(𝑡)≤3𝑉(𝑡)+sin(𝑡)𝑉(𝑡) and 𝐷_𝐻𝑊(𝑡)≥3𝑊(𝑡)+sin(𝑡)𝑊(𝑡). Next, let us construct the set sequences by 𝐷_𝐻𝑉^𝑛+1(𝑡)=3𝑉^𝑛(𝑡)−𝜀𝑛,𝑉^𝑛+1(0)=𝑋₀−𝜀𝑛,𝐷_𝐻𝑊^𝑛+1(𝑡)=4𝑊^𝑛(𝑡)+𝜀𝑛,𝑊^𝑛+1(0)=𝑋₀+𝜀𝑛, (3.24) for all 𝑛=0,1,2,… we verify that monotone sequences of constructions above such that (a){𝑉^𝑛(𝑡)}−−−−^𝑛→∞→𝑀(𝑡)and 𝑀(𝑡) is a minimal of (3.23);(b){𝑊^𝑛(𝑡)}−−−−^𝑛→∞→𝑀(𝑡)and 𝑀(𝑡) is a maximal of (3.23).First, we prove (a). Indeed, let 0<𝜀₂<𝜀₁≤𝜀, then for each positive integer 𝑛, we consider 𝐷_𝐻𝑊^𝑛+1(𝑡)=4𝑊^𝑛(𝑡)+𝜀𝑛,𝑊^𝑛+1(0)=𝑋₀+𝜀𝑛,𝑊(𝑡)=𝑋₀𝑒^4𝑡+𝜀. (3.25) Because 𝑊^𝑛+1(0,𝜀₂)≤𝑊^𝑛+1(0,𝜀₁) and 𝐷_𝐻𝑊^𝑛+1(𝑡,𝜀₂)≤4𝑊^𝑛(𝑡)+𝜀₂/𝑛, otherwise 𝐷_𝐻𝑊^𝑛+1(𝑡,𝜀₁)≥4𝑊^𝑛(𝑡)+𝜀₂/𝑛, hence 𝐷_𝐻𝑊^𝑛+1(𝑡,𝜀₁)≥𝐷_𝐻𝑊^𝑛+1(𝑡,𝜀₂). By using Corollary 3.5, to get 𝑊^𝑛+1(𝑡,𝜀₂)≤𝑊^𝑛+1(𝑡,𝜀₁)for all 𝑡∈[0,𝜋/2] and 𝜀₂<𝜀₁. On the other hand 𝑊^𝑛+1(𝑡)≤𝑊^𝑛(𝑡) with 𝜀 fixed.
Since the family of functions {𝑊^𝑛+1(𝑡,𝜀)} is equicontinuous and uniformly bounded on [0,𝜋/2], it follows Ascoli-Arzela theorem (see [4]) that there exists a decreasing sequence {𝜀/(𝑛+1)_𝑘}−−−−^𝑘→∞→0and uniform limit 𝑀(𝑡)=lim_𝑘→∞𝑊^(𝑛+1)𝑘(𝑡,𝜀)exits on [0,𝜋/2]. Obviously 𝑀(𝑡)=𝑋₀⋅𝑒^4𝑡, the uniform continuity of 𝐹 implies that 𝐹(𝑡,𝑊^(𝑛+1)𝑘(𝑡,𝜀),𝐻(𝑡,𝑊^(𝑛+1)𝑘(𝑡,𝜀))) tends uniformly to 𝐹(𝑡,𝑀(𝑡),𝐻(𝑡,𝑀(𝑡)))as 𝑘→∞, and thus 𝑊^(𝑛+1)𝑘(𝑡,𝜀)=𝑋₀+𝜀𝑛_𝑘+₀^𝜋/24𝑊^(𝑛+1)𝑘(𝑠,𝜀)𝑑𝑠. (3.26) Which in turn yields that the limit 𝑀(𝑡) is a solution of (3.23) on [0,𝜋/2].
Next we will show that 𝑀(𝑡) is a required maximal solution of (3.23) on [0,𝜋/2]. For this purpose, we observe that 𝑋(0)=𝑋₀<𝑋₀+𝜀/𝑛=𝑊^𝑛+1(0,𝜀) and 𝐹 is nondecreasing, hence we get 𝐷_𝐻𝑋(𝑡)(𝑡)(𝑡)𝑋(𝑡)+𝜀<3𝑋+sin𝑛≤3𝑊^𝑛(𝑡,𝜀)+sin(𝑡)𝑊^𝑛()+𝜀𝑡,𝜀𝑛≤𝐷_𝐻𝑊^𝑛+1(𝑡,𝜀). (3.27)
By using Corollary 3.5, then we get 𝑋(𝑡)<𝑊^𝑛+1(𝑡,𝜀) on [0,𝜋/2]. The uniqueness of maximal solution 𝑀(𝑡) show that 𝑊^𝑛+1(𝑡,𝜀)tends uniformly to 𝑀(𝑡)is the maximal solution of (3.23) with 𝑀(𝑡)=lim_𝑘→∞𝑊^(𝑛+1)𝑘(𝑡,𝜀). (3.28) Finally, we will prove (b). Similarly, let 0<𝜀₂<𝜀₁≤𝜀, then for each positive integer 𝑛, we consider 𝐷_𝐻𝑉^𝑛+1(𝑡)=3𝑉^𝑛(𝑡)−𝜀𝑛,𝑉^𝑛+1(0)=𝑋₀−𝜀𝑛,𝑉(𝑡)=𝑋₀𝑒^3𝑡−𝜀. (3.29) Because 𝑉^𝑛+1(0,𝜀₁)≤𝑉^𝑛+1(0,𝜀₂) and 𝐷_𝐻𝑉^𝑛+1(𝑡,𝜀₂)≤3𝑉^𝑛(𝑡)−𝜀₂/𝑛, otherwise 𝐷_𝐻𝑉^𝑛+1(𝑡,𝜀₁)≤3𝑉^𝑛(𝑡)−𝜀₂/𝑛, hence 𝐷_𝐻𝑉^𝑛+1(𝑡,𝜀₁)≤𝐷_𝐻𝑉^𝑛+1(𝑡,𝜀₂). By using Corollary 3.5, to get 𝑉^𝑛+1(𝑡,𝜀₁)≤𝑉^𝑛+1(𝑡,𝜀₂)for all 𝑡∈[0,𝜋/2] and 𝜀₂<𝜀₁. On the other 𝑉^𝑛(𝑡)≤𝑉^𝑛+1(𝑡) with 𝜀 fixed.
Since the family of functions {𝑉^𝑛+1(𝑡,𝜀)} is equicontinuous and uniformly bounded on [0,𝜋/2], it follow by Ascoli-Arzela theorem (see [4]) that there exists a decreasing sequence {𝜀/(𝑛+1)_𝑘}−−−−^𝑘→∞→0and uniform limit 𝑀(𝑡)=lim_𝑘→∞𝑉^(𝑛+1)𝑘(𝑡,𝜀)exits on [0,𝜋/2]. Obviously 𝑀(𝑡)=𝑋₀⋅𝑒^3𝑡, the uniform continuity of 𝐹 implies that 𝐹(𝑡,𝑉^(𝑛+1)𝑘(𝑡,𝜀),𝐻(𝑡,𝑉^(𝑛+1)𝑘(𝑡,𝜀))) tends uniformly to 𝐹(𝑡,𝑀(𝑡),𝐻(𝑡,𝑀(𝑡)))as 𝑘→∞, and thus 𝑉^(𝑛+1)𝑘(𝑡,𝜀)=𝑋₀+𝜀𝑛_𝑘+₀^𝜋/23𝑉^(𝑛+1)𝑘(𝑠,𝜀)𝑑𝑠 (3.30) which in turn yields that the limit 𝑀(𝑡) is a solution of (1) on [0,𝜋/2].
Next we will show that 𝑀(𝑡) is a required minimal solution of (3.23) on [0,𝜋/2]. For this purpose, we observe that 𝑋(0)=𝑋₀>𝑋₀−𝜀/𝑛=𝑉^𝑛+1(0,𝜀) and 𝐹 is nondecreasing, hence we get 𝐷_𝐻𝑋(t)(𝑡)(𝑡)𝑋(𝑡)−𝜀>3𝑋+sin𝑛≥3𝑉^𝑛(𝑡,𝜀)+sin(𝑡)𝑉^𝑛()−𝜀𝑡,𝜀𝑛≥𝐷_𝐻𝑉^𝑛+1(𝑡,𝜀). (3.31)

By using Corollary 3.5, then we get 𝑉^𝑛+1(𝑡,𝜀)<𝑋(𝑡) on [0,𝜋/2]. The uniqueness of minimal solution 𝑀(𝑡) show that 𝑉^𝑛+1(𝑡,𝜀) tends uniformly to 𝑀(𝑡) is the minimal solution of (3.23) with𝑀(𝑡)=lim_𝑘→∞𝑉^(𝑛+1)𝑘(𝑡,𝜀). (3.32)

Based on (3.25) combining (3.28), (3.29) and (3.32), we will solve the minimal and maximal solutions of (3.23). Its graphical representation can be seen in Figure 1.

Figure 1 

The minimal and maximal solutions of (3.23).

Acknowledgments

The authors gratefully acknowledge the referees for their careful reading and many valuable remarks which improved the presentation of the paper.