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

Authors: Ngo Van Hoa; Nguyen Dinh Phu
Journal: Abstract and Applied Analysis
 (2012)

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## 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 formDHX(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.


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## Body

## 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:(1.1)DHX(t)=F(t,X(t)),
where X(t0)=X0, X(t)∈KCC(Rn), 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 (Kc(Rn),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 Section2, 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 KCC(Rn) denote the collection of all nonempty compact convex subsets of Rn. Given A,B∈KCC(Rn), the Hausdorff distance between A and B is defined by(2.1)D[A,B]=max⁡{sup⁡a∈Ainf⁡b∈B∥a-b∥Rn,sup⁡b∈Binf⁡a∈A∥b-a∥Rn},
where ∥·∥Rn denotes the Euclidean norm in Rn and {θn}—the zero points set in KCC(Rn). It is known that (KCC(Rn),D) is a complete metric space and KCC(Rn) is a complete and separable with respect to metric D.We define the magnitude of a nonempty subset ofA:(2.2)D[A,θn]=∥A∥=sup⁡{∥a∥Rn,a∈A},
where θn is the zero element of Rn which is regarded as a one point set. ∥A∥=D[A,θn]-norm in KCC(Rn) is finite when the supremum in (2.2) is attained with A∈KCC(Rn).The Hausdorff metric (1.1) satisfies the properties below:(2.3)D[A+C,B+C]=D[A,B],D[A,B]=D[B,A],D[λA,λB]=λD[A,B],D[A,B]≤D[A,C]+D[C,B],D[A+A′,B+B′]≤D[A,B]+D[A′,B′]for allA,B,C,A′,B′∈KCC(Rn) and λ∈R+. If α,β∈R, and A,B∈KCC(Rn), then(2.4)α(A+B)=αA+αB,α(βA)=(αβ)A,1⋅A=A.
It is known that (KCC(Rn),D) is a complete metric space and if the space KCC(Rn) is equipped with the natural algebraic operations of addition and nonnegative scalar multiplication, then KCC(Rn) becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space.LetA,B∈KCC(Rn). The set C∈KCC(Rn) 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→KCC(Rn) has a Hukuhara derivative DHF(t0) at a point t0∈I, if(2.5)lim⁡h→0+h-1[F(t0+h)-F(t0)],lim⁡h→0+h-1[F(t0)-F(t0-h)]
exist in the topology of KCC(Rn) and are equal to DHF(t0).By embeddingKCC(Rn) as a complete cone in a corresponding Banach space and taking into account result on the differentiation of Bochner integral, we find that if(2.6)F(t)=X0+∫t0tΦ(s)ds,X0∈KCC(Rn),where Φ:I→KCC(Rn) is integrable in the sense of Bochner, then DHF(t) exists and the equality DHF(t)=Φ(t) a.e on I holds.The Hukuhara integral ofF is given by(2.7)∫IF(s)ds=cl⁡[∫If(s)ds:fisacontinuousselectorofF]
for any compact set I⊂R+.Some properties of the Hukuhara integral are in [1]. If F:I→KCC(Rn) is integrable, one has(2.8)∫t0t2F(s)ds=∫t0t1F(s)ds+∫t1t2F(s)ds,t0≥t1≥t2,∫t0tλF(s)ds=λ∫t0tF(s)ds,λ∈R.
If F,G:I→KCC(Rn) are integrable, then D[F(·),G(·)]:I→R is integrable and(2.9)D[∫t0tF(s)ds,∫t0tG(s)ds]≤∫t0tD[F(s),G(s)]ds.


## 3. Main Results

We consider the set-valued differential equations (SSDEs) with feedback control under the form(3.1)DHX(t)=F(t,X(t),H(t,X(t))),X(0)=X0,∀t∈[0,T],
where F:[0,T]×KCC(Rn)×KCC(Rd)→KCC(Rn) and H:[0,T]×KCC(Rn)→KCC(Rd) is a feedback control, state set X0,X(t)∈KCC(Rn).Definition 3.1.
The mapping setX:[0,T]→KCC(Rn) 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],KCC(Rn)] with Hukuhara derivative DH by t;(ii)
X(0)=X0;(iii)
F(t,X(t),H(t,X(t)) is integrable on [0,T];(iv)
for allt∈[0,T],(3.2)X(t)=X0+∫0tF(S,X(s),H(s,X(s)))ds
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)
byK+(K0,+) the subfamily of KCC(Rn) consisting of sets X∈KCC(Rn) such that any x∈X is a nonnegative (positive) vector of n components satisfying xi≥0(xi>0) for i=1,2,…,n,(ii)
byK-(K0,-) the subfamily of KCC(Rn) consisting of sets X∈KCC(Rn) such that any x∈X is a nonpostive (negative) vector of n components satisfying xi≤0(xi<0) for i=1,2,…,n.By Definition 3.2, we notice that K+ is a positive cone in KCC(Rn) and K0,+ is the nonempty interior of K+. K- is a negative cone in KCC(Rn) and K0,- is the nonempty interior of K-. We can therefore induce a partial odering in KCC(Rn). Thus, if X∈K+(K0,+) is X≥θn(X>θn), that is, with any x∈X is satisfying xi≥0(xi>0) for i=1,2,…,n and X∈K-(K0,-) is X≤θn(X<θn), that is, with any x∈X is satisfying xi≤0(xi<0) for i=1,2,…,n. Now we define the ordering in KCC(Rn).Definition 3.3.
For anyX,Y∈KCC(Rn), if there exists a Z∈KCC(Rn) such that Z∈K+(K0,+) and X=Y+Z, then we write X≥Y(X>Y). Similarly, if there exists a Z¯∈KCC(Rn) such that Z¯∈K-(K0,-) and X=Y+Z¯, then we write X≤Y(X<Y).Theorem 3.4.
Assume the following:(H1)
F(·,X(·),U(·))∈C[[0,T],KCC(Rn)] is monotone nondecreasing in (X,U)∈KCC(Rn)×KCC(Rd) 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],KCC(Rn)] is monotone nondecreasing in X∈KCC(Rn) for every t∈[0,T], that is, for fixed t∈[0,T], H(t,X1)≥H(t,X2) wherever X1≥X2;(H2)
there existV,W∈C1[I,KCC(Rn)] such that DHV(t)≤F(t,V(t),H(t,V(t))) and(3.3)DHW(t)≥F(t,W(t),H(t,W(t)));(H3)
for anyX1,X2∈KCC(Rn) with X1≥X2 and some positive number real L>0 such that(3.4)F(t,X1,H(t,X1))≤F(t,X2,H(t,X2))+L(X1-X2);
then V(t)≤W(t) for t∈I provided V(0)≤W(0).Proof.
For anyη=(η1,η2,…,ηn)>θn, we define Wη(t)=W(t)+ηe2Lt and we note that V(0)≤W(0)≤Wη(0). By using (2.5), we infer DHWη(t)=DHW(t)+2Lηe2Lt. Let tδ be the supremum of all positive number δ such that V(0)<Wη(0) implies V(t)<Wη(t) on [0,δ]. Thus tδ>0 and V(tδ)<Wη(tδ). Using (H1)–(H3), we get
(3.5)DHV(tδ)≤F(tδ,V(tδ),H(tδ,V(tδ)))≤F(tδ,Wη(tδ),H(tδ,Wη(tδ)))≤F(tδ,W(tδ),H(tδ,W(tδ)))+L⋅(Wη(tδ)-W(tδ))≤F(tδ,W(tδ),H(tδ,W(tδ)))+Lηe2Ltδ≤DHW(tδ)+2Lηe2Ltδ=DHWη(tδ).
Equation (2.5), together with (3.5), implies that there exists an h>0 such that
(3.6)Wη(tδ)-V(tδ)<Wη(t)-V(t),tδ<t<tδ+h.
This contradicts that tδ>0 is the supremum in view of the continuity of function involved and consequently that the inequality V(t)<Wη(t) holds for t∈I. Taking the limit η→0 yields the desired result. This proof is complete.Corollary 3.5.
LetV,W∈C1[I,KCC(Rn)] such that DHV(t)≤DHW(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 Theorem3.4.Definition 3.6.
V,W∈C1[I,KCC(Rn)] are said to be the lower solution and upper solution of (3.1) respectively if
(3.7)DHV(t)≤F(t,V(t),H(t,V(t))),V(0)≤X(0),DHW(t)≥F(t,W(t),H(t,W(t))),W(0)≥X(0).Theorem 3.7 (existence of solution).
AssumeV,W∈C1[I,KCC(Rn)] 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 anyη=(η1,η2,…,ηn)>θn, we define Wη(t)=W(t)+ηe2Lt and Vη(t)+ηe2Lt=V(t). Let tδ be the supremum of all positive number δ such that V(0)<Wη(0) implies V(t)<Wη(t) on [0,η]. Thus tδ>0 and V(tδ)<Wη(tδ), by putting the above we infer that
(3.8)Wη(tδ)=W(tδ)+ηe2L(tδ)>W(tδ),Wη(0)=W(0)+η>W(0).
Similarly, Vη(tδ)<V(tδ) and Wη(0)<V(0). By using Theorem 3.4, we have Vη(tδ)<V(tδ)<W(tδ)<Wη(tδ). Since V,W are lower and upper solutions of (3.1), we have that
(3.9)Vη(0)<V(0)<X(0)<W(0)<Wη(0),Vη(tδ)<X(tδ)<Wη(tδ),
where X(tδ) is solution of (3.1). Now, we wish to show that Vη(t)<X(t)<Wη(t) on I. If it is not true, then there exists a t1∈I such that X(t1)=Wη(t1) and Vη(t)<X(t)<Wη(t) on I∖{t1}. This implies that X(t1)=Wη(t1)>W(t1) and DHX(t1)>DHW(t1). Equation (2.5), together with DHX(t1)>DHW(t1), implies that there exists an h>0 such that
(3.10)X(t1)-W(t1)<X(t)-W(t),tδ<t1<tδ+h.
This contradicts that X(t1)>W(t1), hence we have that X(t)<Wη(t). Similarly, we can show that Vη(t)<X(t) and hence relation Vη(t)<X(t)<Wη(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.
LetM̲(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 M̲(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,KCC(Rn)] and V(t)≤W(t) for all t∈I;(M2)
hypotheses (H1), (H3) satisfy;(M3)
F is map bounded sets into bounded sets in KCC(Rn).Then there exists monotone sequence {Wn(t)} and {Vn(t)} in KCC(Rn) such that Vn(t)→M̲(t), Wn(t)→M¯(t) as n→∞ in KCC(Rn), where M̲(t), M¯(t) are the minimal and maximal solutions of (3.1), respectively.Proof.
Let us construct the set of integrodifferential sequences by(3.11)Vn+1(t)=X0+∫0tF(s,Vn(s),H(s,Vn(s)))(0.1),Wn+1(t)=X0+∫0tF(s,Wn(s),H(s,Wn(s)))ds(0.2)
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
(3.12)DHVn+1(t)=F(t,Vn(t),H(t,Vn(t))),Vn+1(0)=X0(0.4),DHWn+1(t)=F(t,Wn(t),H(t,Wn(t))),Wn+1(0)=X0(0.5).
First, we claim that the iterations are such that
(3.13)V0≤V1≤⋯≤Vn≤Wn≤⋯≤W1≤W0.
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
(3.14)DHV1(t)=F(t,V0(t),H(t,V0(t))),DHV0(t)≤F(t,V0(t),H(t,V0(t))).
Hence DHV1(t)≥DHV0(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
(3.15)DHV1(t)=F(t,V0(t),H(t,V0(t)))≤F(t,W0(t),H(t,W0(t)))=DHW1(t),
addition V1(0)=X0=W1(0) and Corollary 3.5, then (iii) is proved.
By using inductive method, we assumeV0≤Vn-1≤Vn≤Wn≤Wn-1≤W0 on I, then we have to claim that Vn≤Vn+1≤Wn+1≤Wn, by means of the monotone property of F,G we obtain
(3.16)DHVn(t)=F(t,Vn-1(t),H(t,Vn-1(t)))≤F(t,Vn(t),H(t,Vn(t)))=DHVn+1(t),DHWn(t)=F(t,Wn-1(t),H(t,Wn-1(t)))≥F(t,Wn(t),H(t,Wn(t)))=DHWn+1(t).
From Vn(0)=Vn+1(0)=Wn+1(0)=Wn(0)=X0, and by virtue of Corollary 3.5 we get Vn≤Vn+1 and Wn+1≤Wn for all t∈I. Again, by means of the monotone property of F,G and our assumption, we have
(3.17)DHVn+1(t)=F(t,Vn(t),H(t,Vn(t)))≤F(t,Wn(t),H(t,Wn(t)))=DHWn+1(t),
for all t∈I. Using again Corollary 3.5, we get Vn+1≤Wn+1. Consequently,
(3.18)V0≤V1≤⋯≤Vn≤Wn≤⋯≤W1≤W0.
Combining (3.11) and F is continuous multiplication, it follows that Vn, Wn 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 {Vn} and {Wn} below. Consider for any t<th,t,th∈I, we have
(3.19)D[Vn(th),Vn(t)]≤D[∫tthF(s,Vn(s),H(t,Vn(t)))ds,θn]≤M(th-t).
Hence {Vn} and {Wn} are uniformly bounded and equicontinuity on I. On using Ascoli-Arzela theorem (see [1]) in this setup, we obtain a subsequence {Vnk}which converges uniformly to M̲(t)∈KCC(Rn) on I. Arguing in a similarly to the {Vn}, we conclude that {Wn} 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
(3.20)DHM¯(t)=F(t,M¯(t),H(t,M¯(t))),M¯(0)=X0,DHM̲(t)=F(t,M̲(t),H(t,M̲(t))),M̲(0)=X0.
Moreover, by means of (3.18) we easily get V0(t)≤M̲(t)≤M¯(t)≤W0(t) on I.
Finally, we show thatM̲(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)≤M̲(t)≤X(t)≤M¯(t)≤W0(t) on I. Suppose that for some n, Vn≤X(t)≤Wn on I. By using monotone nondecreasing of F, G, we get
(3.21)DHX(t)=F(t,X(t),H(t,X(t)))≥F(t,Vn(t),H(t,Vn(t)))=DHVn+1(t),
where X0=Vn+1(0). Applying Corollary 3.5, then we get X(t)≤Vn+1(t) on I for all n. Similarly, we get Vn+1≤X(t) for all n. By using assumption V0(t)≤X(t)≤W0(t) from the principle of mathematical induction, we infer that Vn≤X(t)≤Wnfor all n. Taking limit as n→∞, then we obtain M̲(t)≤X(t)≤M¯(t). The proof is complete.Corollary 3.10.
If addition to the assumptions of Theorem3.7 assume that F satisfies
(3.22)F(t,X1,H(t,X1))+L1(X1-X2)≥F(t,X2,H(t,X2))
for X1≥X2∈KCC(Rn) and L1>0, then M̲(t)=X(t)=M¯(t) is the unique solution of (3.1).Example 3.11.
We consider set-valued differential equation with feedback control inKCC(ℝ):
(3.23)DHX(t)=3X(t)+sin⁡(t)X(t),X(0)=[1,2],t∈[0,π2],
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 thatF(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)=X0e3t-ε, W(t)=X0e4t+ε are lower and upper solutions of (3.23) for all ε>0. We note that DHV(t)≤3V(t)+sin(t)V(t) and DHW(t)≥3W(t)+sin(t)W(t). Next, let us construct the set sequences by
(3.24)DHVn+1(t)=3Vn(t)-εn,Vn+1(0)=X0-εn,DHWn+1(t)=4Wn(t)+εn,Wn+1(0)=X0+εn,
for all n=0,1,2,… we verify that monotone sequences of constructions above such that (a)
{Vn(t)}→n→∞M̲(t)and M̲(t) is a minimal of (3.23);(b)
{Wn(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
(3.25)DHWn+1(t)=4Wn(t)+εn,Wn+1(0)=X0+εn,W(t)=X0e4t+ε.
Because Wn+1(0,ε2)≤Wn+1(0,ε1) and DHWn+1(t,ε2)≤4Wn(t)+ε2/n, otherwise DHWn+1(t,ε1)≥4Wn(t)+ε2/n, hence DHWn+1(t,ε1)≥DHWn+1(t,ε2). By using Corollary 3.5, to get Wn+1(t,ε2)≤Wn+1(t,ε1)for all t∈[0,π/2] and ε2<ε1. On the other hand Wn+1(t)≤Wn(t) with ε fixed.
Since the family of functions{Wn+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)=limk→∞W(n+1)k(t,ε)exits on [0,π/2]. Obviously M¯(t)=X0·e4t, 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
(3.26)W(n+1)k(t,ε)=X0+εnk+∫0π/2(4W(n+1)k(s,ε))ds.
Which in turn yields that the limit M¯(t) is a solution of (3.23) on [0,π/2].
Next we will show thatM¯(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=Wn+1(0,ε) and F is nondecreasing, hence we get
(3.27)DHX(t)<3X(t)+sin⁡(t)X(t)+εn≤3Wn(t,ε)+sin⁡(t)Wn(t,ε)+εn≤DHWn+1(t,ε).
By using Corollary3.5, then we get X(t)<Wn+1(t,ε) on [0,π/2]. The uniqueness of maximal solution M¯(t) show that Wn+1(t,ε)tends uniformly to M¯(t)is the maximal solution of (3.23) with
(3.28)M¯(t)=lim⁡k→∞W(n+1)k(t,ε).
Finally, we will prove (b). Similarly, let 0<ε2<ε1≤ε, then for each positive integer n, we consider
(3.29)DHVn+1(t)=3Vn(t)-εn,Vn+1(0)=X0-εn,V(t)=X0e3t-ε.
Because Vn+1(0,ε1)≤Vn+1(0,ε2) and DHVn+1(t,ε2)≤3Vn(t)-ε2/n, otherwise DHVn+1(t,ε1)≤3Vn(t)-ε2/n, hence DHVn+1(t,ε1)≤DHVn+1(t,ε2). By using Corollary 3.5, to get Vn+1(t,ε1)≤Vn+1(t,ε2)for all t∈[0,π/2] and ε2<ε1. On the other Vn(t)≤Vn+1(t) with ε fixed.
Since the family of functions{Vn+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 M̲(t)=limk→∞V(n+1)k(t,ε)exits on [0,π/2]. Obviously M̲(t)=X0·e3t, 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,M̲(t),H(t,M̲(t)))as k→∞, and thus
(3.30)V(n+1)k(t,ε)=X0+εnk+∫0π/2(3V(n+1)k(s,ε))ds
which in turn yields that the limit M̲(t) is a solution of (1) on [0,π/2].
Next we will show thatM̲(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=Vn+1(0,ε) and F is nondecreasing, hence we get
(3.31)DHX(t)>3X(t)+sin⁡(t)X(t)-εn≥3Vn(t,ε)+sin⁡(t)Vn(t,ε)-εn≥DHVn+1(t,ε).By using Corollary3.5, then we get Vn+1(t,ε)<X(t) on [0,π/2]. The uniqueness of minimal solution M̲(t) show that Vn+1(t,ε) tends uniformly to M̲(t) is the minimal solution of (3.23) with(3.32)M̲(t)=lim⁡k→∞V(n+1)k(t,ε).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).



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*Source: 816218-2012-02-21.xml*