BOUNDARY VALUE PROBLEM FOR A FIRST ORDER MULTIVALUED DIFFERENTIAL SYSTEM
AURELIAN CERNEA
The existence of solutions of a boundary value problem for a first order multivalued differential system is investigated. New results are obtained when the set-valued map that defines the problem is lipschitzian or it is lower semicontinuous.
AMS 2000 Subject Classification: 34A60.
Key words: boundary value problem, differential inclusion, fixed point.
1. INTRODUCTION
In this paper we study the following boundary value problem for first order differential inclusions
(1.1) x0 ∈A(t)x+F(t, x), a.e.(I), M x(0) +N x(1) =η,
where I = [0,1], F(·,·) : I × Rn → P(Rn) is a set-valued map, A(·) is a continuous (n×n) matrix function, M and N are (n×n) constant real matrices and η∈Rn.
In the case whenF(·,·) is assumed to be upper semicontinuous, several existence results for problem (1.1) are obtained in [2] by using topological transversality arguments, fixed point theorems and differential inequalities.
In [4] are provided two other existence results obtained by the application of the set-valued contraction principle which allowed to avoid the assumption that the values ofF(·,·) are convex which is an essential hypotheses in [2].
The aim of the present paper is twofold. On one hand, in the case when F(t,·) is lower semicontinuous with nonconvex values, by Schaefer’s fixed point theorem ([7]) combined with a selection theorem of Bressan Colombo ([3]) for lower semicontinuous set-valued maps with decomposable values we establish an existence result for problem (1.1).
On the other hand, when F(t,·) is Lipschitz we show that Filippov’s ideas ([5]) can be suitably adapted in order to obtain the existence of solu- tions to problem (1.1). This approach allows to improve the main result in [4].
Recall that for a differential inclusion defined by a lipschitzian set-valued map
MATH. REPORTS12(62),1 (2010), 1–8
with nonconvex values, Filippov’s theorem ([5]) consists in proving the exis- tence of a solution starting from a given almost solution. Moreover, the result provides an estimate between the starting almost solution and the solution of the differential inclusion.
For the motivation of study of problem (1.1) we refer to [2] and refe- rences therein.
The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the sequel and in Section 3 we prove our main results.
2. PRELIMINARIES
Let (X, d) be a metric space and letI ⊂Rbe a compact interval. Denote by L(I) the σ-algebra of all Lebesgue measurable subsets of I, by P(X) the family of all nonempty subsets of X and byB(X) the family of all Borel subsets of X. If A ⊂I then χA(·) : I → {0,1} denotes the characteristic function of A. For any subsetA⊂X we denote by A the closure ofA.
Recall that the Pompeiu-Hausdorff distance of the closed subsetsA, B⊂ X is defined by
dH(A, B) = max{d∗(A, B),d∗(B, A)}, d∗(A, B) = sup{d(a, B);a∈A}, where d(x, B) = infy∈Bd(x, y).
As usual, if (X,| · |) is a normed space we denote byC(I, X) the Banach space of all continuous functionsx(·) :I →Xendowed with the norm|x(·)|C = supt∈I|x(t)| and by L1(I, X) the Banach space of all (Bochner) integrable functions x(·) :I →X endowed with the norm |x(·)|1=R
I|x(t)|dt.
A subsetD⊂L1(I, X) is said to bedecomposableif for anyu(·), v(·)∈D and any subset A∈ L(I) one hasuχA+vχB ∈D, where B=I\A.
Consider T : X → P(X) a set-valued map. A point x ∈ X is called a fixed point for T(·) if x ∈T(x). T(·) is said to be bounded on bounded sets if T(B) :=S
x∈BT(x) is a bounded subset of X for all bounded sets B inX.
T(·) is said to be compact if T(B) is relatively compact for any bounded sets B inX. T(·) is said to be totally compact ifT(X) is a compact subset ofX.
T(·) is said to be upper semicontinuous if for any open set D ⊂ X, the set {x∈X;T(x)⊂D} is open inX. T(·) is called completely continuous if it is upper semicontinuous and totally bounded on X.
We recall the following fixed point theorem due to Schaefer (e.g., [7]).
Theorem 2.1. Let X be a normed linear space,S ⊂X a convex set and 0∈S. Assume that H:S →S is a completely continuous operator.
If the set
(H) :={x∈S; x=λH(x) for some λ∈(0,1)}
is bounded, then H has at least one fixed point in S.
We recall that a multifunction T(·) : X → P(X) is said to be lower semicontinuous if for any closed subsetC ⊂X, the subset{s∈X;G(s)⊂C}
is closed.
If F(·,·) : I ×Rn → P(Rn) is a set-valued map with compact values and x(·)∈C(I,Rn) we define
SF(x) :={f ∈L1(I,Rn); f(t)∈F(t, x(t)) a.e. (I)}.
We say that F(·,·) is of lower semicontinuous type ifSF(·) is lower semicon- tinuous with closed and decomposable values.
Theorem 2.2 ([3]). Let S be a separable metric space and G(·) :S → P(L1(I,Rn)) be a lower semicontinuous set-valued map with closed decompo- sable values.
ThenG(·)has a continuous selection(i.e., there exists a continuous map- ping g(·) :S→L1(I,Rn) such that g(s)∈G(s) ∀s∈S).
Let I = [0,1], let |x| be the norm of x ∈ Rn and kAk be the norm of any matrix A. As usual, we denote by AC(I,Rn) the space of absolutely continuous from I toRn. OnAC(I,Rn) we consider the norm | · |C.
A functionx(·)∈AC(I,Rn) is called a solution of problem (1.1) if there exists a function f(·)∈L1(I,Rn) withf(t)∈F(t, x(t)) a.e. (I) such that (2.1) x0(t) =A(t)x(t) +f(t), a.e.(0,1), M x(0) +N x(1) =η.
Let Φ(·) be a fundamental matrix solution of the differential equations x0 =A(t)x that satisfy Φ(0) =In, where In is the (n×n) identity matrix.
The next result is well known (e.g., [2]).
Lemma 2.3 ([2]). If f(·) : [0,1]→Rn is an integrable function then the problem
(2.2) x0(t) =A(t)x(t) +f(t), a.e.(0,1), M x(0) +N x(1) = 0 has a unique solution provided det(M+NΦ(1))6= 0. This solution is given by
x(t) = Z 1
0
G(t, s)f(s)ds,
with G(·,·) the Green function associated to problem(2.2). Namely, (2.3) G(t, s) =
( Φ(t)J(s) if 0≤t≤s,
Φ(t)Φ(s)−1+ Φ(t)J(s) if s≤t≤1, where J(t) =−(M+NΦ(1))−1NΦ(1)Φ(t)−1.
If we consider the problem with nonhomogeneous boundary conditions, i.e., problem (2.1), then it is easy to verify that its solution is given by (2.4) x(t) = Φ(t)(M +NΦ(1))−1η+
Z 1 0
G(t, s)f(s)ds.
In the sequel we assume thatA(·) is a continuous (n×n) matrix function, M and N are (n×n) constant real matrices such that det(M+NΦ(1))6= 0 and we set G0:= supt,s∈I|G(t, s)|.
3. THE MAIN RESULTS
We consider first the case when F(·,·) is lower semicontinuous in the second variable.
Hypothesis 3.1. i) F(·,·) : I ×Rn → P(Rn) has compact values, F(·,·) is L(I)⊗ B(Rn) measurable and x → F(t, x) is lower semicontinuous for almost all t∈I.
ii) There exist M(·)∈L1(I,R+) such that
sup{|v|; v∈F(t, x)} ≤M(t) a.e. (I), ∀x∈Rn. Theorem 3.2. Assume that Hypothesis 3.1is satisfied.
Then problem(1.1)has at least one solution on I.
Proof. We note first that if Hypothesis 3.1 is satisfied then F(·,·) is of lower semicontinuous type (e.g., [6]). Therefore, we apply Theorem 2.2 to deduce that there existsf(·) :AC(I,Rn)→L1(I,Rn) such thatf(x)∈SF(x)
∀x∈AC(I,Rn).
We consider the problem
(3.1) x0(t) =A(t)x(t) +f(x(t)), a.e. (0,1), M x(0) +N x(1) =η.
It is clear that if x(·) ∈ AC(I,Rn) is a solution of (3.1) then x(·) is a solution to problem (1.1). We transform problem (3.1) into a fixed point problem. Consider the operator H:AC(I,Rn)→AC(I,Rn) defined by
H(x)(t) := Φ(t)(M +NΦ(1))−1η+ Z 1
0
G(t, s)f(x(s))ds.
We shall show thatH is completely continuous, i.e., it is continuous and maps bounded sets into relatively compact sets.
Sincef(·) is continuous it follows from Lebesgue’s dominated convergence theorem that H is continuous.
We prove now thatHmaps bounded sets into bounded sets inAC(I,Rn).
LetB ⊂AC(I,Rn) be a bounded set. Then there exist q >0 such that
|x|C ≤q ∀x∈B. If we take x∈B, then H(x)(t) = Φ(t)(M+NΦ(1))−1η+
Z 1 0
G(t, s)f(x(s))ds.
One may write for any t∈I
|H(x)(t)| ≤sup
t∈I
|Φ(t)(M+NΦ(1))−1η|+G0 Z 1
0
|f(x(s))|ds
≤sup
t∈I
|Φ(t)(M+NΦ(1))−1η|+G0|M(·)|1.
We show next thatT(·) maps bounded sets into equi-continuous sets.
LetB ⊂AC(I,Rn) as before and x∈B. Then for any t, τ ∈I we have
|H(x)(t)−H(x)(τ)| ≤
Z 1 0
G(t, s)f(x(s))ds− Z 1
0
G(τ, s)f(x(s))ds
≤
≤ Z 1
0
|G(t, s)−G(τ, s)| · |f(x(s))|ds≤ Z 1
0
|G(t, s)−G(τ, s)|M(s)ds.
The right hand side of the above inequality tends to zero ast→τ. We apply now Arzela-Ascoli’s theorem we deduce thatH(·) is completely continuous on AC(I,Rn).
In order to apply Theorem 2.1 one more step is needed. It remains to prove that the set
(H) ={x∈S;x=λH(x) for some λ∈(0,1)}
is bounded.
Indeed, let x ∈(H), then x=λH(x) for some λ∈(0,1) and thus, for each t∈I
|x(t)| ≤sup
t∈I
|Φ(t)(M +NΦ(1))−1η|+G0 Z 1
0
|f(x(s))|ds
≤sup
t∈I
|Φ(t)(M +NΦ(1))−1η|+G0|M(·)|1 and the proof is complete.
We consider next the situation when F(·,·) is Lipschitz in the second variable. In order to study problem (1.1) in this case we introduce the following hypothesis on F.
Hypothesis 3.3. (i) F(·,·) : I ×Rn → P(Rn) has nonempty closed values and for every x∈Rn F(·, x) is measurable.
(ii)There exists L(·)∈L1(I,R+)such that for almost allt∈I, F(t,·) is L(t)-Lipschitz in the sense that
dH(F(t, x), F(t, y))≤L(t)|x−y| ∀x, y∈Rn.
Theorem 3.4. Assume that Hypothesis3.3is satisfied andG0|L(·)|1 <1.
Let y(·)∈AC(I,Rn) be such that there exists q(·)∈L1(I,R+) withd(y0(t)− A(t)y(t), F(t, y(t)))≤q(t), a.e. (I). Denote µ=M y(0) +N y(1).
Then there exists x(·) a solution of problem (1.1)satisfying for all t∈I
|x(t)−y(t)| ≤ 1
1−G0|L(·)|1sup
t∈I
|Φ(t)(M+NΦ(1))−1(η−µ)|+
(3.2)
+ G0
1−G0|L(·)|1 Z 1
0
q(t)dt.
Proof.The set-valued mapt→F(t, y(t)) is measurable with closed values and
F(t, y(t))∩ {y0(t)−A(t)y(t) +q(t)B} 6=∅ a.e.(I), where B ⊂Rn is the closed unit ball.
It follows (e.g., Theorem 1.14.1 in [1]) that there exists a measurable selection f1(t)∈F(t, y(t)) a.e. (I) such that
(3.3) |f1(t)−y0(t) +A(t)y(t)| ≤q(t) a.e.(I).
Define x1(t) = Φ(t)(M+NΦ(1))−1η+R1
0 G(t, s)f1(s)dsand one has
|x1(t)−y(t)| ≤sup
t∈I
|Φ(t)(M+NΦ(1))−1(η−µ)|+G0
Z 1 0
q(t)dt.
Denote m:= supt∈I|Φ(t)(M +NΦ(1))−1(η−µ)|.
We claim that it is enough to construct the sequences xn(·)∈C(I,Rn), fn(·)∈L1(I,Rn),n≥1 with the following properties
(3.4) xn(t) = Φ(t)(M+NΦ(1))−1η+ Z 1
0
G(t, s)fn(s)ds, t∈I, (3.5) fn(t)∈F(t, xn−1(t)) a.e.(I), n≥1,
(3.6) |fn+1(t)−fn(t)| ≤L(t)|xn(t)−xn−1(t)| a.e.(I), n≥1.
If this construction is realized then from (3.3)–(3.6) we have for almost all t∈I
|xn+1(t)−xn(t)| ≤ Z 1
0
|G(t, t1)| · |fn+1(t1)−fn(t1)|dt1 ≤
≤G0
Z 1 0
L(t1)|xn(t1)−xn−1(t1)|dt1 ≤G0
Z 1 0
L(t1) Z 1
0
|G(t1, t2)|·
·|fn(t2)−fn−1(t2)|dt2dt1 ≤G20 Z 1
0
L(t1) Z 1
0
L(t2)|xn−1(t2)−xn−2(t2)|dt2dt1≤
≤Gn0 Z 1
0
L(t1) Z 1
0
L(t2). . . Z 1
0
L(tn)|x1(tn)−y(tn)|dtn. . .dt1 ≤
≤(G0|L(·)|1)n
m+G0
Z 1 0
q(t)dt
.
Therefore {xn(·)} is a Cauchy sequence in the Banach space C(I,Rn), hence converging uniformly to some x(·) ∈ C(I,Rn). Therefore, by (3.6), for almost all t ∈ I, the sequence {fn(t)} is Cauchy in Rn. Let f(·) be the pointwise limit of fn(·).
Moreover, one has
|xn(t)−y(t)| ≤ |x1(t)−y(t)|+
n−1
X
i=1
|xi+1(t)−xi(t)| ≤m+
(3.7)
+G0 Z 1
0
q(t)dt+
n−1
X
i=1
m+G0 Z 1
0
q(t)dt
(G0|L(·)|1)i= m+G0R1 0 q(t)dt 1−G0|L(·)|1 . On the other hand, from (3.3), (3.4) and (3.7) we obtain for almost all t∈I
|fn(t)−y0(t) +A(t)y(t)| ≤
n−1
X
i=1
|fi+1(t)−fi(t)|+
+|f1(t)−y0(t) +A(t)y(t)| ≤L(t)m+G0
R1 0 q(t)dt
1−G0|L(·)|1 +q(t).
Hence the sequence fn(·) is integrably bounded and therefore f(·) ∈ L1(I,Rn).
Using Lebesgue’s dominated convergence theorem and taking the limit in (3.4), (3.5) we deduce that x(·) is a solution of (1.1). Finally, passing to the limit in (3.7) we obtained the desired estimate on x(·).
It remains to construct the sequencesxn(·), fn(·) with the properties in (3.4)–(3.6). The construction will be done by induction.
Since the first step is already realized, assume that for some N ≥1 we already constructed xn(·) ∈C(I,Rn) and fn(·) ∈L1(I,Rn), n= 1,2, . . . , N, satisfying (3.4), (3.6) for n = 1,2, . . . , N and (3.5) for n = 1,2, . . . , N −1.
The set-valued map t → F(t, xN(t)) is measurable. Moreover, the map t → L(t)|xN(t)−xN−1(t)|is measurable. By the lipschitzianity of F(t,·) we have that for almost all t∈I
F(t, xN(t))∩ {fN(t) +L(t)|xN(t)−xN−1(t)|B} 6=∅.
Theorem 1.14.1 in [1] yields that there exists a measurable selection fN+1(·) ofF(·, xN(·)) such that
|fN+1(t)−fN(t)| ≤L(t)|xN(t)−xN−1(t)| a.e.(I).
We define xN+1(·) as in (3.4) with n = N + 1. Thus fN+1(·) satisfies (3.5) and (3.6) and the proof is complete.
Remark3.5. According to Theorem 3.2 in [4], if the assumptions of Theo- rem 3.4 are satisfied then for any ε > 0 there exists xε(·) a solution of (1.1) satisfying for all t∈I
|xε(t)−y(t)| ≤ 1
1−G0|L(·)|1 sup
t∈I
|Φ(t)(M +NΦ(1))−1(η−µ)|+
(3.8)
+ G0
1−G0|L(·)|1 Z 1
0
q(t)dt+ε.
Obviously, the estimation in (3.2) is better than the one in (3.8).
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Received 12 February 2009 Faculty of Mathematics and Informatics University of Bucharest
Str. Academiei nr. 14 010014 Bucharest, Romania
