Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843 - 7265 (print)
Volume 3 (2008), 167 – 175
ON THE SOLUTION SET OF A TWO POINT
BOUNDARY VALUE PROBLEM
Aurelian Cernea
Abstract. We consider a nonconvex and nonclosed second order differential inclusion with
nonlocal boundary conditions and we prove the arcwise connectedness of the solution set.
1
Introduction
In this paper we study the second order differential inclusion
x00 − λx0 ∈ F (t, x, H(t, x)),
a.e. (I)
(1.1)
with boundary conditions of the form
x(0) = c0 ,
x(1) = c1 ,
(1.2)
where I = [0, 1], F (., ., .) : I × R2 → P(R), H(., .) : I × R → P(R), λ > 0 and
ci ∈ R, i = 1, 2.
When F does not depend on the last variable (1.1) reduces to
x00 − λx0 ∈ F (t, x),
a.e. (I).
(1.3)
Qualitative properties of the set of solutions of problem (1.3)-(1.2) may be found
in [1, 2, 7, 9, 10, 11] etc. In all these papers the set-valued map F is assumed to be
at least closed-valued. Such an assumption is quite natural in order to obtain good
properties of the solution set, but it is interesting to investigate the problem when
the right-hand side of the multivalued equation may have nonclosed values.
Following the approach in [13] we consider the problem (1.1)-(1.2), where F and
H are closed-valued multifunctions Lipschitzian with respect to the second variable
and F is contractive in the third variable. Obviously, the right-hand side of the
differential inclusion in (1.1) is in general neither convex nor closed. We prove
2000 Mathematics Subject Classification:34A60
Keywords: differential inclusion; boundary value problem; set-valued contraction; fixed point.
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168
A. Cernea
the arcwise connectedness of the solution set to (1.1)-(1.2). The main tool is a
result ([12]) concerning the arcwise connectedness of the fixed point set of a class of
nonconvex nonclosed set-valued contractions. This idea was already used for similar
results for other classes of differential inclusions ([3, 4, 5, 6, 8, 13]).
The paper is organized as follows: in Section 2 we recall some preliminary results
that we use in the sequel and in Section 3 we prove our main result.
2
Preliminaries
Let Z be a metric space with the distance dZ and let 2Z be the family of all nonempty
closed subsets of Z. For a ∈ Z and A, B ∈ 2Z set dZ (a, B) = inf b∈B dZ (a, b) and
d∗Z (A, B) = supa∈A dZ (a, B). Denote by DZ the Pompeiu-Hausdorff generalized
metric on 2Z defined by
DZ (A, B) = max{d∗Z (A, B), d∗Z (B, A)},
A, B ∈ 2Z .
In what follows, when the product Z = Z1 × Z2 of metric spaces Zi , i = 1, 2, is
considered,
it is assumed that Z is equipped with the distance dZ ((z1 , z2 ), (z10 , z20 )) =
P2
0
i=1 dZi (zi , zi ).
Let X be a nonempty set and let F : X → 2Z be a set-valued map from X to Z.
The range of F is the set F (X) = ∪x∈X F (x). Let (X, F) be a measurable space.
The multifunction F : X → 2Z is called measurable if F −1 (Ω) ∈ F for any open set
Ω ⊂ Z, where F −1 (Ω) = {x ∈ X; F (x)∩Ω 6= ∅}. Let (X, dX ) be a metric space. The
multifunction F is called Hausdorff continuous if for any x0 ∈ X and every > 0
there exists δ > 0 such that x ∈ X, dX (x, x0 ) < δ implies DZ (F (x), F (x0 )) < .
Let (T, F, µ) be a finite, positive, nonatomic measure space and let (X, |.|X ) be
a Banach space. We denote by L1 (T, X) the Banach space of all (equivalence classes
of) Bochner integrable functions u : T → X endowed with the norm
Z
|u|L1 (T,X) =
|u(t)|X dµ
T
A nonempty set K ⊂ L1 (T, X) is called decomposable if, for every u, v ∈ K and
every A ∈ F, one has
χA .u + χT \A .v ∈ K
where χB , B ∈ F indicates the characteristic function of B.
A metric space Z is called an absolute retract if, for any metric space X and
any nonempty closed set X0 ⊂ X, every continuous function g : X0 → Z has a
continuous extension g : X → Z over X. It is obvious that every continuous image
of an absolute retract is an arcwise connected space.
In what follows we recall some preliminary results that are the main tools in the
proof of our result.
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On a two point boundary value problem
169
Let (T, F, µ) be a finite, positive, nonatomic measure space, S a separable Banach space and let (X, |.|X ) be a real Banach space. To simplify the notation we
write E in place of L1 (T, X).
Lemma 1. ([13]) Assume that φ : S × E → 2E and ψ : S × E × E → 2E are
Hausdorff continuous multifunctions with nonempty, closed, decomposable values,
satisfying the following conditions
a) There exists L ∈ [0, 1) such that, for every s ∈ S and every u, u0 ∈ E,
DE (φ(s, u), φ(s, u0 )) ≤ L|u − u0 |E .
b) There exists M ∈ [0, 1) such that L + M < 1 and for every s ∈ S and every
(u, v), (u0 , v 0 ) ∈ E × E,
DE (ψ(s, u, v), ψ(s, u0 , v 0 )) ≤ M (|u − u0 |E + |v − v 0 |E ).
Set F ix(Γ(s, .)) = {u ∈ E; u ∈ Γ(s, u)}, where Γ(s, u) = ψ(s, u, φ(s, u)), (s, u) ∈
S × E. Then
1) For every s ∈ S the set F ix(Γ(s, .)) is nonempty and arcwise connected.
2) For any si ∈ S, and any ui ∈ F ix(Γ(s, .)), i = 1, ..., p there exists a continuous
function γ : S → E such that γ(s) ∈ F ix(Γ(s, .)) for all s ∈ S and γ(si ) = ui , i =
1, ..., p.
Lemma 2. ([13]) Let U : T → 2X and V : T × X → 2X be two nonempty closedvalued multifunctions satisfying the following conditions
a) U is measurable and there exists r ∈ L1 (T ) such that DX (U (t), {0}) ≤ r(t)
for almost all t ∈ T .
b) The multifunction t → V (t, x) is measurable for every x ∈ X.
c) The multifunction x → V (t, x) is Hausdorff continuous for all t ∈ T .
Let v : T → X be a measurable selection from t → V (t, U (t)).
Then there exists a selection u ∈ L1 (T, X) such that v(t) ∈ V (t, u(t)), t ∈ T .
Let I = [0, 1], by C(I) we denote the Banach space of all continuous functions
from I to R with the norm ||x(.)||C = supt∈I |x(t)|, by AC 1 we denote the space
of differentiable functions x(.) : (0, 1) → R whose first derivative x0 (.) is absolutely
continuous and by L1 we denote the Banach space of
R 1Lebesgue integrable functions
x(.) : [0, 1] → R endowed with the norm ||u(.)||1 = 0 |u(t)|dt.
A function x(.) ∈ AC 1 is said to be a solution of (1.3)-(1.2) if there exists a
function v(.) ∈ L1 with v(t) ∈ F (t, x(t)), a.e. (I) such that x00 (t) − λx0 (t) = v(t),
a.e. (I) and x(.) satisfies conditions (1.2).
The next statement is well known (e.g., [1]).
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A. Cernea
Lemma 3. If v(.) : [0, 1] → R is an integrable function then the problem
x00 (t) − λx0 (t) = v(t)
a.e. (I)
x(0) = c0 ,
x(1) = c1 ,
has a unique solution x(.) ∈ AC 1 given by
Z 1
x(t) = Pc (t) +
G(t, s)v(s)ds,
0
where, if c = (c0 , c1 ) ∈
R2 ,
we denote by
1
[(eλ − eλt )c0 + (eλt − 1)c1 ]
−1
the unique solution of the problem
Pc (t) =
(2.1)
eλ
x00 − λx0 = 0
x(0) = c0 ,
x(1) = c1 ,
and
λt
1
(e − 1)(eλs − eλ )
λs
λ
e (1 − e ) (eλs − 1)(eλt − eλ )
is the Green function associated to the problem.
G(t, s) =
x00 − λx0 = 0
x(0) = 0,
if
if
0≤t≤s≤1
0≤s≤t≤1
(2.2)
x(1) = 0.
Note that if a = (a1 , a2 ), b = (b1 , b2 ) ∈ R2 we put ||a|| = |a1 | + |a2 | and
|Pa (t) − Pb (t)| ≤ ||a − b||.
Denote M := supt,s∈I |G(t, s)|.
In order to study problem (1.1)-(1.2) we introduce the following hypothesis.
Hypothesis. Let F : I × R2 → 2R and H : I × R → 2R be two set-valued maps
with nonempty closed values, satisfying the following assumptions:
i) The set-valued maps t → F (t, u, v) and t → H(t, u) are measurable for all
u, v ∈ R.
ii) There exists l ∈ L1 such that, for every u, u0 ∈ R,
D(H(t, u), H(t, u0 )) ≤ l(t)|u − u0 |
a.e. (I).
iii) There exist m ∈ L1 and θ ∈ [0, 1) such that, for every u, v, u0 , v 0 ∈ R,
D(F (t, u, v), F (t, u0 , v 0 )) ≤ m(t)|u − u0 | + θ|v − v 0 |
a.e. (I).
iv) There exist f, g ∈ L1 such that
d(0, F (t, 0, 0)) ≤ f (t),
d(0, H(t, 0)) ≤ g(t)
a.e. (I).
For c = (c0 , c1 ) ∈ R2 we denote by S(c) the solution set of (1.1)-(1.2).
In what follows N (t) := max{l(t), m(t)}.
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On a two point boundary value problem
3
The main result
Even if the multifunction from the right-hand side of (1.1) has, in general, nonclosed
nonconvex values, the solution set S(c) has some meaningful properties, stated in
Theorem 4 below.
R1
Theorem 4. Assume that Hypothesis is satisfied and 2M 0 N (s)ds + θ < 1. Then
1) For every c ∈ R2 , the solution set S(c) of (1.1)-(1.2) is nonempty and arcwise
connected in the space C(I).
2) For any ci ∈ R2 and any ui ∈ S(ci ), i = 1, ..., p, there exists a continuous
function s : R2 → C(I) such that s(c) ∈ S(c) for any c ∈ R2 and s(ci ) = ui , i =
1, ..., p.
3) The set S = ∪c∈R2 S(c) is arcwise connected in C(I).
Proof. 1) For c ∈ R2 and u ∈ L1 , set
Z
uc (t) = Pc (t) +
1
G(t, s)u(s)ds,
t ∈ I,
0
where Pc (.) and G(., .) are defined in (2.1) and (2.2), respectively.
1
1
We prove that the multifunctions φ : R2 × L1 → 2L and ψ : R2 × L1 × L1 → 2L
given by
φ(c, u) = {v ∈ L1 ; v(t) ∈ H(t, uc (t)) a.e. (I)},
ψ(c, u, v) = {w ∈ L1 ; w(t) ∈ F (t, uc (t), v(t)) a.e. (I)},
c ∈ R2 , u, v ∈ L1 satisfy the hypotheses of Lemma 1.
Since uc (.) is measurable and H satisfies Hypothesis i) and ii), the multifunction
t → H(t, uc (t)) is measurable and nonempty closed-valued, it has a measurable
selection. Therefore due to Hypothesis iv), the set φ(c, u) is nonempty. The fact
that the set φ(c, u) is closed and decomposable follows by simple computations. In
the same way we obtain that ψ(c, u, v) is a nonempty closed decomposable set.
Pick (c, u), (c1 , u1 ) ∈ R2 × L1 and choose v ∈ φ(c, u). For each ε > 0 there exists
v1 ∈ φ(c1 , u1 ) such that, for every t ∈ I, one has
|v(t) − v1 (t)| ≤ D(H(t, uc (t)), H(t, uc1 (t))) + ε ≤ N (t)[|Pc (t) − Pc1 (t)|+
1
Z
|G(t, s)|.|u(s) − u1 (s)|ds] + ε ≤ N (t)[||c − c1 || + sup |G(t, s)|.||u − u1 ||1 ] + ε.
t,s∈I
0
Hence
Z
||v − v1 ||1 ≤ ||c − c1 ||.
1
Z
0
1
N (t)dt||u − u1 ||1 + ε
N (t)dt + M
0
for any ε > 0.
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A. Cernea
This implies
1
Z
1
Z
dL1 (v, φ(c1 , u1 )) ≤ ||c − c1 ||.
N (t)dt||u − u1 ||1
N (t)dt + M
0
0
for all v ∈ φ(c, u). Therefore,
d∗L1 (φ(c, u), φ(c1 , u1 )) ≤ ||c − c1 ||.
1
Z
1
Z
N (t)dt||u − u1 ||1 .
N (t)dt + M
0
0
Consequently,
1
Z
DL1 (φ(c, u), φ(c1 , u1 )) ≤ ||c − c1 ||.
Z
1
N (t)dt||u − u1 ||1
N (t)dt + M
0
0
which shows that φ is Hausdorff continuous and satisfies the assumptions of Lemma
2.
Pick (c, u, v), (c1 , u1 , v1 ) ∈ R2 × L1 × L1 and choose w ∈ ψ(c, u, v). Then, as
before, for each ε > 0 there exists w1 ∈ ψ(c1 , u1 , v1 ) such that for every t ∈ I
|w(t) − w1 (t)| ≤ D(F (t, uc (t), v(t)), F (t, uc1 (t), v1 (t))) + ε ≤ N (t)|uc (t)−
Z
1
uc1 (t)| + θ|v(t) − v1 (t)| + ε ≤ N (t)[|Pc (t) − Pc1 (t)| +
|G(t, s)|.|u(s) − u1 (s)|ds]
0
+θ|v(t) − v1 (t)| + ε ≤ N (t)[||c − c1 || + M ||u − u1 ||1 ] + θ|v(t) − v1 (t)| + ε.
Hence
Z
1
||w − w1 ||1 ≤ ||c − c1 ||.
Z
0
1
Z
0
1
Z
≤ ||c − c1 ||.
N (t)dt + (M
0
1
N (t)dt||u − u1 ||1 + θ||v − v1 ||1 + ε
N (t)dt + M
N (t)dt + θ)dL1 ×L1 ((u, v), (u1 , v1 )) + ε.
0
As above, we deduce that
DL1 (ψ(c, u, v), ψ(c1 , u1 , v1 )) ≤
Z
≤ ||c − c1 ||.
1
Z
N (t)dt + (M
0
1
N (t)dt + θ)dL1 ×L1 ((u, v), (u1 , v1 )),
0
namely, the multifunction ψ is Hausdorff continuous and satisfies the hypothesis of
Lemma 1.
Define Γ(c, u) = ψ(c, u, φ(c, u)), (c, u) ∈ R2 × L1 . According to Lemma 1, the
set F ix(Γ(c, .)) = {u ∈ L1 ; u ∈ Γ(c, u)} is nonempty and arcwise connected in
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On a two point boundary value problem
L1 . Moreover, for fixed ci ∈ R2 and vi ∈ F ix(Γ(ci , .)), i = 1, ..., p, there exists a
continuous function γ : R2 → L1 such that
γ(c) ∈ F ix(Γ(c, .)),
γ(ci ) = vi ,
∀c ∈ R2 ,
(3.1)
i = 1, ..., p.
(3.2)
We shall prove that
F ix(Γ(c, .)) = {u ∈ L1 ;
u(t) ∈ F (t, uc (t), H(t, uc (t)))
a.e. (I)}.
(3.3)
Denote by A(c) the right-hand side of (3.3). If u ∈ F ix(Γ(c, .)) then there is
v ∈ φ(c, v) such that u ∈ ψ(c, u, v). Therefore, v(t) ∈ H(t, uc (t)) and
u(t) ∈ F (t, uc (t), v(t)) ⊂ F (t, uc (t), H(t, uc (t))) a.e. (I),
so that F ix(Γ(c, .)) ⊂ A(c).
Let now u ∈ A(c). By Lemma 2, there exists a selection v ∈ L1 of the multifunction t → H(t, uc (t))) satisfying
u(t) ∈ F (t, uc (t), v(t)) a.e. (I).
Hence, v ∈ φ(c, v), u ∈ ψ(c, u, v) and thus u ∈ Γ(c, u), which completes the proof of
(3.3).
We next note that the function T : L1 → C(I),
Z 1
T (u)(t) :=
G(t, s)u(s)ds, t ∈ I
0
is continuous and one has
S(c) = Pc (.) + T (F ix(Γ(c, .))),
c ∈ R2 .
(3.4)
Since F ix(Γ(c, .)) is nonempty and arcwise connected in L1 , the set S(c) has the
same properties in C(I).
2) Let ci ∈ R2 and let ui ∈ S(ci ), i = 1, ..., p be fixed. By (3.4) there exists
vi ∈ F ix(Γ(ci , .)) such that
ui = Pci (.) + T (vi ),
i = 1, ..., p.
If γ : R2 → L1 is a continuous function satisfying (3.1) and (3.2) we define, for every
c ∈ R2 ,
s(c) = Pc (.) + T (γ(c)).
Obviously, the function s : R2 → C(I) is continuous, s(c) ∈ S(c) for all c ∈ R2 , and
s(ci ) = Pci (.) + T (γ(ci )) = Pci (.) + T (vi ) = ui ,
i = 1, ..., p.
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174
A. Cernea
3) Let u1 , u2 ∈ S = ∪c∈R2 S(c) and choose ci ∈ R2 , i = 1, 2 such that ui ∈ S(ci ),
i = 1, 2. From the conclusion of 2) we deduce the existence of a continuous function
s : R2 → C(I) satisfying s(ci ) = ui , i = 1, 2 and s(c) ∈ S(c), c ∈ R2 . Let
h : [0, 1] → R2 be a continuous mapping such that h(0) = c1 and h(1) = c2 . Then
the function s ◦ h : [0, 1] → C(I) is continuous and verifies
s ◦ h(0) = u1 ,
s ◦ h(1) = u2 ,
s ◦ h(τ ) ∈ S(h(τ )) ⊂ S,
τ ∈ R2 .
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Aurelian Cernea
Faculty of Mathematics and Informatics
University of Bucharest,
Academiei 14, 010014 Bucharest, Romania.
e-mail: acernea@fmi.unibuc.ro
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