Directory UMM :Journals:Journal_of_mathematics:EJQTDE:

Electronic Journal of Qualitative Theory of Differential Equations
2005, No. 22, 1-11; http://www.math.u-szeged.hu/ejqtde/

Existence of Solutions for Nonconvex Third Order
Differential Inclusions.
Britney Hopkins
Department of Mathematics and Statistics
University of Arkansas at Little Rock
2801 S. University
Little Rock, AR 72204
bjhopkins1@ualr.edu
Abstract
This paper proves the existence of solutions for a third order initial value nonconvex differential inclusion. We start with an upper semicontinuous compact valued multifunction F
which is contained in a lower semicontinuous convex function ∂V and show that,
x(3) (t) ∈ F (x(t), x′ (t), x′′ (t)), x(0) = x0 , x′ (0) = y0 , x′′ (0) = z0 .

Keywords: Nonconvex Differential Inclusions
AMS Subject Classification: 34G20, 47H20

1

Introduction

The origins of boundary and initial value problems for differential inclusions are in the theory of
differential equations and serve as models for a variety of applications including control theory.
Existence results for the second order differential inclusion,
x′′ ∈ F (x, x′ ), x(0) = x0 , x′ (0) = y0 ,
have been obtained by many authors (see [4], [5] and the references therein). In [5], Lupulescu
showed existence for the problem
x′′ ∈ F (x, x′ ) + f (t, x, x′ ), x(0) = x0 , x′ (0) = y0
for the case in which F is an upper semicontinuous compact valued multifunction such the F (x, y) ⊂
∂V (y) and f is a Carath´eodory function.
In this paper, we prove an existence result for the third order differential inclusion,
x(3) (t) ∈ F (x(t), x′ (t), x′′ (t)), x(0) = x0 , x′ (0) = y0 , x′′ (0) = z0 ,
where F is an upper semicontinuous compact valued multifunction and F (x, y, z) ⊂ ∂V (z) for some
proper lower semicontinuous convex function V. Expounding upon the methods used to establish
existence by Lupulescu in [4] and [5], we define a sequence of approximate solutions on a given
interval and show that the sequence converges to an actual solution.
EJQTDE, 2005 No. 22, p. 1

2

Preliminaries

Let Rm be an m dimensional Euclidean space with an inner product h·, ·i and norm k · k.
Let x ∈ Rm and r > 0. The open ball centered at x with radius r is defined by
Br (x) = {y ∈ Rm : kx − yk < r},
where B r (x) denotes its closure.
For the proper lower semicontinuous convex function V : Rm → R, the multifunction ∂V :
m
R → 2R defined by
m

∂V (x) = {γ ∈ Rm : V (y) − V (x) ≥ hγ, y − xi, ∀y ∈ Rm } ,
is the subdifferential of V.
Let L2 [a, b] be a Hilbert space with the inner product defined by

hx, yi =

Z

b

x(t)y(t)dt,

a

where y(t) denotes the complex conjugate of y(t), and the norm is defined as

kxk =

s
Z

b

|x(t)|2 dt.

a

Let coF (x, y, z) denote the closed convex hull of F and xn ⇒ x denote that xn converges

uniformly to x.
We need the following theorems from Aubin and Cellina [1].
Theorem 0.3.4 Consider a sequence of absolutely continuous functions xk (·) from an interval
I to a Banach Space X satisfying
(i) for every t ∈ I, xk (t)k is a relatively compact subset of X ;
(ii) there exists a positive function c(·) ∈ L2 (I) such that, for almost all t ∈ I, kx′k (t)k ≤ c(t).
Then there exists a subsequence, again denoted by xk (·), converging to an absolutely continuous
function x(·) from I to X in the sense that
(i) xk (·) converges uniformly to x(·) over compact subsets of I ;
(ii) x′k (·) converges weakly to x′k (·) in L2 (I, X).

EJQTDE, 2005 No. 22, p. 2

Theorem 1.1.4 (the Convergence Theorem) Let F be a proper hemicontinuous map from
a Hausdorff locally convex space X to the closed convex subsets of a Banach Space Y. Let I be
an interval of R and xk (·) and yk (·) be measurable functions from I to Y respectively satisfying
for almost all t in I and for every neighborhood ℵ of 0 in X × Y , there exists a k0 = k0 (t, ℵ) such
that for every k0 ≤ k, (xk (t), yk (t)) ∈ graph(F ) + ℵ. If,
(i) xk (·) converges almost everywhere to a function x(·) from I to X ;
(ii) yk (·) belongs to L2 (I, Y ) and converges weakly to y(·) in L2 (I, Y ),

then, for almost all t ∈ I,
(x(t), y(t)) ∈ graph(F ) i.e.y(t) ∈ F (x(t)).
We also need the following lemma from Brezis [2].
Lemma 3.3 Let u ∈ D(V ) almost everywhere on [0, T ] and suppose g ∈ L2 ([0, T ], R) such
that g(t) ∈ ∂V (u(t)) almost everywhere on [0, T ]. Then, the function t 7−→ V (u(t)) is absolutely
continuous on [0, T ].
Also, let t ∈ [0, T ] such that u(t) ∈ D(V ) and let u and V(u) be differentiable. Then for all
t ∈ [0, T ]


du
d
V (u(t)) = h, (t) ∀h ∈ ∂V (u(t)).
dt
dt

3

The Main Result

THEOREM: If F : Ω → 2R

m

and V : Rm → R satisfy the assumptions

(A1) Ω ⊂ R2m where Ω is open and F : Ω → 2R
multifunction;

m

is a compact valued upper semicontinuous

(A2) there exists a lower semicontinuous proper convex function V : Rm → R such that
F (x, y, z) ⊂ ∂V (z) for every (x, y, z) ∈ Ω.
Then, for every (x0 , y0 , z0 ) ∈ Ω, there exists a T > 0 and a solution x : [0, T ] → Rm of
x(3) (t) ∈ F (x(t), x′ (t), x′′ (t)), x(0) = x0 , x′ (0) = y0 , x′′ (0) = z0 .

(1)

EJQTDE, 2005 No. 22, p. 3

By a solution we are referring to an absolutely continuous function x : [0, T ] → Rm with
absolutely continuous first and second derivatives with the initial values x(0) = x0 , x′ (0) = y0 ,
x′′ (0) = z0 , and x(3) (t) ∈ F (x(t), x′ (t), x′′ (t)), a.e. on [0, T ].
¯r (x0 , y0 , z0 ) ⊂ Ω for some r > 0 since Ω is open.
PROOF: Suppose (x0 , y0 , z0 ) ∈ Ω. Then, K = B
By assumption (A1)
F (K) =

[

F (x, y, z)

(x,y,z)∈K

is compact. Then there exists an M > 0 such that
sup{kvk : v ∈ F (x, y, z), (x, y, z) ∈ K)} ≤ M.

(2)

Set

T < min

(

 21

r
r
2r
r  r  21  r  31
,
,
,
,
,
M M
M

2kz0 k 2ky0 k 3kz0k

Let n, j be integers where 1 ≤ j ≤ n. Set tjn =

jT
n

)

.

(3)



j
. For t ∈ tj−1
n , tn define,

2

3
1
1
t − tjn znj +
t − tjn vnj ,
xn (t) = xjn + t − tjn ynj +
2
6

(4)

where x0n = x0 , yn0 = y0 , and zn0 = z0 .



For 0 ≤ j ≤ n − 1 and vnj ∈ F xjn , ynj , znj , define



xj+1

n




ynj+1






znj+1

 
 2
 3
T
1 T
1 T
ynj +
znj +
vnj
n
2 n
6 n
 
 2
T
1 T
j
j
= yn +
zn +
vnj
2 n
n
T
vnj .
= znj +
n

= xjn +

(5)



We claim that x1n , yn1 , zn1 ∈ K. Using (5), we have
 

 

1


z − z0
=
z 0 + T v 0 − z0
≤ T M < r.
n
n

n
n
n

EJQTDE, 2005 No. 22, p. 4

As well as,


 2
 

1

1
T
T


0
0
0
yn − y0
=
yn +
zn +
vn − y0


n
2 n
 2
 
1 T
T
kz0 k +
M
≤
n
2 n
1
1
< r + r = r.
2
2
Also,


 2
 3
 

1

1
T
T
1
T


0
0
0
0
xn − x0
=
xn +
yn +
zn +
vn − x0


n
2 n
6 n
 2
 3
 
1 T
1 T
T
ky0 k +
kz0 k +
M
≤
n
2 n
6 n
1
1
< T ky0 k + T 2 kz0 k + T 3 M
2
6
1
1
1
< r + r + r = r.
2
3
6
Hence the claim holds. Now suppose j ≥ 1. We make the assumption that,

 2
 3
 

0

1

1 jT
1 T
T

0
2
2
0
j
0

y
+
z
+
[
3j
−
3j
+
1
v
+
3j
−
9j
+
7
vn
x
=
x
+
j

n
n
n
n
n

n
2 n
6 n






+ 3j 2 − 15j + 19 vn2 + · · · + 7vnj−2 + vnj−1 ],

 
 2


y j = y 0 + j T z 0 + 1 T
(2j − 1)vn0 + (2j − 3)vn1 + · · · + 3vnj−2 + vnj−1 ,

n
n
n


2 n


 n



 0
T

j
0
z = z +
vn + vn1 + · · · + vnj−1 .

n
n
n

(6)

To see this, let j = 1. Then,
 
T
vn0 = zn0+1 ,
n
 2
T 0 1 T
1
0
yn = yn + zn +
vn0 = yn0+1 ,
n
2 n
 2
 3
T 0 1 T
1 T
1
0
0
xn = xn + yn +
zn +
vn0 = x0+1
n .
n
2 n
6 n
zn1 = zn0 +

EJQTDE, 2005 No. 22, p. 5

Thus, (6) holds when j = 1. Let’s suppose assumption (6) holds for j > 1. Using (5) we see that,
znj+1

 
T
vnj
=
+
n
 
 

T
T  0
1
j−1
0
+
vn + vn + · · · + vn
vjn
= zn +
n
n
 

T  0
0
= zn +
vn + vn1 + · · · + vnj .
n
znj

Thus the assumption holds for znj . Using this and (5) we have,
ynj+1

 2
 
1 T
T
j
zn +
vnj
=
+
n
2 n
 2
 


1 T
T
zn0 +
(2j − 1)vn0 + (2j − 3)vn1 + · · · + 3vnj−2 + vnj−1
= yn0 + j
n
2 n

 2
 

1 T
T  0
T
0
1
j−1
+
zn +
v + vn + · · · + vn
vnj
+
n
n n
2 n
 2 



 

1
3
1
T
T
zn0 +
j−
vn0 + j −
vn1 + · · · + vnj−1 + vn1 + . . . vnj−1
= yn0 + (j + 1)
n
n
2
2
2
 2
1 T
vnj
+
2 n
 

 2 



T
1
1
3
T
1
= yn0 + (j + 1)
zn0 +
j+
vn0 + j −
vn1 + · · · + vnj−1 + vnj
n
n
2
2
2
2
 
 2


1 T
T
= yn0 + (j + 1)
zn0 +
(2j + 1)vn0 + (2j − 1)vn1 + · · · + vnj
n
2 n
 2
 


1 T
T
zn0 +
(2(j + 1) − 1)vn0 + (2(j − 1) − 3)vn1 + · · · + vnj .
= yn0 + (j + 1)
n
2 n
ynj

Thus the assumption holds for ynj . Finally, with this and (5) we have,

 2
 3
 
1 T
1 T
T
znj +
vnj
ynj +
n
2 n
6 n
 2
 3
 
 2

1 jT
1 T
T
yn0 +
zn0 +
(3j − 3j + 1)vn0 + (3j 2 − 9j + 7)vn1 + · · · + vnj−1
= x0n + j
n
2 n
6 n
!
 
 
 2


T
1 T
T
0
0
0
1
j−2
j−1
+
yn + j
zn +
(2j − 1)vn + (2j − 3)vn + · · · + 3vn + vn
n
n
2 n
 2 
 3


1 T
1 T
T  0
+
zn0 +
vnj
vn + · · · + vnj−1 +
2 n
n
6 n

xj+1
= xjn +
n

EJQTDE, 2005 No. 22, p. 6

 
 2
T
1
T
0
2
=
+ (j + 1)
yn + (j + 1)
zn0
n
2
n
 3
 2





1 T
3j − 3j + 1 vn0 + 3j 2 − 9j + 7 vn1 + · · · + vnj−1
+
6 n
 
 3
 1 T 3 j
 0
T
1
j−1
+
jvn + (j − 1)vn + · · · + vn
vn
+
n
6 n
 2
 
1
T
T
yn0 + (j + 1)2
zn0
= x0n + (j + 1)
n
2
n
 3

 2




1 T
3j − 3j + 1 vn0 + 3j 2 − 9j + 7 vn1 + · · · + vnj−1
+
6 n
 3
 
 1 T 3 j
 0
1 T
1
j−1
+
+
6jvn + (6j − 6)vn + · · · + 6vn
vn
6 n
6 n
 2
 
1
T
T
yn0 + (j + 1)2
zn0
= x0n + (j + 1)
n
2
n
 3

 2




1 T
+
3j + 3j + 1 vn0 + 3j 2 − 3j + 1 vn1 + · · · + 7vnj−1 + vnj .
6 n
x0n



Thus the assumption holds for xjn . Using (2), (3) and the relations in (6), we show that xjn , ynj , znj ∈
K.


 

j


 0
j−1
1

zn − z0
=
zn0 + T
−
z
v
+
v
+
·
·
·
+
v
0
n
n
n

n
 
T
≤j
M
n
≤ TM
< r.

And,


 2
 



j


1
T
T


0
0
1
j−2
j−1
0
yn − y0
=
yn + j
zn +
− y0
(2j − 1)vn + (2j − 3)vn + · · · + 3vn + vn


n
2 n
 
 2
T
1 jT
≤j
kz0 k +
M
n
2 n
1
≤ T kz0 k + T 2 M
2
1
1
< r+ r
2
2
= r.
EJQTDE, 2005 No. 22, p. 7

Finally,

 2
 
j

1 jT
T
xn − x0
=
yn0 +
zn0
x0n + j

n
2 n

 3


 2

0


1 T

3j − 3j + 1 vn + 3j 2 − 9j + 7 vn1 + · · · + vnj−1 − x0
+

6 n
 2
 3
jT
1 jT
1 jT
≤
kz0 k +
M
ky0 k +
n
2 n
6 n
1
1
≤ T ky0 k + T 2 kz0 k + T 3 M
2
6
 
1
1
1 2
r + r = r.
< r+
2
2 3
6


Thus, xjn , ynj , znj ∈ K = Br (x0 , y0 , z0 ) for 1 ≤ j ≤ n. Now, from the definition of xn in (4) we
have,




j 1
j 2 j
′
j
j


xn (t) = yn + t − tn zn + 2 t − tn vn ,


(7)
x′′n (t) = znj + t − tjn vnj ,

 (3)

xn (t) = vnj .


(3)


By (2) we have that
xn (t)
=
vnj
≤ M. Similarly, (2) and (3) give the following,



kx′′n (t)k =
znj + t − tjn vnj


 
0

j

T  0
j
1
j−1

v
+
t
−
t
=
+
z
v
+
v
+
·
·
·
+
v
n
n
n
n
n
n
n
 
 
T
jT
M+
M
≤ kz0 k +
n
n
< kz0 k + 2r.

As well as,

kx′n (t)k



j


j 1


j
j 2 j

=
yn + t − tn zn +
t − tn vn
2
 2
 2
 2
 
 
1 jT
jT
1 T
T
jT
kz0 k +
kz0 k +
M+
M+
M
≤ ky0 k +
n
2 n
n
n
2 n
≤ ky0 k + T kz0 k + 2T 2 M + T kz0 k

< ky0 k + 3r.

EJQTDE, 2005 No. 22, p. 8

And finally,



j





j
1
1
j 2 j
j 3 j
j
z
+
v
t
−
t
t
−
t
y
+
kxn (t)k =
x
+
t
−
t
n
n
n
n
n
n
n
2
6
 

1 2
1
1 3
T
≤ kx0 k + T ky0 k + T kz0 k + T M +
ky0 k + T kz0 k + T 2 M
2
6
n
2
 2
 3
1 T
1 T
(kz0 k + T M ) +
M
+
2 n
6 n
1
1
≤ kx0 k + T ky0 k + T 2 kz0 k + T 3 M + T ky0 k + T 2 kz0 k
2
6
1
1
1
1
+ T 3 M + T 2 M kz0 k + T 3 M + T 3 M
2
2
2
6
1
2
3
= kx0 k + 2T ky0 k + 2T kz0 k + T M + T 3 M
3
4
1
< kx0 k + r + r + r + r
3
3
< kx0 k + 4r.


(3)
(3)
Since
xn (t)
≤ M ∀t ∈ [0, T ] the sequence (xn (t)) is bounded in L2 ([0, T ], Rm). Furthermore,
ε
suppose ε > 0 and ∀t ∈ [0, T ], and ∀τ ∈ [0, T ], |t − τ | < M
. Then,
kx′′n (t)

−

x′′n (τ )k


Z t


(3)

kxn (s)kds
≤
τ
Z t



≤ 
M ds
τ

= M |t − τ |
 ε 
≤M
M
= ε.

Thus, (x′′n ) is equicontinuous. Similarly (x′n ) and (xn ) are equicontinuous. Theorem 0.3.4 in [1]
gives the following:
There exists a subsequence, again denoted (xn )n that converges to an absolutely continuous
function x : [0, T ] → Rm such that:
(i) (xn ) ⇒ x on [0, T ],
(ii) (x′n ) ⇒ x′ on [0, T ],
(iii) (x′′n ) ⇒ x′′ on [0, T ],
3
2
m
(iv) (x(3)
n ) converges weakly to x in L ([0, T ], R ) .

EJQTDE, 2005 No. 22, p. 9

By the Convergence Theorem, theorem 1.4.1 in [1], we have that
x(3) (t) ∈ coF (x(t), x′ (t), x′′ (t)) ⊂ ∂V (x′′ (t)) a.e., t ∈ [0, T ].
Also, by the above and lemma 3.3 in [2],
d
dt V

Since,

RT
0

d
dt V

(x′′ (t)) dt =

2



(x′′ (t)) = x(3) (t), x(3) (t) =
x(3) (t)
.

R T
(3)
2
x (t)
dt we have,
0
′′

′′

V (x (T )) − V (x (0)) =

Z

T



(3)
2
x (t)
dt.

0

(8)







(3)
j
However, by (7) we also have xn (t) = vnj ∈ F xjn , ynj , znj ⊂ ∂V x′′n (tjn ) , ∀t ∈ tj−1
n , tn . Which,
from the definition of subdifferential, gives the following,
V x′′n tjn





− V x′′n tj−1
n





D



E
′′
j
′′
j−1
≥ x(3)
n (t), xn tn − xn tn
+
*
Z j
x(3)
n (t),

=

=

Z

=

Z

tjn

tj−1
n
tjn

tj−1
n

D

tn

tj−1
n

x(3)
n (s)ds

E
(3)
x(3)
n (t), xn (t) dt



(3)
2
xn (t)
dt.

Combining the above inequalities with (8), we get the following inequality,

RT

(3)
2
V (x′′n (T )) − V (z0 ) ≥ 0
xn (t)
dt.

If we let n approach infinity, we have

V (x′′ (T )) − V (z0 ) ≥ lim sup
n→∞

Z

0

T

Z

T

0



(3)
2
xn (t)
dt

Z T




(3)
2
(3)
2
(t)
dt
≥
lim
sup
x

xn (t)
dt
n→∞
0

2


(3)

(3)
2
x (t)
≥ lim sup
xn (t)
.
n→∞

However, the weak lower semicontinuity of the norm gives,




2
(3)
2


(t)
x (t)
≤ lim inf
x(3)
.
n
n→∞

EJQTDE, 2005 No. 22, p. 10

Thus we have,



2
(3)
2


(t)
x (t)
= lim
x(3)
.
n
n→∞



(3)
Hence the sequence xn → x(3) pointwise. By assumption (A1), F is closed, implying
x(3) (t) ∈ F (x(t), x′ (t), x′′ (t)), a.e. t ∈ [0, T ].

References
[1] J. P. Aubin and A. Cellina; Differential inclusions, Berlin, Springer-Verlag, 1984.
[2] H. Brezis; Operateurs Maximaux Monotones et Semigroups de Contractions Dans Les Espaces
de Hilbert, Amsterdam, North-Holland, 1973.
[3] K. Deimling; Multivalued Differential Equations, Walter de Gruyter, Berlin, New York, 1992.
[4] V. Lupulescu; Existence of Solutions for Nonconvex Functional Differential Inclusions, Electronic Journal of Differential Equations, Vol. 2004(2004), No. 141, pp.1-6.
[5] V. Lupulescu; Existence of Soutions for Nonconvex Second Order Differential Inclusions, Applied Mathematics E-Notes, 3(2003), 115-123.

(Received September 16, 2005)

EJQTDE, 2005 No. 22, p. 11