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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 dif-ferential 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) =x}

0, 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 whichFis an upper semicontinuous compact valued multifunction such theF(x, y)⊂

∂V(y) andf 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,

2

Preliminaries

LetRmbe anm dimensional Euclidean space with an inner product h·,·iand normk · k.

Letx∈Rmandr >0. The open ball centered atx with radiusr is defined by

Br(x) ={y∈Rm:kx−yk< r},

whereBr(x) denotes its closure.

For the proper lower semicontinuous convex function V : Rm → R, the multifunction ∂V :

Rm→2Rm

defined by

∂V(x) ={γ∈Rm_{:V(y)−V(x)≥ hγ, y−xi,∀y∈}

Rm},

is the subdifferential ofV.

LetL2[a, b] be a Hilbert space with the inner product defined by

hx, yi=

Z b

a

x(t)y(t)dt,

wherey(t) denotes the complex conjugate ofy(t), and the norm is defined as

kxk=

s Z b

a

|x(t)|2_dt.

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

uniformly tox.

We need the following theorems from Aubin and Cellina [1].

Theorem 0.3.4Consider a sequence of absolutely continuous functionsxk(·) from an interval I to a Banach SpaceX satisfying

(i) for everyt∈I,xk(t)k is a relatively compact subset ofX;

(ii) there exists a positive functionc(·)∈L2_{(I) such that, for almost allt∈I,kx}′

k(t)k ≤c(t).

Then there exists a subsequence, again denoted byxk(·), converging to an absolutely continuous

functionx(·) fromI toX in the sense that

(i) xk(·) converges uniformly tox(·) over compact subsets ofI;

(ii) x′

k(·) converges weakly tox

′

Theorem 1.1.4 (the Convergence Theorem)LetF be a proper hemicontinuous map from a Hausdorff locally convex spaceX to the closed convex subsets of a Banach SpaceY. LetI be an interval ofR andxk(·) and yk(·) be measurable functions fromI toY respectively satisfying

for almost allt inI and for every neighborhoodℵof 0 inX×Y, there exists ak0=k0(t,ℵ) such

that for everyk0≤k,(xk(t), yk(t))∈graph(F) +ℵ.If,

(i) xk(·) converges almost everywhere to a functionx(·) fromI to X;

(ii) yk(·) belongs toL2(I, Y) and converges weakly toy(·) inL2(I, Y),

then, for almost allt∈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 functiont 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]

d

dtV(u(t)) =

h,du dt(t)

∀h∈∂V(u(t)).

3

The Main Result

THEOREM:IfF : Ω→2Rm

andV :Rm_{→Rsatisfy the assumptions}

(A1) Ω⊂R2m _{where Ω is open andF : Ω→2}Rm

is a compact valued upper semicontinuous multifunction;

(A2) there exists a lower semicontinuous proper convex functionV :Rm_{→Rsuch that}

F(x, y, z)⊂∂V(z) for every (x, y, z)∈Ω.

Then, for every (x0, y0, z0)∈Ω,there exists aT >0 and a solutionx: [0, T]→Rmof

x(3)(t)∈F(x(t), x′

(t), x′′

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) =z}

0, andx(3)(t)∈F(x(t), x′(t), x′′(t)),a.e. on [0, T].

PROOF:Suppose (x0, y0, z0)∈Ω. Then,K= ¯Br(x0, y0, z0)⊂Ω for somer >0 since Ω is open.

By assumption (A1)

F(K) = [

(x,y,z)∈K

F(x, y, z)

is compact. Then there exists anM >0 such that

sup{kvk: v∈F(x, y, z),(x, y, z)∈K)} ≤M. (2)

Set

T <min

(

r M,

r

M

12 , r

M

13 , r

2kz0k , r

2ky0k ,

2r

3kz0k

12)

. (3)

Letn, jbe integers where 1≤j≤n. Settj

n =

jT

n. Fort∈

tj−1

n , t

j n

define,

xn(t) =xjn+ t−t j n

ynj +

1 2 t−t

j n

2

znj +

1 6 t−t

j n

3

vnj, (4)

wherex0

n=x0, yn0 =y0, andz0n=z0.

For 0≤j≤n−1 andvj

n∈F xjn, ynj, zjn

,define



      

      

xj+1

n =xjn+

T n

yj

n+

1 2

T n

2

zj

n+

1 6

T n

3

vj n

yj+1

n =ynj +

T n

zj

n+

1 2

T n

2

vnj

zj+1

n =zjn+

T n

vj n.

(5)

We claim that x1

n, y1n, zn1

∈K. Using (5), we have

z1_n−z0=

z_n0+

T n

v0_n−z0

≤

T n

As well as,

Hence the claim holds. Now supposej≥1. We make the assumption that,

Thus, (6) holds whenj= 1. Let’s suppose assumption (6) holds forj >1. Using (5) we see that,

Thus the assumption holds forzj

n. Using this and (5) we have,

Thus the assumption holds foryj

=x0n+ (j+ 1)

Thus the assumption holds forxj

Finally,

≤M. Similarly, (2) and (3) give the following,

And finally,

kxn(t)k=

xjn+ t−t j n

yjn+

1 2 t−t

j n

2

znj+

1 6 t−t

j n

3

vjn

≤ kx0k+Tky0k+1

2T

2_kz 0k+1

6T

3_{M +}

T

n ky0k+Tkz0k+

1 2T

2_M

+1 2

T n

2

(kz0k+T M) +1

6

T n

3

M

≤ kx0k+Tky0k+1

2T

2_kz 0k+1

6T

3_{M +Tky}

0k+T2kz0k

+1 2T

3_M+1

2T

2_Mkz 0k+1

2T

3_M+1

6T

3_M

=kx0k+ 2Tky0k+ 2T2kz0k+T3M +1

3T

3_M <kx0k+r+4

3r+r+ 1 3r

<kx0k+ 4r.

Since x

(3)

n (t)

≤M∀t∈[0, T] the sequence (x

(3)

n (t)) is bounded inL2([0, T], Rm). Furthermore,

supposeε >0 and∀t∈[0, T], and∀τ∈[0, T],|t−τ|< _Mε.Then,

kx′′

n(t)−x

′′

n(τ)k ≤

Z t

τ

kx(3)n (s)kds

≤

Z t

τ

M ds

=M|t−τ|

≤M ε 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)⇒xon [0, T],

(ii) (x′

n)⇒x

′

on [0, T],

(iii) (x′′

n)⇒x

′′

on [0, T],

(iv) (x(3)n ) converges weakly tox3 inL2([0, T], R m

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 from the definition of subdifferential, gives the following,

V x′′

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

V(x′′

If we letn approach infinity, we have

V(x′′

However, the weak lower semicontinuity of the norm gives,

Thus we have,

x

(3)_(t)

2

= lim

n→∞

x

(3)

n (t)

2 .

Hence the sequencex(3)n

→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, Elec-tronic Journal of Differential Equations, Vol. 2004(2004), No. 141, pp.1-6.

[5] V. Lupulescu;Existence of Soutions for Nonconvex Second Order Differential Inclusions, Ap-plied Mathematics E-Notes, 3(2003), 115-123.