ON THE EXISTENCE OF VIABLE SOLUTIONS FOR A CLASS OF SECOND ORDER

ON THE EXISTENCE OF VIABLE SOLUTIONS FOR A CLASS OF SECOND ORDER

DIFFERENTIAL INCLUSIONS Aurelian Cernea

Faculty of Mathematics, University of Bucharest Academiei 14, 70109 Bucharest, Romania

Abstract

We prove the existence of viable solutions to the Cauchy problem x

∈ F (x, x

), x(0) = x

, x

(0) = y

, where F is a set-valued map defined on a locally compact set M ⊂ R

, contained in the Fr´echet subdifferential of a φ-convex function of order two.

Keywords: viable solutions, φ-monotone operators, differential inclusions.

2000 Mathematics Subject Classification: 34A60.

1. Introduction

In the present paper, we consider the second order differential inclusion of the form

(1.1) x

∈ F (x, x

), x(0) = x

, x

(0) = y

,

where F (., .) : M ⊂ R

× R

→ P(R

) is a given set-valued map and x

, y

∈ R

.

Existence of viable solutions to problem (1.1) has been studied by many authors, mainly in the case when the multifunction is convex valued ([2], [6], [8], [10] etc.).

Recently in [9], the situation when the multifunction is not convex val-

ued is considered. More exactly, in [9] the existence of viable solutions to

the problem (1.1) is proved when F (., .) is an upper semicontinuous, com-

pact valued multifunction contained in the subdifferential of a proper convex

function.

The aim of this paper is to extend the result of [9] to the case when the multi- function F is contained in the Fr´echet subdifferential of a φ-convex function of order two. Since the class of proper convex functions is strictly contained in the class of φ-convex functions of order two, our result generalizes the one in [9].

On the other hand, our result may be considered as an extension of our previous existence result ([5]) obtained for first order differential inclusion of the form

(1.2) x

∈ F (x), x(0) = x

to the more general problem (1.1). The proof of our main result follows the general ideas of [5] and [9].

The paper is organized as follows: in Section 2, we recall some prelim- inary facts that we need in the sequel and in Section 3, we prove our main result.

2. Preliminaries

We denote by P(R

) the set of all subsets of R

and by R

the set of all positive real numbers. For ² > 0 we put B(x, ²) = {y ∈ R

; ky − xk < ²}.

With B we denote the unit ball in R

. By cl(A) we denote the closure of the set A ⊂ R

, by co(A) we denote the convex hull of A and we put kAk = sup{kak; a ∈ A}.

Let Ω ⊂ R

be an open set and let V : Ω → R ∪ {+∞} be a function with domain D(V ) = {x ∈ R

; V (x) < +∞}.

Definition 2.1. The multifunction ∂

V : Ω → P(R

), defined as:

∂

V (x) = {α ∈ R

, lim inf

y→x

V (y) − V (x)− < α, y − x >

ky − xk ≥ 0} if V (x) < +∞

and ∂

V (x) = ∅ if V (x) = +∞ is called the Fr´echet subdifferential of V . We also put D(∂

V ) = {x ∈ R

; ∂

V (x) 6= ∅}.

According to [4] the values of ∂

are closed and convex.

Definition 2.2. Let V : Ω → R∪{+∞} be a lower semicontinuous function.

We say that V is a φ-convex of order 2 if there exists a continuous map

φ

: (D(V ))

× R

→ R

such that for every x, y ∈ D(∂

V ) and every α ∈ ∂

V (x) we have

(2.1) V (y) ≥ V (x)+ < α, x−y > −φ

(x, y, V (x), V (y))(1+kαk

)kx−yk

. In [4] there are several examples and properties of such maps.

For M ⊂ R

and x ∈ M we recall that the contingent cone to M at x is defined by

K

(x) = {v ∈ R

; lim inf

h→0+

1

h d(x + hv, M ) = 0}

and the second-order contingent set to M at (x, y) ∈ M × R

is defined by:

K

(x, y) = {v ∈ R

; lim inf

h→0+

d(x + hy +

, M ) h

/2 = 0}.

Remark 2.3 ([2], [6], [10]). If F (., .) is upper semicontinuous, compact convex valued and x(.) : [0, T ] → R

is a solution to the Cauchy problem (1.1) such that x(t) ∈ M , ∀t ∈ I, then

(x(t), x

(t)) ∈ graph(K

(.)), ∀t ∈ [0, T ].

In what follows, for a multifunction F : M ⊂ R

× R

→ P(R

) and for any (x

, y

) ∈ M we consider the problem (1.1) under the following assumptions.

Hypothesis 2.4.

(i) M = M

× M

⊂ R

× R

is a locally compact set such that graph(K

(.)) ⊂ M.

(ii) F is upper semicontinuous (i.e., ∀z ∈ M, ∀² > 0 there exists δ > 0 such that kz − z

k < δ implies F (z

) ⊂ F (z) + ²B) with compact values and such that

F (x, y) ∩ K

(x, y) 6= ∅, ∀(x, y) ∈ M.

(iii) There exists a proper lower semicontinuous φ-convex function of order two V : R

→ R ∪ {+∞} such that

F (x, y) ⊂ ∂

V (y), ∀(x, y) ∈ M.

Finally, by a viable solution to problem (1.1) we mean an absolutely con- tinuous function x(.) : [0, T ] → R

with an absolutely continuous derivative x

(.) such that x(0) = x

, x

(0) = y

,

x

(t) ∈ F (x(t), x

(t)) a.e. ([0, T ]) and

(x(t), x

(t)) ∈ M, ∀t ∈ [0, T ].

3. The main result

In order to prove our main result we need the following lemma.

Lemma 3.1 ([9]). Assume that Hypothesis 2.4 is verified and let M

⊂ M be a compact subset such that (x

, y

) ∈ M

. Then, for every k ∈ N there exist h

∈ (h

(k),

] and u

∈ R

such that

x

+ h

y

+ (h

)

2 u

∈ M

, (x

, y

, u

) ∈ graph(F ) + 1

k (B × B × B).

Our main result is the following:

Theorem 3.2. Consider F : M → P(R

) and V : R

→ R∪{∞} satisfying Hypothesis 2.4. Then, for every (x

, y

) ∈ graph(K

(.)) there exist T > 0 and x(.) : [0, T ] → R

a viable solution to problem (1.1).

P roof. Let (x

, y

) ∈ graph(K

(.)). Since M is locally compact, there exists r > 0 such that M

= M ∩ B

(x

, y

) is a compact set. Moreover, by the upper semicontinuity of F and Proposition 1.1.3 in [1], the set

F (B

(x

, y

)) = ∪

0,y0)

F (x, y) is compact, hence there exists L > 0 such that

sup{kvk; v ∈ F (x, y); (x, y) ∈ B

(x

, y

)} ≤ L < +∞.

Let φ

be the continuous function appearing in Definition 2.2.

Since V (.) is continuous on D(V ) (e.g. [7]), by possibly decreasing r one can assume that for all y ∈ B

(y

) ∩ D(V )

|V (y) − V (y

)| ≤ 1.

Set

S := sup{φ

(y

, y

, z

, z

); y

∈ B

(y

), z

∈ [V (y

) − 1, V (y

) + 1], i = 1, 2}, T = min

½ r

2(L + 1) ,

r r

L + 1 , r 2(ky

k + 1)

¾ .

By Lemma 3.1, for (x

, y

) ∈ M

there exist h

∈ (h

(k),

] and u

∈ R

such that

x

+ h

y

+ (h

)

2 u

∈ M

, (x

, y

, u

) ∈ graph(F ) + 1

k (B × B × B).

We define

x

:= x

+ h

y

+ (h

)

2 u

, y

:= y

+ h

u

.

According to the choice of T and if h

< T we have kx

− x

k ≤ h

ky

k + (h

)

2 ku

k

≤ h

ky

k + (h

)

2 (L + 1) < r, ky

− y

k ≤ h

ku

k ≤ h

(L + 1) < r.

Hence (x

, y

) ∈ M

and applying again Lemma 3.1 we find h

∈ (h

(k),

] and u

∈ R

such that

x

+ h

y

+ (h

)

2 u

∈ M

, (x

, y

, u

) ∈ graph(F ) + 1

k (B × B × B).

By recurrence, for each k ∈ N , there exist m

∈ N

and h

, x

, y

, u

such

that for every p = 2, . . . , m

− 1 we have:

(a) ^P

h

≤ T < ^P

h

,

(b) x

= x

+ ( ^P

h

)y

+

^P

(h

)

u

+ ^P

^P

h

h

u

, (c) y

= y

+ ^P

h

u

,

(d) (x

, y

) ∈ M

,

(e) (x

, y

, u

) ∈ graph(F ) +

(B × B × B).

Assume that h

, x

, y

, u

have been constructed for q ≤ p satisfying (a) – (e) and we construct h

, x

, y

, u

.

By Lemma 3.1, since (x

, y

) ∈ M

there exist h

∈ (h

(k),

] and u

∈ R

such that

x

+ h

y

+ (h

)

2 u

∈ M

, (x

, y

, u

) ∈ graph(F ) + 1

k (B × B × B).

If h

+ h

+ . . . + h

≥ T , then we set m

= p. If h

+ h

+ . . . + h

< T we define

x

:= x

+ h

y

+ (h

)

2 u

, y

:= y

+ h

u

.

From (b) and (c) it follows

x

= x

+ h

y

+ (h

)

2 u

= x

+ ^³ X

h

^´ y

+ 1 2

X

(h

)

u

+

p−1

X

i=0

X

h

h

u

,

y

:= y

+ h

u

= y

+ X

h

u

.

We deduce that

kx

− x

k ≤ ^³ X

h

^´ ky

k + L + 1 2

³ X

i=0

h

^´

< r, ky

− y

k ≤

X

h

ku

k ≤ (L + 1) ^³ X

h

^´ < r,

hence (x

, y

) ∈ M

.

Let us note that this iterative process is finite because h

∈ (h

(k),

], implies the existence of m

∈ N such that

h

+ h

+ . . . + h

≤ T < h

+ h

+ . . . + h

.

By (e), for every k ∈ N and any p ∈ {0, 1, . . . , m

} there exists (a

, b

, v

) ∈ graph(F ) such that

kx

− a

k < 1

k , ky

− b

k < 1

k , ku

− v

k < 1 k . It follows that

kx

k ≤ kx

− x

k + kx

k ≤ 1 + kx

k, ky

k ≤ ky

− x

k + ky

k ≤ 1 + ky

k, ku

k ≤ ku

− v

k + kv

k ≤ 1 + L.

For every k ∈ N

, by the upper semicontinuity of F (.) in (x

, y

) ∈ M

, there exists δ = δ(k) such that kx − x

k < δ, ky − y

k < δ implies F (x, y) ⊂ F (x

, y

) +

B. One may take δ ≥

, hence ka

− x

k <

δ, kb

− y

k < δ and then

(3.1)

F (a

, b

) ⊂ F (x

, y

) + 1 k B F (a

, b

) + 1

k B ⊂ F (x

, y

) + 2 k B u

∈ F (x

, y

) + 2

k B

We define t

= h

+ h

+ . . . + h

, t

= 0. Obviously, ∀ k ∈ N , p ∈ {1, . . . , m

}

t

− t

< 1

k and t

≤ T < t

. For k ≥ 1, p ∈ {1, . . . , m

} we define I

:= [t

, t

] and

x

(t) = x

+ (t − t

)y

+ 1

2 (t − t

)

u

, t ∈ I

.

Hence

x

(t) = y

+ (t − t

)u

, t ∈ I

. x

(t) = u

, t ∈ I

,

thus, for all t ∈ [0, T ]

(3.2)

kx

(t)k ≤ ku

k < L + 1,

kx

(t)k ≤ ky

k + (t − t

)ku

k < ky

k + L + 2, kx

(t)k ≤ kx

k + ky

k + L + 3.

On the other hand,

kx

(t) − x

k ≤ 1

k (ky

k + L + 2), ∀t ∈ [0, T ], kx

(t) − y

k ≤ 1

k (L + 1), ∀t ∈ [0, T ], hence from (e) we find that

(x

(t), x

(t), x

(t)) ∈ graphF + ²(k)(B × B × B), where ²(k) → 0 as k → ∞.

Then by (3.2) we find that x

(.) is bounded in L

([0, T ], R

), x

(.), x

(.) are bounded in C([0, T ], R

), and equi-Lipschitzean; therefore, apply- ing Theorem 0.3.4 in [1] we obtain the existence of a subsequence (again denoted by x

(.)) and an absolutely continuous function x(.) : [0, T ] → R

such that

x

(.) converges uniformly to x(.), x

(.) converges uniformly to x

(.),

x

(.) converges weakly in L

([0, T ], R

) to x

(.).

Taking into account Hypothesis 2.4 and Theorem 1.4.1 in [1] we obtain that x

(t) ∈ coF (x(t), x

(t)) ⊂ ∂

V (x

(t)) a.e. ([0, T ]).

Since the mapping x(.) is absolutely continuous and x

(t) ∈ ∂

V (x

(t))

almost everywhere on [0, T ], we apply Theorem 2.2 in [4] and we deduce

that there exists T

> 0 such that the mapping t → V (x

(t)) is absolutely continuous on [0, min{T, T

}] and

(3.3) (V (x

(t)))

=< x

(t), x

(t) > a.e. [0, min{T, T

}].

Without loss of generality we may assume that T = min{T, T

}.

Using the properties of the mapping V (.), the definition of S and (3.3) we find b

∈ B such that

u

− 2

k b

∈ F (x

(t), x

(t)) ⊂ ∂

V (x

(t)) and such that for every t ∈ I

we have

V (x

(t

)) − V (x

(t

)) ≥ hu

− 2 k b

,

Z

k

t^p−1_k

x

(t)dti

− φ

(x

(t

), x

(t

), V (x

(t

)), V (x

(t

)) . ^³ 1 + ku

− 2

k b

k

^´ kx

(t

) − x

(t

)k

. If T ∈ (t

, t

], we obtain

V (x

(T )) − V (x

(t

)) ≥ hu

− 2

k b

, Z

t^mk−1_k

x

(t)dti

− φ

(x

(T ), x

(t

), V (x

(T )), V (x

(t

)) . ^³ 1 + ku

− 2

k b

k

^´ kx

(T ) − x

(t

)k

. By adding on p the last inequalities we get

V (x

(T )) − V (y

) ≥ Z

0

kx

(t)k

dt + a(k) + b(k), where

a(k) = −

mk

X

p=1

2

k hb

, (t

− t

)u

i b(k) = −

mk

X

p=1

φ

(x

(t

), x

(t

)

, V (x

(t

)), V (x

(t

)) . ^³ 1 + ku

− 2

k b

k

^´ kx

(t

) − x

(t

)k

.

On the other hand, one has

|a(k)| ≤ 2 k

mk

X

p=1

kb

k.k Z

k

t^p−1_k

x

(t)dtk

≤ 2 k

Z

0

kx

(s)kds ≤ 1

k 2T (L + 1) and

|b(k)| ≤

mk

X

p=1

S(1 + L

)

mk

X

p=1

k Z

k

t^p−1_k

x

(t)dtk

≤ S(1 + L

)

mk

X

p=1

1 k

Z

k

t^p−1_k

kx

(t)k

dt ≤ S(1 + L

)

mk

X

p=1

1 k

Z

0

kx

(t)k

dt

≤ 1

k S(1 + L

)T (L + 1)

and thus lim

a(k) = lim

b(k) = 0. We deduce that (3.4) V (x

(T )) − V (y

) ≥ lim sup

k→∞

Z

0

kx

(t)k

dt . Using (3.3) we infer that

lim sup

k→∞

Z

0

kx

(t)k

dt ≤ Z

0

kx

(t)k

dt

and, since {x

(.)}

converges weakly in L

([0, T ], R

) to x

(.), by the lower semicontinuity of the norm in L

([0, T ], R

) (e.g. Proposition III.30 in [3]) we obtain that

k→∞

lim Z

0

kx

(t)k

dt = Z

0

kx

(t)k

dt

i.e., x

(.) converges strongly in L

([0, T ], R

). Hence, there exists a subse- quence (still denoted) x

(.) that converges pointwise to x

(.).

On the other hand, from Hypothesis 2.4 graph(F ) is closed and it follows that

k→∞

lim d((x

(t), x

(t), x

(t)), graph(F )) = 0.

Thus

x

(t) ∈ F (x(t), x

(t)) a.e. ([0, T ]).

It remains to prove that (x(t), x

(t)) ∈ M, ∀t ∈ [0, T ]. By the definition of x

(.) and (3.2) we have

kx

(t) − x

k < 1

k (ky

k + L + 2), kx

(t) − y

k < 1

k (L + 1)

and thus lim

d((x

(t), x

(t)), (x

, y

)) = 0. But (x

, y

) ∈ M

, ∀n ∈ N

, so we have that

d((x(t), x

(t)), M

) ≤ d((x(t), x

(t)), (x

(t), x

(t))) + d((x

(t), x

(t)), (x

, y

)) + d((x

, y

), M

) and letting k → ∞ we find that

d((x(t), x

(t)), M

) = 0.

Finally, using the fact that M

is closed we get that (x(t), x

(t)) ∈ M

, ∀t ∈ [0, T ] which completes the proof.

Remark 3.3. If V (.) : R

→ R is a proper lower semicontinuous convex function, then (e.g. [7]) ∂

V (x) = ∂V (x), where ∂V (.) is the subdifferential in the sense of convex analysis of V (.), and Theorem 3.2 yields the result in [9].

References

[1] J.P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin 1984.

[2] A. Auslender and J. Mechler, Second order viability problems for differential inclusions, J. Math. Anal. Appl. 181 (1984), 205–218.

[3] H. Brezis, Analyse fonctionelle, th´eorie et applications, Masson, Paris 1983.

[4] T. Cardinali, G. Colombo, F. Papalini and M. Tosques, On a class of evolution equations without convexity, Nonlinear Anal. 28 (1996), 217–234.

[5] A. Cernea, Existence of viable solutions for a class of nonconvex differential

inclusions, J. Convex Anal., submitted.

[6] B. Cornet and B. Haddad, Th´eoreme de viabilit´e pour inclusions differentielles du second order, Israel J. Math. 57 (1987), 225–238.

[7] M. Degiovanni, A. Marino and M. Tosques, Evolution equations with lack of convexity, Nonlinear Anal. 9 (1995), 1401–1443.

[8] T.X.D. Ha and M. Marques, Nonconvex second order differential inclusions with memory, Set-valued Anal. 5 (1995), 71–86.

[9] V. Lupulescu, Existence of solutions to a class of second order differential inclusions, Cadernos de Matematica, Aveiro Univ., CM01/I-11.

[10] L. Marco and J.A. Murillo, Viability theorems for higher-order differential inclusions, Set-valued Anal. 6 (1998), 21–37.

[11] M. Tosques, Quasi-autonomus parabolic evolution equations associated with a class of non linear operators, Ricerche Mat. 38 (1989), 63–92.

Received 20 February 2002