A NOTE ON CONSTRAINED SECOND-ORDER DIFFERENTIAL INCLUSIONS

A NOTE ON CONSTRAINED SECOND-ORDER DIFFERENTIAL INCLUSIONS

WITHOUT CONVEXITY

AURELIAN CERNEA

We prove the existence of viable solutions to the Cauchy problemx∈F(x, x) + f(t, x, x), x(0) = x0, x(0) = y0,x(t) ∈ K, where K ⊂ Rⁿ is a closed set, F a set-valued map contained in the Clarke subdifferential of a uniformly regular function, andfa Carath´eodory map.

AMS 2000 Subject Classifications: 34A60.

Key words: uniform regular function, Clarke subdifferential, viable solution, dif- ferential inclusion.

1. INTRODUCTION

In this note we consider the second order differential inclusions of the form

(1.1) x∈F(x, x) +f(t, x, x), x(0) =x₀, x(0) =y₀,

where F(·,·) : D ⊂ Rⁿ × Rⁿ → P(Rⁿ) is a given set-valued map, f(·,·,·) :D₁⊂R×Rⁿ×Rⁿ → P(Rⁿ) is a given function and x₀, y₀ ∈Rⁿ.

Existence of solutions of problem (1.1) that satisfy a constraint of the formx(t)∈K,∀t, well known as viable solutions, has been studied by many authors, mainly in the case when the multifunction is convex valued andf ≡0 ([2], [7], [8], [9] etc.).

Recently [1], the situation where the multifunction is not convex valued has been considered. More exactly, in [1] is proved the existence of viable solutions of problem (1.1) when F(·,·) is an upper semicontinuous, compact valued multifunction contained in the subdifferential of a proper convex func- tion.

The aim of this note is to improve on the result in [1]. Namely, we prove existence of viable solutions of problem (1.1) in the case where the set- valued map F(·,·) is upper semicontinuous compact valued and contained in the Clarke subdifferential of a uniformly regular function. Note that an alternative improvement of the result in [1] is provided in [6]. More precisely,

MATH. REPORTS9(59),2 (2007), 175–181

the convexity of the function that contains the set-valued map in [1] is relaxed in [6] in the sense that this function is assumed to beφ-convex of order two.

The proof of our result follows general ideas in [1] and [3]. We note that in the proof we only pointed out the differences that appear with respect to the other approaches.

The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel while in Section 3 we prove our main result.

2. PRELIMINARIES

We denote by P(Rⁿ) the set of all subsets of Rⁿ and by R₊ the set of all positive real numbers. For > 0 we putB(x, ) = {y ∈Rⁿ;y−x< } andB(x, ) ={y∈Rⁿ;y−x ≤}. By B we denote the unit ball inRⁿ, by cl(A) the closure of the set A⊂Rⁿ, by co(A) the convex hull of A, and put A= sup{a;a∈A}.

Consider a locally Lipschitz continuous functionV :Rⁿ→R. For every direction v ∈ Rⁿ its Clarke directional derivative at x in the direction v is defined by

D_CV(x;v) = lim sup

y→x, t→0+

V(y+tv)−V(y)

t .

TheClarke subdifferential (generalized gradient) of V at x is defined by

∂_CV(x) ={q∈Rⁿ, q, v ≤D_CV(x;v)∀v∈Rⁿ}. We recall that theproximal subdifferential ofV(·) is defined by

∂_PV(x) ={q ∈Rⁿ, ∃δ, σ >0 such that V(y)−V(x) +σy−x²≥ q, y−x ∀y∈B(x, δ)}.

Definition 2.1 ([3]). Let V :Rⁿ →R∪ {∞} be a lower semicontinuous function and letO ⊂dom(V) be a nonempty open subset. We say that V is uniformly regularover O if there exists a positive numberβ >0 such that for allx∈O and for all ξ∈∂_PV(x) one has

ξ, y−x ≤V(y)−V(x) +βy−x² ∀y∈O.

We say that V is uniformly regular over the closed set K if there exists an open setO containing K such thatV is uniformly regular over O.

In [3] there are several examples and properties of such maps. According to [3], any lower semicontinuous proper convex functionV is uniformly regular over any nonempty compact subset of its domain with β = 0; any lower-C² functionV is uniformly regular over any nonempty convex compact subset of its domain.

The next results will be used in the proof of our result.

Lemma2.2 ([3]). Let V :Rⁿ→R be a locally Lipschitz function and let

∅ =K⊂dom(V) a closed set. If V is uniformly regular over K, then i) ∂_CV(x) =∂_PV(x) ∀x∈K;

ii) if z(·) : [0, τ] → Rⁿ is absolutely continuous, g(·) : [0, τ] → Rⁿ is measurable and g(t)∈∂_CV(z(t)) a.e. ([0, τ]), then

(V ◦z)(t) =z(t)², a.e.([0, τ]).

In what follows we assume

Hypotheses 2.3. i) Ω = K ×O, where K ⊂ Rⁿ is a closed set and O⊂Rⁿ is a nonempty open set.

ii) F(·,·) : Ω → P(Rⁿ) is upper semicontinuous (i.e., ∀z ∈ Ω, ∀ > 0 there exists δ > 0 such that z−z < δ implies F(z) ⊂ F(z) +B) with compact values.

iii)f(·,·,·) :R×Ω→Rⁿ is a Carath`eodory function, i.e., ∀(x, y)∈Ω, t→f(t, x, y) is measurable, for all t∈Rf(t,·) is continuous and there exists m(·)∈L²(R,R+) such thatf(t, x, y) ≤m(t) ∀(t, x, y)∈R×Ω.

iv) For all (t, x, v) ∈R×Ω, there exists w∈F(x, v) such that lim inf

h→0

1 h²d

x+hv+h² 2 w+

_t+h

t f(s, x, v)ds, K

= 0.

v) There exists a locally Lipschitz function V : Rⁿ → R∪ {+∞} uni- formly regular overO such that

F(x, y)⊂∂_CV(y), ∀(x, y)∈Ω.

3. THE MAIN RESULT

In order to prove our result we need the following lemmas.

Lemma3.1 ([1]). Assume that Hypotheses 2.3 i)–iv) are satisfied. Con- sider (x₀, y₀)∈Ω, r >0 such that B(x₀, r)⊂O, M := sup{F(t, x); (t, x)∈ Ω₀ := [K×B(y₀, r)]∩B((x₀, y₀), r)}, T₁ >0such that_T1

0 (m(s) +M+ 1)ds <

3r,T₂= min{_3(M^r₊₁₎,_3(y^2r

0+r)}andT ∈(0,min{T₁, T₂}). Then for every >0 there exist η ∈ (0, ) and p ≥ 1 such that for all i= 1, . . . , p−1 there exists (h_i,(x_i, y_i), w_i)∈[η, ]×Ω₀×Rⁿ for which

x_i =x_i−1+h_i−1y_i−1+h²_i−1

2 w_i−1+

_{hi−2+hi−1}

hi−2

f(s, x_i−1, y_i−1)ds∈K, y_i =y_i−1+h_i−1w_i−1, w_i ∈F(x_i, y_i) +

TB,

(x_i, y_i)∈Ω₀, p−1

i=0

h_i < T ≤ p

i=0

h_i.

Moreover, for >0 small enough we have _p−1

i=0 h²_i

2 ≤_p−1

i=0 h_i < T.

Fork≥1 and q= 1, . . . , pdenote byh^k_q the real number associated with = ¹_k and (t, x, y) = (h^k_q−1, x_q, y_q) given by Lemma 3.1.

Define t⁰_k = 0, t^p_k =T, t^q_p = h^k₀ +· · ·+h^k_q−1 and consider the sequence x_k(·) : [t^q−1_k , t^q_k]→Rⁿ,k≥1, defined by

x_k(0) =x₀, x_k(t) =x_q−1+(t−t^q−1_k )y_q−1+1

2(t−t^q−1_k )²w_q−1+ _t

t^q−1_k (t−s)f(s, x_q−1, y_q−1)ds.

Lemma 3.2 ([1]). Assume that Hypotheses 2.3 i)–iv) are satisfied and consider the sequencex_k(·) constructed above. Then there exist a subsequence, still denoted byx_k(·), and an absolutely continuous function x(·) : [0, T]→Rⁿ such that

i)x_k(·) converges uniformly to x(·);

ii) x_k(·) converges uniformly to x(·);

iii)x_k(·) converges weakly in L²([0, T],Rⁿ) to x(·);

iv)the sequence(_p

q=1

_t^q_k

t^q−1_k < x_k(s), f(s, x_k(t^q−1_k ), x_k(t^q−1_k ))>ds)_kcon- verges to_T

0 < x(s), f(s, x(s), x(s))>ds;

v)for every t∈(0, T) there existsq∈ {1, . . . , p} such that

k→∞lim d((x_k(t), x_k(t), x_k(t)−f(t, x_k(t^q−1_k ), x_k(t^q−1_k ))),graph(F)) = 0;

vi)x(·) is a solution of the convexified problem

x∈coF(x, x) +f(t, x, x), x(0) =x₀, x(0) =v₀. We are now able to prove our result.

Theorem3.3. Assume that Hypotheses2.3are satisfied. Then for every (x₀, y₀) ∈ Ω there exist T > 0 and a solution x(·) : [0, T] → Rⁿ of problem (1.1) that satisfiesx(t)∈K ∀t∈[0, T].

Proof. Let (x₀, y₀)∈Ω and considerr >0,T >0 as in Lemma 3.1 and x_k(·) : [0, T] → Rⁿ, x(·) : [0, T] → Rⁿ as in Lemma 3.2. Let β > 0 be the constant appearing in Definition 2.1.

It follows from the statement vi) in Lemma 3.2 and Hypothesis 2.3 v) that

(3.1) x(t)−f(t, x(t), x(t))∈∂_CV(x(t))

for almost allt∈[0, T].

Since the mapping x(·) is absolutely continuous, from (3.1) and Lem- ma 2.2 we obtain

(3.2) (V(x(t))) =x(t), x(t)−f(t, x(t), x(t)) a.e.[0, T].

From (3.2) we have (3.3) V(x(T))−V(y₀) =

_T

0 x(s)²ds− _T

0 x(s), f(s, x(s), x(s) ds.

On the other hand, for q= 1, . . . , pand t∈[t^q−1_k , t^q_k),

x_k(t)−f(t, x_k(t^q−1_k ), x_k(t^q−1_k ))∈F(x_k(t^q−1_k ), x_k(t^q−1_k )) + 1 kTB and, therefore,

x_k(t)−f(t, x_k(t^q−1_k ), x_k(t^q−1_k ))∈∂_CV(x_k(t^q−1_k )) + 1 kTB.

We deduce the existence ofb^q_k∈B such that x_k(t)−f(t, x_k(t^q−1_k ), x_k(t^q−1_k ))− b^q_k

kT ∈∂_CV(x_k(t^q−1_k )).

By Definition 2.2 we have V(x_k(t^q_k))−V(x_k(t^q−1_k ))≥

x_k(t)−f(t, x_k(t^q−1_k ), x_k(t^q−1_k ))−

−b^q_k kT,

_t^q

k

t^q−1_k x_k(s)ds

−βx_k(t^q_k)−x_k(t^q−1_k )². Using the fact that x_k(·) is constant on [t^q−1_k , t^q_k], we can write

V(x_k(t^q_k))−V(x_k(t^q−1_k ))≥ _t^q

k

t^q−1_k x_k(s), x_k(s) ds−

− _t^q

k

t^q−1_k x_k(s), f(s, x_k(t^q−1_k ), x_k(t^q−1_k )) ds− _t^q

k

t^q−1_k

x_k(s), b^q_k kT

ds−

−βx_k(t^q_k)−x_k(t^q−1_k )². By summing over q the last inequalities we get (3.4) V(x_k(T))−V(y₀)≥

_T

0 x_k(s)²ds−

− p q=1

_t^q

k

t^q−1_k x_k(s), f(s, x_k(t^q−1_k ), x_k(t^q−1_k ))ds+a(k) +b(k),

where a(k) =−

p q=1

1 kT

_t^q

k

t^q−1_k x_k(s), b^q_k ds, b(k) =−β p q=1

x_k(t^q_k)−x_k(t^q−1_k )². On the other hand, we have

|a(k)| ≤ 1 kT

p q=1

b^q_{k t}^q

k

t^q−1_k x_k(s)ds≤

≤ 1 kT

_T

0 x_k(s)ds≤ 1 kT

_T

0 M + 1

T +m(s)

ds and

|b(k)| ≤β p q=1

_t^q

k

t^q−1_k x_k(s)ds ² ≤β

p q=1

1 k

_t^p

k

t^p−1_k x_k(s)²ds≤

≤ β k

_T

0 x_k(s)²ds≤ β k

_T

0 M+ 1

T +m(s) ₂

ds.

We conclude that

k→∞lim a(k) = lim

k→∞b(k) = 0.

Hence, using statement iv) in Lemma 3.2 and the continuity of the function V(·) and letting k→ ∞in (3.4) yield

(3.5) V(x(T))−V(y₀)≥

≥lim sup

k→∞

_T

0 x_k(s)²ds− _T

0 x(s), f(s, x₍s), x(s)) ds.

Using (3.2) we deduce that lim sup

k→∞

_T

0 x_k(t)²dt≤ _T

0 x(t)²dt

and, sincex_k(·) converges weakly inL²([0, T],Rⁿ) tox(·), by the lower semi- continuity of the norm in L²([0, T],Rⁿ) (e.g. Proposition III.30 in [4]) we obtain that

k→∞lim _T

0 x_k(t)²dt= _T

0 x(t)²dt,

i.e., x_k(·) converges strongly in L²([0, T],Rⁿ). Hence there exists a subse- quence (still denoted)x_k(·) that converges pointwise tox(·).

It follows from statement v) in Lemma 3.2 that

d((x(t), x(t), x(t)−f(t, x(t), x(t))),graph(F)) = 0 a.e. ([0, T]) and, since by Hypothesis 2.4 graph(F) is closed, we obtain

x(t)∈F(x(t), x(t)) +f(t, x(t), x(t)) a.e.([0, T]).

In order to prove the viability constraint satisfied byx(·), fixt∈[0, T].

There exists a sequence (t^q_k)_k such that t= lim_k→∞t^q_k. But lim_k→∞x(t)− x_k(t^q_k)= 0 andx_k(t^q_k)∈K. So, the fact thatK is closed givesx(t)∈K and the proof is complete.

Remark3.4. If V(·) :Rⁿ →R is a proper lower semicontinuous convex function, then∂_cV(x) =∂V(x), where∂V(·) is the subdifferential in the sense of Convex Analysis ofV(·), and Theorem 3.3 yields Theorem 2.1 in [1].

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Received 27 February 2006 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania