Journal of Convex Analysis Volume 7 (2000), No. 1, 183–195

On the Second-Order Contingent Set and Differential Inclusions

Brahim Aghezzaf

Département de mathématiques et d’ informatique, Faculté des Sciences Aïn Chock, BP: 5366 Maarif, Casablanca, Maroc. e-mail: aghezzaf@facsc-achok.ac.ma

Saïd Sajid

Département de mathématiques, F.S.T.M, BP. 146, Mohammadia, Maroc. e-mail: saidsajid@uh2m.ac.ma

Received May 4, 1998 Revised manuscript received December 9, 1999

In this paper, we establish the existence of solutions of a nonconvex second order differential inclusion of the following type:

x^{..} (t) ∈ F (x(t), x^{.} (t)) a.e, x(0) = x_{0} ∈ K, x^{.} (0) = v_{0} ∈ Ω,

such that x(t) ∈ K, where K is a closed subset and Ω is an open subset of ℝ^{n}. When K is in addition convex, we introduce the contingent cone T_{K} to prove the existence of solutions of the differential inclusion:

x^{..} (t) ∈ G(x(t), x^{.} (t)) a.e, x(t) ∈ K and x^{.} (t) ∈ T_{K}(x(t))

Keywords: Continuous multifunctions, compact multifunctions, contingent cone, second-order contingent set

1991 Mathematics Subject Classification: 34A60

1. Introduction

Let K be a nonempty closed subset of ℝ^{n}. We shall denote by d_{K}(.) or d(., K) the distance to K, by T_{K}(x) the contingent cone at x, by gr(T_{K}) the graph of the multifunction x → T_{K}(x) and by A_{K}(x, y) the second-order contingent set to K, i.e

A_{K}(x, y) = \{z ∈ ℝ^{n} : lim inf_{t→0^{+}} \frac{d_{K}(x + ty + \frac{t^{2}}{2} z)}{t^{2}} = 0\}

where (x, y) ∈ K × ℝ^{n}. For more on properties of the second-order contingent set, we refer to [4, 5, 8, 9].

In the present paper, we are concerned with A_{K}(x, y) in the case where the set K is, in addition, convex. More precisely, for all (x, y) ∈ gr(T_{K}), we prove that A_{K}(x, y) and the second-order contingent set

C_{K}(x, y) = \{z ∈ ℝ^{n} : lim_{t→0^{+}} \frac{d_{K}(x + ty + \frac{t^{2}}{2} z)}{t^{2}} = 0\}

ISSN 0944-6532 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

184 B. Aghezzaf, S. Sajid / On the second-order contingent set and differential inclusions

coincide with T_{K}(x).

As application, we establish in a special case where K is defined by constraint equalities that the Aubin’s contingent derivative set D(T_{K})(x, y)(y) at (x, y) in direction y is equal to T_{K}(x).

Another application consists of studying the existence of second-order viable solution for a class of multivalued differential equation with nonconvex right-hand side.

Throughout this paper, we denote by Ω a nonempty open subset of ℝ^{n}, by D a nonempty closed convex subset of ℝ^{n}, by gr(T_{D}) the graph of the multifunction T_{D}(.), by C(ℝ^{n}) the collection of all nonempty compact subsets of ℝ^{n} and by F and G two C(ℝ^{n})- continuous multifunctions defined on K × Ω respectively. Let (x_{0}, v_{0}) be an element of K × Ω or gr(T_{D}). Consider the problems:

\begin{cases}x^{..} (t) ∈ F (x(t), x^{.} (t)) & a.e on [0, T [, \\ x(0) = x_{0}, x^{.} (0)) = v_{0}, \\ x(t) ∈ K & ∀t ∈ [0, T [. \end{cases}  (1.1)

and

\begin{cases}x^{..} (t) ∈ G(x(t), x^{.} (t)) & a.e on [0, T [, \\ x(0) = x_{0}, x^{.} (0)) = v_{0}, \\ (x(t), x^{.} (t)) ∈ gr(T_{D}) & ∀t ∈ [0, T [. \end{cases}  (1.2)

By a solution of (1.1) or (1.2) we mean (T, x(.)) where T > 0 and x : [0, T] → ℝ^{n} is an absolutely continuous trajectory for which x^{.} (.) is also absolutely continuous, which satisfies (1.1) or (1.2).

Similar problems were investigated by [4, 6]. In the present paper we study problem (1.1) (resp. (1.2)) under the assumption H_{1} : ∀ (x, y) ∈ K × Ω, F (x, y) ⊂ C_{K}(x, y) (resp. H_{2} : ∀ (x, y) ∈ gr(T_{D}), G(x, y) ⊂ T_{K}(x)).

2. Notations, definitions and main results

Let X, Y be two metric spaces, R be a closed valued multifunction from X to Y. We denote by gr(R) the graph of R. We say that R is lower semi-continuous if for any open subset V of Y the set {x ∈ X : R(x) ∩ V ≠ ∅} is open. Let (C(Y), h) be the collection of all nonempty closed subsets of Y equipped with the Hausdorff distance h defined by h(A, B) = max{e(A, B), e(B, A)}, where e(A, B) = sup{d(x, B) : x ∈ A}. A C(Y)- valued multifunction R defined on X is continuous if the mapping R : X → (C(Y), h) is continuous.

For any vector normed space S and a nonempty subset A of S, we denote by cl A, co A, χ_{A} the closure, the convex hull and the characteristic function of A. For all x ∈ S, π_{A}(x) stands for the set of all y ∈ A for which ∥x − y∥ = d(x, A).

For r > 0, we denote by B(x, r) the closed with center at x and radius r.

B. Aghezzaf, S. Sajid / On the second-order contingent set and differential inclusions 185

Assumption H_{1} Let K be a nonempty closed subset of ℝ^{n}, Ω be a nonempty open subsets of ℝ^{n} and F be a multifunction from K ×Ω to the space of all nonempty subsets of ℝ^{n}. Let (x_{0}, v_{0}) ∈ K ×Ω. On F we make the following hypotheses:

- F is continuous with compact values.

- ∀(x, y) ∈ K × Ω, F (x, y) ⊂ C_{K}(x, y). Assumption H_{2} Let D be a nonempty closed convex subset of ℝ^{n}, G be a multifunction from gr(T_{D}) to nonempty subsets of ℝ^{n}. Let (x_{0}, v_{0}) ∈ gr(T_{D}). Suppose that:

- G is continuous with compact values.

- ∀ (x, y) ∈ gr(T_{D}), G(x, y) ⊂ T_{D}(x).

- gr(T_{D}) is closed. Here are the main results. Theorem 2.1. Assume that H_{1} is satisfied, and let (x_{0}, v_{0}) ∈ K × Ω, then there exists T_{0} > 0 and an absolutely continuous function x(.) : [0, T_{0}] → ℝ^{n} for which x^{.} (.) is also absolutely continuous such that:

\begin{cases}ẍ(t) ∈ F (x(t), ẋ(t)) & a.e on [0, T_{0}[ \\ x(0) = x_{0} ẋ(0)) = v_{0} \\ x(t) ∈ K & ∀t ∈ [0, T_{0}[ \end{cases}

Theorem 2.2. Assume that H_{2} is satisfied and let (x_{0}, v_{0}) ∈ gr(T_{D}), then there exists T_{1} > 0 and an absolutely continuous function u(.) : [0, T_{1}] → ℝ^{n} for which u^{.} (.) is absolutely continuous and such that:

\begin{cases}u^{..} (t) ∈ G(u(t), u^{.} (t)) & a.e on [0, T_{1}[ \\ u(0) = x_{0}, u^{.} (0)) = v_{0} \\ (u(t), u^{.} (t)) ∈ gr(T_{D}) & ∀t ∈ [0, T_{1}[ \end{cases}

To prove Theorem 2.2, we need the following result: Theorem 2.3. Let C be a nonempty closed convex subset of ℝ^{n}. Then, the second-order contingent set A_{C}(x, y) at (x, y) ∈ gr(T_{C}) coincides with the contingent cone T_{C}(x).

3. Preliminary results In this section, we state some definitions and results collected in Aubin and Cellina [1]. Proposition 3.1. Let A be a nonempty closed convex subset of ℝ^{n} and x ∈ A. Then

T_{A}(x) = cl (∪_{h>0} \frac{1}{h}(A − x))

For the proof see [1, p.174]. Proposition 3.2. The multifunction D ↦ 2^{ℝ^{n}}, x → T_{D}(x) is lower semi-continuous.

Proof. See Aubin and Cellina [1, Th.1, p.220].

Lemma 3.3. Let Q be a lower semi-continuous multifunction from a metric space X to the space of nonempty subsets of a metric space Y. Then for all ε > 0, the set

{x ∈ X : Q(x_{0}) ⊂ Q(x) + B(0, ε)}

is open.

The proof is straighforward and is omitted.

Definition 3.4. Let I be a bounded interval of ℝ, recall that a function f : I → ℝ^{n} is called absolutely continuous if there exists an integrable function g : I → ℝ^{n} such that for all s, t ∈ I we have f(t) = f(s) + ∫_{s}^{t} g(τ)dτ, where g is denoted f^{.}.

Proposition 3.5. Let x, y ∈ ℝ^{n}, ε(.) a mapping with values in ℝ^{n} such that ε(h) → 0 as h → 0. Then

lim inf_{h→0^{+}} \frac{1}{h} [d_{D}(x + h(y + ε(h))) − d_{D}(x)] ≤ d(y, T_{D}(π_{D}(x))).

For the proof, see [1, Prop.1, pp. 202 or Prop.3, pp. 178-179].

Corollary 3.6. Let I be a bounded interval of ℝ and f : I ↦ ℝ^{n} be a lipschitz function. Set g(t) = d_{D}(f(t)). Then g is absolutely continuous and for almost every t ∈ I

g^{.} (t) ≤ d(f^{.} (t), T_{D}(π_{D}(f(t))).

Proof. Let t ∈ I and h be sufficiently small such that t + h ∈ I. We have

f(t + h) = f(t) + ∫_{t}^{t+h} f^{.} (τ) dτ.  (3.1)

Without loss of generality, we may assume that

\frac{1}{h} ∫_{t}^{t+h} f^{.} (s)ds →f^{.} (t) when h → 0.

Then (3.1) implies that

f(t + h) = f(t) + h(f^{.} (t) + ε(h)),  (3.2)

where ε(h) → 0 as h → 0. By using Proposition 3.5, from (3.2) we obtain

g^{.} (t) = lim inf_{h→0^{+}} \frac{1}{h}(g(t + h) − g(t)),
= lim inf_{h→0^{+}} \frac{1}{h} [d_{D}(f(t) + h(f(t) + ε(h))) − d_{D}(f(t))] ,
≤ d((f^{.} (t), T_{D}(π_{D}(f(t))).

Definition 3.7. Let J be an interval of ℝ, x(.): J → ℝ^{n} be a bounded function. We define the oscillation of x(.) on J by

ω_{J}(x(.)) = sup {∥x(t_{1}) − x(t_{2})∥ : t_{1}, t_{2} ∈ J}

Let I be an inteval of ℝ and B(I, ℝ^{n}) the space of all bounded functions from I to ℝ^{n} and H be a nonempty subset of B(I, ℝ^{n}).

Definition 3.8. We say that H is equioscillating if ∀ε > 0, ∃ a finite partition of I into subintervals J_{k} (1 ≤ k ≤ r) such that

∀x(.) ∈ H, ∀k ∈ {1, ..., r} , ω_{J_{k}}(x(.)) ≤ ε.

Theorem 3.9. Assume that H is equioscillating and for all t ∈ I the set H(t) = {x(t) : x(.) ∈ H} is precompact. Then H is precompact in B(I, ℝ^{n}).

For the proof, see Aubin and Cellina [1, Th.5, p.15]

4. Proof of the main results

To begin with, let us prove:

∀(x, v) ∈ gr(T_{D}), A_{D}(x, v) = C_{D}(x, v) = T_{D}(x).

Proof. Let z ∈ A_{D}(x, v). Since D is closed and convex, there exists x_{n} ∈ D, t_{n} → 0 as n → +∞ and a function ε(.) with ε(t) → 0 when t → 0^{+} such that:

x_{n} = x + t_{n}v + \frac{t_{n}^{2}}{2} z + t_{n}^{2}ε(t_{n}), ∀n ∈ ℕ,

hence

\frac{x_{n} − x}{t_{n}} − v = \frac{t_{n}}{2} z + t_{n}ε(t_{n}), ∀n ∈ ℕ.  (4.1)

Since v belongs to T_{D}(x) which is a closed cone using the Proposition 3.1, we deduce from (4.1) that z ∈ T_{D}(x).

Conversely, let z ∈ T_{D}(x). For t > 0, set f(t) = x + tv + \frac{t^{2}}{2} z. By Corollary 3.6, we have that

d(f(t), D) ≤ ∫_{0}^{t} d(v + τz, T_{D}(π_{D}(f(τ)))) dτ,

so that

\frac{1}{t^{2}} d(f(t), D) ≤ \frac{1}{t} ∫_{0}^{t} \frac{1}{τ}d(v + τz, T_{D}(π_{D}(f(τ)))) dτ,  (4.2)

by using Proposition 3.1, it is easy to check that for any τ > 0

\frac{1}{τ}d(v + τz, T_{D}(π_{D}(f(τ)))) = d(\frac{1}{τ}v + z, T_{D}(π_{D}(f(τ)))).

Let ε > 0. Since π_{D}(.) is continuous and z + \frac{1}{τ}v ∈ T_{D}(x), ∀ τ > 0, by Proposition 3.2

and Lemma 3.3, there exists t_{0} > 0 such that

d(\frac{1}{τ}v + z, T_{D}(π_{D}(f(s)))) < ε,  ∀s ≤ t_{0}, ∀τ > 0,  (4.3)

thus by combining (4.2) and (4.3) it follows that

\frac{1}{t^{2}} d(f(t), D) < ε, ∀t ≤ t_{0}.

This implies that z ∈ C_{D}(x, v).

Hence A_{D}(x, v) = C_{D}(x, v) = T_{D}(x). This completes the proof.

Remark 4.1. If C a nonempty compact subset of gr(A_{D}), then

\frac{1}{t^{2}} d_{D}(x + ty + \frac{t^{2}}{2} z) → 0 as t → 0^{+}

uniformly on C.

As application, let us explicit the Aubin’s notion of contingent derivative of T_{L} at point (x_{0}, v_{0}) in direction v_{0} defined by

D(T_{L})(x_{0}, v_{0})(v_{0}) = {w ∈ ℝ^{n} : (x_{0}, w) ∈ T_{gr(T_{L})}(x_{0}, v_{0})},

where L is a set defined by constraint equalities. More precisely, suppose that:

L = {x ∈ Ω : f_{i}(x) = 0, ∀i = 1, ..., m} ,

where the f_{i} are real-valued functions defined and C^{2} on an open subset ℝ^{n}. Suppose that the gradients (∇f_{i}(x_{0}))_{i=1,...,m} are linearly independent. Then, it has been proved respectively in [5, Prop.9] and [6, Prop.2.2] that

A_{L}(x_{0}, v_{0}) = {w ∈ ℝ^{n}, ⟨∇f_{i}(x_{0}), w⟩ + ⟨∇^{2}f_{i}(x_{0})v_{0}, v_{0}⟩ = 0, ∀i = 1, ..., m}

and

D(T_{L})(x_{0}, v_{0})(v_{0}) = A_{L}(x_{0}, v_{0}),

where ∇^{2}f_{i}(x_{0}) denotes the Hessian matrix at x_{0}. Hence if L is in addition convex, then by virtue of Theorem 2.3, we have

D(T_{L})(x_{0}, v_{0})(v_{0}) = T_{L}(x_{0}).

We are able to give the proof of Theorem 2.1.

Proof. Let r_{0} > 0 be such that B(v_{0}, r_{0}) ⊂ Ω. Let w_{0} ∈ F (x_{0}, v_{0}). Set

r = \frac{r_{0}}{2} , K_{0} = B ((x_{0}, v_{0}), r) ∩ (K × B(v_{0}, r)) .

B. Aghezzaf, S. Sajid / On the second-order contingent set and differential inclusions 189

Since F is continuous with compact values, there exists M > max(1, ∥v_{0}∥ + r) such that

h (F (x, y), {0}) ≤ M − 1 ∀(x, y) ∈ K_{0}.  (4.4)

Set

K_{1} = (K_{0} × B(0, M − 1)) ∩ gr(C_{K}), T = \frac{r}{M}

and

η_{k} = \frac{r}{2^{k}} ∀k ∈ ℕ.

Since K_{0} and K_{1} are compact there exists h_{k}, δ_{k} such that

max \{h_{k}, δ_{k}\} < min \{r, 2\frac{M − ∥v_{0}∥ − r}{∥w_{0}∥ + r}\} ,  (4.5)

and for all (x, y), (x′, y′) ∈ K_{0} and (u, v, w) ∈ K_{1} one has

h (F (x, y), F (x′, y′)) < \frac{η_{k}}{3} , ∀(x′, y′) ∈ B((x, y), δ_{k}),  (4.6)

and

d_{K}(u + tv + \frac{t^{2}}{2} w) < \frac{η_{k}}{6} t^{2}, ∀t ∈ [0, h_{k}].  (4.7)

For any number k, choose an integer n_{k} such that n_{k+1} > n_{k} and

max \{\frac{MT}{2^{n_{k}}} , \frac{(∥v_{0}∥ + MT)T}{2^{n_{k}}} \} < min\{h_{k}, δ_{k}\}.  (4.8)

Set

l_{k} = \frac{T}{2^{n_{k}}} , t_{i}^{k} = il_{k}, k = 0, ..., 2^{n_{k}}.

Observe that

∀i ∈ {0, ...2^{n_{k}}}, ∃j ∈ {0, ..., 2^{n_{k}}.}, t_{i}^{k+1} = t_{j}^{k}.  (4.9)

Let us define a sequence of polygonal approximate solution of the problem (1.1). Indeed, let us consider

y ∈ π_{K} (x_{0} + t_{1}^{k}v_{0} + \frac{(t_{1}^{k})^{2}}{2} w_{0}) ,

and

x^{..}_{1}^{k} (0) = 2\frac{y − x_{0} − t_{1}^{k}v_{0}}{(t_{1}^{k})^{2}} .

On [0, t_{1}^{k}], define

x_{1}^{k}(t) = x_{0} + tv_{0} + \frac{t^{2}}{2} x^{..}_{1}^{k} (0),

190 B. Aghezzaf, S. Sajid / On the second-order contingent set and differential inclusions

then x_{1}^{k}(t_{1}^{k}) ∈ K. Moreover for all s ∈ [0, t_{1}^{k}[ we have that

∥x^{..}_{1}^{k} (s) − w_{0}∥ = ∥x^{..}_{1}^{k} (0) − w_{0}∥ ,
≤ \frac{2}{l_{k}^{2}} ∥y − x_{0} − l_{k}v_{0} − \frac{(l_{k})^{2}}{2} w_{0}∥ ,
≤ \frac{2}{l_{k}^{2}} d_{K}(x_{0} + l_{k}v_{0} + \frac{(l_{k})^{2}}{2} w_{0}).

Then by (4.7), for all s ∈ [0, t_{1}^{k}[ it follows that

∥x^{..}_{1}^{k} (s) − w_{0}∥ < \frac{η_{k}}{3} ,  (4.10)

and

d(x^{..}_{1}^{k} (0), F (x_{1}^{k}(0), x^{.}_{1}^{k} (0)) ≤ ∥x^{..}_{1}^{k} (0) − w_{0}∥ ,
< \frac{η_{k}}{3} .

Hence

∥x^{..}_{1}^{k} (0)∥ ≤ sup_{(x,y)∈K_{0}} h (F (x, y), \{0\}) + \frac{η_{k}}{3} ,
≤ M.

Consequently, relation (4.8) implies

∥x^{.}_{1}^{k} (t) − v_{0}∥ ≤ Ml_{k},
< δ_{k}, ∀t ∈ [0, t_{1}^{k}].

On the other hand, from (4.5) and (4.8) we deduce

∥x_{1}^{k}(t) − x_{0}∥ ≤ t_{1}^{k} ∥v_{0}∥ + \frac{(t_{1}^{k})^{2}}{2} ∥w_{0}∥
≤ Ml_{k}
< δ_{k} ∀t ∈ [0, t_{1}^{k}]

Thus (x_{1}^{k}(t_{1}^{k}), x^{.}_{1}^{k} (t_{1}^{k})) ∈ K_{0}.

For each integer p > 1 and t ∈ [0, T [, set s_{p}(t) to be the initial point of the p-th partition to which t belongs. At each nodal point t_{i}^{p} let us define a piecewise function x_{i}^{p}(.) on [0, t_{i}^{p}] with the following properties:

(i) x_{i}^{p}(.) is a continuous function with x_{i}^{p}(0) = x_{0}, x^{.}_{i}^{p} (0) = v_{0} and its second derivative x^{..}_{i}^{p} (.) is constant on each interval [t_{j}^{p}, t_{j+1}^{p}[, j = 0, ...., i − 1.

(ii) At each nodal point t_{j}^{p} ∈ [0, t_{i}^{p}[ , x_{i}^{p}(t_{j}^{p}) ∈ K and (for t_{j}^{p} < t_{i}^{p})

x^{..}_{i}^{p} (t_{j}^{p}) ∈ F (x_{i}^{p}(t_{j}^{p}), x^{.}_{i}^{p} (t_{j}^{p})) + \frac{η_{p}}{3} B(0, 1).

(iii) max\{∥x_{i}^{p}(t_{i}^{p}) − x_{0}∥ , ∥x^{.}_{i}^{p} (t_{i}^{p}) − v_{0}∥\} < \frac{r t_{i}^{p}}{T}

(iv) At each nodal point t_{j}^{p} ∈ [0, t_{i}^{p}[, ∥x^{..}_{i}^{p} (t_{j}^{p})− x^{..}_{i}^{p} (s_{p−1}(t_{j}^{p}))∥ < η_{p}.

Since s_{p−1}(t_{1}^{p}) = 0, it is clear that x_{1}^{p}(.) verifies the previous properties. By induction, assume that has been constructed a function x_{i}^{p}(.) on [0, t_{i}^{p}] with i ∈ {0, ..., 2^{n_{p}} − 1} satisfying (i)–(iv). Set s = s_{p−1}(t_{i}^{p}). Since |s − t_{i}^{p}| < l_{p} we have by (4.8)

∥x_{i}^{p}(t_{i}^{p}) − x_{i}^{p}(s)∥ ≤ (MT + ∥v_{0}∥) l_{p},
< δ_{p},

and

∥x^{.}_{i}^{p} (t_{i}^{p})− x^{.}_{i}^{p} (s)∥ ≤ Ml_{p},
< δ_{p}.

Hence relation (4.6) implies

h(F (x_{i}^{p}(t_{i}^{p}), x^{.}_{i}^{p} (t_{i}^{p})), F (x_{i}^{p}(s), x^{.}_{i}^{p} (s))) < \frac{η_{p}}{3} .  (4.11)

Moreover, by assumption (ii)

d(x^{..}_{p} (s), F (x_{i}^{p}(s), x^{.}_{i}^{p} (s))) < \frac{η_{p}}{3} ,  (4.12)

so that by (4.11), (4.12) and the compactness of the values of F, there exists w ∈

F (x_{i}^{p}(t_{i}^{p}), x^{.}_{i}^{p} (t_{i}^{p})) such that

∥x^{..}_{i}^{p} (s) − w∥ ≤ \frac{η_{p}}{3} + \frac{η_{p}}{3} ,  (4.13)

and therefore, assertion (iii) and triangular inequality imply that

∥w − w_{0}∥ ≤ \frac{η_{p}}{3} + \frac{η_{p}}{3} + \frac{t_{i}^{p}}{3} .  (4.14)

Since t_{i}^{p} ≤ T = \frac{r}{M} and by the choice of η_{p} and M, we have

∥w − w_{0}∥ < r,  (4.15)

hence (x_{i}^{p}(t_{i}^{p}), x^{.}_{i}^{p} (t_{i}^{p}), w) ∈ K_{1}.

On the other hand, given y ∈ π_{K}(x_{i}^{p}(t_{i}^{p}) + l_{p} x^{.}_{i}^{p} (t_{i}^{p}) + \frac{(l_{p})^{2}}{2} w) and consider

x^{..}_{i}^{p} (t_{i}^{p}) = 2\frac{y − x_{i}^{p}(t_{i}^{p}) − l_{p} x^{.}_{i}^{p} (t_{i}^{p})}{(l_{p})^{2}}.

192 B. Aghezzaf, S. Sajid / On the second-order contingent set and differential inclusions

For t ∈ [t_{i}^{p}, t_{i+1}^{p}], set

y_{p}(t) = x_{i}^{p}(t_{i}^{p}) + (t − t_{i}^{p}) x^{.}_{i}^{p} (t_{i}^{p}) + \frac{1}{2}(t − t_{i}^{p})^{2} x^{..}_{i}^{p} (t_{i}^{p}).

Observe that y_{p}(t_{i+1}^{p}) ∈ K. Moreover

\begin{aligned}d(x^{..}_{i}^{p} (t_{i}^{p}), F (x_{i}^{p}(t_{i}^{p}), x^{.}_{i}^{p} (t_{i}^{p})) & ≤ ∥x^{..}_{i}^{p} (t_{i}^{p}) − w∥ , \\ & = \frac{2}{(l_{p})^{2}} d_{K}(x_{i}^{p}(t_{i}^{p}) + l_{p} x^{.}_{i}^{p} (t_{i}^{p}) + \frac{(l_{p})^{2}}{2} w), \\ & < \frac{η_{p}}{3} . \end{aligned}  (4.16)

Hence the function

z_{p} = x_{i}^{p}(.)χ_{[0,t_{i}^{p}]}(.) + y_{p}(.)χ_{[t_{i}^{p},t_{i+1}^{p}]}(.)

satisfies (i) and (ii). Let us prove that z_{p} verifies (iii) and (iv).

∥z_{p}(t_{i+1}^{p}) − x_{0}∥ = ∥y_{p}(t_{i}^{p}) − x_{0}∥ ,
≤ ∥x_{i}^{p}(t_{i}^{p}) − x_{0}∥ + l_{p} (∥ v_{0}∥ + MT) + \frac{(l_{p})^{2}}{2} ∥ x^{..}_{i}^{p} (t_{i}^{p})∥ .

Since MT = r and l_{p} < 2\frac{M−∥ v_{0}∥− r}{M}, then

\begin{aligned}∥z_{p}(t_{i+1}^{p}) − x_{0}∥ & ≤ \frac{rt_{i}^{p}}{T} + \frac{rl_{p}}{T} , \\ & ≤ \frac{rt_{i+1}^{p}}{T} . \end{aligned}  (4.17)

Furthermore

z^{.}_{p} (t_{i}^{p}) =x^{.}_{i}^{p} (t_{i}^{p}) + l_{p} x^{..}_{i}^{p} (t_{i}^{p}).  (4.18)

Therefore

∥z^{.}_{p} (t_{i+1}^{p}) − v_{0}∥ ≤ ∥x^{.}_{i}^{p} (t_{i}^{p}) − v_{0}∥ + Ml_{p},
≤ \frac{r t_{i}^{p}}{T} + \frac{r l_{p}}{T} ,
≤ \frac{rt_{i+1}^{p}}{T} .

Hence, relations (4.15) and (4.17) imply (iii). About (iv), we have

∥z^{..}_{p} (t_{i}^{p})− z^{..}_{p} (s)∥ = ∥x^{..}_{i}^{p} (t_{i}^{p})− x^{..}_{i}^{p} (s)∥ ,

from that we obtain by using (4.13) and (4.16)

∥z^{..}_{p} (t_{i}^{p})− z^{..}_{p} (s)∥ ≤ η_{p}.

This implies (iv).

Consequently, there exists a sequence {u_{m}(.)} of absolutely continuous functions on [0, T] satisfying (i)–(iv).

Claim 4.2. The sequence {u^{..}_{m} (.)} is equioscillating.

Proof. Let p ∈ ℕ and I = [t_{i}^{p}, t_{i+1}^{p}[ an interval of the p-th partition of [0, T]. Let q > p and t_{j}^{q} be a nodal point of I. For i = 0, ..., p, set s^{i} = s_{q−i} ∘ s_{q−i+1} ∘ ... ∘ s_{q}. It is clear that t_{j}^{q} = s^{0}(t_{j}^{q}), s^{1}(t_{j}^{q}), ..., s^{p}(t_{j}^{q}) = t_{i}^{p} are in I. We wish to compute the oscillation of u^{..}_{m} (.) on I. If m < p then the oscillation is zero. Hence consider m > p. Let t ∈ I and t_{j}^{m} be the initial point of the m-th partition to which t belongs. Since u^{..}_{m} (t) = u^{..}_{m} (t_{j}^{m}) then by (iv) we have that:

∥u^{..}_{m} (t_{j}^{m})− u^{..}_{m} (s^{m−1}(t_{j}^{m})∥ ≤ η_{m},
∥u^{..}_{m} (s^{m−1}(t_{j}^{m}))− u^{..}_{m} (s^{m−2}(t_{j}^{m})∥ ≤ η_{m−1},
.
.
.
∥u^{..}_{m} (s^{p−1}(t_{j}^{m}))− u^{..}_{m} (t_{i}^{p})∥ ≤ η_{p}.

Consequently

\begin{aligned}∥u^{..}_{m} (t_{i}^{p})− u^{..}_{m} (s)∥ & ≤ ∑_{i=p}^{m} η_{i}, \\ & ≤ \frac{1}{2^{p−1}} . \end{aligned}

ω(u^{..}_{m} (.)) ≤ \frac{1}{2^{p−1}} ∀m ∈ ℕ.

On the other hand, since

∥u^{..}_{m} (t)∥ ≤ M and ∥u^{.}_{m} (t)∥ ≤ MT + ∥v_{0}∥ , ∀m ∈ ℕ, ∀t ∈ I,

then by Theorem 3.9, {u^{..}_{m} (.)} converges uniformly to a function w(.), so that {u_{m}(.)} and {u^{.}_{m} (.)} converge uniformly to functions u and v respectively. The functions u, v and w are related by the formula

v(t) = u^{.} (t) and w(t) =u^{..} (t) a.e on [0, T]

Let t ∈ [0, T], for each m we denote by t_{i}^{m} the initial point of the m-th partition to which t belongs. We have

d(w(t), F (u(t), v(t))) ≤ ∥w(t)− u^{..}_{m} (t)∥

+ ∥u^{..}_{m} (t)− u^{..}_{m} (t_{i}^{m})∥

+ d(u^{..}_{m} (t_{i}^{m}), F (u_{m}(t_{i}^{m}), u^{.}_{m} (t_{i}^{m}))

+ h(F (u_{m}(t_{i}^{m}), u^{.}_{m} (t_{i}^{m})), F (u_{m}(t), u^{.}_{m} (t))

+ h(F (u_{m}(t), u^{.}_{m} (t)), F (u(t), v(t)).

Since {u^{..}_{m} (t)} converges to w(t), u^{..}_{m} (t) equals u^{..}_{m} (t_{i}^{m}), u^{..} (t_{i}^{m}) belongs to the set F (u_{m}(t_{i}^{m}), u^{.}_{m} (t_{i}^{m})) + \frac{1}{3}η_{m} B(0, 1), u_{m}(.), u(.), u^{.}_{m} (.) and u^{.} (.) are lipschitzean, \{t_{i}^{m} : m ∈ N} converges to t and F is continuous, the right-hand side of the above inequality converges to zero, since F have closed values, we deduce that

u^{..} (t) = w(t) ∈ F (u(t), u^{.} (t)) a.e on [0, T [.

On the other hand

d(u(t), K) ≤ ∥u_{m}(t) − u(t)∥ + ∥u_{m}(t) − u_{m}(t_{i}^{m})∥ + d(u_{m}(t_{i}^{m}), K),

hence

u(t) ∈ K, ∀t ∈ [0, T [.

Finally

\begin{cases}u^{..} (t) ∈ F (u(t), u^{.} (t)) & a.e on [0, T [ \\ u(0) = x_{0,} u^{.} (0) = v_{0} ∈ T_{K}(x_{0}) \\ u(t) ∈ K & ∀t ∈ [0, T [ \end{cases}

To prove Theorem 2.2, by the same reasoning we construct an approximate solution {u_{m}(.)} satisfying at each nodal point t_{i}^{m}, u^{.}_{m} (t_{i+1}^{m}) ∈ T_{D}(u_{m}(t_{i}^{m})).

Relation (4.18) implies that

\begin{aligned}u^{.}_{m} (t_{i+1}^{m}) & = u^{.}_{m} (t_{i}^{m}) + l_{m} u^{..}_{m} (t_{i}^{m}) \\ & = \frac{2}{l_{m}} (y− u^{.}_{m} (t_{i}^{m}) − u_{m}(t_{i}^{m})) . \end{aligned}

Since y ∈ D, then by Proposition 3.1

(u_{m}(t_{i}^{m}), u^{.}_{m} (t_{i}^{m})) ∈ gr(T_{D}), ∀m ∈ ℕ.

Since gr(T_{D}) is closed then by passing to the limit there exists an absolutely continuous function x : [0, T] → ℝ^{n} for which x^{.} is also absolutely continuous which is a solution of the problem (1.2). This completes the proof of Theorem 2.2.

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