Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 21, 1-10; http://www.math.u-szeged.hu/ejqtde/

GENERAL EXISTENCE RESULTS FOR NONCONVEX THIRD ORDER DIFFERENTIAL INCLUSIONS

MESSAOUD BOUNKHEL AND BUSHRA AL-SENAN

DEPARTMENT OF MATHEMATICS KING SAUD UNIVERSITY RIYADH, SAUD ARABIA

Abstract. In this paper we prove the existence of solutions to the following third order differential inclusion:

\{ \begin{matrix}x^{(3)}(t) ∈ F (t, x(t), ẋ(t), ẍ(t)) + G(x(t), ẋ(t), ẍ(t)), a.e. on [0, T] \\ x(0) = x_{0}, ẋ(0) = u_{0}, ẍ(0) = v_{0}, and ẍ(t) ∈ S, ∀t ∈ [0, T], \end{matrix}

where F : [0, T] × ℍ × ℍ × ℍ → ℍ is a continuous set-valued mapping, G : ℍ × ℍ × ℍ → ℍ is an upper semi-continuous set-valued mapping with G(x, y, z) ⊂ ∂^{C}g(z) where g : ℍ → ℝ is a uniformly regular function over S and locally Lipschitz and S is a ball compact subset of a separable Hilbert space ℍ.

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. In [6] Hopkins studied an existence result for the third order differential inclusion

(ThODI) \{ \begin{matrix}x^{(3)}(t) ∈ G(x(t), ẋ(t), ẍ(t)), \\ x(0) = x_{0}, ẋ(0) = u_{0}, ẍ(0) = v_{0}, \end{matrix}

where G is set-valued mapping with an upper semi-continuous compact valued included in the subdifferential of a convex lower semi-continuous function g : ℝ^{d} → ℝ, that is, G(x, y, z) ⊂ ∂^{C}g(z). In this paper we prove the existence of viable solutions for the general form of the third order differential inclusion

(GThODI) \{ \begin{matrix}x^{(3)}(t) ∈ F (t, x(t), ẋ(t), ẍ(t)) + G(x(t), ẋ(t), ẍ(t)), a.e. on [0, T] \\ x(0) = x_{0}, ẋ(0) = u_{0}, ẍ(0) = v_{0}, and ẍ(t) ∈ S, ∀t ∈ [0, T], \end{matrix}

where F : [0, T] × ℍ × ℍ × ℍ → ℍ is a continuous set-valued mapping, G : ℍ × ℍ × ℍ → ℍ is an upper semi-continuous set-valued mapping with G(x, y, z) ⊂ ∂^{C}g(z) where g : ℍ → ℝ is a uniformly regular function over S and locally Lipschitz, and S is a ball compact subset of a separable Hilbert space ℍ. This general problem covers (T hODI) in three different ways. First, it extends (T hODI) from finite dimensional setting to separable Hilbert spaces. Secondly, it extends g to the case of uniformly regular function (not necessary convex) and also it covers (ThODI)

EJQTDE, 2010 No. 21, p. 1 by taking F = 0. Problem (GThODI) includes as a special case the following differential variational inequality: Given T > 0 and three points x_{0}, u_{0}, v_{0} ∈ ℍ.

AMS subject classification: 34G20, 47H20

Key Words: Nonconvex Differential Inclusions, Uniformly Regular Functions.

(DVI) \{ \begin{matrix}Find b ∈ (0, T), x : [0, b] → ℍ such that ẍ(t) ∈ S, on [0, b] and ∀w ∈ S \\ ⟨x^{(3)}(t), w − ẍ(t)⟩ ≤ a(x(t) + ẋ(t) − ẍ(t), w − ẍ(t)), a.e. on [0, b] \\ x(0) = x_{0}, ẋ(0) = u_{0}, ẍ(0) = v_{0}. \end{matrix}

where S = {x ∈ ℍ : Λ(x) ≤ 0} (Λ : ℍ → ℝ is a C^{1} convex function), a(·, ·) is a real bilinear, symmetric, bounded, and elliptic form on ℍ × ℍ. We use our main theorem to prove that (DVI) has at least one solution.

This paper is organized as follows. In Section 2, we recall some definitions and results that will be needed in the paper. In Section 3, we prove our main existence theorem, by constructing a sequence of approximate solutions and showing its convergence to the solution of the given problem. Section 4 contains the application to differential variational inequalities.

2. Preliminaries

Throughout the paper ℍ will denote a separable Hilbert space. We need to recall, from [1], some notation and definitions that will be used in all the paper.

Definition 2.1. ([1]) Let f : ℍ → R ∪ {+∞} be a l.s.c. function and O ⊂ domf be a nonempty open subset. We will say that f is uniformly regular over O with respect to β ≥ 0 (we will also say β-uniformly regular) if for all x̄ ∈ O and for all ξ ∈ ∂^{P} f(x̄) one has

⟨ξ, x − x̄⟩ ≤ f(x) − f(x̄) + β∥x − x̄∥^{2} ∀x ∈ O.

Here ∂^{P} f(x̄) denotes the proximal subdifferential of f at x (for its definition the reader is refereed for instance to [3]). We say that f is uniformly regular over a closed set S if there exists an open set O containing S such that f is uniformly regular over O.

The class of functions that are uniformly regular over sets is so large, it contains convex sets, p-convex sets and epigraph of lower-C^{2} functions. The following proposition gives some properties for uniformly regular locally Lipschitz functions over sets needed in the sequel. For the proof of these results we refer the reader to [1, 4].

Proposition 2.2. Let f : ℍ → R be a locally Lipschitz function and ∅ ≠ S ⊂ domf. If f is uniformly regular over S, then the following hold:

(i) The proximal subdifferential of f is closed over S as a set-valued mapping, that is, for every x_{n} → x with x_{n} ∈ S and every ζ_{n} → ζ (weakly) with ζ_{n} ∈ ∂^{P} f(x_{n}) one has ζ ∈ ∂^{P} f(x);

(ii) The proximal subdifferential of f coincides with ∂^{C}f(x) the Clark subdifferential of f (see for instance [3] for the definition of ∂^{C}f(x)), i.e., ∂^{C}f(x) = ∂^{P} f(x) for all x ∈ S;

(iii) The proximal subdifferential of f is upper hemicontinuous over S, i.e, the support function x → σ(v, ∂^{P} f(x)) is u.s.c. over S for every v ∈ ℍ (where σ(v, S) = sup_{s∈S}⟨v, s⟩);

EJQTDE, 2010 No. 21, p. 2 (iv) For any absolutely continuous map x : [0, T] → S one has

\frac{d}{dt}(f ∘ x)(t) = ⟨∂^{C}f(x(t)); ẋ(t)⟩.

3. Existence results for third order differential inclusions

We start with the following technical lemmas. Their proofs follow the same lines as in the proof of Theorem 2.3 in [2].

Lemma 3.1. Assume that

(1) S is nonempty subset in ℍ, υ_{0} ∈ S and K_{0} = S ∩ (υ_{0} + ρB) is a compact set for some ρ > 0.

(2) P : [0, T] × ℍ × ℍ × ℍ → ℍ is an u.s.c. set valued mapping with nonempty compact values.

(3) For any (t, x, y, z) ∈ [0, T]×S ×S ×S the following tangent condition holds

lim inf_{h→0^{+}} \frac{1}{h}e(υ + hP (t, x, y, z); S) = 0,

where e(A, S) := sup_{a∈A}d_{S}(a).

Let α = min\{T, \frac{ρ}{M+1}, 1\} where M = sup\{∥P (t, x, y, z)∥ : (t, x, y, z) ∈ [0, T] × K_{0} × K_{0} × K_{0}}. Then, there exists a > 0 such that we can construct sequences {w_{i}^{m}}, {t_{i}^{m}}, {λ_{i}^{m}}, {x_{i}^{m}}, {u_{i}^{m}} and {υ_{i}^{m}} satisfying for some rank ν_{m} ≥ 0 the following assertions:

(1) 0 = t_{0}^{m}, t_{ν_{m}}^{m} ≤ a < T with a < α and t_{i}^{m} = Σ_{k=0}^{i−1}λ_{k}^{m} for all i ∈ {1, 2, ..., ν_{m}},

(2) υ_{i}^{m} = υ_{0} + Σ_{k=0}^{i−1}λ_{k}^{m}w_{k}^{m} and (t_{i}^{m}, υ_{i}^{m}) ∈ [0, T] × K_{0} for all i ∈ {1, 2, ..., ν_{m}},

(3) w_{i}^{m} ∈ P (t_{i}^{m}, x_{i}^{m}, u_{i}^{m}, υ_{i}^{m}) + \frac{1}{m}B with w_{i}^{m} = \frac{ψ_{i}^{m}−υ_{i}^{m}}{λ_{i}^{m}} and ψ_{i}^{m} ∈ S ∩ B(υ_{i}^{m} + λ_{i}^{m}b_{i}^{m}, M + 1) for all i ∈ {1, 2, ..., ν_{m}}, where

λ_{i}^{m} = max\{ζ ∈ (0, \frac{α}{2}] : ζ ≤ T − t_{i}^{m} and d_{S}(υ_{i}^{m} + ζb_{i}) < \frac{1}{m}ζ\},

(4) u_{i}^{m} = u_{0} + υ_{i}^{m}t_{i}^{m} − w_{i}^{m} \frac{t_{i}^{m}^{2}}{2} ,

(5) x_{i}^{m} = x_{0} + u_{0}t_{i}^{m} + υ_{i}^{m} \frac{t_{i}^{m}^{2}}{2} − w_{i}^{m} \frac{t_{i}^{m}^{3}}{3} .

Lemma 3.2. Let P (t, x, y, z) = F (t, x, y, z) + G(x, y, z). Under the same assumptions in Lemma 3.1, we can construct sequence of the step functions υ_{m}, u_{m}, x_{m}, f_{m}, c_{m} and θ_{m} with the following properties:

(1) υ_{m}(t) = υ_{i}^{m} + (t − t_{i}^{m})w_{i}^{m} on [t_{i}^{m}, t_{i+1}^{m}) for all i ∈ {1, 2, ..., ν_{m}},

(2) u_{m}(t) = u_{0} + ∫_{0}^{t} υ_{m}(s)ds, x_{m}(t) = x_{0} + ∫_{0}^{t} u_{m}(s)ds on [0, a],

(3) f_{m}(t) = f_{i}^{m} ∈ F (θ_{m}(t), x_{m}(θ_{m}(t)), u_{m}(θ_{m}(t)), υ_{m}(θ_{m}(t))) on [t_{i}^{m}, t_{i+1}^{m}), with θ_{m}(t) = t_{i}^{m} if t ∈ [t_{i}^{m}, t_{i+1}^{m}) for all i ∈ {1, 2, ..., ν_{m}}, θ_{m}(a) = a,

(4) y_{m}(t) = y_{m}(θ_{m}(t)) ∈ G(x_{m}(θ_{m}(t)), u_{m}(θ_{m}(t)), υ_{m}(θ_{m}(t))) on [t_{i}^{m}, t_{i+1}^{m}), where y_{i}^{m} = w_{i}^{m} − f_{i}^{m} − c_{i}^{m},

(5) c_{m}(t) = c_{i}^{m} ∈ \frac{1}{m}B if t ∈ [t_{i}^{m}, t_{i+1}^{m}) for all i ∈ \{1, 2, ..., ν_{m}\}, and

lim_{h→∞} sup_{t∈[0,a]} ∥c_{m}(t)∥ = 0.

Also,

∥υ_{m}(t_{i+1}^{m}) − υ_{m}(t_{i}^{m})∥ ≤ (M + 1)(t_{i+1}^{m} − t_{i}^{m}),
∥υ̇_{m}(t)∥ = ∥w_{i}^{m}∥ ≤ M + 1, a. e. on [0, a],

and

∥υ_{m}(t)∥ < ρ + a(M + 1) < 2ρ,

with M as in Lemma 3.1. Furthermore, ψ_{i}^{m} ∈ K_{1} = S ∩ B(0, R) with R = ∥υ_{0}∥ + 2ρ + 2M + 1.

Now we are in position to state and prove the main result in this section.

Theorem 3.3. Let S be a nonempty subset of ℍ and let g : ℍ → R be a locally Lipschitz function which is uniformly regular over S with constant β ≥ 0. Assume that

(1) S is ball compact.

(2) F : [0, T] × ℍ × ℍ × ℍ → ℍ is uniformly continuous set-valued mapping with compact values.

(3) G : ℍ × ℍ × ℍ → ℍ is an u.s.c. set valued mapping with compact values and G(x, y, z) ⊂ ∂^{P} g(z), for all x, y, z ∈ S.

(4) For any (t, x, y, z) ∈ [0, T]×S ×S ×S the following tangent condition holds

lim inf_{h→0} \frac{1}{h}e(x + h(F (t, x, y, z) + G(x, y, z)); S) = 0,

where e(A, S) := sup_{a∈A} d_{S}(a).

Then, for any v_{0} ∈ S, u_{0}, x_{0} ∈ ℍ, there exists a ∈ (0, T) such that

\{ \begin{matrix}υ̇(t) ∈ F (t, x(t), u(t), υ(t)) + G(x(t), u(t), υ(t)) a.e. on [0, a], \\ υ(t) ∈ S on [0, a], x(t) = x_{0} + ∫_{0}^{t} u(s)ds, u(t) = u_{0} + ∫_{0}^{t} υ(s)ds, \\ x(0) = x_{0}, u(0) = u_{0}, v(0) = v_{0}, \end{matrix}

has an absolutely continuous solution on [0, a]. In other words, there exists a ∈ (0, T) such that (GThODI) has an absolutely continuous solution on [0, a].

Proof. Let L > 0 and ρ > 0 be two positive scalars such that g is Lipschitz over υ_{0} + ρB with ratio L. Since S is ball compact, K_{0} = S ∩ (υ_{0} + ρB) is compact in ℍ. Let M and a be two positive scalars such that

∥F (t, x, y, z) + G(x, y, z) + H(t, x, y, z)∥ ≤ M,

for all (t, x, y, z) ∈ [0, T] × K_{0} × K_{0} × K_{0} and a = min\{T, \frac{ρ}{M+1}, 1\}. By applying Lemma 3.2, there exist sequences of step functions υ_{m}, u_{m}, x_{m}, f_{m}, y_{m}, c_{m} and θ_{m} with the following properties:

(1) υ_{m}(t) = υ_{i}^{m} + (t − t_{i}^{m})w_{i}^{m} on [t_{i}^{m}, t_{i+1}^{m}) for all i ∈ {1, 2, ..., ν_{m}},

(2) u_{m}(t) = u_{0} + ∫_{0}^{t} υ_{m}(s)ds, x_{m}(t) = x_{0} + ∫_{0}^{t} u_{m}(s)ds on [0, a],

(3) f_{m}(t) = f_{m}(θ_{m}(t)) ∈ F (θ_{m}(t), x_{m}(θ_{m}(t)), u_{m}(θ_{m}(t)), υ_{m}(θ_{m}(t))) on [t_{i}^{m}, t_{i+1}^{m}),

with θ_{m}(t) = t_{i}^{m} if t ∈ [t_{i}^{m}, t_{i+1}^{m}) for all i ∈ {1, 2, ..., ν_{m}}, θ_{m}(a) = a,

(4) y_{m}(t) = y_{m}(θ_{m}(t)) ∈ G(x_{m}(θ_{m}(t)), u_{m}(θ_{m}(t)), υ_{m}(θ_{m}(t))) on [t_{i}^{m}, t_{i+1}^{m}),

where y_{i}^{m} = w_{i}^{m} − f_{i}^{m} − h_{i}^{m} − c_{i}^{m}

EJQTDE, 2010 No. 21, p. 4 (5) c_{m}(t) = c_{i}^{m} ∈ \frac{1}{m}B if t ∈ [t_{i}^{m}, t_{i+1}^{m}) for all i ∈ \{1, 2, ..., ν_{m}\}, and

lim_{m→∞} sup_{t∈[0,a]} ∥c_{m}(t)∥ = 0.

Also,

∥υ_{m}(t_{i+1}^{m}) − υ_{m}(t_{i}^{m})∥ ≤ (M + 1)(t_{i+1}^{m} − t_{i}^{m}),
∥υ̇_{m}(t)∥ = ∥w_{i}^{m}∥ ≤ M + 1,

and

∥υ_{m}(t)∥ < ρ + a(M + 1) < 2ρ.

We want to prove that υ_{m} converges to a solution of the given differential inclusion. First, we mention that the sequence f_{m} can be constructed with the relative compactness property in the space of bounded functions (see [7]). Therefore, without loss of generality we can suppose that there is a bounded function f such that

lim_{m→∞} sup_{t∈[0,a]} ∥f_{m}(t) − f(t)∥ = 0.

We note that

υ_{m}(t) = υ_{i}^{m} + (t − t_{i}^{m})w_{i}^{m} = υ_{i}^{m} + \frac{t − t_{i}^{m}}{λ_{i}^{m}} (ψ_{i}^{m} − υ_{i}^{m}), for all t ∈ [t_{i}^{m}, t_{i+1}^{m}).

Clearly υ_{m} is continuous on all the interval [0, a]. Indeed, it is continuous on

[t_{i}^{m}, t_{i+1}^{m}) and υ_{m}(t_{i}^{m}) = lim_{t \underrightarrow{>} t_{i}^{m}} υ_{m}(t) = v_{i}^{m} and

lim_{t \underrightarrow{<} t_{i}^{m}} υ_{m}(t) = lim_{t \underrightarrow{<} t_{i}^{m}} [υ_{i−1}^{m} + (t − t_{i−1}^{m})w_{i−1}^{m}] = υ_{i−1}^{m} + λ_{i−1}^{m}w_{i−1}^{m} = v_{i}^{m},

and hence υ_{m} is continuous on the nodes t_{i}^{m}. Therefore, the sequence of mappings υ_{m} is equi-Lipschitz with ratio M + 1 on all [0, a]. On the other hand, we have 0 ≤ t − t_{i}^{m} ≤ t_{i+1}^{m} − t_{i}^{m} = λ_{i}^{m} and so 0 ≤ \frac{t−t_{i}^{m}}{λ_{i}^{m}} ≤ 1, and hence we get

\frac{t − t_{i}^{m}}{λ_{i}^{m}} (ψ_{i}^{m} − υ_{i}^{m}) ∈ \overline{co}[\{0\} ∪ \{K_{1} − K_{0}\}].

Thus

υ_{m}(t) ∈ K := K_{0} + \overline{co}[\{0\} ∪ \{K_{1} − K_{0}\}].

Therefore, since the set K is compact (because K_{0} and K_{1} are compact) and ∥υ̇_{m}(t)∥ ≤ M + 1, for all t ∈ [0, a], then, the assumptions of Arzela-Ascoli theorem are satisfied. Hence a subsequence of υ_{m} may be extracted (still denoted υ_{m}) that converges to a Lipschitz function υ : [0, a] → ℍ such that

lim_{m→∞} max_{t∈[0,a]} ∥υ_{m}(t) − υ(t)∥ = 0,

and

υ̇_{m} → υ̇ in the weak topology of L^{2}([0, a]; ℍ).

Since ∥υ_{m}(t)∥ < 2ρ we have

∥u̇_{m}(t)∥ = ∥υ_{m}(t)∥ < 2ρ,

and

∥ẋ_{m}(t)∥ = ∥u_{m}(t)∥ = ∥u_{0} + ∫_{0}^{t} υ_{m}(s)ds∥ ≤ ∥u_{0}∥ + 2ρT,

EJQTDE, 2010 No. 21, p. 5 On the other hand

u_{m}(t) = u_{0} + ∫_{0}^{t} υ_{m}(s)ds ∈ K_{0} + aK := K_{2},

and

x_{m}(t) = x_{0} + ∫_{0}^{t} u_{m}(s)ds ∈ K_{0} + aK_{2} := K_{3}.

Clearly, K_{2} and K_{3} are compact sets in ℍ, then by Arzela-Ascoli theorem, there are subsequences of u_{m} and x_{m} (still denoted u_{m} and x_{m} respectively) that converge to absolutely continuous mappings u : [0, a] → ℍ and x : [0, a] → ℍ respectively, such that

lim_{m→∞} sup_{t∈[0,a]} ∥u_{m}(t) − u(t)∥ = 0,

lim_{m→∞} sup_{t∈[0,a]} ∥x_{m}(t) − x(t)∥ = 0,

and

u̇_{m} → u̇, ẋ_{m} → ẋ in the weak topology of L^{2}([0, a]; ℍ).

Also, we have that (υ_{m} ∘ θ_{m}), (u_{m} ∘ θ_{m}) and (x_{m} ∘ θ_{m}) converge uniformly on [0, a] to υ, u, and x respectively. Since υ_{m}(θ_{m}(t)) = υ_{i}^{m} ∈ K_{0}, by closedness of K_{0} we get, υ(t) ∈ K_{0} ⊂ S. Indeed,

d(υ(t); K_{0}) ≤ d(υ_{m}(θ_{m}(t)); K_{0}) + ∥υ(t) − υ_{m}(θ_{m}(t))∥ → 0, as m → ∞.

By construction, we have

f_{m}(t) = f_{m}(θ_{m}(t)) ∈ F (θ_{m}(t), x_{m}(θ_{m}(t)), u_{m}(θ_{m}(t)), υ_{m}(θ_{m}(t))),

and so by the continuity of F and the closedness of its values we obtain

f(t) ∈ F (t, x(t), u(t), υ(t)).

Now, put y(t) = υ̇(t) − f(t), we must prove that

y(t) ∈ G(x(t), u(t), υ(t)) a.e. on [0, a].

By construction, we have for a.e. t ∈ [0, a]

y_{m}(t) ∈ G(x_{m}(θ_{m}(t)), u_{m}(θ_{m}(t)), υ_{m}(θ_{m}(t))) ⊂ ∂^{C}g(υ_{m}(θ_{m}(t)) = ∂^{P} g(υ_{m}(θ_{m}(t)),

where the above equality follows from the uniform regularity of g over S and the part (ii) in Proposition 2.2. The weak convergence of υ̇_{m} and Mazur’s lemma entail for almost all t ∈ [0, a]

υ̇(t) ∈ ⋂_{m} \overline{co}\{υ̇_{k}(t) : k ≥ m\}.

So, for any ξ ∈ ℍ we have

⟨ξ, υ̇(t)⟩ ≤ inf_{m} sup_{k≥m} ⟨ξ, υ̇_{k}(t)⟩
≤ lim sup_{m} [σ(ξ, ∂^{P} g(υ_{m}(θ_{m}(t)))) + ⟨ξ, f_{m}(t) + c_{m}(t)⟩]
≤ σ(ξ, ∂^{P} g(υ(t))) + ⟨ξ, f(t)⟩ = σ(ξ, ∂^{P} g(υ(t)) + f(t)),

where the last inequality follows from upper hemicontinuity of the proximal subdifferential of uniformly regular functions (part (iii) in Proposition 2.2) and the

EJQTDE, 2010 No. 21, p. 6 uniform convergence on [0, a] of f_{m} and c_{m} to f and 0 respectively, and the fact that υ_{m}(θ_{m}(t)) → υ(t) on K_{0}. Thus, by the convexity and the closedness of proximal subdifferential of uniformly regular functions, we have

y(t) = υ̇(t) − f(t) ∈ ∂^{P} g(υ(t)).

As υ is absolutely continuous and g is uniformly regular locally Lipschitz function over S we get by part (iv) in Proposition 2.2

\frac{d}{dt}(g ∘ υ)(t) = ⟨∂^{P} g(υ(t)), υ̇(t)⟩
= ⟨υ̇(t) − f(t), υ̇(t)⟩
= ∥υ̇(t)∥^{2} − ⟨f(t), υ̇(t)⟩.

Consequently

(3.1)  g(υ(a)) − g(υ_{0}) = ∫_{0}^{a} ∥υ̇(t)∥^{2}dt − ∫_{0}^{a} ⟨f(t), υ̇(t)⟩dt.

On the other hand, since y_{i}^{m} ∈ G(x_{i}^{m}, u_{i}^{m}, υ_{i}^{m}) ⊂ ∂^{P} g(υ_{i}^{m}) then

g(υ_{i+1}^{m}) − g(υ_{i}^{m}) ≥ ⟨y_{i}^{m}, υ_{i+1}^{m} − υ_{i}^{m}⟩ − β∥υ_{i+1}^{m} − υ_{i}^{m}∥^{2}
= ⟨υ̇_{m}(t) − f_{m}(t) − c_{m}(t), ∫_{t_{i}^{m}}^{t_{i+1}^{m}} υ̇_{m}(s)ds⟩
− β∥υ_{i+1}^{m} − υ_{i}^{m}∥^{2}
≥ ∫_{t_{i}^{m}}^{t_{i+1}^{m}} ∥υ̇_{m}(s)∥^{2}ds − ∫_{t_{i}^{m}}^{t_{i+1}^{m}} ⟨f_{m}(s), υ̇_{m}(s)⟩ds
− ∫_{t_{i}^{m}}^{t_{i+1}^{m}} ⟨c_{m}(s), υ̇_{m}(s)⟩ds − β(M + 1)^{2}(t_{i+1}^{m} − t_{i}^{m})^{2}
≥ ∫_{t_{i}^{m}}^{t_{i+1}^{m}} ∥υ̇_{m}(s)∥^{2}ds − ∫_{t_{i}^{m}}^{t_{i+1}^{m}} ⟨f_{m}(s), υ̇_{m}(s)⟩ds
− ∫_{t_{i}^{m}}^{t_{i+1}^{m}} ⟨c_{m}(s), υ̇_{m}(s)⟩ds − \frac{β(M + 1)^{2}}{m} (t_{i+1}^{m} − t_{i}^{m}).

By adding, we obtain

g(υ_{m}(a)) − g(υ_{0}) ≥ ∫_{0}^{a} ∥υ̇_{m}(s)∥^{2}ds − ∫_{0}^{a} ⟨f_{m}(s), υ̇_{m}(s)⟩ds
− ∫_{0}^{a} ⟨c_{m}(s), υ̇_{m}(s)⟩ds − \frac{βa(M + 1)^{2}}{m} .

Passing to the limit superior as m → ∞

g(υ(a)) − g(υ_{0}) ≥ lim sup_{m} ∫_{0}^{a} ∥υ̇_{m}(s)∥^{2}ds − lim_{m} ∫_{0}^{a} ⟨f_{m}(s), υ̇_{m}(s)⟩ds
≥ lim sup_{m} ∫_{0}^{a} ∥υ̇_{m}(s)∥^{2}ds − ∫_{0}^{a} ⟨f(s), υ̇(s)⟩ds.

EJQTDE, 2010 No. 21, p. 7 This inequality compared with (3.1) yields

∫_{0}^{a} ∥υ̇(t)∥^{2}dt ≥ lim sup_{m} ∫_{0}^{a} ∥υ̇_{m}(s)∥^{2}ds,

and so

∥υ̇∥_{L^{2}}^{2} ≥ lim sup_{m} ∥υ̇_{m}∥_{L^{2}}^{2}.

On the other hand, the weak lower semi-continuity of the norm ensures

∥υ̇∥_{L^{2}} ≤ lim inf_{m} ∥υ̇_{m}∥_{L^{2}}.

Therefore, we get

∥υ̇∥_{L^{2}} = lim_{m} ∥υ̇_{m}∥_{L^{2}},

which ensures that υ̇_{m} converges uniformly to υ̇ in L^{2}([0, a]; ℍ). By construction we have for almost all t in [0, a]

((x_{m}(θ_{m}(t)), u_{m}(θ_{m}(t)), υ_{m}(θ_{m}(t))), υ̇_{m}(t) − f_{m}(t) − c_{m}(t)) ∈ graph G,

and since G has closed graph, we conclude that

((x(t), u(t), υ(t)), υ̇(t) − f(t)) ∈ graph G,

that is,

υ̇(t) − f(t) ∈ G(x(t), u(t), υ(t)) a.e. on [0, a].

Thus, for almost all t in [0, a]

υ̇(t) ∈ G(x(t), u(t), υ(t)) + f(t) ⊂ G(x(t), u(t), υ(t)) + F (t, x(t), u(t), υ(t)).

The proof then is complete. □

We end this section with some important corollaries. The first one is an extension of the main result in [6] from finite dimensional spaces to separable Hilbert spaces and from the case of convex functions to uniform regular functions. Our proof is completely different to the one given in [6].

Corollary 3.4. Let S be a nonempty closed subset of ℍ and let g : ℍ → R be a locally Lipschitz function which is uniformly regular over S with constant β ≥ 0. Assume that

(1) S is ball compact.

(2) G : ℍ × ℍ × ℍ → ℍ is a u.s.c. set valued mapping with compact values and G(x, y, z) ⊂ g(z), for all x, y, z ∈ S.

(3) For any (t, x, y, z) ∈ [0, T]×S ×S ×S the following tangent condition holds

lim_{h→0^{+}} inf \frac{1}{h}e(x + h(G(x, y, z)); S) = 0.

Then, for every x_{0}, u_{0} ∈ ℍ, and every v_{0} ∈ S, there exists a ∈ (0, T) and an absolutely continuous solution of the following third order differential inclusion

\{ \begin{matrix}x^{(3)}(t) ∈ G(x(t), ẋ(t), ẍ(t)), a.e. on [0, a], \\ x(0) = x_{0}, ẋ(0) = u_{0}, ẍ(0) = v_{0}, and ẍ(t) ∈ S \end{matrix}

Proof. It is a direct application of Theorem 3.3 with F ≡ {0}. □

Now, we are going to prove the existence of solution for third order nonconvex sweeping processes with a perturbation in separable Hilbert spaces, that is,

(ThOSPP) \{ \begin{matrix}x^{(3)}(t) ∈ N_{S}^{C}(ẍ(t)) + F (t, x(t), ẋ(t), ẍ(t)), \\ x(0) = x_{0}, ẋ(0) = u_{0}, ẍ(0) = v_{0}. \end{matrix}

Corollary 3.5. Let ℍ be a separable Hilbert. Assume that (1) S is a nonempty uniformly prox-regular closed subset in ℍ;

(2) F : [0, T] × ℍ × ℍ × ℍ → ℍ is an uniformly continuous set-valued mapping with compact values;

(3) For any (t, x, y, z) ∈ [0, T]×S ×S ×S the following tangent condition holds

lim inf_{h→0^{+}} \frac{1}{h}e(x + h(∂^{C}d_{S}(z) + F (t, x, y, z); S) = 0.

Then, for any x_{0}, u_{0} ∈ ℍ and every v_{0} ∈ S, there exists a ∈ (0, T) such that (ThOSPP) has at least one absolutely continuous solution on [0, a].

Proof. In [1], the author proved in Theorem 4.1 that the function d_{S} is uniformly regular over S. Thus, we can apply Theorem 3.3 with g := d_{S} and the set-valued mapping G := ∂^{C}d_{S} which satisfies the hypothesis of Theorem 3.3 then we get a solution x of the third order differential inclusion

\{ \begin{matrix}x^{(3)}(t) ∈ ∂^{C}d_{S}(ẍ(t)) + F (t, x(t), ẋ(t), ẍ(t)), \\ x(0) = x_{0}, ẋ(0) = u_{0}, ẍ(0) = v_{0}, and ẍ(t) ∈ S, ∀t ∈ [0, a]. \end{matrix}

Now, since ẍ(t) ∈ S we have ∂^{C}d_{S}(ẍ(t)) ⊂ N_{S}^{C}(ẍ(t)) and so x is a solution of (T hOSP P). □

4. Application to differential variational inequalities

In this section we are interested with the following differential variational inequality: Given T > 0 and three points x_{0}, u_{0}, v_{0} ∈ ℍ.

(DVI) \{ \begin{matrix}Find b ∈ (0, T), x : [0, b] → ℍ such that ẍ(t) ∈ S, on [0, b] and ∀w ∈ S \\ ⟨x^{(3)}(t), w − ẍ(t)⟩ ≤ a(x(t) + ẋ(t) − ẍ(t), w − ẍ(t)), a.e. on [0, b], \\ x(0) = x_{0}, ẋ(0) = u_{0}, ẍ(0) = v_{0}. \end{matrix}

where S = {x ∈ ℍ : Λ(x) ≤ 0} (Λ : ℍ → ℝ is a C^{1} convex function), a(·, ·) is a real bilinear, symmetric, bounded, and elliptic form on ℍ × ℍ. Let A be a linear and bounded operator on ℍ associated with a(·, ·), that is, a(u, v) = ⟨Au, v⟩ , ∀u, v ∈ ℍ. We prove the existence of solutions of (DVI). To do that, we recall first (see for example [5]) that

(4.1)  T (C; x) = {v ∈ ℍ : lim_{h→0^{+}} h^{−1}d_{C}(x + hv) = 0}, ∀x ∈ C,

for any closed convex set C. In our case we can check (see for instance [5]) that

(4.2)  T (S; x) = {v ∈ ℍ : ⟨▽Λ(x), v⟩ = 0}, ∀x ∈ S.

EJQTDE, 2010 No. 21, p. 9 Proposition 4.1. Assume that ℍ is a separable Hilbert space and that Λ, and A satisfy

(4.3)  ⟨▽Λ(x), ∂d_{S}(z) + A(y − z)⟩ = 0, ∀x, y, z ∈ ℍ.

Then, for any x_{0}, u_{0}, v_{0} ∈ ℍ with Λ(v_{0}) = 0, there exists b ∈ (0, T) such that (DVI) has at least one absolutely continuous solution on [0, b].

Proof. Since S is a closed convex set, the variational inequality of type (DVI) can be rewritten in the form of (ThOSPP) as follows

\{ \begin{matrix}x^{(3)}(t) − A(x(t) + ẋ(t) − ẍ(t)) ∈ N_{S}^{C}(ẍ(t)), \\ x(0) = x_{0}, ẋ(0) = u_{0}, ẍ(0) = v_{0}, and ẍ(t) ∈ S on [0, b]. \end{matrix}

Take F (t, x, y, z) = {A(x + y − z)}. Clearly F is uniformly continuous (since A is bounded linear operator) with compact values. By (4.3), we have

⟨▽Λ(x), ∂d_{S}(z) + A(x + y − z)⟩ = 0, ∀x, y, z ∈ ℍ.

and so (4.2) yields

∂d_{S}(z) + A(x + y − z) ⊂ T (x; S)

and hence we obtain by (4.1)

lim inf_{h→0^{+}} \frac{1}{h}e(x + h(∂d_{S}(z) + A(x + y − z)); S) = 0.

Consequently, all the assumptions of Corollary 3.5 are satisfied and so the proof is complete. □

Acknowledgements. The authors thank the referee for the valuable suggestions. The first author would like to thank the Abdus Salam International Center for Theoretical Physics (ICTP) for providing excellent facilities for finalizing the paper during his short visit to ICTP in July-August 2009.

References

1. M. Bounkhel; Existence results of nonconvex differential inclusions, J. Portugaliae Mathematica, Vol. 59 (2002), No. 3, pp. 283 − 310.

2. M. Bounkhel and T. Haddad; Existence of viable solutions for nonconvex differential inclusions, Electronic Journal of Differential Equations, Vol. 2005(2005), No. 50, pp. 1 − 10.

3. M. Bounkhel and L. Thibault; On various notions of regularity of sets, Nonlinear Anal.: Theory, Methods and Applications, Vol. 48(2002), No. 2, pp. 223 − 246.

4. M. Bounkhel; On arc-wise essentially smooth mappings between Banach spaces, J. Optimization, Vol. 51(2002), No.1, pp. 11 − 29.

5. F. H. Clarke; Optimization and Nonsmooth Analysis, John Wiley & Sons Inc., New York, 1983.

6. B. Hopkins; Existence of solutions for nonconvex third order differential inclusions, EJQTDE, 2005, No. 22, pp. 1 − 11.

7. X. D. H. Truong; Existence of viable solutions of nonconvex differential inclusions, Atti. Semi. Mat. Fis. Univ. Modena, 47(1999), no. 2, pp. 457-471.

(Received July 10, 2009)