Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 11, 1-22; http://www.math.u-szeged.hu/ejqtde/

On a time-dependent subdifferential evolution inclusion with a nonconvex upper-semicontinuous perturbation

S. Guillaume^{1}, A. Syam^{2}

1 Département de mathématiques, Université d’Avignon, 33, rue Pasteur, 84000 Avignon, France, sophie.guillaume@univ-avignon.fr

2 Département de mathématiques et informatique, Université Cadi Ayyad, 618 Marrakech, Maroc, syam@fstg-marrakech.ac.ma

Abstract. We investigate the existence of local approximate and strong solutions for a time-dependent subdifferential evolution inclusion with a nonconvex upper-semicontinuous perturbation.

Keywords. Convex function, Evolution equation, Nonconvex perturbation, Subdifferential, Upper-semicontinuous operator.

AMS (MOS) Subject Classifications. 35G25, 47J35, 47H14

1 Introduction

For a given family of convex lower-semicontinuous functions (f^{t})_{t∈[0,T]}, defined on a separable real Hilbert space X with range in ℝ ∪ {∞}, and a family of multivalued operators (B(t, .))_{t∈[0,T]} on X, we shall prove an existence theorem for evolution equations of type:

u′(t) + ∂f^{t}(u(t)) + B(t, u(t)) ∋ 0 , t ∈ [0, T].  (1)

For each t, ∂f^{t} denotes the ordinary subdifferential of convex analysis. The operator B(t, .) : X \overrightarrow{→} X is a multivalued perturbation of ∂f^{t}, dependent on the time t.

When the perturbation B(t, .) is single valued and monotone, many existence, uniqueness and regularity results have been established, see Brezis [3] (if f^{t} is independent of t), Attouch-Damlamian [2] and Yamada [18]. The study of case B(t, .) nonmonotone and upper-semicontinuous with convex closed values has been developed under some assumptions of compactness on dom f^{t} = {x ∈ X | f^{t}(x) < ∞} the effective domain of f^{t}. For example, Attouch-Damlamian [1] have studied the case f independent of time. Otani [15] has extended this result with more general assumptions (the convex function f^{t} depends on time). He has also studied the case where −B(t, .) is the subdifferential of a lower semicontinuous convex function, see [14]. In this article, the operator B(t, .) will be assumed upper-semicontinuous with compact values which are not necessary convex, and it is not assumed be a contraction map. Nevertheless, −B(t, .) will be assumed cyclically monotone. Cellina and Staicu [7] have studied this type of inclusion when f^{t} and B(t, .) are not dependent on t.

This paper is organized as follows. In Section 2 we recall some definitions and results on time-dependent subdifferential evolution inclusions and upper-semicontinuity of operators which will be used in the sequel. We also introduce the assumptions of our main result. In Section 3 we obtain existence of approximate solutions for the problem (1) and give properties of these solutions. In Section 4 we establish existence theorem for the problem (1). We particularly study two cases where the family (f^{t})_{t} satisfies more restricted assumptions. Examples illustrate our results in Section 5.

2 Perturbed problem

Assume that X is a real separable Hilbert space. We denote by ∥.∥ the norm associated with the inner product ⟨., .⟩ and the topological dual space is identified with the Hilbert space. Let T > 0 and (f^{t})_{t∈[0,T]} be a family of convex lower-semicontinuous (lsc, in short) proper functions on X. We will denote by ∂f^{t} the ordinary subdifferential of convex analysis.

Definition 2.1 A function u : [0, T] → X is said strong solution of

u′ + ∂f^{t}(u) + B(t, u) ∋ 0

if ^{1}: (i) there exists β ∈ L^{2}(0, T; X) such that β(t) ∈ B(t, u(t)) for a.e. t ∈ [0, T], (ii) u is a solution of \{ \begin{matrix}u′(t) + ∂f^{t}(u(t)) + β(t) ∋ 0 & for a.e. t ∈ [0, T] \\ u(t) ∈ dom f^{t} & for any t ∈ [0, T]. \end{matrix}

The aim result of this article is, for each u_{0} ∈ dom f^{0}, the existence of a local strong solution u of u′ + ∂f^{t}(u) + B(t, u) ∋ 0 with u(0) = u_{0}, when the values of the uppersemicontinuous multiapplication B(t, .) are not convex.

We shall consider the following assumption on (f^{t})_{t∈[0,T]}, see Kenmochi [10, 11]:

1As usual, L^{r}(0, T; X) (T ∈]0, ∞]) denotes the space of X-valued measurable functions on [0, T) which are r^{th} power integrable (if r = ∞, then essentially bounded). For r = 2, L^{2}(0, T; X) is a Hilbert space, in which ∥.∥_{L^{2}(0,T;X)} and ⟨., .⟩_{L^{2}(0,T;X)} are the norm and the scalar product.

(H_{0}): for each r ⩾ 0, there are absolutely continuous real-valued functions h_{r} and k_{r} on [0, T] such that:

(i) h′_{r} ∈ L^{2}(0, T) and k′_{r} ∈ L^{1}(0, T),
(ii) for each s, t ∈ [0, T] with s ⩽ t and each x_{s} ∈ dom f^{s} with ∥x_{s}∥ ⩽ r there
exists x_{t} ∈ dom f^{t} satisfying
\{ \begin{matrix}∥x_{t} − x_{s}∥ ⩽ |h_{r}(t) − h_{r}(s)|(1 + |f^{s}(x_{s})|^{1/2}) \\ f^{t}(x_{t}) ⩽ f^{s}(x_{s}) + |k_{r}(t) − k_{r}(s)|(1 + |f^{s}(x_{s})|). \end{matrix}

or the slightly stronger assumption, see Yamada [18], denoted by (H), when (ii) holds for any s, t in [0, T].

The following existence theorem have been proved in [19]:

Theorem 2.1 Let T > 0 and β ∈ L^{2}(0, T; X). Let u_{0} ∈ dom f^{0}. If (H_{0}) holds, then the problem

\{ \begin{matrix}u′(t) + ∂f^{t}(u(t)) + β(t) ∋ 0 , a.e. t ∈ [0, T] \\ u(t) ∈ dom f^{t} , t ∈ [0, T] \\ u(0) = u_{0} \end{matrix}

has a unique solution u : [0, T] → X which is absolutely continuous.

Furthermore, we have the following type of energy inequality, see [11, Chapter 1]: if ∥u(t)∥ < r for t ∈ [0, T], then

f^{t}(u(t)) − f^{s}(u(s)) + \frac{1}{2} ∫_{s}^{t} ∥u′(τ)∥^{2} dτ ⩽ \frac{1}{2} ∫_{s}^{t} ∥β(τ)∥^{2} dτ + ∫_{s}^{t} c_{r}(τ) [ 1 + |f^{τ}(u(τ))| ] dτ

(2)

for any s ⩽ t in [0, T], where c_{r} : τ ↦ 4|h′_{r}(τ)|^{2} + |k′_{r}(τ)| is an element of L^{1}(0, T).

Let us add a compactness assumption on each f^{t} by using the following definition:

Definition 2.2 A function f : X → ℝ ∪ {+∞} is said of compact type if the set {x ∈ X | |f(x)| + ∥x∥^{2} ≤ c} is compact at each level c.

Denote by L_{w}^{2}(0, T; X) the space L^{2}(0, T; X) endowed with the weak topology. Under this compactness assumption on each f^{t}, the map

p : ( \begin{matrix}L_{w}^{2}(0, T; X) & → & C([0, T]; X) \\ β & ↦ & u \end{matrix} )

is continuous and maps bounded set into relatively compact sets following [9, proposition 3.3], β and u being defined in Theorem 2.1.

Recall the definition of upper-semicontinuity of operators.

Definition 2.3 Let E_{1} and E_{2} be two Hausdorff topological sets. A multivalued operator B : E_{1} \overrightarrow{→} E_{2} is said upper-semicontinuous (usc in short) at x ∈ Dom B if for all neighborhood V_{2} of the subset Bx of E_{2}, there exists a neighborhood V_{1} of x in E_{1} such that B(V_{1}) ⊂ V_{2}.

Furthermore, if E_{1} and E_{2} are two Hausdorff topological spaces with E_{2} compact and B : E_{1} \overrightarrow{→} E_{2} is a multivalued map with Bx closed for any x ∈ E_{1}, then B is usc if and only if the graph of B is closed in E_{1} × E_{2}. We introduce following conditions on the multifunction B : [0, T] × X \overrightarrow{⟶} X:

(B_{o}) : (i) Dom(∂f^{t}) ⊂ Dom B(t, .) for any t ∈ [0, T],

(ii) there exist nonnegative constants ρ, M such that ∥x − u_{0}∥ ⩽ ρ implies B(t, x) ⊂ M𝔹_{X} for any t ∈ [0, T] and x ∈ Dom ∂f^{t}.

(B) :

(i) Dom(∂f^{t}) ⊂ Dom B(t, .) and the set B(t, x) is compact for any t ∈ [0, T] and x ∈ Dom(∂f^{t}),

(ii) there exist a nonnegative real ρ and a convex lsc function φ : X → ℝ such that ∥x − u_{0}∥ ⩽ ρ implies B(t, x) ⊂ −∂φ(x) for any t ∈ [0, T] and x ∈ Dom(∂f^{t}),

(iii) for a.e. t ∈ [0, T], the restriction of B(t, .) to Dom(∂f^{t}) is usc,

(iv) for each r ⩾ 0, there is a nonnegative real-valued function g_{r} on [0, T]^{2} such

that (a) lim_{t→s_{−}} g_{r}(t, s) = 0,

(b) for each s, t ∈ [0, T] with t ⩽ s and each x_{s} ∈ Dom ∂f^{s} and β_{s} ∈ B(s, x_{s}) with ∥x_{s}∥ ∨ ∥β_{s}∥ ⩽ r there exists x_{t} ∈ Dom B(t, .) and β_{t} ∈ B(t, x_{t}) satisfying

∥x_{t} − x_{s}∥ ∨ ∥β_{t} − β_{s}∥ ⩽ g_{r}(t, s).

By convexity, the function φ of (B)(ii) is M-Lipschitz continuous on some closed ball u_{0} + ρ𝔹_{X} and the inclusion ∂φ(x) ⊂ M𝔹_{X} holds for any x ∈ u_{0} + ρ𝔹_{X}. In fact, we could take φ with extended real values and u_{0} in the interior of the effective domain of φ. Thus, (B)(ii) implies (B_{o})(ii). The condition (B)(ii) means that −B(t, .) is cyclically monotone uniformly in t. An example is the multiapplication B(t, .) : ℝ^{n} \overrightarrow{→} ℝ^{n} defined by

β = (β_{1}, …, β_{n}) ∈ B(t, x) ⟺ β_{1} ∈ \{ \begin{matrix}\{1\} & if x_{1} < 0 \\ \{−1, 1\} & if x_{1} = 0 \\ \{−1\} & if x_{1} > 0 \end{matrix} and β_{2} = ⋯ = β_{n} = 0

for any x = (x_{1}, …, x_{n}) ∈ ℝ^{n}. When B(t, .) = −∂ψ^{t} with ψ^{t} : X → ℝ ∪ {+∞} a lsc proper function, then ψ^{t} is convex if this operator is monotone and (B)(ii) is equivalent to the existence of a real constant α_{t} with ψ^{t} = φ + α_{t}. In this case we deals with the problem u′ + ∂f^{t}(u) − ∂φ(u) ∋ 0, see Otani [14] when f^{t} is not dependent on t. This condition (ii) could be extended to a function φ^{t} which depends on the time t, and also with a nonconvex function: for example, a convex composite function, see [8].

The condition (B)(iii) is always satisfied if B(t, .) or −B(t, .) is a maximal monotone operator of X, and more generally if they are ϕ-monotone of order 2.

The condition (B)(iv) is always satisfied if B(t, .) = B : X \overrightarrow{→} X is not depending on the time t. It can also be written for any t ⩽ s in [0, T]:

lim_{t→s_{−}} e(gphB(t, .) ∩ r𝔹_{X^{2}}, gphB(s, .)) = 0,

e standing for the excess between two sets. When B(t, .) or −B(t, .) is the subdiffferential of a convex lsc function ψ^{t} which satisfies (H_{0}), the condition (iv) is satisfied.

3 Existence of approximate solutions

For any real λ > 0 and t ∈ [0, T], the function f_{λ}^{t} shall denote the Moreau-Yosida proximal function of index λ of f^{t}, and we set

J_{λ}^{t} = (I + λ∂f^{t})^{−1} , Df_{λ}^{t} = λ^{−1}(I − J_{λ}^{t}).

We first prove the approximate result of existence :

Theorem 3.1 Let (f^{t})_{t∈[0,T]} be a family of proper convex lsc functions on X with each f^{t} of compact type. Assume that (H) and (B_{o}) are satisfied. For each u_{0} ∈ dom f^{0}, there exists T_{0} ∈]0, T] such that u′ + ∂f^{t}(u) + B(t, u) ∋ 0 has at least an approximate solution x : [0, T_{0}] → X with x(0) = u_{0} in the following sense: there exist sequences (x_{n})_{n} of absolutely continuous functions from [0, T_{0}] to X, (u_{n})_{n} and (β_{n})_{n} of piecewise constant functions from [0, T_{0}] to X which satisfy:

1. for a.e. t ∈ [0, T_{0}]

\{ \begin{matrix}x′_{n}(t) + ∂f^{t}(x_{n}(t)) + β_{n}(t) ∋ 0 \\ x_{n}(0) = u_{0} \end{matrix} and β_{n}(t) ∈ B(θ_{n}(t), u_{n}(t))

where 0 ⩽ t − θ_{n}(t) ⩽ 2^{−n}T,

2. there exists N ∈ ℕ such that for any n ⩾ N:

∀t ∈ [0, T] ∥x_{n}(t) − u_{0}∥ ⩽ ρ and ∥β_{n}(t)∥ ⩽ M,

3. (x_{n})_{n} and (u_{n})_{n} converge uniformly to x on [0, T_{0}], (β_{n})_{n} converges weakly to β in L^{2}(0, T_{0}; X), (x′_{n})_{n} converges weakly to x′ in L^{2}(0, T_{0}; X) and x is the solution of x′(t) + ∂f^{t}(x(t)) + β(t) ∋ 0, x(0) = u_{0} on [0, T_{0}].

3.1 Proof of Theorem 3.1

Lemma 3.1 We can find a set {z_{t} : t ∈ [0, T]} and ρ_{0} > 0 such that z_{t} ∈ ρ_{0}𝔹, f^{t}(z_{t}) ≤ ρ_{0} for every t ∈ [0, T].

Proof. Let z_{0} ∈ dom f^{0} and r > 0 such that r ⩾ ∥z_{0}∥ ∨ |f^{0}(z_{0})|. For all t ∈ [0, T], there exists z_{t} ∈ dom f^{t} satisfying

\{ \begin{matrix}∥z_{t} − z_{0}∥ ⩽ |h_{r}(t) − h_{r}(0)|(1 + |f^{0}(z_{0})|^{1/2}) \\ f^{t}(z_{t}) ⩽ f^{0}(z_{0}) + |k_{r}(t) − k_{r}(0)|(1 + |f^{0}(z_{0})|). \end{matrix}

The lemma holds with ρ_{0} = ( r + ∥h′_{r}∥_{L^{1}}(1 + r^{1/2}) ) ∨ ( r + ∥k′_{r}∥_{L^{1}}(1 + r) ). □

Lemma 3.2 [18, Proposition 3.1]. Let x ∈ X and λ > 0. The map t ↦ J_{λ}^{t}x is continuous on [0, T].

Proof. ¿From Kenmochi [11, Chapter 1, Section 1.5, Theorem 1.5.1], there is a nonnegative constant α such that f^{t}(x) ⩾ −α(∥x∥ + 1) for all x ∈ X and t ∈ [0, T]. Thus,

f_{λ}^{t}(x) − \frac{1}{2λ}∥x − J_{λ}^{t}x∥^{2} = f^{t}(J_{λ}^{t}x) ⩾ −α(1 + ∥J_{λ}^{t}x∥),

which implies

∥x − J_{λ}^{t}x∥^{2} ⩽ 2λα(1 + ∥J_{λ}^{t}x − x∥ + ∥x∥) + 2λf_{λ}^{t}(x).  (3)

Since 2λf_{λ}^{t}(x) ⩽ 2λf^{t}(z_{t})+∥z_{t} −x∥^{2} ⩽ 2λρ_{0} +(ρ_{0} +∥x∥)^{2} by Lemma 3.1, we can conclude:

sup{∥J_{λ}^{t}x∥ | t ∈ [0, T] , λ ∈]0, 1] , x ∈ r𝔹} < ∞
sup{|f^{t}(J_{λ}^{t}x)| | t ∈ [0, T] , x ∈ r𝔹} < ∞

for any r > 0.

Let t ∈ [0, T] and r ⩾ ∥J_{λ}^{t}x∥. By assumption (H_{0}), for each s ∈ [0, T] with s ⩾ t there exists x_{s} ∈ dom f^{s} satisfying

\{ \begin{matrix}∥J_{λ}^{t}x − x_{s}∥ ⩽ |h_{r}(t) − h_{r}(s)|(1 + |f^{t}(J_{λ}^{t}x)|^{1/2}) \\ f^{s}(x_{s}) ⩽ f^{t}(J_{λ}^{t}x) + |k_{r}(t) − k_{r}(s)|(1 + |f^{t}(J_{λ}^{t}x)|). \end{matrix}

Since λ^{−1}(x − J_{λ}^{s}x) ∈ ∂f^{s}(J_{λ}^{s}x), we have

f^{s}(J_{λ}^{s}x) + \frac{1}{λ}⟨x − J_{λ}^{s}x, x_{s} − J_{λ}^{s}x⟩ ⩽ f^{s}(x_{s}) ⩽ f^{t}(J_{λ}^{t}x) + |k_{r}(t) − k_{r}(s)|(1 + |f^{t}(J_{λ}^{t}x)|).

Hence, for any s ⩾ t, we have

\frac{1}{λ}⟨x − J_{λ}^{s}x, J_{λ}^{t}x − J_{λ}^{s}x⟩

⩽ \frac{1}{λ}⟨x − J_{λ}^{s}x, J_{λ}^{t}x − x_{s}⟩ + f^{t}(J_{λ}^{t}x) − f^{s}(J_{λ}^{s}x) + |k_{r}(t) − k_{r}(s)|(1 + |f^{t}(J_{λ}^{t}x)|)
⩽ ∥Df_{λ}^{s}(x)∥|h_{r}(t) − h_{r}(s)|(1 + |f^{t}(J_{λ}^{t}x)|^{1/2}) + f^{t}(J_{λ}^{t}x) − f^{s}(J_{λ}^{s}x)

+|k_{r}(t) − k_{r}(s)|(1 + |f^{t}(J_{λ}^{t}x)|).

By symmetry it is true for any s ∈ [0, T]. In the same way for t, s in [0, T], we have

\frac{1}{λ}⟨x − J_{λ}^{t}x, J_{λ}^{s}x − J_{λ}^{t}x⟩ ⩽ ∥Df_{λ}^{t}(x)∥|h_{r}(t) − h_{r}(s)|(1 + |f^{s}(J_{λ}^{s}x)|^{1/2}) + f^{s}(J_{λ}^{s}x) − f^{t}(J_{λ}^{t}x)

+|k_{r}(t) − k_{r}(s)|(1 + |f^{s}(J_{λ}^{s}x)|).

Adding these two inequalities we obtain

\frac{1}{λ}∥J_{λ}^{s}x − J_{λ}^{t}x∥^{2} ⩽ [ ∥Df_{λ}^{t}(x)∥ ∨ ∥Df_{λ}^{s}(x)∥ ] |h_{r}(t) − h_{r}(s)|(1 + |f^{s}(J_{λ}^{s}x)|^{1/2} ∨ |f^{t}(J_{λ}^{t}x)|^{1/2})
+|k_{r}(t) − k_{r}(s)|(1 + |f^{s}(J_{λ}^{s}x)| ∨ |f^{t}(J_{λ}^{t}x)|).

Since both ∥Df_{λ}^{t}(x)∥ and |f^{t}(J_{λ}^{t}x)| are bounded, t ↦ J_{λ}^{t}x is continuous on [0, T]. □

By [11, Lemma 1.5.3], for r ⩾ ∥u_{0}∥ + 1, M_{1} ⩾ |f^{0}(u_{0})| + αr + α + 1 and T_{1} ∈]0, T [ such that

[1 + M_{1} exp ∫_{0}^{T} |k′_{r}|] ∫_{0}^{T_{1}} |h′_{r}| ⩽ 1,

there exists an absolutely continuous function v on [0, T_{1}] satisfying:

* v(0) = u_{0} and lim sup_{t→0_{+}} f^{t}(v(t)) ⩽ f^{0}(u_{0})
* ∥v(t)∥ ⩽ r for any t ∈ [0, T_{1}]
* for any t ∈ [0, T_{1}], |f^{t}(v(t))| ⩽ M_{1} + M_{1} exp ∫_{0}^{T} |k′_{r}| ∫_{0}^{t} |h′_{r}|
* for almost any t ∈ [0, T_{1}], ∥v′(t)∥ ⩽ [1 + M_{1} exp ∫_{0}^{T} |k′_{r}|] |h′_{r}(t)|.

For r ⩾ ∥u_{0}∥ + ρ, let us choose T_{2} > 0 such that

(|f^{0}(u_{0})| + \frac{M^{2}}{2} T_{2} + ∫_{0}^{T_{2}} c_{r}(τ) dτ) (1 + T_{2} exp ∫_{0}^{T_{2}} c_{r}(τ) dτ) ⩽ |f^{0}(u_{0})| + ρ.

Let r ⩾ (∥u_{0}∥ ∨ |f^{0}(u_{0})|) + ρ + 1 be fixed. Let us choose T_{0} > 0 small enough in order to have

(1 + r^{1/2})^{2}T_{0} ∫_{0}^{T_{0}} |h′_{r}| ⩽ \frac{ρ^{2}}{32} , T_{0} ⩽ T_{1} ∧ T_{2} and

M_{T} \sqrt{T_{0}} + [M + α]T_{0} + [1 + M_{1} exp(∫_{0}^{T} |k′_{r}|)] ∫_{0}^{T_{0}} |h′_{r}(s)| ds ⩽ \frac{ρ}{4}

where M_{T} = 2 [M_{1} + M_{1} (exp ∫_{0}^{T} |k′_{r}|) ∫_{0}^{T} |h′_{r}(s)| ds + αr + α]^{1/2}.

Lemma 3.3 Let β : [0, T] → X be a measurable function with ∥β(t)∥ ⩽ M for a.e. t ∈ [0, T]. Then,

∀t ∈ [0, T_{0}] ∥p(β)(t) − u_{0}∥ ⩽ \frac{ρ}{2},

the map p being defined in Section 2.

Proof. The curve u = p(β) exists on [0, T] following Theorem 2.1. We have for a.e. t ∈ [0, T_{0}]:

\frac{d}{dt} \frac{1}{2}∥u(t) − v(t)∥^{2} ⩽ f^{t}(v(t)) − f^{t}(u(t)) + [M + ∥v′(t)∥] ∥u(t) − v(t)∥
⩽ \frac{1}{2}M_{T}^{2} + [M + ∥v′(t)∥ + α] ∥u(t) − v(t)∥.

We thus obtain for any t ∈ [0, T_{0}]

\frac{1}{2}∥u(t) − v(t)∥^{2} ⩽ \frac{1}{2}M_{T}^{2}T_{0} + ∫_{0}^{t}[M + ∥v′(s)∥ + α] ∥u(s) − v(s)∥ ds.

Gronwall’s lemma yields for any t ∈ [0, T_{0}]

∥u(t) − v(t)∥ ⩽ M_{T} \sqrt{T_{0}} + ∫_{0}^{t}[M + ∥v′(s)∥ + α] ds
⩽ M_{T} \sqrt{T_{0}} + [M + α]T_{0} + [1 + M_{1} exp ∫_{0}^{T} |k′_{r}|] ∫_{0}^{T_{0}} |h′_{r}(s)| ds ⩽ \frac{ρ}{4}.

Furthermore,

∥v(t) − u_{0}∥ ⩽ ∫_{0}^{t} ∥v′(s)∥ ds ⩽ [1 + M_{1} exp ∫_{0}^{T} |k′_{r}|] ∫_{0}^{T_{0}} |h′_{r}(s)| ds ⩽ \frac{ρ}{4}.

By choice of T_{0} > 0, we obtain ∥u(t) − u_{0}∥ ⩽ ρ/2 for any t ∈ [0, T_{0}]. □

For simplicity of notation, we now write T instead of T_{0}. We also assume that f^{t}(x) ⩾ 0 for any x ∈ X with ∥x − u_{0}∥ ⩽ ρ, since we have f^{t}(x) ⩾ −α(∥u_{0}∥ + ρ + 1).

Let n ∈ ℕ^{⋆} such that :

α^{2}2^{−6n} + 2^{−3n+1} [r + (1 + r)(∫_{0}^{T} |k′_{r}| + α)] ⩽ \frac{ρ^{2}}{32}.

Let us set f_{n}^{t} = f_{2^{−3n}}^{t} and J_{n}^{t} = J_{2^{−3n}}^{t}. Let P be a partition of [0, T]:

P = {0 = t_{0}^{n} < t_{1}^{n} < ⋯ < t_{2^{n}}^{n} = T }

where t_{k}^{n} = k2^{−n}T for k = 0, …, 2^{n}.

Let us set u_{0}^{n} = J_{n}^{t_{0}^{n}}u_{0}. By assumption (B_{o})(i), B(t_{0}^{n}, u_{0}^{n}) is non empty and contains an element β_{0}^{n}. Let t ∈ [0, T]. Under the assumption (H_{0}), there exists u_{n,t} ∈ dom f^{t} satisfying

\{ \begin{matrix}∥u_{n,t} − u_{0}∥ ⩽ |h_{r}(t) − h_{r}(0)|(1 + r^{1/2}) \\ f^{t}(u_{n,t}) ⩽ r + |k_{r}(t) − k_{r}(0)|(1 + r). \end{matrix}

Using the definition of the Moreau-Yosida approximate, we obtain

\frac{2^{3n}}{2} ∥J_{n}^{t}u_{0} − u_{0}∥^{2} = f_{n}^{t}(u_{0}) − f^{t}(J_{n}^{t}u_{0})
⩽ f^{t}(u_{n,t}) + \frac{2^{3n}}{2} ∥u_{n,t} − u_{0}∥^{2} + α∥J_{n}^{t}u_{0} − u_{0}∥ + α(1 + ∥u_{0}∥)
⩽ r + (1 + r) ∫_{0}^{t} |k′_{r}| + \frac{2^{3n}}{2} (1 + r^{1/2})^{2}t ∫_{0}^{t} |h′_{r}|^{2} + α∥J_{n}^{t}u_{0} − u_{0}∥ + α(1 + r).

Thus, by choice of r, T and n we obtain

∥J_{n}^{t}u_{0} − u_{0}∥
⩽ α2^{−3n} + \sqrt{α^{2}2^{−6n} + 2r2^{−3n} + 2(1 + r)2^{−3n}(∫_{0}^{T} |k′_{r}| + α) + (1 + \sqrt{r})^{2}T ∫_{0}^{T} |h′_{r}|^{2}}
⩽ \frac{ρ}{2}.

In particular, ∥u_{0}^{n}−u_{0}∥ ⩽ ρ/2. Under (B_{o})(ii) it follows ∥β_{0}^{n}∥ ⩽ M. Let us set x_{0}^{n} = p(β_{0}^{n}). By Lemma 3.3,

∀t ∈ [0, T] ∥x_{0}^{n}(t) − u_{0}∥ ⩽ \frac{ρ}{2}.

Let us set u_{1}^{n} = J_{n}^{t_{1}^{n}}x_{0}^{n}(t_{1}^{n}) and take β_{1}^{n} ∈ B(t_{1}^{n}, u_{1}^{n}). Since J_{n}^{t_{1}^{n}} is 1-Lipschitz continuous, we have

∥u_{1}^{n} − u_{0}∥ ⩽ ∥x_{0}^{n}(t_{1}^{n}) − u_{0}∥ + ∥J_{n}^{t_{1}^{n}}u_{0} − u_{0}∥ ⩽ ρ.

Next, (B_{o})(ii) implies ∥β_{1}^{n}∥ ⩽ M. We then set

β_{1}^{n}(t) = \{ \begin{matrix}β_{0}^{n} if t ∈ [t_{0}^{n}, t_{1}^{n}[ \\ β_{1}^{n} if t ∈ [t_{1}^{n}, T] \end{matrix}

Let us set x_{1}^{n} = p(β_{1}^{n}). By unicity and continuity of the curve it follows x_{1}^{n}(t) = x_{0}^{n}(t) if t ∈ [t_{0}^{n}, t_{1}^{n}]. Furthermore, ∥β_{1}^{n}(t)∥ ⩽ M for any t ∈ [0, T]. By Lemma 3.3,

∀t ∈ [0, T] ∥x_{1}^{n}(t) − u_{0}∥ ⩽ \frac{ρ}{2}.

Let k ∈ ℕ^{⋆}. Assume that there exists a map β_{k−1}^{n} : [0, T] → X which is constant on each [t_{k−1}^{n}, t_{k}^{n}[ with ∥β_{k−1}^{n}(t)∥ ⩽ M for any t ∈ [0, T]. Set x_{k−1}^{n} = p(β_{k−1}^{n}). Then,

∀t ∈ [0, T] ∥x_{k−1}^{n}(t) − u_{0}∥ ⩽ \frac{ρ}{2}.

Let us set u_{k}^{n} = J_{n}^{t_{k}^{n}}x_{k−1}^{n}(t_{k}^{n}) and take β_{k}^{n} ∈ B(t_{k}^{n}, u_{k}^{n}). Since

∥u_{k}^{n} − u_{0}∥ ⩽ ∥x_{k−1}^{n}(t_{k}^{n}) − u_{0}∥ + ∥J_{n}^{t_{k}^{n}}u_{0} − u_{0}∥ ⩽ ρ

we have ∥β_{k}^{n}∥ ⩽ M. We then set

β_{k}^{n}(t) = \{ \begin{matrix}β_{k−1}^{n}(t) if t ∈ [t_{0}^{n}, t_{k}^{n}[ \\ β_{k}^{n} if t ∈ [t_{k}^{n}, T] \end{matrix}

Let us set x_{k}^{n} = p(β_{k}^{n}). By unicity it follows x_{k}^{n}(t) = x_{k−1}^{n}(t) if t ∈ [0, t_{k}^{n}]. Furthermore, ∥β_{k}^{n}(t)∥ ⩽ M for any t ∈ [0, T]. By Lemma 3.3,

∀t ∈ [0, T] ∥x_{k}^{n}(t) − u_{0}∥ ⩽ \frac{ρ}{2}.

We then set

x_{n} := x_{2^{n}−1}^{n} = ∑_{k=0}^{2^{n}} x_{k}^{n} χ_{[t_{k}^{n},t_{k+1}^{n}[} and β_{n} := β_{2^{n}−1}^{n} = ∑_{k=0}^{2^{n}} β_{k}^{n} χ_{[t_{k}^{n},t_{k+1}^{n}[},

where χ_{[t_{k}^{n},t_{k+1}^{n}[}(t) = 1 if t ∈ [t_{k}^{n}, t_{k+1}^{n}[ , and = 0 otherwise. For all t ∈ [0, T [, there exists 0 ⩽ k ⩽ 2^{n} with t ∈ [t_{k}^{n}, t_{k+1}^{n}[ and we set

θ_{n}(t) = t_{k}^{n} and θ_{n}(T) = T.

So, x_{n} : [0, T] → X is an absolutely continuous function and β_{n} : [0, T] → X is a measurable map which satisfy for a.e. t ∈ [0, T]

\{ \begin{matrix}x′_{n}(t) + ∂f^{t}(x_{n}(t)) + β_{n}(t) ∋ 0 \\ x_{n}(0) = u_{0} \end{matrix} and β_{n}(t) ∈ B(θ_{n}(t), u_{n}(t))

where we set u_{n}(t) = J_{n}^{θ_{n}(t)}x_{n}(θ_{n}(t)). By construct, there exists N ∈ ℕ such that for any n ⩾ N:

∀t ∈ [0, T] ∥x_{n}(t) − u_{0}∥ ⩽ ρ and ∥β_{n}(t)∥ ⩽ M.

A subsequence of (β_{n})_{n}, again denoted by (β_{n})_{n}, converges weakly to β in L^{2}(0, T; X). By continuity of the map p, the sequence x_{n} = p(β_{n}) converges uniformly to a curve x = p(β) on [0, T] and a subsequence of (x′_{n})_{n} converges weakly to x′ in L^{2}(0, T; X). In other words, the curve x is the solution of x′(t) + ∂f^{t}(x(t)) + β(t) ∋ 0, x(0) = u_{0} on [0, T].

Let n ∈ ℕ^{⋆} and t ∈ [0, T]. We have

∥u_{n}(t) − x(t)∥ ⩽ ∥x_{n}(θ_{n}(t)) − x(θ_{n}(t))∥ + ∥x(θ_{n}(t)) − x(t)∥ + ∥J_{n}^{θ_{n}(t)}x(t) − x(t)∥. (4)

Under the assumption (H_{0}), there exists u_{n,t} ∈ dom f^{θ_{n}(t)} satisfying

\{ \begin{matrix}∥u_{n,t} − x(t)∥ ⩽ |h_{r}(θ_{n}(t)) − h_{r}(t)|(1 + r^{1/2}) \\ f^{θ_{n}(t)}(u_{n,t}) ⩽ r + |k_{r}(θ_{n}(t)) − k_{r}(t)|(1 + r). \end{matrix}

\frac{2^{3n}}{2} ∥J_{n}^{θ_{n}(t)}x(t) − x(t)∥^{2} = f_{n}^{θ_{n}(t)}(x(t)) − f^{t}(J_{n}^{θ_{n}(t)}x(t))
⩽ f^{θ_{n}(t)}(u_{n,t}) + \frac{2^{3n}}{2} ∥u_{n,t} − x(t)∥^{2} + α∥J_{n}^{θ_{n}(t)}x(t) − x(t)∥ + α(1 + r)
⩽ r + (1 + r) ∫_{θ_{n}(t)}^{t} |k′_{r}| + \frac{2^{3n}}{2} (1 + r^{1/2})^{2}(t − θ_{n}(t)) ∫_{θ_{n}(t)}^{t} |h′_{r}|^{2}
+α∥J_{n}^{θ_{n}(t)}x(t) − x(t)∥ + α(1 + r).

Thus,

∥J_{n}^{θ_{n}(t)}x(t) − x(t)∥ ⩽

α2^{−3n} + \sqrt{α^{2}2^{−6n} + 2r2^{−3n} + 2(1 + r)2^{−3n}(∫_{0}^{T} |k′_{r}| + α) + (1 + r^{1/2})^{2}2^{−n} ∫_{0}^{T} |h′_{r}|^{2}}

and (J_{n}^{θ_{n}(.)}x)_{n} converges uniformly to x on [0, T]. Since (x_{n})_{n} converges uniformly to x on [0, T] and x is continuous on [0, T], (4) assures the uniform convergence of (u_{n})_{n} to x on [0, T]. □

3.2 Properties of approximate solutions

Lemma 3.4 We have ∥x(t)∥ ∨ |f^{t}(x(t))| ⩽ r for all t ∈ [0, T]. Under the assumption (B)(ii), the element β(t) belongs to −∂φ(x(t)) for a.e. t ∈ [0, T].

Proof. By inequality (2), we have for any s ⩽ t in [0, T]

f^{t}(x(t)) − f^{s}(x(s)) + \frac{1}{2} ∫_{s}^{t} ∥x′(τ)∥^{2} dτ ⩽ \frac{1}{2}M^{2}T + ∫_{s}^{t} c_{r}(τ) [ 1 + |f^{τ}(x(τ))| ] dτ.

Since ∥x(t) − u_{0}∥ ⩽ ρ, we have assumed for simplicity that f^{t}(x(t)) ⩾ 0 for any t ∈ [0, T]. Gronwall’s lemma yields for any t ∈ [0, T]

f^{t}(x(t)) ⩽ (f^{0}(u_{0}) + \frac{M^{2}}{2} T + ∫_{0}^{T} c_{r}(τ) dτ) (1 + T exp ∫_{0}^{T} c_{r}(τ) dτ) ⩽ |f^{0}(u_{0})|+ρ. (5)

by assumption on T.

Next, β_{n}(t) belongs to −∂φ(u_{n}(t)) with the uniform convergence of (u_{n})_{n} to x on [0, T]. Let us define φ̃ : L^{2}(0, T; X) → ℝ ∪ {+∞} by φ̃(u) = ∫_{0}^{T} φ(u(t)) dt. It is known that φ̃ is proper lsc convex and

α ∈ ∂φ̃(u) ⟺ α(t) ∈ ∂φ(u(t)) for a.e. t ∈ [0, T].

Thus, −β_{n} ∈ ∂φ̃(u_{n}). Passing to the limit we obtain −β ∈ ∂φ̃(x). Hence, β(t) belongs to −∂φ(x(t)) for a.e. t ∈ [0, T]. □

Lemma 3.5 For almost any t ∈ [0, T] we have f^{t}(x(t)) = lim inf_{n→+∞} f^{t}(x_{n}(t)). Furthermore,

lim_{n→+∞} ∫_{0}^{T} f^{t}(x_{n}(t)) dt = ∫_{0}^{T} f^{t}(x(t)) dt and lim_{n→+∞} ∫_{0}^{T} (f^{t})^{⋆}(y_{n}(t)) dt = ∫_{0}^{T} (f^{t})^{⋆}(y(t)) dt,

where we set y_{n}(t) = −x′_{n}(t) − β_{n}(t) and y(t) = −x′(t) − β(t) for a.e. t in [0, T].

Proof. By lower semicontinuity of f^{t}, the inequality

f^{t}(x(t)) ⩽ lim inf_{n→+∞} f^{t}(x_{n}(t))

holds for any t ∈ [0, T]. The maps v ↦ ∫_{0}^{T} f^{t}(v(t)) dt and w ↦ ∫_{0}^{T} (f^{t})^{⋆}(w(t)) dt are proper lsc convex on L^{2}(0, T; X). So,

lim inf_{n→+∞} ∫_{0}^{T} f^{t}(x_{n}(t)) dt ⩾ ∫_{0}^{T} f^{t}(x(t)) dt and lim inf_{n→+∞} ∫_{0}^{T} (f^{t})^{⋆}(y_{n}(t)) dt ⩾ ∫_{0}^{T} (f^{t})^{⋆}(y(t)) dt.

But, f^{t}(x_{n}(t)) + (f^{t})^{⋆}(y_{n}(t)) = ⟨y_{n}(t), x_{n}(t)⟩ for any t ∈ [0, T], with

lim_{n→+∞} ∫_{0}^{T} ⟨y_{n}(t), x_{n}(t)⟩ dt = ∫_{0}^{T} ⟨y(t), x(t)⟩ dt.

□

Lemma 3.6 We have the inequality

∫_{0}^{T} ⟨β(s), x′(s)⟩ ds ⩽ lim inf_{n→+∞} ∫_{0}^{T} ⟨β_{n}(s), x′_{n}(s)⟩ ds.  (6)

Proof. Let n ∈ ℕ^{⋆}. The maps x_{n}, β_{n} and u_{n} are constant on [t_{k}^{n}, t_{k+1}^{n}[, k = 0, …, 2^{n} − 1. Hence,

∫_{0}^{T} ⟨β_{n}(s), x′_{n}(s)⟩ ds = ∑_{k=0}^{2^{n}−1} ∫_{t_{k}^{n}}^{t_{k+1}^{n}}⟨β_{k}^{n}, (x_{k}^{n})′(s)⟩ ds = ∑_{k=0}^{2^{n}−1}⟨β_{k}^{n}, x_{k}^{n}(t_{k+1}^{n}) − x_{k}^{n}(t_{k}^{n})⟩.

Since β_{k}^{n} ∈ −∂φ(u_{k}^{n}) for any k = 0, …, 2^{n} − 1, we obtain:

∫_{0}^{T} ⟨β_{n}(s), x′_{n}(s)⟩ ds ⩾ ∑_{k=0}^{2^{n}−1} φ(u_{k}^{n}) − φ(x_{k}^{n}(t_{k+1}^{n})) − M∥u_{k}^{n} − x_{k}^{n}(t_{k}^{n})∥

= φ(u_{0}^{n}) − φ(x_{2^{n}−1}^{n}(t_{2^{n}}^{n})) + ∑_{k=1}^{2^{n}−1} φ(u_{k}^{n}) − φ(x_{k}^{n}(t_{k}^{n})) − M∥u_{k}^{n} − x_{k}^{n}(t_{k}^{n})∥

⩾ φ(J_{n}^{0}u_{0}) − φ(x_{n}(T)) − 2M ∑_{k=1}^{2^{n}−1} ∥u_{k}^{n} − x_{k}^{n}(t_{k}^{n})∥.

Since ∥u_{k}^{n} − u_{0}∥ ⩽ ∥u_{n}(t_{k}^{n}) − x_{n}(t_{k}^{n})∥ + ρ/2 ⩽ ρ for n large enough, we have f^{t_{k}^{n}}(u_{k}^{n}) ⩾ 0. Furthermore, inequality (5) assures that f^{t_{k}^{n}}(x_{k}^{n}(t_{k}^{n})) ⩽ f^{0}(u_{0})+ρ ⩽ r. Using the definition of the Moreau-Yosida approximate, we obtain

\frac{2^{3n}}{2} ∥u_{k}^{n} − x_{k}^{n}(t_{k}^{n})∥^{2} = f_{n}^{t_{k}^{n}}(x_{k}^{n}(t_{k}^{n})) − f^{t_{k}^{n}}(u_{k}^{n}) ⩽ r.

Thus, ∥u_{k}^{n} − x_{k}^{n}(t_{k}^{n})∥ ⩽ \sqrt{r2^{−3n+1}} and

∑_{k=1}^{2^{n}−1} ∥u_{k}^{n} − x_{k}^{n}(t_{k}^{n})∥ ⩽ \sqrt{r2^{−n+1}}.

Consequently,

lim_{n→+∞} ∑_{k=1}^{2^{n}−1} ∥u_{k}^{n} − x_{k}^{n}(t_{k}^{n})∥ = 0.

By continuity of φ and convergence of (x_{n})_{n} to x, we obtain

lim inf_{n→+∞} ∫_{0}^{T} ⟨β_{n}(s), x′_{n}(s)⟩ ds ⩾ φ(u_{0}) − φ(x(T)).

Since β(s) ∈ −∂φ(x(s)) almost everywhere, ⟨β(s), x′(s)⟩ = −(φ ∘ x)′(s) holds for a.e. s and we obtain the inequality (6) . □

4 Existence of strong solutions

We now prove the existence of strong solutions.

4.1 General case

Let us set

Φ(x, y) = ∫_{0}^{T} ⟨y(t), x′(t)⟩ dt − f^{T} (x(T)) + f^{0}(u_{0})

for any absolutely continuous function x : [0, T] → X with x′ ∈ L^{2}(0, T; X) and any fonction y ∈ L^{2}(0, T; X).

Theorem 4.1 Let (f^{t})_{t∈[0,T]} be a family of proper convex lsc functions on X with each f^{t} of compact type which satifies the assumption (H). Assume that for any sequence (x_{n})_{n} in H^{1}(0, T; X) which converges uniformly to the absolutely continuous function x with the weak convergence of (x′_{n})_{n} to x′ in L^{2}, and for any (y_{n})_{n} which converges weakly to y in L^{2} with y_{n}(t) ∈ ∂f^{t}(x_{n}(t)) for almost all t, there exists n_{k} → +∞ such that

lim inf_{k→+∞} Φ(x_{n_{k}}, y_{n_{k}}) ⩾ Φ(x, y).

Then, for each u_{0} ∈ dom f^{0}, there exists T_{0} ∈]0, T] such that u′ + ∂f^{t}(u) + B(t, u) ∋ 0 has at least a strong solution u : [0, T_{0}] → X with u(0) = u_{0}.

Proof. Consider x an approximate solution. We prove x′(t) + ∂f^{t}(x(t)) + B(t, x(t)) ∋ 0 for a.e. t in [0, T]. So, we begin by prove that (x′_{n})_{n} converges strongly to x′ in L^{2}(0, T; X). \underline{Step 1.} - Let us set y_{n}(t) = −x′_{n}(t) − β_{n}(t) and y(t) = −x′(t) − β(t) for a.e. t in [0, T]. It is easy to see that for any n ∈ ℕ and almost any t ∈ [0, T]:

∥x′_{n}(t)∥^{2}+⟨y_{n}(t), x′_{n}(t)⟩+⟨β_{n}(t), x′_{n}(t)⟩ = 0 and ∥x′(t)∥^{2}+⟨y(t), x′(t)⟩+⟨β(t), x′(t)⟩ = 0.

The sequence (x′_{n})_{n} converges weakly to x′ in L^{2}(0, T; X). The strong convergence of (x′_{n})_{n} to x′ in L^{2}(0, T; X) is equivalent to

lim sup_{n→+∞} ∫_{0}^{T} ∥x′_{n}(t)∥^{2} dt ⩽ ∫_{0}^{T} ∥x′(t)∥^{2} dt.

¿From Lemma 3.6 it follows:

lim sup_{n→+∞} ∫_{0}^{T} ∥x′_{n}(t)∥^{2} dt ⩽ − lim inf_{n→+∞} ∫_{0}^{T} ⟨y_{n}(t), x′_{n}(t)⟩ dt − lim inf_{n→+∞} ∫_{0}^{T} ⟨β_{n}(t), x′_{n}(t)⟩ dt
⩽ − lim inf_{n→+∞} ∫_{0}^{T} ⟨y_{n}(t), x′_{n}(t)⟩ dt − ∫_{0}^{T} ⟨β(t), x′(t)⟩ dt.

Since ∫_{0}^{T} ⟨β(t), x′(t)⟩ dt = − ∫_{0}^{T} ∥x′(t)∥^{2} dt + ∫_{0}^{T} ⟨y(t), x′(t)⟩ dt, it suffices to show that

∫_{0}^{T} ⟨y(t), x′(t)⟩ dt ⩽ lim inf_{n→+∞} ∫_{0}^{T} ⟨y_{n}(t), x′_{n}(t)⟩ dt.  (7)

\underline{Step 2.} - Under the assumption on Φ, it follows by lower semicontinuity of f^{T} :

lim inf_{k→+∞} ∫_{0}^{T} ⟨y_{n_{k}}(t), x′_{n_{k}}(t)⟩ dt ⩾ f^{T} (x(T))−f^{0}(u_{0})+lim inf_{k→+∞} Φ(x_{n_{k}}, y_{n_{k}}) ⩾ ∫_{0}^{T} ⟨y(t), x′(t)⟩ dt.

\underline{Step 3.} - Let N be the negligeable subset of [0, T] such that, for any t ∈ [0, T]\N, we have x′_{n}(t) + ∂f^{t}(x_{n}(t)) + β_{n}(t) ∋ 0, β_{n}(t) ∈ B(θ_{n}(t), u_{n}(t)) and (x′_{n}(t))_{n} converges to x′(t). Since each f^{t} are of compact type, the sets X(t) := cl{x_{n}(t) | n ∈ ℕ^{⋆}} and U(t) := cl{u_{n}(t) | n ∈ ℕ^{⋆}} are compact in Dom(∂f^{t}). Let r ⩾ M ∨ (ρ + ∥u_{0}∥). Under the assumption (B)(iv), for each n ⩾ N and t ∈ [0, T] with t ≠ θ_{n}(t), there exists z_{n}^{t} ∈ Dom B(t, .) and α_{n}^{t} ∈ B(t, z_{n}^{t}) satisfying

∥z_{n}^{t} − u_{n}(t)∥ ∨ ∥α_{n}^{t} − β_{n}(t)∥ ⩽ g_{r}(θ_{n}(t), t).

When t = θ_{n}(t), we simply take z_{n}^{t} = u_{n}(t) and α_{n}^{t} = β_{n}(t). Then, (z_{n}^{t})_{n} converges to x(t) and Z(t) := cl{z_{n}^{t} | n ∈ ℕ^{⋆}} is compact in Dom(∂f^{t}).

The restriction of B(t, .) to Dom(∂f^{t}) being usc, the set {B(t, z)|z ∈ Z(t)} is compact in X. Hence, Γ(t) := cl{α_{n}^{t} | n ∈ ℕ^{⋆}}, and thus cl{β_{n}(t) | n ∈ ℕ^{⋆}}, are compact. So, Y (t) := cl{y_{n}(t) | n ∈ ℕ^{⋆}} is compact in X.

Let us set F^{t}(x) = ∂f^{t}(x) ∩ Y (t) and G^{t}(x) = B(t, x) ∩ Γ(t) for any x ∈ Dom(∂f^{t}) and t ∈ [0, T]. The multimaps F^{t} and G^{t} are upper semicontinuous on Dom(∂f^{t}) with compact values in X. Let us denote by e the excess between two sets. We have:

d(−x′_{n}(t), F^{t}(x(t)) + G^{t}(x(t))) ⩽ d(y_{n}(t), F^{t}(x(t))) + d(β_{n}(t), G^{t}(x(t)))
⩽ e(F^{t}(x_{n}(t)), F^{t}(x(t))) + ∥β_{n}(t) − α_{n}^{t}∥ + d(α_{n}^{t}, G^{t}(x(t)))
⩽ e(F^{t}(x_{n}(t)), F^{t}(x(t))) + g_{r}(θ_{n}(t), t) + e(G^{t}(z_{n}^{t}), G^{t}(x(t))).

The upper-semicontinuity of F^{t} and G^{t} assures that

lim_{n→+∞} d(−x′_{n}(t), F^{t}(x(t)) + G^{t}(x(t))) = 0.

Since (x′_{n})_{n} converges to x′ a.e. on [0, T], the equality d(−x′(t), F^{t}(x(t)) + G^{t}(x(t))) = 0 holds for a.e. t ∈ [0, T] and we obtain by closedness of F^{t}(x(t)) + G^{t}(x(t)):

−x′(t) ∈ F^{t}(x(t)) + G^{t}(x(t)) for a.e. t ∈ [0, T].

Consequently, x is a local solution to x′ + ∂f^{t}(x) + B(t, x) ∋ 0 with x(0) = u_{0}. □

4.2 Two particular cases

We consider two particular cases for which we can apply Theorem 4.1. These cases contains those of f^{t} not depending on t. First,

Corollary 4.1 Let (f^{t})_{t∈[0,T]} be a family of proper convex lsc functions on X with each f^{t} of compact type. Let u_{0} ∈ dom f^{0}. Assume that f^{t} = g ∘ F^{t} where g is a proper convex lsc function on a Hilbert space Y and (F^{t})_{t∈[0,T]} is a family of differentiable maps from X to Y such that (DF^{t})_{t} is equilipschitz continuous on a neighborhood of u_{0} and such that:

1. for each r ⩾ 0, there is absolutely continuous real-valued function b_{r} on [0, T] such that:

(a) b′_{r} ∈ L^{2}(0, T),

(b) for each s, t ∈ [0, T], sup_{∥x∥⩽r} ∥F^{t}(x) − F^{s}(x)∥ ⩽ |b_{r}(t) − b_{r}(s)|,

2. the qualification condition ℝ_{+}[dom g − F^{0}(u_{0})] − DF^{0}(u_{0})X = Y holds,

3. for each r ⩾ 0, there exists a negligible subset N of [0, T] such that the mapping t ↦ F^{t}(x) admits a derivative Δ^{t}(x) on [0, T]\N for any x ∈ r𝔹_{X} and Δ^{t} is continuous on r𝔹_{X} for any t ∈ [0, T]\N,

4. the mapping (t, x) ↦ DF^{t}(x) is bounded on [0, T] × r𝔹_{X} for each r > 0 and it is continuous at t for each x.

Assume that (H) and (B) are satisfied. Then, there exists T_{0} ∈]0, T] such that u′+∂f^{t}(u)+ B(t, u) ∋ 0 has at least a strong solution u : [0, T_{0}] → X with u(0) = u_{0}.

Remark under assumption 1., the mapping t ↦ F^{t}(x) is absolutely continuous and admits a derivative at a.e. t ∈ [0, T] for each x. With the uniformly inequality 1.(b), we can hope that the almost derivability of t ↦ F^{t}(x) at t is uniform in x ∈ r𝔹_{X} thanks to the regularity of F^{t} at x. Illustrate the importance of differentiability of F^{t} by the following example : F (t, x) = h(t − x) where X = Y = ℝ and the real function h is convex, Lipschitz continuous and non differentiable on [0, T].

Proof of Corollary 4.1. Consider x : [0, T] → X an approximate solution. Let us set y_{n}(t) = −x′_{n}(t) − β_{n}(t) and y(t) = −x′(t) − β(t) for a.e. t in [0, T], z_{n}(t) = F^{t}(x_{n}(t)) and z(t) = F^{t}(x(t)) for a.e. t ∈ [0, T]. Then, z_{n} and z are absolutely continuous on [0, T], thus are derivable at a.e. t ∈ [0, T] and

z′_{n}(t) = Δ^{t}(x_{n}(t)) + DF^{t}(x_{n}(t))x′_{n}(t) , z′(t) = Δ^{t}(x(t)) + DF^{t}(x(t))x′(t).

Under the qualification condition, we have for any x ∈ X

∂f^{t}(x) = DF^{t}(x)^{⋆}∂g(F^{t}(x)).

Let us write y_{n}(t) = DF^{t}(x_{n}(t))^{⋆}w_{n}(t) and y(t) = DF^{t}(x(t))^{⋆}w(t) where w_{n}(t) ∈ ∂g(z_{n}(t)) and w(t) ∈ ∂g(z(t)) for almost all t ∈ [0, T]. Hence, g ∘ z_{n} and g ∘ z are absolutely continuous with ⟨w_{n}(t), z′_{n}(t)⟩ = (g ∘ z_{n})′(t) for almost all t ∈ [0, T]. So,

⟨y_{n}(t), x′_{n}(t)⟩ = \frac{d}{dt}(f^{t} ∘ x_{n})(t) − ⟨w_{n}(t), Δ^{t}(x_{n}(t)).

and Φ(x_{n}, y_{n}) = − ∫_{0}^{T} ⟨w_{n}(t), Δ^{t}(x_{n}(t))⟩ dt. Let r ⩾ ∥x_{n}(t)∥ ∨ ∥x(t)∥ for any n ∈ ℕ and t ∈ [0, T]. By continuity of Δ^{t} on r𝔹_{X}, (Δ^{t}(x_{n}(t)))_{n} converges to Δ^{t}(x(t)) for a.e t ∈ [0, T]. Next, for a.e. t and any x ∈ r𝔹_{X}, we have ∥Δ^{t}(x)∥ ⩽ |b′_{r}(t)|. By Lebesgue’s theorem Δ^{.}(x_{n}(.)) converges to Δ^{.}x(.)) in L^{2}(0, T, Y). Since a subsequence of (w_{n})_{n} converges weakly to w, we can apply Theorem 4.1. □

For example, if F^{t} is the affine mapping x ↦ A(t)x + b(t) where A(t) : X → Y is linear continuous and b(t) ∈ Y, the assumption of Corollary 4.1 becomes :

1. b is absolutely continuous on [0, T] and there is absolutely continuous real-valued function a on [0, T] such that:

(a) a′ ∈ L^{2}(0, T),

(b) for each s, t ∈ [0, T], ∥A(t) − A(s)∥ ⩽ |a(t) − a(s)|.

2. the qualification condition ℝ_{+} dom g − A(0)X = Y holds.

3. for each r ⩾ 0, there exists a negligible subset N of [0, T] such that A′(t) is continuous on r𝔹_{X} for any t ∈ [0, T]\N.

Second, we use the conjugate of f^{t}.

Lemma 4.1 Let (f^{t})_{t∈[0,T]} be a family of proper convex lsc functions on X satisfying (H). Assume that :

for each r ⩾ 0, there exists a negligible subset N of [0, T] such that for any t ∈ [0, T]\N, the mapping s ↦ (f^{s})^{⋆}(y) admits a derivative γ̇(t, y) at t for any y ∈ Dom ∂(f^{t})^{⋆}.

Let x : [0, T] → X be an absolutely continuous function and y : [0, T] → Y be such that y(t) ∈ ∂f^{t}(x(t)) for a.e. t ∈ [0, T]. For almost all t ∈ [0, T], we have

γ̇(t, y(t)) + \frac{d}{dt}f^{t}(x(t)) = ⟨y(t), x′(t)⟩.  (8)

Proof. Let s and t be in [0, T]\N where N is a suitable negligible subset of [0, T]. We have :

(f^{s})^{⋆}(y(s)) − (f^{t})^{⋆}(y(s)) ⩽ (f^{s})^{⋆}(y(s)) − (f^{t})^{⋆}(y(t)) − ⟨y(s) − y(t), x(t)⟩

since x(t) ∈ ∂(f^{t})^{⋆}(y(t)). From f^{t}(x(t)) + (f^{t})^{⋆}(y(t)) = ⟨y(t), x(t)⟩, we deduce

(f^{s})^{⋆}(y(s)) − (f^{t})^{⋆}(y(s)) ⩽ f^{t}(x(t)) − f^{s}(x(s)) + ⟨y(s), x(s) − x(t)⟩.

In the same way, for almost every t, s in [0, T] we have

(f^{s})^{⋆}(y(s)) − (f^{t})^{⋆}(y(s)) ⩽ f^{t}(x(t)) − f^{s}(x(s)) + ⟨y(s), x(s) − x(t)⟩.

Changing the role of s and t, we also have:

(f^{t})^{⋆}(y(t)) − (f^{s})^{⋆}(y(t)) ⩽ f^{s}(x(s)) − f^{t}(x(t)) + ⟨y(t), x(t) − x(s)⟩
⩽ (f^{t})^{⋆}(y(s)) − (f^{s})^{⋆}(y(s)) + ⟨y(t) − y(s), x(t) − x(s)⟩.

The function t ↦ f^{t}(x(t)) being absolutely continuous on [0, T], see [11, Chapter 1], we obtain (8). □

The existence of γ̇ implies some regularity on the domain of (f^{t})^{⋆}. For example, consider (f^{t})^{⋆}(y) = h(t − y) where X = Y = ℝ and the real function h is convex, Lipschitz continuous and non differentiable on [0, T]. Then, we can not apply above lemma. The domain of (f^{t})^{⋆} changes with t. But, we can apply Corollary 4.1 since f^{t}(x) = tx+h^{⋆}(−x). However, we have the absolutely continuity of s ↦ (f^{s})^{⋆}(y) in the following sense :

Proposition 4.1 Let t ∈ [0, T], y ∈ Y, η > 0 and r > 0 such that if |t − s| ⩽ η, the set ∂(f^{s})^{⋆}(y)∩r𝔹_{X} is nonempty. Then, s ↦ (f^{s})^{⋆}(y) is absolutely continuous on ]t−η, t+η[.

Proof. 1) Lemma 3.1 with β = ρ_{o} assures that (f^{t})^{⋆}(y) ⩾ ⟨y, z_{t}⟩ − f^{t}(z_{t}) ⩾ −∥y∥β − β for any t ∈ [0, T] and y ∈ X. For y ∈ ∂f^{t}(x), it follows

−α(∥x∥ + 1) ⩽ f^{t}(x) = ⟨y, x⟩ − (f^{t})^{⋆}(y) ⩽ ∥y∥[ ∥x∥ + β ] + β.

So, there is a nonnegative constant β such that (f^{t})^{⋆}(y) ⩾ −β(∥y∥ + 1) for all y ∈ X and t ∈ [0, T]. Furthermore, for each r > 0, there is a nonnegative constant c such that |f^{t}(x)| ⩽ c(∥y∥ + 1) for all x ∈ r𝔹_{X}, t ∈ [0, T] and y ∈ ∂f^{t}(x).

2) Let t be fixed in [0, T] and y ∈ ∂f^{t}(x). Let r ⩾ ∥x∥ and s ∈ [t, T]. Under the assumption (H_{0}), there exists x_{s} ∈ dom f^{s} satisfying

\{ \begin{matrix}∥x − x_{s}∥ ⩽ |h_{r}(t) − h_{r}(s)|(1 + |f^{t}(x)|^{1/2}) \\ f^{s}(x_{s}) ⩽ f^{t}(x) + |k_{r}(t) − k_{r}(s)|(1 + |f^{t}(x)|). \end{matrix}

By definition of conjugate of a convex function, it follows

(f^{t})^{⋆}(y) − (f^{s})^{⋆}(y) ⩽ ⟨y, x − x_{s}⟩ + f^{s}(x_{s}) − f^{t}(x)
⩽ ∥y∥|h_{r}(t) − h_{r}(s)|(1 + |f^{t}(x)|^{1/2}) + |k_{r}(t) − k_{r}(s)|(1 + |f^{t}(x)|).

We conclude thanks to 1):

(f^{t})^{⋆}(y) − (f^{s})^{⋆}(y) ⩽ ∥y∥|h_{r}(t) − h_{r}(s)|(1 + |f^{t}(x)|^{1/2}) + |k_{r}(t) − k_{r}(s)|(1 + |f^{t}(x)|)
⩽ ∥y∥|h_{r}(t) − h_{r}(s)|(1 + \sqrt{c} + \sqrt{c∥y∥}) + |k_{r}(t) − k_{r}(s)|(1 + c + c∥y∥)

3) Let y ∈ Y, r > 0 and s, t ∈ [0, T]. If the intersections of ∂(f^{t})^{⋆}(y) and ∂(f^{s})^{⋆}(y) with r𝔹_{X} are non empty, let x_{s} ∈ ∂(f^{s})^{⋆}(y) and x_{t} ∈ ∂(f^{t})^{⋆}(y) with r ⩾ ∥x_{s}∥ ∨ ∥x_{t}∥. Step 2) implies

|(f^{s})^{⋆}(y) − (f^{t})^{⋆}(y)| ⩽

∥y∥|h_{r}(t) − h_{r}(s)|(1 + |f^{s}(x_{s})|^{1/2} ∨ |f^{t}(x_{t})|^{1/2}) + |k_{r}(t) − k_{r}(s)|(1 + |f^{s}(x_{s})| ∨ |f^{t}(x_{t})|).

By 1) we conlude:

|(f^{s})^{⋆}(y)−(f^{t})^{⋆}(y)| ⩽ ∥y∥|h_{r}(t)−h_{r}(s)|(1+c^{1/2} +(c∥y∥)^{1/2})+|k_{r}(t)−k_{r}(s)|(1+c+c∥y∥).

□

Corollary 4.2 Let (f^{t})_{t∈[0,T]} be a family of proper convex lsc functions on X with each f^{t} of compact type. Assume that (H) and (B) are satisfied and that:

1. for each r ⩾ 0, there exists a negligible subset N of [0, T] such that for any t in [0, T]\N, the mapping s ↦ (f^{s})^{⋆}(y) admits a derivative γ̇(t, y) at t for any y ∈ Dom ∂(f^{t})^{⋆}.

2. for any (y_{n})_{n} which converges weakly to y in L^{2}(0, T; X) with y_{n}(t) ∈ ∂f^{t}(x_{n}(t)) where (x_{n})_{n} converges uniformly, there exists n_{k} → +∞ such that

lim inf_{k→+∞} ∫_{0}^{T} γ̇(t, y_{n_{k}}(t)) dt ⩾ ∫_{0}^{T} γ̇(t, y(t)) dt.

Then, for each u_{0} ∈ dom f^{0}, there exists T_{0} ∈]0, T] such that u′ + ∂f^{t}(u) + B(t, u) ∋ 0 has at least a strong solution u : [0, T_{0}] → X with u(0) = u_{0}.

The assumption 2. is true when γ̇(t, .) is lsc and convex on Dom ∂(f^{t})^{⋆}.

Proof. Lemma 4.1 implies Φ(x, y) = ∫_{0}^{T} γ̇(t, y(t)) dt. By assumption 2., we can apply

Theorem 4.1. □

5 Examples of families (f^{t})_{t}

5.1 Rafle

See Castaing, Valadier and Moreau [4, 17, 13]. Let (C(t))_{t∈[0,T]} a family of nonempty closed convex subsets of X whose intersection with bounded closed sets is compact. Consider the indicator function f^{t} = δ_{C(t)} of C(t). Assume that for each r ⩾ 0, there is an absolutely continuous real-valued function a_{r} on [0, T] such that:

(i) a′_{r} ∈ L^{2}(0, T);
(ii) for each s, t in [0, T], we have e(C(s) ∩ r𝔹_{X}, C(t)) ⩽ |a_{r}(s) − a_{r}(t)|.

Under the assumption (B), we can apply Theorem 4.1: Assume that for any sequence (x_{n})_{n} in H^{1}(0, T; X) which converges uniformly to the absolutely continuous function x with the weak convergence of (x′_{n})_{n} to x′ in L^{2}, and for any (y_{n})_{n} which converges weakly to y in L^{2} with y_{n}(t) ∈ N_{C(t)}(x_{n}(t)) for almost all t, there exists n_{k} → +∞ such that

lim inf_{k→+∞} ∫_{0}^{T} ⟨x′_{n_{k}}(t), y_{n_{k}}(t)⟩ dt ⩾ ∫_{0}^{T} ⟨x′(t), y(t)⟩ dt.

Then, for each u_{0} ∈ C(0), there exists T_{0} ∈]0, T] such that u′ + N_{C(t)}(u) + B(t, u) ∋ 0 has at least a strong solution u : [0, T_{0}] → X with u(0) = u_{0}.

Corollary 4.1 becomes:

Corollary 5.1 Let u_{0} ∈ C(0) and (F^{t})_{t∈[0,T]} be a family of differentiable maps from X to an other Hilbert space Y such that (DF^{t})_{t} is equilipschitz continuous on a neighborhood of u_{0}. Assume that C(t) = (F^{t})^{−1}(C), C being a nonempty closed convex set. Under the assumptions (B) and:

1. for each r ⩾ 0, there is absolutely continuous real-valued function b_{r} on [0, T] such that:

(a) b′_{r} ∈ L^{2}(0, T),

(b) for each s, t ∈ [0, T], sup_{∥x∥⩽r} ∥F^{t}(x) − F^{s}(x)∥ ⩽ |b_{r}(t) − b_{r}(s)|,

2. the qualification condition ℝ_{+}[C − F^{0}(u_{0})] − DF^{0}(u_{0})X = Y holds,

3. for each r ⩾ 0, there exists a negligible subset N of [0, T] such that the mapping t ↦ F^{t}(x) admits a derivative Δ^{t}(x) on [0, T]\N for any x ∈ r𝔹_{X} and Δ^{t} is continuous on r𝔹_{X} for any t ∈ [0, T]\N,

4. the mapping (t, x) ↦ DF^{t}(x) is bounded on [0, T] × r𝔹_{X} for each r > 0 and it is continuous at t for each x,

there exists T_{0} ∈]0, T] such that u′ + N_{C(t)}(u) + B(t, u) ∋ 0 has at least a strong solution u : [0, T_{0}] → X with u(0) = u_{0}.

On the other hand, (f^{t})^{⋆} is the support function of C(t), denoted by σ_{C(t)}. Moreau have proved in [13] that when C is absolutely continuous, the map t ↦ σ_{C(t)}(y) is absolutely continuous on [0, T] for any y ∈ D, where D is the domain of σ_{C(t)} which is not dependent of t. Corollary 4.2 becomes:

Corollary 5.2 Assume that (B) is satisfied and that:

1. for each r ⩾ 0, there exists a negligible subset N of [0, T] such that for any t in [0, T]\N, the mapping s ↦ σ_{C(s)}(y) admits a derivative γ̇(t, y) at t for any y ∈ Dom ∂σ_{C(t)}.

2. for any (y_{n})_{n} which converges weakly to y in L^{2}(0, T; X) with y_{n}(t) ∈ N_{C(t)}(x_{n}(t)) where (x_{n})_{n} converges uniformly, there exists n_{k} → +∞ such that

lim inf_{k→+∞} ∫_{0}^{T} γ̇(t, y_{n_{k}}(t)) dt ⩾ ∫_{0}^{T} γ̇(t, y(t)) dt.

Then, for each u_{0} ∈ dom f^{0}, there exists T_{0} ∈]0, T] such that u′ + N_{C(t)}(u) + B(t, u) ∋ 0 has at least a strong solution u : [0, T_{0}] → X with u(0) = u_{0}.

By example, consider the affine map F^{t}(x) = a(t)x + b(t) where a(t) ∈ ℝ_{+}^{⋆} is derivable nonincreasing at t ∈ [0, T] and b(t) ∈ Y is absolutely continuous at t ∈ [0, T]. We can apply both corollary 4.1 and 4.2 since

γ̇(t, y) = \frac{−1}{a(t)^{2}} [a(t)⟨y, b′(t)⟩ + a′(t)(σ_{C}(y) − ⟨y, b(t)⟩)]

for a.e. t ∈ [0, T] and any y ∈ Y. So, γ̇(t, .) is convex l.s.c. on X and

lim_{n→+∞} ∫_{0}^{T} γ̇(t, y_{n}(t)) dt = ∫_{0}^{T} γ̇(t, y(t)) dt.

5.2 Viscosity

Let f : X → ℝ ∪ {+∞} be a convex lsc proper function of compact type. Consider

f^{t}(x) = f(x) + \frac{ε(t)}{2}∥x∥^{2} where ε is an absolutely continuous real-valued function on [0, T] with nonnegative values and ε′ ∈ L^{1}(0, T). We can write f^{t} = g ∘ F^{t} with g(x, r) = f(x) + r for any (x, r) ∈ X × ℝ and F^{t}(x) = (x, \frac{ε(t)}{2}∥x∥^{2}) for any x ∈ X. By absolutely continuity of ε, Δ^{t} exists and is continuous on X for a.e. t ∈ [0, T] and

Δ^{t}(x) = (0, \frac{ε′(t)}{2}∥x∥^{2}).

Furthermore,

DF^{t}(x)y = (y, ε(t)⟨x, y⟩)

and DF^{t} satifies assumption 5. We can apply Corollary 4.1.

On the other hand, (f^{t})^{⋆} = (f^{⋆})_{ε(t)} is a C^{1}-function on X and, for any y ∈ X, the map t ↦ (f^{t})^{⋆}(y) is absolutely continuous on [0, T] with for a.e. t ∈ [0, T]

γ̇(t, y) = −\frac{ε′(t)}{2}∥D(f^{t})^{⋆}(y)∥^{2}.

By definition of y_{n}, x_{n}(t) = D(f^{t})^{⋆}(y_{n}(t)) holds for a.e. t ∈ [0, T] and any n ∈ ℕ, hence

γ̇(t, y_{n}(t)) = −\frac{ε′(t)}{2}∥x_{n}(t)∥^{2}.

In the same way,

γ̇(t, y(t)) = −\frac{ε′(t)}{2}∥x(t)∥^{2}.

By uniform convergence of (x_{n})_{n} to x on [0, T] it follows

lim_{n→+∞} ∫_{0}^{T} γ̇(t, y_{n}(t)) dt = ∫_{0}^{T} γ̇(t, y(t)) dt.

We can apply Corollary 4.2.

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(Received January 22, 2004)