Samir Adly
email: samir.adly@unilim.fr

Tahar Haddad

Ba Khiet Le

B Samir

State-Dependent Implicit Sweeping Process in the Framework of Quasistatic Evolution Quasi-Variational Inequalities

Keywords: Moreau's sweeping process, Evolution variational inequalities, Unilateral constraints, Quasistatic frictional contact problems Mathematics Subject Classification 49J40, 47J20, 47J22, 58E35, 74M15, 74M10, 74G25

This paper deals with the existence and uniqueness of solutions for a class of statedependent sweeping processes with constrained velocity in Hilbert spaces without using any compactness assumption, which is known to be an open problem. To overcome the difficulty, we introduce a new notion called hypomonotonicity-like of the normal cone to the moving set, which is satisfied by many important cases. Combining this latter notion with an adapted Moreau's catching-up algorithm and a Cauchy technique, we obtain the strong convergence of approximate solutions to the unique solution, which is a fundamental property. Using standard tools from convex analysis, we show the equivalence between this implicit state-dependent sweeping processes and quasistatic evolution quasi-variational inequalities. As an application, we study the state-dependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics.

Introduction

In the seventies, J. J. Moreau introduced and thoroughly studied the so-called sweeping processes [START_REF] Moreau | Sur l'evolution d'un système élastoplastique[END_REF][START_REF] Moreau | Rafle par un convexe variable I, Sém[END_REF][START_REF] Moreau | Rafle par un convexe variable II, Sém[END_REF], which have a particular form of differential inclusions [START_REF] Adly | A Variational Approach to Nonsmooth dynamics[END_REF] involving the outward normal cone to a moving closed and convex set in a Hilbert space. The original motivation is to model quasistatic evolution in elastoplasticity, friction dynamics, granular material, contact dynamics. However, many applications of the sweeping processes can be also found nowadays in nonsmooth mechanics, convex optimization, modeling of crowd motion, mathematical economics, dynamic networks, switched electrical circuits, etc (see, e.g., [START_REF] Addi | A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems[END_REF][START_REF] Adly | A Variational Approach to Nonsmooth dynamics[END_REF][START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF][START_REF] Goeleven | Complementarity and Variational Inequalities in Electronics[END_REF][START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF] and the references therein). Existence and uniqueness of solutions of such systems and their classical variants (subjected to perturbation forces, state-dependent, second order sweeping processes, etc) have been studied fruitfully in the literature (see, e.g., [START_REF] Moreau | Rafle par un convexe variable I, Sém[END_REF][START_REF] Moreau | Rafle par un convexe variable II, Sém[END_REF][START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF][START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF][START_REF] Adly | Unbounded second-order state-dependent Moreau's sweeping processes in Hilbert spaces[END_REF][START_REF] Brogliato | Nonsmooth Mechanics. Models, Dynamics and Control. Communications and Control Engineering Series[END_REF][START_REF] Bounkhel | State dependent sweeping process in p-uniformly smooth and q-uniformly convex banach spaces[END_REF]).

Recently in [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF], the authors proposed a new variant with velocity in the constraint [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF] and provided various applications in electrical circuits. Under the coercivity of the bounded symmetric linear monotone involved operators and the Lipschitz continuity of the moving set (assumed to be bounded) with respect to the Hausdorff distance, the authors showed in [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF] an existence and uniqueness of solution of problem [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF]. Later the case of unbounded moving set is relaxed in [START_REF] Adly | On semicoercive sweeping process with velocity constraint[END_REF], by assuming only the semicoercivity of the involved linear operators. Continuing the idea, in [START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF] the authors showed the existence and uniqueness of solutions for a new variant [START_REF] Goeleven | Complementarity and Variational Inequalities in Electronics[END_REF], by taking into account a linear combination of the velocity and state in Hilbert spaces. As an application, the quasistatic antiplane frictional contact problem for linear elastic materials with short memory was formulated as an implicit sweeping process with velocity constraint. This work creates a bridge between the different mathematical abstract results in the literature and their applications in nonsmooth mechanics, which was the initial motivations of Moreau [START_REF] Moreau | Sur l'evolution d'un système élastoplastique[END_REF].

In this paper, we are interested in a generalization of the variant studied in [START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF]tothe case of a state-dependent moving set [START_REF] Bounkhel | State dependent sweeping process in p-uniformly smooth and q-uniformly convex banach spaces[END_REF] without using any compactness assumption, which is known to be an open problem. The additional initial viability condition [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF]is no longer required here, which allows nonzero initial velocity. The set of constraints is supposed to move in an absolutely continuous way with respect to time and Lipschitz continuous with respect to the state. The normal cone to the moving set is assumed to be hypomonotone-like (see Assumption 2 for the definition). Let us mention that the class of set-valued maps satisfying the latter property is very large for applications (see Lemmas 3.1, 3.2 for more details).

To conclude this introduction, we mention that the present work can be considered as an improvement and an extension of the research initiated by the authors in [START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF]. In fact, the technique utilized in the current paper differs from the one used in [START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF]. The proof here is based on an implicit time discretization combined with the construction of a Cauchy sequence of functions via linear interpolation that converges strongly to a unique solution of [START_REF] Bounkhel | State dependent sweeping process in p-uniformly smooth and q-uniformly convex banach spaces[END_REF]. Clearly, the uniqueness of a solution as well as the strong convergence of approximate solutions to the unique solution, obtained in this paper, are important properties, compared to the weak convergence of a subsequence obtained in [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF][START_REF] Adly | On semicoercive sweeping process with velocity constraint[END_REF][START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF]. Note that the compatibility condition [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF] is no longer necessary here, which is in concordance with the result obtained in [START_REF] Sofonea | Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems[END_REF] for the quasistatic antiplane frictional contact problem (see Theorem 10.3 page 193 and Remark 4.6 in [START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF]).

The paper is organized as follows. We recall in Sect. 2 some basic notation, definitions and results which are used throughout the paper. In Sect. 3, the well-posedness of ( 11) is thoroughly studied by using an implicit time discretization. An application to the quasistatic frictional contact problem is given in Sect. 4. Finally, Sect. 5 closes the paper with some concluding remarks.

Notation, Preliminaries and General Framework

We begin with some notation used in the paper. Let H be a Hilbert space. Denote by

• the norm in H ,by •, • the scalar product and by B the closed unit ball in H .Let be given a closed, convex set C ⊂ H . The distance from a point s to C is defined by

d(s, C) := inf x∈C s -x . The Hausdorff distance between two closed, convex sets C 1 , C 2 ⊂ H is defined by d H (C 1 , C 2 ) := max{ sup x 1 ∈C 1 d(x 1 , C 2 ), sup x 2 ∈C 2 d(x 2 , C 1 )}.
It is easy to see that

d H (C 1 , C 2 ) = ε ⇒ C 1 ⊂ C 2 + εB and C 2 ⊂ C 1 + εB.
The normal cone of a closed and convex set S is defined as follows

N S (x) := {x * ∈ H : x * , y -x ≤0, ∀y ∈ S}. It is not difficult to see that x * ∈ N S (x) ⇔ σ(S, x * ) = x * , x and x ∈ S, ( 1
)
where σ is the support function, defined by

σ(S, x * ) := sup ξ ∈S x * ,ξ , x * ∈ H .
Given a continuous function v : R → R, the modulus of continuity of v is the function ω :[0, +∞] → [0, +∞], defined by

ω(h) := sup |x-y|≤h |v(x) -v(y)|, h > 0.
In the following, we summarize some known definitions and results concerning maximally monotone operators. The domain and graph of a set-valued operator T : H ⇒ H are defined, respectively, by

D(T ) := {x ∈ H : T (x) =∅}, gph(T ) := {(x, y) : x ∈ H , y ∈ T (x)}.
The inverse of T is defined by

T -1 (x * ) := {x ∈ H : x * ∈ T (x)}, x * ∈ H .
The operator T is said to be monotone, if for all x, y ∈ H , x * ∈ T (x), y * ∈ T (y),we have

x *y * , xy ≥0.

In addition, if there is no monotone mapping T ′ such that gph(T ) is contained strictly in gph(T ′ ), then T is called maximally monotone.

The operator T is said to be coercive, if for all

(x n , y n ) ∈ gph(T ) such that lim n→+ ∞ x n =+∞, we have lim n→+ ∞ x n -x 0 , y n x n =+∞,
for some x 0 ∈ H . It is known that coercive maximally monotone operators in reflexive Banach spaces are surjective (see, e.g, [START_REF] Barbu | Nonlinear Differential Equations of Monotone Types in Banach Spaces[END_REF]). We close this section with some versions of discrete and continuous Gronwall's inequalities [START_REF] Showalter | Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations[END_REF].

Then, for all n, we have

u n ≤ α exp n-1 k=0 β k . Lemma 2.2 Let T > 0 be given and a(•), b(•) ∈ L 1 ([0, T ]; R) with b(t) ≥ 0 for almost all t ∈[0, T ]. Let the absolutely continuous function w :[0, T ]→R + satisfy (1 -α)w ′ (t) ≤ a(t)w(t) + b(t)w α (t), a.e. t ∈[0, T ], (3)
where 0 ≤ α<1. Then for all t ∈[0, T ], one has

w 1-α (t) ≤ w 1-α (0)exp t 0 a(τ )dτ + t 0 exp t s a(τ )dτ b(s)ds. ( 4
)
The mathematical formulation of the sweeping process introduced by J.J. Moreau is given by

ẋ(t) ∈-N C(t) (x(t)) a.e. t ∈[0, T ], x(0) = x 0 ∈ C(0), (5)
where N C(t) (•) denotes the normal cone operator associated to the closed and convex set C(t) in a Hilbert space H . In [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF], the authors studied the following new variant

A 1 u(t) + A 0 u(t) -f (t) ∈-N C(t) ( u(t)) a.e. t ∈[0, T ], u(0) = u 0 , (6)
where C(•) is a bounded moving set, A 1 , A 0 : H → H are two bounded, symmetric, linear and monotone operators and f :[0, T ]→H is a continuous mapping.

In [START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF], the authors proposed the following model

u(t) ∈-N C(t) (A u(t) + Bu(t)) a.e. t ∈[0, T ], u(0) = u 0 , (7)
with additional initial condition

Bu 0 ∈ C(0) (8)
where A, B : H → H are two linear, bounded, symmetric and monotone operators and A is coercive, i.e., there exists some β>0 such that Ax, x ≥β x 2 , ∀x ∈ H .

The moving set t → C(t) varies in an absolutely continuous way, i.e., there exists some absolutely continuous function v :[0, T ]→R such that

d H (C(s), C(t)) ≤|v(s) -v(t)| for all s, t ∈[0, T ], (10)
where d H denotes the Hausdorff distance.

In this paper, we are interested in a generalization of the variant studied in [START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF]tothe case of a state-dependent moving set without requiring the additional initial viability condition, which allows nonzero initial velocity. More precisely, let A, B : H → H be two linear, bounded and symmetric operators with coercive A and C :[0, T ]×H ⇒ H be a set-valued mapping with nonempty, closed and convex values. Our aim is to study the well-posedness of the following sweeping process

u(t) ∈-N C(t,u(t)) (A u(t) + Bu(t)) a.e. t ∈[0, T ], u(0) = u 0 , (11)
without requiring the additional initial viability condition Bu 0 ∈ C(0, u 0 ), which allows nonzero initial velocity.

Main Result

In this section, the well-posedness (in the sense of existence and uniqueness of solutions) of the state-dependent sweeping processes in Hilbert spaces is studied. Note that the generalization for reflexive Banach spaces can be done in a similar way. First, let us make the following assumptions.

Assumption 1

The set-valued mapping C :[ 0, T ]×H ⇒ H has nonempty, closed and convex values and there exists an absolutely continuous function v :[0, T ]→R and L ≥ 0 such that for all s, t ∈[0, T ] and u,w

∈ H d H C(s, u), C(t,w) ≤|v(s) -v(t)|+L u -w . ( 12
)
Assumption 2 The normal cone of C is hypomonotone-like, in the sense that for given R > 0, there exist L ≥ 0 and an absolutely continuous function ṽ :[0, T ]→R such that if

a i ∈ N C(t i ,u i ) (b i ) for a i ∈ H , u i , b i ∈ RB, t i ∈[0, T ], i = 1, 2 then a 1 -a 2 , b 1 -b 2 ≥-L a 1 -a 2 u 1 -u 2 -( a 1 + a 2 )|ṽ(t 1 ) -ṽ(t 2 )|. (13)

Assumption 3 Let A, B : H → H be symmetric, linear and bounded operators satisfying

Ax, x ≥β x 2 and Bx, x ≥0forsomeβ>0.

Remark 3.1 If C(t, u)

≡ C is a constant, closed and convex set, then the normal cone of C is monotone and obviously is hypomonotone-like. Now we provide some important non-constant cases to show that the class of the moving set C satisfying Assumptions 1 and 2 is large for applications.

Lemma 3.1 Let C(t, u) = f (t, u) + C 1 (t) for all t ∈[ 0, T ], x ∈ H ,
v 1 ,v 2 :[0, T ]→R and L ≥ 0 such that d H C 1 (t), C 1 (s) ≤|v 1 (t) -v 1 (s)| and f (t, u) -f (s,w) ≤|v 2 (t) -v 2 (s)|+L u -w , (14)
for all t, s ∈[0, T ] and u,w ∈ H . Then Assumptions 1 and 2 are satisfied.

Proof It is easy to see that C has nonempty, closed and convex values and

d H C(s, u), C(t,w) ≤|v(s) -v(t)|+L u -w , for all s, t ∈[0, T ], u,w ∈ H , where v(t) := t 0 (|v 1 (s)|+|v 2 (s)|)ds.
To check the hypomonotonicity-like of the normal cone of C, let us note that

a i ∈ N C(t i ,u i ) (b i ) = N C 1 (t i )+ f (t i ,u i ) (b i ) = N C 1 (t i ) (b i -f (t i , u i )), i = 1, 2. (15) Hence a 1 , z -b 1 + f (t 1 , u 1 ) ≤0, for all z ∈ C 1 (t 1 ). (16)
In addition, one has b

2 ∈ C 1 (t 2 )+ f (t 2 , u 2 ) ⊂ C 1 (t 1 )+|v 1 (t 1 )-v 1 (t 2 )|B+ f (t 2 , u 2 ).
Combining with the last inequality, we obtain

a 1 , b 2 -b 1 + f (t 1 , u 1 ) -f (t 2 , u 2 ) ≤|v 1 (t 1 ) -v 1 (t 2 )| a 1 . (17)
Similarly, one has

a 2 , b 1 -b 2 + f (t 2 , u 2 ) -f (t 1 , u 1 ) ≤|v 1 (t 1 ) -v 1 (t 2 )| a 2 . (18)
From ( 17) and ( 18), we deduce that

= a 1 -a 2 , b 1 -b 2 ≥ a 1 -a 2 , f (t 1 , u 1 ) -f (t 2 , u 2 ) -(|v 1 (t 1 ) -v 1 (t 2 )|)( a 1 + a 2 ) ≥-L a 1 -a 2 u 1 -u 2 -(|v 1 (t 1 ) -v 1 (t 2 )|+(|v 2 (t 1 ) -v 2 (t 2 )|)( a 1 + a 2 ) ≥-L a 1 -a 2 u 1 -u 2 -(|v(t 1 ) -v(t 2 )|)( a 1 + a 2 ),
which completes the proof of Lemma 3.1.

⊓ ⊔ Lemma 3.2 Let C(t, u) = f 1 (t, u) + f 2 (t, u)B, where f 1 :[ 0, T ]×H → H , f 2 : [0, T ]×H →[α, +∞), (t, u) → f i (t, u
) is supposed to be v i -absolutely continuous in t, L i -Lipschitz continuous in u for some L i ≥ 0,i = 1, 2 and for some α>0. Then Assumptions 1 and 2 are satisfied.

Proof Clearly C has nonempty, closed and convex values and

d H C(s, u), C(t,w) ≤|v(s) -v(t)|+L u -w , for all s, t ∈[0, T ], u,w ∈ H , where v(t) := t 0 (|v 1 (s)|+|v 2 (s)|)ds and L = L 1 + L 2 .
It remains to check the hypomonotonicity-like property. Let be given R > 0 and

a i ∈ N f 1 (t i ,u i )+ f 2 (t i ,u i )B (b i ) = N f 2 (t i ,u i )B (b i -f 1 (t i , u i )), b i , u i ∈ RB, i = 1, 2.
Hence

a 1 ∈ N f 2 (t 1 ,u 1 )B (b 1 -f 1 (t 1 , u 1 )), (19)
a 2 ∈ N f 2 (t 2 ,u 2 )B (b 2 -f 1 (t 2 , u 2 )) = N f 2 (t 1 ,u 1 )B (b 2 -f 1 (t 2 , u 2 )) f 2 (t 1 , u 1 ) f 2 (t 2 , u 2 ) , (20)
where the second part follows from the fact that f 2 (t 1 , u 1 )>0, f 2 (t 2 , u 2 )>0 and the definition of the normal cone. From ( 19) and ( 20), we obtain

a 1 -a 2 , b 1 -f 1 (t 1 , u 1 ) - (b 2 -f 1 (t 2 , u 2 )) f 2 (t 1 , u 1 ) f 2 (t 2 , u 2 ) ≥0,
which implies that

a 1 -a 2 , b 1 -f 1 (t 1 , u 1 ) ≥ a 1 -a 2 , (b 2 -f 1 (t 2 , u 2 )) f 2 (t 1 , u 1 ) f 2 (t 2 , u 2 ) .
Hence

Φ = a 1 -a 2 , b 1 -f 1 (t 1 , u 1 ) -b 2 + f 1 (t 2 , u 2 ) ≥ a 1 -a 2 , (b 2 -f 1 (t 2 , u 2 )) f 2 (t 1 , u 1 ) f 2 (t 2 , u 2 ) -a 1 -a 2 , (b 2 -f 1 (t 2 , u 2 )) f 2 (t 2 , u 2 ) f 2 (t 2 , u 2 ) = a 1 -a 2 , (b 2 -f 1 (t 2 , u 2 ))( f 2 (t 1 , u 1 ) -f 2 (t 2 , u 2 )) f 2 (t 2 , u 2 ) ≥-k a 1 -a 2 f 2 (t 1 , u 1 ) -f 2 (t 2 , u 2 )
where k :=

R+ sup t∈[0,T ],u∈RB f 1 (t, u) α < +∞. Thus a 1 -a 2 , b 1 -b 2 ≥-a 1 -a 2 (k f 2 (t 1 , u 1 ) -f 2 (t 2 , u 2 ) + f 1 (t 1 , u 1 ) -f 1 (t 2 , u 2 ) ) ≥-L a 1 -a 2 u 1 -u 2 -( a 1 + a 2 )|ṽ(t 1 ) -ṽ(t 2 )|,
where L := L 1 + kL 2 and ṽ(t) :

= t 0 (|v 1 (s)|+k|v 2 (s)|)ds. ⊓ ⊔
The following lemma is a consequence of a general chain rule concerning the composition of a convex function with a linear operator (see, e.g., [START_REF] Clarke | Functional Analysis. Calculus of Variations and Optimal Control[END_REF]Theorem 4.13]). For this particular case, we can provide a simple and direct proof.

Lemma 3.3 Let Assumption 3 hold. Then for a given closed and convex set D ⊂ H, one has AN

D (Ax) = N A -1 (D) (x), ∀x ∈ A -1 (D). (21)
Proof For a given x ∈ A -1 (D), we have

ζ ∈ AN D (Ax) ⇔∃ζ ′ ∈ N D (Ax) s.t.ζ = Aζ ′ ⇔ ζ = Aζ ′ and ζ ′ , y -Ax ≤0, ∀y ∈ D ⇔ ζ = Aζ ′ and ζ ′ , Ay ′ -Ax ≤0, ∀y ′ ∈ A -1 (D) ⇔ ζ, y ′ -x ≤0, ∀y ′ ∈ A -1 (D) ⇔ ζ ∈ N A -1 (D) (x),
and the proof is thereby completed.

⊓ ⊔

The next result extends [16, Proposition 1] toward the infinite dimensional setting of this paper. Let us note that the surjectivity of the operator N D + A plays an important role in the well-posedness of the algorithm proposed in Theorem 3.1.

Lemma 3.4

Let Assumption 3 hold. Then for a given closed and convex set D ⊂ H, the set-valued mapping x → N D (x) + Ax is surjective. In addition, the mapping (N D + A) -1 is single-valued and 1/β-Lipschitz continuous, with β is the coercivity constant of A.

Proof Note that the set-valued mapping N D + A is maximally monotone and coercive. Hence, it is surjective (see, e.g, [START_REF] Barbu | Nonlinear Differential Equations of Monotone Types in Banach Spaces[END_REF]). Let

x i ∈ (N D + A) -1 (y i ), i = 1, 2. Then, one has y i ∈ N D (x i ) + Ax i and thus y 2 -y 1 , x 2 -x 1 ≥ A(x 1 -x 2 ), x 1 -x 2 ) ≥ β x 1 -x 2 2 ,
which implies that

x 1 -x 2 ≤ 1 β y 1 -y 2 .
Therefore, (N D + A) -1 is single-valued and 1/β-Lipschitz continuous. ⊓ ⊔ Lemma 3.5 Assume Assumption 1. Then, there exists a constant c > 0 which depends only on the initial data such that for all t ∈[0, T ], u ∈ H , we can find some y ∈ C(t, u) satisfying y ≤c + L uu 0 .

Proof Fix some y 0 ∈ C(0, u 0 ). From Assumption 1, one has

y 0 ∈ C(0, u 0 ) ⊂ C(t, u) + (|v(t) -v(0)|+L u -u 0 )B.
Hence, we can choose some y ∈ C(t, u) such that Proof Let be given some positive integer n ≥ 1, define h n := T /n and t n i := ih for 0 ≤ i ≤ n -1. We approximate [START_REF] Bounkhel | State dependent sweeping process in p-uniformly smooth and q-uniformly convex banach spaces[END_REF] by the following adapted Moreau's catching-up algorithm:

y -y 0 ≤|v(t) -v(0)|+L u -u 0 .
-Set u n 0 = u 0 . -F o r0≤ i ≤ n -1, we want to find v n i+1 and u n i+1 such that v n i+1 ∈-N C(t n i+1 ,u n i ) Av n i+1 + Bu n i , u n i+1 = u n i + h n v n i+1 . (22)
The first line of ( 22) can be rewritten as follows

v n i+1 ∈-N C(t n i+1 ,u n i )-Bu n i Av n i+1 , (23)
or equivalently by using Lemma 3.3,

Av n i+1 ∈-AN D (Av n i+1 ) =-N A -1 (D) v n i+1 , (24)
where

D = C(t n i+1 , u n i ) -Bu n i . Thanks to Lemma 3.4, one can compute uniquely v n i+1 by v n i+1 = (N A -1 (D) + A) -1 (0), (25)
and the algorithm is well defined.

(a) Boundedness of the iterates From ( 22) and the coerciveness of A, one has, for all v ∈ C(t n i+1 , u n i ), that

β v n i+1 2 ≤ Av n i+1 ,v n i+1 ≤ Av n i+1 + Bu n i -v + v -Bu n i ,v n i+1 ≤ v -Bu n i ,v n i+1 = v -Bu n i v n i+1 which implies that v n i+1 ≤ 1 β v -Bu n i = 1 β v -Bu 0 -h n i k=1 Bv n k . (26)
Using Lemma 3.5, we can always choose v ∈ C(t n i+1 , u n i ) such that v ≤c + L u n iu 0 ≤c + h n L i k=1 v n k for some constant c > 0. The last inequality allows us to deduce

v n i+1 ≤ 1 β (c + Bu 0 ) + 1 β h n ( B +L) i k=1 v n k . (27)
By using a discrete version of Gronwall's inequality, one has

v n i+1 ≤ 1 β (c + Bu 0 )exp 1 β ( B +L)ih n . (28)
Consequently, the sequence (v n i ) 1≤i≤n is uniformly bounded by

M := 1 β (c + Bu 0 )exp 1 β ( B +L)T .

(b) Convergence of the approximate solutions

Now we construct the functions u n (•), θ n (•), η n (•) as follow: on

[t n i , t n i+1 ] for 0 ≤ i ≤ n -1, we set u n (t) := u n i + u n i+1 -u n i h n t -t n i ,θ n (t) := t n i ,η n (t) := t n i+1 .
Then, for all t ∈]t n i ,

t n i+1 [ un (t) = u n i+1 -u n i h n = v n i+1 ,
Σ = v n j+1 -v m i+1 , Av n j+1 + Bu n j -Av m i+1 -Bu m i = v n j+1 -v m i+1 , A(v n j+1 -v m i+1 ) + v n j+1 -v m i+1 , B(u n j -u m i ) ≤|ṽ t m i+1 -ṽ t n j+1 | v m i+1 + v n j+1 + L u m i -u n j v m i+1 -v n j+1 ≤|ṽ t m i+1 -ṽ(t) +ṽ(t) -ṽ(t n j+1 )|( v m i+1 + v n j+1 ) + L u m i -u n j v m i+1 -v n j+1 ≤ 2M(ω(h m ) + ω(h n )) + L u m i -u n j v m i+1 -v n j+1 = ε 1 (h m , h n ) + L u m i -u n j v m i+1 -v n j+1
where ε 1 (h m , h n ) := 2M(ω(h m )+ω(h n )) and ω(•) denotes the modulus of continuity of ṽ(•). Using the coerciveness of A, the last inequality allows us to deduce that

△= β 2 v m i+1 -v n j+1 2 ≤ ε 1 (h m , h n ) + ( L + B ) u m i -u n j v m i+1 -v n j+1 - β 2 v m i+1 -v n j+1 2 ≤ ε 1 (h m , h n ) + L2 1 2β u m i -u n j 2 where L1 = L + B
(use the fact that x → αxβx 2 is concave and reaches its maximum at x = α 2β )

≤ ε 1 (h m , h n ) + L2 1 2β ( u m (θ m (t)) -u m (t) + u m (t) -u n (t) + u n (t) -u n (θ n (t)) ) 2 ≤ ε 1 (h m , h n ) + 3 L2 1 2β ( u m (θ m (t)) -u m (t) 2 + u m (t) -u n (t) 2 + u n (t) -u n (θ n (t)) 2 ) (use the inequality (a + b + c) 2 ≤ 3(a 2 + b 2 + c 2 ), ∀a, b, c ∈ R) ≤ ε 1 (h m , h n ) + 3M 2 L2 1 2β (h 2 m + h 2 n ) + 3 L2 1 2β u m (t) -u n (t) 2 . ( 30
) Let ε 2 (h m , h n ) := ε 1 (h m , h n ) + 3M 2 L1 2 2β (h 2 m + h 2 n ) then ε 2 (h m , h n ) → 0asm, n →+∞since ω(h n ) → 0asn →+∞ . Hence, from (30)
, we obtain

β 2 um (t) -u n (t) 2 ≤ ε 2 (h m , h n ) + 3 L2 1 2β u m (t) -u n (t) 2 (31) which implies that um (t) -u n (t) ≤ε(h m , h n ) + 2 L1 β u m (t) -u n (t)
where

ε(h m , h n ) := 2ε 2 (h m , h n ) β .
By using Gronwall's inequality, one has

u m (t) -u n (t) ≤exp( 2 L1 T β )ε(h m , h n )t ≤ exp( 2 L1 T β )ε(h m , h n )T .
Hence the sequence of functions (u n (•)) n is a Cauchy sequence in C(0, T ; H ) and there exists u ∈ C(0, T ; H ) such that u n converges to u uniformly. c) u is a solution of [START_REF] Bounkhel | State dependent sweeping process in p-uniformly smooth and q-uniformly convex banach spaces[END_REF] Clearly, u(0) = u 0 and u(•) is M-Lipschitz continuous since for all s, t ∈[ 0, T ], we have

u(t) -u(s) = lim n→+ ∞ u n (t) -u n (s) ≤M|t -s|.
Consequently, u(•) is differentiable for almost every t ∈[ 0, T ]. By using Banach-Alaoglu theorem, there exist some ξ(•) in L 2 (0, T ; H ) and a subsequence, still denoted by ( un (•)) n such that un (•) converges weakly to ξ(•) in L 2 (0, T ; H ).

Fix t ∈[0, T ] and let z(t) := u(t) -u 0 - t 0 ξ(s)ds.W ehave z(t), u(t) -u 0 - t 0 ξ(s)ds = lim n→+ ∞ z(t), u n (t) -u 0 - t 0 ξ(s)ds = lim n→+ ∞ z(t), t 0 ( un (s) -ξ(s))ds = lim n→+ ∞ T 0 1 [0,t] (s)z(t), un (s) -ξ(s) ds = 0.
D ε := ϕ ∈ L 2 (0, T ; H ) : Aϕ(t) + Bu(t) ⊂ C(t, u(t)) + εB a.e. t ∈[0, T ] .
It is easy to see that D ε is a weakly closed set in L 2 (0, T ; H ) since it is closed and convex. Note that, we have 22), we have

A un (t) + Bu n (θ n (t)) ∈ C(η n (t), u(θ n (t))) ⊂ C(t, u(t)) + (|v(η n (t)) -v(t)|+L u(θ n (t)) -u(t) )B. Since u(θ n (t)) -u(t) →0 and |v(η n (t)) -v(t)
u(t) ∈-N C(t,u(t)) (A u(t) + Bu(t)) a.e. t ∈[0, T ]. Given v ∈ C(t, u(t)), since C(t, u(t)) ⊂ C(η n (t), u(θ n (t))) + (|v(η n (t)) -v(t)|+ L u(θ n (t)) -u(t) )B, we can find some v ′ ∈ C(η n (t), u(θ n (t))) such that v -v ′ ≤ |v(η n (t)) -v(t)|+L u(θ n (t)) -u(t) .From(
un (t) ∈-N C(η n (t),u n (θ n (t))) (A un (t) + Bu n (θ n (t))) a.e. t ∈[0, T ],
which implies that

-un (t), v ′ -A un (t) -Bu n (θ n (t)) ≤0. Thus -un (t), v -A un (t) -Bu n (t) ≤ -u n (t), v -v ′ -Bu n (t) + Bu n (θ n (t)) ≤α n (t),
where

α n (t) = M(|v(η n (t)) -v(t)|+L u(θ n (t)) -u(t) + B M|θ n (t) -t|) → 0asn →+∞.
Since it is true for all v ∈ C(t, u(t)), we deduce that

σ(C(t, u(t)), -u n (t)) + un (t), A un (t) + Bu n (t) ≤α n (t).
Integrating both sides of the last inequality from 0 to T and taking the lower limit, one obtains

lim inf n→∞ T 0 {σ(C(t, u(t)), -u n (t)) + un (t), A un (t) + Bu n (t) }dt ≤ 0. (32)
Let D(t) = C(t, u(t)) for each t then D has nonempty, closed convex values and varies in an absolutely continuous way. Recall that the convex mapping

x → T 0 σ(D(t), x(t)
)dt is also weakly lower semicontinuous on L 2 (0, T ; H ) [START_REF] Rockafellar | Conjugate Duality and Optimization[END_REF],seealso [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF]). Therefore,

([
T 0 σ(D(t), -u(t))dt ≤ lim inf n→∞ T 0 σ(D(t), -u n (t))dt. (33)
In addition, the function x → T 0 x(t), Ax(t) dt is also weakly lower semicontinuous on L 2 (0, T ; X ) and un converges weakly to u in L 2 (0, T ; X ). Thus, we have

T 0 u(t), A u(t) dt ≤ lim inf n→∞ T 0 un (t), A un (t) . (34)
We note that B =∇ϕ B where the function ϕ B (x) := 1 2 Bx, x is convex and continuous. Therefore, one has

T 0 u(t), Bu(t) dt = T 0 d dt ϕ B (u(t))dt = ϕ B (u(T )) -ϕ B (u(0)) = lim n→+ ∞ (ϕ B (u n (T )) -ϕ B (u n (0))) = lim n→+ ∞ T 0 d dt ϕ B (u n (t))dt = lim n→+ ∞ T 0 un (t), Bu n (t) dt. (35)
From ( 32), ( 33), (34), (35), we obtain that

T 0 σ(D(t), -u(t)) + u(t), A u(t) + Bu(t) dt ≤ 0. ( 36
) Since A u(t) + Bu(t) ∈ D(t) for almost t ∈[0, T ],wehave σ(D(t), -u(t)) + u(t), A u(t) + Bu(t) ≥0, a.e. t ∈[0, T ]. (37)
From ( 36) and (37), one gets

σ(D(t), -u(t)) + u(t), A u(t) + Bu(t) =0, a.e. t ∈[0, T ],
or equivalently,

u(t) ∈-N D(t) (A u(t) + Bu(t)), a.e. t ∈[0, T ].
Therefore, u(•) is a solution of [START_REF] Bounkhel | State dependent sweeping process in p-uniformly smooth and q-uniformly convex banach spaces[END_REF]. d) Uniqueness: Let u 1 (•), u 2 (•) be two solutions of [START_REF] Bounkhel | State dependent sweeping process in p-uniformly smooth and q-uniformly convex banach spaces[END_REF] with the same initial condition

u 1 (0) = u 2 (0) = u 0 .First,weprovethat ui (•) is uniformly bounded, i = 1, 2. Indeed, since ui (t) ∈-N C(t,u i (t)) (A ui (t) + Bu i (t)), one has ui (t), y -A ui (t) -Bu i (t)) ≥0, ∀y ∈ C(t, u i (t)), a.e. t ∈[0, T ] (38)
which implies that

β ui (t) 2 ≤ ui (t), A ui (t) ≤ ui (t) y -Bu i (t)) . (39)
Lemma 3.5 allows us to choose some y t ∈ C(t, u i (t)) such that y t ≤c + L u i (t)u 0 . Combining with (39), we deduce that

ui (t) ≤ 1 β y t -Bu i (t)) ≤ 1 β [(L + B ) t 0 ui (s) ds + c + (L + B ) u 0 ]. (40)
Using Gronwall's inequality, one obtains that

ui (t) ≤ 1 β [(L + B )(c + (L + B ) u 0 )exp (L + B )T β .
Therefore, u i (t) and A ui (t) + Bu i (t) are uniformly bounded by some R > 0. From the hypomonotonicity-like of the normal cone of C, there exists L ≥ 0 such that

u1 (t) -u 2 (t), A u1 (t) + Bu 1 (t) -A u2 (t) -Bu 2 (t) ≤ L u 1 (t) -u 2 (t) u1 (t) -u 2 (t) a.e. t ∈[0, T ].
Using the coerciveness of A, the last inequality implies that

β 2 u1 (t) -u 2 (t) 2 ≤ ( L + B ) u 1 (t) -u 2 (t) u1 (t) -u 2 (t) - β 2 u1 (t) -u 2 (t) 2 ≤ L2 1 2β u 1 (t) -u 2 (t) 2 where L1 := L + B . (41)
Hence

u1 (t) -u 2 (t) ≤ L1 β u 1 (t) -u 2 (t) .
From the Gronwall's inequality, we obtain that

u 1 (t) -u 2 (t) 2 ≤ u 1 (0) -u 2 (0) 2 exp L1 t β = 0 for all t ∈[0, T ].
Consequently, one has u 1 ≡ u 2 and the uniqueness of solutions is obtained. ⊓ ⊔ Remark 3.2 (i) The existence and uniqueness result in Theorem 3.1 can be generalized similarly for reflexive Banach spaces without any difficulty. In this case, the set-valued mapping C :[0, T ]×X ⇒ X * and the linear operators A, B : X → X * , where X is a reflexive Banach space with its dual X * .

(ii) If H ≡ R n or if some compactness assumption is supposed, then Assumption 2 and the Cauchy convergence can be skipped since in these cases, the boundedness of the iterates is sufficient to obtain the strong convergence of certain subsequence of the approximate solutions.

(iii) Although our assumptions are more general than the ones used in [START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF] (we consider the state-dependent case without using the additional initial viability condition), we even get the strong convergence of the approximate solutions comparing with the weak convergence of a sub-approximate solutions obtained in [START_REF] Adly | An implicit sweeping process approach to quasistatic evolution variational inequalities[END_REF]. The coerciveness of A plays an essential role in the Cauchy property of the approximate solutions.

Proof First we note that the problem (42) is equivalent to (45) and ( 45) can be rewritten into the form of [START_REF] Bounkhel | State dependent sweeping process in p-uniformly smooth and q-uniformly convex banach spaces[END_REF]. Indeed, since J is proper convex lower semicontinuous, positively homogeneous of degree 1 with J (0) = 0, it is easy to see that

J (y) = σ(C, y) = I * C (y),
where C = ∂ J (0) is a closed, convex set. Thus

y ∈ ∂ J (w) ⇔ w ∈ ∂ J * (y) = ∂ I C (y) = N C (y).
Consequently, one has

f (t, u(t)) -A u(t) -Bu(t) ∈ ∂ J ( u(t)) ⇔u(t) ∈ N C ( f (t, u(t)) -A u(t) -Bu(t)) = N C-f (t,u(t)) (-A u(t) -Bu(t)) ⇔u(t) ∈-N C(t,u(t)) (A u(t) + Bu(t)),
where C(t, u) := f (t, u) -C = f (t, u) -∂ J (0), which is a particular case of Lemma 3.1. Therefore, all assumptions of Theorem 3.1 are satisfied and the conclusion follows.

⊓ ⊔ Now we give an example for the quasistatic frictional contact problem involving viscoelastic materials with short memory under state-dependent perturbation forces.

Example 4.1

The physical setting consists of a deformable body Ω ⊂ R d (d = 2, 3) which is in contact with a rigid foundation. The boundary Ŵ = ∂Ω is supposed to be piecewise smooth composed of three parts: 1). On the boundary Ŵ U , the body is assumed to be fixed, which means that the displacement is given and on Ŵ C the body is in bilateral frictional contact with a foundation, i.e., there is no loss of contact between the body and the foundation. The tractions act on Ŵ N .In the quasistatic case, where all acceleration effects are neglected, the equilibrium of the body in this physical setting is based on the following constitutive law and boundary conditions: find a displacement field u :[0, T ]×Ω → R d and stress field such that:

Ŵ = Ŵ U ∪Ŵ N ∪Ŵ C , with Ŵ U ∩Ŵ N = Ŵ U ∩Ŵ C = Ŵ N ∩Ŵ C =∅(see Fig.
- ∂ ∂t ∂ ∂ x j a ijkh ε kh (u) - ∂ ∂ x j b ijkh ε kh (u) = f 0i (t, u), in ]0, T [×Ω (46)
u i = 0on]0, T [×Ŵ U (47) σν = f 2 on ]0, T [×Ŵ N (48) |σ τ |≤g on Ŵ C , u ν = 0onŴ C (49) |σ τ | < g ⇒ ∂u τ ∂t = 0, |σ τ |=g ⇒ ∃ λ>0 such that ∂u τ ∂t =-λσ τ (50) u(0, x) = u 0 (x), x ∈ Ω. (51)
In the equilibrium Eq. ( 46), f 0 (t, u) represents the density of a body forces, assumed to be dependent on t and u. We suppose that the coefficients (a ijkh ) and (b ijkh ) have For the variational formulation, we introduce the following notations:

H = (H 1 (Ω)) 3 (52) K ={v ∈ (H 1 (Ω)) 3 : v = 0onŴ U and v ν = 0onŴ C }, (53)
a(u,v) = Ω a ijkh ε kh (u)ε ij (v)dx, u,v∈ H (54) b(u,v) = Ω b ijkh ε kh (u)ε ij (v)dx, u,v∈ H (55) J (v) = Ŵ C g|v τ |dŴ, v ∈ H (56) f (t, u), v = Ω f 0 (t, u)vdx + Ŵ N f 2 (t)vdŴ. (57)
Using standard arguments, we are led to the following variational formulation, which consists of finding u(t) such that Find u :[0, T ]→H such that u(t) ∈ K a.e. t ∈[0, T ] and ∀v ∈ K a( u(t), v -u(t)) + b(u(t), v -u(t)) + J (v) -J ( u(t)) ≥ f (t, u(t)), v -u(t) , u(0) = u 0 ∈ H .

(58)

For more details about the variational formulation of problem ( 46)-(51), we refer to [START_REF] Duvaut | Inequalities in Mechanics and Physics[END_REF][START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF] for more details. We recall that if Ω is a bounded open set with a piecewise smooth boundary Ŵ, then the following Korn's inequality holds:

Ω ε ij (v)ε ij (v)dx + Ω v i v i dx ≥ c v 2 H , ∀v ∈ H , (59)
where c > 0 is a constant depending only on Ω. Hence, the coercivity of the bilinear forms a and b is a consequence of the Korn's inequality. The symmetry of a and b follows also from the symmetry of the corresponding elasticity coefficients in (46). We suppose that (t, u) → f 0 (t, u) is absolutely continuous with respect to the time t and Lipschitz continuous with respect to u and that t → f 2 (t) is absolutely continuous.

The following result is a consequence of Proposition 4.1.

Corollary 4.1 Under the above assumptions, for each u 0 ∈ H , problem (58) has a unique solution which is Lipschitz continuous.

Conclusions

In this paper, we study the existence and the uniqueness of solution for a class of state-dependent implicit sweeping process in Hilbert spaces without any compactness assumption. The new notion, introduced in this paper, of hypomonotonicity-like of the normal cone to the moving set turns out to be a fundamental property, which is satisfied in many concrete situations. By using an implicit time discretization and a Cauchy technique, we obtain the strong convergence of approximate functions to the unique solution, which is an important property in view of the construction of adequate numerical algorithms. The equivalence between such implicit state-dependent sweeping processes and quasistatic evolution quasi-variational inequalities is also shown. The theoretical result is applied to analyze the state-dependent quasistatic frictional contact problem in nonsmooth mechanics. Many other open questions need further investigations. The numerical analysis of the quasistatic evolution quasi-variational inequalities using the discretization scheme studied in this paper could be also of great interest. Quasi-variational inequalities are also used to formulate many other problems in many areas such as for instance traffic transportation, Nash equilibrium in game theory and impulse control problems. It would be interesting to investigate other applications in this context. This is out of the scope of the current version and will be carried out in a future study. We expect that the ideas and tools developed in this paper will be applied in the same manner to other fields, where problems can be modeled as state-dependent implicit sweeping process with constrained velocity.

Lemma 2 . 1 1 k=0β

 211 Let α>0 and (u n ), (β n ) be nonnegative sequences satisfying u n ≤ α + nk u k ∀n = 0, 1, 2,... (with β -1 := 0).

Theorem 3 . 1

 31 By setting c := |v(0)|+max t∈[0,T ] |v(t)|+ y 0 , the conclusion follows.⊓ ⊔ Now, we are ready to state the main result. Assume Assumptions 1, 2 and 3 satisfied. Then problem[START_REF] Bounkhel | State dependent sweeping process in p-uniformly smooth and q-uniformly convex banach spaces[END_REF] has a unique solution which is Lipschitz continuous.

  It implies that u(t) = u 0 + t 0 ξ(s)ds for all t ∈[ 0, T ] and hence u(t) = ξ(t) for almost every t ∈[ 0, T ]. Now, we prove that A u(t) + Bu(t) ∈ C(t, u(t)) for almost all t ∈[0, T ].Givenε>0, let us define the set

Fig. 1 A

 1 Fig. 1 A body in contact with a rigid foundation

  |→0asn →+∞, one deduces that un ∈ D ε for n large enough. Thus u ∈ D ε since un converges weakly to u in L

2

(0, T ; H ). As a result, we obtain that A u(t) + Bu(t) ⊂ C(t, u(t)) + εB for every ε>0.

Since C(t, u(t)) is closed and convex, we get A u(t) + Bu(t) ∈ C(t, u(t)).N o wi t remains to prove that

Application to Quasistatic Frictional Contact Problem

Let us consider the following evolution variational inequality

Suppose that the following assumptions hold: (A3) j : K → R is convex, positively homogeneous of degree 1 (i.e., j(λx) = λ j(x), ∀λ>0) and Lipschitz continuous with j(0

for some L ≥ 0 and some absolutely continuous function v :[0, T ]→R. Now, we define the functional J : H → R ∪{+∞}as follows:

Then J is proper, positively homogeneous of degree 1, convex, and lower semicontinuous with J (0) = 0. Let A, B : H → H be the linear bounded and symmetric operators such that Au,v =a(u,v), Bu,v =b(u,v).

Then (42) becomes

Find u :[0, T ]→H such that for a.e. t ∈[0, T ] and ∀v ∈ H

which is equivalent to the following differential inclusion:


Nicolas Robin¹

Frederic Anciaux²

Guillaume R Coudevylle¹

Nicolas Robin
email: robin.nicolas@hotmail.fr

Effects of Tropical Climate and Language Format on Imagery Ability Among French-Creole Bilinguals

Keywords: Imagery ability, bilingual, French-Creole, tropical environment Effects of Tropical Climate and Language

This study examined how a tropical climate (i.e., hot and wet climatic environment) and different language instructions, in French and in Creole, could affect imagery ability on French-Creole bilingual in the French West Indies. Participants were divided into two groups:

Half of them completed the French Movement Imagery Questionnaire (MIQ-F), the others realized the Creole version (MIQ-C), in Neutral Climate (NC) then in Tropical Climate (TC).

The results of this original study showed that participants had lower visual and kinaesthetic MIQ-F and kinaesthetic MIQ-C imagery scores, but had higher visual MIQ-C imagery scores, in TC than in NC. Poor imagers had significant lower kinaesthetic MIQ-F, and better visual MIQ-C imagery scores in TC than in NC. The results of this study suggested that imagery language instruction can modulate the influence of tropical climate on imagery ability. The difference between Creole and French language effects on cognition as imagery is discussed.

kinaesthetic (i.e., feeling or polysensory experience of the body while performing a movement). A comprehensive yet inexpensive method of screening participants' visual and kinaesthetic imagery abilities is the use of self-report questionnaires. Using the [START_REF] Hall | Movement Imagery Questionnaire[END_REF] Movement Imagery Questionnaire (MIQ), which allows differentiating poor from good imagers on the basis on simple movements (e.g., put the arm above the head and lower it with the closed fist), many authors observed that good imagers improve their acquisition in learning tasks [START_REF] Goss | Imagery ability and the acquisition and retention of movements[END_REF]. The authors proposed that good imagery ability facilitates the memory information encoded in the representation of actions. Individuals with higher levels of imagery ability "good imagers" experience greater benefits from imagery compared with their lower level counterparts "poor imagers" (e.g., [START_REF] Toussaint | On the content of sensorimotor representations after actual and motor imagery practice[END_REF][START_REF] Robin | The influences of tropical climate and load on mental chronometry moderated by imagery ability[END_REF][START_REF] Williams | Layered stimulus response training improves motor imagery ability and movement execution[END_REF]. Indeed [START_REF] Robin | Effects of motor imagery training on returning serve accuracy in tennis: The role of imagery ability[END_REF] observed that motor imagery (i.e., to visually imagine realizing a tennis serving return focusing attention on the ball trajectory and the target) improved the precision of the task, and that this improvement was better in good imagers than in poor imagers. Moreover, the language is a factor that can influence movement imagery ability. For example, Anciaux, Alin, Leher, and Mondor (2002), while using a Creole version of the MIQ (MIQ-C), observed that bilingual subjects aged from 14 to 16 reported more vivid visual images than participants aged from 11 to 13 whereas there was no difference in French with the French version of the MIQ (MIQ-F). Indeed, the language system is deeply intertwined to the motor system [START_REF] Stins | Words that move us: The effects of sentences on body sway[END_REF], its processing is embodied (Rueschemeyr, Lindemann, van Rooij, van Dam, & Bekkering, 2010) and shares neural structures with other cognitive activities as mental imagery or conceptual knowledge [START_REF] Gallese | The brain's concepts: The role of the sensory-motor system in conceptual knowledge[END_REF].

In Guadeloupe, the communication is made in French and in Creole. The "Creole" term, at first, referred to the inhabitants of colonial territories, born in the islands. Afterwards, it applied to the manners to speak [START_REF] Véronique | Créole, créoles français et théories de la créolisation[END_REF]. Every language occupies a particular status and plays a specific role in the daily sparring matches [START_REF] Anciaux | L'influence de la langue sur la capacité d'imagerie du mouvement[END_REF]. French is the official language, that of the information and the education [START_REF] Anciaux | Langue de présentation des consignes et performances motrices chez des bilingues français/créole aux Antilles françaises[END_REF]. Creole represents the vernacular language, in other words, the spoken language inside the West Indian community (Ferguson, 1959). According to [START_REF] Anciaux | L'enfant, le créole et l'EPS aux Antilles françaises : Une approche pluridisciplinaire du bilinguisme dans les apprentissages moteurs[END_REF], it is cultural and dominated by the people language, the symbol of the identity and the partner of the private or familiar conversations. The coexistence in the same community of two linguistic codes of which the statutes are uneven, as it is the case in the French overseas departments, is translated by a diglossia [START_REF] Fergusson | Diglossia[END_REF] or bilingualism [START_REF] Hamers | Bilinguality and bilingualism[END_REF].

Nevertheless, that deprives nothing of the fact that the great majority of the inhabitants of Guadeloupe are French-Creole bilinguals [START_REF] Massina | Une observation de la récupération différentielle des deux langues chez une patiente aphasique bilingue français/créole guadeloupéen[END_REF]. In this specific context, Guadeloupean people build themselves linguistic and driving behavior, attitudes and mental representations in correlation with, on one hand French, and on the other hand Creole [START_REF] Anciaux | L'influence de la langue sur la capacité d'imagerie du mouvement[END_REF]. The bilingualism allows the organization of a system of the reality and the information appropriate to each of the languages [START_REF] Massina | Une observation de la récupération différentielle des deux langues chez une patiente aphasique bilingue français/créole guadeloupéen[END_REF][START_REF] Mishra | Bilingualism and cognitive control[END_REF]. In Guadeloupe, the education is mainly driven in French. However, psycholinguistic studies highlighted the positive effects of using Creole on learning and performances (Durizot Jno-Baptiste, 1996), participation of the pupils [START_REF] Giraud | L'école aux Antilles[END_REF], clarification of the instructions [START_REF] Vasseur | Le cas de l'adulte bilingue dans les dispositifs de formation en Guadeloupe : un exemple des tensions à l'oeuvre dans une communauté créole[END_REF], application of the acquired knowledge [START_REF] Gauvin | Du créole opprimé au créole libéré, défense de la langue réunionnaise[END_REF] and engender faster and more successful cognitive learning [START_REF] Giraud | L'école aux Antilles[END_REF]. The Creole officially entered the school by obtaining in 2001 the status of regional language and that of the object of optional education [START_REF] Anciaux | Langue de présentation des consignes et performances motrices chez des bilingues français/créole aux Antilles françaises[END_REF]. A study on the linguistic imagination of Creole speakers supports the idea that the Creole is a language made by images or embellished with images because it is mainly transmitted orally [START_REF] Michelot | La situation sociolinguistique de la Martinique : enquête[END_REF]. Previous studies showed an effect of the language on the mental imagery movement capacities, the value of imaging of the words and the performances of motor reminding with French-Creole bilinguals [START_REF] Anciaux | L'enfant, le créole et l'EPS aux Antilles françaises : Une approche pluridisciplinaire du bilinguisme dans les apprentissages moteurs[END_REF][START_REF] Anciaux | L'influence de la langue sur la capacité d'imagerie du mouvement[END_REF][START_REF] Anciaux | Imagerie visuelle et rappel moteur : Effet du bilinguisme français/créole[END_REF]. For example, the language of presentation of the instructions seems to influence the evocation of mental images to certain French-Creole bilinguals, and therefore, to increase the memory capacities and the motor reminding of series of actions [START_REF] Anciaux | Langue de présentation des consignes et performances motrices chez des bilingues français/créole aux Antilles françaises[END_REF]. These singular language effects could also be situated at the level of the mobilization of the motivation, attention, emotion, memory, and cognition. Language mediates not only the social relationship systems, but also the control of cognitive processes [START_REF] Ardila | Bilingualism and aging. Perspectives on communication disorders and sciences in culturally and linguistically diverse populations[END_REF]. The latest authors proposed that using one additional language to communicate is reflected in the pattern of brain organization and activity. Indeed, changes in the brain of bilingual individuals have been reported by Mechelli and collaborators (2004) who found that learning a second language increases the density of grey matter in the left inferior parietal cortex and influence cognition (see Bialystok, 2017 for a review). For examples, studies demonstrated higher anticipatory control in the occulomotor domain [START_REF] Singh | Effect of bilingualism on anticipatory oculomotor control[END_REF], inhibitory control and attention in proficient bilinguals [START_REF] Mishra | Bilingualism and cognitive control[END_REF]. Bilingual people also have to be able to switch between languages when required [START_REF] De Baene | Brain circuit for cognitive control is shared by task and language switching[END_REF]. The ability to select the appropriate language (i.e., Creole or French) and switch between theses languages involves general executive processing [START_REF] Bialystok | The bilingual adaptation: How minds accommodate experience[END_REF], is regarded as a linguistic ability dependent upon frontal lobe activity [START_REF] Abutalebi | Bilingual language production: The neurocognition of language representation and control[END_REF][START_REF] Hernandez | Language switching and language representation in Spanish-English bilinguals: An fMRI study[END_REF] and is increased with second language proficiency (Zhao et al., in press). The Creole being a language that is spoken most of the time in TC, it seems important to take this particularly hot environment into account.

The impact of the environmental temperature on the cognitive performances is fairly well documented (for a review, see [START_REF] Gaoua | Cognitive function in hot environments: A question of methodology[END_REF]. Many authors observed that heat stress is associated with decreased performance across multiple tasks [START_REF] Berg | The impact of heat stress on operative performance and cognitive function during simulated laparoscopic operative tasks[END_REF]Coudevylle, Sinnapah, Robin, Collado, & Hue, 2019;[START_REF] Qian | Environmental heat stress enhances mental fatigue during sustained attention task performing: Evidence from an ASL perfusion study[END_REF]. For example, [START_REF] Ramsey | Recommended alert limits for perceptual motor loss in hot environments[END_REF] confirmed that simple mental tasks show little, if any, decrement in the heat, and added that more difficult tasks (perceptual motor tasks) show the onset of decrements in the range between 30-33°C. The global workspace theory [START_REF] Baars | In the theatre of consciousness. Global Workspace Theory. A rigorous scientific theory of consciousness[END_REF]) can be proposed as a possible explanation of these results. This theory suggests that consciousness enables multiple neuronal networks to compete or cooperate in solving problems in different environmental conditions [START_REF] Gaoua | Cognitive function in hot environments: A question of methodology[END_REF]. Humans have a limited cognitive capacity due to different stimuli competing for the limited conscious access to the vast global workspace. For example, [START_REF] Hocking | Evaluation of cognitive performance in the heat by functional brain imaging and psychometric testing[END_REF] suggested that the performance in the cognitive tasks would deteriorate when the total resources are insufficient for both the task and the thermal stress. Moreover, [START_REF] Qian | Environmental heat stress enhances mental fatigue during sustained attention task performing: Evidence from an ASL perfusion study[END_REF] proposed that heat stress has a potential fatigue-enhancing effect when individual is performing highly cognition demanding attention task. In addition, numerous works on the TC showed that this climate could be particularly disabling and increases thermal discomfort (e.g., Coudevylle, Sinnapah, Popa-Roch, Hue, & Robin, in press).

Thermal comfort/discomfort is the condition of mind that expresses satisfaction or dissatisfaction with the thermal environment, which is generally assessed by scales [START_REF] Xiong | Effects of temperature steps on human health and thermal comfort[END_REF]. [START_REF] Gaoua | Cognitive function in hot environments: A question of methodology[END_REF] observed that impairment in working memory without alteration in attentional process during heat exposure was shown. The latter author suggested that exposure to a hot environment is a competing variable to the cognitive processes. Moreover, we noted that some studies reported gender differences related to heat.

For example, [START_REF] Wyon | Spontaneous magnitude estimation of thermal discomfort during changes in the ambient temperature[END_REF] reported that females could better withstand the negative effects of heat stress than males, and [START_REF] Lan | Investigation of gender difference in thermal comfort for Chinese people[END_REF] found that males prefer cooler room temperature than females who furthermore are less sensitive to humidity. It seems that heat stress is associated with decreased cognitive and psychomotor performance across multiple tasks, sometimes with gender effects, but to date, the effect of a particularly hot environment on mental imagery ability and different language instructions remains unstudied to our knowledge.

The aim of this original study was to evaluate whether TC differently influences MIQ scores (Creole vs. French versions) upon participants' imagery ability. On the basis of the global workspace theory, on the fact that the performance in the cognitive tasks could deteriorate due to the thermal stress [START_REF] Hocking | Evaluation of cognitive performance in the heat by functional brain imaging and psychometric testing[END_REF], and on [START_REF] Guillot | Neural mechanisms for expertise in mental imagery[END_REF] who observed that poor imagers need to recruit more mental resources in order to build a vivid representation of the movements, we predicted that the imagery ability of good imagers would be less affected by the deleterious effect of hot and wet climatic environment than that of poor imagers. Moreover, we presumed that because the Creole is an image language learned in tropical place, the representation of action would be easier in TC than in Neutral Climate (NC) when using Creole imagery instructions. We also expected that MIQ scores would be higher in TC than in NC when instructions were given in Creole rather than French. Finally, although gender effect is generally not observed in the MIQ, TC and more particularly heat is likely to produce one and to enhance sensation of fatigue in males.

Therefore, the evaluation of the influence of TC on MI should be accompanied by the measures of the fatigue, comfort and thermal sensations in order to control that these factors do not differently influence participants, according to their gender, in each environmental condition (i.e., NC and TC).

Materials and methods

Participants

One hundred and ten self-declared right-handed participants gave their informed consent to participate in the study (M age = 31.01 years, age range: 26-35 years, SD = 2.47). All were French-Creole bilingual Guadeloupean primary school teachers (in the French West Indies) living in the tropical environment throughout the year. They teach all subjects to pupils from 2 to 11 years old and have the habilitation to teach Creole. As many factors can influence mental performance, before each session, participants completed a control selfreport questionnaire evaluating their consumption of medicine, drug, alcoholic and nonalcoholic drinks in the 24h preceding each experimental trial. Ninety-six participants were retained (good and poor imagers) and divided into two groups. Fourteen participants were excluded being in intermediate categories (see [START_REF] Robin | Effects of motor imagery training on returning serve accuracy in tennis: The role of imagery ability[END_REF]. In the first group, 48 participants (33 females and 15 males) were retained (M age = 31.52 years, SD = 2.46 years) with regard to their MIQ-C scores: Poor imager (N = 24; scores < 23, mean < 2.5) and good imager (N = 24; scores > 41, mean > 4.5) groups. Forty-eight other participants (33 females and 15 males) were retained (M age = 30.41 years, SD = 2.37 years) with regard to their MIQ-F scores: Poor imager (N = 24; scores < 23, mean < 2.5) and good imager (N = 24; scores > 41, mean > 4.5) groups.

Material

Questionnaire. During each experimental session, all the participants completed the French: MIQ-F [START_REF] Lorant | Validation de la version française du Movement Imagery Questionnaire (MIQ) [Validation of the French version of the Movement Imagery Questionnaire (MIQ)[END_REF] or Creole: MIQ-C [START_REF] Anciaux | L'influence de la langue sur la capacité d'imagerie du mouvement[END_REF] translated and validated versions of the Movement Imagery Questionnaire (MIQ, [START_REF] Hall | Movement Imagery Questionnaire[END_REF]. The construct validity (internal consistency of visual and kinaesthetic scales were respectively 0.87 and 0.91 for the MIQ-F; 0.89 and 0.88 for the MIQ-C and corresponded to composite reliability scores, see McNeish, 2018 for similar procedure) and the test-retest reliability (0.83 for the two) of the available French and Creole versions were duly validated.

Each version of the MIQ consists in 18 items that assess individual visual (9 items) and kinaesthetic (9 items) imagery abilities. The actions to be performed, which are similar in the two scales, involve a variety of arm, leg, and whole body movements (e.g., raising and lowering the right knee or performing a front roll on a mat and finishing in standing position).

At the outset and for each item, participants listen to an audio description of the movement to be carried out. After that, the participants actually perform the movement and are then asked to either visually or kinaesthetically imagine it. Following this step, it was written in the MIQ: "Now, consider the ease or difficulty with which you were able to do this mental task", and participants rate their ease of imaging on a 7-point Likert-type scales for the visual imagery (from 1 = very hard to see to 7 = very easy to see) or on an other 7-point Likert-type scales for the kinaesthetic imagery (from 1 = very hard to feel to 7 = very easy to feel). Then, the items for each subscale are added, a higher score representing a greater ease of imaging.

Procedure

The present experiment was conducted on a rectangular room (5 × 7 meters). Sessions were realized in NC (mean temperature = 24.07°C, SD = 0.69; and humidity = 47.2% rH, SD = 3.7) or TC (mean temperature = 31.12°C, SD = 0.29; and humidity = 73.4% rH, SD = 6.3).

Temperature and humidity were recorded with a thermohygrometer Fisherbrand (humidity precision ± 5% rH from 35 to 75% rH, and ± 8% rH beyond; temperature precision ± 0.1°C).

TC is characterized by consistently high monthly temperatures, often exceeding 18°C throughout the year, and rainfall that exceeds evapotranspiration for at least 270 days per year [START_REF] Salati | Precipitation and water recycling in tropical rain forests with special reference to the amazon basim[END_REF]. According to the annual climatic bulletin 2015, these values varied across the day. For example, between 10:00 am and 04:00 pm, indoor mean temperature can frequently reach 31°C ± 2° (rH 80% ± 10%), but in air-conditioned rooms the mean temperature generally measured is 24°C ± 1° (rH 45% ± 5%).

All participants performed the procedure with a one-week interval between sessions (i.e., one in NC and one in TC). These conditions were presented in a randomized order.

During each session, the participants completed the control questionnaire, indicated their feeling of fatigue, thermal and comfort sensations and realized the MIQ-F or the MIQ-C.

Data analysis

For each participant, visual and kinaesthetic imagery scores at the MIQ-C or MIQ-F were calculated and verified that all variables were normally distributed (Shapiro-Wilk test).

The dependent variables for the MIQ-C and MIQ-F were submitted to separate ANOVAs (with repeated measures on the last factor) contrasting 2 imagery ability (good imagers vs. poor imagers) × 2 gender (female vs. male) × 2 temperature conditions (TC vs. NC) , with imagery ability and gender as within-subjects factors and temperature condition as a betweensubjects factor. As sensation of fatigue and thermal comfort and sensation can influence cognitive process in function of environmental conditions, assessment then analyses on these 3 control variables were realized. Feeling of fatigue was recorded on 7-point scale ranging respectively from 1 "fully alert" to 7 "completely exhausted". Moreover, participants were asked to rate both their thermal comfort on a 4-point scale (from 1, comfortable, to 4, very uncomfortable) and their thermal sensation on a 7-point scale (from -3, cold, to +3, hot). The averaged scale scores for the feeling of fatigue and thermal comfort and sensation were submitted to ANOVAs (with repeated measures on the second factor) using the gender (female vs. male) as between-participants factor and the temperature conditions (TC vs. NC) as a within-participants factor. All significant main effects and interactions were broken down using the post-hoc Newman-Keuls technique. Alpha was set at .05 for all analyses and effect sizes (η p 2 ) were indicated. The software package Statistica (12, 64-bit) was used for all analyzes.

Results

Preliminary analysis on control variables: feeling of fatigue and thermal comfort and sensation

The ANOVAs revealed that there was no gender main effects or interactions between gender and temperature condition (TC vs. NC) for feeling of fatigue, thermal comfort and sensation. However, main effects of temperature condition were observed. Post-hoc Newman-Keuls tests revealed that the participants had higher feeling of fatigue, lower thermal comfort and felt hotter in TC than in NC (main results are summaries in Table 1).

________________________

Table 1 near here ________________________

MIQ-F visual imagery analysis

The ANOVA revealed a significant main effect of imagery ability, F(1, 44) = 1519.1, p < .01, η p 2 = 0.97. Post-hoc Newman-Keuls test revealed that poor imagers had worse visual imagery MIQ-F mean scores than good imagers, which correspond to our criteria of inclusion.

Moreover, the ANOVA revealed a significant main effect of climate, F(1, 44) = 121.33, p < .01, η p 2 = 0.73. Post-hoc Newman-Keuls test revealed that participants had worse visual imagery MIQ-F mean scores in TC than in NC (see Figure 1). However, the ANOVA did not reveal any significant main effect of gender, F(1, 44) = 0.006, p = .

MIQ-C kinaesthetic imagery analysis

The ANOVA revealed a significant main effect of imagery ability,

F

Discussion

The aim of this original study was to evaluate whether tropical climate differently influences participants' imagery ability when using Creole versus French MIQ validated versions. As sensation of fatigue, thermal comfort and sensation do not statistically differently influence gender or participants in each specific environmental condition (NC vs. TC), the influence of the latter variable can be easily interpreted. The results of this original study showed that participants had worse visual and kinaesthetic MIQ-F and kinaesthetic MIQ-C imagery scores in TC than in NC, but had better visual imagery MIQ-C scores in TC than in NC. Moreover, poor imagers had significant lower kinaesthetic imagery MIQ-F scores in TC than in NC, and had significant better visual imagery MIQ-C scores in TC than in NC. The results are discussed in the following sections.

Climate and mental imagery

The results evidenced a significant main effect of the climate: Participants had worse visual and kinaesthetic MIQ-F and kinaesthetic MIQ-C imagery scores in TC than in NC.

This result confirmed those recently obtained by [START_REF] Robin | Effects of tropical climate on mental rotation: The role of imagery ability[END_REF] during which participants fulfilled the Movement Imagery Questionnaire Revised [START_REF] Hall | Measuring movement imagery abilities: A revision of the Movement Imagery Questionnaire[END_REF] and the a mental rotation test and that showed that TC negatively influenced mental rotation, motor imagery and imagery ability. Similar results were observed during heat exposure in studies using different complex tasks [START_REF] Hocking | Evaluation of cognitive performance in the heat by functional brain imaging and psychometric testing[END_REF][START_REF] Robin | The influences of tropical climate and load on mental chronometry moderated by imagery ability[END_REF][START_REF] Robin | The Influence of Tropical Climate on Cognitive Task Performance and Aiming Accuracy in Young International Fencers[END_REF]. The Maximal Adaptability Model [START_REF] Hancock | A dynamic model of stress and sustained attention[END_REF]) that assumes that heat exerts its detrimental effects on performance by competing for and eventually draining attentional resources could explain these results. Indeed, it has been proposed that the normative zone is the zone of comfort wherein cognitive adjustments to task demands are easily accomplished and performance is at near optimal level [START_REF] Hancock | Effects of heat stress on cognitive performance: The current state of knowledge[END_REF]. As the level of environmental stress increases, by increasing temperature and humidity for examples, attentional resources are progressively drained. For example, Chase, Karwowski, Benedict, Quesade, and Irwin-Chase ( 2005) reported a poor dual-task performance at 30°C or 35°C and observed that participants were being unable to successfully allocate their attention to the tasks. In line with the Global Workspace theory [START_REF] Baars | In the theatre of consciousness. Global Workspace Theory. A rigorous scientific theory of consciousness[END_REF], Hocking and collaborators (2001) proposed that the performance would decrease when the total resources are insufficient for both the thermal stress (i.e., TC) and the task. It was probably the case when participants had to realize visual or kinaesthetic imagery in the MIQ-F and kinaesthetic imagery in the MIQ-C. However, the fact that participants had better visual imagery MIQ-C scores in TC than in NC might seem surprising. This result will be discussed in the following section.

Language and mental imagery

Participants of this study were French-Creole bilingual Guadeloupean teachers. The fact that they had better visual imagery MIQ-C scores in TC than in NC whereas the opposite was observed for the visual imagery with the MIQ-F and kinaesthetic imagery for the MIQ-C and the MIQ-F seems to indicate that Creole and French, although sharing common brain structures also activates different neural pathways specific to each language. These results seem coherent with the case study of [START_REF] Massina | Une observation de la récupération différentielle des deux langues chez une patiente aphasique bilingue français/créole guadeloupéen[END_REF] who noticed a differential recovery of French and Creole for a patient who followed six months of reeducation in French. The authors observed a better recovery of Creole than re-educated French and found an interpretation in linguistic structures (lexical, morphophonologic, morphosyntaxic), but also in the distribution of main areas for language of the Creole. According to [START_REF] Bialystok | The bilingual adaptation: How minds accommodate experience[END_REF], bilingualism changes the brain structures that underlie language processing, and also has the potential to modify brain and cognitive systems more generally. The difference between bilinguals was also showed in [START_REF] Ardila | Bilingualism and aging. Perspectives on communication disorders and sciences in culturally and linguistically diverse populations[END_REF] study who observed that during normal aging there is a more rapid decline of the second than the first language, and elder ones tend clearly to prefer using the first language. They proposed that life experiences may be associated with the use of a specific language, and life memories are also associated with the contextual language existing when the memory was acquired. Finally, [START_REF] Monaco | Embodied Semantics in a Second Language: Critical Review and Clinical Implications[END_REF] evoked that understanding how the first and second languages are represented in the bilingual's brain and how they are intertwined will favour diagnosing and rehabilitation strategies.

We can wonder about why visual imagery was facilitated in TC when imagery instructions were given in Creole, a climate that normally could cause a decrease of performance. According to [START_REF] Michelot | La situation sociolinguistique de la Martinique : enquête[END_REF], Creole is a language made by images. We might presume that Creole could particularly facilitate mental visual imagery particularly because of shared neural structures [START_REF] Gallese | The brain's concepts: The role of the sensory-motor system in conceptual knowledge[END_REF]. Indeed, Anciaux andcollaborators (2002, 2005) showed an effect of the language on mental movement imagery capacities, the value of imaging the words and the performances of motor reminding with French-Creole bilinguals.

For example, the language of presentation of the instructions seems to influence the evocation of mental images and therefore, to increase the memory capacities and the motor reminding of a series of actions [START_REF] Anciaux | Langue de présentation des consignes et performances motrices chez des bilingues français/créole aux Antilles françaises[END_REF]. ). Indeed, as previously evoked, the language embodied (Rueschemeyr et al. 2010) and intertwined to the motor system [START_REF] Stins | Words that move us: The effects of sentences on body sway[END_REF]. These singular language effects could also be situated at the level of the mobilization of the motivation, attention, emotion, memory, and vision (Byalistok, 2017;[START_REF] Mishra | Attention and vision in language processing[END_REF]. Language mediates not only the social relationship systems but also the control of cognitive processes [START_REF] Ardila | Bilingualism and aging. Perspectives on communication disorders and sciences in culturally and linguistically diverse populations[END_REF][START_REF] Mishra | Bilingualism and cognitive control[END_REF] and the use of one additional language is reflected in the pattern of brain organization and activity (Byalistok, 2017). Moreover, we can suppose that when participants learned Creole, they built a representation including environmental elements as climate and the fact to be in TC during visual imagery with the MIQ-C may have facilitated or not limited the access to this representation. Indeed, in the sport domain, [START_REF] Guillot | Influence of environmental context on motor imagery quality[END_REF] investigated the effectiveness of motor imagery when performed in a context close to actual practice situations rather than in a neutral environment in a table tennis task. They observed that environmental conditions (i.e., tactile, proprioceptive and auditory information) helped athletes to visualize and feel sensations elicited by actual practice. Some environmental conditions could therefore have a beneficial effect on motor imagery as NC for mental imagery with French visual and kinaesthetic imagery instructions or TC as seem to indicate the result of the participants who realized visual imagery with Creole instructions. Participants of the current study were born in Guadeloupe except for 5 of them who arrived when they were less than 5 years old. Therefore, statistical analysis including this element as between factors can't be done (given the very small size of the latest group). However, it would be interesting to evaluate if French-Creole bilingual people who learned Creole out of a tropical climate would benefit from this climate condition when realizing visual imagery with Creole instructions in TC. Moreover, according to [START_REF] Ardila | Bilingualism and aging. Perspectives on communication disorders and sciences in culturally and linguistically diverse populations[END_REF], research has supported the assumption that using two or more different languages can frequently provide cognitive advantages. [START_REF] Costa | Bilingualism aids conflict resolution: Evidence from the ANT task[END_REF] considered that three different attentional networks can be distinguished: Alerting, orienting and executive control.

Comparing monolingual and bilingual subjects, the authors found that bilingual participants were not only faster in performing different attention tasks, but also more efficient in the alerting and executive control networks. As bilingualism can exert beneficial influence in the attainment of efficient attentional mechanisms for example [START_REF] Mishra | Attention and vision in language processing[END_REF], it should also be interesting to compare the visual imagery MIQ-C scores of the participants of the current study with those of monolingual Creole speakers as Haitians living in TC in order to assess the effect of bilingualism on visual imagery. We could also suggest that Creole should appear, for bilinguals, as a cognitive load reduction. TC should represent a cognitive load (e.g., thermal) whereas Creole seems to bring a kind of comfort zone or cognitive facilitation that reduces this load. Further research is needed to evaluate the influence of TC on attentional resource.

Climate, language and imagery ability

Despite the impact that imagery can have on performance and learning, an individual ability to create and control images can determine the effectiveness of its use [START_REF] Guillot | Construction of the motor imagery integrative model in sport: A review and theoretical investigation of motor imagery use[END_REF]. Besides showing main effect of the climate, two significant interactions between imagery ability and climate condition were observed in the current study. First, the results revealed that poor imagers had significant lower kinaesthetic imagery MIQ-F scores in TC than in NC. This imagery ability effect is consistent with those of [START_REF] Logie | Low and high imagers activate networks differentially in mental rotation[END_REF] in a mental rotation task, [START_REF] Guillot | Neural mechanisms for expertise in mental imagery[END_REF] recording autonomic nervous response and measuring the activation of cerebral structures and [START_REF] Robin | Effects of motor imagery training on returning serve accuracy in tennis: The role of imagery ability[END_REF] in a tennis task, for examples. In the latter study, the authors proposed that poor imagers can benefit an imagery practice, as kinaesthetic imagery. However, it is supposed that they needed more trials to become as effective as good imagers as observed in the study of Goss and collaborators (1986) and to previously developed a sensory-specific representation of actions to successfully employ motor imagery as suggested by [START_REF] Toussaint | On the content of sensorimotor representations after actual and motor imagery practice[END_REF]. The fact that poor imagers had lower kinaesthetic imagery MIQ-F scores in TC than in NC, whereas good imagers performance were not affected should be explained by the addition of more difficulties to generate a vivid and accurate kinaesthetic images of action (for poor imagers) in TC which imposes to them a supplementary constraint in term of cognitive resources. We supposed that TC represent a higher level of stress for poor imagers realizing kinaesthetic imagery and the depletion of cognitive resources results in a progressive decline of performance efficiency in mental imagery. On the contrary, the good imagers have probably more efficiently used the remaining resources using adaptive strategies such as attentional focus [START_REF] Robin | The influences of tropical climate and load on mental chronometry moderated by imagery ability[END_REF].

Secondly, another significant interaction between imagery ability and climate condition revealed that poor imagers had significant better visual imagery MIQ-C scores in TC than in NC. As evoked in the language section, Creole is a language made by images [START_REF] Michelot | La situation sociolinguistique de la Martinique : enquête[END_REF] that seems to facilitate the use of mental visual imagery. The Creole is most of the time learnt in TC places. As proposed by [START_REF] Guillot | Influence of environmental context on motor imagery quality[END_REF], an environment closed to the one in which the task was learnt seems to favor the benefits of imagery. That's why we can presume that TC can favor the use of visual imagery when the imagery instructions are in Creole in particular for poor imagers. Indeed, good imagers performance was not influenced by the climate when realizing visual imagery with Creole instructions. It is likely that the remaining resources of good imagers were sufficient to obtain a successful outcome. The performance in mental imagery would have been deteriorated only if the total resources had been insufficient for both the task and the climate effect.

Finally, the cognitive dissonance theory that concerns the dynamic interplay of cognition, emotion, and motivation [START_REF] Festinger | A theory of cognitive dissonance[END_REF] could be used to explain the results.

Indeed, according to Harmon-Jones and Harmon-Jones ( 2002), when dissonant information is encountered, a negative emotive state causes the person to engage in cognitive work to support the commitment. We can presume that Creole being associated with TC contains less dissonance (as French in NC), particularly during visual imagery, and thus disturb less the participants than Creole in NC or French in TC.

The results of the present study reporting that visual and kinaesthetic imagery MIQ-F scores and kinaesthetic imagery MIQ-C scores were negatively affected by TC, whereas visual imagery MIQ-C was positively affected by TC, prompt us to question of the imagery ability stability depending on environmental conditions and imagery language instructions. Further research will be realized in our laboratory in order to examine how these might predict the success of an imagery intervention in tropical climate: For example, in school task (e.g., mental rotation in mathematics) or in sports (e.g., tennis) that are realized in TC condition, the instruction to realize visual imagery, should permit better acquisition, learning or precision, if it is done in Creole than in French on French-Creole bilinguals.

Gender effect

For decades, gender has aroused considerable debate [START_REF] Koslow | Sex-related differences and visual-spatial mental imagery as factors affecting symbolic motor skill acquisition[END_REF]. Many authors evidenced that females and males differ in their imagery generating capabilities [START_REF] Ozel | Influence of sport on gender differences in spatial imagery[END_REF][START_REF] Taktek | The effects of mental imagery on the acquisition of motor skills and performance: A literature review[END_REF]) and reported differences in cortical activation patterns and most especially hemispheric lateralization of brain activity [START_REF] Seurinck | Does egocentric mental rotation elicit sex differences?[END_REF]. The "psychosocial" variety (stereotype threat, sexrole identification, experience and socialization) and the "biological" variety (e.g., sexual hormones) are the two main causes used to explain these gender differences [START_REF] Titze | Mental rotation performance and the effect of gender in fourth graders and adults[END_REF]. It is possible that because all the participants were teachers, the "psychosocial" variety was limited. Moreover, the fact that gender effect was not observed could be explained by the number of the participants selected in this study (i.e., 96) with more women (N = 66) than men (N = 30). Indeed, in primary schools, women represent 80% of all the teachers. However, the results of the literature are inconsistent and many authors using the MIQ [START_REF] Williams | Further validation and development of the movement imagery questionnaire[END_REF], the MIQ-R [START_REF] Lorant | Validation de la traduction française du Movement Imagery Questionnaire -Revised (MIQ-R) [Validation of the French translation of the Movement Imagery Questionnaire-Revised (MIQ-R)[END_REF], the MIQ-F [START_REF] Lorant | Validation de la version française du Movement Imagery Questionnaire (MIQ) [Validation of the French version of the Movement Imagery Questionnaire (MIQ)[END_REF], or the MIQ-C [START_REF] Anciaux | L'influence de la langue sur la capacité d'imagerie du mouvement[END_REF] have already observed that it does not exist a significant difference between men and women as regards motor imagery ability. This finding was corroborated in our sample.

Limitations of this study

The fact that fatigue, thermal comfort and sensation were assessed by only a single self-report item, unless being the most frequently used method in the literature [START_REF] Lan | Investigation of gender difference in thermal comfort for Chinese people[END_REF], is a limit. Further research using more complex fatigue subjective estimation or thermal comfort evaluation is needed.

Another limit could be the use of self-reported questionnaire. The choice of the MIQ-F and MIQ-C was done because they can be fulfilled everywhere (school or playground).

Moreover the two questionnaires are validated and the most commonly used French and Creole tests [START_REF] Anciaux | L'influence de la langue sur la capacité d'imagerie du mouvement[END_REF][START_REF] Robin | The influences of tropical climate and load on mental chronometry moderated by imagery ability[END_REF]. Further research using more objective measures will be realized in our laboratory.

Conclusion

This original study highlights the necessity to measure participants' imagery ability, to take the environmental conditions (e.g., climate), and to consider the language instruction when realizing mental imagery interventions. By showing significant differences in the MIQ-C scores for kinaesthetic imagery and in the MIQ-F scores for visual and kinaesthetic imagery between TC and NC, the present study confirms the results observed in the literature [START_REF] Gaoua | Cognitive function in hot environments: A question of methodology[END_REF] and could be interpreted as additional support for the relative resistance of working memory to heat stress. However, the fact that visual imagery MIQ-C scores were positively affected by TC seems to indicate that it is important to take the environment in which a language of imagery instruction was learnt into account. Imagery ability stability seems to depend of environmental condition and also language instruction. Notes. The results shown are averages, standard deviation (in brackets) and Confidence Level Cronbach's alpha with a 95% confidence interval [CI]. ** Significant differences (p < .05) of the main effects of climate condition in ANOVAs using the gender (female vs. male) as between-participants factor and the climate condition (TC vs. NC) as within-participants factor.

  Figure 1 near here

Figure 1 .

 1 Figure 1

Figure 2 .

 2 Figure 2. Main effects of imagery ability (A) and climate condition (B) and Imagery ability ×

Figure 3 .

 3 Figure 3. Main effects of imagery ability (A) and climate condition (B) and Imagery ability ×

Figure 4 .

 4 Figure 4. Main effects of imagery ability (A) and climate condition (B) on kinaesthetic

Table 1

 1 Subjective assessments on feeling of fatigue and thermal comfort and in neutral and tropical

	climates
	Climate condition	Tropical climate	Neutral climate
	Feeling of fatigue [fully alert (1); very lively (2); okay (3); a little tired (4); moderately tired
	(5); extremely tired (6); completely exhausted (7)]
	Mean, (SD), [CI]	4.32 (1.01) [4.10, 4.56]	3.18 (1.09) [2.93, 3.43]
	Anova, (effect size)	F(1, 94) = 70.81**, (η2 = 0.42)
	Thermal comfort [comfortable (1); slightly uncomfortable; (2) uncomfortable; (3) very
	uncomfortable (4)]
	Mean, (SD), [CI]	3.74 (0.81) [3.65, 3.85]	1.04 (0.22) [1.00, 1.11]
	Anova, (effect size)	F(1, 94) = 2457.90**, (η2 = 0.99)
	Thermal sensation [hot (3); warm (2); slightly warm (1); neutral (0); slightly cool (-1); cool
	(-2); cold (-3)]
	Mean, (SD), [CI]	2.71 (0.57) [2.56, 2.82]	-0.39 (0.69) [-0.53, -0.22]
	Anova, (effect size)	F(1, 94) = 945.63**, (η2 = 0.90)

Acknowledgments

The authors would like to acknowledge all the participants of the study.

Conflicts of Interest

None declared

Figure 4