History-dependent operators and prox-regular sweeping processes

We consider an abstract inclusion in a real Hilbert space, governed by an almost history-dependent operator and a time-dependent multimapping with prox-regular values. We establish the unique solvability of the inclusion under appropriate assumptions on the data. The proof is based on the arguments of monotonicity, fixed point, and prox-regularity. We then use our result in order to deduce some direct consequences, including an existence and uniqueness result for a class of sweeping processes associated with prox-regular sets. Finally, we provide an example in a finite dimensional case inspired by a rheological model in solid mechanics.


Introduction
A large variety of boundary valued problems arising in mechanics, physics, and engineering sciences lead, in a weak formulation, to nonlinear inclusions. Their solvability involves arguments coming from nonlinear, set-valued, convex, and nonsmooth analysis, among others. Currently, there is a growing interest in the solution of inclusions governed by a special class of operators, the so-called almost history-dependent operators. Such kind of problems arise in the study of different constitutive laws used in the viscoelasticity and viscoplasticity. They also describe the frictional or frictionless contact between a deformable body and an obstacle. References in the field include [1,5,19,20]. There, existence and uniqueness results have been provided by using a fixed point theorem for almost historydependent operators.
Nevertheless, the inclusions studied in the previously cited papers have been associated with a family of convex sets. Removing the convexity in the study of the corresponding inclusions leads to important mathematical difficulties and gives rise to new and challenging mathematical problems. This can be achieved through the class of prox-regular sets (also known as positively reached, weakly convex, O(2)-convex, ϕ-convex, proximally smooth (see, e.g., [11] and the references therein)). Recall that a closed set is said to be prox-regular [23] provided that its metric projection is single-valued and continuous on a suitable enlargement of the set. Prox-regular sets share important properties with con- where the indices i, j run between 1 and d and the summation convention over repeated indices is used. Consider a constitutive law derived by using the following rheological arguments.
1) The model is obtained by connecting in series an elastic model with a viscoelastic (or viscoplastic) model. Then, at each moment t in the interval of interest I, the strain field satisfies the equality ε(t) = ε 1 where ε 1 and ε 2 represent the strain field in the elastic and the viscoelastic (or viscoplastic) model, respectively. We refer to ε 1 and ε 2 as the "regular" and "irregular" strain.
2) The regular strain satisfies the equality where σ = σ (t) denotes the stress field and B : S d → S d represents the compliance operator which could be nonlinear. This operator is supposed to be inversible, and its inverse will be denoted by A, i.e., B -1 = A.
3) On the other hand, we assume that the irregular strain field is such that ε 2 (t) ∈ N C(t); σ (t) + Rσ (t) , where C(t) is a subset of S d to be defined and R is a memory operator. A popular example of such an operator is given by where D(·) denotes a given relaxation tensor. Moreover, for any ω ∈ S d , notation N(C(t), ω) represents a set of S d which depends on C(t). Note that (3) shows that at each time moment t the irregular strain depends on the current value of the stress (i.e., σ (t)) and the history of the stress process (described by the term Rσ (t)). We now combine relations (1)-(3) to deduce that A concrete example of constitutive law of the form (4) can be obtained by taking R ≡ 0 and N(C(t), ω) = ∂ψ C(t) (ω) for any ω ∈ S d , where C(t) ⊂ S d is a given convex set (say the von Mises convex) and ∂ψ C(t) represents the Moreau-Rockafellar subdifferential of the indicator function (in the sense of convex analysis) ψ C(t) of the set C(t). This leads to the well-known Hencky law see, e.g., [22,26] and the references therein. Note that in (5) we assume that the convex C is time-dependent, and this could arise when C depends on the temperature field, for instance. Now, we are looking for a stress function σ which, applied to the deformable body, keeps it in equilibrium, i.e., the strain field vanishes. Then, at each moment t ∈ I, we have -Bσ (t) ∈ N C(t); σ (t) + Rσ (t) and, using the notation ω = ε 1 combined with equalities (2) and B -1 = A, we find that -ω(t) ∈ N C(t); Aω(t) + RAω(t) . Therefore, with the notation RAω(t) = Sω(t) we arrive at the following inclusion problem.

Problem 1 Find a regular strain function
Motivated by the above mechanical problem, in this paper we shall study inclusions of the form (6) in the abstract framework of real Hilbert spaces, under the assumption that C(t) represents a family of prox-regular sets and S is an almost history-dependent operator.
The rest of the manuscript is organized as follows. In Sect. 2 we recall some notation and preliminaries which are used in the rest of the paper. In Sect. 3 we introduce the abstract history-dependent inclusion and state the main existence and uniqueness result,

Notation and preliminaries
In the whole paper, all vector spaces will be real vector spaces. We use R + for the set of nonnegative reals, that is, R + := [0, +∞). The letter T stands for an extended nonnegative real, i.e., T ∈ R + ∪ {+∞} and I := [0, T] ∩ R + . In what follows X is a (real) Hilbert space endowed with its inner product (·, ·) X and its associated norm · X . The open (resp. closed) ball with respect to the norm · X centered at x ∈ X with radius r > 0 is denoted by The letter U X (resp. B X ) stands for the open (resp. closed) unit ball of X centered at the origin 0 X , that is, U X := B(0 X , 1) (resp. B X := B[0 X , 1]). The strong and weak convergences in X will be denoted by → and , respectively, and are considered as n → ∞, even if we do not mention it explicitly. Recall that · X enjoys the so-called sequential Kadec-Klee property, that is, every sequence (x n ) n ⊂ X satisfying x n x along with x n X → x X for some x ∈ X converges strongly to x.
Projections and nonlinear operators The metric projection multimapping Proj S : X ⇒ X associated with a nonempty subset S ⊂ X is defined as where d S (·) (or d(·, S)) is the distance function from S, that is, When the set Proj S (x) is reduced to a singleton for some vector x ∈ X, we say that the metric projection of x on S is well defined. In such a case, the unique element of Proj S (x) is denoted by proj S (x) or P S (x). It is an exercise to check that, for any x, x ∈ X, for all y ∈ S. (7) It is known (and not difficult to establish) that the multimapping Proj S (·) is monotone, that is, for every x 1 , x 2 ∈ X, p 1 ∈ Proj S (x 1 ), and p 2 ∈ Proj S (x 2 ).
In the development below, the concept of strong monotonicity of operators will be needed. Recall that an operator A : X → X is said to be strongly monotone with constant m A > 0 provided that Operators enjoying the Lipschitz property will be also used. We say that the operator A : The following result on strongly monotone Lipschitz continuous operators will be crucial in our study. Proposition 2.1 Let A : X → X be a strongly monotone Lipschitz continuous operator with respective constants m A > 0 and L A > 0. Then A : X → X is invertible, and its inverse A -1 : X → X is also strongly monotone and Lipschitz continuous with respective m A

respectively.
For the proof of Proposition 2.1, we refer to [25,Theorem 1.24].
Proximal normal cone We now assume that S ⊂ X and U is a nonempty open subset of X. Then the proximal normal cone to S at x ∈ X is defined as the set otherwise.
For each x ∈ S, it is known that N(S; x) is a convex cone (not necessarily closed), containing 0 X . Further, it can be checked that, for given (v, for all x ∈ S. From the above definition, it is not difficult to see that that for any v ∈ X with Proj S (v) = ∅ the following inclusion holds: vw ∈ N(S; w) for all w ∈ Proj S (v).
We conclude this part devoted to the proximal normal cone by recalling that if S is convex, then the following equality holds: Prox-regular sets We now recall the notion of prox-regular sets in Hilbert spaces. For historical comments, proofs, and further results, we refer to the survey by G. Colombo and L. Thibault [11] (see also the forthcoming monograph [28]) and the references therein.

Definition 2.2
Let S be a nonempty closed subset of X, and let r ∈ (0, +∞]. One says that S is r-prox-regular (or uniformly prox-regular with constant r) whenever for all x ∈ S, v ∈ N(S; x) ∩ B X and t ∈ (0, r] one has Concerning this definition we have the following comments. First, note that if S is rprox-regular, then it is r -prox-regular for any 0 < r < r. Further, it is known that the class of ∞-prox-regular subsets of X is nothing but the class of nonempty closed convex sets of X. Given a nonempty subset S ⊂ X, we denote by U r (S) and Enl r (S) the r-open and closed enlargement of S, that is, Moreover, if r := +∞, we set 1/r := 0 and U r (S) := X.
The following theorem provides some useful characterizations and properties of uniform prox-regular sets.

Theorem 2.3
Let S be a nonempty closed subset of X. The following assertions are equivalent for any extended real r ∈ (0, +∞]. (a) The set S is r-prox-regular.
(d) For any 0 < s < r, proj S (x) is well defined for every x ∈ U s (S) and the mapping proj S (·) is (1s/r) -1 -Lipschitz continuous therein, i.e., Let N be any of the normal cones in the sense of the Fréchet, Mordukhovich, or Clarke (see, e.g., [9,16,28] for the definitions and basic properties). It is known that assertions (b) and (c) with the truncated normal cone N (S; ·) ∩ B X in place of the truncated proximal one N(S; ·) ∩ B X are also equivalent to the r-prox-regularity of S. Further, any r-prox-regular set S enjoys the following normal regularity: Moreover, taking r = ∞ in Theorem 2.3 leads to the following result. (a) The set S is convex.
We now proceed with two results strongly involved in the proof of our main theorem below. The first one is related to inclusion (10) for prox-regular sets.
The second result deals with some convergence properties of prox-regular sets. Lemma 2.6 Let (S n ) n∈N be a sequence of r-prox-regular subsets of X for some r ∈ (0, +∞], and let also S be an r-prox-regular subset of X. Then, for every x ∈ U r (S) such that d(x, S n ) → d(x, S), one has that proj S n (x) is well defined for n ∈ N large enough and According to Theorem 2.3, the functions f n and f are convex and Fréchet differentiable on V . Moreover, for each y ∈ V . Set v n := x -proj S n (x) for every n ≥ N . Note that the sequence (v n ) n≥N is bounded since, by assumption, (d S n (x)) n≥N converges in R. Let (v s(n) ) n≥N be any weakly convergent subsequence of the sequence (v n ) n≥N . Let v be its limit with respect to the weak topology on X. Fix any z ∈ V . Keeping in mind that f s(n) (·) is a convex function, for each integer n ≥ N , we may write Then, passing to the limit as n → ∞, we find that Coming back to (11), we see that v = x -proj S (x). Therefore, the whole sequence (v n ) n≥N converges weakly in X to x -proj S (x). On the other hand, we obviously have These two ingredients allow us to apply the Kadec-Klee property of the norm of X to obtain the strong convergence v n → x -proj S (x) in X. It results from the above that the whole sequence (v n ) n≥N converges to x -proj S (x) in X, which means that proj S n (x) → proj S (x) in X. The proof is then complete.
Examples and counter-examples Theorem 2.3 shows that prox-regular and convex sets share many properties, including the differentiability of distance function, the existence of nearest points, and (hypo)monotonicity of normals, among others. This naturally led several authors to study preservation of prox-regularity under various set operations. In what follows we shall use an example based on the following general result.
Remark 2.8 For the convenience of the reader, we also provide the following counterexamples. a) [11] Given real r > 0, there is an r-prox-regular set S of R 2 such that Q := S ∩ R × {0} fails to be uniformly (even locally !) prox-regular. b) The inverse image of a uniformly prox-regular set by a continuous linear mapping may fail to be prox-regular. Indeed, the above sets Q and S satisfy A -1 (Q) = S with A : R → R 2 defined by c) [4, Example 7] The direct image of a uniformly prox-regular set by a continuous linear mapping may fail to be prox-regular. d) [2] The sublevel {f ≤ 0} (resp. the level {f = 0}) is not prox-regular even for smooth functions. This can be seen in a straightforward way with the function f : Finally, we recall that, despite the above counter-examples, sufficient conditions ensuring the prox-regularity of C := {f 1 ≤ 0, . . . , f p ≤ 0, h 1 = 0, . . . , h q = 0} are developed in [2] in the framework of Hilbert spaces.

History-dependent and almost history-dependent operators
It is well known that C([0, T]; Y ) is a Banach space whenever Y is a Banach space. The case I = R + leads to the space C(R + ; Y ). If Y is a Banach space, then C(R + ; Y ) can be organized in a canonical way as a Fréchet space, i.e., a complete metric space in which the corresponding topology is induced by a countable family of seminorms.
The vector space of continuously differentiable functions on I with values in Y is denoted by C 1 (I; Y ). Obviously, for any function v : I → Y , the inclusion v ∈ C 1 (I; Y ) holds if and only if v ∈ C(I; Y ) andv ∈ C(I; Y ). Here and in what follows,v(·) stands for the derivative of the function v(·). For a function v ∈ C 1 (I; Y ), the following equality will be used in the next section of this manuscript: Everywhere below, given two normed spaces Y and Z and an operator S : C(I; Y ) → C(I; Z), for any function u ∈ C(I; X), we use the shorthand notation Su(t) to represent the value of the function Su at the point t ∈ I, that is, Su(t) := (Su)(t).
We end this section with two important classes of operators defined on the space of continuous functions.
The next fixed point result makes clear the interest of such operators.

Problem statement and main results
In this section we state an existence and uniqueness result for a time-dependent inclusion involving nonlinear operators. To this end we consider a nonempty closed bounded subset K ⊂ X, a multimapping C : I ⇒ X, and two operators A : X → X and S : C(I; X) → C(I; K). As usual, Im(C) denotes the range of C(·), that is, With the above data and notation at hand, we introduce the following inclusion problem.

Problem 2 Find a continuous function u
In the study of (14) we consider the following assumptions. (C) The multimapping C : I ⇒ X has r-prox-regular values for some real r ∈ (0, +∞] and, for every t ∈ I and every sequence (t n ) n≥1 of I converging to t, one has d u, C(t n ) → d u, C(t) for all u ∈ U r C(t) . (S) For any nonempty compact set J ⊂ I, there exist l S J > 0 and L S J > 0 such that, for all u 1 , u 2 ∈ C(I; X) and t ∈ J , inequality (13) holds. Note that, using Lemma 2.6, it follows that the Wijsman-type convergence (15) is equivalent to the convergence in X proj S n (x) → proj S (x) for all x ∈ U r (S).
Our main result in the study of Problem 2 that we state here and prove in the next section is the following.
In addition, assume that for any nonempty compact set J ⊂ I the following smallness condition holds: Then Problem 2 has at least a solution u(·). Moreover, the solution takes values in sB X := {sb : b ∈ B X } and is the unique solution of Problem 2 with this property.
Remark 3.2 As mentioned in the introduction, Problem 2 has been already studied in [19] under the assumption that C(t) is a nonempty closed convex moving set (that is, ∞-proxregular). It should be noted that if r = ∞, then estimate (18) obviously holds, (17) means that the operator A -1 is bounded on the set Im(C) -K , and (19) becomes This inequality plays a crucial role for the well-posedness of Problem 2 driven by a convex set C(t) studied in [19]. There, the set K is possibly unbounded (say K = X), and there is no need to assume the boundedness of the operator A -1 .
In the rest of this section we present some consequences of Theorem 3.1. Proof Since S is a history-dependent operator, Definition 2.9(a) guarantees that condition (S) holds with l S J = 0 for any compact J ⊂ I. We deduce from here that in this case the smallness condition (19) is satisfied. Therefore, Corollary 3.3 is a direct consequence of Theorem 3.1.
Theorem 3.1 allows us to obtain an existence and uniqueness result for a first order sweeping process. To present it, besides the data C, A, and S and their associated assumptions (C), (A), and (S), respectively, we consider an operator B : X → K and an element u 0 such that: We are now in a position to introduce the following sweeping process.

Problem 3 Find a continuously differentiable function
Our first result in this section is the following.
Next, we consider the auxiliary problem of finding a function v : Let L B > 0 be a Lipschitz constant of the operator B. We use assumptions (S) and (B) to see that, for any nonempty compact set J ⊂ I, any functions v 1 , v 2 ∈ C(I; X), and any t ∈ I, the following inequality holds: Therefore, we are in a position to apply Theorem 3.1 in order to obtain the existence of a unique function v ∈ C(I; X) with values in sB X , which satisfies the time-dependent inclusion (21). Denote by u : I → X the function defined by Then (20)- (22) imply that u is a solution of Problem 3 with regularity u ∈ C 1 (I; X). This proves the existence part of the theorem. The uniqueness part follows from the unique solvability of auxiliary problem (21), guaranteed by Theorem 3.1.
A direct consequence of Corollary 3.4 is the following. The proof of Corollary 3.5 follows from arguments similar to those used in the proof of Corollary 3.3 and, therefore, we skip it.

Proof of Theorem 3.1
The proof of Theorem 3.1 will be carried out in several steps that we present below. We start with a fixed point result for the projection mapping on a prox-regular set.
Then the following statements hold for any η ∈ K .
(b) We claim that is a contraction on C. First, note that On the other hand, an elementary computation shows that Combining the above equivalences with inclusion s ∈ (0, β], we deduce that κ 2 (1δ) < 1, which is the desired inequality. Therefore, the mapping is a contraction on the nonempty closed subset C of the Hilbert space X with constant κ(1δ) 1 2 . From the arbitrariness of s ∈ (ρα, β], it is easy to see that is a contraction on C with constant (1ρα/r) -1 (1δ) 1 2 . (c) The classical Banach fixed point theorem then guarantees the existence of unique z η ∈ C such that z η = z η . Now, putting together this equality and the inclusion (10), we get -ρB(z ηη) ∈ N(C; z η ) or, equivalently (keeping in mind that N(C; z η ) is a cone in X), It remains to observe that (23) and assumption ρ ∈ (0, 1) (coming from inequality m B < L 2 B ) imply that This inclusion and Lemma 2.5 entail that which concludes the proof.
Then, for any η ∈ C(I; K), there exists a unique continuous function z η : I → X such that Proof Let η ∈ C(I; K). Thanks to Lemma 4.1, we know that for every t ∈ I there exists a unique element z η (t) ∈ C(t) such that This justifies the claimed existence and uniqueness property. It remains to establish that z η (·) is a continuous function. Fix t ∈ I and consider a sequence (t n ) n∈N of elements of I which converges to t. Due to the closedness of I, we obviously have t ∈ I. For each n ∈ N, denote C n := C(t n ), η n := η(t n ), ζ n := z η (t n ), and ω n := ζ n -ρB(ζ nη n ). Set also C ∞ := C(t), η ∞ := η(t), ζ ∞ := z η (t), and ω ∞ := ζ ∞ -ρB(ζ ∞η ∞ ). With the above notation at hand, it is clear that for every integer n ∈ N we have ζ ∞ = proj C ∞ (ω ∞ ) and ζ n = proj C n (ω n ), hence, We now estimate each of the two terms in the right-hand side of (27). We start by setting u ∞ := ζ ∞η ∞ and u n := ζ nη n for all n ∈ N.
It is readily seen that Set δ := ρm B . Fix any real ε > 0 with ρα + ε < β and let s ∈ (ρα + ε, β). Since ζ ∞ ∈ C ∞ ⊂ U r (C ∞ ), we can use assumption (C) to see that d(ζ ∞ , C n ) → d(ζ ∞ , C ∞ ) = 0. Thus, we can find some integer N ≥ 1 such that Fix for a moment an integer n ≥ N . We easily observe that On the other hand, Lemma 4.1(a) guarantees that d C n (ω n ) ≤ ρα < s and the r-prox-regularity of C n implies that with κ := (1s/r) -1 . Using this inequality, the definition of ω ∞ , ω n , and (28), we see that Moreover, using the m B -strong monotonicity of B and its L B -Lipschitz property yields or, equivalently, Finally, letting L := κ √ 1δ and taking into account inequalities (31), (30), and (29) it follows that Noting that L ∈ (0, 1) (see Lemma 4.1(b)) and coming back to inequality (27), we see that Next, using inequality it follows that ω ∞ ∈ U r (C ∞ ). Therefore, using assumption (C) and Theorem 2.6, we obtain that It remains to use (32) and the continuity of the function η : I → K to see that ζ n = z(t n ) → z(t) = ζ ∞ in X, as n → ∞. This shows that the function z η : I → X is continuous and concludes the proof.
The next step is the following. -Lipschitz continuous. Note that m A -1 < min{L A -1 , L 2 A -1 } and, moreover, Let η ∈ C(I; K) and denote by z η ∈ C(I; X) the function obtained in Lemma 4.2 with B := A -1 . Then It follows from the definition of proximal normal cone that z η (t)u η (t)z η (t) = -u η (t) ∈ N C(t); z η (t) for all t ∈ I, and this concludes the proof of the existence part of the lemma. Now, let u 1 , u 2 : I → X be two functions such that -u 1 (t) ∈ N C(t); Au 1 (t) + η(t) ∩ sB X andu 2 (t) ∈ N C(t); Au 2 (t) + η(t) ∩ sB X for every t ∈ I. Fix any t ∈ I. Then, for each i ∈ {1, 2}, we have This implies that and adding these inequalities yields Proof Let η * ∈ C(I; K) be the fixed point of the operator , and let u := u η ∈ C(I; X) be the function given by Lemma 4.3 with η := η . So, we have -u (t) ∈ N C(t); Au(t) + η(t) ∩ sB X for all t ∈ I.
This inclusion combined with equality η = η = Su implies that -u (t) ∈ N C(t); Au (t) + Su (t) ∩ sB X for all t ∈ I, which shows that u is a solution to Problem 2. This proves the existence part of Theorem 3.1. The uniqueness part is a direct consequence of the uniqueness of the fixed point of the operator .

An example
In this section we provide an example of Problem 2 for which our abstract results work. To this end we consider two elements a 1 , a 2 ∈ S d and three real constants g 1 , g 2 , and k such that a 1a 2 > g 1 + k, (38) a 1 + g 1 a 1 -a 2g 1g 2 < √ 7 32 .
We now introduce the sets For the sake of simplicity, we only consider the setting where C(t) is autonomous, i.e., independent of time t. Nevertheless, we mention that the result below in this section can be easily extended to the case when g 1 , g 2 are real-valued positive functions depending on t ∈ I and, in this case, the set C(t) will depend on t. We denote by P K : S d → K the projection operator on the closed convex set K and consider the following inclusion problem.
Problem 4 Find a continuous function u : I → X such that -u(t) ∈ N C(t); u(t) + P K t 0 u(s) ds for all t ∈ I.
We have the following existence and uniqueness result.
Therefore, Lemma 2.7 guarantees that for each t ∈ I the set C(t) is r-prox-regular with r = 1 2 a 1 -a 2g 1g 2 .