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Historydependent operators and proxregular sweeping processes
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2022, Article number: 5 (2022)
Abstract
We consider an abstract inclusion in a real Hilbert space, governed by an almost historydependent operator and a timedependent multimapping with proxregular 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 proxregularity. 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 proxregular sets. Finally, we provide an example in a finite dimensional case inspired by a rheological model in solid mechanics.
1 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, setvalued, 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 socalled almost historydependent 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 proxregular sets (also known as positively reached, weakly convex, \(\mathcal{O}(2)\)convex, φconvex, proximally smooth (see, e.g., [11] and the references therein)). Recall that a closed set is said to be proxregular [23] provided that its metric projection is singlevalued and continuous on a suitable enlargement of the set. Proxregular sets share important properties with convex sets, including smoothness of distance function, hypomonotonicity of normals, and global Lipschitz property of metric projection (see Sect. 2). Proxregularity property has been recognized as a key concept in variational analysis and its applications [16, 24, 28]. The good behavior of metric projection and distance function for proxregular sets is extensively involved in the study of Moreau’s sweeping processes (that is, differential inclusions driven by the normal cone of a moving set [15, 17, 18]) as shown in [3, 6–8, 10, 12–14, 21, 27, 29] and the references therein. However, at the best of our knowledge, there are few literature works concerning historydependent inclusions governed by proxregular sets.
The aim of this current paper is to fill this gap. Indeed, here we consider a version of the historydependent inclusion considered in [19], associated with a family of proxregular sets. We provide an existence and uniqueness result and, to this end, we use various properties of the proxregular sets, including the properties of the projection operators and proximal normal cone. Extending the results in [19] to the nonconvex case represents the main trait of novelty in our work.
The structure of the inclusion we consider in this paper is motivated by the study of constitutive laws in solid mechanics, as we show in what follows. Let \(d\in \{2,3\}\) and denote by \(\mathbb{S}^{d}\) the space of second order symmetric tensors on \(\mathbb{R}^{d}\) or, equivalently, the space of symmetric matrices of order d. Recall that the inner product and the Euclidean norm on \(\mathbb{S}^{d}\) are defined by
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
where \(\boldsymbol{\varepsilon }_{1}\) and \(\boldsymbol{\varepsilon }_{2}\) represent the strain field in the elastic and the viscoelastic (or viscoplastic) model, respectively. We refer to \(\boldsymbol{\varepsilon }_{1}\) and \(\boldsymbol{\varepsilon }_{2}\) as the “regular” and “irregular” strain.
2) The regular strain satisfies the equality
where \(\boldsymbol{\sigma }=\boldsymbol{\sigma }(t)\) denotes the stress field and \(B:\mathbb{S}^{d}\to \mathbb{S}^{d}\) 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
where \(C(t)\) is a subset of \(\mathbb{S}^{d}\) to be defined and \({\mathcal{R}}\) is a memory operator. A popular example of such an operator is given by
where \({\mathcal{D}}(\cdot )\) denotes a given relaxation tensor. Moreover, for any \(\boldsymbol{\omega }\in \mathbb{S}^{d}\), notation \(N(C(t),\boldsymbol{\omega })\) represents a set of \(\mathbb{S}^{d}\) 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., \(\boldsymbol{\sigma }(t)\)) and the history of the stress process (described by the term \(\mathcal{R}\boldsymbol{\sigma }(t)\)).
We now combine relations (1)–(3) to deduce that
A concrete example of constitutive law of the form (4) can be obtained by taking \(\mathcal{R}\equiv 0\) and \(N(C(t),\boldsymbol{\omega })=\partial \psi _{C(t)}( \boldsymbol{\omega })\) for any \(\boldsymbol{\omega }\in \mathbb{S}^{d}\), where \(C(t)\subset \mathbb{S}^{d}\) is a given convex set (say the von Mises convex) and \(\partial \psi _{C(t)}\) represents the Moreau–Rockafellar subdifferential of the indicator function (in the sense of convex analysis) \(\psi _{C(t)}\) of the set \(C(t)\). This leads to the wellknown Hencky law
see, e.g., [22, 26] and the references therein. Note that in (5) we assume that the convex C is timedependent, 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 \(t\in I\), we have
and, using the notation \(\boldsymbol{\omega }=\boldsymbol{\varepsilon }_{1}\) combined with equalities (2) and \(B^{1}=A\), we find that
Therefore, with the notation \(\mathcal{R}A\boldsymbol{\omega }(t)= \mathcal{S}\boldsymbol{\omega }(t)\) we arrive at the following inclusion problem.
Problem 1
Find a regular strain function \(\boldsymbol{\omega }:I\to \mathbb{S}^{d}\) such that
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 proxregular sets and \(\mathcal{S}\) is an almost historydependent 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 historydependent inclusion and state the main existence and uniqueness result, Theorem 3.1, together with its consequences. The proof of the theorem is given in Sect. 4 based on a fixed point argument. Finally, in Sect. 5 we provide an example of Problem 1 for which our abstract results work.
2 Notation and preliminaries
In the whole paper, all vector spaces will be real vector spaces. We use \(\mathbb{R}_{+}\) for the set of nonnegative reals, that is, \(\mathbb{R}_{+}:=[0,+\infty )\). The letter T stands for an extended nonnegative real, i.e., \(T\in \mathbb{R}_{+}\cup \{ +\infty \} \) and \(I:=[0,T]\cap \mathbb{R_{+}}\). In what follows X is a (real) Hilbert space endowed with its inner product \((\cdot ,\cdot )_{X}\) and its associated norm \(\ \cdot \_{X}\). The open (resp. closed) ball with respect to the norm \(\\cdot \_{X}\) centered at \(x\in X\) with radius \(r>0\) is denoted by \(B(x,r)\) (resp. \(B[x,r]\)). The letter \(\mathbb{U}_{X}\) (resp. \(\mathbb{B}_{X}\)) stands for the open (resp. closed) unit ball of X centered at the origin \(0_{X}\), that is, \(\mathbb{U}_{X}:=B(0_{X},1)\) (resp. \(\mathbb{B}_{X}:=B[0_{X},1]\)). The strong and weak convergences in X will be denoted by → and ⇀, respectively, and are considered as \(n\to \infty \), even if we do not mention it explicitly. Recall that \(\\cdot \_{X}\) enjoys the socalled sequential Kadec–Klee property, that is, every sequence \((x_{n})_{n}\subset X\) satisfying \(x_{n}\rightharpoonup x\) along with \(\x_{n}\_{X}\rightarrow \x\_{X}\) for some \(x\in X\) converges strongly to x.
Projections and nonlinear operators
The metric projection multimapping \(\mathrm{Proj}_{S}:X\rightrightarrows X\) associated with a nonempty subset \(S\subset X\) is defined as
where \(d_{S}(\cdot )\) (or \(d(\cdot ,S)\)) is the distance function from S, that is,
When the set \(\mathrm{Proj}_{S}(\overline{x})\) is reduced to a singleton for some vector \(\overline{x}\in X\), we say that the metric projection of x̅ on S is well defined. In such a case, the unique element of \(\mathrm{Proj}_{S}(\overline{x})\) is denoted by \(\mathrm{proj}_{S}(\overline{x})\) or \(P_{S}(\overline{x})\). It is an exercise to check that, for any \(x,x'\in X\),
It is known (and not difficult to establish) that the multimapping \(\mathrm{Proj}_{S}(\cdot )\) is monotone, that is,
for every \(x_{1},x_{2}\in X\), \(p_{1}\in \mathrm{Proj}_{S}(x_{1})\), and \(p_{2}\in \mathrm{Proj}_{S}(x_{2})\).
In the development below, the concept of strong monotonicity of operators will be needed. Recall that an operator \(A:X\rightarrow X\) 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 \(A:Y\rightarrow X\) is Lipschitz continuous with constant \(L_{A}>0\) on \(Y\subset X\) provided that
The following result on strongly monotone Lipschitz continuous operators will be crucial in our study.
Proposition 2.1
Let \(A:X\to X\) be a strongly monotone Lipschitz continuous operator with respective constants \(m_{A}>0\) and \(L_{A}>0\). Then \(A:X\to X\) is invertible, and its inverse \(A^{1}:X\to X\) is also strongly monotone and Lipschitz continuous with respective \(\frac{m_{A}}{L_{A}^{2}}\) and \(\frac{1}{m_{A}}\), respectively.
For the proof of Proposition 2.1, we refer to [25, Theorem 1.24].
Proximal normal cone
We now assume that \(S\subset X\) and U is a nonempty open subset of X. Then the proximal normal cone to S at \(x\in X\) is defined as the set
For each \(x\in S\), 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 \((v,x)\in X\times S\), the inclusion \(v\in N(S;x)\) holds if and only if there is some real \(\sigma \geq 0\) such that
From the above definition, it is not difficult to see that that for any \(v \in X\) with \(\mathrm{Proj}_{S}(v)\neq \emptyset \) the following inclusion holds:
We conclude this part devoted to the proximal normal cone by recalling that if S is convex, then the following equality holds:
Proxregular sets
We now recall the notion of proxregular 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 \(r\in (0,+\infty ]\). One says that S is rproxregular (or uniformly proxregular with constant r) whenever for all \(x\in S\), \(v\in N(S;x)\cap \mathbb{B}_{X}\) and \(t\in (0, r]\) one has
Concerning this definition we have the following comments. First, note that if S is rproxregular, then it is \(r'\)proxregular for any \(0< r'< r\). Further, it is known that the class of ∞proxregular subsets of X is nothing but the class of nonempty closed convex sets of X.
Given a nonempty subset \(S\subset X\), we denote by \(U_{r}(S)\) and \(\operatorname{Enl}_{r}(S)\) the ropen and closed enlargement of S, that is,
and
Moreover, if \(r:=+\infty \), we set \(1/r:=0\) and \(U_{r}(S):=X\).
The following theorem provides some useful characterizations and properties of uniform proxregular sets.
Theorem 2.3
Let S be a nonempty closed subset of X. The following assertions are equivalent for any extended real \(r\in (0,+\infty ]\).

(a)
The set S is rproxregular.

(b)
For all \(x_{1},x_{2}\in S\) and \(\zeta \in N(S;x_{1})\cap \mathbb{B}_{X}\), one has
$$ ( \zeta ,x_{2}x_{1} )_{X} \leq \frac{1}{2r} \Vert x_{1}x_{2} \Vert _{X}^{2}. $$ 
(c)
For all \(x_{1},x_{2}\in S\), \(\zeta _{1}\in N(S;x_{1})\cap \mathbb{B}_{X}\), and \(\zeta _{2}\in N(S;x_{2})\cap \mathbb{B}_{X}\), one has
$$ ( \zeta _{1}\zeta _{2},x_{1}x_{2} )_{X} \geq \frac{1}{r} \Vert x_{1}x_{2} \Vert _{X}^{2}. $$ 
(d)
For any \(0< s< r\), \(\mathrm{proj}_{S}(x)\) is well defined for every \(x\in U_{s}(S)\) and the mapping \(\mathrm{proj}_{S}(\cdot )\) is \((1s/r)^{1}\)Lipschitz continuous therein, i.e.,
$$ \bigl\Vert \mathrm{proj}_{S}(x)\mathrm{proj}_{S} \bigl(x'\bigr) \bigr\Vert _{X} \leq \frac{1}{1s/r} \bigl\Vert xx' \bigr\Vert _{X}\quad \textit{for all } x,x'\in U_{s}(S). $$ 
(e)
The function \(x\mapsto d_{S}^{2}(x)\) is \(C^{1,1}\) on \(U_{r}(S)\) and
$$ \nabla d_{S}^{2}(x)=2 \bigl(x\mathrm{proj}_{S}(x) \bigr)\quad \textit{for all } x\in U_{r}(S). $$ 
(f)
For any \(0< s< r\), the function \(x\mapsto d_{S}^{2}(x)+\frac{s}{rs}\x\_{X}^{2}\) is convex on any open convex subset of \(U_{s}(S)\).
Let \(\mathcal{N}\) 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 \(\mathcal{N}(S;\cdot )\cap \mathbb{B}_{X}\) in place of the truncated proximal one \(N(S;\cdot )\cap \mathbb{B}_{X}\) are also equivalent to the rproxregularity of S. Further, any rproxregular set S enjoys the following normal regularity:
Moreover, taking \(r=\infty \) in Theorem 2.3 leads to the following result.
Corollary 2.4
Let S be a nonempty closed subset of X. The following assertions are equivalent.

(a)
The set S is convex.

(b)
For all \(x_{1},x_{2}\in S\), \(\zeta _{1}\in N(S;x_{1})\), and \(\zeta _{2}\in N(S;x_{2})\), one has
$$ ( \zeta _{1}\zeta _{2},x_{1}x_{2} )_{X} \geq 0. $$ 
(c)
For every \(x\in X\), \(\mathrm{proj}_{S}(x)\) is well defined and the mapping \(\mathrm{proj}_{S}(\cdot )\) is 1Lipschitz continuous, i.e.,
$$ \bigl\Vert \mathrm{proj}_{S}(x)\mathrm{proj}_{S} \bigl(x'\bigr) \bigr\Vert _{X} \leq \bigl\Vert xx' \bigr\Vert _{X}. $$ 
(d)
The function \(x\mapsto d_{S}^{2}(x)\) is \(C^{1,1}\) on X and
$$ \nabla d_{S}^{2}(x)=2 \bigl(x\mathrm{proj}_{S}(x) \bigr) \quad \textit{for all } x\in X. $$ 
(e)
The function \(x\mapsto d_{S}^{2}(x)\) is convex on X.
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 proxregular sets.
Lemma 2.5
([3])
Let \(r\in (0,+\infty ]\), S be an rproxregular set of X, and let \(x,x'\in X\). If \(xx'\in N(S;x')\) and \(\Vert xx' \Vert _{X} \leq r\) (resp. \(\Vert xx' \Vert _{X}< r\)), then \(x'\in \mathrm{Proj}_{S}(x)\) (resp. \(x'=\mathrm{proj}_{S}(x)\)).
The second result deals with some convergence properties of proxregular sets.
Lemma 2.6
Let \((S_{n})_{n\in \mathbb{N}}\) be a sequence of rproxregular subsets of X for some \(r\in (0,+\infty ]\), and let also S be an rproxregular subset of X. Then, for every \(x\in U_{r}(S)\) such that \(d(x,S_{n})\rightarrow d(x,S)\), one has that \(\mathrm{proj}_{S_{n}}(x)\) is well defined for \(n\in \mathbb{N}\) large enough and
Proof
Fix any \(x\in U_{r}(S)\) and assume that \(d(x,S_{n})\rightarrow d(x,S)\). Take real \(0< s'< r\) such that \(d(x,S)< s'\) and take \(s\in \mathbb{R}\) such that \(s'< s< r\). Since \(d(x,S)< s'\), there is an integer \(N\in \mathbb{N}\) such that for every integer \(n\geq N\) one has \(d(x,S_{n})< s'\). Set \(V:=B(x,ss')\) and fix any integer \(n\geq N\). For every \(y\in V\), we have
so \(V\subset U_{s}(S_{n})\) and, similarly, \(V\subset U_{s}(S)\). Define the function \(f_{n}:V\rightarrow \mathbb{R}\) by
and the function \(f:V\rightarrow \mathbb{R}\) by
According to Theorem 2.3, the functions \(f_{n}\) and f are convex and Fréchet differentiable on V. Moreover,
for each \(y\in V\). Set \(v_{n}:=x\mathrm{proj}_{S_{n}}(x)\) for every \(n\geq N\). Note that the sequence \((v_{n})_{n\geq N}\) is bounded since, by assumption, \((d_{S_{n}}(x))_{n\geq N}\) converges in \(\mathbb{R}\). Let \((v_{s(n)})_{n\geq N}\) be any weakly convergent subsequence of the sequence \((v_{n})_{n\geq N}\). Let v be its limit with respect to the weak topology on X. Fix any \(z\in V\). Keeping in mind that \(f_{s(n)}(\cdot )\) is a convex function, for each integer \(n\geq N\), we may write
Then, passing to the limit as \(n\to \infty \), we find that
Since \(z\in V\) is arbitrary, we get \(2v+\frac{2s}{rs}x\in \partial f(x)= \{ \nabla f(x) \} \), i.e.,
Coming back to (11), we see that \(v=x\mathrm{proj}_{S}(x)\). Therefore, the whole sequence \((v_{n})_{n\geq N}\) converges weakly in X to \(x\mathrm{proj}_{S}(x)\). 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 \(v_{n} \to x\mathrm{proj}_{S}(x)\) in X. It results from the above that the whole sequence \((v_{n})_{n\geq N}\) converges to \(x\mathrm{proj}_{S}(x)\) in X, which means that \(\mathrm{proj}_{S_{n}}(x)\to \mathrm{proj}_{S}(x)\) in X. The proof is then complete. □
Examples and counterexamples
Theorem 2.3 shows that proxregular 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 proxregularity under various set operations. In what follows we shall use an example based on the following general result.
Lemma 2.7
([28])
If \(S_{1}\) and \(S_{2}\) are rproxregular sets of X for some \(r>0\), then \(S_{1}\cup S_{2}\) is \(\min \{g/2,r\}\)proxregular whenever \(g:=\inf_{(c_{1},c_{2})\in S_{1}\times S_{2}}\c_{1}c_{2}\>0\).
Remark 2.8
For the convenience of the reader, we also provide the following counterexamples.

a)
[11] Given real \(r>0\), there is an rproxregular set S of \(\mathbb{R}^{2}\) such that \(Q:=S\cap \mathbb{R}\times \{0\}\) fails to be uniformly (even locally !) proxregular.

b)
The inverse image of a uniformly proxregular set by a continuous linear mapping may fail to be proxregular. Indeed, the above sets Q and S satisfy \(A^{1}(Q)=S\) with \(A:\mathbb{R}\rightarrow \mathbb{R}^{2}\) defined by
$$ A(x):=(x,0) \quad \text{for all } x\in \mathbb{R}. $$ 
c)
[4, Example 7] The direct image of a uniformly proxregular set by a continuous linear mapping may fail to be proxregular.

d)
[2] The sublevel \(\{f\leq 0\}\) (resp. the level \(\{f=0\}\)) is not proxregular even for smooth functions. This can be seen in a straightforward way with the function \(f:\mathbb{R}^{2}\rightarrow \mathbb{R}\) defined by
$$ f(x,y):=xy\quad \text{for all } (x,y)\in \mathbb{R}^{2}. $$
Finally, we recall that, despite the above counterexamples, sufficient conditions ensuring the proxregularity of \(C:=\{f_{1}\leq 0,\ldots ,f_{p}\leq 0,h_{1}=0,\ldots ,h_{q}=0\}\) are developed in [2] in the framework of Hilbert spaces.
Historydependent and almost historydependent operators
For a normed space \((Y,\\cdot \_{Y})\), we denote by \(C(I;Y)\) the space of continuous functions defined on I with values in Y, i.e.,
The case \(T>0\) (i.e., \(I=[0,T]\)) leads to the space \(C([0,T];Y)\) which is a normed space equipped with the norm \(\\cdot \_{C([0,T];Y)}\) defined by
It is well known that \(C([0,T];Y)\) is a Banach space whenever Y is a Banach space. The case \(I=\mathbb{R}_{+}\) leads to the space \(C(\mathbb{R}_{+};Y)\). If Y is a Banach space, then \(C(\mathbb{R}_{+};Y)\) 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 \(v:I\rightarrow Y\), the inclusion \(v\in C^{1}(I;Y)\) holds if and only if \(v\in C(I;Y)\) and \(\dot{v}\in C(I;Y)\). Here and in what follows, \(\dot{v}(\cdot )\) stands for the derivative of the function \(v(\cdot )\). For a function \(v\in C^{1}(I;Y)\), 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 \(\mathcal{S}\colon C(I;Y)\to C(I;Z)\), for any function \(u\in C(I;X)\), we use the shorthand notation \(\mathcal{S}u(t)\) to represent the value of the function \(\mathcal{S}u\) at the point \(t\in I\), that is, \(\mathcal{S}u(t):=(\mathcal{S}u)(t)\).
We end this section with two important classes of operators defined on the space of continuous functions.
Definition 2.9
Let \((Y,\\cdot \_{Y})\) (resp. \((Z,\\cdot \_{Z})\)) be a normed space, and let K (resp. \(K'\)) be a nonempty closed subset of Y (resp. Z). An operator \(\mathcal{S}\colon C(I;K)\to C(I;K')\) is called:

a)
historydependent (h.d. for short), if for any nonempty compact set \(\mathcal{J}\subset I\), there exists \(L_{\mathcal{J}}^{\mathcal{S}}>0\) such that, for all \(u_{1},u_{2}\in C(I;K)\) and all \(t\in \mathcal{J}\),
$$ \bigl\Vert \mathcal{S}u_{1}(t)\mathcal{S}u_{2}(t) \bigr\Vert _{Z}\le L_{\mathcal{J}}^{\mathcal{S}} \int _{0}^{t} \bigl\Vert u_{1}(s)u_{2}(s) \bigr\Vert _{Y} \,ds. $$(12) 
b)
almost historydependent (a.h.d. for short), if for any nonempty compact set \(\mathcal{J}\subset I\), there exist \(l_{\mathcal{J}}^{\mathcal{S}}\in [0,1)\) and \(L_{\mathcal{J}}^{\mathcal{S}}>0\) such that, for all \(u_{1},u_{2}\in C(I;K)\) and all \(t\in \mathcal{J}\),
$$ \bigl\Vert \mathcal{S}u_{1}(t)\mathcal{S}u_{2}(t) \bigr\Vert _{Z}\le l_{\mathcal{J}}^{\mathcal{S}} \bigl\Vert u_{1}(t)u_{2}(t) \bigr\Vert _{Y}+L_{\mathcal{J}}^{\mathcal{S}} \int _{0}^{t} \bigl\Vert u_{1}(s)u_{2}(s) \bigr\Vert _{Y} \,ds. $$(13)
The next fixed point result makes clear the interest of such operators.
Theorem 2.10
Let K be a nonempty closed subset of a Banach space Y, and let \(\Lambda : C(I;K)\to C(I;K)\) be an almost historydependent operator. Then Λ has a unique fixed point, i.e., there exists a unique element \(\eta ^{*}\in C(I;K)\) such that \(\Lambda \eta ^{*}=\eta ^{*}\).
A proof of Theorem 2.10 can be found in [26, pp. 41–45]. There, the main properties of historydependent and almost historydependent operators are presented together with various examples and applications.
3 Problem statement and main results
In this section we state an existence and uniqueness result for a timedependent inclusion involving nonlinear operators. To this end we consider a nonempty closed bounded subset \(K\subset X\), a multimapping \(C:I\rightrightarrows X\), and two operators \(A:X\rightarrow X\) and \(\mathcal{S}:C(I;X)\rightarrow C(I;K)\). As usual, \(\operatorname{Im}(C)\) denotes the range of \(C(\cdot )\), that is,
With the above data and notation at hand, we introduce the following inclusion problem.
Problem 2
Find a continuous function \(u:I\to X\) such that
In the study of (14) we consider the following assumptions.
 \((\mathcal{C})\):

The multimapping \(C:I\rightrightarrows X\) has rproxregular values for some real \(r\in (0,+\infty ]\) and, for every \(t\in I\) and every sequence \((t_{n})_{n\geq 1}\) of I converging to t, one has
$$ d \bigl(u,C(t_{n}) \bigr)\rightarrow d \bigl(u,C(t) \bigr) \quad \text{for all } u\in U_{r} \bigl(C(t) \bigr). $$(15)  \((\mathcal{A})\):

The mapping A is \(m_{A}\)strongly monotone and \(L_{A}\)Lipschitz continuous for some reals \(m_{A},L_{A}>0\).
 \((\mathcal{S})\):

For any nonempty compact set \(\mathcal{J}\subset I\), there exist \(l_{\mathcal{J}}^{\mathcal{S}}>0\) and \(L_{\mathcal{J}}^{\mathcal{S}}>0\) such that, for all \(u_{1},u_{2}\in C(I;X)\) and \(t\in \mathcal{J}\), inequality (13) holds.
Note that, using Lemma 2.6, it follows that the Wijsmantype convergence (15) is equivalent to the convergence in X
Our main result in the study of Problem 2 that we state here and prove in the next section is the following.
Theorem 3.1
Assume \((\mathcal{C})\), \((\mathcal{A})\), and \((\mathcal{S})\). Assume also that
along with
In addition, assume that for any nonempty compact set \(\mathcal{J}\subset I\) the following smallness condition holds:
Then Problem 2has at least a solution \(u(\cdot )\). Moreover, the solution takes values in \(s\mathbb{B}_{X}:=\{sb:b\in \mathbb{B}_{X}\}\) and is the unique solution of Problem 2with 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 \(r=\infty \), then estimate (18) obviously holds, (17) means that the operator \(A^{1}\) is bounded on the set \(\operatorname{Im}(C)K\), and (19) becomes
This inequality plays a crucial role for the wellposedness 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.
Corollary 3.3
Assume \((\mathcal{C})\), \((\mathcal{A})\), (16), (17), and (18). Assume also that \(\mathcal{S}:C(I;X)\to C(I;K)\) is a historydependent operator. Then Problem 2has at least a solution \(u(\cdot )\). Moreover, the solution takes values in \(s\mathbb{B}_{X}:=\{sb:b\in \mathbb{B}_{X}\}\) and is the unique solution of Problem 2with this property.
Proof
Since \(\mathcal{S}\) is a historydependent operator, Definition 2.9(a) guarantees that condition \((\mathcal{S})\) holds with \(l_{\mathcal{J}}^{\mathcal{S}}=0\) for any compact \({\mathcal{J}}\subset I\). 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 \(\mathcal{S}\) and their associated assumptions (\(\mathcal{C}\)), (\(\mathcal{A}\)), and (\(\mathcal{S}\)), respectively, we consider an operator \(B:X\rightarrow K\) and an element \(u_{0}\) such that:
 \((\mathcal{B})\):

\(B:X\rightarrow X\) is a Lipschitz continuous operator with values in \(Y\subset X\).
 \((\mathcal{K})\):

\(Y+K\subset K\).
 \((\mathcal{U})\):

\(u_{0}\in X\).
We are now in a position to introduce the following sweeping process.
Problem 3
Find a continuously differentiable function \(u:I\to X\) such that
Our first result in this section is the following.
Corollary 3.4
Assume that \((\mathcal{C})\), \((\mathcal{A})\), \((\mathcal{S})\), \((\mathcal{B})\), \((\mathcal{K})\), \((\mathcal{U})\), (16), (17), (18), and (19). Then Problem 2has at least a solution \(u(\cdot )\). Moreover, its derivative \(\dot{u}(\cdot )\) takes values in \(s\mathbb{B}_{X}:=\{sb:b\in \mathbb{B}_{X}\}\) and \(u(\cdot )\) is the unique solution of Problem 3with this property.
Proof
We use assumption \((\mathcal{K})\) to introduce the operator \(\widetilde{\mathcal{S}}:C(I;X)\to C(I;K)\) defined by
Next, we consider the auxiliary problem of finding a function \(v:I\to X\) such that
Let \(L_{B}>0\) be a Lipschitz constant of the operator B. We use assumptions \((\mathcal{S})\) and \((\mathcal{B})\) to see that, for any nonempty compact set \(\mathcal{J}\subset I\), any functions \(v_{1}, v_{2}\in C(I;X)\), and any \(t\in I\), 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 \(v\in C(I;X)\) with values in \(s\mathbb{B}_{X}\), which satisfies the timedependent inclusion (21). Denote by \(u:I\to X\) the function defined by
Then (20)–(22) imply that u is a solution of Problem 3 with regularity \(u\in C^{1}(I;X)\). 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.
Corollary 3.5
Assume \((\mathcal{K})\), \((\mathcal{A})\), \((\mathcal{B})\), \((\mathcal{U})\), (16), (17), and (18). Assume also that \(\mathcal{S}:C(I;X)\to C(I;K)\) is a historydependent operator. Then Problem 2has at least a solution \(u(\cdot )\). Moreover, its derivative \(\dot{u}(\cdot )\) takes values in \(s\mathbb{B}_{X}:=\{sb:b\in \mathbb{B}_{X}\}\) and \(u(\cdot )\) is the unique solution of Problem 2with this property.
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.
4 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 proxregular set.
Lemma 4.1
Let C be an rproxregular subset of X for some \(r\in (0,+\infty )\), and let \(B:X\rightarrow X\) be an \(m_{B}\)strongly monotone and \(L_{B}\)Lipschitz continuous operator with \(m_{B}<\min \{L_{B},L_{B}^{2}\}\). Assume that
Then the following statements hold for any \(\eta \in K\).

(a)
For all \(z\in C\) and all \(\theta \in (0,1]\), one has \(z\theta B(z\eta )\in \mathrm{Enl}_{\theta \alpha }(C)\subset U_{r}(C)\).

(b)
For every \(s\in [\rho \alpha ,\beta ]\), the mapping \(C\ni z\mapsto \mathrm{proj}_{C} (z\rho B(z\eta ) )\) is a contraction on C of constant \((1s/r)^{1}(1m_{B}\rho )^{1/2}\).

(c)
There exists a unique element \(z_{\eta }\in C\) such that
$$ z_{\eta }=\mathrm{proj}_{C} \bigl(z_{\eta }\rho B(z_{\eta }\eta ) \bigr)= \mathrm{proj}_{C} \bigl(z_{\eta }B(z_{\eta }\eta ) \bigr). $$(24)
Proof
(a) Fix \(\eta \in K\). Note that inclusion \((a)\) follows directly from the estimates
valid for every \(z\in C\) and every \(\theta \in (0,1]\). This allows us to consider the mapping \(\Lambda :C\to C\) defined by
where \(\delta :=m_{B} \rho =\frac{m_{B}^{2}}{L_{B}^{2}}\in (0,1)\).
(b) We claim that Λ is a contraction on C. First, note that
Pick any \(s\in (\rho \alpha ,\beta ]\). We have \(z\rho B(z\eta )\in \operatorname{Enl}_{\rho \alpha }(C)\subset U_{s}(C)\) for all \(z\in C\). Fix any \(z_{1}, z_{2}\in C\) and set \(u_{i}:=z_{i}\eta \) for each \(i\in \{1,2\}\). Using the definition of Λ in (25) combined with the \(\kappa :=(1s/r)^{1}\)Lipschitz property of \(\mathrm{proj}_{C}(\cdot )\) on the sopen enlargement \(U_{s}(C)\) of C, the \(m_{B}\)strong monotonicity of B, and (26), we see that
On the other hand, an elementary computation shows that
with \(\gamma _{1}:=r(1+\sqrt{1\delta })>r(1\sqrt{1\delta })=:\gamma _{2}> \beta \). Combining the above equivalences with inclusion \(s\in (0,\beta ]\), we deduce that \(\kappa ^{2}(1\delta )<1\), which is the desired inequality. Therefore, the mapping Λ is a contraction on the nonempty closed subset C of the Hilbert space X with constant \(\kappa (1\delta )^{\frac{1}{2}}\). From the arbitrariness of \(s\in (\rho \alpha ,\beta ]\), it is easy to see that Λ is a contraction on C with constant \((1\rho \alpha /r)^{1} (1\delta )^{\frac{1}{2}}\).
(c) The classical Banach fixed point theorem then guarantees the existence of unique \(z_{\eta }\in C\) such that \(\Lambda z_{\eta }= z_{\eta }\). Now, putting together this equality and the inclusion (10), we get
or, equivalently (keeping in mind that \(N(C;z_{\eta })\) is a cone in X),
It remains to observe that (23) and assumption \(\rho \in (0,1)\) (coming from inequality \(m_{B}< L_{B}^{2}\)) imply that
This inclusion and Lemma 2.5 entail that
which concludes the proof. □
We now use Lemma 4.1 to deduce the following result.
Lemma 4.2
Assume \((\mathcal{C})\) and let \(B:X\rightarrow X\) be an \(m_{B}\)strongly monotone and \(L_{B}\)Lipschitz continuous operator with \(m_{B}<\min \{L_{B},L_{B}^{2}\}\). Moreover, assume that
Then, for any \(\eta \in C(I;K)\), there exists a unique continuous function \(z_{\eta }:I\rightarrow X\) such that
Proof
Let \(\eta \in C(I;K)\). Thanks to Lemma 4.1, we know that for every \(t\in I\) there exists a unique element \(z_{\eta }(t)\in C(t)\) such that
This justifies the claimed existence and uniqueness property. It remains to establish that \(z_{\eta }(\cdot )\) is a continuous function. Fix \({t}\in I\) and consider a sequence \((t_{n})_{n\in \mathbb{N}}\) of elements of I which converges to t. Due to the closedness of I, we obviously have \(t\in I\). For each \(n\in \mathbb{N}\), denote \(C_{n}:=C(t_{n})\), \(\eta _{n}:=\eta (t_{n})\), \(\zeta _{n}:=z_{\eta }(t_{n})\), and \(\omega _{n}:=\zeta _{n}\rho B(\zeta _{n}\eta _{n})\). Set also \(C_{\infty }:=C(t)\), \(\eta _{\infty }:={\eta }(t)\), \(\zeta _{\infty }:=z_{\eta }({t})\), and \({\omega }_{\infty }:=\zeta _{\infty }\rho B(\zeta _{\infty }\eta _{ \infty })\). With the above notation at hand, it is clear that for every integer \(n\in \mathbb{N}\) we have
hence,
We now estimate each of the two terms in the righthand side of (27). We start by setting
It is readily seen that
Set \(\delta :=\rho m_{B}\). Fix any real \(\varepsilon >0\) with \(\rho \alpha +\varepsilon <\beta \) and let \(s\in (\rho \alpha +\varepsilon ,\beta )\). Since \(\zeta _{\infty }\in C_{\infty }\subset U_{r}(C_{\infty })\), we can use assumption \((\mathcal{C})\) to see that \(d(\zeta _{\infty },C_{n})\rightarrow d(\zeta _{\infty },C_{\infty })=0\). Thus, we can find some integer \(N\geq 1\) such that
Fix for a moment an integer \(n\geq N\). We easily observe that
On the other hand, Lemma 4.1(a) guarantees that
and the rproxregularity of \(C_{n}\) implies that
with \(\kappa :=(1s/r)^{1}\). Using this inequality, the definition of \(\omega _{\infty }\), \(\omega _{n}\), 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 \(L:=\kappa \sqrt{1\delta }\) and taking into account inequalities (31), (30), and (29) it follows that
Noting that \(L\in (0,1)\) (see Lemma 4.1(b)) and coming back to inequality (27), we see that
Next, using inequality
it follows that \(\omega _{\infty } \in U_{r}(C_{\infty })\). Therefore, using assumption \((\mathcal{C})\) and Theorem 2.6, we obtain that
It remains to use (32) and the continuity of the function \(\eta :I\to K\) to see that \(\zeta _{n}=z(t_{n})\to z({t})=\zeta _{\infty }\) in X, as \(n\rightarrow \infty \). This shows that the function \(z_{\eta }:I\to X\) is continuous and concludes the proof. □
The next step is the following.
Lemma 4.3
Assume \((\mathcal{C})\), \((\mathcal{A})\), (16), (17), and (18). Then, for any \(\eta \in C(I;K)\), there exists a unique continuous function \(u_{\eta }:I\rightarrow X\) such that
Proof
Using Proposition 2.1, it follows that the operator \(A^{1}\) is \(m_{A^{1}}:=\frac{m_{A}}{L_{A}^{2}}\)strongly monotone and \(L_{A^{1}}:=\frac{1}{m_{A}}\)Lipschitz continuous. Note that \(m_{A^{1}}<\min \{L_{A^{1}},L_{A^{1}}^{2}\}\) and, moreover,
Let \(\eta \in C(I;K)\) and denote by \(z_{\eta }\in C(I;X)\) the function obtained in Lemma 4.2 with \(B:=A^{1}\). Then
with \(u_{\eta }\in C(I;X)\) defined by
It follows from the definition of proximal normal cone that
and this concludes the proof of the existence part of the lemma.
Now, let \(u_{1}, u_{2}:I\rightarrow X\) be two functions such that
for every \(t\in I\). Fix any \(t\in I\). Then, for each \(i\in \{1,2\}\), we have
along with
This implies that
and adding these inequalities yields
Using now the \(m_{A}\)strong monotonicity and the \(L_{A}\)Lipschitz property of the operator A, we get
We now use the the assumption \(m_{A}>\frac{s L_{A}^{2}}{r}\) to deduce that \(u_{1}(t)=u_{2}(t)\). □
Lemma 4.3 allows us to consider the operator \(\Lambda :C(I;K)\to C(I;K)\) defined by
We have the following result.
Lemma 4.4
Assume \((\mathcal{C})\), \((\mathcal{A})\), \((\mathcal{H})\), (16), (17), (18), and (19). Then the operator Λ has a unique fixed point \(\eta ^{*}\in C(I;K)\).
Proof
According to Theorem 2.10, it is enough to prove that the operator \(\Lambda :C(I;K)\rightarrow C(I;K)\) is an almost historydependent operator. Let \(\eta _{1}\), \(\eta _{2}\in C(I;K)\). Using Lemma 4.3, we find two continuous functions \(u_{1}:=u_{\eta _{1}}:I\rightarrow s \mathbb{B}_{X}\) and \(u_{2}:=u_{\eta _{2}}:I\rightarrow s \mathbb{B}_{X}\) such that
for all \(t\in I\). Let \(\mathcal{J}\) be a nonempty compact subset of I and let \(t\in \mathcal{J}\). Using (33) and assumption (\(\mathcal{S}\)) yields
On the other hand, from (34) we see that
for each \(i\in \{1,2\}\) and, therefore,
Taking \(i=1\) and \(v:=Au_{2}(t)+\eta _{2}(t)\) in the above estimate yields
Similarly, taking \(i=2\) and \(v:=Au_{1}(t)+\eta _{1}(t)\), we get
Adding the previous two inequalities, we find that
Fix \(\varepsilon >0\) such that \(m_{A}\frac{\varepsilon ^{2}}{2}\frac{2s L_{A}^{2}}{r}>0\). Then, using the Cauchy–Schwarz inequality and the elementary inequality \(2ab\leq (\varepsilon a)^{2}+(\frac{b}{\varepsilon })^{2}\) valid for every real a, b, we obtain that
On the other hand, note that
Therefore, setting \(a_{\varepsilon }:=m_{A}\frac{\varepsilon ^{2}}{2} \frac{2s L_{A}^{2}}{r}\), \(b_{\varepsilon }:=\frac{1}{2\varepsilon ^{2}}+\frac{2s}{r}\) and taking into account the strong monotonicity of the operator A, we find that
Now, choosing \(\varepsilon :=\sqrt{m_{A}}\), we get
with \(c:= (\frac{1}{2m_{A}}+\frac{2s}{r} )^{1/2} (\frac{m_{A}}{2} \frac{2sL_{A}^{2}}{r} )^{1/2}\). Substituting this inequality in (35) yields
We now invoke the smallness assumption (19) to obtain that the operator Λ is an almost historydependent operator. It remains to apply Theorem 2.10 to complete the proof. □
We are now in a position to provide the proof of Theorem 3.1.
Proof
Let \(\eta ^{*}\in C(I;K)\) be the fixed point of the operator Λ, and let \(u^{\star }:=u_{\eta ^{\star }}\in C(I;X)\) be the function given by Lemma 4.3 with \(\eta :=\eta ^{\star }\). So, we have
This inclusion combined with equality \(\eta ^{\star }=\Lambda \eta ^{\star }=\mathcal{S}u^{\star }\) implies that
which shows that \(u^{\star }\) 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 Λ. □
5 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 \(\boldsymbol{a}_{1}, \boldsymbol{a}_{2}\in \mathbb{S}^{d}\) and three real constants \(g_{1}\), \(g_{2}\), and k such that
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 realvalued positive functions depending on \(t\in I\) and, in this case, the set \(C(t)\) will depend on t. We denote by \(P_{K}:\mathbb{S}^{d}\to K\) the projection operator on the closed convex set K and consider the following inclusion problem.
Problem 4
Find a continuous function \(\boldsymbol{u}:I\to X\) such that
We have the following existence and uniqueness result.
Theorem 5.1
Assume (36)–(39). Then Problem 4has at least a solution \(\boldsymbol{u}(\cdot )\). Moreover, the solution takes values in \(s\mathbb{B}_{\mathbb{S}^{d}}:=\{s\boldsymbol{b}: \boldsymbol{b}\in \mathbb{B}_{\mathbb{S}^{d}}\}\) with \(s:=\\boldsymbol{a}_{1}\+g_{1}\) and is the unique solution of Problem 4with this property.
Proof
We apply Corollary 3.3 on the space \(X=\mathbb{S}^{d}\) with \(C(t)\) defined by (40)–(42), K defined by (43), and the operators A, \(\mathcal{S}\) given by
First, we remark that the sets \(C_{1}\) and \(C_{2}\) are convex and, therefore they are ∞prox regular. Moreover, using assumptions (37), (38), we see that
Therefore, Lemma 2.7 guarantees that for each \(t\in I\) the set \(C(t)\) is rproxregular with
In addition, since \(C(t)\) does not depend on t, it follows that convergence (15) holds. We conclude from here that assumption \((\mathcal{C})\) is satisfied.
On the other hand, it is obvious to see that the operator A satisfies condition \(({\mathcal{A}})\) with \(m_{A}=1\) and \(L_{A}=1+\varepsilon \) for any \(\varepsilon >0\). Moreover, the operator \(\mathcal{S}\) defined by (45) is historydependent operator (keeping in mind that \(P_{K}\) is 1Lipschitz on \(\mathbb{S}^{d}\)). In addition, inequality (16) is obviously satisfied.
We now show that with a convenient choice of ε, conditions (17) and (18) are satisfied. To this end we use (37) and (38) to see that \(\operatorname{Im} (C)K=(C_{1}\cup C_{2})K=C_{1}\) and, therefore,
Hence, using (46), (47) and equalities \(m_{A}=1\), \(L_{A}=1+\varepsilon \), it follows that conditions (17), (18) are equivalent with the inequalities
respectively. Next, an elementary calculus reveals that
and, using this inequality with \(x=\frac{1}{(1+\varepsilon )^{2}}\), we deduce that
Choose now \(0<\varepsilon <\frac{2}{\sqrt[4]{7}}1\). Then, using (39) and (50), we find that (48) and (49) hold, which implies that conditions (17) and (18) are satisfied, too. Theorem 5.1 is now a direct consequence of Corollary 3.3. □
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Acknowledgements
The first author is indebted to Assalé Adjé for helpful discussions on proxregularity.
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This research was supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie SklodowskaCurie Grant Agreement No. 823731 CONMECH.
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Nacry, F., Sofonea, M. Historydependent operators and proxregular sweeping processes. Fixed Point Theory Algorithms Sci Eng 2022, 5 (2022). https://doi.org/10.1186/s1366302200715w
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DOI: https://doi.org/10.1186/s1366302200715w