Abstract
The initial value problem for a multivalued differential equation is studied, which is governed by the sum of a monotone, hemicontinuous, coercive operator fulfilling a certain growth condition and a Volterra integral operator in time of convolution type with exponential decay. The two operators act on different Banach spaces where one is not embedded in the other. The setvalued righthand side is measurable and satisfies certain continuity and growth conditions. Existence of a solution is shown via a generalisation of the Kakutani fixedpoint theorem.
Introduction
Problem Statement and Main Result
We consider the multivalued differential equation^{Footnote 1}
where
Here, T > 0 defines the considered time interval, λ > 0 is a given parameter and v_{0}, u_{0} are the given initial data of the problem.
The operator \(A\colon V_{A}\to V_{A}^{\ast }\) is a monotone, hemicontinuous, coercive operator satisfying a certain growth condition, where V_{A} is a real, reflexive Banach space. The operator \(B\colon V_{B}\to V_{B}^{\ast }\) is linear, bounded, strongly positive, and symmetric, where V_{B} denotes a real Hilbert space. The space V_{A} shall be compactly and densely embedded in a real Hilbert space H, whereas V_{B} shall be only continuously and densely embedded in H. The dual of H is identified with H itself, such that both V_{A}, H, \(V_{A}^{\ast }\) and V_{B}, H, \(V_{B}^{\ast }\) form a socalled Gelfand triple. However, we do not assume any relation between V_{A} and V_{B} apart from V = V_{A} ∩ V_{B} being separable and densely embedded in both V_{A} and V_{B}. We do not assume that V_{A} is embedded into V_{B} or the other way around. Overall, we have the scale
of Banach and Hilbert spaces, where all embeddings are meant to be continuous and dense and the embedding V_{A} ⊂ H is even meant to be compact.
The operator F : [0, T] × H → P_{fc}(H) is measurable, fulfils a certain growth condition in the second argument and the graph of v↦F(t, v) is sequentially closed in H × H_{w} for almost all t ∈ (0, T), where H_{w} denotes the Hilbert space H equipped with the weak topology. The set P_{fc}(H) denotes the set of all nonempty, closed, and convex subsets of H.
Multivalued differential equations appear, e.g., in the formulation of optimal feedback control problems. If we consider the inclusion as an equation with a side condition on the righthand side, i.e.,
we can consider f as the control of our system with the corresponding state v and F as the set of admissible controls, which, in the case of F depending on v, leads to a feedback control system.
Physical applications of the system we are considering in this work are, e.g., heat flow in materials with memory (see, e.g., MacCamy [25], Miller [30]) or viscoelastic fluid flow (see, e.g., Desch, Grimmer, and Schappacher [11], MacCamy [26]). Another application related to that are nonFickian diffusion models which describe diffusion processes of a penetrant through a viscoelastic material (see, e.g., Edwards [13], Edwards and Cohen [14], Shaw and Whiteman [39]). They also appear, e.g., in mathematical biology (see, e.g., Cushing [8], Fedotov and Iomin [18], Mehrabian and Abousleiman [27]).
Due to the specific form of the kernel k given in (2), we can rewrite our system into the coupled system
where C and D are suitably chosen linear operators such that B = CD.
Instead of the kernel k(z) = λe^{−λz}, we might also consider k(z) = ce^{−λz} with c, λ > 0. However, for simplicity, we will stick to the first type of kernel. Actually, this type appears naturally in many applications. In these applications, \(\frac {1}{\lambda }\) is often describing a relaxation or averaged delay time. If we consider the limit λ → 0, the system (4) decouples such that u(t) = Du_{0}, t ∈ [0, T], is the solution of the second equation. In the case \(\lambda \to \infty \), the system reduces to a single firstorder equation for v without memory.
In the case of the kernel k(z) = ce^{−λz}, the behaviour for λ → 0 is slightly different. The limit then yields a secondorder in time equation for u (see, e.g., Emmrich and Thalhammer [16]).
Literature Overview
This work is a continuation of Eikmeier, Emmrich, and Kreusler [15]. There, the singlevalued instead of the multivalued differential equation is considered in the same setting concerning the spaces V_{A} and V_{B}. However, due to the structure of the proof in the present work, we additionally need the compact embedding V_{A} ⊂ H and we have to assume that the righthand side is pointwisely Hvalued.
Nonlinear integrodifferential equations have been considered by many authors through the years. Results on wellposedness for more general classes of nonlinear evolution equations including Volterra operators, but only in the case of Hilbert spaces V_{A} = V_{B}, can be found in, e.g., Gajewski, Gröger, and Zacharias [19]. In contrast to this, Crandall, Londen, and Nohel [7] study the case of a doubly nonlinear problem, where both nonlinear operators are assumed to be (possibly multivalued) maximal monotone subdifferential operators and the domain of definition of one of them has to be continuously and densely embedded in the domain of definition of the other one. For more references on nonlinear and also linear evolutionary integrodifferential equations see Eikmeier, Emmrich, and Kreusler [15, Section 1.2].
Multivalued differential equations have also been studied by various authors. Basic results, also for setvalued analysis, can be found in, e.g., Aubin and Cellina [2], Aubin and Frankowska [3], or Deimling [9]. In O’Regan [32], some extensions of the results shown in Deimling [9] are presented. A semilinear multivalued differential equation with a linear, bounded, and strongly positive operator and a setvalued nonlinear operator is, e.g., considered in Beyn, Emmrich, and Rieger [4].
In particular, integrodifferential equations in the multivalued case have been studied by, e.g., Papageorgiou [33,34,35,36]. The equations are considered under different assumptions with the setvalued operator appearing in the integral term. In most of the works mentioned, examples of applications in the theory of optimal control are given.
In the context of viscoelastic contact problems, evolution inclusions and hemivariational inequalities have been discussed in, e.g., Migórski [28] and Migórski, Ochal, and Sofonea [29].
The optimal feedback control of a motion of a viscoelastic fluid via a multivalued differential equation is, e.g., considered in Gori et al. [21] and Obukhovskiı̆, Zecca, and Zvyagin [31]. Existence of solutions for the equation are shown via topological degree theory.
Organisation of the Paper
The paper is organised as follows: In Section 2, we introduce the general notation and some basic results from setvalued analysis. In Section 3, we state our assumptions on the operators A, B, and F and some preliminary results concerning properties we need in the following Section 4, where we prove existence of a solution to problem (1). This is done via a generalisation of the Kakutani fixedpoint theorem.
Notation
Let X be a Banach space with its dual X^{∗}. The norm in X and the standard norm in X^{∗} are denoted by ∥⋅∥_{X} and \(\\cdot \_{X^{\ast }}\), respectively. The duality pairing between X and X^{∗} is denoted by 〈⋅,⋅〉. If X is a Hilbert space, the inner product in X is denoted by (⋅,⋅). For the intersection X ∩ Y of two Banach spaces X and Y, we consider the norm ∥⋅∥_{X∩Y} = ∥⋅∥_{X} + ∥⋅∥_{Y}, and for the sum X + Y, we consider the norm
Note that (X ∩ Y )^{∗} = X^{∗} + Y^{∗} if X and Y are embedded in a locally convex space and X ∩ Y is dense in X and Y with respect to the norm above, see, e.g., Gajewski et al. [19, pp. 12ff.].
Now, let X be a real, reflexive, and separable Banach space and \(1\leq p\leq \infty \). By L^{p}(0, T;X), we denote the usual space of Bochner measurable (sometimes also called strongly measurable), pintegrable functions equipped with the standard norm. For \(1\leq p<\infty \), the duality pairing between L^{p}(0, T;X) and its dual space L^{q}(0, T;X^{∗}), where \(\frac {1}{p}+\frac {1}{q}=1\) for p > 1 and \(q=\infty \) for p = 1, is also denoted by 〈⋅,⋅〉, and it is given by
see, e.g., Diestel and Uhl [12, Theorem 1 on p. 98, Corollary 13 on p. 76, Theorem 1 on p. 79].
By W^{1, p}(0, T;X), \(1\leq p\leq \infty \), we denote the usual space of weakly differentiable functions u ∈ L^{p}(0, T;X) with \(u^{\prime }\in L^{p}(0,T;X)\), equipped with the standard norm. By \({\mathscr{C}}([0,T];X)\), we denote the space of functions that are continuous on [0, T] with values in X, whereas \({\mathscr{C}}_{w}([0,T];X)\) denotes the space of functions that are continuous on [0, T] with respect to the weak topology in X. We have the continuous embedding \(W^{1,1}(0,T;X) \subset {\mathscr{C}}([0,T];X)\), see, e.g., Roubíček [38, Lemma 7.1]. Furthermore, a function u ∈ W^{1,1}(0, T;X) is almost everywhere equal to a function that is absolutely continuous on [0, T] with values in X, see, e.g., Brézis [5, Theorem 8.2]. We denote the set of all these functions by \(\mathscr {AC}([0,T];X)\). By \({\mathscr{C}}^{1}([0,T])\), we denote the space of realvalued functions that are continuously differentiable on [0, T]. By c, we denote a generic positive constant.
Now, let us recall some definitions from setvalued analysis. Let (Ω,Σ) be a measurable space and let X be a complete separable metric space. By \({\mathscr{L}}([a,b])\) and \({\mathscr{B}}(X)\), we denote the Lebesgue σalgebra on the interval \([a,b]\subset \mathbb {R}\) and the Borel σalgebra on X, respectively. By P_{f}(X), we denote the set of all nonempty and closed subsets U ⊂ X, and by P_{fc}(X), we denote the set of all nonempty, closed, and convex subsets U ⊂ X.
For a setvalued function F : Ω → 2^{X} ∖{∅}, let
The graph of such a setvalued function is defined as
A function F : Ω → P_{f}(X) is called measurable (sometimes also called weakly measurable) if the preimage of each open set is measurable, i.e.,
for every open U ⊂ X.^{Footnote 2} A function f : Ω → X is called measurable selection of F if f(ω) ∈ F(ω) for all ω ∈Ω and f is measurable. Each measurable setvalued function has a measurable selection, see, e.g., Aubin and Frankowska [3, Theorem 8.1.3].
Now, let (Ω,Σ, μ) be a complete σfinite measure space and let X be a separable Banach space. For a setvalued function F : Ω → P_{f}(X) and \(p\in [1,\infty )\), we denote by \(\mathcal {F}^{p}\) the set of all pintegrable selections of F, i.e.,
where L^{p}(Ω;X, μ) denotes the space of Bochner measurable, pintegrable functions with respect to μ.^{Footnote 3} If F is integrably bounded, i.e., there exists a nonnegative function \(m\in L^{p}({\Omega };\mathbb {R},\mu )\) such that F(ω) ⊂ m(ω)B_{X} for μalmost all ω ∈Ω, where B_{X} denotes the unit ball in X, each measurable selection of F is in \(\mathcal {F}^{p}\) due to Lebesgue’s theorem on dominated convergence. The integral of F is defined as
For properties of this integral, see, e.g., Aubin and Frankowska [3, Chapter 8.6].
For a setvalued function F : Ω × X → P_{f}(X), a function v: Ω → X and \(p\in [1,\infty )\), we denote by \(\mathcal {F}^{p}(v)\) the set of all pintegrable selections of the mapping ω↦F(ω, v(ω)), i.e.,
Finally, let X, Y be Banach spaces and Ω ⊂ Y. A setvalued function F : Ω → 2^{X} ∖{∅} is called upper semicontinuous if F^{− 1}(U) is closed in Ω for all closed U ⊂ X.
Main Assumptions and Preliminary Results
Throughout this paper, let V_{A} be a real, reflexive Banach space and let V_{B} and H be real Hilbert spaces, respectively. As mentioned in Section 1, we require that V = V_{A} ∩ V_{B} is separable and the embeddings stated in (3) are fulfilled (with the embedding V_{A} ⊂ H meant to be compact). Let also \(2\leq p <\infty \), 1 < q ≤ 2 with \(\frac {1}{p}+\frac {1}{q}=1\).
For \(A\colon V_{A} \to V_{A}^{\ast }\), we say the assumptions (A) are fulfilled if

i)
A is monotone,

ii)
A is hemicontinuous, i.e., 𝜃↦〈A(u + 𝜃v), w〉 is continuous on [0,1] for all u, v, w ∈ V_{A},

iii)
A fulfils a growth condition of order p − 1, i.e., there exists β_{A} > 0 such that
$$ \Av\_{V_{A}^{\ast}} \leq {\upbeta}_{A} \left( 1+\v\_{V_{A}}^{p1}\right) $$for all v ∈ V_{A},

iv)
A is pcoercive, i.e., there exist μ_{A} > 0, c_{A} ≥ 0 such that
$$ \langle Av,v\rangle \geq \mu_{A} \v\^{p}_{V_{A}}c_{A} $$for all v ∈ V_{A}.
One operator satisfying these assumptions is, e.g., the pLaplacian −Δ_{p} = −∇⋅ (∇^{p− 2}∇) acting between the standard Sobolev spaces \(W_{0}^{1,p}({\Omega })\) and W^{− 1, p}(Ω) for a bounded Lipschitz domain Ω, see, e.g., Zeidler [40, p. 489]. For \(B\colon V_{B}\to V_{B}^{\ast }\), we say the assumptions (B) are fulfilled if

i)
B is linear,

ii)
B is bounded, i.e., there exists β_{B} > 0 such that
$$ \Bv\_{\ast} \leq {\upbeta}_{B} \v\ $$for all v ∈ V_{B},

iii)
B is strongly positive, i.e., there exists μ_{B} > 0 such that
$$ \langle Bv,v\rangle \geq \mu_{B} \v\^{2} $$for all v ∈ V_{B},

iv)
B is symmetric.
Following these assumptions, B induces a norm ∥⋅∥_{B} := 〈B⋅,⋅〉^{1/2} in V_{B} that is equivalent to \(\\cdot \_{V_{B}}\). Therefore, we denote the space L^{2}(0, T;(V_{B},∥⋅∥_{B})) by L^{2}(0, T;B). An example for an operator satisfying these assumptions is the Laplacian −Δ acting between the standard Sobolev spaces \({H_{0}^{1}}({\Omega })\) and H^{− 1}(Ω), again for a bounded Lipschitz domain Ω, as well as the fractional Laplacian (−Δ)^{s} for \(\frac {1}{2}<s<1\), acting between the standard Sobolev–Slobodeckiı̆ spaces \({H_{0}^{s}}({\Omega })\) and H^{−s}(Ω).
Finally, we say that F : [0, T] × H → P_{fc}(H) fulfils the assumptions (F) if

i)
F is measurable,

ii)
for almost all t ∈ (0, T), the graph of the mapping v↦F(t, v) is sequentially closed in H × H_{w}, where H_{w} denotes the Hilbert space H equipped with the weak topology,

iii)
\(F(t,v) \leq a(t) + b\v\_{H}^{2/q}\) a.e. with a ∈ L^{q}(0, T), a(t) ≥ 0 a.e. and b > 0.
Note that it is also possible to consider \(A\colon [0,T]\times V_{A}\to V_{A}^{\ast }\) and \(B\colon [0,T]\times V_{B}\to V_{B}^{\ast }\), where the mappings t↦A(t, v), v ∈ V_{A}, and t↦B(t, v), v ∈ V_{B}, are assumed to be measurable and all the assumptions above are assumed to hold uniformly in t. However, for simplicity, we will only consider the case of autonomous operators A and B.
These operators can be extended to operators defined on L^{p}(0, T;V_{A}) and L^{1}(0, T;V_{B}), respectively. The monotonicity and hemicontinuity of \(A\colon V_{A}\to V_{A}^{\ast }\) imply demicontinuity (see, e.g., Zeidler [40, Proposition 26.4 on p. 555]). Due to the separability of \(V_{A}^{\ast }\), the theorem of Pettis (see, e.g., Diestel and Uhl [12, Theorem 2 on p. 42]) then implies that A maps Bochner measurable functions v: [0, T] → V_{A} into Bochner measurable functions \(Av\colon [0,T]\to V_{A}^{\ast }\), where (Av)(t) = Av(t) for almost all t ∈ (0, T). Due to the growth condition, we have the estimate
for all v ∈ L^{p}(0, T;V_{A}), i.e., A maps L^{p}(0, T;V_{A}) into \(L^{q}(0,T;V_{A}^{\ast })\).
Via the same definition (Bv)(t) = Bv(t) for a function v: [0, T] → V_{B}, we can extend the operator \(B\colon V_{B}\to V_{B}^{\ast }\) to a linear, bounded, strongly positive, and symmetric operator mapping L^{2}(0, T;V_{B}) into its dual or to a linear, bounded operator mapping L^{r}(0, T;V_{B}) into \(L^{r}(0,T;V_{B}^{\ast })\), \(1\leq r \leq \infty \), respectively.
Due to the definition (2) of the operator K, we have the following lemma.
Lemma 1
Let X be an arbitrary Banach space, k(z) = λe^{−λz}, λ > 0, u_{0} ∈ X. The operator K : L^{2}(0, T; X) → L^{2}(0, T; X) is welldefined, affinelinear, and bounded. The estimate
is satisfied for all v ∈ L^{2}(0, T; X), where \(\k\_{L^{1}(0,T)}= 1e^{\lambda T}\). Further, the estimate
is satisfied for all v ∈ L^{1}(0, T; X), i.e., K is also an affinelinear, bounded operator of L^{1}(0, T; X) into \({\mathscr{C}}([0,T];X)\) (even \(\mathscr {AC}([0,T];X)\)).
The proof is based on simple calculations, therefore we omit it here. Following this lemma, we obtain the following properties of the operator BK.
Corollary 1
Let the assumptions of Lemma 1 (with X = V_{B}) and assumption (B) be fulfilled. Then the operator \(BK\colon L^{2}(0,T;V_{B})\to L^{2}(0,T;V_{B}^{\ast })\) is welldefined, affinelinear, and bounded. The same holds for \(BK\colon L^{1}(0,T;V_{B})\to {\mathscr{C}}([0,T];V_{B}^{\ast })\).
One crucial relation in this setting, resulting from the exponential kernel, is the following one. Let X be an arbitrary Banach space and v ∈ L^{1}(0, T;X). Then we have
for almost all t ∈ (0, T).
Concerning the operator F, we need a measurability result in order to be able to extract measurable selections of the multivalued mapping t↦F(t, u(t)), where u itself is a measurable function.
Lemma 2
Let X be a separable Banach space, let F : [0, T] × X → P_{f}(X) be measurable and let v: [0, T] → X be Bochner measurable. Then the mapping \(\tilde {F}_{v}\colon [0,T]\to P_{f}(X)\), t ↦ F(t, v(t)), is measurable.
Proof
Let U ⊂ X be open. Consider
where π_{[0, T]} denotes the projection onto [0, T]. Since v is Bochner measurable, its graph belongs to \({\mathscr{L}}([0,T])\otimes {\mathscr{B}}(X)\), see, e.g., Castaing and Valadier [6, Theorem III.36]. Note again that for a separable Banach space X, Bochner measurability and \({\mathscr{L}}([0,T])\)\({\mathscr{B}}(X)\)measurability are equivalent, see, e.g., Denkowski, Migórski, and Papageorgiou [10, Corollary 3.10.5]. Due to the measurability of F, the intersection space in the equation above also belongs to \({\mathscr{L}}([0,T])\otimes {\mathscr{B}}(X)\). Since the projection maps measurable sets into measurable sets (at least in this setting, see, e.g., Castaing and Valadier [6, Theorem III.23]), we have \(\tilde {F}^{1}_{v}(U)\in {\mathscr{L}}([0,T])\), which finishes the proof. □
Finally, we need an integrationbyparts formula similar to the one provided in Roubíček [38, Lemma 7.3] for functions in the spaces \(\mathcal {W}(0,T):=\{v\in L^{p}(0,T;V_{A}) \mid v^{\prime }\in L^{q}(0,T;V^{\ast })\}\) and \({\mathscr{C}}^{1}([0,T];V)\), respectively.
Lemma 3
Let \(v\in \mathcal {W}(0,T)\), \(w\in {\mathscr{C}}^{1}([0,T];V)\). Then the integrationbyparts formula
holds for all t_{1}, t_{2} ∈ [0, T].
Proof
Due to the density of \({\mathscr{C}}^{1}([0,T];V_{A})\) in \(\mathcal {W}(0,T)\) (see, e.g., Roubíček [38, Lemma 7.2]), there exists a sequence \(\{v_{n}\}\subset {\mathscr{C}}^{1}([0,T];V_{A})\) such that \(v_{n}\to v \in \mathcal {W}(0,T)\). The formula (7) obviously holds for v_{n}, \(n\in \mathbb {N}\), and w due to classical calculus. Therefore, we have
for all \(n\in \mathbb {N}\). The convergence v_{n} → v in \(\mathcal {W}(0,T)\) yields
On the other hand, the continuous embedding \(\mathcal {W}(0,T)\subset {\mathscr{C}}([0,T];V^{\ast })\) (see, e.g., Roubíček [38, Lemma 7.1]) yields
for all t ∈ [0, T], in particular for t_{1}, t_{2}. This finishes the proof. □
Existence of a Solution
Theorem 1
Let the assumptions (A), (B), and (F) be fulfilled and let u_{0} ∈ V_{B}, v_{0} ∈ H be given. Then there exists a solution \(v\in L^{p}(0,T;V_{A})\cap {\mathscr{C}}_{w}([0,T];H)\) to (1) with \(Kv\in {\mathscr{C}}_{w}([0,T];V_{B})\) and \(v^{\prime }\in L^{q}(0,T;V_{A}^{\ast })+L^{\infty }(0,T;V_{B}^{\ast })\), i.e., the initial condition is fulfilled in H and there exists \(f\in \mathcal {F}^{1}(v)\) such that the equation
holds in the sense of \(L^{q}(0,T;V_{A}^{\ast })\), i.e.,
for all φ ∈ L^{p}(0, T;V_{A}).
Proof
Following the proof of [35, Theorem 3.1], we want to apply the Kakutani fixedpoint theorem, generalised by Glicksberg [20] and Fan [17] to infinitedimensional locally convex topological vector spaces. The same fixedpoint theorem has also been applied in, e.g., Kalita, Migórski, and Sofonea [22] and Kalita, Szafraniec, and Shillor [23] to prove existence of solutions to partial differential inclusions.
First, we need a priori estimates for the solution. Assume \(v\in L^{p}(0,T;V_{A})\cap {\mathscr{C}}_{w}([0,T];H)\) solves problem (1) with the regularity stated in the theorem. Due to Lemma 2, there exists a measurable selection f : [0, T] → H of the mapping t↦F(t, v(t)). The growth condition of F implies
for almost all t ∈ (0, T), and since a ∈ L^{q}(0, T) and \(v\in {\mathscr{C}}_{w}([0,T];H)\), we have f ∈ L^{q}(0, T;H). Now, test the equation
with v and integrate the resulting equation over (0, t), t ∈ [0, T], which yields
Since we neither know \(v^{\prime }\in L^{q}(0,T;V_{A}^{\ast })\) nor \(BKv\in L^{q}(0,T;V_{A}^{\ast })\), it is not possible to do integration by parts for each term separately. However, [15, Lemma 4.3] yields
Due to the coercivity of A and Young’s inequality, we have
After rearranging, the estimate on F yields
Applying Gronwall’s lemma, we obtain
for all t ∈ [0, T], where M_{1} > 0 depends on the problem data. This also immediately yields
as well as
for all t ∈ [0, T], where M_{2} > 0 also depends on the problem data.
We also need a priori estimates for the derivative \(v^{\prime }\). Due to the estimate (5) and the assumptions (F), we have
The a priori estimates (8), (9), and (10) above yield
where M_{3} depends on the same parameters as M_{1} and M_{2} as well as on β_{B}.
Next, we define the truncation \(\hat {F}\) of F via
This setvalued function \(\hat {F}\) has the same measurability and continuity properties as F: In order to prove the measurability, consider an arbitrary open subset U ⊂ H and
where \(r_{M_{1}}\) is the radial retraction in H to the ball \(B_{M_{1}}^{H}\) of radius M_{1}. Due to the measurability of F, the first set is an element of \({\mathscr{L}}([0,T])\otimes {\mathscr{B}}(H)\), and since the second set is obviously an element of the same σalgebra, \(\hat {F}\) is measurable.
For proving that \(\hat {F}\) fulfils the same continuity condition as F, let N ⊂ [0, T] be the set of Lebesguemeasure 0 such that the graph of v↦F(t, v) is sequentially closed in H × H_{w} for all t ∈ [0, T] ∖ N. Now, for arbitrary t ∈ [0, T] ∖ N, consider a sequence \(\{(v_{n},w_{n})\}\subset \text {graph}(\hat {F}(t,\cdot ))\) with v_{n} → v and \(w_{n} \rightharpoonup w\) for some v, w ∈ H. We have to show \(w\in \hat {F}(t,v)\). Since the radial retraction \(r_{M_{1}}\) in H is continuous, we have \(r_{M_{1}}(v_{n})\to r_{M_{1}}(v)\) in H. Then, \(w_{n}\in \hat {F}(t,v_{n})=F(t,r_{M_{1}}(v_{n}))\) and the continuity condition on F immediately yield \(w\in F(t,r_{M_{1}}(v))=\hat {F}(t,v)\).
Due to the estimate on F in the assumptions (F), we have
for almost all t ∈ (0, T) and all v ∈ H. Now, set
We define the solution operator
with G(f) = v, where v is the unique solution to the problem
in the sense of \(L^{q}(0,T;V_{A}^{\ast })\), which exists due to [15, Theorem 4.2, Corollary 5.2]. Note that
in this case. Note also that \(v\in {\mathscr{C}}_{w}([0,T];H)\) and \(Kv\in {\mathscr{C}}_{w}([0,T];V_{B})\) by [15, Theorem 4.2]. Now, the aim is to show that G is sequentially weakly continuous.
We therefore consider a sequence {f_{n}}⊂ E and f ∈ E with \(f_{n}\rightharpoonup f\) in L^{q}(0, T;H). Analogously to the proof of the a priori estimates (8), (9), (10), and (12), it can be shown that the sequence {v_{n}} of corresponding solutions, i.e., v_{n} = G(f_{n}), the sequence \(\{v_{n}^{\prime }\}\) of derivatives, and the sequence {Kv_{n}} are bounded in \(L^{p}(0,T;V_{A})\cap L^{\infty }(0,T;H)\), \(L^{q}(0,T;V_{A}^{\ast })+L^{\infty }(0,T;V_{B}^{\ast })\), and \(L^{\infty }(0,T;V_{B})\), respectively. Due to the estimate (5) on A, this implies the boundedness of the sequences {Av_{n}} in \(L^{q}(0,T;V_{A}^{\ast })\), see also (11). Since L^{p}(0, T;V_{A}) is a reflexive Banach space and \(L^{\infty }(0,T;H)\), \(L^{q}(0,T;V_{A}^{\ast })\), \(L^{\infty }(0,T;V_{B})\) as well as \(L^{q}(0,T;V_{A}^{\ast })+L^{\infty }(0,T;V_{B}^{\ast })\) are duals of separable normed spaces, there exists a subsequence (again denoted by n) and \(v\in L^{p}(0,T;V_{A})\cap L^{\infty }(0,T;H)\), \(\hat {v}\in L^{q}(0,T;V_{A}^{\ast })+L^{\infty }(0,T;V_{B}^{\ast })\), \(\tilde {a}\in L^{q}(0,T;V_{A}^{\ast })\), and \(u\in L^{\infty }(0,T;V_{B})\) such that
as \(n\to \infty \). We obviously have \(\hat {v}=v^{\prime }\). As v ∈ L^{p}(0, T;V_{A}) and \(v^{\prime }\in L^{q}(0,T;V^{\ast })\), we have \(v\in {\mathscr{C}}([0,T];V^{\ast })\), see, e.g., Roubíček [38, Lemma 7.1]. Together with \(v\in L^{\infty }(0,T;H)\), this yields \(v\in {\mathscr{C}}_{w}([0,T];H)\), see, e.g., Lions and Magenes [24, Chapter 3, Lemma 8.1].
In order to show that G is sequentially weakly continuous, we have to pass to the limit in the equation
and show that v is a solution to problem (13). First, we want to show u = Kv. We know that the operator \(\hat {K}\colon L^{2}(0,T;H)\to L^{2}(0,T;H)\) with \(\hat {K}w:=Kwu_{0}\) is welldefined, linear, and bounded, see Lemma 1. Thus it is weaklyweakly continuous and \(v_{n}\overset {\ast }{\rightharpoonup } v\) in \(L^{\infty }(0,T;H)\) (and therefore \(v_{n}\rightharpoonup v\) in L^{2}(0, T;H)) implies \(Kv_{n}Kv=\hat {K}v_{n}\hat {K}v \rightharpoonup 0\) in L^{2}(0, T;H). This yields u = Kv. Due to the linearity and boundedness of \(B\colon V_{B}\to V_{B}^{\ast }\) and thus its weakly*weakly* continuity, we also have \(BKv_{n} \overset {\ast }{\rightharpoonup } Bu=BKv\) in \(L^{\infty }(0,T;V_{B}^{\ast })\).
Next, let us show v(0) = v_{0} and \(v_{n}(T)\rightharpoonup v(T)\) in H. Due to estimate (8), the sequence {v_{n}(T)} is bounded in H, so there exists v_{T} ∈ H such that, up to a subsequence, \(v_{n}(T)\rightharpoonup v_{T}\) in H. Now, consider \(\phi \in {\mathscr{C}}^{1}([0,T])\), w ∈ V (recall that V = V_{A} ∩ V_{B}). Since v_{n} solves (14) in the sense of \(L^{q}(0,T;V_{A}^{\ast })\) and v solves
in the sense of \(L^{q}(0,T;V_{A}^{\ast })\), the integrationbyparts formula in Lemma 3 and the regularity \(v\in {\mathscr{C}}_{w}([0,T];H)\) yield
as \(n\to \infty \). Choosing \(\phi (t)=1\frac {t}{T}\), this yields (v_{n}(0), w) → (v(0), w) for all w ∈ V. Due to v_{n}(0) = v_{0} for all \(n\in \mathbb {N}\), we have v(0) = v_{0}. Choosing \(\phi (t)=\frac {t}{T}\), we get (v_{n}(T), w) → (v(T), w) for all w ∈ V and therefore also v_{T} = v(T) in H.
Next, let us show \((Kv_{n})(T) \rightharpoonup (Kv)(T)\) in V_{B}. Estimate (10) implies the boundedness of the sequence {(Kv_{n})(T)} in V_{B}, therefore there exists u_{T} ∈ V_{B} such that, up to a subsequence, \((Kv_{n})(T) \rightharpoonup u_{T}\) in V_{B}. Consider again \(\phi (t)=\frac {t}{T}\), w ∈ V. Due to relation (6), we have^{Footnote 4}
as \(n\to \infty \). This immediately yields u_{T} = (Kv)(T).
In order to show that v is a solution to problem (13), it remains to show \(\tilde {a}=Av\). Using the integrationbyparts formula [15, Lemma 4.3], we obtain
Since we have \(v_{n} \rightharpoonup v\) in \(\overline {W}(0,T)\) and since the embedding \(\overline {W}(0,T)\subset L^{p}(0,T;H)\) is compact (see, e.g., Roubíček [38, Lemma 7.7]), there exists a subsequence, again denoted by n, such that v_{n} → v in L^{p}(0, T;H). This yields 〈f_{n}, v_{n}〉→〈f, v〉.^{Footnote 5} We also obviously have
Due to the convergences \(v_{n}(T) \rightharpoonup v(T)\) in H, \((Kv_{n})(T)\rightharpoonup (Kv)(T)\) in V_{B} as well as \(Kv_{n} \overset {\ast }{\rightharpoonup } Kv\) in \(L^{\infty }(0,T;V_{B})\) (and thus \(Kv_{n} \rightharpoonup Kv\) in L^{2}(0, T;V_{B})) and the lower semicontinuity of the norm, we obtain
using again the integrationbyparts formula [15, Lemma 4.3]. As v solves (15) in the sense of \(L^{q}(0,T;V_{A}^{\ast })\), we have
Now, for arbitrary w ∈ L^{p}(0, T;V_{A}), the monotonicity of A implies
Therefore, we obtain
and, together with (16),
Choosing w = v ± rz for an arbitrary z ∈ L^{p}(0, T;V_{A}) and r > 0 and using the hemicontinuity as well as the growth condition of \(A\colon V_{A}\to V_{A}^{\ast }\), Lebesgue’s theorem on dominated convergence yields for r → 0
for all z ∈ L^{p}(0, T;V_{A}), which implies \(\tilde {a}=Av\).
As the last step of this proof, consider the operator R: E → P_{fc}(E) with \(R(f)=\mathcal {F}^{1}(G(f))\), where the set \(\mathcal {F}^{1}(G(f))\) is meant with respect to the truncation \(\hat {F}\) instead of F, i.e.,
Following the proof of [35, Theorem 3.1], this operator is upper semicontinuous on E equipped with the weak topology. Thus, we can apply the generalisation of the Kakutani fixedpoint theorem (see Glicksberg [20] and Fan [17]) to obtain the existence of f ∈ E such that \(f\in R(f)=\mathcal {F}^{1}(G(f))\). This implies that v = G(f) solves (1) with the righthand side \(\hat {F}\). However, due to the a priori estimate (8), we have \(\hat {F}(t,v(t))=F(t,v(t))\) for almost all t ∈ [0, T] which proves the assertion. □
Example
We present an example for the problem discussed in this work. Let \({\Omega } \subseteq \mathbb {R}^{3}\) be a bounded domain with boundary of class \({\mathscr{C}}^{1,1}\) or a convex polyhedral and let p > 6. Consider
where K is given by (2) and
with \(g(v)=\v\_{L^{2}({\Omega })}^{12/p}v\) and a certain function a ∈ L^{q}(0, T), a(t) ≥ 0 a.e.
Here, we have A given by −Δ_{p} with \(V_{A}=W_{0}^{1,p}({\Omega })\) and B given by Δ^{2} with \(V_{B}=H^{2}({\Omega })\cap {H_{0}^{1}}({\Omega })\). Obviously, V_{A} is not embedded in V_{B}. As we have d = 3 and p > 6, V_{B} is also not embedded in V_{A}. However, \(V_{A}\cap V_{B}=H^{2}({\Omega })\cap W_{0}^{1,p}({\Omega })\) is separable and densely embedded in both V_{A} and V_{B}. With H = L^{2}(Ω), we obtain the scale (3) of Banach and Hilbert spaces. As already mentioned, the pLaplacian fulfils the assumptions (A). It is easy to see that Δ^{2} fulfils the assumptions (B). Thus, it remains to prove that F fulfils the assumptions (F).
First, we prove the measurability of the mapping (t, v)↦d(x, F(t, v)) for arbitrary fixed x ∈ L^{2}(Ω), which is equivalent to the measurability of F, see, e.g., Denkowski, Migórski, and Papageorgiou [10, Theorem 4.2.11]. Here, d is the distance function in L^{2}(Ω), i.e., \(d(x,A)= \inf _{y\in A} \xy\_{L^{2}({\Omega })}\) for x ∈ L^{2}(Ω), A ⊂ L^{2}(Ω). For x∉F(t, v), we have
This is obviously measurable.
To prove the continuity condition, consider {v_{n}},{w_{n}}⊂ L^{2}(Ω) and v, w ∈ L^{2}(Ω) with v_{n} → v, \(w_{n}\rightharpoonup w\), and w_{n} ∈ F(t, v_{n}). We have to show w ∈ F(t, v). As w_{n} ∈ F(t, v_{n}), we have
The continuity of g and the lower semicontinuity of the norm yield
i.e., w ∈ F(t, v).
Finally, we have
As \(2\frac {2}{p}=\frac {2}{q}\), this yields the desired growth condition on F.
Notes
Also named differential inclusion by many authors. However, in order to distinguish this kind of problem from the ones containing subdifferentials or setvalued (maximal monotone) operators, we chose the name multivalued differential equation.
Depending on the assumptions on (Ω,Σ) and X, there are many equivalent definitions of measurability for setvalued functions, see, e.g., Denkowski, Migórski, and Papageorgiou [10, Theorem 4.3.4].
Note that in the case of a separable Banach space X, the Bochner measurability of f coincides with the Σ\({\mathscr{B}}(X)\)measurability, see, e.g., Amann and Escher [1, Chapter X, Theorem 1.4], Denkowski, Migórski, and Papageorgiou [10, Corollary 3.10.5], or Papageorgiou and Winkert [37, Theorem 4.2.4]
Note that u_{0} cancels out in relation (6), so we consider X = V_{A}.
Here, we need the (in comparison to the singlevalued case stronger) assumptions that the embedding V_{A} ⊂ H is compact and that f(t) ∈ H in order to identify the limit of the sequence {〈f_{n}, v_{n}〉}.
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Eikmeier, A., Emmrich, E. On a Multivalued Differential Equation with Nonlocality in Time. Vietnam J. Math. 48, 703–718 (2020). https://doi.org/10.1007/s10013020004124
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DOI: https://doi.org/10.1007/s10013020004124
Keywords
 Nonlinear evolution equation
 Multivalued differential equation
 Differential inclusion
 Monotone operator
 Volterra operator
 Exponentially decaying memory
 Existence
 Kakutani fixedpoint theorem
Mathematics Subject Classification (2010)
 47J35
 34G25
 45K05
 35R70