Introduction

Problem Statement and Main Result

We consider the multivalued differential equationFootnote 1

$$ \begin{array}{rl} v^{\prime}(t) + Av(t) +(BKv)(t) &\in F(t,v(t)), \quad t\in(0,T),\\ v(0)&=v_{0}, \end{array} $$
(1)

where

$$ (Kv)(t) = u_{0}+{{\int}_{0}^{t}} k(t-s) v(s) \mathrm{d}s, \quad k(z)=\lambda e^{-\lambda z}. $$
(2)

Here, T > 0 defines the considered time interval, λ > 0 is a given parameter and v0, u0 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 VA is a real, reflexive Banach space. The operator \(B\colon V_{B}\to V_{B}^{\ast }\) is linear, bounded, strongly positive, and symmetric, where VB denotes a real Hilbert space. The space VA shall be compactly and densely embedded in a real Hilbert space H, whereas VB shall be only continuously and densely embedded in H. The dual of H is identified with H itself, such that both VA, H, \(V_{A}^{\ast }\) and VB, H, \(V_{B}^{\ast }\) form a so-called Gelfand triple. However, we do not assume any relation between VA and VB apart from V = VAVB being separable and densely embedded in both VA and VB. We do not assume that VA is embedded into VB or the other way around. Overall, we have the scale

$$ V_{A}\cap V_{B}=V\subset V_{C} \subset H = H^{\ast} \subset V_{C}^{\ast} \subset V^{\ast} = V_{A}^{\ast} + V_{B}^{\ast},\quad C\in \{A,B\}, $$
(3)

of Banach and Hilbert spaces, where all embeddings are meant to be continuous and dense and the embedding VAH is even meant to be compact.

The operator F : [0, T] × HPfc(H) is measurable, fulfils a certain growth condition in the second argument and the graph of vF(t, v) is sequentially closed in H × Hw for almost all t ∈ (0, T), where Hw denotes the Hilbert space H equipped with the weak topology. The set Pfc(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 right-hand side, i.e.,

$$ \begin{array}{@{}rcl@{}} v^{\prime}(t) + Av(t) + (BKv)(t) &=&f(t), \qquad\quad t\in(0,T), \\ f(t)&\in& F(t,v(t)), \quad t\in(0,T),\\ v(0)&=&v_{0}, \end{array} $$

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 non-Fickian 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

$$ \begin{array}{rll} v^{\prime}(t) + Av(t) + Cu(t) &\in F(t,v(t)), \quad& t\in(0,T),\\ (u-Du_{0})'(t) + \lambda (u-Du_{0})(t) &=\lambda Dv(t), & t\in(0,T),\\ v(0)&=v_{0},& \\ u(0)&=Du_{0},& \end{array} $$
(4)

where C and D are suitably chosen linear operators such that B = CD.

Instead of the kernel k(z) = λeλz, we might also consider k(z) = ceλz 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) = Du0, t ∈ [0, T], is the solution of the second equation. In the case \(\lambda \to \infty \), the system reduces to a single first-order equation for v without memory.

In the case of the kernel k(z) = ceλz, the behaviour for λ → 0 is slightly different. The limit then yields a second-order 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 single-valued instead of the multivalued differential equation is considered in the same setting concerning the spaces VA and VB. However, due to the structure of the proof in the present work, we additionally need the compact embedding VAH and we have to assume that the right-hand side is pointwisely H-valued.

Nonlinear integro-differential equations have been considered by many authors through the years. Results on well-posedness for more general classes of nonlinear evolution equations including Volterra operators, but only in the case of Hilbert spaces VA = VB, 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 integro-differential equations see Eikmeier, Emmrich, and Kreusler [15, Section 1.2].

Multivalued differential equations have also been studied by various authors. Basic results, also for set-valued 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 set-valued nonlinear operator is, e.g., considered in Beyn, Emmrich, and Rieger [4].

In particular, integro-differential 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 set-valued 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 set-valued 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 fixed-point 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 XY of two Banach spaces X and Y, we consider the norm ∥⋅∥XY = ∥⋅∥X + ∥⋅∥Y, and for the sum X + Y, we consider the norm

$$ \|z\|_{X+Y} = \inf \left\{\max \left( \|z_{X}\|_{X}, \|z_{Y}\|_{Y}\right) ~|~ z=z_{X}+z_{Y} \text{ with } z_{X}\in X,~z_{Y}\in Y\right\}. $$

Note that (XY ) = X + Y if X and Y are embedded in a locally convex space and XY 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 Lp(0, T;X), we denote the usual space of Bochner measurable (sometimes also called strongly measurable), p-integrable functions equipped with the standard norm. For \(1\leq p<\infty \), the duality pairing between Lp(0, T;X) and its dual space Lq(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

$$ \langle g,f\rangle = {{\int}_{0}^{T}} \langle g(t),f(t)\rangle\mathrm{d} t, $$

see, e.g., Diestel and Uhl [12, Theorem 1 on p. 98, Corollary 13 on p. 76, Theorem 1 on p. 79].

By W1, p(0, T;X), \(1\leq p\leq \infty \), we denote the usual space of weakly differentiable functions uLp(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 uW1,1(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 real-valued functions that are continuously differentiable on [0, T]. By c, we denote a generic positive constant.

Now, let us recall some definitions from set-valued 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 σ-algebra on the interval \([a,b]\subset \mathbb {R}\) and the Borel σ-algebra on X, respectively. By Pf(X), we denote the set of all nonempty and closed subsets UX, and by Pfc(X), we denote the set of all nonempty, closed, and convex subsets UX.

For a set-valued function F : Ω → 2X ∖{}, let

$$ |F(\omega)| := \sup\left\{\|x\|_{X} ~|~ x\in F(\omega)\right\},\quad \omega\in {\Omega}. $$

The graph of such a set-valued function is defined as

$$ \text{graph}(F)=\{(\omega,x)\in {\Omega}\times X ~|~ x\in F(\omega)\}. $$

A function F : Ω → Pf(X) is called measurable (sometimes also called weakly measurable) if the preimage of each open set is measurable, i.e.,

$$ F^{-1}(U):=\{\omega\in {\Omega} ~|~ F(\omega)\cap U \neq \emptyset\} \in {\Sigma} $$

for every open UX.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 set-valued function has a measurable selection, see, e.g., Aubin and Frankowska [3, Theorem 8.1.3].

Now, let (Ω,Σ, μ) be a complete σ-finite measure space and let X be a separable Banach space. For a set-valued function F : Ω → Pf(X) and \(p\in [1,\infty )\), we denote by \(\mathcal {F}^{p}\) the set of all p-integrable selections of F, i.e.,

$$ \mathcal{F}^{p}:=\left\{ f\in L^{p}({\Omega};X,\mu) ~|~ f(\omega)\in F(\omega) \text{ a.e. in } {\Omega}\right\}, $$

where Lp(Ω;X, μ) denotes the space of Bochner measurable, p-integrable 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)BX for μ-almost all ω ∈Ω, where BX 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

$$ {\int}_{\Omega} F \mathrm{d} \mu := \left\{{\int}_{\Omega} f \mathrm{d} \mu ~|~ f\in \mathcal{F}^{1}\right\}. $$

For properties of this integral, see, e.g., Aubin and Frankowska [3, Chapter 8.6].

For a set-valued function F : Ω × XPf(X), a function v: Ω → X and \(p\in [1,\infty )\), we denote by \(\mathcal {F}^{p}(v)\) the set of all p-integrable selections of the mapping ωF(ω, v(ω)), i.e.,

$$ \mathcal{F}^{p}(v):=\left\{ f\in L^{p}({\Omega};X,\mu) ~|~ f(\omega)\in F(\omega, v(\omega)) \text{ a.e. in } {\Omega}\right\}. $$

Finally, let X, Y be Banach spaces and Ω ⊂ Y. A set-valued function F : Ω → 2X ∖{} is called upper semicontinuous if F− 1(U) is closed in Ω for all closed UX.

Main Assumptions and Preliminary Results

Throughout this paper, let VA be a real, reflexive Banach space and let VB and H be real Hilbert spaces, respectively. As mentioned in Section 1, we require that V = VAVB is separable and the embeddings stated in (3) are fulfilled (with the embedding VAH 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

  1. i)

    A is monotone,

  2. ii)

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

  3. iii)

    A fulfils a growth condition of order p − 1, i.e., there exists βA > 0 such that

    $$ \|Av\|_{V_{A}^{\ast}} \leq {\upbeta}_{A} \left( 1+\|v\|_{V_{A}}^{p-1}\right) $$

    for all vVA,

  4. iv)

    A is p-coercive, i.e., there exist μA > 0, cA ≥ 0 such that

    $$ \langle Av,v\rangle \geq \mu_{A} \|v\|^{p}_{V_{A}}-c_{A} $$

    for all vVA.

One operator satisfying these assumptions is, e.g., the p-Laplacian −Δp = −∇⋅ (|∇|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

  1. i)

    B is linear,

  2. ii)

    B is bounded, i.e., there exists βB > 0 such that

    $$ \|Bv\|_{\ast} \leq {\upbeta}_{B} \|v\| $$

    for all vVB,

  3. iii)

    B is strongly positive, i.e., there exists μB > 0 such that

    $$ \langle Bv,v\rangle \geq \mu_{B} \|v\|^{2} $$

    for all vVB,

  4. iv)

    B is symmetric.

Following these assumptions, B induces a norm ∥⋅∥B := 〈B⋅,⋅〉1/2 in VB that is equivalent to \(\|\cdot \|_{V_{B}}\). Therefore, we denote the space L2(0, T;(VB,∥⋅∥B)) by L2(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 Hs(Ω).

Finally, we say that F : [0, T] × HPfc(H) fulfils the assumptions (F) if

  1. i)

    F is measurable,

  2. ii)

    for almost all t ∈ (0, T), the graph of the mapping vF(t, v) is sequentially closed in H × Hw, where Hw denotes the Hilbert space H equipped with the weak topology,

  3. iii)

    \(|F(t,v)| \leq a(t) + b\|v\|_{H}^{2/q}\) a.e. with aLq(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 tA(t, v), vVA, and tB(t, v), vVB, 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 Lp(0, T;VA) and L1(0, T;VB), 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] → VA 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

$$ \| Av\|_{L^{q}(0,T;V_{A}^{\ast})}\leq c \left( 1+ \|v\|^{p-1}_{L^{p}(0,T;V_{A})}\right) $$
(5)

for all vLp(0, T;VA), i.e., A maps Lp(0, T;VA) into \(L^{q}(0,T;V_{A}^{\ast })\).

Via the same definition (Bv)(t) = Bv(t) for a function v: [0, T] → VB, we can extend the operator \(B\colon V_{B}\to V_{B}^{\ast }\) to a linear, bounded, strongly positive, and symmetric operator mapping L2(0, T;VB) into its dual or to a linear, bounded operator mapping Lr(0, T;VB) 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) = λeλz, λ > 0, u0X. The operator K : L2(0, T; X) → L2(0, T; X) is well-defined, affine-linear, and bounded. The estimate

$$ \|Kv-u_{0}\|_{L^{2}(0,T;X)} \le \|k\|_{L^{1}(0,T)} \|v\|_{L^{2}(0,T;X)} $$

is satisfied for all vL2(0, T; X), where \(\|k\|_{L^{1}(0,T)}= 1-e^{-\lambda T}\). Further, the estimate

$$ \|Kv-u_{0}\|_{\mathscr{C}([0,T];X)} \le \lambda \|v\|_{L^{1}(0,T;X)} $$

is satisfied for all vL1(0, T; X), i.e., K is also an affine-linear, bounded operator of L1(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 = VB) 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 well-defined, affine-linear, 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 vL1(0, T;X). Then we have

$$ (Kv)'(t)=\lambda \big(v(t) - \left( (Kv)(t) - u_{0}\right) \big) $$
(6)

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 tF(t, u(t)), where u itself is a measurable function.

Lemma 2

Let X be a separable Banach space, let F : [0, T] × XPf(X) be measurable and let v: [0, T] → X be Bochner measurable. Then the mapping \(\tilde {F}_{v}\colon [0,T]\to P_{f}(X)\), tF(t, v(t)), is measurable.

Proof

Let UX be open. Consider

$$ \begin{array}{@{}rcl@{}} \tilde{F}^{-1}_{v}(U)&=& \{t\in [0,T] ~|~ F(t,v(t))\cap U\neq \emptyset \} \\ &=& \pi_{[0,T]}(\{(t,x)\in [0,T]\times X ~|~ F(t,x)\cap U \neq \emptyset , x=v(t)\})\\ &=& \pi_{[0,T]}(\{(t,x)\in [0,T]\times X ~|~ F(t,x)\cap U \neq \emptyset \} \cap \text{graph}(v)), \end{array} $$

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 integration-by-parts 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 integration-by-parts formula

$$ {\int}_{t_{1}}^{t_{2}}\left( \langle v^{\prime}(t),w(t)\rangle + \langle w^{\prime}(t),v(t)\rangle \right) \mathrm{d} t= \langle v(t_{2}),w(t_{2})\rangle-\langle v(t_{1}),w(t_{1})\rangle $$
(7)

holds for all t1, t2 ∈ [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 vn, \(n\in \mathbb {N}\), and w due to classical calculus. Therefore, we have

$$ {\int}_{t_{1}}^{t_{2}}\left( \langle v_{n}^{\prime}(t),w(t)\rangle +\langle w^{\prime}(t),v_{n}(t)\rangle \right)\mathrm{d} t= \langle v_{n}(t_{2}),w(t_{2})\rangle- \langle v_{n}(t_{1}),w(t_{1})\rangle $$

for all \(n\in \mathbb {N}\). The convergence vnv in \(\mathcal {W}(0,T)\) yields

$$ {\int}_{t_{1}}^{t_{2}}\left( \langle v_{n}^{\prime}(t),w(t)\rangle +\langle w^{\prime}(t),v_{n}(t)\rangle\right) \mathrm{d} t \to {\int}_{t_{1}}^{t_{2}}\left( \langle v^{\prime}(t),w(t)\rangle +\langle w^{\prime}(t),v(t)\rangle\right) \mathrm{d} t. $$

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

$$ \langle v_{n}(t),w(t)\rangle \to \langle v(t), w(t)\rangle $$

for all t ∈ [0, T], in particular for t1, t2. This finishes the proof. □

Existence of a Solution

Theorem 1

Let the assumptions (A), (B), and (F) be fulfilled and let u0VB, v0H 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

$$ v^{\prime}+Av+BKv=f $$

holds in the sense of \(L^{q}(0,T;V_{A}^{\ast })\), i.e.,

$$ \langle v^{\prime}+BKv, \varphi\rangle + \langle Av, \varphi\rangle = \langle f,\varphi\rangle $$

for all φLp(0, T;VA).

Proof

Following the proof of [35, Theorem 3.1], we want to apply the Kakutani fixed-point theorem, generalised by Glicksberg [20] and Fan [17] to infinite-dimensional locally convex topological vector spaces. The same fixed-point 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 tF(t, v(t)). The growth condition of F implies

$$ \|f(t)\|_{H}\leq a(t)+b\|v(t)\|_{H}^{2/q} $$

for almost all t ∈ (0, T), and since aLq(0, T) and \(v\in {\mathscr{C}}_{w}([0,T];H)\), we have fLq(0, T;H). Now, test the equation

$$ v^{\prime} + Av +BKv = f $$

with v and integrate the resulting equation over (0, t), t ∈ [0, T], which yields

$$ {{\int}_{0}^{t}}\langle v^{\prime}(s)+(BKv)(s),v(s)\rangle \mathrm{d} s + {{\int}_{0}^{t}} \langle Av(s),v(s)\rangle \mathrm{d} s ={{\int}_{0}^{t}} \langle f(s),v(s)\rangle \mathrm{d} s. $$

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

$$ \begin{array}{@{}rcl@{}} {{\int}^{t}_{0}} \langle v^{\prime}(s) + (BKv)(s), v(s)\rangle \mathrm{d} s &=& \frac{1}{2}\|v(t)\|^{2}_{H} - \frac{1}{2}\|v_{0}\|^{2}_{H} + \frac{1}{2\lambda}\|(Kv)(t)\|^{2}_{B} -\frac{1}{2\lambda}\|u_{0}\|^{2}_{B} \\ && - {{\int}_{0}^{t}} \langle (BKv)(s), u_{0}\rangle \mathrm{d} s + {{\int}_{0}^{t}} \|(Kv)(s)\|^{2}_{B}\mathrm{d} s. \end{array} $$

Due to the coercivity of A and Young’s inequality, we have

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{2}\|v(t)\|^{2}_{H} + \mu_{A} {{\int}_{0}^{t}} \|v(s)\|^{p}_{V_{A}} \mathrm{d} s + \frac{1}{2\lambda}\|(Kv)(t)\|^{2}_{B} + {{\int}_{0}^{t}} \|(Kv)(s)\|^{2}_{B}\mathrm{d} s \\ &\leq& c_{A} T + \frac{1}{2} \|v_{0}\|^{2}_{H} + \frac{1}{2\lambda}\|u_{0}\|_{B}^{2} + {{\int}_{0}^{t}} \|f(s)\|_{V_{A}^{\ast}} \|v(s)\|_{V_{A}} \mathrm{d} s + {{\int}_{0}^{t}} \|(Kv)(s)\|_{B} \|u_{0}\|_{B} \mathrm{d} s \\ &\leq& c_{A} T + \frac{1}{2} \|v_{0}\|^{2}_{H} + \frac{1}{2\lambda}\|u_{0}\|_{B}^{2} + c{{\int}_{0}^{t}} \|f(s)\|_{V_{A}^{\ast}}^{q}\mathrm{d} s + \frac{\mu_{A}}{2} {{\int}_{0}^{t}} \|v(s)\|^{p}_{V_{A}} \mathrm{d} s \\ && + \frac{1}{2} {{\int}_{0}^{t}} \|(Kv)(s)\|_{B}^{2} \mathrm{d} s +\frac{T}{2} \| u_{0}\|^{2}_{B}. \end{array} $$

After rearranging, the estimate on F yields

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{2}\|v(t)\|^{2}_{H} + \frac{\mu_{A}}{2} {{\int}_{0}^{t}} \|v(s)\|^{p}_{V_{A}} \mathrm{d} s + \frac{1}{2\lambda}\|(Kv)(t)\|^{2}_{B} + \frac{1}{2} {{\int}_{0}^{t}} \|(Kv)(s)\|_{B}^{2} \mathrm{d} s \\ &\leq& c\left( 1+ \|v_{0}\|_{H}^{2} + \|u_{0}\|_{B}^{2} + {{\int}_{0}^{t}} \|f(s)\|_{V_{A}^{\ast}}^{q}\mathrm{d} s \right) \\ &\leq& c\left( 1+ \|v_{0}\|_{H}^{2} + \|u_{0}\|_{B}^{2} + {{\int}_{0}^{t}} \left( a(s) + b \|v(s)\|_{H}^{2/q}\right)^{q}\mathrm{d} s \right) \\ &\leq& c\left( 1+ \|v_{0}\|_{H}^{2} + \|u_{0}\|_{B}^{2} + \|a\|_{L^{q}(0,T)}^{q} + {{\int}_{0}^{t}}\|v(s)\|_{H}^{2}\mathrm{d} s \right). \end{array} $$

Applying Gronwall’s lemma, we obtain

$$ \|v(t)\|^{2}_{H} \leq M_{1} $$
(8)

for all t ∈ [0, T], where M1 > 0 depends on the problem data. This also immediately yields

$$ {{\int}_{0}^{t}}\|v(s)\|^{2}_{V_{A}} \mathrm{d} s \leq M_{2} $$
(9)

as well as

$$ \|(Kv)(t)\|^{2}_{B} \leq M_{2} $$
(10)

for all t ∈ [0, T], where M2 > 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

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\!\!&&\|v^{\prime}\|_{L^{q}(0,T;V_{A}^{\ast}) + L^{\infty}(0,T;V_{B}^{\ast})} \\ \!\!\!\!\!\!\!\!\!\!\!\!\!& \leq& \max\!\left( \|Av\|_{L^{q}(0,T;V_{A}^{\ast})} + \|f\|_{L^{q}(0,T;V_{A}^{\ast})},~\|BKv\|_{L^{\infty}(0,T;V_{B}^{\ast})} \right) \\ \!\!\!\!\!\!\!\!\!\!\!\!\!& \leq& \max\!\left( c\!\left( 1+ \|v\|^{p-1}_{L^{p}(0,T;V_{A})}\right) + \|a\|_{L^{q}(0,T)} + b \|v\|_{L^{2}(0,T;H)}^{2/q}, {\upbeta}_{B} \|Kv\|_{L^{\infty}(0,T;V_{B})} \!\right)\!. \end{array} $$
(11)

The a priori estimates (8), (9), and (10) above yield

$$ \|v^{\prime}\|_{L^{q}(0,T;V_{A}^{\ast})+ L^{\infty}(0,T;V_{B}^{\ast})} \leq M_{3}, $$
(12)

where M3 depends on the same parameters as M1 and M2 as well as on βB.

Next, we define the truncation \(\hat {F}\) of F via

$$ \hat{F}(t,w)= \left\{\begin{array}{llll} F(t,w) & \text{ if }~\|w\|_{H} \leq M_{1}, \\ F\left( t,\frac{M_{1}}{\|w\|_{H}} w\right) & \text{ if }~\|w\|_{H} > M_{1}. \end{array}\right. $$

This set-valued function \(\hat {F}\) has the same measurability and continuity properties as F: In order to prove the measurability, consider an arbitrary open subset UH and

$$ \begin{array}{@{}rcl@{}} \hat{F}^{-1}(U) &=& \left\{ (t,v)\in [0,T]\times H \mid F(t, r_{M_{1}}(v))\cap U \neq \emptyset \right\} \\ & =& \left\{ (t,v)\in [0,T]\times H \mid F(t, v)\cap U \neq \emptyset \right\} \cap \left( [0,T] \times B_{M_{1}}^{H} \right), \end{array} $$

where \(r_{M_{1}}\) is the radial retraction in H to the ball \(B_{M_{1}}^{H}\) of radius M1. 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 σ-algebra, \(\hat {F}\) is measurable.

For proving that \(\hat {F}\) fulfils the same continuity condition as F, let N ⊂ [0, T] be the set of Lebesgue-measure 0 such that the graph of vF(t, v) is sequentially closed in H × Hw 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 vnv and \(w_{n} \rightharpoonup w\) for some v, wH. 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

$$ |\hat{F}(t,v)| \leq \hat{a}(t):=a(t) +b M_{1}^{2/q} $$

for almost all t ∈ (0, T) and all vH. Now, set

$$ E:= \{f\in L^{q}(0,T;H)\mid \|f(t)\|_{H} \leq \hat{a}(t) \text{ a.e.}\}. $$

We define the solution operator

$$ G\colon E\to \overline{W}(0,T):=\left\{ v\in L^{p}(0,T;V_{A}) \mid v^{\prime}\in L^{q}(0,T;V_{A}^{\ast})+L^{\infty}(0,T;V_{B}^{\ast}) \right\} $$

with G(f) = v, where v is the unique solution to the problem

$$ \begin{array}{lllll} v^{\prime}+Av+BKv&=f, \\ v(0)&=v_{0}, \end{array} $$
(13)

in the sense of \(L^{q}(0,T;V_{A}^{\ast })\), which exists due to [15, Theorem 4.2, Corollary 5.2]. Note that

$$ \begin{array}{@{}rcl@{}} v^{\prime}=f-Av-BKv &\in& L^{q}(0,T;H)+L^{q}(0,T;V_{A}^{\ast})+L^{\infty}(0,T;V_{B}^{\ast}) \\ & \subset& L^{q}(0,T;V_{A}^{\ast}) + L^{\infty}(0,T;V_{B}^{\ast}) \end{array} $$

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 {fn}⊂ E and fE with \(f_{n}\rightharpoonup f\) in Lq(0, T;H). Analogously to the proof of the a priori estimates (8), (9), (10), and (12), it can be shown that the sequence {vn} of corresponding solutions, i.e., vn = G(fn), the sequence \(\{v_{n}^{\prime }\}\) of derivatives, and the sequence {Kvn} 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 {Avn} in \(L^{q}(0,T;V_{A}^{\ast })\), see also (11). Since Lp(0, T;VA) 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

$$ \begin{array}{@{}rcl@{}} v_{n} &\rightharpoonup& v \qquad \text{ in }~L^{p}(0,T;V_{A}), \\ v_{n} &\overset{\ast}\rightharpoonup& v \qquad \text{ in }~L^{\infty}(0,T;H), \\ v_{n}^{\prime} &\overset{\ast}\rightharpoonup& \hat{v} \qquad \text{ in }~L^{q}(0,T;V_{A}^{\ast})+L^{\infty}(0,T;V_{B}^{\ast}), \\ Av_{n} &\overset{\ast}\rightharpoonup& \tilde{a} \qquad \text{ in }~L^{q}(0,T;V_{A}^{\ast}), \\ Kv_{n} &\overset{\ast}\rightharpoonup& u \qquad \text{ in }~L^{\infty}(0,T;V_{B}), \end{array} $$

as \(n\to \infty \). We obviously have \(\hat {v}=v^{\prime }\). As vLp(0, T;VA) 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

$$ v_{n}^{\prime} + Av_{n} +BKv_{n} = f_{n} $$
(14)

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 \(\hat {K}w:=Kw-u_{0}\) is well-defined, linear, and bounded, see Lemma 1. Thus it is weakly-weakly continuous and \(v_{n}\overset {\ast }{\rightharpoonup } v\) in \(L^{\infty }(0,T;H)\) (and therefore \(v_{n}\rightharpoonup v\) in L2(0, T;H)) implies \(Kv_{n}-Kv=\hat {K}v_{n}-\hat {K}v \rightharpoonup 0\) in L2(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) = v0 and \(v_{n}(T)\rightharpoonup v(T)\) in H. Due to estimate (8), the sequence {vn(T)} is bounded in H, so there exists vTH such that, up to a subsequence, \(v_{n}(T)\rightharpoonup v_{T}\) in H. Now, consider \(\phi \in {\mathscr{C}}^{1}([0,T])\), wV (recall that V = VAVB). Since vn solves (14) in the sense of \(L^{q}(0,T;V_{A}^{\ast })\) and v solves

$$ v^{\prime}+\tilde{a}+BKv=f $$
(15)

in the sense of \(L^{q}(0,T;V_{A}^{\ast })\), the integration-by-parts formula in Lemma 3 and the regularity \(v\in {\mathscr{C}}_{w}([0,T];H)\) yield

$$ \begin{array}{@{}rcl@{}} &&\left( v_{n}(T),w\right) \phi(T) - \left( v_{n}(0),w\right)\phi(0)\\ &=& {{\int}_{0}^{T}} \langle v_{n}^{\prime}(t), w\rangle \phi(t)\mathrm{d} t + {{\int}_{0}^{T}} \langle v_{n}(t),w\rangle \phi^{\prime}(t)\mathrm{d} t \\ & =& {{\int}_{0}^{T}} \langle f_{n} - Av_{n} - BKv_{n}, w \rangle \phi(t) \mathrm{d} t + {{\int}_{0}^{T}} \langle v_{n}(t),w\rangle \phi^{\prime}(t)\mathrm{d} t \\ &\rightarrow& {{\int}_{0}^{T}} \langle f - \tilde{a} - BKv, w \rangle \phi(t) \mathrm{d} t + {{\int}_{0}^{T}} \langle v(t),w\rangle \phi^{\prime}(t)\mathrm{d} t \\ &=& {{\int}_{0}^{T}} \langle v^{\prime}, w \rangle \phi(t) \mathrm{d} t + {{\int}_{0}^{T}} \langle v(t),w\rangle \phi^{\prime}(t)\mathrm{d} t \\ &=& \big(v(T),w\big) \phi(T) - \left( v(0),w\right)\phi(0) \end{array} $$

as \(n\to \infty \). Choosing \(\phi (t)=1-\frac {t}{T}\), this yields (vn(0), w) → (v(0), w) for all wV. Due to vn(0) = v0 for all \(n\in \mathbb {N}\), we have v(0) = v0. Choosing \(\phi (t)=\frac {t}{T}\), we get (vn(T), w) → (v(T), w) for all wV and therefore also vT = v(T) in H.

Next, let us show \((Kv_{n})(T) \rightharpoonup (Kv)(T)\) in VB. Estimate (10) implies the boundedness of the sequence {(Kvn)(T)} in VB, therefore there exists uTVB such that, up to a subsequence, \((Kv_{n})(T) \rightharpoonup u_{T}\) in VB. Consider again \(\phi (t)=\frac {t}{T}\), wV. Due to relation (6), we haveFootnote 4

$$ \begin{array}{@{}rcl@{}} \left( (Kv_{n})(T), w\right)&=& {{\int}_{0}^{T}} \langle (Kv_{n})'(t), w\rangle \frac{t}{T} \mathrm{d} t + {{\int}_{0}^{T}} \langle (Kv_{n})(t),w\rangle \frac{1}{T} \mathrm{d} t \\ &=&\lambda {{\int}_{0}^{T}} \langle v_{n}(t)-((Kv_{n})(t)-u_{0}),w\rangle \frac{t}{T} \mathrm{d} t + {{\int}_{0}^{T}} \langle (Kv_{n})(t),w\rangle \frac{1}{T} \mathrm{d} t \\ &\rightarrow& \lambda {{\int}_{0}^{T}} \langle v(t)-((Kv)(t)-u_{0}),w\rangle \frac{t}{T} \mathrm{d} t + {{\int}_{0}^{T}} \langle (Kv)(t),w\rangle \frac{1}{T} \mathrm{d} t \\ &=& {{\int}_{0}^{T}} \langle (Kv)'(t), w\rangle \frac{t}{T} \mathrm{d} t + {{\int}_{0}^{T}} \langle (Kv)(t),w\rangle \frac{1}{T} \mathrm{d} t \\ &=& \left( (Kv)(T), w\right) \end{array} $$

as \(n\to \infty \). This immediately yields uT = (Kv)(T).

In order to show that v is a solution to problem (13), it remains to show \(\tilde {a}=Av\). Using the integration-by-parts formula [15, Lemma 4.3], we obtain

$$ \begin{array}{@{}rcl@{}} \langle Av_{n},v_{n}\rangle &=& \langle f_{n}, v_{n}\rangle - \langle v_{n}^{\prime}+BKv_{n}, v_{n}\rangle \\ &=& \langle f_{n}, v_{n}\rangle - \frac{1}{2}\|v_{n}(T)\|^{2}_{H} + \frac{1}{2}\|v_{0}\|^{2}_{H} - \frac{1}{2\lambda}\|(Kv_{n})(T)\|^{2}_{B} +\frac{1}{2\lambda}\|u_{0}\|^{2}_{B} \\ && + {{\int}_{0}^{T}} \langle (BKv_{n})(s), u_{0}\rangle \mathrm{d} s - \|Kv_{n}\|^{2}_{L^{2}(0,T;B)}. \end{array} $$

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 vnv in Lp(0, T;H). This yields 〈fn, vn〉→〈f, v〉.Footnote 5 We also obviously have

$$ {{\int}_{0}^{T}} \langle (BKv_{n})(s), u_{0}\rangle \mathrm{d} s \to {{\int}_{0}^{T}} \langle (BKv)(s), u_{0}\rangle \mathrm{d} s. $$

Due to the convergences \(v_{n}(T) \rightharpoonup v(T)\) in H, \((Kv_{n})(T)\rightharpoonup (Kv)(T)\) in VB as well as \(Kv_{n} \overset {\ast }{\rightharpoonup } Kv\) in \(L^{\infty }(0,T;V_{B})\) (and thus \(Kv_{n} \rightharpoonup Kv\) in L2(0, T;VB)) and the lower semicontinuity of the norm, we obtain

$$ \begin{array}{@{}rcl@{}} \limsup_{n\to\infty} \langle Av_{n},v_{n}\rangle &\leq& \langle f, v\rangle - \frac{1}{2}\|v(T)\|^{2}_{H} + \frac{1}{2}\|v_{0}\|^{2}_{H} - \frac{1}{2\lambda}\|(Kv)(T)\|^{2}_{B} +\frac{1}{2\lambda}\|u_{0}\|^{2}_{B} \\ && + {{\int}_{0}^{T}} \langle (BKv)(s), u_{0}\rangle \mathrm{d} s - \| Kv\|^{2}_{L^{2}(0,T;B)} \\ &=& \langle f,v\rangle - \langle v^{\prime}+BKv,v\rangle, \end{array} $$

using again the integration-by-parts formula [15, Lemma 4.3]. As v solves (15) in the sense of \(L^{q}(0,T;V_{A}^{\ast })\), we have

$$ \limsup_{n\to\infty} \langle Av_{n},v_{n}\rangle \leq \langle \tilde{a},v\rangle. $$
(16)

Now, for arbitrary wLp(0, T;VA), the monotonicity of A implies

$$ \begin{array}{@{}rcl@{}} \langle Av_{n}, v_{n}\rangle &=& \langle Av_{n}-Aw,v_{n}-w\rangle + \langle Aw, v_{n}-w\rangle +\langle Av_{n}, w\rangle \\ &\geq& \langle Aw, v_{n}-w \rangle + \langle Av_{n}, w\rangle. \end{array} $$

Therefore, we obtain

$$ \liminf_{n\to\infty} \langle Av_{n}, v_{n}\rangle \geq \langle Aw, v-w\rangle + \langle \tilde{a},w\rangle $$

and, together with (16),

$$ \langle Aw-\tilde{a}, v-w\rangle\leq 0. $$

Choosing w = v ± rz for an arbitrary zLp(0, T;VA) 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

$$ \langle Av -\tilde{a} , z\rangle =0 $$

for all zLp(0, T;VA), which implies \(\tilde {a}=Av\).

As the last step of this proof, consider the operator R: EPfc(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.,

$$ \mathcal{F}^{1}(G(f))=\left\{ f\in L^{1}(0,T;H) \mid f(t)\in \hat{F}(t,(G(f))(t)) \text{ a.e. in } (0,T) \right\}. $$

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 fixed-point theorem (see Glicksberg [20] and Fan [17]) to obtain the existence of fE such that \(f\in R(f)=\mathcal {F}^{1}(G(f))\). This implies that v = G(f) solves (1) with the right-hand 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

$$ \begin{array}{@{}rcl@{}} \frac{\partial}{\partial t}v(x,t) -{\Delta}_{p}v(x,t) + ({\Delta}^{2}Kv)(x,t)&\in& (F(t,v))(x,t), \quad\quad (x,t)\in {\Omega}\times (0,T), \\ v(x,t)={\Delta} v(x,t)&=&0, \quad\quad\quad\quad\quad\quad~\quad (x,t)\in \partial{\Omega}\times (0,T), \\ v(x,0)&=&v_{0}(x),\quad\quad\quad\quad\quad~~ x\in{\Omega}, \end{array} $$

where K is given by (2) and

$$ F(t,v)= \left\{ w\in L^{2}({\Omega})\mid \|g(v)-w\|_{L^{2}({\Omega})} \leq a(t)\right\} = B^{L^{2}({\Omega})}_{a(t)}(g(v)) $$

with \(g(v)=\|v\|_{L^{2}({\Omega })}^{1-2/p}v\) and a certain function aLq(0, T), a(t) ≥ 0 a.e.

Here, we have A given by −Δp with \(V_{A}=W_{0}^{1,p}({\Omega })\) and B given by Δ2 with \(V_{B}=H^{2}({\Omega })\cap {H_{0}^{1}}({\Omega })\). Obviously, VA is not embedded in VB. As we have d = 3 and p > 6, VB is also not embedded in VA. However, \(V_{A}\cap V_{B}=H^{2}({\Omega })\cap W_{0}^{1,p}({\Omega })\) is separable and densely embedded in both VA and VB. With H = L2(Ω), we obtain the scale (3) of Banach and Hilbert spaces. As already mentioned, the p-Laplacian fulfils the assumptions (A). It is easy to see that Δ2 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 xL2(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 L2(Ω), i.e., \(d(x,A)= \inf _{y\in A} \|x-y\|_{L^{2}({\Omega })}\) for xL2(Ω), AL2(Ω). For xF(t, v), we have

$$ d(x, F(t,v))= d\left( x, B^{L^{2}({\Omega})}_{a(t)}(g(v))\right) = d(x, g(v))-a(t). $$

This is obviously measurable.

To prove the continuity condition, consider {vn},{wn}⊂ L2(Ω) and v, wL2(Ω) with vnv, \(w_{n}\rightharpoonup w\), and wnF(t, vn). We have to show wF(t, v). As wnF(t, vn), we have

$$ \|g(v_{n}) -w_{n}\|_{L^{2}({\Omega})} \leq a(t). $$

The continuity of g and the lower semicontinuity of the norm yield

$$ \|g(v)-w\|_{L^{2}({\Omega})} \leq \liminf_{n\to \infty} \|g(v_{n})-w_{n}\|_{L^{2}({\Omega})} \leq a(t), $$

i.e., wF(t, v).

Finally, we have

$$ \begin{array}{@{}rcl@{}} |F(t,v)| &=& \sup_{w\in F(t,v)} \|w\|_{L^{2}({\Omega})} \\ & \leq& \sup_{w\in F(t,v)} \|g(v) -w\|_{L^{2}({\Omega})} + \|g(v)\|_{L^{2}({\Omega})} \\ & \leq& a(t) + \|v\|_{L^{2}({\Omega})}^{2-2/p}. \end{array} $$

As \(2-\frac {2}{p}=\frac {2}{q}\), this yields the desired growth condition on F.