On a Multivalued Differential Equation with Nonlocality in Time

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 set-valued right-hand side is measurable and satisfies certain continuity and growth conditions. Existence of a solution is shown via a generalisation of the Kakutani fixed-point theorem.


Problem Statement and Main Result
We consider the multivalued differential equation 1 v (t) + Av(t) + (BKv)(t) ∈ F (t, v(t)), t ∈ (0, T ), 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 : V A → V * A is a monotone, hemicontinuous, coercive operator satisfying a certain growth condition, where V A is a real, reflexive Banach space. The operator B : V B → V * B 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 and V B , H , V * B form a so-called 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 f c (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 f c (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.,

v (t) + Av(t) + (BKv)(t) = f (t),
t ∈ (0, T ), 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 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, 1 λ 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 λ → ∞, the system reduces to a single first-order 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 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 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 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 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 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.
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 Obukhovskiȋ, 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 · X * , 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 ≤ p ≤ ∞. By L p (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 ≤ p < ∞, the duality pairing between L p (0, T ; X) and its dual space L q (0, T ; X * ), where 1 p + 1 q = 1 for p > 1 and q = ∞ for p = 1, is also denoted by ·, · , and it is given by Now, let us recall some definitions from set-valued analysis. Let ( , ) be a measurable space and let X be a complete separable metric space. By L([a, b]) and B(X), we denote the Lebesgue σ -algebra on the interval [a, b] ⊂ R and the Borel σ -algebra on X, respectively. By P f (X), we denote the set of all nonempty and closed subsets U ⊂ X, and by P f c (X), we denote the set of all nonempty, closed, and convex subsets U ⊂ X.
For a set-valued function F : The graph of such a set-valued function is defined as A function F : → P f (X) is called measurable (sometimes also called weakly measurable) if the preimage of each open set is measurable, i.e., all 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 ( , , μ) be a complete σ -finite measure space and let X be a separable Banach space. For a set-valued function F : → P f (X) and p ∈ [1, ∞), we denote by F p the set of all p-integrable selections of F , i.e., where L p ( ; X, μ) denotes the space of Bochner measurable, p-integrable functions with respect to μ. 3 If F is integrably bounded, i.e., there exists a nonnegative function m ∈ For properties of this integral, see, e.g., Aubin and Frankowska [3,Chapter 8.6].

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 ≤ p < ∞, 1 < q ≤ 2 with 1 One operator satisfying these assumptions is, e.g., the p-Laplacian − p = −∇ ·(|∇| p−2 ∇) acting between the standard Sobolev spaces W 1,p 0 ( ) and W −1,p ( ) for a bounded Lipschitz domain , see, e.g., Zeidler [40, p. 489 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 well-defined, affine-linear, and bounded. The estimate is satisfied for all v ∈ L 1 (0, T ; X), i.e., K is also an affine-linear, bounded operator of The proof is based on simple calculations, therefore we omit it here. Following this lemma, we obtain the following properties of the operator BK. 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.
The formula (7) obviously holds for v n , n ∈ N, and w due to classical calculus. Therefore, we have On the other hand, the continuous embedding for all t ∈ [0, T ], in particular for t 1 , t 2 . This finishes the proof.

the initial condition is fulfilled in H and there exists
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 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.
, it is not possible to do integration by parts for each term separately. However, [15,Lemma 4.3] 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 . Due to the estimate (5) and the assumptions (F), we have v L q (0,T ;V * 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 truncationF of F viâ This set-valued functionF 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 H M 1 of radius M 1 . Due to the measurability of F , the first set is an element of L([0, T ]) ⊗ B(H ), and since the second set is obviously an element of the same σ -algebra,F is measurable.
For proving thatF fulfils the same continuity condition as F , let N ⊂ [0, T ] be the set of Lebesgue-measure 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 )} ⊂ graph(F (t, ·)) with v n → v and w n w for some v, w ∈ H . We have to show w ∈F (t, v). Since the radial retraction r M 1 in H is continuous, we have r M 1 (v n ) → r M 1 (v) in H . Then, w n ∈F (t, v n ) = F (t, r M 1 (v n )) and the continuity condition on F immediately yield w ∈ F (t, 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 in the sense of L q (0, T ; V * A ), which exists due to [15,Theorem 4.2,Corollary 5.2]. Note that in this case. Note also that v ∈ C w ([0, T ]; H ) and Kv ∈ 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 f in L q (0, T ; H ). Analogously to the proof of the a priori estimates (8)  In order to show that G is sequentially weakly continuous, we have to pass to the limit in the equation v n + 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 operatorK : by n, such that v n → v in L p (0, T ; H ). This yields f n , v n → f, v . 5 We also obviously have 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 ), we have lim sup n→∞ Av n , v n ≤ ã, v .
Now, for arbitrary w ∈ L p (0, T ; V A ), the monotonicity of A implies Therefore, we obtain lim inf n→∞ Av n , v n ≥ Aw, v − w + ã, w 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 : V A → V * A , Lebesgue's theorem on dominated convergence yields for r → 0 Av −ã, z = 0 for all z ∈ L p (0, T ; V A ), which impliesã = Av. As the last step of this proof, consider the operator R : is meant with respect to the truncationF 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 fixed-point theorem (see Glicksberg [20] and Fan [17]) to obtain the existence of f ∈ E such that f ∈ R(f ) = F 1 (G(f )). This implies that v = G(f ) solves (1) with the righthand sideF . However, due to the a priori estimate (8), we haveF (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 ⊆ R 3 be a bounded domain with boundary of class C 1,1 or a convex polyhedral and let p > 6. Consider where K is given by (2) and with g(v) = v 1−2/p L 2 ( ) v and a certain function a ∈ L q (0, T ), a(t) ≥ 0 a.e. Here, we have A given by − p with V A = W It is easy to see that 2 fulfils the assumptions (B). Thus, it remains to prove that F fulfils the assumptions (F).
To prove the continuity condition, consider {v n }, {w n } ⊂ L 2 ( ) and v, w ∈ L 2 ( ) with v n → v, w n 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 g(v n ) − w n L 2 ( ) ≤ a(t). The continuity of g and the lower semicontinuity of the norm yield i.e., w ∈ F (t, v).
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