Nonlocal problems in Hilbert spaces

An existence result for differential inclusions in a separable Hilbert space is furnished. A wide family of nonlocal boundary value problems is treated, including periodic, anti-periodic, mean value and multipoint conditions. The study is based on an approximation solvability method. Advanced topological methods are used as well as a Scorza Dragoni-type result for multivalued maps. The conclusions are original also in the single-valued setting. An application to a nonlocal dispersal model is given.

1. Introduction. The paper deals with the nonlocal problem x (t) ∈ F (t, x(t)) for a.e. t ∈ [0, T ] in a separable Hilbert space H where F : [0, T ] × H H is a multivalued map (multimap) and M : C([0, T ], H) → H is a bounded linear operator. The investigation of periodic, antiperiodic, mean value and multi-point solutions is included. The multivalued framework can be motivated by the introduction of control terms into the process, by the appearance of jump discontinuities or by an incomplete knowledge of the model as in Section 3. Advanced topological methods were recently used for the study of (1), based on suitable topological degrees (see e.g. [1], [8] and [10]). Recent results in this context can be found in [3], [4], [6], [7], [8] and [12]. In particular, in [4] and [6] the assumptions involve the weak topology in the state space; an abstract homotopy invariant is introduced in [8], in order to detect steady-states solutions of (1) with an additional m-accreative term appearing also in [12]. A new approach was proposed in [3], based on the approximation solvability method and it was showed there that a quite general family of nonlinear terms can be considered. While the transversality condition in [3] (see [3, (3.1)]) is taken on all an open set, we refine here that technique by assuming a strictly located condition (see condition (3) below), i.e. only on a suitable boundary. This change is not marginal since it requires the use of a Scorza Dragoni type result for multivalued functions (see e.g. [2] and [9, Proposition 5.1]) and the introduction of a sequence of auxiliary problems (see (6)). The main result is Theorem 1.1 below; we point out that it is new also in a single-valued framework.
We denote by H ω the topological space H equipped with its weak topology and assume the following conditions 104 IRENE BENEDETTI, LUISA MALAGUTI AND VALENTINA TADDEI (H) (H, · H ) is a separable Hilbert space which is compactly embedded in a Banach space (E, · E ) with the relation of norms: for some q > 0; (F 1) F takes nonempty, convex, closed and bounded values and for every w ∈ H the multifunction F (·, w) : H is a linear bounded operator satisfying M ≤ 1.
Its proof appears in Section 2. Section 3 contains an application to a nonlocal dispersal model given by an integro-differential equation; its multivalued nature is given by the possible uncertainty of the integral kernel which is not determined, but belongs to a prescribed family of functions.
Step 1. Introduction of a sequence of problems in a finite dimensional space. Denote by {e n } ∞ n=1 an orthonormal basis of H and for every n ∈ N, let H n be an n−dimensional subspace of H with the basis {e k } n k=1 and P n be the projection of H onto H n . For the sake of simplicity, we denote by F and M also their restrictions F/ [0,T ]×Hn and M/ [0,T ]×Hn . We first prove that, for every fixed n ∈ N, the problem has a solution. By a solution to (4) we mean a function x ∈ W 1,1 ([0, T ], H n ) such that x(0) = P n M x and there exists f ∈ S x such that x (t) = P n f (t) for a.e. t ∈ [0, T ]. Let us denote by K = {x ∈ H : x H < R 0 } and by Q n the closed and convex set C([0, T ], K ∩ H n ).
Step 2. Introduction of a sequence of approximating problems. To prove that problems (4) are solvable we use an approximation result of Scorza-Dragoni type (see e.g. [9, Proposition 5.1]). According to Urisohn lemma, given ε ∈ (0, R 0 ), there exists a continuous function is well-defined, continuous and bounded in H.
Since P n is a linear operator, from (F1) it easily follows that P n F : [0, T ]×(K ∩H n ) H n has nonempty, convex and bounded values. We prove now that the map P n F has closed values too.
x) converging to w and w m ∈ F (t, x) with w m = P n w m for all m ∈ N. From (F1), without loss of generality, we get the existence of v ∈ F (t, x) such that w m v, hence that w m P n v, for the linearity and continuity of P n . From the uniqueness of the weak limit we then obtain w = P n v, i.e. that P n F has closed values. Since P n is continuous, (F1) also implies that yield the existence of a subsequence, denoted as the sequence, such that w m v ∈ F (t, x). According to the linearity and continuity of P n , we then get w = P n v, i.e. that P n F (t, ·) is closed, hence u.s.c.. Let τ be the Lebesgue measure in R. Applying [9, Proposition 5.1 ] to P n F : H n with closed, bounded, convex and possibly empty values satisfying G n (t, q) ⊂ P n F (t, q) for every (t, q) and a monotone decreasing sequence Step 3. Solvability of problem (6). To prove that problem (6) has a solution, we shall apply a classical continuation principle (see, e.g. [2, Proposition 2]). Fix q ∈ Q n ; according to the properties of G n and since S q = ∅ we have that moreover, it is well known that, for each q ∈ Q n , λ ∈ [0, 1] and f ∈ R nq , the linear initial value problem has a unique solution denoted by H nm (f, λ). Let us introduce now the multimap T nm : According to (F1) it has convex values. We prove now that T nm has a closed graph in According to the convergence of {q k }, condition (F4) and by the Dunford-Pettis Theorem there exists f ∈ L 1 ([0, T ], H n ) and a suitable subsequence of {f k } denoted as the sequence, such that f k f. Moreover, the continuity of φ implies that in L 1 ([0, T ], H n ). From (M) we get that x k (0) = λ k P n M q k → λP n M q and by the finite dimension of H n we then obtain that for all t ∈ [0, T ]. According to the uniqueness of the limit we get x = y. Finally, the upper semicontinuity of G n in [0, T ] \ θ m × (K ∩ H n ) for every m implies that f ∈ R nq , i.e. the closure of the graph of T nm for every m ∈ N.
Given now x ∈ T nm (Q n × [0, 1]), we have that m )]dt, for some q ∈ Q n , λ ∈ [0, 1] and f ∈ R nq . The boundedness and equicontinuity of T nm (Q n × [0, 1]) follow from (F4), the boundedness of Q n and the continuity of φ, thus the Ascoli-Arzelá theorem implies the compactness of T nm .
Therefore, j ≤ − 1 m R 0 < 0 for every j ∈ J (t 0 , q(t 0 )) and since the multimap G n is u.s.c. in [0, T ] \ θ m × (K ∩ H n ) and q is continuous, we can choose h sufficiently small such that (8) gives again a contradiction. Therefore it follows the existence of q nm ∈ Q n with q nm ∈ T nm (q nm , 1), i.e. of a solution of problem (6).
For every n ∈ N, we got a sequence {q nm } ⊂ AC([0, T ], K ∩ H n ) such that with f m ∈ R nqnm . Hence, {q nm } m is bounded. Moreover, according to (F4), {f m } m is integrably bounded, thus the boundedness of φ implies the integrable boundedness of {q nm }. The Ascoli-Arzelà Theorem then implies the existence of q n ∈ AC([0, T ], K ∩ H n ) such that {q nm } has a subsequence, again denoted as the sequence, with q nm → q n uniformly in [0, T ], and q nm q n in L 1 [0, T ]. Notice, moreover, that since φ is bounded and lim m→∞ χ θm (t) = 0 Consequently, a standard limiting argument (see e.g. [13, page 88]) implies that q n is a solution of (4).
3. Applications. We apply the developed abstract theory to the Cauchy multi-point problem (10) associated to a nonlocal diffusion process. The multivalued nature of the nonlinear integro-differential equation in (10) depends on the integral kernel, which can be unknown and can only be chosen in a suitable family of functions.
Assume that (f ) the partial derivative ∂f ∂z : [0, 1] × R → R is continuous and there is a positive constant N , such that We assume that b = N + 6δ|Ω|β, where δ = max 1] |f (t, 0)|. The symbol D stands for the derivative (i.e. the gradient) with respect to the variables in the vector ξ and for (t, z) ∈ [0, 1] × R, we denote f 2 (t, z) = ∂f (t,z) ∂z . By a solution to (10) we mean a continuous function u : [0, 1] × Ω → R whose partial derivative ∂u(t,ξ) ∂t exists, for a.a. t ∈ [0, 1], and it satisfies (10). Let H = W 1,2 (Ω, R) and E = L 2 (Ω, R). It is clear that H is a separable Hilbert space which is compactly embedded in E. By means of a reformulation of this problem we will prove the existence of a continuous function u(t, ξ) such that at every value t the function u(t, ·) belongs to the Sobolev space W 1,2 (Ω, R). To this aim, for each t ∈ [0, 1], set x(t) = u(t, ·). Then we can substitute (10) with the following problem where F : w(ξ)) for a.a. ξ ∈ Ω.

IRENE BENEDETTI, LUISA MALAGUTI AND VALENTINA TADDEI
First we remark that G is well-defined. Indeed for w ∈ H and v ∈ S, by Fubini's Theorem we have that {v(ξ, ·)} ∈ L 2 (Ω, R) for a.e. ξ ∈ Ω; so g can be defined and for a.e. ξ ∈ Ω it holds Trivially, by the definition of the set S it follows that g ∈ H.
From (f ) it follows that for all (t, z) ∈ [0, 1] × Ω. Moreover So f is well-defined as well. Trivially G has bounded and convex values. To prove that G has closed values we have to show that given w ∈ H and {g n } ⊂ H such that g n ∈ G(w) for any n ∈ N and g n H → g, it follows g ∈ G(w). First notice that w.l.o.g. {g n } almost pointwise converges to g. By the definition of the multimap G there exists a sequence of functions {v n } ⊂ W 1,2 (Ω × Ω, R) such that g n satisfies By the weak compactness of the set S there exists a subsequence, denoted as the sequence, such that v n v 0 , v 0 ∈ S in W 1,2 (Ω × Ω, R). By the compact embedding of W 1,2 (Ω × Ω, R) into L 2 (Ω × Ω, R), the weak convergence of {v n } in W 1,2 (Ω × Ω, R) implies its strong convergence in L 2 (Ω × Ω, R) and hence the almost pointwise convergence of a suitable subsequence. Denote with h : Ω → R the function: From the dominated almost pointwise convergence of {v n } to v 0 it follows that |g n (ξ)−h(ξ)| goes to zero as n goes to ∞ for a.e. ξ ∈ Ω. Therefore by the uniqueness of the limit we have that g(ξ) = h(ξ) for a.e. ξ ∈ Ω, hence g ∈ G(w). Now let w n E → w 0 . We have Hence f (t, w n ) E → f (t, w 0 ) and then f (t, ·) is E − E continuous. Moreover the multimap G is E − E u.s.c.. Indeed, if w n E → w 0 , w.l.o.g. {w n } almost pointwise converges to w 0 and the convergence is dominated in E. Let moreover g n ∈ G(w n ), implying the existence of {v n } ⊂ S such that g n (ξ) = w n (ξ) Ω v n (ξ, η)w n (η) dη − bw n (ξ), ξ ∈ Ω.