Nonlocal semilinear evolution equations without strong compactness: theory and applications

A semilinear multivalued evolution equation is considered in a reflexive Banach space. The nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. No strong compactness is assumed, neither on the evolution operator generated by the linear part, or on the nonlinear term. A wide family of nonlocal associated boundary value problems is investigated by means of a fixed point technique. Applications are given to an optimal feedback control problem, to a nonlinear hyperbolic integro-differential equation arising in age-structure population models, and to a multipoint boundary value problem associated to a parabolic partial differential equation.

MSC:34G25, 34B10, 34B15, 47H04, 28B20, 34H05.

The paper deals with the following semilinear evolution inclusion in a reflexive Banach space E:

where {\{A(t)\}}_{t\in [a,b]} is a family of linear not necessarily bounded operators, F:[a,b]\times E⊸E is a multivalued map (multimap for short) and L:C([a,b];E)\to E, M:C([a,b];E)\to E are a linear and a nonlinear operator respectively. See Section 3 for detailed assumptions.

The boundary condition considered is very general and includes the initial value problem, the two-point problem as well as several nonlocal conditions. For instance, the following two cases are covered by our general framework when Lx=x(a):

1. (i)

   M(y)=\frac{1}{b-a}{\int }_a^{b}g(y(t))dt with g:E\to E such that \parallel g(x_1)-g(x_2)\parallel \le \kappa \parallel x_1-x_2\parallel for every x_1,x_2\in E and ;

2. (ii)

   M(y)={\sum }_{i=1}^n{\alpha }_{i}y(t_i)+y_0, with y_0\in E, {\alpha }_i\in \mathbb{R}\setminus \{0\}, t_i\in [a,b], i=1,\dots ,n and ,

where D is the uniform bound for the evolution operator U generated by {\{A(t)\}}_{t\in [a,b]} (see formula (2.1)) and K is the invertible linear operator LU(\cdot ,a) (see hypothesis (L2)).

Since the pioneering work of Byszewsky [1], nonlocal problems have been extensively studied due to being of interest in several contexts. For instance, the multipoint boundary value problem (case (ii) above) has better application in physics than the classical initial problem because it allows measurements at t=t_i\in [a,b], i=1,\dots ,n rather than just at t=0. Abstract problems similar to (1.1) are considered, for example, in [2, 3] and [4] with the linear part generator of a C_0-semigroup or of a strongly continuous evolution operator; the semigroup and the evolution operator are always assumed to be compact there. In [5] a semilinear functional differential inclusion associated with a multivalued nonlocal boundary condition is considered, and in [6] an abstract problem as (1.1) is studied with a lower Scorza-Dragoni type nonlinearity; in both papers, regularity conditions by means of a measure of non-compactness on the nonlinearity are assumed. However, in [7] and [8] we proved the existence of classical solutions for the inclusion (1.1) associated with a two-point boundary condition by means of weak topology, thus avoiding hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term F. The same is done in the present paper, for a wider class of boundary conditions, in order to find a result of existence for mild solutions of (1.1) (see Theorem 3.1). In particular, unlike [5] and [6], this allows to treat a class of nonlinear terms F which are not necessarily compact-valued.

Problems with a nonlinear nonlocal boundary condition of type (1.1) occur in many fields like pharmacokinetics, neural networks, epidemiology models etc, see e.g. [[9], Section 10.2]. In Section 4 we present several applications of the abstract result Theorem 3.1. More precisely, with our techniques we can handle diffusion problems of the form

joined with several boundary conditions and, in example (4.1), associated with a feedback control. This model arises from population dynamics theory in the case of intraspecific competition for resources (see [10]), while we refer to [11] for recent achievements for its controllability.

Moreover, we can consider the following class of nonlinear hyperbolic integro-differential problems arising in population biology:

They describe the time evolution of the age-structure of a population. The first models studying the age-structure of a population were the linear ones by McKendrick and Von Forster [12, 13]. The nonlinear model was introduced by Gurtin-MacCamy [14]. We are able to solve problem (1.2), writing it as an abstract problem of type (1.1). A similar approach for this type of models, but in a different setting, can be found in [15].

In Section 3 the abstract existence result is proved by means of a linearization argument and the application of the Ky Fan fixed point theorem. We investigate problem (1.1) in the space of continuous functions C([a,b];E) endowed with the weak topology. As was done in [4], on the nonlinear term F we assume a growth condition (see (F3′)) stronger than the sublinear growth, but weaker than the global integrably boundedness; instead, the sublinear growth condition is enough for studying the Cauchy problem and we have, as a by-product, the weak compactness of the solution set (see [16] and also Remark 3.2).

In Section 2 some definitions and results used in the sequel of the paper are recalled.

As far as we know, all the results showed are new even for the case of a single-valued nonlinear term F.

Let (E,\parallel \cdot \parallel ) be a reflexive Banach space, and let E_w be the space E endowed with the weak topology. We denote by nB the closed ball centered at the origin and of radius n of E, and for a set A\subset E, the symbol {\overline{A}}^w denotes the weak closure of A. We take for granted that a bounded subset A of a reflexive Banach space E is weakly relatively compact. In the whole paper, we denote by {\parallel \cdot \parallel }_1 and {\parallel \cdot \parallel }_0 the L^1([a,b];E)-norm and the C([a,b];E)-norm respectively and by ν the Lebesgue measure on [a,b]. We recall (see [[17], Theorem 4.3]) that a sequence \{x_n\} weakly converges to an element x\in C([a,b];E) if and only if

1. 1.

   there exists N>0 such that for every n\in \mathbb{N} and t\in [a,b], \parallel x_n(t)\parallel \le N;

2. 2.

   for every t\in [a,b], x_n(t)\rightharpoonup x(t);

Denoting \mathrm{\Delta }=\{(t,s)\in [a,b]\times [a,b]:a\le s\le t\le b\}, we recall that a two-parameter family {\{U(t,s)\}}_{(t,s)\in \mathrm{\Delta }}, where U(t,s):E\to E is a bounded linear operator and (t,s)\in \mathrm{\Delta }, is called an evolution system if the following conditions are satisfied:

1. 1.

   U(s,s)=I, a\le s\le b; U(t,r)U(r,s)=U(t,s), a\le s\le r\le t\le b;

2. 2.

   (t,s)\mapsto U(t,s) is strongly continuous on Δ, i.e., the map (t,s)\to U(t,s)x is continuous on Δ for every x\in E.

For every evolution system, we can consider the respective evolution operator U:\mathrm{\Delta }\to \mathcal{L}(E), where \mathcal{L}(E) is the space of all bounded linear operators in E.

We observe that since the evolution operator U is strongly continuous on the compact set Δ, by the uniform boundedness theorem, there exists a constant D=D_{\mathrm{\Delta }}>0 such that

for details about evolution systems, we refer to [18, 19].

Let G:L^1([a,b];E)\to C([a,b];E) be the Cauchy operator defined by

Finally, for the sake of completeness, we recall some well-known results that we will need in the main section.

Firstly, we state the Ky Fan fixed point theorem.

Theorem 2.1 ([[20], Theorem 1])

Let X be a Hausdorff locally convex topological vector space, let V be a compact convex subset of X, and let G:V⊸V be an upper semicontinuous multimap with closed, convex values. Then G has a fixed point.

Then we mention two results that are contained in the so-called Eberlein-Smulian theory.

Theorem 2.2 ([[21], Theorem 1, p.219])

Let Ω be a subset of a Banach space X. The following statements are equivalent:

1. 1.

   Ω is relatively weakly compact;

2. 2.

   Ω is relatively weakly sequentially compact.

Corollary 2.1 ([[21], p.219])

Let Ω be a subset of a Banach space X. The following statements are equivalent:

1. 1.

   Ω is weakly compact;

2. 2.

   Ω is weakly sequentially compact.

We recall the Krein-Smulian theorem.

Theorem 2.3 ([[22], p.434])

The convex hull of a weakly compact set in a Banach space E is weakly compact.

In conclusion, we recall the Pettis measurability theorem, which we use in Section 4.

Theorem 2.4 ([[23], p.278])

Let (X,\mathrm{\Sigma }) be a measure space, and let E be a separable Banach space. Then f:X\to E is measurable if and only if for every e\in E^{\mathrm{\prime }} the function e\circ f:X\to \mathbb{R} is measurable with respect to Σ and the Borel σ-algebra in ℝ.

3.1 Problem setting

We consider the semilinear differential inclusion (1.1) under the following hypotheses:

1. (A)

   {\{A(t)\}}_{t\in [a,b]} is a family of linear not necessarily bounded operators with A(t):D(A)\subset E\to E, D(A) not depending on t and dense in E, generating an evolution operator U:\mathrm{\Delta }\to \mathcal{L}(E);

(F1) for every t\in [a,b] and x\in E, F(t,x) is nonempty, convex and weakly compact, and for every x\in E, F(\cdot ,x):[a,b]⊸E has a measurable selection;

(F2) for a.a. t\in [a,b], F(t,\cdot ):E_w⊸E_w is weakly sequentially closed;

(L1) L:C([a,b];E)\to E is a linear continuous operator;

(L2) the operator K:E\to E defined as Kx=LU(\cdot ,a)x is invertible;

1. (M)

   M:C([a,b];E)\to E is a weakly sequentially continuous operator mapping bounded sets into bounded sets and such that

   (3.1)

We observe that if Lx=x(a), we have that K equals the identity operator, thus condition (L2) is trivially satisfied. When A(t)\equiv A is the abstract formulation of the Laplace operator, it generates a strongly continuous compact semigroup of contractions S(t) such that S(t)x=x if and only if x=0 (see, e.g., [4] and [24]). According to the Fredholm alternative, S(t)-I is invertible for all t\in (a,b], where I denotes the identity operator. Hence, the associated periodic problem satisfies condition (L2).

In the remaining part of this section, we always assume the following condition on the multivalued map F.

(F3) for each bounded subset Ω of E, there exists {\eta }_{\mathrm{\Omega }}\in L^1([a,b];\mathbb{R}) such that, for a.a. t\in [a,b], {sup}_{x\in \mathrm{\Omega }}\parallel F(t,x)\parallel \le {\eta }_{\mathrm{\Omega }}(t).

Instead, for our main result (see Theorem 3.1), we need the stronger assumption as follows.

(F3′) For every n\in \mathbb{N}, there exists {\phi }_n\in L^1([a,b];\mathbb{R}) such that, for a.a. t\in [a,b], {sup}_{\parallel x\parallel \le n}\parallel F(t,x)\parallel \le {\phi }_n(t) and

The previous boundedness condition is anyway weaker than the one that is often used in literature in relation with boundary value problems, i.e., the following one.

(F3^′′) There exists \eta \in L^1([a,b];\mathbb{R}) such that, for every x\in E and a.a. t\in [a,b], \parallel F(t,x)\parallel \le \eta (t).

We look for mild solutions, i.e., for functions that satisfy the following definition.

Definition 3.1 A continuous function x:[a,b]\to E is said to be a mild solution of problem (1.1) if there exists a function f\in L^1([a,b];E) such that f(t)\in F(t,x(t)) for a.a. t\in [a,b] and

By the assumption of the invertibility of the operator K, it is possible to write the solution as

Remark 3.1 If E is a separable Banach space, according to the Kuratowski-Ryll-Nardzewski theorem (see [[25], Theorem A]), the measurability of F(\cdot ,x) for every x\in E is sufficient to obtain condition (F1). Moreover, in [25] sufficient conditions are given to obtain the existence of a strongly measurable selection for the multimap F(\cdot ,x) in not necessarily separable Banach spaces.

3.2 Existence result

Given q\in C([a,b];E), let us denote

Note that with our hypotheses on F, the set S_q is always nonempty as the following proposition shows.

Proposition 3.1 (see [[16], Proposition 2.2])

Let F:[a,b]\times E⊸E be a multimap satisfying properties (F1), (F2) and (F3). Then, for every q\in C([a,b];E), the set S_q is nonempty.

We denote by T:C([a,b];E)⊸C([a,b];E) the multioperator defined as

The fixed points of the multioperator T are mild solutions of problem (1.1).

Fix n\in \mathbb{N}, let Q_n be the closed ball centered at the origin and of radius n of C([a,b];E), and denote by T_n=T{\lfloor }_{Q_n}:Q_n⊸C([a,b];E) the restriction of the operator T on the set Q_n. We show some properties of T_n.

Proposition 3.2 The multioperator T_n has a weakly sequentially closed graph.

Proof Let \{q_m\}\subset Q_n and \{x_m\}\subset C([a,b];E) satisfying x_m\in T_n(q_m) for all m and q_m\rightharpoonup q, x_m\rightharpoonup x in C([a,b];E); we will prove that x\in T_n(q).

Since q_m\in Q_n for all m and q_m(t)\rightharpoonup q(t) for every t\in [a,b], it follows that \parallel q(t)\parallel \le {lim\hspace{0.17em}inf}_{m\to \mathrm{\infty }}\parallel q_m(t)\parallel \le n for all t. Moreover, M is weakly sequentially continuous, and hence M(q_m)\rightharpoonup M(q) in E. The fact that x_m\in T_n(q_m) means that there exists a sequence \{f_m\}, f_m\in S_{q_m}, such that

We observe that, according to (F3), \parallel f_m(t)\parallel \le {\eta }_{nB}(t) for a.a. t and every m, i.e., \{f_m\} is bounded and uniformly integrable and \{f_m(t)\} is bounded in E for a.a. t\in [a,b]. Hence, by the reflexivity of the space E and by the Dunford-Pettis theorem (see [[22], p.294]), we have the existence of a subsequence, denoted as the sequence, and a function g such that f_m\rightharpoonup g in L^1([a,b];E).

Moreover, we have

Indeed, let e^{\mathrm{\prime }}:E\to \mathbb{R} be a linear continuous operator. By the linearity and continuity of the integral and the evolution operator U(t,s), we have that the operator

is a linear and continuous operator from L^1([a,t];E) to ℝ for all t\in [a,b]. Then, from the definition of the weak convergence, we have for every t\in [a,b]

Moreover, \parallel G{f}_m(t)\parallel \le D{\parallel {\eta }_{nB}\parallel }_1 for all m and every t\in [a,b]. Consequently, G{f}_m\rightharpoonup Gg weakly in C([a,b];E). Hence, for the linearity and continuity of the operator L, we have that LG{f}_m\rightharpoonup LGg. Finally, by the linearity and continuity of K^{-1} (due to hypotheses (L1) and (L2)) and U(t,s), we have that

implying, for the uniqueness of the weak limit in E, that x_0(t)=x(t) for all t\in [a,b].

To conclude, we only have to prove that g(t)\in F(t,q(t)) for a.a. t\in [a,b].

By Mazur’s convexity theorem, we have the existence of a sequence

satisfying {\tilde{f}}_m\to g in L^1([a,b];\mathbb{R}) and, up to a subsequence, there is N_0\subset [a,b] with Lebesgue measure zero such that {\tilde{f}}_m(t)\to g(t) for all t\in [a,b]\setminus N_0. With no loss of generality, we can also assume that F(t,\cdot ):E_w⊸E_w is weakly sequentially closed and {sup}_{\parallel x\parallel \le n}\parallel F(t,x)\parallel \le {\eta }_{nB}(t) for every t\notin N_0.

Fix t_0\notin N_0 and assume, by contradiction, that g(t_0)\notin F(t_0,q(t_0)). By the reflexivity of the space E, the restriction F_{nB}(t_0,\cdot ) of the multimap F(t_0,\cdot ) on the set nB is weakly compact. Hence, by Corollary 2.1, F_{nB}(t_0,\cdot ) is a weakly closed multimap, and by [[26], Theorem 1.1.5], it is weakly u.s.c. Since \parallel q(t_0)\parallel \le n and since F_{nB}(t_0,q(t_0)) is closed and convex, from the Hahn-Banach theorem, there is a weakly open convex set V\supset F_{nB}(t_0,q(t_0)) satisfying q(t_0)\notin \overline{V}. Since F_{nB}(t_0,\cdot ) is u.s.c., we can also find a weak neighborhood V_1 of q(t_0) such that F_{nB}(t_0,x)\subset V for all x\in V_1 with \parallel x\parallel \le n. Note that \parallel q_m(t_0)\parallel \le n for all m. The convergence q_m(t_0)\rightharpoonup q(t_0) as m\to \mathrm{\infty } then implies the existence of m_0\in \mathbb{N} such that q_m(t_0)\in V_1 for all m>m_0. Therefore f_m(t_0)\in F_{nB}(t_0,q_m(t_0))\subset V for all m>m_0. The convexity of V implies that {\tilde{f}}_m(t_0)\in V for all m>m_0 and, by the convergence, we arrive at the contradictory conclusion that g(t_0)\in \overline{V}. We conclude that g(t)\in F(t,q(t)) for a.a. t\in [a,b]. □

Proposition 3.3 The multioperator T_n is weakly compact.

Proof We first prove that T_n(Q_n) is weakly relatively sequentially compact.

Let \{q_m\}\subset Q_n and \{x_m\}\subset C([a,b];E) satisfying x_m\in T_n(q_m) for all m. By the definition of the multioperator T_n, there exists a sequence \{f_m\}, f_m\in S_{q_m}, such that

Moreover, reasoning as in Proposition 3.2, we have that there exist a subsequence, denoted as the sequence, and a function g such that f_m\rightharpoonup g in L^1([a,b];E). Since the operator M maps bounded sets into bounded sets and Q_n is bounded, we obtain that M(q_m)\rightharpoonup \overline{x}\in E up to subsequence. Therefore

Furthermore, by the weak convergence of \{f_m\} and \{M(q_m)\}, we have

for all m\in \mathbb{N}, for all t\in [a,b], and for some N>0. Reasoning again like in Proposition 3.2, it is then easy to prove that x_m\rightharpoonup l in C([a,b];E). Thus T_n(Q_n) is weakly relatively sequentially compact, hence weakly relatively compact by Theorem 2.2. □

Proposition 3.4 The multioperator T_n has convex and weakly compact values.

Proof Fix q\in Q_n, since F is convex-valued, the set T_n(q) is convex from the linearity of the integral and of the operators K^{-1}, L, and U(t,s) for all (t,s)\in \mathrm{\Delta }. The weak compactness of T_n(q) follows by Propositions 3.3 and 3.2. □

Theorem 3.1 Under the assumptions (A), (F1), (F2), (F3′), (L1), (L2) and (M), problem (1.1) has at least a mild solution.

Proof We show that there exists n\in \mathbb{N} such that the operator T_n maps the ball Q_n into itself.

According to (3.2), there exists a subsequence, still denoted as the sequence, such that

Assume by contradiction that there exist two sequences \{q_n\} and \{x_n\} such that q_n\in Q_n, x_n\in T_n(q_n) and x_n\notin Q_n for all n\in \mathbb{N}. By the definition of T_n, there exists a sequence \{g_n\}\subset L^1([a,b];E), g_n(s)\in F(s,q_n(s)) \mathrm{\forall }n\in \mathbb{N} and q.o. s\in [a,b], such that

By the assumption x_n\notin Q_n, we must have, for every n,

Moreover q_n\in Q_n implies by (F3′) that \parallel g_n(t)\parallel \le {\phi }_n(t) for a.a. t\in [a,b], hence {\parallel g_n\parallel }_1\le {\parallel {\phi }_n\parallel }_1. Consequently,

Therefore

Notice that if for every n\in \mathbb{N}, then {lim}_{n\to \mathrm{\infty }}\frac{\parallel M(q_n)\parallel }{n}=0 because M maps bounded sets into bounded sets; if {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\parallel q_n\parallel }_0=+\mathrm{\infty }, then by hypothesis we have

In both cases . Moreover, by (3.4)

Hence

obtaining a contradiction.

Fix now n\in \mathbb{N} such that T_n(Q_n)\subseteq Q_n. By Proposition 3.3 the set V_n={\overline{T_n(Q_n)}}^w is a weakly compact set. Let now W_n=\overline{co}(V_n), where \overline{co}(V_n) denotes the closed convex hull of V_n. By Theorem 2.3, W_n is a weakly compact set. Moreover, from the fact that T_n(Q_n)\subset Q_n and that Q_n is a convex closed set, we have that W_n\subset Q_n and hence

Therefore, from Proposition 3.2 and from Corollary 2.1, we obtain that the restriction of the multimap T_n on W_n has a closed graph, and hence, by Proposition 3.4, it is weakly u.s.c (see [[26], Theorem 1.1.5]). The conclusion then follows by Theorem 2.1. □

When dealing with Cauchy problems, it is possible to weaken condition (F3′) and to obtain the weak compactness of the solution set, as pointed out in the following remark.

Remark 3.2 In [[27], Theorem 4.1], under condition (F3), an existence result for local classical solutions for the Cauchy problem associated to a semilinear differential inclusion is proved. Following the proof’s outline of the cited theorem and combing it with Propositions 3.2, 3.3, 3.4, it is easy to obtain the existence of local mild solutions for (1.1) with a Cauchy initial condition under the same boundedness assumption (F3). Moreover, in the same paper (see Theorem 4.2), a global classical solution is obtained for the initial value problem under the following classical assumption.

(F3^′′′) There exists \alpha \in L^1([a,b];\mathbb{R}) such that, for every x\in E and a.a. t\in [a,b], \parallel F(t,x)\parallel \le \alpha (t)(1+\parallel x\parallel ).

Again, under (F3^′′′), in [[16], Theorem 4.2] the existence of a nonempty weakly compact set of mild solutions is showed for the initial value problem.

We point out the fact that the boundedness hypotheses considered are related as follows: condition (F3^′′) implies (F3′) which implies (F3^′′′) which implies (F3).

4.1 Optimal feedback control problems of population diffusion models

Let Ω be a bounded domain in {\mathbb{R}}^n with boundary of class C^{\mathrm{\infty }} and u:[0,d]\times \mathrm{\Omega }\to \mathbb{R}. We consider the following feedback control problem:

with W(r)=\{s\in \mathbb{R}:\ell r+m_1\le s\le \ell r+m_2\}, where \ell >0 and .

We assume the following hypotheses:

1. (i)

   a and b are globally measurable in [0,d]\times \mathrm{\Omega }, and there exist two functions {\phi }^1,{\phi }^2\in L^1([0,d];\mathbb{R}) such that, for a.a. x\in \mathrm{\Omega } and every t\in [0,d],

   

the map p:[0,d]\times \mathrm{\Omega }\times \mathbb{R}\to \mathbb{R} satisfies the following conditions:

1. (ii)

   for all r\in \mathbb{R}, p(\cdot ,\cdot ,r):[0,d]\times \mathrm{\Omega }\to \mathbb{R} is measurable;

2. (iii)

   for a.a. t\in [0,d] and x\in \mathrm{\Omega }, p(t,x,\cdot ):\mathbb{R}\to \mathbb{R} is continuous;

3. (iv)

   there exists {\phi }^3\in L^1([0,d];\mathbb{R}) such that, for a.a. x\in \mathrm{\Omega } and every t\in [0,d] and r\in \mathbb{R},

   |p(t,x,r)|\le {\phi }^3(t).

Let y:[0,d]\to L^2(\mathrm{\Omega };\mathbb{R}), v:[0,d]\to L^2(\mathrm{\Omega };\mathbb{R}), f:[0,d]\times L^2(\mathrm{\Omega };\mathbb{R})\times L^2(\mathrm{\Omega };\mathbb{R})\to L^2(\mathrm{\Omega };\mathbb{R}), and V:L^2(\mathrm{\Omega };\mathbb{R})⊸L^2(\mathrm{\Omega };\mathbb{R}) be the maps defined by

We can write the feedback control problem (4.1) as a first-order inclusion in the Hilbert space E=L^2(\mathrm{\Omega };\mathbb{R})

where F(t,z)=f(t,z,V(z)), A:W^{2,2}(\mathrm{\Omega };\mathbb{R})\cap W_0^{1,2}(\mathrm{\Omega };\mathbb{R})\to L^2(\mathrm{\Omega };\mathbb{R}) is the linear operator defined as Ay=\mathrm{\Delta }y and y_0=u_0(\cdot ).

The operators L:C([0,d];E)\to E and M:C([0,d];E)\to E are defined as Ly=y(0) and M(y)=y_0. Trivially, L is a linear and bounded operator, and M is sequentially continuous with respect to the weak topology, and the operator K of condition (L2) is the identity operator, in particular it is invertible. Moreover, it is known that A generates a semigroup of contractions U(t) on E with the constant D of (2.1) equal to 1 (see, e.g., [19] p.209-210). Hence hypotheses (A),(L1),(L2) and (M) are satisfied with l=0.

We prove that there exists a mild solution

where v(s)\in V(y(s)), s\in [0,d].

We show now that all the hypotheses of Theorem 3.1 are satisfied.

The map g:[a,b]\to L^2(\mathrm{\Omega };\mathbb{R}) defined as

is a selection of F(\cdot ,z). Moreover, by the separability of the space L^2(\mathrm{\Omega };\mathbb{R}), conditions (i), (ii), (iii), and by the Pettis measurability theorem (see Theorem 2.4), we obtain that g is a measurable map, and so we have obtained for every z\in L^2(\mathrm{\Omega };\mathbb{R}) the existence of a measurable selection of F(\cdot ,z).

We prove now that the map F verifies condition (F2), i.e., that the map F(t,\cdot ):L^2(\mathrm{\Omega };\mathbb{R})\to L^2(\mathrm{\Omega };\mathbb{R}) is weakly sequentially closed for a.a. t\in [0,d].

We start proving the sequential closedness with respect to the weak topology of the multimap V.

Let \{{\alpha }_j\}\in L^2(\mathrm{\Omega };\mathbb{R}), {\alpha }_j\rightharpoonup \alpha in L^2(\mathrm{\Omega };\mathbb{R}), \{{\omega }_j\}\in L^2(\mathrm{\Omega };\mathbb{R}), {\omega }_j\in V({\alpha }_j), {\omega }_j\rightharpoonup \omega. According to Mazur’s convexity lemma, for each j there exist k_j\in \mathbb{N} and positive numbers {\lambda }_{ji}, i=0,\dots ,k_j such that {\sum }_{i=0}^{k_j}{\lambda }_{ji}=1 and the sequence {\tilde{\alpha }}_j={\sum }_{i=0}^{k_j}{\lambda }_{ji}{\alpha }_{j+i}\to \alpha in L^2(\mathrm{\Omega };\mathbb{R}). Then we can extract a subsequence, denoted as the sequence, satisfying {\tilde{\alpha }}_j(x)\to \alpha (x) for a.a. x\in \mathrm{\Omega }. We have that the convex combination {\tilde{\omega }}_j={\sum }_{i=0}^{k_j}{\lambda }_{ji}{\omega }_{j+i} converges weakly to ω. Moreover, by the definition of V, we have that

Applying again Mazur’s convexity lemma, for each j there exist h_j\in \mathbb{N} and positive numbers {\nu }_{ji}, i=0,\dots ,h_j such that {\sum }_{i=0}^{h_j}{\nu }_{ji}=1 and the sequence {\tilde{\tilde{\omega }}}_j={\sum }_{i=0}^{h_j}{\nu }_{ji}{\tilde{\omega }}_{j+i}\to \omega in L^2(\mathrm{\Omega };\mathbb{R}). Then we can extract a subsequence, denoted as the sequence, satisfying {\tilde{\tilde{\omega }}}_j(x)\to \omega (x) for a.a. x\in \mathrm{\Omega }. As before,

Then passing to the limit, we obtain

i.e., \omega \in V(\alpha ).

Let now t\in [0,d] be fixed, let \{{\alpha }_n\}\subset L^2(\mathrm{\Omega };\mathbb{R}), weakly convergent to \alpha \in L^2(\mathrm{\Omega };\mathbb{R}), and let \{w_n\}\in L^2(\mathrm{\Omega };\mathbb{R}) with w_n\in F(t,{\alpha }_n) for every n\in \mathbb{N}, weakly convergent to w\in L^2(\mathrm{\Omega };\mathbb{R}). By the definition of the multimap F, we have

where f_1,f_2:[0,d]\times L^2(\mathrm{\Omega };\mathbb{R})\to L^2(\mathrm{\Omega };\mathbb{R}), f_1(t,\alpha )(x)=p(t,x,{\int }_{\mathrm{\Omega }}\alpha (\xi )d\xi )\alpha (x), f_2(t,\beta )(x)=a(t,x)\beta (x)+b(t,x).

By the definition of the multimap V and the weak convergence of \{{\alpha }_n\}, we have that the sequence \{{\beta }_n\} is norm bounded. Hence, by the reflexivity of the space L^2(\mathrm{\Omega };\mathbb{R}), up to a subsequence, \{{\beta }_n\} weakly converges to \beta \in L^2(\mathrm{\Omega };\mathbb{R}) and the weak closure of the multimap V implies \beta \in V(\alpha ). Moreover, by the continuity of the map p and condition (iv), we have that \{f_1(t,{\alpha }_n)\} converges weakly to f_1(t,\alpha ) and it is easy to see that \{f_2(t,{\beta }_n)\} converges weakly to f_2(t,\beta ). In conclusion, we have obtained

Furthermore, easily, V has convex and closed values; thus, by the linearity of the map f_2 and following the same reasonings as above, F is convex closed-valued as well.

Finally, the multimap F:[0,d]\times L^2(\mathrm{\Omega };\mathbb{R})⊸L^2(\mathrm{\Omega };\mathbb{R}) verifies all the hypotheses of Theorem 3.1. Indeed from (i), (iv) and again from the definition of V, we have that

Denoting with \phi (t)={\phi }^3(t)+2\ell {\phi }^1(t)+{|\mathrm{\Omega }|}^{1/2}[(m_1+m_2){\phi }^1(t)+{\phi }^2(t)], we have

obtaining both that for every t\in [0,d] and \alpha \in L^2(\mathrm{\Omega };\mathbb{R}) the set F(t,\alpha ) is bounded (hence relatively compact by the reflexivity of L^2(\mathrm{\Omega };\mathbb{R})) and that condition (F3^′′′) is satisfied. Then applying Theorem 3.1 (see also Remark 3.2), we obtain the existence of a function such that

with h(s)\in F(s,y(s))=f(s,y(s),V(y(s))).

Finally, applying the implicit function theorem of Filippov type (see [[28], Theorem 7.2]), we have that there exists v:[0,d]\to L^2(\mathrm{\Omega };\mathbb{R}) such that v(t)\in V(y(t)) and h(t)=f(t,y(t),v(t)), t\in [0,d].

Theorem 4.1 Let j:C([0,d];L^2(\mathrm{\Omega };\mathbb{R}))\to \mathbb{R} be a lower semicontinuous (l.s.c. for short) functional with respect to the weak topology. Then, under conditions (i)-(iv), there exists a mild solution (u^{\ast },{\omega }^{\ast }) of problem (4.1) such that

where \mathcal{S}(u_0) is the set of all mild trajectories of the system (4.1) with the initial value u_0.

Proof Under the hypotheses (i)-(iv), the set \mathcal{S}(u_0)\ne \mathrm{\varnothing }; see the proof above. Moreover, according to [[16], Theorem 4.2] (see also Remark 3.2), the solution set \mathcal{S}(u_0) is weakly compact in C([0,d],L^2(\mathrm{\Omega };\mathbb{R})). Let u^{\ast } be the minimizer of j on \mathcal{S}(u_0), and let {\omega }^{\ast } be the corresponding control, then the pair (u^{\ast },{\omega }^{\ast }) is the required optimal solution. □

4.2 Age-structure population model

We consider the nonlinear hyperbolic integro-differential problem (1.2) which was already introduced in Section 1. This model arises in population biology and describes the time evolution of the age-structure of a population. Here we consider the case of a nonlinear equation with linear boundary conditions. The independent variables t and a denote respectively time and age, and u(t,a) represents the density of individuals of age a at time t. The death rate f is a nonnegative term depending on the time, the age and the total size of the population {\int }_0^{B}u(t,a)da. The boundary condition accounts for the birth in the population. The weight function b measures the fertility at age a.

We consider problem (1.2) under the following hypothesis:

1. (i)

   for all c\in \mathbb{R}, f(\cdot ,\cdot ,c):[0,T]\times [0,B]\to \mathbb{R} is measurable;

2. (ii)

   for a.a. t\in [0,T] and a\in [0,B], f(t,a,\cdot ):\mathbb{R}\to \mathbb{R} is continuous;

3. (iii)

   there exists \alpha \in L^1([0,T];\mathbb{R}) such that, for every t\in [0,T], a\in [0,B] and c\in \mathbb{R}, |f(t,a,c)|\le \alpha (t);

4. (iv)

   b\in L^2([0,B];\mathbb{R}).

Problem (1.2) can be written as the following Cauchy problem in the Banach space E=L^2([0,B];\mathbb{R}):

where y:[0,T]\to E is defined as y(t)=u(t,\cdot ), y_0=u(\cdot ), F:[0,T]\times E\to E is the single-valued map F(t,y)(a)=-f(t,a,{\int }_0^{B}y(s)ds)y(a) and A:D(A)\to E, with

is the linear operator Ay=-y^{\prime }. A is the generator of the translation semigroup, which satisfies the identity

This semigroup is intensely studied in [29, 30], thus we do not give any details here. We just recall that the translation semigroup is not compact.

As showed in Section 4.1, the initial condition y(0)=u_0 satisfies (L1), (L2) and (M).

According to (iii), for every y\in L^2([0,B];\mathbb{R}), the function a\to f(t,a{\int }_0^{B}y(s)ds)y(a) belongs to L^2([0,B];\mathbb{R}), hence F is nonempty-valued. Moreover, the Pettis measurability theorem (see Theorem 2.4), the separability of L^2([0,B];\mathbb{R}) and conditions (i) and (ii) imply that F is globally measurable (see [[26], Corollary 1.3.1]), and hence, according to Remark 3.1, since F is single-valued, condition (F1) is also satisfied.

We now prove that F(t,\cdot ) is weakly sequentially continuous for a.a. t, so we take y_n\rightharpoonup y in L^2([0,B],\mathbb{R}). Then {\int }_0^B{y}_n(s)ds\to {\int }_0^{B}y(s)ds, thus (ii) implies that f(t,a,{\int }_0^B{y}_n(s)ds)\to f(t,a,{\int }_0^{B}y(s)ds) for a.a. a\in [0,B]. Moreover, it is possible to show that f(t,a,{\int }_0^B{y}_n(s)ds)y_n\rightharpoonup f(t,a,{\int }_0^{B}y(s)ds)y in L^2([0,B];\mathbb{R}) obtaining condition (F2).

Finally, according to (iii), we have, for a.a. t\in [0,T] and every y\in L^2([0,B];\mathbb{R}),

and so the growth condition (F3^′′′) is satisfied. Recalling Remark 3.2, this condition is sufficient for the existence of at least a solution for the Cauchy problem.

Hence we find a mild solution y\in C([0,T];L^2([0,B];\mathbb{R})). We stress that by (4.4) the solution u(t,\cdot )=y(t) is a mild solution of (1.2) satisfying the required boundary conditions. Indeed,

Remark 4.1

With our techniques, we can attach also problems of the following general form: