Nonlocal semilinear evolution equations without strong compactness: theory and applications

3.1 Problem setting

(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;

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.

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

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

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.

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 $x\in T_n(q)$.

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 $L^1([a,b];E)$.

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. $t\in [a,b]$.

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.

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.

obtaining a contradiction.

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

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

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

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.

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.

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

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

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.

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 $h(t)=f(t,y(t),v(t))$, $t\in [0,d]$.