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:
\{\begin{array}{c}{u}_{t}=\mathrm{\Delta}u+p(t,x,{\int}_{\mathrm{\Omega}}u(t,\xi )\phantom{\rule{0.2em}{0ex}}d\xi )u(t,x)+a(t,x)w(t,x)+b(t,x),\hfill \\ \phantom{\rule{1em}{0ex}}t\in [0,d],x\in \mathrm{\Omega},w(t,x)\in W(u(t,x)),\hfill \\ u(t,x)=0,\phantom{\rule{1em}{0ex}}t\in [0,d],x\in \partial \mathrm{\Omega},\hfill \\ u(0,x)={u}_{0}(x),\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},\hfill \end{array}
(4.1)
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}.
We assume the following hypotheses:

(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:

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

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

(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})\u22b8{L}^{2}(\mathrm{\Omega};\mathbb{R}) be the maps defined by
We can write the feedback control problem (4.1) as a firstorder inclusion in the Hilbert space E={L}^{2}(\mathrm{\Omega};\mathbb{R})
\{\begin{array}{c}{y}^{\mathrm{\prime}}(t)\in Ay(t)+F(t,y(t)),\phantom{\rule{1em}{0ex}}t\in [0,d],y(t)\in {L}^{2}(\mathrm{\Omega};\mathbb{R}),\hfill \\ y(0)={y}_{0},\hfill \end{array}
(4.2)
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.209210). 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
g(t)(x)=p(t,x,{\int}_{\mathrm{\Omega}}z(\xi )\phantom{\rule{0.2em}{0ex}}d\xi )z(x)+a(t,x)(\ell z(x)+{m}_{1})+b(t,x)
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
\ell {\tilde{\alpha}}_{j}(x)+{m}_{1}\le {\tilde{\omega}}_{j}(x)\le \ell {\tilde{\alpha}}_{j}(x)+{m}_{2}\phantom{\rule{1em}{0ex}}\text{a.a.}x\in \mathrm{\Omega}.
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,
\ell \sum _{i=0}^{{h}_{j}}{\nu}_{ji}{\tilde{\alpha}}_{j+i}(x)+{m}_{1}\le \sum _{i=0}^{{h}_{j}}{\nu}_{ji}{\tilde{\omega}}_{j+i}(x)\le \ell \sum _{i=0}^{{h}_{j}}{\nu}_{ji}{\tilde{\alpha}}_{j+i}(x)+{m}_{2}.
Then passing to the limit, we obtain
\ell \alpha (x)+{m}_{1}\le \omega (x)\le \ell \alpha (x)+{m}_{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}j,i=0,\dots ,{h}_{j},\text{and a.a.}x\in \mathrm{\Omega},
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
{w}_{n}={f}_{1}(t,{\alpha}_{n})+{f}_{2}(t,{\beta}_{n}),\phantom{\rule{1em}{0ex}}\text{with}{\beta}_{n}\in V({\alpha}_{n})\text{for every}n\in \mathbb{N},
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 )\phantom{\rule{0.2em}{0ex}}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
w={f}_{1}(t,\alpha )+{f}_{2}(t,\beta )\in f(t,\alpha ,V(\alpha ))=F(t,\alpha ).
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 closedvalued as well.
Finally, the multimap F:[0,d]\times {L}^{2}(\mathrm{\Omega};\mathbb{R})\u22b8{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
\begin{array}{rcl}{\parallel F(t,\alpha )\parallel}_{2}& \le & {\parallel {f}_{1}(t,\alpha )\parallel}_{2}+{\parallel {f}_{2}(t,V(\alpha ))\parallel}_{2}\\ \le & {\phi}^{3}(t){\parallel \alpha \parallel}_{2}+{\phi}^{1}(t)(\ell {\parallel \alpha \parallel}_{2}+{m}_{1}{\mathrm{\Omega}}^{1/2}+\ell {\parallel \alpha \parallel}_{2}+{m}_{2}{\mathrm{\Omega}}^{1/2})+{\phi}^{2}(t){\mathrm{\Omega}}^{1/2}\\ \le & ({\phi}^{3}(t)+2\ell {\phi}^{1}(t)){\parallel \alpha \parallel}_{2}+{\mathrm{\Omega}}^{1/2}[({m}_{1}+{m}_{2}){\phi}^{1}(t)+{\phi}^{2}(t)].\end{array}
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
{\parallel F(t,\alpha )\parallel}_{2}\le \phi (t)(1+{\parallel \alpha \parallel}_{2}),
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
y(t)=U(t){y}_{0}+{\int}_{0}^{t}U(ts)h(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in [0,d]
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
j\left({u}^{\ast}\right)=\underset{x\in \mathcal{S}({u}_{0})}{min}j(x),
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 Agestructure population model
We consider the nonlinear hyperbolic integrodifferential problem (1.2) which was already introduced in Section 1. This model arises in population biology and describes the time evolution of the agestructure 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)\phantom{\rule{0.2em}{0ex}}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:

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

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

(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);

(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}):
\{\begin{array}{c}{y}^{\prime}(t)=Ay(t)+F(t,y(t)),\hfill \\ y(0)={y}_{0},\hfill \end{array}
(4.3)
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 singlevalued map F(t,y)(a)=f(t,a,{\int}_{0}^{B}y(s)\phantom{\rule{0.2em}{0ex}}ds)y(a) and A:D(A)\to E, with
D(A)=\{y\in {W}^{1,2}([0,B];\mathbb{R}):y(0)={\int}_{0}^{B}b(a)y(a)\phantom{\rule{0.2em}{0ex}}da\},
is the linear operator Ay={y}^{\prime}. A is the generator of the translation semigroup, which satisfies the identity
U(t)y(a)=\{\begin{array}{cc}y(at)\hfill & \text{if}ta,\hfill \\ {\int}_{0}^{B}b(s)U(ta)y(s)\phantom{\rule{0.2em}{0ex}}ds\hfill & \text{if}t\ge a.\hfill \end{array}
(4.4)
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)\phantom{\rule{0.2em}{0ex}}ds)y(a) belongs to {L}^{2}([0,B];\mathbb{R}), hence F is nonemptyvalued. 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 singlevalued, 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)\phantom{\rule{0.2em}{0ex}}ds\to {\int}_{0}^{B}y(s)\phantom{\rule{0.2em}{0ex}}ds, thus (ii) implies that f(t,a,{\int}_{0}^{B}{y}_{n}(s)\phantom{\rule{0.2em}{0ex}}ds)\to f(t,a,{\int}_{0}^{B}y(s)\phantom{\rule{0.2em}{0ex}}ds) for a.a. a\in [0,B]. Moreover, it is possible to show that f(t,a,{\int}_{0}^{B}{y}_{n}(s)\phantom{\rule{0.2em}{0ex}}ds){y}_{n}\rightharpoonup f(t,a,{\int}_{0}^{B}y(s)\phantom{\rule{0.2em}{0ex}}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}),
{\parallel F(t,y)\parallel}_{2}^{2}={\int}_{0}^{B}{f}^{2}(t,a,{\int}_{0}^{B}u(t,a)\phantom{\rule{0.2em}{0ex}}da){y}^{2}(a)\phantom{\rule{0.2em}{0ex}}da\le {\alpha}^{2}(t){\parallel y\parallel}_{2}^{2},
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,
\begin{array}{rcl}u(t,0)& =& y(t)(0)=[U(0)y(t)](0)={\int}_{0}^{B}b(s)[U(0)y(t)](s)\phantom{\rule{0.2em}{0ex}}ds={\int}_{0}^{B}b(s)y(t)(s)\phantom{\rule{0.2em}{0ex}}ds\\ =& {\int}_{0}^{B}b(s)u(t,s)\phantom{\rule{0.2em}{0ex}}ds.\end{array}
Remark 4.1
With our techniques, we can attach also problems of the following general form:
\{\begin{array}{c}{u}_{t}={u}_{a}+\mathrm{\Delta}uf(t,a,x,{\int}_{0}^{B}u(t,a,x)\phantom{\rule{0.2em}{0ex}}da)u,\phantom{\rule{1em}{0ex}}0\le a\le B,0\le t\le T,x\in \mathrm{\Omega},\hfill \\ u(0,a,x)={u}_{0}(a,x),\phantom{\rule{1em}{0ex}}0\le a\le B,x\in \mathrm{\Omega},\hfill \\ u(t,a,x)=0,\phantom{\rule{1em}{0ex}}0\le a\le B,0\le t\le T,x\in \partial \mathrm{\Omega},\hfill \\ u(t,0,x)={\int}_{0}^{B}b(a,x)u(t,a,x)\phantom{\rule{0.2em}{0ex}}da,\phantom{\rule{1em}{0ex}}0\le t\le T,x\in \mathrm{\Omega},\hfill \end{array}
(4.5)
where the Laplacian operator is understood only with respect to the space variable x and \mathrm{\Omega}\subset {\mathbb{R}}^{2} is a bounded closed domain with Lipschitz boundary. This model represents the agestructure of a population with spatial diffusion which takes into account the consequences of the environment changes on populations dynamics (see [31, 32]). The same arguments as above show that Theorem 3.1 applies to problem (4.5), taking again into account Remark 3.2. In fact, in this case, given the Banach space E={L}^{2}([0,B]\times \mathrm{\Omega};\mathbb{R}), we define the operator Ay=\frac{\partial y}{\partial a}+\mathrm{\Delta}y on
D(A)=\{y:y(a,\cdot )\in {W}^{2,2}(\mathrm{\Omega};\mathbb{R}),y(\cdot ,x)\in {W}^{1,2}([0,B];\mathbb{R}),y(0,x)={\int}_{0}^{B}b(a,x)y(a,x)\phantom{\rule{0.2em}{0ex}}da\}.
This operator generates a strongly continuous semigroup (see [33]), and all the other conditions can be shown as in the previous example.
4.3 Multipoint boundary value problem
Let t\in [a,b] and \mathrm{\Omega}\subset {\mathbb{R}}^{m} be a bounded set with a sufficiently regular boundary. Consider the multipoint boundary value problem
\{\begin{array}{c}{u}_{t}\in \mathrm{\Delta}u+[{p}_{1}(t,x,{\int}_{\mathrm{\Omega}}k(x,\xi )u(t,\xi )\phantom{\rule{0.2em}{0ex}}d\xi ),{p}_{2}(t,x,{\int}_{\mathrm{\Omega}}k(x,\xi )u(t,\xi )\phantom{\rule{0.2em}{0ex}}d\xi )]\phi (t,x),\hfill \\ \phantom{\rule{1em}{0ex}}t\in [a,b],x\in \mathrm{\Omega},\hfill \\ u(t,x)=0,\phantom{\rule{1em}{0ex}}t\in [a,b],x\in \partial \mathrm{\Omega},\hfill \\ u(a,x)+{\sum}_{i=1}^{n}{c}_{i}u({t}_{i},x)={u}_{0}(x),\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},\hfill \end{array}
(4.6)
under the following hypotheses:

(i)
k:\mathrm{\Omega}\times \mathrm{\Omega}\to \mathbb{R} is measurable with k(x,\cdot )\in {L}^{2}(\mathrm{\Omega};\mathbb{R}) and {\parallel k(x,\cdot )\parallel}_{2}\le 1 for all x\in \mathrm{\Omega};

(ii)
{p}_{i}:[a,b]\times \mathrm{\Omega}\times \mathbb{R}\to \mathbb{R}, i=1,2, satisfy the following conditions:

(a)
for every r\in \mathbb{R} and i=1,2, {p}_{i}(\cdot ,\cdot ,r) is measurable;

(b)
for a.a. t\in [a,b] and all x\in \mathrm{\Omega}, {p}_{1}(t,x,\cdot ) is l.s.c. and {p}_{2}(t,x,\cdot ) is u.s.c.;

(c)
there exist \eta \in {L}^{1}([a,b];\mathbb{R}) and \lambda :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) increasing such that, for a.a. t\in [a,b] and every x\in \mathrm{\Omega}, r\in \mathbb{R} and i=1,2, {p}_{i}(t,x,r)\le \eta (t)\lambda (r) and
\underset{r\to \mathrm{\infty}}{lim\hspace{0.17em}inf}\frac{\lambda (r)}{r}=0;
(4.7)

(d)
for every t\in [a,b], x\in \mathrm{\Omega} and r\in \mathbb{R}, {p}_{1}(t,x,r)\le {p}_{2}(t,x,r);

(iii)
\phi :[a,b]\times \mathrm{\Omega}\to \mathbb{R} is measurable with \phi (t,\cdot )\in {L}^{2}(\mathrm{\Omega};\mathbb{R}) for a.a. t\in [a,b] and {\parallel \phi (t,\cdot )\parallel}_{2}\in {L}^{\mathrm{\infty}}([a,b];\mathbb{R});

(iv)
{u}_{0}\in {L}^{2}(\mathrm{\Omega};\mathbb{R});

(v)
{t}_{i}\in [a,b], {c}_{i}\in \mathbb{R}\setminus \{0\}, i=1,2,\dots ,n satisfy a<{t}_{1}<{t}_{2}<\cdots <{t}_{n}\le b and {\sum}_{i=1}^{n}{c}_{i}<1.
We search for solutions u\in C([a,b];{L}^{2}(\mathrm{\Omega};\mathbb{R})), and we can transform problem (4.6) into the nonlocal boundary value problem
\{\begin{array}{c}{y}^{\mathrm{\prime}}(t)\in Ay(t)+F(t,y(t)),\hfill \\ y(a)+{\sum}_{i=1}^{n}{c}_{1}y({t}_{i})={y}_{0}\hfill \end{array}
(4.8)
in the reflexive Hilbert space {L}^{2}(\mathrm{\Omega};\mathbb{R}), where y:[a,b]\to {L}^{2}(\mathrm{\Omega};\mathbb{R}), A and {y}_{0} are as in Section 4.1. For \alpha \in {L}^{2}(\mathrm{\Omega};\mathbb{R}), the function {I}_{\alpha}:\mathrm{\Omega}\to \mathbb{R} defined by {I}_{\alpha}(x)={\int}_{\mathrm{\Omega}}k(x,\xi )\alpha (\xi )\phantom{\rule{0.2em}{0ex}}d\xi is well defined and measurable according to (i). Moreover, {I}_{\alpha}(x)\le {\parallel \alpha \parallel}_{2} for every x\in \mathrm{\Omega}. The multimap F:[a,b]\times {L}^{2}(\mathrm{\Omega};\mathbb{R})\u22b8{L}^{2}(\mathrm{\Omega};\mathbb{R}) is defined by y\in F(t,\alpha ) if and only if there is a measurable function \beta :\mathrm{\Omega}\to \mathbb{R} satisfying {p}_{1}(t,x,{I}_{\alpha}(x))\le \beta (x)\le {p}_{2}(t,x,{I}_{\alpha}(x)) and y(x)=\beta (x)\phi (t,x) for a.a. x\in \mathrm{\Omega}. This definition is well posed, i.e., F is nonemptyvalued according to (ii) and (iii), and it is easy to see that F is also convexvalued.
Given (t,\alpha )\in [a,b]\times {L}^{2}(\mathrm{\Omega};\mathbb{R}), from (ii) we have that
\begin{array}{rcl}{\parallel y\parallel}_{2}& =& \sqrt{{\int}_{\mathrm{\Omega}}{\beta}^{2}(x){\phi}^{2}(t,x)\phantom{\rule{0.2em}{0ex}}dx}\\ \le & \sqrt{{\int}_{\mathrm{\Omega}}max\{{p}_{1}^{2}(t,x,{I}_{\alpha}(x)),{p}_{2}^{2}(t,x,{I}_{\alpha}(x))\}{\phi}^{2}(t,x)\phantom{\rule{0.2em}{0ex}}dx}\\ \le & \eta (t)\lambda \left({\parallel \alpha \parallel}_{2}\right){\parallel \phi (t,\cdot )\parallel}_{2},\end{array}
(4.9)
for every y\in F(t,\alpha ). Hence F has weakly relatively compact values. Moreover, according to (4.9) and (ii)(c), we have that
\underset{{\parallel \alpha \parallel}_{2}\le n}{sup}\parallel F(t,\alpha )\parallel \le \eta (t)\lambda (n){\parallel \phi (t,\cdot )\parallel}_{2}:={\phi}_{n}(t)\phantom{\rule{1em}{0ex}}\text{for}n\in \mathbb{N};
and therefore (F3′) follows from (4.7).
Now we investigate (F2), and hence we fix a value t in [a,b] for which (ii)(b) is satisfied and consider the sequences \{{\alpha}_{n}\},\{{y}_{n}\}\subset {L}^{2}(\mathrm{\Omega};\mathbb{R}) such that {\alpha}_{n}\rightharpoonup \alpha, {y}_{n}\rightharpoonup y in {L}^{2}(\mathrm{\Omega};\mathbb{R}) and {y}_{n}\in F(t,{\alpha}_{n}) for all n\in \mathbb{N}. Notice that the weak convergence of {\alpha}_{n} implies both the existence of \sigma >0 such that {\parallel {\alpha}_{n}\parallel}_{2}\le \sigma and {I}_{{\alpha}_{n}}(x)\to {I}_{\alpha}(x) for all x\in \mathrm{\Omega}. Moreover, {y}_{n}={\beta}_{n}\phi (t,\cdot ) for all n and some measurable {\beta}_{n}:\mathrm{\Omega}\to \mathbb{R} satisfying {p}_{1}(t,x,{I}_{{\alpha}_{n}}(x))\le {\beta}_{n}(x)\le {p}_{2}(t,x,{I}_{{\alpha}_{n}}(x)) for a.a. x\in \mathrm{\Omega}. According to (ii)(c), {\beta}_{n}(x)\le \eta (t)\lambda (\sigma ) a.e. in Ω for all n. Since Ω is bounded in {\mathbb{R}}^{m}, the set \{{\beta}_{n}\}\subset {L}^{2}(\mathrm{\Omega};\mathbb{R}) and it is weakly relatively compact. Hence we can find a subsequence, again denoted as the sequence, satisfying {\beta}_{n}\rightharpoonup \beta \in {L}^{2}(\mathrm{\Omega};\mathbb{R}). Consequently, applying Mazur’s convexity lemma, for each n there exist {k}_{n}\in \mathbb{N} and positive numbers {\delta}_{ni}, i=0,1,\dots ,{k}_{n}, such that {\sum}_{i=0}^{{k}_{n}}{\delta}_{ni}=1 and the sequence {\tilde{\beta}}_{n}:={\sum}_{i=0}^{{k}_{n}}{\delta}_{ni}{\beta}_{n+i}\to \beta. Up to a subsequence, denoted again as the sequence, {\tilde{\beta}}_{n}(x)\to \beta (x) for a.a. x\in \mathrm{\Omega}. Since
\sum _{i=0}^{{k}_{n}}{\delta}_{ni}{p}_{1}(t,x,{I}_{{\alpha}_{n+i}}(x))\le {\tilde{\beta}}_{n}(x)\le \sum _{i=0}^{{k}_{n}}{\delta}_{ni}{p}_{2}(t,x,{I}_{{\alpha}_{n+i}}(x))\phantom{\rule{1em}{0ex}}\text{for a.a.}x\in \mathrm{\Omega},
passing to the limit as n\to \mathrm{\infty} and according to (ii)(b), we obtain that {p}_{1}(t,x,{I}_{\alpha}(x))\le \beta (x)\le {p}_{2}(t,x,{I}_{\alpha}(x)) a.e. in Ω. Let {\tilde{y}}_{n}:={\sum}_{i=0}^{{k}_{n}}{\delta}_{ni}{y}_{n+i}. It is possible to see that {\tilde{y}}_{n}\rightharpoonup y. However, {\tilde{y}}_{n}(x)={\tilde{\beta}}_{n}(x)\phi (t,x), and hence {\tilde{y}}_{n}(x)\to \beta (x)\phi (t,x) for a.a. x\in \mathrm{\Omega} and the convergence is dominated since {\tilde{\beta}}_{n}(x)\le {\sum}_{i=0}^{{k}_{n}}{\delta}_{ni}{\beta}_{n}(x)\le \eta (t)\lambda (\sigma ) for a.a. x\in \mathrm{\Omega}. Therefore {\tilde{y}}_{n}\to \beta \phi (t,\cdot ) in {L}^{2}(\mathrm{\Omega};\mathbb{R}), and the uniqueness of the weak limit implies that y=\beta \phi (t,\cdot ). We have showed that F(t,\cdot ) is weakly sequentially closed, i.e., that (F2) is satisfied, for a.a. t\in [a,b]. Then by (4.9) F has weakly sequentially compact values since {L}^{2}(\mathrm{\Omega};\mathbb{R}) is reflexive, and according to Corollary 2.1, it is weakly compactvalued. Furthermore, according to the Pettis measurability theorem (see Theorem 2.4), it is possible to show that the maps t\mapsto {p}_{i}(t,\cdot ,{I}_{\alpha}(\cdot ))\phi (t,\cdot ), i=1,2, are measurable selection of F(\cdot ,\alpha ) for every \alpha \in {L}^{2}(\mathrm{\Omega};\mathbb{R}); hence condition (F1) is satisfied.
For y\in {L}^{2}(\mathrm{\Omega};\mathbb{R}) we have that Ly=y(a) and then, as showed in Section 3, it satisfies both (L1) and (L2). Whereas M:C([a,b];{L}^{2}(\mathrm{\Omega};\mathbb{R}))\to {L}^{2}(\mathrm{\Omega};\mathbb{R}) is such that M(y)={y}_{0}{\sum}_{i=0}^{n}{c}_{i}y({t}_{i}) and it satisfies condition (M) (see Introduction); estimate (3.1), in particular, depends on (v). All the assumptions of Theorem 3.1 are then satisfied, and hence problem (4.8) is solvable. It implies that the multipoint boundary value problem (4.6) has a solution u\in C([a,b];{L}^{2}(\mathrm{\Omega};\mathbb{R})).
Remark 4.2 Similarly as before, it is possible to show that also the nonlocal boundary value problem given by the differential inclusion in (4.6) associated with the boundary condition
u(a,x)=\frac{1}{ba}{\int}_{a}^{b}g(u(s,x))\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},
is solvable, provided that g:\mathbb{R}\to \mathbb{R} is Lipschitzian of some constant k<1.
Remark 4.3 In (4.1) and (4.6) it is possible to substitute the Laplacian operator with a more general operator L:{W}^{2,2}(\mathrm{\Omega};\mathbb{R})\cap {W}_{0}^{1,2}(\mathrm{\Omega};\mathbb{R})\to {L}^{2}(\mathrm{\Omega};\mathbb{R}) in divergence form of the following type:
(Lu)(x)=\sum _{i,j=1}^{n}\frac{\partial}{\partial {x}_{j}}({a}_{i,j}(x)\frac{\partial}{\partial {x}_{i}}u(x))
with t\in [a,b], x\in \mathrm{\Omega} and

(a)
{a}_{i,j}\in {L}^{\mathrm{\infty}}(\mathrm{\Omega}), {a}_{i,j}={a}_{j,i}, i,j=1,2,\dots ,n;

(b)
there exists c>0 such that, for a.a. x\in \mathrm{\Omega} and every \xi \in {\mathbb{R}}^{n}, c{\parallel \xi \parallel}^{2}\le {\sum}_{i,i=1}^{n}{a}_{i,j}(x){\xi}_{i}{\xi}_{j}.