Combining the foregoing continuation principle with the Scorza-Dragoni-type technique (cf. Proposition 2.2), we are ready to state the main result of the paper concerning the solvability and localization of a solution of the multivalued Dirichlet problem (2).
Theorem 3.1 Consider the Dirichlet b.v.p. (2). Suppose that is an upper-Carathéodory mapping which is either globally measurable or quasi-compact. Furthermore, let be a non-empty, open, convex, bounded subset containing 0 of a separable Banach space E satisfying the Radon-Nikodym property. Let the following conditions ()-() be satisfied:
() , for a.a. and each , and each bounded , where and γ is the Hausdorff m.n.c. in E.
() For every non-empty, bounded , there exists such that
for a.a. and all .
Furthermore, let there exist and a function , i.e. a twice continuously differentiable function in the sense of Fréchet, satisfying (H1)-(H3) (cf. Proposition 2.3) with Fréchet derivative Lipschitzian in .b Let there still exist such that
where denotes the second Fréchet derivative of V at x in the direction . Finally, let
for a.a. and all , , and .
Then the Dirichlet b.v.p. (2) admits a solution whose values are located in . If, moreover, , for a.a. , then the obtained solution is non-trivial.
Proof Since the proof of this result is rather technical, it will be divided into several steps. At first, let us define the sequence of approximating problems. For this purpose, let k be as in Proposition 2.3 and consider a continuous function such that , for all , and , for all . According to Proposition 2.3 (see also Remark 2.1), the function , where
is well defined, continuous and bounded.
Since the mapping has, according to Proposition 2.2, the Scorza-Dragoni property, we are able to find a decreasing sequence of subsets of and a mapping with compact, convex values such that, for all ,
is u.s.c. on ,
is continuous in (cf. e.g.).
If we put , then , , for all , the mapping is u.s.c. on and is continuous in .
For each , let us define the mapping with compact, convex values by the formula
Let us consider the b.v.p.
Now, let us verify the solvability of problems . Let be fixed. Since is globally u.s.c. on , is measurable, for each , and, due to the continuity of , is u.s.c., for all . Therefore, is an upper-Carathéodory mapping. Moreover, let us define the upper-Carathéodory mapping by the formula
Let us show that, when is sufficiently large, all assumptions of Proposition 2.4 (for ) are satisfied.
For this purpose, let us define the closed set by
and let the set Q of candidate solutions be defined as . Because of the convexity of K, the set Q is closed and convex.
For all and , consider still the associated fully linearized problem
and denote by the solution mapping which assigns to each the set of solutions of .
ad (i) In order to verify condition (i) in Proposition 2.4, we need to show that, for each , the problem is solvable with a convex set of solutions. So, let be arbitrary and let be a measurable selection of , which surely exists (see, e.g., [, Theorem 1.3.5]). According to () and the definition of , it is also easy to see that . The homogeneous problem corresponding to b.v.p. ,
has only the trivial solution, and therefore the single-valued Dirichlet problem
admits a unique solution which is one of solutions of . This is given, for a.a. , by , where G is the Green function associated to the homogeneous problem (12). The Green function G and its partial derivative are defined by (cf. e.g. [, pp.170-171])
Thus, the set of solutions of is non-empty. The convexity of the solution sets follows immediately from the definition of and the fact that problems are fully linearized.
ad (ii) Let be bounded. Then, there exists a bounded such that and, according to () and the definition of , there exists with such that, for all , and ,
Therefore, the mapping satisfies condition (ii) from Proposition 2.4.
ad (iii) Since the verification of condition (iii) in Proposition 2.4 is technically the most complicated, it will be split into two parts: (iii1) the quasi-compactness of the solution operator , (iii2) the condensity of w.r.t. the monotone and non-singular m.n.c. α defined by (7).
ad (iii1) Let us firstly prove that the solution mapping is quasi-compact. Since is a complete metric space, it is sufficient to prove the sequential quasi-compactness of . Hence, let us consider the sequences , , , , for all , such that in and . Moreover, let , for all . Then there exists, for all , such that
Since and in , there exists a bounded such that , for all and . Therefore, there exists, according to condition (), such that , for every and a.a. , where .
Moreover, for every and a.a. ,
Thus, satisfies, for every and a.a. , and , where
Furthermore, for every and a.a. , we have
Hence, the sequences and are bounded and is uniformly integrable.
For each , the properties of the Hausdorff m.n.c. yield
Since , for all and all , it follows from condition () that, for a.a. ,
Since the function is Lipschitzian on with some Lipschitz constant (see Remark 2.1), we get
Since and in , we get, for all , , which implies that , for all .
For all , the sequence is relatively compact as well since, according to the semi-homogeneity of the Hausdorff m.n.c.,
Moreover, by means of (4) and (18),
By similar reasoning, we also get
by which , are relatively compact, for all .
Moreover, since satisfies for all (13), is relatively compact, for a.a. . Thus, according to [, Lemma III.1.30], there exist a subsequence of , for the sake of simplicity denoted in the same way as the sequence, and such that converges to in and converges weakly to in . According to the classical closure results (cf. e.g. [, Lemma 5.1.1]), , which implies the quasi-compactness of .
ad (iii2) In order to show that, for sufficiently large, is α-condensing with respect to the m.n.c. α defined by (7), let us consider a bounded subset such that . Let be a sequence such that
At first, let us show that the set is bounded. If , then there exist , and such that
with , for a.a. .
Since Θ is bounded, there exists such that , for all and all . Hence, according to (), there exists such that , for a.a. . Consequently
Thus, the set is bounded.
Moreover, we can find , and satisfying, for a.a. , , such that, for all , and are defined by (15) and (16), respectively, where is defined by (14).
By similar reasoning as in the part ad (iii1), we obtain
for a.a. , and that
Since , for a.a. , and , for all , where Θ is a bounded subset of , there exists such that , for all and . Hence, it follows from condition () that
This implies , for a.a. and all .
Moreover, by virtue of the semi-homogeneity of the Hausdorff m.n.c., for all , we have
Let us denote
According to (4) and (15) we thus obtain for each ,
By similar reasonings, we can see that, for each ,
when starting from condition (16). Subsequently,
Since we assume that and , we get
Since we have, according to (), , we can choose such that, for all , , we have
Therefore, we get, for sufficiently large , the contradiction which ensures the validity of condition (iii) in Proposition 2.4.
ad (iv) For all , the set coincides with the unique solution of the linear system
According to (15) and (16), for all ,
we have, for all ,
Let us now consider such that . Then it follows from (21) that we are able to find such that, for all , , and , . Therefore, for all , , , for all , which ensures the validity of condition (iv) in Proposition 2.4.
ad (v) The validity of the transversality condition (v) in Proposition 2.4 can be proven quite analogously as in  (see pp.40-43 in ) with the following differences:
due to the Dirichlet boundary conditions, belongs to the open interval ,
since , we have .
In this way, we can prove that there exists such that every problem , where , satisfies all the assumptions of Proposition 2.4. This implies that every such admits a solution, denoted by , with , for all . By similar arguments as in , but with the expression replaced by , according to condition (), we can obtain the result that there exists a subsequence, denoted as the sequence, and a function such that and in and also in , when . Thus, a classical closure result (see e.g. [, Lemma 5.1.1]) guarantees that x is a solution of (2) satisfying , for all , and the sketch of proof is so complete. □
The case when , with to be completely continuous and to be Lipschitzian, for a.a. , represents the most classical example of a map which is γ-regular w.r.t. the Hausdorff measure of non-compactness γ. The following corollary of Theorem 3.1 can be proved quite analogously as in [, Example 6.1 and Remark 6.1].
Corollary 3.1 Let be a separable Hilbert space and let us consider the Dirichlet b.v.p.:
is an upper-Carathéodory, globally measurable, multivalued mapping and is completely continuous, for a.a. , such that
for a.a. , all with , where is an arbitrary constant, , and all ,
is a Carathéodory multivalued mapping such that
where , and is Lipschitzian, for a.a. , with the Lipschitz constant
Moreover, suppose that
there exists such that, for all with , , and , we have
Then the Dirichlet problem (22) admits, according to Theorem 3.1, a solution such that , for all .
Remark 3.1 For , the completely continuous mapping allows us to make a comparison with classical single-valued results recalled in the Introduction. Unfortunately, our in (i) (see also () in Theorem 3.1) is the only mapping which is (unlike in [, Example 6.1 and Remark 6.1], where under some additional restrictions quite liberal growth restrictions were permitted) globally bounded w.r.t. . Furthermore, our sign condition in (iii) is also (unlike again in [, Example 6.1 and Remark 6.1], where under some additional restrictions the Hartman-type condition like (iH) in the Introduction was employed) the most restrictive among their analogies in [6–13]. On the other hand, because of multivalued upper-Carathéodory maps in a Hilbert space which are γ-regular, our result has still, as far as we know, no analogy at all.