The existence and localization result is obtained for a multivalued Dirichlet problem in a Banach space. The upper-Carathéodory and Marchaud right-hand sides are treated separately because in the latter case, the transversality conditions derived by means of bounding functions can be strictly localized on the boundaries of bound sets.
MSC: 34A60, 34B15, 47H04.
Keywords:Dirichlet problem; bounding functions; solutions in a given set; condensing multivalued operators
The Dirichlet problem and its special case with homogeneous boundary conditions, usually called the Picard problem, belong to the most frequently studied boundary value problems. A lot of results concerning the standard problem for scalar second-order ordinary differential equations were generalized in various directions.
In Euclidean spaces, besides many extensions to vector equations, vector inclusions were under consideration, e.g., in [1-4]. In abstract spaces, usually in Banach and Hilbert spaces, equations, e.g., in [5-11] and inclusions, e.g., in [9,12,13] were treated.
Sadovskii’s or Darbo’s fixed point theorems, jointly with the usage of a measure of noncompactness, were applied in [5,8,9,11]. Kakutani’s or Ky Fan’s fixed point theorems were applied with the upper and lower solutions technique in  and with a measure of noncompactness in . On the other hand, continuation principles were employed in [2,4,7].
The main aim of our present paper is an extension of the finite-dimensional results in [2,4] into infinite-dimensional ones. We were also stimulated by the work of Jean Mawhin in , where degree arguments were applied to the Dirichlet problem in a Hilbert space probably for the first time, and in , where a bound sets approach was systematically developed. Hence, besides these two approaches, our extension consists in the consideration of differential inclusions in rather general Banach spaces and the usage of a measure of noncompactness. Similar results were already obtained in an analogous way by ourselves for Floquet problems in [15-18].
Besides the existence, the localization of solutions will be obtained in our main theorems (see Theorem 5.1 and Theorem 5.2). Unlike in , where the solutions belong to a positively invariant set, in our paper, some trajectories can escape from the prescribed set of candidate solutions. Moreover, the associated bound set need not be compact as in . Similarly, the main difference between our results and those in [9,13] consists in the application of a continuation principle jointly with a bound sets approach, which allows us to check fixed point free boundaries of given bound sets. This, in particular, means that, unlike in [9,13], some trajectories can again escape from the prescribed set of candidate solutions in a transversal way.
Let E be a Banach space (with the norm ) satisfying the Radon-Nikodym property (e.g., reflexivity) and let us consider the Dirichlet boundary value problem (b.v.p.)
where is an upper-Carathéodory mapping or a globally upper semicontinuous mapping with compact, convex values (for the related definitions, see Section 2).
The main purpose of the present paper is to prove the existence of a Carathéodory solution to problem (1) in a given set Q. This will be achieved by means of a suitable continuation principle. The crucial condition of the continuation principle described in Section 3 consists in guaranteeing the fixed point free boundary of Q w.r.t. an admissible homotopical bridge starting from (1) (see condition (v) in Proposition 3.1 below). This requirement will be verified by means of Lyapunov-like bounding functions, i.e., via a bound sets technique. That is also why the whole Section 4 is devoted to this technique applied to Dirichlet problem (1). We will distinguish two cases, namely when F is an upper-Carathéodory mapping and when F is globally upper semicontinuous (i.e., a Marchaud mapping). Unlike in the first case, the second one allows us to apply bounding functions which can be strictly localized at the boundaries of given bound sets.
Let E be a Banach space having the Radon-Nikodym property (see, e.g., [, pp.694-695]), i.e., if for every finite measure space and every vector measure of bounded variation, which is absolutely continuous w.r.t. μ, we can find a Bochner integrable function such that
for each . Let be a closed interval. By the symbol , we will mean the set of all Bochner integrable functions . For the definition and properties, see, e.g., [, pp.693-701].
The symbol will denote the set of functions whose first derivative is absolutely continuous. Then and the fundamental theorem of calculus (the Newton-Leibniz formula) holds (see, e.g., [, pp.243-244], [, pp.695-696]). In the sequel, we will always consider as a subspace of the Banach space .
Given and , the symbol will denote, as usually, the set , where B is the open unit ball in E, i.e., .
We will also need the following definitions and notions from multivalued analysis. Let X, Y be two metric spaces. We say that F is a multivalued mapping from X to Y (written ) if for every , a nonempty subset of Y is given. We associate with F its graph , the subset of , defined by
A multivalued mapping is called upper semicontinuous (shortly, u.s.c.) if for each open subset , the set is open in X.
A multivalued mapping is called compact if the set is contained in a compact subset of Y; it is called quasi-compact if it maps compact sets onto relatively compact sets; and completely continuous if it maps bounded sets onto relatively compact sets.
We say that a multivalued mapping with closed values is a step multivalued mapping if there exists a finite family of disjoint measurable subsets , such that and F is constant on every . A multivalued mapping with closed values is called strongly measurable if there exists a sequence of step multivalued mappings such that as for a.a. , where stands for the Hausdorff distance.
A multivalued mapping is called an upper-Carathéodory mapping if the map is strongly measurable for all , the map is u.s.c. for almost all and the set is compact and convex for all .
Let us note that if are Banach spaces, then an upper-Carathéodory mapping is weakly superpositionally measurable, i.e., that for each continuous , the composition possesses a single-valued measurable selection (see, e.g., [12,20]).
A multivalued mapping is called Lipschitzian in if there exists a constant such that
for a.a. and for all .
In the sequel, the measure of noncompactness will also be employed.
Definition 2.1 Let N be a partially ordered set, E be a Banach space and let denote the family of all nonempty subsets of E. A function is called a measure of noncompactness (m.n.c.) in E if for all , where denotes the closed convex hull of Ω.
An m.n.c. β is called:
(i) monotone if for all ,
(ii) nonsingular if for all and ,
(iii) invariant with respect to the union with compact sets if for every relatively compact and every ,
(iv) regular when if and only if Ω is relatively compact,
(v) algebraically semi-additive if for all .
Definition 2.2 An m.n.c. β with values in a cone of a Banach space has the semi-homogeneity property if for all and all .
It is obvious that an m.n.c. which is invariant with respect to the union with compact sets is also nonsingular.
The typical example of an m.n.c. is the Hausdorff measure of noncompactnessγ defined, for all , by
The Hausdorff measure of noncompactness is monotone, nonsingular, algebraically semi-additive and has the semi-homogeneity property.
Let be such that , for a.a. , all and suitable , then (cf.)
Moreover, for all subsets Ω of E (see, e.g., ),
Let us now introduce the function
defined on the bounded set , where the ordering is induced by the positive cone in and where denotes the modulus of continuity of a subset .a Such a μ is an m.n.c. in , as shown in the following lemma (proven in ), where the properties of μ will be also discussed.
Lemma 2.1The functionμgiven by (4) defines an m.n.c. in ; such an m.n.c. μis monotone, invariant with respect to the union with compact sets and regular.
The m.n.c. μ defined by (4) will be used in order to solve problem (1) (cf. Theorem 5.1).
Definition 2.3 Let E be a Banach space and . A multivalued mapping with compact values is called condensing with respect to an m.n.c.β (shortly, β-condensing) if for every such that , it holds that Ω is relatively compact.
A family of mappings with compact values is called β-condensing if for every such that , it holds that Ω is relatively compact.
The following convergence result will be also employed.
Lemma 2.2 (cf. [, Lemma III.1.30])
LetEbe a Banach space and assume that the sequence of absolutely continuous functions satisfies the following conditions:
(i) the set is relatively compact for every ,
(ii) there exists such that for a.a. and for all ,
(iii) the set is weakly relatively compact for a.a. .
Then there exists a subsequence of (for the sake of simplicity denoted in the same way as the sequence) converging to an absolutely continuous function in the following way:
1. converges uniformly toxin ,
2. converges weakly in to .
Lemma 2.3Let be a compact interval, let , be Banach spaces and let be a multivalued mapping satisfying the following conditions:
(i) has a strongly measurable selection for every ,
(ii) is u.s.c. for a.a. ,
(iii) the set is compact and convex for all .
Assume in addition that for every nonempty, bounded set , there exists such that
for a.a. and every . Let us define the Nemytskiǐ operator in the following way: for every . Then, if sequences and , , , are such that in and weakly in , then .
3 Continuation principle
The proof of the main result (cf. Theorem 5.1 below) will be based on the combination of a bound sets technique together with the following continuation principle developed in .
Proposition 3.1Let us consider the general multivalued b.v.p.
where is an upper-Carathéodory mapping and . Let be an upper-Carathéodory mapping such that
for all . Moreover, assume that the following conditions hold:
(i) There exist a closed set and a closed, convex set with a nonempty interior IntQsuch that each associated problem
where and , has a nonempty, convex set of solutions (denoted by ).
(ii) For every nonempty, bounded set , there exists such that
for a.a. and all , and .
(iii) The solution mapping is quasi-compact andμ-condensing with respect to a monotone and nonsingular m.n.c. μdefined on .
(iv) For each , the set of solutions of the problem is a subset of IntQ, i.e., for all .
(v) For each , the solution mapping has no fixed points on the boundary∂QofQ.
Then the b.v.p. (5) has a solution inQ.
The proof of the continuation principle is based on the fact that the family of problems depending on two parameters and is associated to the original b.v.p. (5). This family is defined in such a way that if is its corresponding solution mapping, then all fixed points of the map are solutions of (5) (see condition (6)).
4 Bound sets technique
The continuation principle formulated in Proposition 3.1 requires, in particular, the existence of a suitable set of candidate solutions. The set Q should satisfy the transversality condition (v), i.e., it should have a fixed-point free boundary with respect to the solution mapping . Since the direct verification of the transversality condition is usually a difficult task, we will devote this section to a bound sets technique which can be used for guaranteeing such a condition. For this purpose, we will define the set Q as , where K is nonempty and open in E and denotes its closure.
Hence, let us consider the Dirichlet boundary value problem (1) and let be a -function satisfying
(H2) for all .
Definition 4.1 A nonempty open set is called a bound set for the b.v.p. (1) if every solution x of (1) such that for each does not satisfy for any .
Let be the Banach space dual to E and let us denote by the pairing (the duality relation) between E and , i.e., for all and , we put . The proof of the following proposition is quite analogous to the finite-dimensional case considered in . Nevertheless, for the sake of completeness, we present it here, too.
Proposition 4.1Let be an open set such that and be an upper-Carathéodory mapping. Assume that the function has a locally Lipschitzian Fréchet derivative and satisfies conditions (H1) and (H2). Suppose, moreover, that there exists such that, for all , and , at least one of the following conditions:
holds for all . ThenKis a bound set for the Dirichlet problem (1).
Proof Let be a solution of problem (1). We assume, by a contradiction, that there exists such that . The point must lie in according to the Dirichlet boundary conditions and the fact that .
Since is locally Lipschitzian, there exist a neighborhood U of and a constant such that is Lipschitzian with a constant L. Let be such that for each .
In order to get the desired contradiction, let us define the function as the composition . According to the regularity properties of x and V, . Since and for all , is a local maximum point for g. Therefore, . Moreover, there exist points , such that and .
Since , where is locally Lipschitzian and is absolutely continuous on , exists for a.a. . Consequently,
At first, let us assume that condition (7) holds and let be such that and exist. Then
and so there exists a function as , such that for each h,
Moreover, since , there exists a function as , such that for each h,
Consequently, we obtain
Since as ,
Moreover, for every and , we have that
According to the Lipschitzianity of , when is sufficiently small, we have that
where L denotes the local Lipschitz constant of in a neighborhood of x. It implies that
according to assumption (7), it leads to a contradiction with inequality (9).
Secondly, let us assume that condition (8) holds and let be such that and exist. Then it is possible to show, using the same procedure as before, that according to assumption (8),
which leads to a contradiction with inequality (10).
Therefore, we get the contradiction in case that at least one of conditions (7), (8) holds which completes the proof. □
If the mapping is globally u.s.c. in , then the transversality conditions can be localized directly on the boundary of K, as will be shown in the following proposition, whose proof is again quite analogous to the finite-dimensional case considered in .
Proposition 4.2Let be a nonempty open set such that and be an upper semicontinuous multivalued mapping with compact, convex values. Assume that there exists a function with a locally Lipschitzian Fréchet derivative which satisfies conditions (H1) and (H2). Suppose, moreover, that for all , and with
the following condition holds:
for all . ThenKis a bound set for problem (1).
Proof Let be a solution of problem (1). We assume, by a contradiction, that there exists such that . Since and x satisfies Dirichlet boundary conditions, .
Let us define the function as the composition . Then and for all , i.e., there is a local maximum for g at the point 0, and so . Consequently, satisfies condition (11).
Since is locally Lipschitzian, there exist a neighborhood U of and a constant such that is Lipschitzian with a constant L.
Let be an arbitrary decreasing sequence of positive numbers such that as , for all .
Since and for all , there exists, for each , such that .
Since , for each ,
where as .
and let be given. As a consequence of the regularity assumptions imposed on F and of the continuity of both x and , there exists such that for each , , it follows that
Subsequently, according to the mean-value theorem (see, e.g., [, Theorem 0.5.3]), there exists such that for each ,
Since F has compact values and ε is arbitrary, we obtain that ζ is a relatively compact set. Thus, there exist a subsequence, for the sake of simplicity denoted as the sequence, of and such that
as implying, for the arbitrariness of ,
As a consequence of the property (14), there exists a sequence , as , such that
for each . Since and , in view of (13) and (15),
Since for all , we have, according to (13), that for each . Since as , it is possible to find such that for all , it holds that . By means of the local Lipschitzianity of , for all ,
Since as ,
If we consider, instead of the sequence , an increasing sequence of negative numbers such that as , for all , we are able to find, for each , such that . Therefore, using the same procedure as in the first part of the proof, we obtain, for sufficiently large, that
where , as and .
This means that as , which implies
Inequalities (16) and (17) are in a contradiction with condition (12), because , satisfies condition (11) and . □
Remark 4.1 One can readily check that for , inequalities (7) and (8), as well as (12), become
with t, x, y, w as in Proposition 4.1 or in Proposition 4.2.
The typical case occurs when is a Hilbert space, denotes the scalar product and
for some . In this case, and it is not difficult to see that conditions (7) and (8), as well as (12), become
with t, x, y and w as in Proposition 4.1 or in Proposition 4.2, where .
Definition 4.2 A -function with a locally Lipschitzian Fréchet derivative which satisfies conditions (H1), (H2) and all assumptions in Proposition 4.1 or Proposition 4.2 is called a bounding function for problem (1).
5 Existence and localization results
Combining the continuation principle with the bound sets technique, we are ready to state the main result of the paper concerning the solvability and localization of a solution of the multivalued Dirichlet problem (1).
Theorem 5.1Consider the Dirichlet b.v.p. (1), where is an upper-Carathéodory multivalued mapping. Assume that is an open, convex set containing 0. Furthermore, let the following conditions be satisfied:
( ) for a.a. and each bounded , where andγis the Hausdorff measure of noncompactness in E.
( ) For every nonempty, bounded set , there exists such that
for a.a. and all ,
( ) .
Finally, let there exist a function with a locally Lipschitzian Fréchet derivative satisfying conditions (H1), (H2), and at least one of conditions (7), (8) for a suitable , all , , , and . Then the Dirichlet b.v.p. (1) admits a solution whose values are located in .
Proof 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 a solution mapping which assigns to each the set of solutions of . We will show that the family of the above b.v.p.s satisfies all assumptions of Proposition 3.1.
In this case, which, together with the definition of , ensures the validity of (6).
ad (i) In order to verify condition (i) in Proposition 3.1, 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 strongly measurable selection of . 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 (19). The Green function G and its partial derivative are defined by (cf., e.g., [, pp.170-171])
Thus, the set of solutions of is nonempty. The convexity of the solution sets follows immediately from the properties of a mapping F and the fact that problems are fully linearized.
ad (ii) Assuming that is defined by , condition (ii) in Proposition 3.1 is ensured directly by assumption (5).
ad (iii) Since the verification of condition (iii) in Proposition 3.1 is technically the most complicated, it will be subdivided into two parts: (iii1) the quasi-compactness of the solution operator , (iii2) the condensity of w.r.t. the monotone and nonsingular (cf. Lemma 2.1) m.n.c. μ defined by (4).
ad (iii1) Let us firstly prove that the solution mapping is quasi-compact. Since is a 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
and 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. .
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.
Since the sequences , are converging, we obtain, in view of ( ),
for a.a. , which implies that is relatively compact.
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 (2), (3), (21) and the semi-homogeneity of the Hausdorff m.n.c.,
By similar reasonings, we can also get
by which , are relatively compact for a.a. . Moreover, since satisfies for all equation (20), is relatively compact for a.a. . Thus, according to Lemma 2.2, 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 . Therefore, the mapping is quasi-compact.
ad (iii2) In order to show that is μ-condensing, where μ is defined by (4), we will prove that any bounded subset such that is relatively compact. Let be a sequence such that
Then we can find , satisfying for a.a. and such that for all ,
In view of ( ), we have, for all ,
Since and Θ is bounded in , by means of ( ), we get the existence of such that for a.a. and all . This implies for a.a. and all .
Moreover, by virtue of the semi-homogeneity of the Hausdorff m.n.c., for all , we have
According to (2), (3) and (22), we so obtain for each ,
By the similar reasonings, we can obtain that for each ,
when starting from condition (23). Subsequently,
Since and , we so get
and, in view of (24) and ( ), we have that
Inequality (24) implies that
Now, we show that both the sequences and are equi-continuous. Let be such that and for all and . Thus, we get that , where comes from ( ), and so is uniformly integrable. This implies that is equi-continuous. Moreover, according to (23), we obtain that
for all and , implying that is bounded; consequently, also is equi-continuous. Therefore,
In view of (25), we have so obtained that
Hence, also and since μ is regular, we have that Θ is relatively compact. Therefore, condition (iii) in Proposition 3.1 holds.
ad (iv) For all , the problem has only the trivial solution. Since , condition (iv) in Proposition 3.1 is satisfied.
ad (v) Let be a solution of the b.v.p. for some , i.e., a fixed point of the solution mapping . In view of conditions (7), (8) (see Proposition 4.1), K is, for all , a bound set for the problem
This implies that , which ensures condition (v) in Proposition 3.1. □
If the mapping is globally u.s.c. in (i.e., a Marchaud map), then we are able to improve Theorem 5.1 in the following way.
Theorem 5.2Consider the Dirichlet b.v.p. (1), where is an upper semicontinuous mapping with compact, convex values. Assume that is an open, convex set containing 0. Moreover, let conditions ( ), ( ), ( ) from Theorem 5.1 be satisfied.
Furthermore, let there exist a function with a locally Lipschitz Frechét derivative satisfying (H1) and (H2). Moreover, let, for all , , and satisfying (11), condition (12) hold for all . Then the Dirichlet b.v.p. (1) admits a solution whose values are located in .
Proof The verification is quite analogous as in Theorem 5.1 when just replacing the usage of Proposition 4.1 by Proposition 4.2. □
6 Illustrative example
Example 6.1 Let be a Hilbert space and let us consider the Dirichlet b.v.p.
(i) is an upper-Carathéodory multivalued mapping and is completely continuous for a.a. such that
for a.a. and all with , ,
(ii) is a Carathéodory multivalued mapping such that
and is Lipschitzian for a.a. with the Lipschitz constant
Moreover, suppose that
(iii) there exist and such that, for all with , , , and , we have
Then the Dirichlet problem (26) admits, according to Theorem 5.1, a solution such that for all .
Indeed. The properties of guarantee that satisfies the inequality (cf., e.g., )
for a.a. and every bounded , where γ stands for the Hausdorff measure of noncompactness in H.
Since is completely continuous and thanks to the algebraic semi-additivity of γ, inequality (27) can be rewritten into
for a.a. and every bounded , i.e., , for (cf. ( )).
Moreover, according to the Lipschitzianity of , the following inequalities take place:
for a.a. and all .
Thus, for , , we arrive at
i.e., (18) in ( ).
Finally, in view of Remark 4.1, we can define the bounding function by the formula
and the bound set K as