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.
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 .
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 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.
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]).
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:
It is obvious that an m.n.c. which is invariant with respect to the union with compact sets is also nonsingular.
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.
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.
The following convergence result will be also employed.
Lemma 2.2 (cf. [, Lemma III.1.30])
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.
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.
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 4.1Letbe an open set such thatandbe an upper-Carathéodory mapping. Assume that the functionhas a locally Lipschitzian Fréchet derivativeand satisfies conditions (H1) and (H2). Suppose, moreover, that there existssuch that, for all, and, at least one of the following conditions:
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 .
Consequently, we obtain
according to assumption (7), it leads to a contradiction with inequality (9).
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 4.2Letbe a nonempty open set such thatandbe an upper semicontinuous multivalued mapping with compact, convex values. Assume that there exists a functionwith a locally Lipschitzian Fréchet derivativewhich satisfies conditions (H1) and (H2). Suppose, moreover, that for all, andwith
the following condition holds:
Subsequently, according to the mean-value theorem (see, e.g., [, Theorem 0.5.3]), there exists such that for each ,
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
with t, x, y, w as in Proposition 4.1 or in Proposition 4.2.
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), whereis an upper-Carathéodory multivalued mapping. Assume thatis an open, convex set containing 0. Furthermore, let the following conditions be satisfied:
Finally, let there exist a functionwith a locally Lipschitzian Fréchet derivativesatisfying 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.
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])
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
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.
when starting from condition (23). Subsequently,
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
In view of (25), we have so obtained that
Theorem 5.2Consider the Dirichlet b.v.p. (1), whereis an upper semicontinuous mapping with compact, convex values. Assume thatis an open, convex set containing 0. Moreover, let conditions (), (), () from Theorem 5.1 be satisfied.
Furthermore, let there exist a functionwith a locally Lipschitz Frechét derivativesatisfying (H1) and (H2). Moreover, let, for all, , andsatisfying (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
Moreover, suppose that
Indeed. The properties of guarantee that satisfies the inequality (cf., e.g., )
Remark 6.1 Consider again (26) in a Hilbert space H, but let this time , be globally u.s.c. mappings with compact, convex values ( is compact (cf., e.g., [, Proposition I.3.20]) and, in particular, bounded) such that
Because of the Dirichlet boundary conditions for , there exists a zero point of , i.e., , by which the same estimates can be also obtained without an explicit usage of the Green function above. Otherwise, it is not so easy to obtain such estimates, because Rolle’s theorem fails in general.
For obtaining the estimation of the solution derivative in a Hilbert space H, one can also apply, under natural assumptions, the p-Nagumo condition derived in .
The authors declare that they have no competing interests.
All authors contributed equally in this article. They read and approved the final manuscript.
The first and third authors were supported by the grant PrF_2012_017. The second author was supported by the national research project PRIN “Ordinary Differential Equations and Applications”.
The m.n.c. is monotone, nonsingular and algebraically subadditive on (cf., e.g., ) and it is equal to zero if and only if all the elements are equi-continuous.
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