Hartman-type conditions are presented for the solvability of a multivalued Dirichlet problem in a Banach space by means of topological degree arguments, bounding functions, and a Scorza-Dragoni approximation technique. The required transversality conditions are strictly localized on the boundaries of given bound sets. The main existence and localization result is applied to a partial integro-differential equation involving possible discontinuities in state variables. Two illustrative examples are supplied. The comparison with classical single-valued results in this field is also made.
MSC: 34A60, 34B15, 47H04.
Keywords:Dirichlet problem; Scorza-Dragoni-type technique; strictly localized bounding functions; solutions in a given set; condensing multivalued operators
In this paper, we will establish sufficient conditions for the existence and localization of strong solutions to a multivalued Dirichlet problem in a Banach space via degree arguments combined with a bound sets technique. More precisely, Hartman-type conditions (cf.), i.e. sign conditions w.r.t. the first state variable and growth conditions w.r.t. the second state variable, will be presented, provided the right-hand side is a multivalued upper-Carathéodory mapping which is γ-regular w.r.t. the Hausdorff measure of non-compactness γ.
The main aim will be two-fold: (i) strict localization of sign conditions on the boundaries of bound sets by means of a technique originated by Scorza-Dragoni , and (ii) the application of the obtained abstract result (see Theorem 3.1 below) to an integro-differential equation involving possible discontinuities in a state variable. The first aim allows us, under some additional restrictions, to extend our earlier results obtained for globally upper semicontinuous right-hand sides and partly improve those for upper-Carathéodory right-hand sides (see ). As we shall see, the latter aim justifies such an abstract setting, because the problem can be transformed into the form of a differential inclusion in a Hilbert -space. Roughly speaking, problems of this type naturally require such an abstract setting. In order to understand in a deeper way what we did and why, let us briefly recall classical results in this field and some of their extensions.
Hence, consider firstly the Dirichlet problem in the simplest vector form:
The first existence results, for a bounded f in (1), are due to Scorza-Dragoni [4,5]. Let us note that his name in the title is nevertheless related to the technique developed in  rather than to the existence results in [4,5].
(2) Lasota and Yorke  improved condition (isign) with suitable constants and in the following way:
Since (iLY) implies (cf.) the existence of a constant such that
If in (iLY), then constant can be even equal to zero, i.e., in (iLY) (see e.g. [, Corollary V.26 on p.74]). Moreover, the related Bernstein-Nagumo-Hartman condition can only hold for x in a suitable convex, closed, bounded subset of (see again e.g. ).
(3) Following the ideas of Mawhin in [7,11,12], Amster and Haddad  demonstrated that an open, bounded subset of , say , need not be convex, provided it has a -boundary ∂D such that condition (iH) can be generalized as follows:
condition (iAH) is obviously more general than the original Hartman condition (iH).
which is another well-known generalization of (isign).
(4) In a Hilbert space H, for a completely continuous mapping f, Mawhin  has shown that, for real constants a, b, c such that , condition (isign) can be replaced in particular by
and (iigrowth) by an appropriate version of the Bernstein-Nagumo-Hartman condition.
(5) In a Banach space E, Schmitt and Thompson  improved, for a completely continuous mapping f, condition (iconv) in the sense that the strict inequality in (iconv) can be replaced by a non-strict one. More concretely, if there exists a convex, open, bounded subset of E with such that
where denotes this time the pairing between E and its dual , jointly with the appropriate Bernstein-Nagumo-Hartman condition, then the problem (1) admits a solution whose values are located in (see [, Theorem 4.1]).
In the Carathéodory case of in (1), for instance, the strict inequality in condition (isign) can be replaced, according to [, Theorem 6.1], by a non-strict one and the constants , can be replaced without the requirement , but globally in , by functions , which are bounded on bounded sets. Moreover, system (1) can be additively perturbed, for the same goal, by another Carathéodory function which is sublinear in both states variables x and y.
On the other hand, the Carathéodory case brings about some obstructions in a strict localization of sign conditions on the boundaries of bound sets (see e.g.[3,15]). The same is also true for other boundary value problems (for Floquet problems, see e.g.[16-18]). Therefore, there naturally exist some extensions of classical results in this way. Further extensions concern problems in abstract spaces, functional problems, multivalued problems, etc. For the panorama of results in abstract spaces, see e.g., where multivalued problems are also considered.
Nevertheless, let us note that in abstract spaces, it is extremely difficult (if not impossible) to avoid the convexity of given bound sets, provided the degree arguments are applied for non-compact maps (for more details, see ).
In this light, we would like to modify in the present paper the Hartman-type conditions (isign), (iigrowth) at least in the following way:
• the given space E to be Banach (or, more practically, Hilbert),
• condition (iigrowth) to be replaced by a suitable growth condition which would allow us reasonable applications (the usage of the Bernstein-Nagumo-Hartman-type condition will be employed in this context by ourselves elsewhere).
Hence, let E be a separable Banach space (with the norm ) satisfying the Radon-Nikodym property (e.g. reflexivity, see e.g. [, pp.694-695]) and let us consider the Dirichlet boundary value problem (b.v.p.)
Let us note that in the entire paper all derivatives will be always understood in the sense of Fréchet and, by the measurability, we mean the one with respect to the Lebesgue σ-algebra in and the Borel σ-algebra in E.
The notion of a solution will be understood in a strong (i.e. Carathéodory) sense. Namely, by a solution of problem (2) we mean a function whose first derivative is absolutely continuous and satisfies (2), for almost all .
The solution of the b.v.p. (2) will be obtained as the limit of a sequence of solutions of approximating problems that we construct by means of a Scorza-Dragoni-type result developed in . The approximating problems will be treated by means of the continuation principle developed in .
Let E be as above and be a closed interval. By the symbol , we shall mean the set of all Bochner integrable functions . For the definition and properties of Bochner integrals, see e.g. [, pp.693-701]. The symbol will be reserved for 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.695-696], [, pp.243-244]). In the sequel, we shall always consider as a subspace of the Banach space and by the symbol we shall mean the Banach space of all linear, bounded transformations endowed with the sup-norm.
We recall also the Pettis measurability theorem which will be used in Section 4 and which we state here in the form of proposition.
Proposition 2.1 [, p.278]
We shall 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 non-empty subset of Y is given. We associate with F its graph , the subset of , defined by .
Let be a given compact interval. A multivalued mapping , where Y is a separable Banach space, is called an upper-Carathéodory mapping if the map is measurable, for all , the map is u.s.c., for almost all , and the set is compact and convex, for all .
The technique that will be used for proving the existence and localization result consists in constructing a sequence of approximating problems. This construction will be made on the basis of the Scorza-Dragoni-type result developed in  (cf. also ).
The following two propositions are crucial in our investigation. The first one is almost a direct consequence of the main result in  (cf. and [, Proposition 2]). The second one allows us to construct a sequence of approximating problems of (2).
Proposition 2.3 (cf. [, Theorem 2.2])
Example 2.1 If V satisfies all the assumptions of Proposition 2.3, then it is easy to prove the existence of such that , for all . Consequently, when E is an arbitrary Hilbert space, we can define by the formula
which satisfies all the properties mentioned in Proposition 2.3.
Definition 2.2 Let N be a partially ordered set, E be a Banach space and let denote the family of all non-empty bounded subsets of E. A function is called a measure of non-compactness (m.n.c.) in E if , for all , where denotes the closed convex hull of Ω.
A m.n.c. β is called:
If N is a cone in a Banach space, then a m.n.c. β is called:
The Hausdorff m.n.c. is monotone, non-singular, semi-homogeneous and regular. Moreover, if and , then (see, e.g., )
Let E be a separable Banach space and be such that , , for a.a. , all and suitable , then (cf.)
Furthermore, for all subsets Ω of E (see e.g.),
Let us now introduce the function
defined on the bounded , where the ordering is induced by the positive cone in and where denotes the modulus of continuity of a subset .a It was proved in  that the function α given by (7) is an m.n.c. in that is monotone, non-singular and regular.
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 bounded such that , we see that Ω is relatively compact.
The proof of the main result (cf. Theorem 3.1 below) will be based on the following slight modification of the continuation principle developed in . Since the proof of this modified version differs from the one in  only slightly in technical details, we omit it here.
Proposition 2.4Let us consider the b.v.p.
Moreover, assume that the following conditions hold:
Then the b.v.p. (8) has a solution inQ.
3 Main result
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 3.1Consider the Dirichlet b.v.p. (2). Suppose thatis an upper-Carathéodory mapping which is either globally measurable or quasi-compact. Furthermore, letbe a non-empty, open, convex, bounded subset containing 0 of a separable Banach spaceEsatisfying the Radon-Nikodym property. Let the following conditions ()-() be satisfied:
Furthermore, let there existand a function, i.e. a twice continuously differentiable function in the sense of Fréchet, satisfying (H1)-(H3) (cf. Proposition 2.3) with Fréchet derivativeLipschitzian in.bLet there still existsuch that
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 continuous in (cf.e.g.).
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
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])
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
Moreover, by means of (4) and (18),
By similar reasoning, we also get
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 .
By similar reasoning as in the part ad (iii1), we obtain
Let us denote
when starting from condition (16). Subsequently,
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.
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].
Moreover, suppose that
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.
4 Illustrative examples
The first illustrative example of the application of Theorem 3.1 concerns the integro-differential equation
involving discontinuities in a state variable. In this equation, the non-local diffusion term replaces the classical diffusion behavior given by . In dispersal models such an integral term takes into account the long-distance interactions between individuals (see e.g.). Moreover, when φ is linear in , (23) can be considered as an alternative version of the classical telegraph equation (see e.g. and the references therein), where the classical diffusivity is replaced by the present non-local diffusivity.
Telegraph equations appear in many fields such as modeling of an anomalous diffusion, a wave propagation phenomenon, sub-diffusive systems or modeling of a pulsate blood flow in arteries (see e.g.[31,32]).
For the sake of simplicity, we will discuss here only the case when φ is globally bounded w.r.t. . On the other hand, for non-strictly localized transversality conditions as in , for instance, a suitable linear growth estimate w.r.t. can be permitted.
Since the function p can have some discontinuities, a solution of (23) satisfying the Dirichlet conditions
and if it satisfies (24).
In fact, problem (25), (24) can be transformed into the abstract setting
Let us now examine the properties of F. According to (a), is well defined. Given , let us show that is measurable. For this purpose, let Ψ be an arbitrary element in the dual space of . Hence, there exists such that , for all , and consequently the composition is such that . Since φ is Carathéodory, it is globally measurable, and so the mapping is globally measurable as well. This implies that, according to the Fubini Theorem, the mapping is measurable, too. Finally, since Ψ was arbitrary, according to the Pettis Theorem (see Proposition 2.1), is measurable.
Moreover, according to (a) and (c), is a Carathéodory mapping such that is -Lipschitzian, for all , and K is well defined and 1-Lipschitzian. It can also be shown that, according to (d) and (e), has compact and convex values. Therefore, the mapping F is globally measurable, and so has the Scorza-Dragoni property (cf. Proposition 2.2).
Let us now verify particular assumptions of Theorem 3.1.
where m is defined by (28).
and so condition () is satisfied with . The obtained form of together with assumption (27) directly guarantee the condition (). It can also be easily shown that properties of F ensure the validity of condition ().
In order to verify conditions imposed on a bounding function, let us define , . The function with obviously satisfies (10), so it is only necessary to check condition (11). Thus, let , , , and . Then there exists such that
we see that
The properties (a)-(f) together with the well-known Hölder inequality then yield
in view of condition (26), (30), and (31).
Hence, the Dirichlet problem (29) admits, according to Theorem 3.1, a solution y satisfying , for a.a. . If , then u is a solution of (24), (25) which is the Filippov solution of the original problem (23), (24).
Finally, we can sum up the above result in the form of the following theorem.
Remark 4.1 In [, Example 5.2], the following formally simpler integro-differential equation in ℝ:
with non-homogeneous Dirichlet conditions
After the homogenization of boundary conditions, the Dirichlet problem takes the form
The result in [, Example 5.2] cannot be, however, deduced from Theorem 3.1, because condition (b) in Example 4.1 cannot be satisfied in this way.
On the other hand, the linear term with coefficient b could not be implemented in their equation, because it is not completely continuous in (36) below, as required in .
In view of the arguments in Remark 4.1, we can conclude by the second illustrative example.
Example 4.2 Consider the following non-homogeneous Dirichlet problem in ℝ:
We will show that, under (33) and (34), problem (32) is solvable, in the abstract setting, by means of Corollary 3.1.
Problem (32) can be homogenized as follows:
Since the Hilbert-Schmidt operator
where is well known to be completely continuous (cf. [, Example 5.2]) and is, according to (33), L-Lipschitzian with , conditions (i), (ii) in Corollary 3.1 can be easily satisfied, for , ,
In this setting, problem (35) takes the abstract form as (22), namely
Since holds, for all (see [, Example 5.2]) one can check that the strict inequality in (iii) in Corollary 3.1 can be easily satisfied, for (32), whenever
where R satisfies (37), and subsequently the same is true for (32), i.e.
• condition (37) holds.
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 grant ‘Singularities and impulses in boundary value problems for non-linear ordinary differential equations’. The second author was supported by the national research project PRIN ‘Ordinary Differential Equations and Applications’.
The m.n.c. is a monotone, non-singular and algebraically subadditive on (cf.e.g.) and it is equal to zero if and only if all the elements are equi-continuous.
Since a -function V has only a locally Lipschitzian Fréchet derivative (cf.e.g.), we had to assume explicitly the global Lipschitzianity of in a non-compact set .
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