### Abstract

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### 1 Introduction

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.*[1]), *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 [2], 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 [3]). 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

Hence, consider firstly the Dirichlet problem in the simplest vector form:

where

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 [2] rather than to the existence results in [4,5].

It is well known (see *e.g.*[3,6-13]) that the problem (1) is solvable on various levels of generality provided:

(i_{sign})

(ii_{growth})

Let us note that the existence of the same constant
_{sign}) and (ii_{growth}) can be assumed either explicitly as in [6,7,9,11,13] or it follows from the assumptions as those in [8,10,12].

(1) Hartmann [9] (*cf.* also [1]) generalized both conditions as follows:

(i_{H})

(i_{H}) the well-known Bernstein-Nagumo-Hartman condition (for its definition and more details,
see *e.g.*[1,14]).

Let us note that the strict inequality in (i_{H}) can be replaced by a non-strict one (see *e.g.* [[1], Chapter XII,II,5], [[11], Corollary 6.2]).

(2) Lasota and Yorke [10] improved condition (i_{sign}) with suitable constants

(i_{LY})

but for
_{growth}) by the Bernstein-Nagumo-Hartman condition.

Since (i_{LY}) implies (*cf.*[10]) the existence of a constant

for
_{LY}) is obviously more liberal than (i_{sign}) as well as than (i_{H}), on the intersection of their domains.

If
_{LY}), then constant
*i.e.*
_{LY}) (see *e.g.* [[7], 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
*e.g.* [7]).

(3) Following the ideas of Mawhin in [7,11,12], Amster and Haddad [6] demonstrated that an open, bounded subset of
*∂D* such that condition (i_{H}) can be generalized as follows:

(i_{AH})

where

Since for the ball

condition (i_{AH}) is obviously more general than the original Hartman condition (i_{H}).

Nevertheless, the growth condition takes there only the form (ii_{growth}), namely with
*R* denotes, this time, the radius of *D*.

For a convex, open, bounded subset
_{AH}) can read as follows:

(i_{conv})

which is another well-known generalization of (i_{sign}).

(4) In a Hilbert space *H*, for a completely continuous mapping *f*, Mawhin [12] has shown that, for real constants *a*, *b*, *c* such that
_{sign}) can be replaced in particular by

(i_{M})

and (ii_{growth}) by an appropriate version of the Bernstein-Nagumo-Hartman condition.

(5) In a Banach space *E*, Schmitt and Thompson [13] improved, for a completely continuous mapping *f*, condition (i_{conv}) in the sense that the strict inequality in (i_{conv}) can be replaced by a non-strict one. More concretely, if there exists a convex,
open, bounded subset
*E* with

(i_{ST})

where
*E* and its dual

In the Carathéodory case of
_{sign}) can be replaced, according to [[8], Theorem 6.1], by a non-strict one and the constants
*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.*[19], 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 [20]).

In this light, we would like to modify in the present paper the Hartman-type conditions
(i_{sign}), (ii_{growth}) at least in the following way:

• the given space *E* to be Banach (or, more practically, Hilbert),

• the right-hand side to be a multivalued upper-Carathéodory mapping *F* which is *γ*-regular w.r.t.

• the inequality in (i_{sign}) to hold w.r.t. *x* strictly on the boundary *∂D* of a convex, bounded subset

• condition (ii_{growth}) 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
*e.g.* reflexivity, see *e.g.* [[21], pp.694-695]) and let us consider the Dirichlet boundary value problem (b.v.p.)

where

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
*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

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 [22]. The approximating problems will be treated by means of the continuation principle developed in [19].

### 2 Preliminaries

Let *E* be as above and
*e.g.* [[21], pp.693-701]. The symbol
*e.g.* [[21], pp.695-696], [[23], pp.243-244]). In the sequel, we shall always consider

Given
*B* is the open unit ball in *E* centered at 0, *i.e.*
*μ* will denote the Lebesgue measure on ℝ.

Let
*E* and let us denote by
*E* and
*i.e.*, for all

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** [[24], p.278]

*Let*
*be a measure space*, *E**be a separable Banach space*. *Then*
*is measurable if and only if for every*
*the function*
*is measurable with respect to* Σ *and the Borel**σ*-*algebra in* ℝ.

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
*Y* is given. We associate with *F* its graph

A multivalued mapping
*upper semicontinuous* (shortly, u.s.c.) if, for each open subset
*X*.

Let
*Y* is a separable metric space, is called *measurable* if, for each open subset
*σ*-algebra of subsets of *J*.

A multivalued mapping
*compact* if the set
*Y* and it is called *quasi*-*compact* if it maps compact sets onto relatively compact sets.

Let
*Y* is a separable Banach space, is called an *upper*-*Carathéodory mapping* if the map

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 [22] (*cf.* also [25]).

For more details concerning multivalued analysis, see *e.g.*[23,26,27].

**Definition 2.1** An upper-Carathéodory mapping
*Scorza*-*Dragoni property* if there exists a multivalued mapping

(i)

(ii) if

(iii) for every

The following two propositions are crucial in our investigation. The first one is
almost a direct consequence of the main result in [22] (*cf.*[25] and [[16], Proposition 2]). The second one allows us to construct a sequence of approximating
problems of (2).

**Proposition 2.2***Let**E**be a separable Banach space and*
*be an upper*-*Carathéodory mapping*. *If**F**is globally measurable or quasi*-*compact*, *then**F**has the Scorza*-*Dragoni property*.

**Proposition 2.3** (*cf.* [[18], Theorem 2.2])

*Let**E**be a Banach space and*
*a non*-*empty*, *open*, *convex*, *bounded set such that*
*Moreover*, *let*
*and*
*be a Fréchet differentiable function with*
*Lipschitzian in*
*satisfying*

(H1)

(H2)
*for all*

(H3)
*for all*
*where*
*is given*.

*Then there exist*
*and a bounded Lipschitzian function*
*such that*
*for every*

**Remark 2.1** Let us note that the function
*ϕ* and
*V* at *x*.

**Example 2.1** If *V* satisfies all the assumptions of Proposition 2.3, then it is easy to prove the existence
of
*E* is an arbitrary Hilbert space, we can define

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
*E*. A function
*measure of non*-*compactness* (m.n.c.) in *E* if

A m.n.c. *β* is called:

(i) *monotone* if

(ii) *non*-*singular* if

If *N* is a cone in a Banach space, then a m.n.c. *β* is called:

(iii) *semi*-*homogeneous* if

(iv) *regular* when

(v) *algebraically subadditive* if

The typical example of an m.n.c. is the *Hausdorff measure of non*-*compactness**γ* defined, for all

The Hausdorff m.n.c. is monotone, non-singular, semi-homogeneous and regular. Moreover,
if
*e.g.*, [27])

Let *E* be a separable Banach space and
*cf.*[27])

Moreover, if
*L*-Lipschitzian, then

for all bounded

Furthermore, for all subsets Ω of *E* (see *e.g.*[17]),

Let us now introduce the function

defined on the bounded
^{a} It was proved in [19] that the function *α* given by (7) is an m.n.c. in

**Definition 2.3** Let *E* be a Banach space and
*condensing with respect to an m.n.c.**β* (shortly, *β*-*condensing*) if, for every bounded

A family of mappings
*β*-*condensing* if, for every bounded

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 [19]. Since the proof of this modified version differs from the one in [19] only slightly in technical details, we omit it here.

**Proposition 2.4***Let us consider the b*.*v*.*p*.

*where*
*is an upper*-*Carathéodory mapping and*
*Let*
*be an upper*-*Carathéodory mapping such that*

*Moreover*, *assume that the following conditions hold*:

(i) *There exist a closed set*
*and a closed*, *convex set*
*with a non*-*empty interior* Int*Q**such that each associated problem*

*where*
*and*
*has a non*-*empty*, *convex set of solutions* (*denoted by*

(ii) *For every non*-*empty*, *bounded set*
*there exists*
*such that*

*for a*.*a*.
*and all*
*and*

(iii) *The solution mapping**is quasi*-*compact and**μ*-*condensing with respect to a monotone and non*-*singular m*.*n*.*c*. *μ**defined on*

(iv) *For each*
*the set of solutions of problem*
*is a subset of* Int*Q*, *i*.*e*.
*for all*

(v) *For each*
*the solution mapping*
*has no fixed points on the boundary**∂Q**of**Q*.

*Then the b*.*v*.*p*. (8) *has a solution in**Q*.

### 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 3.1***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

is well defined, continuous and bounded.

Since the mapping

•

•

•

•
*cf.**e.g.*[2]).

If we put

For each

Let us consider the b.v.p.

Now, let us verify the solvability of problems

Let us show that, when

For this purpose, let us define the closed set

and let the set *Q* of candidate solutions be defined as
*K*, the set *Q* is closed and convex.

For all

and denote by

*ad* (i) In order to verify condition (i) in Proposition 2.4, we need to show that, for
each
*e.g.*, [[27], Theorem 1.3.5]). According to (

has only the trivial solution, and therefore the single-valued Dirichlet problem

admits a unique solution
*G* is the Green function associated to the homogeneous problem (12). The Green function
*G* and its partial derivative
*cf.**e.g.* [[28], pp.170-171])

Thus, the set of solutions of

*ad* (ii) Let

Therefore, the mapping

*ad* (iii) Since the verification of condition (iii) in Proposition 2.4 is technically
the most complicated, it will be split into two parts: (iii_{1}) the quasi-compactness of the solution operator
_{2}) the condensity of
*α* defined by (7).

*ad* (iii_{1}) Let us firstly prove that the solution mapping

where

and that

Since

Moreover, for every

and

Thus,

Furthermore, for every

Hence, the sequences

For each

Since

Since the function

Since

For all

Moreover, by means of (4) and (18),

By similar reasoning, we also get

by which

Moreover, since
*cf.**e.g.* [[27], Lemma 5.1.1]),

*ad* (iii_{2}) In order to show that, for
*α*-condensing with respect to the m.n.c. *α* defined by (7), let us consider a bounded subset

At first, let us show that the set

with

Since Θ is bounded, there exists

Similarly,

Thus, the set

Moreover, we can find

By similar reasoning as in the part *ad* (iii_{1}), we obtain

for a.a.

Since

This implies

Moreover, by virtue of the semi-homogeneity of the Hausdorff m.n.c., for all

Let us denote

and

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

Since we have, according to (

Therefore, we get, for sufficiently large

*ad* (iv) For all

According to (15) and (16), for all

and

where

Since

we have, for all

Let us now consider

*ad* (v) The validity of the transversality condition (v) in Proposition 2.4 can be proven
quite analogously as in [16] (see pp.40-43 in [16]) with the following differences:

– due to the Dirichlet boundary conditions,

– since

In this way, we can prove that there exists
*e.g.* [[27], Lemma 5.1.1]) guarantees that *x* is a solution of (2) satisfying

The case when
*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 [[3], 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*.:

*where*

(i)
*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*

(ii)
*is a Carathéodory multivalued mapping such that*

*where*
*and*
*is Lipschitzian*, *for a*.*a*.
*with the Lipschitz constant*

*Moreover*, *suppose that*

(iii) *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
_{H}) in the Introduction was employed) the most restrictive among their analogies in
[6-13]. On the other hand, because of multivalued upper-Carathéodory maps
*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
*e.g.*[29]). Moreover, when *φ* is linear in
*e.g.*[30] 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.

**Example 4.1** Let us consider the integro-differential equation (23) with

(a) *φ* is Carathéodory, *i.e.*

(b)

(c)

(d)