### Abstract

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### 1 Introduction

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 [9] and with a measure of noncompactness in [13]. 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 [7], where degree arguments were applied to the Dirichlet problem in a Hilbert space probably for the first time, and in [14], 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 [10], 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 [10]. 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
*e.g.*, reflexivity) and let us consider the Dirichlet boundary value problem (b.v.p.)

where

The main purpose of the present paper is to prove the existence of a Carathéodory
solution
*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.

### 2 Preliminaries

Let *E* be a Banach space having the *Radon-Nikodym property* (see, *e.g.*, [[19], pp.694-695]), *i.e.*, if for every finite measure space
*μ*, we can find a Bochner integrable function

for each
*e.g.*, [[19], pp.693-701].

The symbol
*e.g.*, [[15], pp.243-244], [[19], pp.695-696]). In the sequel, we will always consider

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

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

A multivalued mapping
*compact* if the set
*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
*step multivalued mapping* if there exists a finite family of disjoint measurable subsets
*F* is constant on every
*strongly measurable* if there exists a sequence of step multivalued mappings

It is well known that if *Y* is a Banach space, then a strongly measurable mapping
*e.g.*, [12,20]).

A multivalued mapping
*upper-Carathéodory mapping* if the map

Let us note that if
*i.e.*, that for each continuous
*e.g.*, [12,20]).

A multivalued mapping
*Lipschitzian* in

for a.a.

For more details concerning multivalued analysis, see, *e.g.*, [12,15,20,21].

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
*E*. A function
*measure of noncompactness* (m.n.c.) in *E* if

An m.n.c. *β* is called:

(i) *monotone* if

(ii) *nonsingular* if

(iii) *invariant with respect to the union with compact sets* if

(iv) *regular* when

(v) *algebraically semi-additive* if

**Definition 2.2** An m.n.c. *β* with values in a cone of a Banach space has the *semi-homogeneity* property if

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 noncompactness**γ* defined, for all

The Hausdorff measure of noncompactness is monotone, nonsingular, algebraically semi-additive and has the semi-homogeneity property.

Let
*cf.*[20])

Moreover, for all subsets Ω of *E* (see, *e.g.*, [18]),

Let us now introduce the function

defined on the bounded set
^{a} Such a *μ* is an m.n.c. in
*μ* will be also discussed.

**Lemma 2.1***The function**μ**given by* (4) *defines an m*.*n*.*c*. *in*
*such an m*.*n*.*c*. *μ**is 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
*condensing with respect to an m.n.c.**β* (shortly, *β*-*condensing*) if for every

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

The following convergence result will be also employed.

**Lemma 2.2** (*cf.* [[15], Lemma III.1.30])

*Let**E**be 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 to**x**in*

2.
*converges weakly in*
*to*

The following lemma is well known when the Banach spaces
*e.g.*, [[22], p.88]). The present slight modification for

**Lemma 2.3***Let*
*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 [16].

**Proposition 3.1***Let 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* Int*Q**such 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 and**μ*-*condensing with respect to a monotone and nonsingular m*.*n*.*c*. *μ**defined on*

(iv) *For each*
*the set of solutions of the 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*. (5) *has a solution in**Q*.

The proof of the continuation principle is based on the fact that the family

### 4 Bound sets technique

The continuation principle formulated in Proposition 3.1 requires, in particular,
the existence of a suitable set
*Q* should satisfy the transversality condition (v), *i.e.*, it should have a fixed-point free boundary with respect to the solution mapping
*Q* as
*K* is nonempty and open in *E* and

Hence, let us consider the Dirichlet boundary value problem (1) and let

(H1)

(H2)

**Definition 4.1** A nonempty open set
*bound set* for the b.v.p. (1) if every solution *x* of (1) such that

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

**Proposition 4.1***Let*
*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*
*Then**K**is a bound set for the Dirichlet problem* (1).

*Proof* Let

Since
*U* of
*L*. Let

In order to get the desired contradiction, let us define the function
*x* and *V*,
*g*. Therefore,

Since

and

At first, let us assume that condition (7) holds and let

and so there exists a function
*h*,

Moreover, since
*h*,

Consequently, we obtain

Since

Moreover, for every

According to the Lipschitzianity of

where *L* denotes the local Lipschitz constant of
*x*. It implies that

and then

Therefore,

according to assumption (7), it leads to a contradiction with inequality (9).

Secondly, let us assume that condition (8) holds and let

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
*K*, as will be shown in the following proposition, whose proof is again quite analogous
to the finite-dimensional case considered in [2].

**Proposition 4.2***Let*
*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*
*Then**K**is a bound set for problem* (1).

*Proof* Let
*x* satisfies Dirichlet boundary conditions,

Let us define the function
*i.e.*, there is a local maximum for *g* at the point 0, and so

Since
*U* of
*L*.

Let

Since

Since

where

Let

and let
*F* and of the continuity of both *x* and

Subsequently, according to the mean-value theorem (see, *e.g.*, [[24], Theorem 0.5.3]), there exists

Therefore,

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

as

As a consequence of the property (14), there exists a sequence

for each

Since

Since

If we consider, instead of the sequence

where

This means that

Inequalities (16) and (17) are in a contradiction with condition (12), because

**Remark 4.1** One can readily check that for

with *t*, *x*, *y*, *w* as in Proposition 4.1 or in Proposition 4.2.

The typical case occurs when

for some

with *t*, *x*, *y* and *w* as in Proposition 4.1 or in Proposition 4.2, where

**Definition 4.2** 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 5.1***Consider 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*
*and**γ**is 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

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

For all

and denote by

In this case,

ad (i) In order to verify condition (i) in Proposition 3.1, we need to show that for
each

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 (19). The Green function
*G* and its partial derivative
*cf.*, *e.g.*, [[12], pp.170-171])

Thus, the set of solutions of
*F* and the fact that problems

ad (ii) Assuming that

ad (iii) Since the verification of condition (iii) in Proposition 3.1 is technically
the most complicated, it will be subdivided into two parts: (iii_{1}) the quasi-compactness of the solution operator
_{2}) the condensity of
*cf.* Lemma 2.1) m.n.c. *μ* defined by (4).

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

and that

Since

Moreover, for every

and

Thus,

and

Furthermore, for every

Hence, the sequences

Since the sequences

for a.a.

For all

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

ad (iii_{2}) In order to show that
*μ*-condensing, where *μ* is defined by (4), we will prove that any bounded subset

Then we can find

and

In view of (

Since

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

According to (2), (3) and (22), we so obtain for each

where

By the similar reasonings, we can obtain that for each

when starting from condition (23). Subsequently,

yielding

Since

and, in view of (24) and (

Inequality (24) implies that

Now, we show that both the sequences

for all

In view of (25), we have so obtained that

Hence, also
*μ* is regular, we have that Θ is relatively compact. Therefore, condition (iii) in Proposition 3.1
holds.

ad (iv) For all

ad (v) Let
*i.e.*, a fixed point of the solution mapping
*K* is, for all

This implies that

If the mapping
*i.e.*, a Marchaud map), then we are able to improve Theorem 5.1 in the following way.

**Theorem 5.2***Consider 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

where

(i)

for a.a.

(ii)

and

Moreover, suppose that

(iii) there exist

Then the Dirichlet problem (26) admits, according to Theorem 5.1, a solution

Indeed. The properties of
*cf.*, *e.g.*, [20])

for a.a.
*γ* stands for the Hausdorff measure of noncompactness in *H*.

Since
*γ*, inequality (27) can be rewritten into

for a.a.
*i.e.*,
*cf.* (

Moreover, according to the Lipschitzianity of

for a.a.

Thus, for

*i.e.*, (18) in (

Finally, in view of Remark 4.1, we can define the bounding function

and the bound set *K* as