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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Scorza-Dragoni approach to Dirichlet problem in Banach spaces

Jan Andres1*, Luisa Malaguti2 and Martina Pavlačková1

Author Affiliations

1 Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 771 46, Czech Republic

2 Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Via G. Amendola, 2 - pad. Morselli, Reggio Emilia, 42122, Italy

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Boundary Value Problems 2014, 2014:23  doi:10.1186/1687-2770-2014-23

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/23


Received:16 October 2013
Accepted:9 January 2014
Published:27 January 2014

© 2014 Andres et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 L 2 -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:

x ¨ ( t ) = f ( t , x ( t ) , x ˙ ( t ) ) , t [ 0 , 1 ] , x ( 1 ) = x ( 0 ) = 0 , } (1)

where f : [ 0 , 1 ] × R n × R n R n is, for the sake of simplicity allowing the comparison of the related results, a continuous function.

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:

(isign) R > 0 such that f ( t , x , y ) , x > 0 , for ( t , x , y ) [ 0 , 1 ] × R n × R n with x = R ,

(iigrowth) C 1 0 , C 2 0 such that C 1 R < 1 and f ( t , x , y ) C 1 y 2 + C 2 , for ( t , x , y ) [ 0 , 1 ] × R n × R n with x R .

Let us note that the existence of the same constant R > 0 in (isign) and (iigrowth) 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:

(iH) R > 0 such that f ( t , x , y ) , x + y 2 > 0 , for t [ 0 , 1 ] and ( x , y ) R n × R n such that x = R and x , y = 0 ,

(iH) 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 (iH) 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 (isign) with suitable constants K 1 0 and K 2 > 0 in the following way:

(iLY) f ( t , x , y ) , x + y 2 K 1 ( 1 + x + x , y ) + K 2 y ,

but for t [ 0 , 1 ] , ( x , y ) R n × R n , and replaced (iigrowth) by the Bernstein-Nagumo-Hartman condition.

Since (iLY) implies (cf.[10]) the existence of a constant K 0 such that

f ( t , x , y ) , x + y 2 K ( 1 + x + | x , y | ) ,

for ( t , x , y ) [ 0 , 1 ] × R n × R n , the sign condition (iLY) is obviously more liberal than (isign) as well as than (iH), on the intersection of their domains.

If K 1 > 0 in (iLY), then constant K 2 can be even equal to zero, i.e. K 2 = 0 , in (iLY) (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 R n (see again e.g. [7]).

(3) Following the ideas of Mawhin in [7,11,12], Amster and Haddad [6] demonstrated that an open, bounded subset of R n , say D R n , need not be convex, provided it has a C 2 -boundary ∂D such that condition (iH) can be generalized as follows:

(iAH) f ( t , x , y ) , n x I x ( y ) , ( t , x , y ) [ 0 , 1 ] × T D × R n , with n x , y = 0 ,

where n x is the outer-pointing normal unit vector field, T D denotes the tangent vector bundle and I x ( y ) stands for the second fundamental form of the hypersurface.

Since for the ball D : = B ( 0 , R ) , R > 0 , we can have

I x ( y ) = y 2 R and n x = x R ,

condition (iAH) is obviously more general than the original Hartman condition (iH).

Nevertheless, the growth condition takes there only the form (iigrowth), namely with x R replaced by x D ¯ , where R denotes, this time, the radius of D.

For a convex, open, bounded subset D R n , the particular case of (iAH) can read as follows:

(iconv) f ( t , x , y ) , n x > 0 , for ( t , x , y ) [ 0 , 1 ] × R n × R n with x D and n x , y = 0 ,

which is another well-known generalization of (isign).

(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 a + b < 1 , condition (isign) can be replaced in particular by

(iM) f ( t , x , y ) , x ( a x 2 + b x y + c x ) , ( t , x , y ) [ 0 , 1 ] × H × H ,

and (iigrowth) 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 (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 D E of E with 0 D such that

(iST) f ( t , x , y ) , n x 0 , for ( t , x , y ) [ 0 , 1 ] × E × E , with x D and n x , y = 0 ,

where , denotes this time the pairing between E and its dual E , jointly with the appropriate Bernstein-Nagumo-Hartman condition, then the problem (1) admits a solution whose values are located in D ¯ (see [[13], Theorem 4.1]).

In the Carathéodory case of f : [ 0 , 1 ] × R n × R n R n in (1), for instance, the strict inequality in condition (isign) can be replaced, according to [[8], Theorem 6.1], by a non-strict one and the constants C 1 , C 2 can be replaced without the requirement C 1 R < 1 , but globally in [ 0 , 1 ] × R n × R n , by functions c 1 ( t , x ) , c 2 ( t , x ) 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.[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 (isign), (iigrowth) 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. ( x , y ) E × E and either globally measurable or globally quasi-compact,

• the inequality in (isign) to hold w.r.t. x strictly on the boundary ∂D of a convex, bounded subset D E (or, more practically, of the ball B ( 0 , R ) E ),

• 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. [[21], pp.694-695]) and let us consider the Dirichlet boundary value problem (b.v.p.)

x ¨ ( t ) F ( t , x ( t ) , x ˙ ( t ) ) , for a.a.  t [ 0 , T ] , x ( T ) = x ( 0 ) = 0 , } (2)

where F : [ 0 , T ] × E × E E is an upper-Carathéodory multivalued mapping.

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 [ 0 , T ] 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 x : [ 0 , T ] E whose first derivative x ˙ ( ) is absolutely continuous and satisfies (2), for almost all t [ 0 , T ] .

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 [ 0 , T ] R be a closed interval. By the symbol L 1 ( [ 0 , T ] , E ) , we shall mean the set of all Bochner integrable functions x : [ 0 , T ] E . For the definition and properties of Bochner integrals, see e.g. [[21], pp.693-701]. The symbol A C 1 ( [ 0 , T ] , E ) will be reserved for the set of functions x : [ 0 , T ] E whose first derivative x ˙ ( ) is absolutely continuous. Then x ¨ L 1 ( [ 0 , T ] , E ) and the fundamental theorem of calculus (the Newton-Leibniz formula) holds (see e.g. [[21], pp.695-696], [[23], pp.243-244]). In the sequel, we shall always consider A C 1 ( [ 0 , T ] , E ) as a subspace of the Banach space C 1 ( [ 0 , T ] , E ) and by the symbol L ( E ) we shall mean the Banach space of all linear, bounded transformations L : E E endowed with the sup-norm.

Given C E and ε > 0 , the symbol B ( C , ε ) will denote, as usually, the set C + ε B , where B is the open unit ball in E centered at 0, i.e. B = { x E x < 1 } . In what follows, the symbol μ will denote the Lebesgue measure on ℝ.

Let E be the Banach space dual to E and let us denote by , the pairing (the duality relation) between E and E , i.e., for all Φ E and x E , we put Φ ( x ) = : Φ , x .

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 ( X , Σ ) be a measure space, Ebe a separable Banach space. Then f : X E is measurable if and only if for every e E the function e f : X R is measurable with respect to Σ and the Borelσ-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 F : X Y ) if, for every x X , a non-empty subset F ( x ) of Y is given. We associate with F its graph Γ F , the subset of X × Y , defined by Γ F : = { ( x , y ) X × Y y F ( x ) } .

A multivalued mapping F : X Y is called upper semicontinuous (shortly, u.s.c.) if, for each open subset U Y , the set { x X F ( x ) U } is open in X.

Let J R be a compact interval. A mapping F : J Y , where Y is a separable metric space, is called measurable if, for each open subset U Y , the set { t J F ( t ) U } belongs to a σ-algebra of subsets of J.

A multivalued mapping F : X Y is called compact if the set F ( X ) = x X F ( x ) is contained in a compact subset of Y and it is called quasi-compact if it maps compact sets onto relatively compact sets.

Let J R be a given compact interval. A multivalued mapping F : J × X Y , where Y is a separable Banach space, is called an upper-Carathéodory mapping if the map F ( , x ) : J Y is measurable, for all x X , the map F ( t , ) : X Y is u.s.c., for almost all t J , and the set F ( t , x ) is compact and convex, for all ( t , x ) J × X .

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 F : [ 0 , T ] × X × X X is said to have the Scorza-Dragoni property if there exists a multivalued mapping F 0 : [ 0 , T ] × X × X X { } with compact, convex values having the following properties:

(i) F 0 ( t , x , y ) F ( t , x , y ) , for all ( t , x , y ) [ 0 , T ] × X × X ,

(ii) if u , v : [ 0 , T ] X are measurable functions with v ( t ) F ( t , u ( t ) , u ˙ ( t ) ) , for a.a. t [ 0 , T ] , then also v ( t ) F 0 ( t , u ( t ) , u ˙ ( t ) ) , for a.a. t [ 0 , T ] ,

(iii) for every ε > 0 , there exists a closed I ε [ 0 , T ] such that μ ( [ 0 , T ] I ε ) < ε , F 0 ( t , x , y ) , for all ( t , x , y ) I ε × X × X , and F 0 is u.s.c. on I ε × X × X .

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 2.2LetEbe a separable Banach space and F : [ 0 , T ] × E × E E be an upper-Carathéodory mapping. IfFis globally measurable or quasi-compact, thenFhas the Scorza-Dragoni property.

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

LetEbe a Banach space and K E a non-empty, open, convex, bounded set such that 0 K . Moreover, let ε > 0 and V : E R be a Fréchet differentiable function with V ˙ Lipschitzian in B ( K , ε ) ¯ satisfying

(H1) V | K = 0 ,

(H2) V ( x ) 0 , for all x K ¯ ,

(H3) V ˙ ( x ) δ , for all x K , where δ > 0 is given.

Then there exist k ( 0 , ε ] and a bounded Lipschitzian function ϕ : B ( K , k ) ¯ E such that V ˙ x , ϕ ( x ) = 1 , for every x B ( K , k ) ¯ .

Remark 2.1 Let us note that the function x ϕ ( x ) V ˙ x , where ϕ and V ˙ x are the same as in Proposition 2.3, is Lipschitzian and bounded in B ( K , k ) ¯ . The symbol V ˙ x denotes as usually the first Fréchet derivative of 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 σ ( 0 , ε ] such that V ˙ x δ 2 , for all x B ( K , σ ) ¯ . Consequently, when E is an arbitrary Hilbert space, we can define ϕ : B ( K , σ ) ¯ E by the formula

ϕ ( x ) : = V ( x ) V ( x ) 2

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 P ( E ) denote the family of all non-empty bounded subsets of E. A function β : P ( E ) N is called a measure of non-compactness (m.n.c.) in E if β ( co Ω ¯ ) = β ( Ω ) , for all Ω P ( E ) , where co Ω ¯ denotes the closed convex hull of Ω.

A m.n.c. β is called:

(i) monotone if β ( Ω 1 ) β ( Ω 2 ) , for all Ω 1 Ω 2 E ,

(ii) non-singular if β ( { x } Ω ) = β ( Ω ) , for all x E and Ω E .

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

(iii) semi-homogeneous if β ( t Ω ) = | t | β ( Ω ) , for every t R and every Ω E ,

(iv) regular when β ( Ω ) = 0 if and only if Ω is relatively compact,

(v) algebraically subadditive if γ ( Ω 1 + Ω 2 ) γ ( Ω 1 ) + γ ( Ω 2 ) , for all Ω 1 , Ω 2 E .

The typical example of an m.n.c. is the Hausdorff measure of non-compactnessγ defined, for all Ω E by

γ ( Ω ) : = inf { ε > 0 : n 1 x 1 , , x n E : Ω i = 1 n B ( { x i } , ε ) } .

The Hausdorff m.n.c. is monotone, non-singular, semi-homogeneous and regular. Moreover, if M L ( E ) and Ω E , then (see, e.g., [27])

γ ( M Ω ) M L ( E ) γ ( Ω ) . (3)

Let E be a separable Banach space and { f n } n L 1 ( [ 0 , T ] , E ) be such that f n ( t ) α ( t ) , γ ( { f n ( t ) } n ) c ( t ) , for a.a. t [ 0 , T ] , all n N and suitable α , c L 1 ( [ 0 , T ] , R ) , then (cf.[27])

γ ( { 0 T f n ( t ) d t } n ) 0 T c ( t ) d t . (4)

Moreover, if h : E E is L-Lipschitzian, then

γ ( h ( Ω ) ) L γ ( Ω ) , (5)

for all bounded Ω E .

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

γ ( λ [ 0 , 1 ] λ Ω ) = γ ( Ω ) . (6)

Let us now introduce the function

α ( Ω ) : = max { w n } n Ω ( sup t [ 0 , T ] [ γ ( { w n ( t ) } n ) + γ ( { w ˙ n ( t ) } n ) ] , mod C ( { w n } n ) + mod C ( { w ˙ n } n ) ) , (7)

defined on the bounded Ω C 1 ( [ 0 , T ] , E ) , where the ordering is induced by the positive cone in R 2 and where mod C ( Ω ) denotes the modulus of continuity of a subset Ω C ( [ 0 , T ] , E ) .a It was proved in [19] that the function α given by (7) is an m.n.c. in C 1 ( [ 0 , T ] , E ) that is monotone, non-singular and regular.

Definition 2.3 Let E be a Banach space and X E . A multivalued mapping F : X E with compact values is called condensing with respect to an m.n.c.β (shortly, β-condensing) if, for every bounded Ω X such that β ( F ( Ω ) ) β ( Ω ) , we see that Ω is relatively compact.

A family of mappings G : X × [ 0 , 1 ] E with compact values is called β-condensing if, for every bounded Ω X such that β ( G ( Ω × [ 0 , 1 ] ) ) β ( Ω ) , 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 [19]. Since the proof of this modified version differs from the one in [19] only slightly in technical details, we omit it here.

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

x ¨ ( t ) φ ( t , x ( t ) , x ˙ ( t ) ) , for a.a.  t [ 0 , T ] , x S , } (8)

where φ : [ 0 , T ] × E × E E is an upper-Carathéodory mapping and S A C 1 ( [ 0 , T ] , E ) . Let H : [ 0 , T ] × E × E × E × E × [ 0 , 1 ] E be an upper-Carathéodory mapping such that

H ( t , c , d , c , d , 1 ) φ ( t , c , d ) , for all  ( t , c , d ) [ 0 , T ] × E × E .

Moreover, assume that the following conditions hold:

(i) There exist a closed set S 1 S and a closed, convex set Q C 1 ( [ 0 , T ] , E ) with a non-empty interior IntQsuch that each associated problem

P ( q , λ ) x ¨ ( t ) H ( t , x ( t ) , x ˙ ( t ) , q ( t ) , q ˙ ( t ) , λ ) , for a.a.  t [ 0 , T ] , x S 1 , }

where q Q and λ [ 0 , 1 ] , has a non-empty, convex set of solutions (denoted by T ( q , λ ) ).

(ii) For every non-empty, bounded set Ω E × E × E × E , there exists ν Ω L 1 ( [ 0 , T ] , [ 0 , ) ) such that

H ( t , x , y , u , v , λ ) ν Ω ( t ) ,

for a.a. t [ 0 , T ] and all ( x , y , u , v ) Ω and λ [ 0 , 1 ] .

(iii) The solution mappingis quasi-compact andμ-condensing with respect to a monotone and non-singular m.n.c. μdefined on C 1 ( [ 0 , T ] , E ) .

(iv) For each q Q , the set of solutions of problem P ( q , 0 ) is a subset of IntQ, i.e. T ( q , 0 ) Int Q , for all q Q .

(v) For each λ ( 0 , 1 ) , the solution mapping T ( , λ ) has no fixed points on the boundary∂QofQ.

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 that F : [ 0 , T ] × E × E E is an upper-Carathéodory mapping which is either globally measurable or quasi-compact. Furthermore, let K E be a non-empty, open, convex, bounded subset containing 0 of a separable Banach spaceEsatisfying the Radon-Nikodym property. Let the following conditions ( 2 i )-( 2 iii ) be satisfied:

( 2 i ) γ ( F ( t , Ω 1 × Ω 2 ) ) g ( t ) ( γ ( Ω 1 ) + γ ( Ω 2 ) ) , for a.a. t [ 0 , T ] and each Ω 1 K ¯ , and each bounded Ω 2 E , where g L 1 ( [ 0 , T ] , [ 0 , ) ) andγis the Hausdorff m.n.c. inE.

( 2 ii ) For every non-empty, bounded Ω E , there exists ν Ω L 1 ( [ 0 , T ] , [ 0 , ) ) such that

F ( t , x , y ) ν Ω ( t ) , (9)

for a.a. t [ 0 , T ] and all ( x , y ) Ω × E .

( 2 iii )

( T + 4 ) g L 1 ( [ 0 , T ] , R ) < 4 .

Furthermore, let there exist ε > 0 and a function V C 2 ( E , R ) , i.e. a twice continuously differentiable function in the sense of Fréchet, satisfying (H1)-(H3) (cf. Proposition 2.3) with Fréchet derivative V ˙ Lipschitzian in B ( K , ε ) ¯ .bLet there still exist h > 0 such that

V ¨ x ( v ) , v 0 , for all  x B ( K , h ) , v E , (10)

where V ¨ x ( v ) denotes the second Fréchet derivative ofVatxin the direction ( v , v ) E × E . Finally, let

V ˙ x , w > 0 , (11)

for a.a. t ( 0 , T ) and all x K , v E , and w F ( t , x , v ) .

Then the Dirichlet b.v.p. (2) admits a solution whose values are located in K ¯ . If, moreover, 0 F ( t , 0 , 0 ) , for a.a. t [ 0 , T ] , 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 τ : E [ 0 , 1 ] such that τ ( x ) = 0 , for all x E B ( K , k ) , and τ ( x ) = 1 , for all x B ( K , k 2 ) ¯ . According to Proposition 2.3 (see also Remark 2.1), the function ϕ ˆ : E E , where

ϕ ˆ ( x ) = { τ ( x ) ϕ ( x ) V ˙ x , for all  x B ( K , k ) ¯ , 0 , for all  x E B ( K , k ) ¯ ,

is well defined, continuous and bounded.

Since the mapping ( t , x , y ) F ( t , x , y ) has, according to Proposition 2.2, the Scorza-Dragoni property, we are able to find a decreasing sequence { J m } m of subsets of [ 0 , T ] and a mapping F 0 : [ 0 , T ] × E × E E { } with compact, convex values such that, for all m N ,

μ ( J m ) < 1 m ,

[ 0 , T ] J m is closed,

( t , x , y ) F 0 ( t , x , y ) is u.s.c. on [ 0 , T ] J m × E × E ,

ν K ¯ is continuous in [ 0 , T ] J m (cf.e.g.[2]).

If we put J = m = 1 J m , then μ ( J ) = 0 , F 0 ( t , x , y ) , for all t [ 0 , T ] J , the mapping ( t , x , y ) F 0 ( t , x , y ) is u.s.c. on [ 0 , T ] J × E × E and ν K ¯ is continuous in [ 0 , T ] J .

For each m N , let us define the mapping F m : [ 0 , T ] × E × E E with compact, convex values by the formula

F m ( t , x , y ) : = { F 0 ( t , x , y ) + ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ϕ ˆ ( x ) , for all  ( t , x , y ) [ 0 , T ] J × E × E , ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ϕ ˆ ( x ) , for all  ( t , x , y ) J × E × E .

Let us consider the b.v.p.

( P m ) x ¨ ( t ) F m ( t , x ( t ) , x ˙ ( t ) ) , for a.a.  t [ 0 , T ] , x ( T ) = x ( 0 ) = 0 . }

Now, let us verify the solvability of problems ( P m ) . Let m N be fixed. Since F 0 is globally u.s.c. on [ 0 , T ] J × E × E , F m ( , x , y ) is measurable, for each ( x , y ) E × E , and, due to the continuity of ϕ ˆ , F m ( t , , ) is u.s.c., for all t [ 0 , T ] J . Therefore, F m is an upper-Carathéodory mapping. Moreover, let us define the upper-Carathéodory mapping H m : [ 0 , T ] × E × E × E × E × [ 0 , 1 ] E by the formula

H m ( t , x , y , u , v , λ ) H m ( t , u , v , λ ) : = { λ F 0 ( t , u , v ) + ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ϕ ˆ ( u ) , for all  ( t , x , y , u , v , λ ) [ 0 , T ] J for all  ( t , x , y , u , v , λ ) × E 4 × [ 0 , 1 ] , ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ϕ ˆ ( u ) , for all  ( t , x , y , u , v , λ ) J × E 4 × [ 0 , 1 ] .

Let us show that, when m N is sufficiently large, all assumptions of Proposition 2.4 (for φ ( t , x , x ˙ ) : = F m ( t , x , x ˙ ) ) are satisfied.

For this purpose, let us define the closed set S = S 1 by

S : = { x A C 1 ( [ 0 , T ] , E ) : x ( T ) = x ( 0 ) = 0 }

and let the set Q of candidate solutions be defined as Q : = C 1 ( [ 0 , T ] , K ¯ ) . Because of the convexity of K, the set Q is closed and convex.

For all q Q and λ [ 0 , 1 ] , consider still the associated fully linearized problem

P m ( q , λ ) x ¨ ( t ) H m ( t , q ( t ) , q ˙ ( t ) , λ ) , for a.a.  t [ 0 , T ] , x ( T ) = x ( 0 ) = 0 , }

and denote by T m the solution mapping which assigns to each ( q , λ ) Q × [ 0 , 1 ] the set of solutions of P m ( q , λ ) .

ad (i) In order to verify condition (i) in Proposition 2.4, we need to show that, for each ( q , λ ) Q × [ 0 , 1 ] , the problem P m ( q , λ ) is solvable with a convex set of solutions. So, let ( q , λ ) Q × [ 0 , 1 ] be arbitrary and let f q , λ ( ) be a measurable selection of H m ( , q ( ) , q ˙ ( ) , λ ) , which surely exists (see, e.g., [[27], Theorem 1.3.5]). According to ( 2 ii ) and the definition of H m , it is also easy to see that f q , λ L 1 ( [ 0 , T ] , E ) . The homogeneous problem corresponding to b.v.p. P m ( q , λ ) ,

x ¨ ( t ) = 0 , for a.a.  t [ 0 , T ] , x ( T ) = x ( 0 ) = 0 , } (12)

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

x ¨ ( t ) = f q , λ ( t ) , for a.a.  t [ 0 , T ] , x ( T ) = x ( 0 ) = 0 }

admits a unique solution x q , λ ( ) which is one of solutions of P m ( q , λ ) . This is given, for a.a. t [ 0 , T ] , by x q , λ ( t ) = 0 T G ( t , s ) f q , λ ( s ) d s , where G is the Green function associated to the homogeneous problem (12). The Green function G and its partial derivative t G are defined by (cf.e.g. [[28], pp.170-171])

G ( t , s ) = { ( s T ) t T , for all  0 t s T , ( t T ) s T , for all  0 s t T , t G ( t , s ) = { ( s T ) T , for all  0 t < s T , s T , for all  0 s < t T .

Thus, the set of solutions of P m ( q , λ ) is non-empty. The convexity of the solution sets follows immediately from the definition of H m and the fact that problems P m ( q , λ ) are fully linearized.

ad (ii) Let Ω E × E × E × E be bounded. Then, there exists a bounded Ω 1 E such that Ω Ω 1 × Ω 1 × Ω 1 × Ω 1 and, according to ( 2 ii ) and the definition of H m , there exists J ˆ [ 0 , T ] with μ ( J ˆ ) = 0 such that, for all t [ 0 , T ] ( J J ˆ ) , ( x , y , u , v ) Ω and λ [ 0 , 1 ] ,

H m ( t , u , v , λ ) ν Ω 1 ( t ) + 2 ν K ¯ ( t ) max x B ( K , k ) ¯ ϕ ˆ ( x ) .

Therefore, the mapping H m ( t , q ( t ) , q ˙ ( t ) , λ ) satisfies condition (ii) from Proposition 2.4.

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 T m , (iii2) the condensity of T m 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 T m is quasi-compact. Since C 1 ( [ 0 , T ] , E ) is a complete metric space, it is sufficient to prove the sequential quasi-compactness of T m . Hence, let us consider the sequences { q n } , { λ n } , q n Q , λ n [ 0 , 1 ] , for all n N , such that q n q in C 1 ( [ 0 , T ] , E ) and λ n λ . Moreover, let x n T m ( q n , λ n ) , for all n N . Then there exists, for all n N , k n ( ) F 0 ( , q n ( ) , q ˙ n ( ) ) such that

x ¨ n ( t ) = f n ( t ) , for a.a.  t [ 0 , T ] , (13)

where

f n ( t ) = λ n k n ( t ) + ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ϕ ˆ ( q n ( t ) ) , (14)

and that

x n ( T ) = x n ( 0 ) = 0 .

Since q n q and q ˙ n q ˙ in C ( [ 0 , T ] , E ) , there exists a bounded Ω × Ω E × E such that ( q n ( t ) , q ˙ n ( t ) ) Ω × Ω , for all t [ 0 , T ] and n N . Therefore, there exists, according to condition ( 2 ii ), ν Ω L 1 ( [ 0 , T ] , [ 0 , ) ) such that f n ( t ) ϖ ( t ) , for every n N and a.a. t [ 0 , T ] , where ϖ ( t ) : = ν Ω ( t ) + 2 ν K ¯ ( t ) max x B ( K , ε ) ¯ ϕ ˆ ( x ) .

Moreover, for every n N and a.a. t [ 0 , T ] ,

x n ( t ) = 0 T G ( t , s ) f n ( s ) d s (15)

and

x ˙ n ( t ) = 0 T t G ( t , s ) f n ( s ) d s . (16)

Thus, x n satisfies, for every n N and a.a. t [ 0 , T ] , x n ( t ) a and x ˙ n ( t ) b , where

a : = T 4 0 T ϖ ( s ) d s and b : = 0 T ϖ ( s ) d s .

Furthermore, for every n N and a.a. t [ 0 , T ] , we have

x ¨ n ( t ) ϖ ( t ) .

Hence, the sequences { x n } and { x ˙ n } are bounded and { x ¨ n } is uniformly integrable.

For each t [ 0 , T ] , the properties of the Hausdorff m.n.c. yield

γ ( { f n ( t ) } n ) γ ( { λ n k n ( t ) } n ) + ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) γ ( { ϕ ˆ ( q n ( t ) ) } n ) γ ( { k n ( t ) } n ) + ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) × γ ( { ϕ ( q n ( t ) ) V ˙ q n ( t ) : q n ( t ) B ( K , ε ) ¯ } ) .

Since q n ( t ) K ¯ , for all t [ 0 , T ] and all n N , it follows from condition ( 2 i ) that, for a.a. t [ 0 , T ] ,

γ ( { f n ( t ) } n ) g ( t ) ( γ ( { q n ( t ) } n ) + γ ( { q ˙ n ( t ) } n ) ) + ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) γ ( { ϕ ( q n ( t ) ) V ˙ q n ( t ) : q n ( t ) B ( K , ε ) ¯ } ) g ( t ) sup t [ 0 , T ] ( γ ( { q n ( t ) } n ) + γ ( { q ˙ n ( t ) } n ) ) + ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) γ ( { ϕ ( q n ( t ) ) V ˙ q n ( t ) : q n ( t ) B ( K , ε ) ¯ } ) .

Since the function x ϕ ( x ) V ˙ x is Lipschitzian on B ( K , ε ) ¯ with some Lipschitz constant L ˆ > 0 (see Remark 2.1), we get

γ ( { f n ( t ) } n ) ( g ( t ) + L ˆ ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ) sup t [ 0 , T ] ( γ ( { q n ( t ) } n ) + γ ( { q ˙ n ( t ) } n ) ) . (17)

Since q n q and q ˙ n q ˙ in C ( [ 0 , T ] , E ) , we get, for all t [ 0 , T ] , γ ( { q n ( t ) } n ) = γ ( { q ˙ n ( t ) } n ) = 0 , which implies that γ ( { f n ( t ) } n ) = 0 , for all t [ 0 , T ] .

For all ( t , s ) [ 0 , T ] × [ 0 , T ] , the sequence { G ( t , s ) f n ( s ) } is relatively compact as well since, according to the semi-homogeneity of the Hausdorff m.n.c.,

γ ( { G ( t , s ) f n ( s ) } ) | G ( t , s ) | γ ( { f n ( s ) } ) = 0 , for all  ( t , s ) [ 0 , T ] × [ 0 , T ] . (18)

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

γ ( { x n ( t ) } ) = γ ( { 0 T G ( t , s ) f n ( s ) d s } ) = 0 , for all  t [ 0 , T ] .

By similar reasoning, we also get

γ ( { x ˙ n ( t ) } ) = 0 , for all  t [ 0 , T ] ,

by which { x n ( t ) } , { x ˙ n ( t ) } are relatively compact, for all t [ 0 , T ] .

Moreover, since x n satisfies for all n N (13), { x ¨ n ( t ) } is relatively compact, for a.a. t [ 0 , T ] . Thus, according to [[23], Lemma III.1.30], there exist a subsequence of { x ˙ n } , for the sake of simplicity denoted in the same way as the sequence, and x C 1 ( [ 0 , T ] , E ) such that { x ˙ n } converges to x ˙ in C ( [ 0 , T ] , E ) and { x ¨ n } converges weakly to x ¨ in L 1 ( [ 0 , T ] , E ) . According to the classical closure results (cf.e.g. [[27], Lemma 5.1.1]), x T m ( q , λ ) , which implies the quasi-compactness of T m .

ad (iii2) In order to show that, for m N sufficiently large, T m is α-condensing with respect to the m.n.c. α defined by (7), let us consider a bounded subset Θ Q such that α ( T m ( Θ × [ 0 , 1 ] ) ) α ( Θ ) . Let { x n } T m ( Θ × [ 0 , 1 ] ) be a sequence such that

α ( T m ( Θ × [ 0 , 1 ] ) ) = ( sup t [ 0 , T ] [ γ ( { x n ( t ) } n ) + γ ( { x ˙ n ( t ) } n ) ] , mod C ( { x n } n ) + mod C ( { x ˙ n } n ) ) .

At first, let us show that the set T m ( Θ × [ 0 , 1 ] ) is bounded. If x T m ( Θ × [ 0 , 1 ] ) , then there exist q Θ , λ [ 0 , 1 ] and k ( ) F 0 ( , q ( ) , q ˙ ( ) ) such that

x ( t ) = 0 T G ( t , s ) f ( s ) d s , x ˙ ( t ) = 0 T G ( t , s ) t f ( s ) d s , for all  t [ 0 , T ] ,

with f ( t ) = λ k ( t ) + ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ϕ ˆ ( q ( t ) ) , for a.a. t [ 0 , T ] .

Since Θ is bounded, there exists Ω E such that q ( t ) Ω , for all q Θ and all t [ 0 , T ] . Hence, according to ( 2 ii ), there exists ν Ω L 1 ( [ 0 , T ] ) such that k ( t ) ν Ω ( t ) , for a.a. t [ 0 , T ] . Consequently

x ( t ) E max ( t , s ) [ 0 , 1 ] × [ 0 , 1 ] | G ( t , s ) | [ 0 T ν Ω ( s ) d s + 2 max x B ( K , k ) ¯ ϕ ˆ ( x ) 0 T ν K ¯ ( t ) ] T 4 ν Ω + 2 max x B ( K , k ) ¯ ϕ ˆ ( x ) ν K ¯ .

Similarly,

x ˙ ( t ) E max ( t , s ) [ 0 , 1 ] × [ 0 , 1 ] | G ( t , s ) | [ 0 T k ( s ) d s + 2 max x B ( K , k ) ¯ ϕ ˆ ( x ) 0 T ν K ¯ ( t ) ] ν Ω + 2 max x B ( K , k ) ¯ ϕ ˆ ( x ) ν K ¯ .

Thus, the set T m ( Θ × [ 0 , 1 ] ) is bounded.

Moreover, we can find { q n } Θ , { λ n } [ 0 , 1 ] and { k n } satisfying, for a.a. t [ 0 , T ] , k n ( t ) F 0 ( t , q n ( t ) , q ˙ n ( t ) ) , such that, for all t [ 0 , T ] , x n ( t ) and x ˙ n ( t ) are defined by (15) and (16), respectively, where f n ( t ) is defined by (14).

By similar reasoning as in the part ad (iii1), we obtain

γ ( { f n ( t ) } n ) ( g ( t ) + L ˆ ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ) sup t [ 0 , T ] ( γ ( { q n ( t ) } n ) + γ ( { q ˙ n ( t ) } n ) ) ,

for a.a. t [ 0 , T ] , and that

f n ( t ) k n ( t ) + 2 max x B ( K , ε ) ¯ ϕ ˆ ( x ) ν K ¯ ( t ) , for a.a.  t [ 0 , T ]  and all  n N .

Since k n ( t ) F 0 ( t , q n ( t ) , q ˙ n ( t ) ) , for a.a. t [ 0 , T ] , and q n Θ , for all n N , where Θ is a bounded subset of C 1 ( [ 0 , T ] , E ) , there exists Ω K ¯ such that q n ( t ) Ω , for all n N and t [ 0 , T ] . Hence, it follows from condition ( 2 ii ) that

f n ( t ) ν Ω ( t ) + 2 ν K ¯ ( t ) max x B ( K , ε ) ¯ ϕ ˆ ( x ) , for a.a.  t [ 0 , T ] . (19)

This implies G ( t , s ) f n ( t ) | G ( t , s ) | ( ν Ω ( t ) + 2 ν K ¯ ( t ) max x B ( K , ε ) ¯ ϕ ˆ ( x ) ) , for a.a. t , s [ 0 , T ] and all n N .

Moreover, by virtue of the semi-homogeneity of the Hausdorff m.n.c., for all ( t , s ) [ 0 , T ] × [ 0 , T ] , we have

γ ( { G ( t , s ) f n ( s ) } n ) | G ( t , s ) | γ ( { f n ( s ) } n ) T 4 γ ( { f n ( s ) } n ) T 4 ( g ( t ) + L ˆ ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ) × sup t [ 0 , T ] ( γ ( { q n ( t ) } n ) + γ ( { q ˙ n ( t ) } n ) ) .

Let us denote

S : = sup t [ 0 , T ] ( γ ( { q n ( t ) } n ) + γ ( { q ˙ n ( t ) } n ) )

and

S : = sup t [ 0 , T ] ( γ ( { x n ( t ) } n ) + γ ( { x ˙ n ( t ) } n ) ) .

According to (4) and (15) we thus obtain for each t [ 0 , T ] ,

γ ( { x n ( t ) } n ) = γ ( { 0 T G ( t , s ) f n ( s ) d s } n ) T 4 ( g L 1 + L ˆ ( ν K ¯ L 1 ( J m ) + 1 m ν K ¯ L 1 ) ) S .

By similar reasonings, we can see that, for each t [ 0 , T ] ,

γ ( { x ˙ n ( t ) } n ) ( g L 1 + L ˆ ( ν K ¯ L 1 ( J m ) + 1 m ν K ¯ L 1 ) ) S ,

when starting from condition (16). Subsequently,

S T + 4 4 ( g L 1 + L ˆ ( ν K ¯ L 1 ( J m ) + 1 m ν K ¯ L 1 ) ) S . (20)

Since we assume that α ( T m ( Θ × [ 0 , 1 ] ) ) α ( Θ ) and { q n } n Θ , we get

S S T + 4 4 ( g L 1 + L ˆ ( ν K ¯ L 1 ( J m ) + 1 m ν K ¯ L 1 ) ) S .

Since we have, according to ( 2 iii ), T + 4 4 g L 1 < 1 , we can choose m 0 N such that, for all m N , m m 0 , we have

T + 4 4 ( g L 1 + L ˆ ( ν K ¯ L 1 ( J m ) + 1 m ν K ¯ L 1 ) ) < 1 .

Therefore, we get, for sufficiently large m N , the contradiction S < S which ensures the validity of condition (iii) in Proposition 2.4.

ad (iv) For all q Q , the set T m ( q , 0 ) coincides with the unique solution x m of the linear system

x ¨ ( t ) = ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ϕ ˆ ( q ( t ) ) , for a.a.  t [ 0 , T ] , x ( T ) = x ( 0 ) = 0 . }

According to (15) and (16), for all t [ 0 , T ] ,

x m ( t ) = 0 T G ( t , s ) φ m ( s ) d s

and

x ˙ m ( t ) = 0 T t G ( t , s ) φ m ( s ) d s ,

where φ m ( t ) : = ν K ¯ ( t ) ( χ J m ( t ) + 1 m ) ϕ ˆ ( q ( t ) ) .

Since

φ m L 1 max x B ( K , ε ) ¯ ϕ ˆ ( x ) ( ν K ¯ L 1 ( J m ) + ν K ¯ L 1 m ) ,

we have, for all t [ 0 , T ] ,

x m ( t ) T 2 4 max x B ( K , ε ) ¯ ϕ ˆ ( x ) ( ν K ¯ L 1 ( J m ) + ν K ¯ L 1 m ) . (21)

Let us now consider r > 0 such that r B K . Then it follows from (21) that we are able to find m 0 N such that, for all m N , m m 0 , and t [ 0 , T ] , x m r . Therefore, for all m N , m m 0 , T m ( q , 0 ) Int Q , for all q Q , which ensures the validity of condition (iv) in Proposition 2.4.

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, t 0 belongs to the open interval ( 0 , T ) ,

– since A ( t ) = B ( t ) = 0 , we have p ( t ) = ν K ¯ ( t ) .

In this way, we can prove that there exists m 0 N such that every problem ( P m ) , where m m 0 , satisfies all the assumptions of Proposition 2.4. This implies that every such ( P m ) admits a solution, denoted by x m , with x m ( t ) K ¯ , for all t [ 0 , T ] . By similar arguments as in [16], but with the expression Z ( 4 Z k + 1 ) replaced by T 4 , according to condition ( 2 ii ), we can obtain the result that there exists a subsequence, denoted as the sequence, and a function x A C 1 ( [ 0 , T ] , E ) such that x m x and x ˙ m x ˙ in C ( [ 0 , T ] , E ) and also x ¨ m x in L 1 ( [ 0 , T ] , E ) , when m . Thus, a classical closure result (see e.g. [[27], Lemma 5.1.1]) guarantees that x is a solution of (2) satisfying x ( t ) K ¯ , for all t [ 0 , T ] , and the sketch of proof is so complete. □

The case when F = F 1 + F 2 , with F 1 ( t , , ) to be completely continuous and F 2 ( t , , ) to be Lipschitzian, for a.a. t [ 0 , T ] , 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 [[3], Example 6.1 and Remark 6.1].

Corollary 3.1Let E = H be a separable Hilbert space and let us consider the Dirichlet b.v.p.:

x ¨ ( t ) F 1 ( t , x ( t ) , x ˙ ( t ) ) + F 2 ( t , x ( t ) , x ˙ ( t ) ) , for a.a.  t [ 0 , T ] , x ( 0 ) = x ( T ) = 0 , } (22)

where

(i) F 1 : [ 0 , T ] × H × H H is an upper-Carathéodory, globally measurable, multivalued mapping and F 1 ( t , , ) : H × H H is completely continuous, for a.a. t [ 0 , T ] , such that

F 1 ( t , x , y ) ν 1 ( t , D 0 ) ,

for a.a. t [ 0 , T ] , all x H with x D 0 , where D 0 > 0 is an arbitrary constant, ν 1 L 1 ( [ 0 , T ] , [ 0 , ) ) , and all y H ,

(ii) F 2 : [ 0 , T ] × H × H H is a Carathéodory multivalued mapping such that

F 2 ( t , 0 , 0 ) ν 2 ( t ) , for a.a.  t [ 0 , T ] ,

where ν 2 L 1 ( [ 0 , T ] , [ 0 , ) ) , and F 2 ( t , , ) : H × H H is Lipschitzian, for a.a. t [ 0 , T ] , with the Lipschitz constant

L < 4 T ( T + 4 ) .

Moreover, suppose that

(iii) there exists R > 0 such that, for all x H with x = R , t ( 0 , T ) , y H and w F 1 ( t , x , y ) + F 2 ( t , x , y ) , we have

x , w > 0 .

Then the Dirichlet problem (22) admits, according to Theorem 3.1, a solution x ( ) such that x ( t ) R , for all t [ 0 , T ] .

Remark 3.1 For F 2 ( t , x , y ) 0 , the completely continuous mapping F 1 ( t , x , y ) allows us to make a comparison with classical single-valued results recalled in the Introduction. Unfortunately, our F 1 in (i) (see also ( 2 ii ) in Theorem 3.1) is the only mapping which is (unlike in [[3], Example 6.1 and Remark 6.1], where under some additional restrictions quite liberal growth restrictions were permitted) globally bounded w.r.t. y H . Furthermore, our sign condition in (iii) is also (unlike again in [[3], 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 F 1 + F 2 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

u t t ( t , x ) + φ ( t , x , u t ( t , x ) ) = b ( t ) u ( t , x ) + R k ( x , y ) u ( t , y ) d y + p ( R ψ ( x ) u ( t , x ) d x ) f ( u ( t , x ) ) , = t [ 0 , T ] , x R , (23)

involving discontinuities in a state variable. In this equation, the non-local diffusion term R k ( x , y ) u ( t , y ) d y replaces the classical diffusion behavior given by u x x ( t , x ) . In dispersal models such an integral term takes into account the long-distance interactions between individuals (see e.g.[29]). Moreover, when φ is linear in u t , (23) can be considered as an alternative version of the classical telegraph equation (see 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. u t . On the other hand, for non-strictly localized transversality conditions as in [3], for instance, a suitable linear growth estimate w.r.t. u t can be permitted.

Example 4.1 Let us consider the integro-differential equation (23) with φ : [ 0 , T ] × R × R R , b : [ 0 , T ] R , k : R × R R , ψ : R R and p : R R . We assume that

(a) φ is Carathéodory, i.e. φ ( , x , y ) is measurable, for all x , y R , and φ ( t , , ) is continuous, for a.a. t [ 0 , T ] ; φ ( t , x , ) is L ( t ) -Lipschitzian with L L 1 ( [ 0 , T ] ) ; | φ ( t , x , y ) | φ 1 ( t ) φ 2 ( x ) , for a.a. t [ 0 , T ] and all x , y R , where φ 1 L 1 ( [ 0 , T ] ) and φ 2 L 2 ( R ) ; φ ( t , x , 0 ) 0 , for all a.a. t [ 0 , T ] and all x R ,

(b) b L 1 ( [ 0 , T ] ) and satisfies b ( t ) b 0 > 1 , for a.a. t [ 0 , T ] ,

(c) k L 2 ( R × R ) with k L 2 ( R × R ) = 1 ,

(d) p ( r ) 0 , for all r R ; and there can exist r 1 < r 2 < < r k such that p ( ) is continuous, for r r i