Positive solutions for second-order boundary-value problems with sign changing Green's functions

Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 245, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu POSITIVE SOLUTIONS FOR SECOND-ORDER BOUNDARY-VALUE PROBLEMS WITH SIGN CHANGING GREEN’S FUNCTIONS ALBERTO CABADA, RICARDO ENGUIC¸A, LUC´ IA L´ OPEZ-SOMOZA Communicated by Pavel Drabek Abstract. In this article we analyze some possibilities of finding positive solutions for second-order boundary-value problems with the Dirichlet and periodic boundary conditions, for which the corresponding Green’s functions change sign. The obtained results can also be adapted to Neumann and mixed boundary conditions. 1. Introduction In the literature, the existence of positive solutions for boundary-value problems (BVP) has been widely studied, in particular for second-order BVP with periodic and Dirichlet boundary conditions. A standard technique consists in obtaining the existence of positive solutions through Krasnoselskii’s fixed point theorem on cones, or to use fixed point index theory. In these cases, the positivity of the associated Green’s functions is usually fundamental to prove such results. In this paper we are able to prove existence of solutions for several problems where the associated Green’s function changes sign. Hill’s operator properties have been described in several papers, where existence and multiplicity results, comparison principles, Green’s functions and spectral analysis were studied. Some of these results can be originally found in [4, 5, 6, 12, 15]. Positivity results for BVP where the Green’s function can vanish are treated for example in [8, 13]. Graef, Kong and Wang [8] studied the periodic BVP (with T= 1) u00(t) + a(t)u(t) = g(t)f(u(t)), t ∈(0, T), u(0) = u(T), u0(0) = u0(T), with fand gnonnegative continuous functions and gsatisfying the condition mint∈[0,T ]g(t)>0. They assumed the Green’s function to be nonnegative and to satisfy the condition min 0≤s≤TZT 0 G(t, s)dt > 0.(1.1) 2010 Mathematics Subject Classification. 34B15, 34A40. Key words and phrases. Second order differential equations; Dirichlet boundary conditions; periodic boundary conditions; sign changing Green’s function. c 2017 Texas State University. Submitted June 19, 2017. Published October 6, 2017. 1 2 A. CABADA, R. ENGUIC¸ A, L. L´ OPEZ-SOMOZA EJDE-2017/245 Webb [13] considered weaker assumptions to prove the existence of positive solutions of the previous problem, but he still assumed Green’s function to be nonnegative. Despite our results do not require the Green’s function to be nonnegative, they could be applied to this particular case, obtaining positive solutions assuming an integral condition weaker than (1.1) (see Remarks 3.6 and 3.11). On the other hand, some existence results for BVP with sign-changing Green’s function have been considered in [7, 10], where the authors asked for the existence of a subinterval [c, d]⊂[0, T], a function φ∈L1([0, T]) and a constant c∈(0,1] such that the Green’s function Gsatisfies the condition |G(t, s)| ≤ φ(s) for all t∈[0, T] and almost every s∈[0, T], G(t, s)≥c φ(s) for all t∈[c, d] and almost every s∈[0, T].(1.2) It must be pointed out that, if we consider a periodic problem with constant potential a(t) = ρ2for which the related Green’s function changes its sign (i.e. ρ > π/T ,ρ6= 2kπ/T ,k= 1,2, . . .), condition (1.2) is never fulfilled for any strictly positive function φ. This is due to the fact that in such situation the Green’s function is constant along the straight lines of slope equals to one (see [2, 3] for details). Meanwhile, as we will prove on Section 4, our results can be applied without further complications for this case. Moreover, for the Dirichlet BVP with constant potential a(t) = ρ2with signchanging Green’s function (i.e. ρ > π/T,ρ6=kπ/T ,k= 1,2, . . .), as a direct consequence of expression (5.1) below, it is immediate to verify that condition (1.2) holds if and only if ρ2lies between the first and the second eigenvalues of the problem ( π T< ρ < 2π T) but it is never satisfied for ρ > 2π T. However, as we will point out in Section 5, our results can be applied for any nonresonant value of ρ > π/T . Despite this, we must note that the imposed restrictions increase with ρ. Furthermore, in [7, 10] the authors proved the existence of solutions in the cone K0=u∈ C[0, T] : min t∈[c,d]u(t)≥ckuk, that is, they ensured the positivity of the solutions on the subinterval [c, d] but such solutions were allowed to change sign when considering the whole interval [0, T]. As far as we know, positive solutions for BVP with sign-changing Green’s function can be tracked only as back as 2011 in the papers [11, 16]. In the first of these papers, Ma considers the one-parameter family of problems u00(t) + a(t)u(t) = λ g(t)f(u(t)), t ∈(0, T), u(0) = u(T), u0(0) = u0(T).(1.3) By using the Schauder’s fixed point Theorem, the author obtains the existence of a positive solution for sufficiently small values of λ. These existence results are not comparable with the ones we will obtain in this paper. In the second paper, Zhong and An [16] study the following autonomous periodic BVP, with constant potential ρ∈(0,3π 2T]: u00 +ρ2u=f(u), t ∈(0, T ), u(0) = u(T), u0(0) = u0(T).(1.4) In this case, it is very well known that the related Green’s function GP(t, s)≥0 for all ρ∈(0,π T] and it changes sign for ρ∈(π T,3π 2T] (see [2, 3]). With this, it can EJDE-2017/245 POSITIVE SOLUTIONS 3 be defined the constant δ=   ∞if ρ∈(0,π T], inft∈IRT 0G+ P(t,s)ds RT 0G− P(t,s)ds ifρ∈(π T,3π 2T], and using the Krasnoselskii’s fixed point Theorem, the authors prove the following existence result: Theorem 1.1. [16, Theorem 3] Suppose that the following assumptions are fulfilled: (1) f: [0,∞)→[0,∞)is continuous. (2) 0 ≤m= infu≥0{f(u)}and M= supu≥0{f(u)} ≤ M≤ ∞. (3) M/m ≤δ, with M/m =∞when m= 0. Moreover, if δ=∞assume that lim x→∞ f(x) x< ρ2<lim x→0+ f(x) x. Then problem (1.4) has a positive solution on [0, T]. Concerning this specific case, along this paper we improve the range of the values ρfor which the result is still valid. Furthermore, we apply our study to nonconstant potentials and nonautonomous nonlinear parts. As we will see, some of the positivity conditions imposed for the periodic BVP cannot be adapted for the Dirichlet BVP, so the approach that must be used needs to be considerably modified, by using, in this case, a different type of cones. The rest of this article is organized the following way: In Section 2 we state some preliminary results considering the Hill’s operator. In Section 3 some new results concerning the existence of a positive solution for the Hill’s periodic BVP in the case that the Green’s function may change sign are proved. Moreover, in this section, such existence results are generalized to other boundary conditions. In Section 4 we improve Theorem 1.1 for the periodic problem with a constant potential. In Section 5 we approach the Dirichlet BVP, also in the case of a constant potential, where as far as we know, no results for sign changing Green’s function were proved before. 2. Preliminaries Let L[a] be the Hill’s operator related to the potential a L[a]u(t)≡u00(t) + a(t)u(t), t ∈[0, T]≡I, where a:I→R,a∈Lα(I), α≥1. Let X⊂W2,1(I) be a Banach space such that the homogeneous problem L[a]u(t) = 0,for a. e. t∈I, u ∈X(2.1) has only the trivial solution. This condition is known as operator L[a] being nonresonant in X. Moreover, it is very well known that if this condition is satisfied and σ∈L1(I), the nonhomogeneous problem L[a]u(t) = σ(t),for a. e. t∈I, u ∈X has a unique solution u(t) = ZT 0 G(t, s)σ(s)ds, t ∈I, 4 A. CABADA, R. ENGUIC¸ A, L. L ´ OPEZ-SOMOZA EJDE-2017/245 where Gis the corresponding Green’s function. We denote x0 on Iif x≥0 on Iand RT 0x(s)ds > 0. It is said that operator L[a] satisfies a strong maximum principle (MP) in Xif u∈X, L[a]u0 on I⇒u < 0 in (0, T). Analogously, L[a] satisfies the antimaximum principle (AMP) in Xif u∈X, L[a]u0 on I⇒u > 0 in (0, T). The next result is a direct consequence of [3, Corollaries 1.6.6 and 1.6.12], and it ensures that the maximum and anti-maximum principles for the periodic problem are equivalent to the constant sign of the Green’s function. Lemma 2.1. The following claims are equivalent: (1) The related Green’s function Gof problem (2.1) satisfies G(t, s)≥0 (≤0) on I×I. (2) Operator L[a]satisfies a strong maximum (antimaximum) principle in X. We will consider now the periodic boundary-value problem u00(t) + a(t)u(t) = 0, t ∈I, u(0) = u(T), u0(0) = u0(T),(2.2) and we will denote its related Green’s function as GP. Now, let λPbe the smallest eigenvalue of the periodic problem u00(t)+(a(t) + λ)u(t) = 0,for a. e. t∈I, u(0) = u(T), u0(0) = u0(T), and let λAbe the smallest eigenvalue of the anti-periodic problem u00(t)+(a(t) + λ)u(t)=0,for a. e. t∈I, u(0) = −u(T), u0(0) = −u0(T). In [15] it is proved that λP< λA. The following result relates the constant sign of the periodic Green’s function with the sign of these eigenvalues: Lemma 2.2. [15, Theorem 1.1] Suppose that a∈L1(I), then: (1) GP(t, s)≤0on I×Iif and only if λP>0. (2) GP(t, s)≥0on I×Iif and only if λP<0≤λA. If we consider other boundary-value problems, such as the Neumann problem u00(t) + a(t)u(t)=0, t ∈I, u0(0) = u0(T) = 0; (2.3) the Dirichlet problem u00(t) + a(t)u(t)=0, t ∈I, u(0) = u(T) = 0; (2.4) and the mixed problems u00(t) + a(t)u(t) = 0, t ∈I, u0(0) = u(T) = 0; (2.5) u00(t) + a(t)u(t) = 0, t ∈I, u(0) = u0(T) = 0; (2.6) denoting by GN,GD,GM1and GM2the related Green’s functions and λN,λD, λM1and λM2the corresponding smallest eigenvalue of each of the problems, we know that the following results are satisfied (see [6]): Lemma 2.3. (1) GN(t, s)<0on I×Iif and only if λN>0. (2) GN(t, s)≥0on I×Iif and only if λN<0,λM1≥0and λM2≥0. (3) GNchanges sign if and only if min{λM1, λM2}<0. (4) GD(t, s)<0on (0, T)×(0, T)if and only if λD>0. EJDE-2017/245 POSITIVE SOLUTIONS 5 (5) GDchanges sign if and only if λD<0. (6) GM1(t, s)<0on [0, T)×[0, T)if and only if λM1>0. (7) GM1changes sign if and only if λM1<0. (8) GM2(t, s)<0on (0, T]×(0, T]if and only if λM2>0. (9) GM2changes sign if and only if λM2<0. 3. Periodic boundary-value problems Consider now the nonlinear and nonautonomous periodic boundary value problem u00(t) + a(t)u(t) = f(t, u(t)), t ∈I, u(0) = u(T), u0(0) = u0(T).(3.1) We will assume that problem (2.2) is nonresonant and λA<0. From Lemma 2.2, it is clear that in this case the related Green’s function changes its sign on I×I. On the other hand, it is well-known that there exists vP, a positive eigenfunction on I, unique up to a constant, related to λP; that is, vPis such that v00 P(t) + a(t)vP(t) = −λPvP(t),a. e. t∈I, vP(0) = vP(T), v0 P(0) = v0 P(T). Therefore, vP(t) = −λPZT 0 GP(t, s)vP(s)ds and, since vPis positive and λP<0, we have that ZT 0 GP(t, s)vP(s)ds > 0∀t∈I and, consequently, ZT 0 G+ P(t, s)vP(s)ds > ZT 0 G− P(t, s)vP(s)ds ∀t∈I, where G+ Pand G− Pare the positive and negative parts of GP. Since the Green’s function changes sign, it makes sense to define γ= inf t∈IRT 0G+ P(t, s)vP(s)ds RT 0G− P(t, s)vP(s)ds (>1). Moreover, to ensure the existence of solutions of problem (3.1), we will make the following assumptions: (H1) f:I×[0,∞)→[0,∞) satisfies L1-Carath´eodory conditions, that is, f(·, u) is measurable for every u∈R,f(t, ·) is continuous for a. e. t∈Iand for each r > 0 there exists φr∈L1(I) such that f(t, u)≤φr(t) for all u∈[−r, r] and a. e. t∈I. (H2) There exist two positive constants mand Msuch that mvP(t)≤f(t, x)≤ MvP(t) for every t∈Iand x≥0. Moreover, these constants satisfy that M m≤γ. (H3) There exists [c, d]⊂Isuch that Rd cGP(t, s)dt ≥0, for all s∈Iand Rd cGP(t, s)dt > 0, for all s∈[c, d]. 6 A. CABADA, R. ENGUIC¸ A, L. L ´ OPEZ-SOMOZA EJDE-2017/245 Remark 3.1. We note that condition (H2) includes, as particular cases, hypotheses (2) and (3) in Theorem 1.1 imposed in [16]. This is so because if a(t) = ρ2, as in problem (1.4), we have that λP=−ρ2and vP(t) = 1 for all t∈I. Moreover, as we will point out in Section 4, we have that if a(t) = ρ2then ZT 0 GP(t, s)ds =1 ρ2, and condition (H3) is trivially fulfilled for [c, d] = I. Moreover, we note that in (H2) we are not considering the possibility of m= 0. Theorem 1.1 includes this case, but only when γ= +∞, which only happens when the Green’s function is nonnegative. In [16] the authors consider this possibility because they are assuming that ρ∈0,3π 2Tand when ρ∈0,π T,GPis nonnegative. As we will see in Corollary 3.5, hypothesis (H2) is not necessary in this case, so this is the reason why we do not consider the possibility m= 0. We will consider the Banach space (C(I, R),k · k) coupled with the supremum norm kuk≡kuk∞, and define the cone K=u∈ C(I, R) : u≥0 on I, ZT 0 u(s)ds ≥σkuk, where σ=η maxt, s∈I{GP(t, s)}, with η= min s∈[c,d]Zd c GP(t, s)dt.(3.2) Now, it is clear that uis a solution of the periodic problem (3.1) if and only if it is a fixed point of the following operator: Tu(t) = ZT 0 GP(t, s)f(s, u(s)) ds. Lemma 3.2. Assume hypothesis (H1)–(H3). Then T:C(I)→ C(I)is a completely continuous operator which maps the cone Kto itself. Proof. The proof that operator Tis a completely continuous operator follows standard arguments and we omit it. Let us see now that Tmaps the cone to itself. Considering u∈K, then, for all t∈I, the following inequalities are fulfilled: Tu(t) = ZT 0 GP(t, s)f(s, u(s)) ds =ZT 0G+ P(t, s)−G− P(t, s)f(s, u(s)) ds ≥ZT 0m vP(s)G+ P(t, s)−MvP(s)G− P(t, s)ds ≥mZT 0 G+ P(t, s)vP(s)ds −γZT 0 G− P(t, s)vP(s)ds≥0. EJDE-2017/245 POSITIVE SOLUTIONS 7 Moreover, ZT 0Tu(t)dt ≥Zd cTu(t)dt =Zd cZT 0 GP(t, s)f(s, u(s)) ds dt =ZT 0 f(s, u(s)) Zd c GP(t, s)dt ds ≥ηZT 0 f(s, u(s)) ds, and since Tu(t)≤max t,s∈I{GP(t, s)}ZT 0 f(s, u(s)) ds, we deduce that RT 0Tu(t)dt ≥σTu(t) for all t∈I, that is ZT 0Tu(t)dt ≥σkTuk, and the result is concluded.  Now, to prove the existence of solutions for problem (3.1), we use some classical results regarding the fixed point index. We compile them in the following lemma. Let Ω be an open bounded subset of C(I) and let us denote ¯ Ω and ∂Ω its closure and boundary, respectively. Moreover, let us denote ΩK= Ω ∩K. Lemma 3.3. [1, Lemma 12.1] Let ΩKbe an open bounded set with 0∈ΩKand ¯ ΩK6=K. Assume that F:¯ ΩK→Kis a completely continuous map such that x6=Fx for all x∈∂ΩK. Then the fixed point index iK(F, ΩK)has the following properties: (1) If there exists e∈K\{0}such that x6=Fx +λe for all x∈∂ΩKand all λ > 0, then iK(F, ΩK)=0. (2) If x6=µ Fx for all x∈∂ΩKand for every µ≤1, then iK(F, ΩK)=1. (3) If iK(F, ΩK)6= 0, then Fhas a fixed point in ΩK. (4) Let Ω1 Kbe an open set with ¯ Ω1 K⊂ΩK. If iK(F, ΩK)=1and iK(F, Ω1 K) = 0, then Fhas a fixed point in ΩK\¯ Ω1 K. The same result holds if iK(F, ΩK) = 0and iK(F, Ω1 K)=1. Now we are in a position to prove the existence results concerning the periodic problem (3.1) as follows. First, we note that, as an immediate consequence of condition (H2), we deduce the following properties: f0= lim x→0+min t∈[c,d] f(t, x) x=∞, f∞= lim x→∞ max t∈I f(t, x) x= 0, where the interval [c, d] is given in (H3). These properties will let us prove the following theorem. Theorem 3.4. Assume that λA<0and hypothesis (H1)–(H3) hold. Then there exists at least one positive solution of problem (3.1) in the cone K. Proof. Taking into account the definition of f0, we know that there exists δ1>0 such that when kuk ≤ δ1, then f(t, u(t)) >u(t) η,∀t∈[c, d], 8 A. CABADA, R. ENGUIC¸ A, L. L ´ OPEZ-SOMOZA EJDE-2017/245 with ηdefined in (3.2). Let Ω1={u∈K:kuk< δ1} and choose u∈∂Ω1and e∈K\{0}. We will prove that u6=Tu+λ e for every λ > 0. Assume, on the contrary, that there exists some λ > 0 such that u=Tu+λ e, that is, u(t) = Tu(t) + λ e(t)≥ Tu(t)∀t∈I. Then Zd c u(t)dt ≥Zd cTu(t)dt =Zd cZT 0 GP(t, s)f(s, u(s)) ds dt =ZT 0Zd c GP(t, s)dtf(s, u(s)) ds ≥Zd cZd c GP(t, s)dtf(s, u(s)) ds > Zd c u(s)ds, which is a contradiction. Therefore iK(T, Ω1) = 0. Proceeding in an analogous way to [5, 8, 9], we define ˜ f(t, u) = max0≤z≤uf(t, z). Clearly ˜ f(t, ·) is a nondecreasing function on [0,∞). Moreover, since f∞= 0 it is obvious that lim x→∞ max t∈I ˜ f(t, x) x= 0. As a consequence, there exists δ2>0 such that if kuk ≥ δ2then ˜ f(t, kuk)<σ2 T2ηkuk ∀t∈I. Let Ω2={u∈K;kuk< δ2} and choose u∈∂Ω2. We will prove that u6=µTufor every µ≤1. Assume, on the contrary, that there exists some µ≤1 such that u(t) = µTu(t) for all t∈I. Then σkuk ≤ ZT 0 u(t)dt =µZT 0Tu(t)dt =µZT 0ZT 0 GP(t, s)f(s, u(s)) ds dt =µZT 0ZT 0 GP(t, s)dtf(s, u(s)) ds ≤µT max t,s∈I{GP(t, s)}ZT 0 f(s, u(s)) ds ≤µT max t,s∈I{GP(t, s)}ZT 0 ˜ f(s, u(s)) ds ≤µT max t,s∈I{GP(t, s)}ZT 0 ˜ f(s, kuk)ds < µT2η σ σ2 T2ηkuk ≤ σkuk, EJDE-2017/245 POSITIVE SOLUTIONS 9 which is a contradiction. As a consequence, iK(T, Ω2) = 1. We conclude that operator Thas a fixed point, that is, there exists at least a nontrivial solution of problem (3.1).  The previous theorem is also valid if the Green’s function is nonnegative. In this case, hypothesis (H3) would be trivially fulfilled and hypothesis (H2) is not necessary since it is only used to prove that Tmaps the cone to itself, which is obvious (since fis nonnegative) when GPis nonnegative. On the other hand, we would need to add the hypothesis that f0=∞and f∞= 0 (which can not be deduced if we eliminate (H2)). The result reads as follows: Corollary 3.5. Assume that λP<0≤λAand hypothesis (H1) is fulfilled. Then, if f0=∞and f∞= 0 there exists at least one positive solution of problem (3.1) in the cone K. Remark 3.6. We note that for a nonnegative Green’s function, we generalize the results of Graef, Kong and Wang [8, 9] and Webb [13] since our condition (H3) is weaker than condition (1.1) considered by them. Corollary 3.7. If f(t, x)≡f(t)∈L1(I)satisfies (H2), then the unique solution of (3.1) is a nonnegative function on [0, T]. Remark 3.8. We note that u(t)≡1 is the unique solution of the periodic problem u00(t) + a(t)u(t) = a(t), t ∈I, u(0) = u(T), u0(0) = u0(T). Therefore it is clear that ZT 0 GP(t, s)a(s)ds = 1 >0 (3.3) and so the previous reasoning is also valid if a≥0, a > 0 on [c, d], and we change the definition of γby γ∗= inf t∈IRT 0G+ P(t, s)a(s)ds RT 0G− P(t, s)a(s)ds. In this case, assumption (H2) would be substituted by (H2’) There exist two positive constants mand Msuch that ma(t)≤f(t, u)≤ Ma(t) for every t∈I,u > 0. Moreover, these constants satisfy that M m≤γ∗. 3.1. Neumann, Dirichlet and mixed boundary value problems. From the classical spectral theory [14], it is very well know that, as in the periodic case, for any of the boundary conditions introduced in Lemma 2.3, there exists a positive eigenfunction on (0, T) related to the corresponding smallest eigenvalue. Therefore, if we are in the case in which L[a] operator coupled with the associated boundary conditions is nonresonant and the related Green’s function changes sign (different cases are characterized in Lemma 2.3), we could follow the same argument as in the previous section to define γand we would obtain analogous existence results. Hypothesis (H1)–(H3) would be the same with the suitable notation for each of the problems (that is, considering in each case the appropriate Green’s function and eigenfunction). 16 A. CABADA, R. ENGUIC¸ A, L. L ´ OPEZ-SOMOZA EJDE-2017/245 Π 2Π 3Π 4Π 5Π 6Π 1 2 3 4 Figure 3. Graph of γfor the Dirichlet problem. 0.2 0.4 0.6 0.8 1.0 0.001 0.002 0.003 0.004 0.005 Figure 4. Solution of problem (5.2) . 0.2 0.4 0.6 0.8 1.0 -0.01 0.01 0.02 Figure 5. Solution of problem (5.3). Remark 5.2. Analogous arguments and calculations can be done for the Neumann and mixed problems. EJDE-2017/245 POSITIVE SOLUTIONS 17 Acknowledgments. A. Cabada and L. L´opez-Somoza were partially supported by Ministerio de Econom´ıa y Competitividad, Spain, and FEDER, project MTM2013- 43014-P, and by the Agencia Estatal de Investigaci´on (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER. L. L´opez-Somoza was spartially supported by FPU scholarship, Ministerio de Educaci´on, Cultura y Deporte, Spain. R. Engui¸ca was partially supported by Funda¸cao para a Ciˆencia e a Tecnologia, Portugal, UID/MAT/04561/2013. References [1] H. Amann; Fixed point equations and Nonlinear Problems in Ordered Banach Spaces, SIAM Review, Vol. 18, 4 (1976), pp. 620–709. 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Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005. [15] M. Zhang; Optimal conditions for maximum and anti-maximum principles of the periodic solution problem, Bound. Value Prob. 2010 (2010), 26 pages. [16] S. Zhong, Y. An; Existence of positive solutions to periodic boundary value problems with sign-changing Green’s function, Bound. Value Prob. 2011, 2011:8 (2011), 6 pages. Alberto Cabada Instituto de Matem´ aticas, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Galicia, Spain E-mail address:[email protected] Ricardo Enguic¸a Departamento de Matem´ atica, Instituto Polit´ ecnico de Lisboa, Lisboa, Portugal E-mail address:[email protected] Luc´ ıa L´ opez-Somoza Instituto de Matem´ aticas, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Galicia, Spain E-mail address:[email protected]