Spectral characterization of the constant sign Green’s functions for periodic and Neumann boundary value problems of even order

Differential Equations &Applications Volume 14, Number 2 (2022), 335–347 doi:10.7153/dea-2022-14-24 SPECTRAL CHARACTERIZATION OF THE CONSTANT SIGN GREEN’S FUNCTIONS FOR PERIODIC AND NEUMANN BOUNDARY VALUE PROBLEMS OF EVEN ORDER ALBERTO CABADA∗AND LUC´ IA L´ OPEZ-SOMOZA Dedicated to Professor Paul Eloe on his retirement (Communicated by J. Henderson) Abstract. In this paper we will characterize the interval of real parameters Min which the Green’s function GM, related to the operator T2n[M]u(t):=u(2n)(t)+Mu(t)coupled to periodic, u(i)(0)=u(i)(T),i=0,...,2n−1, or Neumann, u(2i+1)(0)=u(2i+1)(T)=0, i= 0,...,n−1 , boundary conditions, has constant sign on its square of definition. More concisely, we will prove that the optimal values are given as the first zeros of GM(0,0)or GM(T/2,0), depending both on the sign of GMand on the fact whether 2nis, or is not, a multiple of 4. Such values will be characterized as the eigenvalues of the operator u(2n)related to suitable boundary conditions. This characterization allows us to obtain such values without calculating the exact expression of the Green’s function. 1. Introduction In the study of nonlinear boundary value problems, a classical and fruitful method to ensure the existence of solutions consists on the construction of a related integral operator, whose fixed points coincide with the solutions of the considered problem. The kernel of the integral operator is known as the Green’s function related to the linear part of the equation. To ensure the existence of solutions that have the same (or opposite) sign than the external force, we need to ensure that the Green’s function has constant sign on its square of definition. Such property is equivalent to have comparison principles and allows to develop monotone iterative techniques (see [12,14]), lower and upper solutions method (see [1,3,10]) or to ensure the existence of solutions in suitable cones (see [13,17]). The paper is organized as follows: in Section 2 it is presented a short survey of the main properties satisfied by the Green’s functions and that allow us to study its constant sign. We particularize such properties to the periodic case and compile the main known Mathematics subject classification (2020): 34B05, 34B08, 34B09, 34B27. Keywords and phrases: Green’s function, periodic boundary conditions, Neumann boundary conditions, constant sign, spectral characterization. ∗Corresponding author. c    ,Zagreb Paper DEA-14-24 335 336 A. CABADA AND L. L´ OPEZ-SOMOZA results till now that ensure the constant sign of the Green’s function related to operator Tn[M]u:=u(n)+Mu defined on the space of periodic functions. In Section 3 the optimal values of M, in which the Green’s function related to operator Tn[M], with n even, has constant sign, are obtained via spectral theory of related problems. Using an equivalence with the periodic case, obtained in [8], of the sign of the Green’s function related to the operator Tn[M]coupled to Neumann boundary conditions, such values are also characterized for Neumann boundary conditions in Section 4. Moreover, in the last section, we present a direct method to obtain explicitly such eigenvalues, via the construction of suitable Wronskians, that allow us to get the exact values without calculating the expression of the Green’s function. As a direct application, we calculate the value of all of them for n⩽12, neven. 2. A survey on constant sign Green’s functions Consider the n-th order linear operator Tn[M]u(t):=u(n)(t)+Mu(t),t∈I,(1) with I≡[0,T]and M∈R. We will consider non homogeneous linear problems, for which the solutions that we are looking for satisfy suitable boundary conditions and belong to the space Wn,1(I)=u∈Cn−1(I):u(n−1)∈AC(I), where AC(I)denotes the set of absolutely continuous functions on I. In particular, we will consider X⊂Wn,1(I)a Banach space such that the following definition is satisfied. DEFINITION 1. Given a Banach space X, an operator Tn[M]is said to be nonresonant in Xif and only if the homogeneous equation Tn[M]u(t)=0a.e.t∈I,u∈X, has only the trivial solution. Let us now introduce the concept of eigenvalue of the corresponding linear equation. DEFINITION 2. Given a Banach space Xand a real value M, we say that λ ∈R is an eigenvalue of operator Tn[M]in Xif and only if the homogeneous equation Tn[M+ λ ]u(t)=0a.e.t∈I,u∈X, has non trivial solutions. It is very well known that if σ ∈L1(I)and operator Tn[M]is nonresonant in X,then the non homogeneous problem Tn[M]u(t)= σ (t)a. e. t∈I,u∈X, Differ. Equ. Appl. 14, No. 2 (2022), 335–347. 337 has a unique solution given by u(t)=T 0 G[M,T](t,s) σ (s)ds,∀t∈I, where G[M,T]denotes the, so-called, Green’s function related to operator Tn[M]on X and it is uniquely determined. See [3] for details. Now, by using the notation h0 for a function h∈L1(I)such that h(t)⩾0for a. e. t∈Iand T 0h(t)dt >0, we introduce the following definitions. DEFINITION 3. Operator Tn[M]admits a maximum principle (MP) in Xif and only if every function u∈Xsuch that Tn[M]u0onIsatisfies that u<0on(0,T). DEFINITION 4. Operator Tn[M]admits an antimaximum principle (AMP) in Xif and only if every function u∈Xsuch that Tn[M]u0onIsatisfies that u>0on (0,T). Clearly, we have that if Tn[M]satisfies either MP or AMP on Xthen it is nonresonant in X. The previously defined comparison principles are equivalent to the constant sign of the Green’s function. See [18, Theorem 4.1] for the case n=2 and the Green’s function related to the periodic problem and [7, Lemma 10] for n=2 and arbitrary boundary conditions. The proof for arbitrary n⩾1 is analogous. The result is the following one: THEOREM 1. The following equivalences hold: •Operator Tn[M]satisfies MP on X if and only if the related Green’s function is nonpositive on I ×I. •Operator Tn[M]satisfies AMP on X if and only if the related Green’s function is nonnegative on I ×I. The following results are a direct adaptation of [3, Theorems 1.8.5 and 1.8.9, Lemmas 1.8.25 and 1.8.33] to this operator. LEMMA 1. Suppose that operator Tn[M]is nonresonant in a Banach space X , its related Green’s function G[M,T]is nonpositive on I ×I , and satisfies condition (Ng)There is a continuous function φ (t)>0for all t ∈(0,T)and k1,k2∈L1(I), such that k1(s)<k2(s)<0for a. e. s ∈I , satisfying φ (t)k1(s)⩽G[M,T](t,s)⩽ φ (t)k2(s),for a. e. (t,s)∈I×I. Then G[M+ λ ,T]is nonpositive on I ×I if and only if λ ∈NT,whereN T:= (−∞, λ 1(T)) or [− μ (T), λ 1(T)), with λ 1(T)>0the first eigenvalue of operator Tn[M] in X and μ (T)⩾0such that Tn[− μ (T)] is nonresonant in X and its related nonpositive Green’s function G[− μ (T),T]vanishes at some point of I ×I. Moreover, G[M+ λ ,T]is monotone decreasing in λ ∈NT. 338 A. CABADA AND L. L´ OPEZ-SOMOZA LEMMA 2. Suppose that operator Tn[M]is nonresonant in a Banach space X , its related Green’s function G[M,T]is nonnegative on I ×I , and satisfies condition (Pg)There is a continuous function φ (t)>0for all t ∈(0,T)and k1,k2∈L1(I), such that 0<k1(s)<k2(s)for a. e. s ∈I , satisfying φ (t)k1(s)⩽G[M,T](t,s)⩽ φ (t)k2(s),for a. e. (t,s)∈I×I. Then G[M+ λ ,T]is nonnegative on I ×I if and only if λ ∈PT,whereP T:= ( λ 1(T),∞)or ( λ 1(T), μ (T)], with λ 1(T)<0the first eigenvalue of operator Tn[M]in X and μ (T)⩾0such that Tn[ μ (T)] is nonresonant in X and the related nonnegative Green’s function G[ μ (T),T]vanishes at some point of the square I ×I. Moreover, G[M+ λ ,T]is monotone decreasing in λ ∈PT. Now, if we fix as boundary conditions the periodic ones, the Banach space Xis denoted as Xn P,T=u∈Wn,1(I):u(k)(0)=u(k)(T),k=0,...,n−1. It is immediate to verify that operator Tn[0]is resonant on Xn P,Tfor all n∈N.Since the associated eigenfunctions are the constants and, obviously, they have constant sign, we have that λ =0 is the main eigenvalue related to operator u(n)on the space Xn P,T. As a consequence, from Lemmas 1and 2, we deduce that, if they are non empty, 0 is the supremum of the interval NTand the infimum of PT. There are many works in which maximum and antimaximum principles have been studied for different operators coupled to this kind of boundary conditions. As a consequence, it is very well known [14]thatT1[M]is inverse positive on X1 P,Tif and only if M>0 and inverse negative if and only if M<0. Moreover, T2[M]is inverse negative on X2 P,Tif and only if M<0. In [1], it is showed that T2[M]is inverse positive on X2 P,T if and only if M∈(0,( π /T)2]. We note that ( π /T)2is the first eigenvalue of operator u coupled to Dirichlet boundary conditions. Similar spectral characterization could be extended to a more general operators with non constant coefficients, see for instance [4,7,8,16,19], and relies on the idea of this paper in order to consider an eigenvalue characterization of the regular extremes μ (T)of the intervals NTand PT.Wealso mention that in [1]thesetNTfor operator T4[M]on T4 P,Tand the sets NTand PTfor operator T3[M]on the space X3 P,Tare also characterized. Finally, we point out that the sets PTfor T4[M]and NTfor the operator T6[M], for the corresponding periodic spaces, are characterized in [6]. It is important to mention that, as far as the authors know, at any of the previous works (except for the case n=2andM>0), the optimal extremes of the intervals NTand PT, for which the Green’s function has constant sign on I×I, haven’t been characterized as an eigenvalue of the same operator related to different boundary conditions. For higher order equations, the optimal values that describe the sets NTand PT are not known. In this situation, for arbitrary n, only non optimal estimations were givenuptodate.In[15], by using the disconjugacy theory [9], the authors obtained the following result. Differ. Equ. Appl. 14, No. 2 (2022), 335–347. 339 LEMMA 3. Let M >0(M <0) be such that |M|<nnn! n 2nTn(n−1)n−1, with [x]the greatest integer smaller than or equal to the real number x. Then operator Tn[M]is inverse positive (inverse negative) on Xn P,T. By using the decomposition of the n-th order operator Tn[M]as a composition of first and second order operators, better estimations on the values of the parameter M for which the operator Tn[M]is either inverse positive or inverse negative in Xn P,Tare obtained in [2]. The results are the following ones. LEMMA 4. [2, Lemma 2.4] Operator Tn[M]is inverse positive on Xn P,Tif one of the following properties is fulfilled 1. n =4k, k∈{1,2,...}and 0<M⩽ π Tsinn+2 2n π n . 2. n =2+4k, k∈{1,2,...}and 0<M⩽ π Tn . 3. n is odd and 0<M⩽ π Tsinn+1 2n π n . LEMMA 5. [2, Lemma 2.5] Operator Tn[M]is inverse negative on Xn P,Tif one of the following properties is fulfilled 1. n =4k, k∈{1,2,...}and − π Tn ⩽M<0. 2. n =2+4k, k∈{1,2,...}and − π Tsinn+2 2n π n ⩽M<0. 3. n is odd and − π Tsinn+1 2n π n ⩽M<0. Obviously, previous results ensure that for any n∈N,thesetsNTand PTare non empty. By checking the proofs of the two previous results, given in [2], it is immediate to verify that the Green’s function GP[M,T]is bounded and strictly positive for any M>0 on the interior of the intervals given in Lemma 4and bounded and strictly negative for M<0 on the interior of the intervals given in Lemma 5. Thus, it is obvious that Lemmas 1and 2hold in these cases. So, we are interested in obtaining the values of μ (T)introduced in Lemmas 1and 2that characterize the interval in Mof the constant sign of the Green’s function. Such values will be given as the first Mfor which GP[M,T]vanishes at some point on I×I(or they are +∞or −∞). 340 A. CABADA AND L. L´ OPEZ-SOMOZA To this end, we will use the following result [1](seealso[3, Section 1.4]), that ensures us that in the particular case of a constant’s coefficient operator, the Green’s function related to the periodic problem is constant over the straight lines of slope equals to 1. The result, particularized to this situation, is the following. LEMMA 6. [1, Lemma 2.1] The Green’s function GP[M,T]related to the operator Tn[M]on the space of T -periodic functions Xn P,Tis given by the following expression: GP[M,T](t,s)=GP[M,T](t−s,0),0⩽s⩽t⩽T, GP[M,T](T+t−s,0),0⩽t<s⩽T. Moreover, the function GP[M,T](t,0)is the unique solution of the following problem: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Tn[M]rM(t)=0,t∈I, r(i) M(0)−r(i) M(T)=0,i=0,...,n−2, r(n−1) M(0)−r(n−1) M(T)=1. (2) As it is stated on the proof of [3, Corollary 1.4.12] for a more general situation, it is immediate to verify that if n=2kis even, then rM(t)=rM(T−t)for all t∈I. Notice that, as a direct consequence, r(j) M(t)=(−1)jr(j) M(T−t)for all t∈Iand j∈{0,1,,...,2k}.(3) In particular, r(2j+1) M(T/2)=0forallj∈{0,1,,...,k−1}.(4) Moreover, we have that for all odd number i⩽2k−3, it is satisfied that r(i) M(0)=r(i) M(T)=−r(i) M(0), which implies that r(2j+1) M(0)=r(2j+1) M(T)=0,j∈{0,1,,...,k−2}.(5) On the other hand, −r(2k−1) M(T)=r(2k−1) M(0)=r(2k−1) M(T)+1, so we conclude that r(2k−1) M(0)=1/2andr(2k−1) M(T)=−1/2.(6) 3. Periodic problem In this section we will obtain an spectral characterization of the constant sign of the Green’s function related to the operator Tn[M]on the space of T-periodic functions Xn P,Twhen nis even. To avoid possible confusions, along the section we will denote n=2k, with k=1,2,.... The obtained result is the following. Differ. Equ. Appl. 14, No. 2 (2022), 335–347. 341 THEOREM 2. Let n =2k with k ∈N.Then 1. The Green’s function related to operator Tn[M]on the space of T -periodic functions Xn P,Tis nonnegative on I ×I (and strictly positive on I ×I if M is on the interior of the intervals) if and only if the following conditions are fulfilled: (a) k =2l+1for some l ∈{0,1,...}, and M ∈(0,Mn],whereMnis the least positive eigenvalue of the following two-point boundary value problem: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r(n)(t)=0,t∈[0,T/2], r(0)=0, r(2j+1)(0)=0,j∈{0,1,,...,k−2}, r(2j+1)(T/2)=0,j∈{0,1,,...,k−1}. (7) (b) k =2lforsomel∈{1,...}, and M ∈(0,Mn],whereMnis the least positive eigenvalue of the following two-point boundary value problem: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r(n)(t)=0,t∈[0,T/2], r(T/2)=0, r(2j+1)(0)=0,j∈{0,1,,...,k−2}, r(2j+1)(T/2)=0,j∈{0,1,,...,k−1}. (8) 2. The Green’s function related to operator Tn[M]on the space of T -periodic functions Xn P,Tis nonpositive on I ×I (and strictly negative on I ×I if M is on the interior of the intervals) if and only if the following conditions are fulfilled: (a) n =2and M ∈(−∞,0). (b) k =2l+1for some l ∈{1,...}, and M ∈[˜ Mn,0),where ˜ Mnis the biggest negative eigenvalue of Problem (8). (c) k =2lforsomel∈{1,...}, and M ∈[˜ Mn,0),where ˜ Mnis the biggest negative eigenvalue of Problem (7). Proof. As a direct consequence of Lemma 6, we have that the sign of the Green’s function on I×Iis given by the one of function rM(t):=GP[M,T](t,0),defined in (2), on I. Moreover, from Lemmas 4and 5we have that there is M0>0 such that rM>0onIfor all M∈(0,M0)and rM<0onIfor all M∈(−M0,0). Finally, using Lemmas 1and 2, we have that our problem is reduced to find the unique negative and positive values of Mfor which rMhas constant sign and vanishes at some point in I. It is important to point out that if Mis not an eigenvalue of operator u(n)on the space of periodic functions Xn P,T(which always holds when either M∈PTor M∈ NT), then identities (4)and(5) warrant that rMsatisfies the two last sets of boundary conditions imposed in Problems (7)and(8). Thus, take M=0forwhichrM>0orrM<0onI. It is clear, from Lemmas 1 and 2,thatMr M(t)>0forallt∈I. 342 A. CABADA AND L. L´ OPEZ-SOMOZA Then, we have r(2k) M(t)=−Mr M(t)<0forallt∈I. This implies that r(2k−1) Mis strictly decreasing on I. As a consequence, from (4)and(6), we deduce that r(2k−1) M(t)>0forallt∈[0,T/2)and r(2k−1) M(t)<0forallt∈(T/2,T]. Thus, we have that r(2k−2) Mis strictly increasing on (0,T/2)and strictly decreasing in (T/2,T). In particular, from (3), we deduce that max t∈I{r(2k−2) M(t)}=r(2k−2) M(T/2)and min t∈I{r(2k−2) M(t)}=r(2k−2) M(0)=r(2k−2) M(T). In case of k=1, we have that the unique M2>0forwhichrM2⩾0onIand rM2takes the value zero at some point of Iis given as the least positive eigenvalue of the following mixed problem r(t)=0,t∈[0,T/2],r(0)=r(T/2)=0. It is very well known that such value is ( π /T)2and coincides with the least eigenvalue of the Dirichlet problem in I:r(t)=0,t∈I,r(0)=r(T)=0(see[7,16,19]). On the other hand, in this case (k=1), we have that the unique ˜ M2<0forwhich r˜ M2⩽0onIand r˜ M2takes the value zero at some point of Iwould be given as the biggest negative eigenvalue of the following problem r(t)=0,t∈[0,T/2],r(T/2)=r(T/2)=0. However, since terminal conditions are considered, the problem r(t)+ λ r(t)=0,t∈[0,T/2],r(T/2)=r(T/2)=0, has only the trivial solution for any real value of λ . This property ensures us that the Green’s function is negative for all M<0, which is, also, a very well known result (see [1,3,14]). Now, if k>1, from (5) we deduce that function r(2k−2) Mcannot have constant sign on I. As a consequence, we have that there is t0∈(0,T/2)such that r(2k−2) M(t)>0forallt∈(t0,T−t0)and r(2k−2) M(t)<0forallt∈[0,t0)∪(T−t0,T]. From this, (3)and(5), we deduce that r(2k−3) M(t)<0forallt∈(0,T/2)and r(2k−3) M(t)>0forallt∈(T/2,T), which implies that r(2k−4) Mis strictly decreasing on (0,T/2)and strictly increasing in (T/2,T). In particular, min t∈I{r(2k−4) M(t)}=r(2k−4) M(T/2)and max t∈I{r(2k−4) M(t)}=r(2k−4) M(0)=r(2k−4) M(T). So, if k=2 we have that the unique M4>0forwhichrM4⩾0onIand rM4takes the value zero at some point of Iis given as the least positive eigenvalue of the following problem r(4)(t)=0,t∈[0,T/2],r(T/2)=r(0)=r(T/2)=r(T/2)=0. Differ. Equ. Appl. 14, No. 2 (2022), 335–347. 343 Moreover, the unique ˜ M4<0forwhichr˜ M4⩽0onIand r˜ M4takes the value zero at some point of Iwould be given as the biggest negative eigenvalue of the following problem r(4)(t)=0,t∈[0,T/2],r(0)=r(0)=r(T/2)=r(T/2)=0. For k>2, using (5) again, we deduce that function r(2k−4) Mcannot have constant sign on I. As a consequence, we have that there is t1∈(0,T/2)such that r(2k−4) M(t)<0forallt∈(t1,T−t1)and r(2k−4) M(t)>0forallt∈[0,t1)∪(T−t1,T], which, arguing as in the previous situations, ensures us that r(2k−5) M(t)>0forallt∈(0,T/2)and r(2k−5) M(t)<0forallt∈(T/2,T). Thus, we are in the same situation as r(2k−1) Mand the result holds by recurrence.  4. Neumann problem Now, for any even natural number n=2k, we deal with the constant sign of the Green’s function, that we will denote as GN[M,T], related to the operator Tn[M] coupled to the so-called Neumann boundary conditions: Xn N,T=u∈Wn,1(I):u(2j+1)(0)=u(2j+1)(T)=0,j=0,...,k−1. We will use a particular case of [8, Theorem 3], where it has been proved an equivalence on the sign of the Green’s functions (for operators with constant coefficients) related to periodic and Neumann boundary conditions on different intervals. THEOREM 3. The following properties hold: 1. GP[M,2T]⩽0on [0,2T]×[0,2T]if and only if GN[M,T]⩽0on I ×I. 2. GP[M,2T]⩾0on [0,2T]×[0,2T]if and only if GN[M,T]⩾0on I ×I. So, as a direct application of previous result and Theorem 2, we characterize the intervals of constant sign of the Green’s function related to the Neumann boundary conditions as follows. THEOREM 4. Let n =2k with k ∈N.Then 1. The Green’s function related to operator Tn[M]on the space Xn N,T, of functions satisfying T –Neumann boundary conditions, is nonnegative on I ×I (and strictly positive on I ×I if M is on the interior of the intervals) if and only if the following conditions are fulfilled: