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Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 298, pp. 1–49.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

STURM-LIOUVILLE BVPS WITH CARATHEODORY NONLINEARITIES

ABDELHAMID BENMEZA¨I, WASSILA ESSERHANE, JOHNNY HENDERSON

Abstract. In this article we study the existence and multiplicity of solu- tions for several classes of Sturm-Liouville boundary value problems having Caratheodory nonlinearities. Many results existing in the literature for such boundary value problems in the continuous framework will find in this work their extensions to the Caratheodory setting.

1. Introduction

Sturm-Liouville boundary value problems (BVP for short) have been the sub- ject of hundreds of articles during the previous five decades, where existence and multiplicity of solutions have been investigated. Often, these works are considered in the continuous framework. For this reason, we are concerned here with existence and multiplicity of solutions for Sturm-Liouville BVPs posed in the Caratheodory framework given by,

£u=f(t, u, µ) in (ξ, η) a.e., au(ξ) +bpu0(ξ) = 0, cu(η) +dpu0(η) = 0,

(1.1) where −∞ ≤ξ < η≤+∞, £u=−(pu0)0+quforu∈dom(£), 1/p, q ∈L1(ξ, η), p >0 in (ξ, η) a.e., (a2+b2)(c2+d2)6= 0 andf : (ξ, η)×R×R→Ris a Caratheodory function, that is,

(i) f(t,·,·) is continuous for a.e. t∈(ξ, η), (ii) f(·, u, µ) is measurable for allu, µ∈R.

In what follows, we letm: (ξ, η)→[0,+∞) be inL1(ξ, η) such thatmis positive on a subset of positive measure,α, β∈L1(ξ, η) andg: (ξ, η)×R→Ris a Caratheodory function. Our first contribution in this work concerns the linear version of (1.1), namely the case wheref(t, u, µ) =µm(t)uand (1.1) takes the form

£u=µmu in (ξ, η) a.e., au(ξ) +bpu0(ξ) = 0, cu(η) +dpu0(η) = 0.

(1.2)

2010Mathematics Subject Classification. 34B15, 34B16, 34B18.

Key words and phrases. Sturm-Liouville BVPs; Half-eigenvalue; Caratheodory nonlinearities;

Jumping nonlinearities.

c

2016 Texas State University.

Submitted August 14, 2016. Published November 22, 2016.

1

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So far we know, the best result existing in the literature (see [41, Theorem 4.9.1]) states that (1.1) admits an increasing sequence of simple eigenvalues (µk)k≥1such that limk→∞µk = +∞and ifφk is the eigenfunction associated withµk and zk is its number of zeros, then zk+1 = zk + 1. Moreover, if m > 0 in (ξ, η) a.e., then z1 = 0. We obtain in this work (see Corollary 3.14) that although m(t)>0 a.e.

t∈(ξ0, η0) (ξ, η), we have alwaysz1= 0.

In fact Corollary 3.14 is a consequence of Theorem 3.10 which is the second contribution in this work. This result concerns the case wheref(t, u, µ) =µm(t)u+

α(t)u+−β(t)u, and the BVP (1.1) takes the form

£u=µmu+αu+−βu in (ξ, η) a.e., au(ξ) +bpu0(ξ) = 0,

cu(η) +dpu0(η) = 0.

(1.3) Note that such a nonlinearity f is positively 1-homogeneous and it is linear on [0,+∞) and on (−∞,0]. For this reason, the BVP (1.3) is said to be half-linear and if (µ, u) is a nontrivial solution, we say that µ is a half-eigenvalue of BVP (1.3). Clearly, if α= β = 0 then BVP (1.3) coincides with the linear eigenvalue BVP (1.2) and this exhibits that the concept of half-eigenvalue generalizes that of eigenvalue. Such types of BVPs have been considered for the first time in [6], where the author introduced the concept of half-eigenvalue. He proved in the case where −∞< ξ < η < +∞, p∈ C1[ξ, η], q, m, α, β ∈ C[ξ, η] and m > 0 in [ξ, η], that BVP (1.3) admits two increasing sequences of simple half-eigenvalues (µ+k)k≥1 and (µk)k≥1. Theorem 3.10 states that the Berestycki’s result holds for our more general case. In [9], Binding and Rynne studied existence of half-eigenvalues and their properties for the periodic version of BVP ((1.3). The importance of the concept of half-eigenvalue in the theory of Sturm-Liouville BVPs appears clearly in all existence and multiplicity results (see [9, Theorems 5.1, 5.3, 5.4]).

Our third contribution consists in Theorem 4.3 of Section 4, where is examined the perturbed version of the BVP (1.3),

£u=µmu+ug(t, u) in (ξ, η) a.e., au(ξ) +bpu0(ξ) = 0,

cu(η) +dpu0(η) = 0,

(1.4)

where g(t,0) = 0, limu→+∞g(t, u) = α(t), limu→−∞g(t, u) =β(t) a.e. t ∈(ξ, η).

Theorem 4.3 concerns the bifurcation diagram of the BVP (1.4). It describes the asymptotic behavior of the two components ζk+ andζk bifurcating from the kth- eigenvalue µk of the BVP (1.2). More precisely, it states that each one of the componentsζk+ andζk rejoins respectively the points (µ+k,∞) and (µk,∞) where (µ+k)k≥1and (µk)k≥1are the two sequences of half-eigenvalues of BVP (1.3). Note that if eitherµκk <1< µk, orµk <1< µκk withκ= + or−, then the BVP

£u=ueg(t, u) in (ξ, η) a.e., au(ξ) +bpu0(ξ) = 0, cu(η) +dpu0(η) = 0,

(1.5)

where eg(t, u) = m(t) +g(t, u), admits a nontrivial solution. Thus, in Section 5, we present situations where this is the case and our contribution consists in The- orem 5.1 and its corollary (Corollary 5.2). In fact, Theorem 5.1 is composed of

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four assertions and each assertion presents a situation where (1.5) admits nodal solutions. The first two assertions generalize and improve many results existing in the literature and so far we know, the last two ones presents new existence results.

In the last section, we consider the case where f(t, u, µ) = g(t, u)−µφ+h, φ, h∈L1(ξ, η), and the BVP (1.1) takes the form

£u=g(t, u)−µφ+h in (ξ, η) a.e., au(ξ) +bpu0(ξ) = 0,

cu(η) +dpu0(η) = 0,

(1.6)

Such a class of nonlinearities is known in the literature by jumping nonlinearities, and the particular case of BVP (1.6) having such a nonlinearity

−u00=ψ(u)−µsin(t)−h in (0, π),

u(0) =u(π) = 0, (1.7)

where h ∈ C[0, π] and Rη

ξ h(t) sin(t)dt = 0, has been widely investigated in the literature. Denote by (λk)k≥1 the sequence of eigenvalues of the BVP

−u00=λu in (0, π), u(0) =u(π) = 0,

and note that sin(t) is the eigenfunction associated with the first eigenvalue λ1. Suppose thatψ∈C1(R) and seta±= limu→±∞ψ0(u), the first existence result for BVP (1.7) was obtained by Hammerstein in [20], where he proved that ifa, a+<

λ1 then BVP (1.7) admits at least one solution. Moreover, if ψ0(u) < λ1 for all u∈R, then the solution is unique. Dolph extended Hammerstein’s result in [18], to the case where λk < a, a+ < λk+1 for some integerk≥1 and he proved that the solution is unique wheneverλk< ψ0(u)< λk+1. The nonlinearityψ under the hypothesis a, a+ < λ1 or µk < a, a+ < λk+1 is said to be without jump since there is no eigenvalue in the intervalI= (min(a, a+),max(a, a+)).

The case where I contains exactly one eigenvalue, has been considered for the first time in [2], under the assumptions thatψ∈C2(R) is convex and 0< a< λ1<

a+< λ2, in which case the authors proved by means of a generalized version of the global inversion theorem to operators having singularities, existence of a manifold Γ inC[0, π] such thatC[0, π]rΓ consists of two components Γ0and Γ2, and (1.7) has no solution if ˜h=µsin(t) +h∈Γ0, exactly two solutions if ˜h∈Γ2, and a unique solution if ˜h∈Γ. In [32], the authors relaxed the condition 0< a< λ1< a+ < µ2 to that −∞< a < λ1 < a+ < µ2, and in [8] the authors proved existence of ¯µ such that Γ = {˜h =µsin(t) +h : µ = ¯µ}, Γ0 ={˜h=µsin(t) +h: µ < µ}¯ and Γ2 ={h˜ =µsin(t) +h:µ >µ}. Many other extensions of the Ambrosetti-Prodi¯ result are obtained in [1, 3, 11, 17, 22, 38]. The case where I contains more than one eigenvalue is considered in [10, 12, 15, 21, 24, 25, 26, 27, 36, 35, 37, 39]. The best result obtained for the minorant of the number of solutions to BVP (1.7) in the above cited references is: if λj−1 < a < λj < · · · < λi < a+ < λi+1 for some integersi, j ≥1 withi≥2(j−1), then the BVP (1.7) admits 2(i−(j−1)) nontrivial solutions forµlarge.

In this section, we assume that g and ∂g∂u are Caratheodory functions and the nonlinearity g has the linear behavior at ±∞, limu→+∞g(t, u)/u = α(t), and

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limu→−∞g(t, u)/u=β(t) a.e. t ∈(ξ, η). Our first contribution consists in Theo- rem 6.1 and its corollary (Corollary 6.3). This theorem provide an existence and uniqueness result of a solution to (1.6) for allµ∈Randφ, h∈L1(ξ, η), and Corol- lary 6.3 consider the case where the nonlinearitygis a separated variables function and shows that Theorem 6.1 is an extension of Hammerstein’s and Dolph’s results to the case of Sturm-Liouville BVPs posed in the Caratheodory frame-work. Theo- rem 6.1 is proved by means of degree theory and eigenvalue properties. The second contribution in this section consists in Theorem 6.7 and its corollary (Corollary 6.11). Theorem 6.7 provides a multiplicity result for BVP (1.6) and Corollary 6.11 consider the case where the nonlinearity g is a separated variables function and shows that Theorem 6.7 recuperates the minorant of the number of solutions to (1.7) obtained in [10, 12, 15, 21, 24, 25, 26, 27, 36, 35, 37, 39] for our general case of Sturm-Liouville BVPs posed in the Caratheodory frame-work.

In the last part of the last section, we present a result (Theorem 6.14) which states that the Ambrosetti-Prodi situation holds for the particular case of BVP (1.7) where the nonlinearitygis a separated variables function; Namely we consider the BVP

£u=m(t)g1(u)−µφ+h in (ξ, η) a.e., au(ξ) +bpu0(ξ) = 0,

cu(η) +dpu0(η) = 0,

(1.8)

where g1∈C2(R,R) and limu→±∞g10(u) =g±. We prove by means of a shooting method that ifg100 >0 andg < µ1 < g+ < µ2 where µ1 and µ2 are respectively and the second eigenvalues of (1.2), then there existsµ such that (1.8) admits

(a) no solution ifµ < µ,

(b) a unique solution ifθ=µ, and (c) exactly two solutions ifθ > µ.

The main tool used in this article to obtain multiplicity results, is the global bifurca- tion theory established by Rabinowitz in [34] on which Dancer gives more precision in [16]. This theory remains a very powerful tool to prove existence and multiplicity results for BVP (1.1), see for example [4, 5, 13, 14, 19, 28, 29, 30, 31, 40].

All the above contributions are presented in Sections 3-6 and Section 2 is devoted to some preliminary results. All these results are not original and we can find in the literature similar utterances, for example the case whereτ∈Rof Theorem 2.2 can be easily found in the literature, although its extension to the caseτ =±∞is easy to prove, we haven’t find in the literature a result providing this situation. Also, we met the spirit of Lemmas 2.8 and 2.9 in [6] but these two results are not clearly stated in the above cited wok. For this reason and for sake of completeness, some results in Section Preleminaries are stated and proved in the manner which agree with the spirit of this work. We end this introduction with the following useful lemma:

Lemma 1.1([23, Corollary 4.7]). Letp∈[1,∞),f ∈Lp(Ω)and(fn)be a sequence inLp(Ω) whereΩis a measurable set inRN. Iffn→f a.e. in Ωandlimkfnkp= kfkp, thenlimkf−fnkp= 0.

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2. Preliminaries 2.1. Notation.

1={(ξ, η) :−∞ ≤ξ < η≤+∞}=R×R,

2={(ξ, η, p) :ρ1= (ξ, η)∈∆1 and 1/p∈Kρ+},

3={(ξ, η, p, q) :ρ1= (ξ, η)∈∆1, ; (ξ, η, p)∈∆2 andq∈L1ρ

1},

4={(ξ, η, p,0, a, b, c, d) : (ξ, η, p)∈∆2and (a2+b2)(c2+d2)6= 0},

∆ ={(ξ, η, p, q, a, b, c, d) : (ξ, η, p, q)∈∆3 and (ξ, η, p,0, a, b, c, d)∈∆4}. Forρ1= (ξ, η)∈∆1, we define

L1ρ1=

m: (ξ, η)→Rmeasurable Z η

ξ

|m(s)|ds <∞ , Kρ1 ={m∈L1ρ

1 : m≥0 a.e. in (ξ, η)}, Kρ

1={m∈Kρ1 : mis positive in a subset of positive measure}, Kρ+

1 ={m∈Kρ1:m >0 a.e. in (ξ, η)}, Cρ1 =n

u: (ξ, η)→R:uis continuous and lim

t→ξu(t), lim

t→ηu(t) exist and are finiteo , ACρ1 ={u∈Cρ1:u0 ∈L1ρ1}.

Forρ2= (ξ, η, p)∈∆2, we define the linear spaces

Wρ2 ={u∈ACρ1 :u[p]∈Cρ1}, W˜ρ2 ={u∈Wρ2 :u[p]∈ACρ1},

where ρ1 = (ξ, η) and u[p] = pu0 is the quasi-derivative of u. These two spaces, respectively, with the norms

kuk1= sup

t∈(ξ,η)

|u(t)|+ sup

t∈(ξ,η)

|u[p](t)|, kuk2=kuk1+ Z η

ξ

|u[p](t)|dt become Banach spaces.

For the sake of simplicity, we write for u∈Wρ2, u(+∞), u[p](+∞) instead of limt→+∞u(t), limt→+∞u[p](t) when η = +∞ and u(−∞), u[p](−∞) instead of limt→−∞u(t), limt→−∞u[p](t) when ξ = −∞. Let u ∈ Wρ2 and t0 be such that ξ≤t0≤η. Ifu(t0) = 0 and u[p](t0)6= 0, thent0is said to be a simple zero ofu.

Throughout this paper, forρ3= (ξ, η, p, q)∈∆3ρ3 is the differential operator defined foru∈Wfρ2 whereρ2= (ξ, η, p) by

£ρ3u(x) =−(u[p])0(x) +q(x)u(x).

Forρ4= (ξ, η, p,0, a, b, c, d)∈∆4, Bρl

4, Bρr

4 are the operators given, for u∈W˜ρ2 whereρ2= (ξ, η, p), by

Bρl4u=au(ξ) +bu[p](ξ), Bρr4u=cu(η) +du[p](η), andEρ4 is the subspace ofWρ2 defined by

Eρ4 ={u∈Wρ2:Bρl

4u=Bρr

4u= 0}.

For integers k ≥ 1, Sk,+ρ4 denotes the set of functions u ∈ Eρ4 having exactly (k−1) zeros in (ξ, η), all are simple and u is positive in a right neighbourhood

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of ξ. It is well known that Sρk,+4 , Sρk,−4 =−Sk,+ρ4 and Sρk4 =Sk,+ρ4 ∪Sρk,−4 are open sets in Eρ4 and if u∈ ∂Sρk,κ4 , (κ = +,−), then there exists τ ∈ (ξ, η) such that u(τ) = u[p](τ) = 0. For u ∈ Sρk

4, (zj)j=kj=0 with ξ = z0 < z1 < · · · < zk = η and u(zj) = 0 forj= 1, . . . , k−1, is said to be the sequence of zeros ofu.

For ρ1 ∈ ∆1 and κ = + or −, let Iκ : Cρ1 → Cρ1 be defined by Iκu(x) = max(κu(x),0).

For allu∈E, we have

u=I+u−Iu, |u|=I+u+Iu.

This implies that, for allu, v∈E,

|I+u−I+v| ≤ |u−v|

2 +||u| − |v||

2 ≤ |u−v|,

|Iu−Iv| ≤ |u−v|

2 +||u| − |v||

2 ≤ |u−v|,

(2.1)

and the operatorsI+, I are continuous.

Remark 2.1. Throughout this paper, when there is no confusion, we write for ρ= (ξ, η, p, q, a, b, c, d)∈∆,L1ρ, Kρ,Kρ, Kρ+,Cρ, ACρ,Wρ, ˜Wρ, Eρ, Sρk,+, Sρk,−, Sρk, £ρ, Bρl, Brρ instead of L1ρ1, Kρ1, Kρ1, Kρ+1, Cρ1, ACρ1, Wρ2, ˜Wρ2, £ρ3, Bρl4, Bρr4, Eρ4, Sρk,+4 , Sρk,−4 , Sρk4, where fori∈ {1,2,3,4}, ρi is the projection ofρonto

i.

2.2. Initial value problem. In this subsection we let ρ3= (ξ, η, p, q)∈∆3, ρ1= (ξ, η), ρ2 = (ξ, η, p), γ, δ ∈ Rand τ is such that ξ ≤τ ≤η. Consider the initial value problem (IVP for short);

£ρ3u=f(t, u), u(τ) =γ, u[p](τ) =δ,

(2.2) wheref : (ξ, η)×R→Ris a Caratheodory function; that is,

(1) f(·, u) is measurable for allu∈R, (2) f(t,·) is continuous for a.e.t∈(ξ, η).

Suppose that

f(·,0)∈L1ρ

1. (2.3)

By a solution to (2.2), we mean a functionφ∈W˜ρ2 such that £ρ3φ=f(t, φ) and φ(τ) =γ,φ[p](τ) =δ.

Theorem 2.2. Assume that Hypothesis (2.3)holds and there exists ψ∈L1ρ

1 such that for all x, y∈Rand a.e. t∈(ξ, η),

|f(t, x)−f(t, y)| ≤ψ(t)|x−y|.

Then (2.2)admits a unique solution.

Proof. Clearly, u is a solution to (2.2) if and only if (u, u[p]) is a solution to the first-order IVP

U0=F(t, U)

U(τ) = (γ, δ) (2.4)

where forU = (u, v) andt∈(ξ, η),F(t, U) = p(t)v , q(t)u−f(t, u) .

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Letκ >1 andX=Cρ1×Cρ1 be equipped with the norm, k(u, v)kκ= sup

t∈(ξ,η)

exp(−κ

Z t τ

ω(r)dr

)(|u(t)|+|v(t)|)

whereω =|q|+ψ+1p. Note that the normk · kκ is equivalent to the normk · k

defined for (u, v)∈X byk(u, v)k= supt∈(ξ,η)|u(t)|+ supt∈(ξ,η)|v(t)|.

At this stage, we have that U = (u, v)∈X is a solution to (2.4) if and only if U(t) =T U(t) whereT U(t) = (γ, δ) +Rt

τF(s, U(s))ds. Since

|F(s, U(s))| ≤ |F(s, U(s))−F(s,0)|+|F(s,0)|

≤ 1

p(s)|v(s)|+ (|q(s)|+ψ(s))|u(s)|+|f(s,0)|

the operatorT :X →X is well defined. Therefore, it suffices to prove thatT is a contraction.

To this aim letU1= (u1, v1), U2= (u2, v2)∈X, we have

|F(s, U1(s))−F(s, U2(s))|

≤ |v1(s)−v2(s)|

p(s) + (|q(s)|+ψ(s))|u1(s)−u2(s)|

≤ω(s)|U1(s)−U2(s)|

(2.5)

and

S(t) = exp(−κ|

Z t τ

ω(r)dr|)|T U1(t)−T U2(t)|

=| Z t

τ

e−κ|Rstω(r)dr|(F(s, U1(s))−F(s, U2(s)))e−κ|Rτsω(r)dr|ds|.

(2.6)

Hence, we obtain from (2.5) and (2.6) that ift > τ, then S(t)≤

Z t τ

e−κRstω(r)dr|F(s, U1(s))−F(s, U2(s))|e−κRτsω(r)drds

≤ Z t

τ

e−κRstω(r)drω(s)|U1(s)−U2(s)|e−κRτsω(r)drds

≤( Z t

τ

e−κRstω(r)drω(s)ds)kU1−U2kκ

≤ 1

κkU1−U2kκ

and ift < τ, then S(t)≤

Z τ t

e−κRtsω(r)dr|F(s, U1(s))−F(s, U2(s))|e−κRsτω(r)drds

≤ Z τ

t

e−κRtsω(r)drω(s)|U1(s)−U2(s)|e−κRsτω(r)drds

≤Z τ t

ω(s)e−κRtsω(r)drds

kU1−U2kκ

≤ 1

κkU1−U2kκ.

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The above estimates onS(t) lead tokT U1−T U2kκ1κkU1−U2kκand (2.2) admits

a unique solution, thus completing the proof.

The following corollary is obtained from Theorem 2.2 and is an extension of [41, Theorem 2.2.1] to the case whereτ can be infinite.

Corollary 2.3. For allρ3= (ξ, η, p, q)∈∆3,γ, δ∈Randξ≤τ ≤η andf ∈L1ρ1 withρ1= (ξ, η), the IVP

£ρ3u=f, u(τ) =γ, u[p](τ) =δ, admits a unique solution.

Now consider the IVP

£ρ3u=ug(t, u), u(τ) = 0, u[p](τ) = 0,

(2.7)

whereg: (ξ, η)×R→Ris a Caratheodory function.

Corollary 2.4. Assume that

|g(t, u)| ≤ψ(t) for allu∈Rand a.e. t∈(ξ, η) for someψ∈L1ρ

1. Then the trivial function is the unique solution for (2.7).

Proof. Indeed, if (λ, u) is a solution to (2.7) thenuis a solution of the IVP

−(pv0)0+ (q+qu)v= 0, v(τ) = 0, v[p](τ) = 0,

where qu(t) = −g(t, u(t)). Since the hypothesis in Corollary 2.4 guarantees that qu∈L1ρ1, we have from Corollary 2.3 thatuis the unique solution of (2.7).

2.3. Comparison results.

Definition 2.5. Letρ2= (ξ, η, p)∈∆2 andu, v∈Wρ2. The functionW r(u, v) = uv[p]−u[p]v is called the Wronksian ofu, v.

It is easy to prove the following lemma.

Lemma 2.6. Let ρ= (ξ, η, p, q, a, b, c, d)∈∆ andu, v∈Wρ. We have [i) If Bρlu=Bρlv= 0, thenW r(u, v)(ξ) = 0;

(ii) If Bρru=Bρrv= 0, thenW r(u, v)(η) = 0;

(iii) If W r(u, v)(t0)6= 0 for some t0 ∈(ξ, η) and £ρu=£ρv = 0, then{u, v}

form a basis of the space of solutions to the differential equation £ρw= 0.

The proof of the following lemma is similar to that of [6, Lemma 2], so it is omitted.

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Lemma 2.7. Let j andk be two integers such thatj ≥k≥2. Suppose that there exist two families of real numbers

ξ0=ξ < ξ1< ξ2<· · ·< ξk−1< ξk =η, η0=ξ < η1< η2<· · ·< ηj−1< ηj =η.

Then, ifξ1< η1, there exist two integersmandnhaving the same parity,1≤m≤ k−1and1≤n≤j−1 such that

ξm< ηn≤ηn+1≤ξm+1.

Lemma 2.8. Letρ= (ξ, η, p, q, a, b, c, d)∈∆ and let fori= 1,2,φi∈Sρkihaving a sequence of zeros(zji)j=kj=0i. If for some integersm, nwithm≤k1−1andn≤k2−1 we have φ1φ2>0andzm1 ≤z2n< z2n+1≤z1m+1, thenRzn+12

zn2 φ1£ρφ2−φ2£ρφ1≥0.

Moreover,Rz2n+1

zn2 φ1£ρφ2−φ2£ρφ1= 0 if and only ifzm1 =zn2 < zn+12 =zm+11 . Proof. Without loss of generality, suppose thatφ1, φ2>0 in (z2n, zn+12 ) and letW r be the Wronksian of φ1 and φ2. Set I = Rz2n+1

z2n φ1£ρφ2−φ2£ρφ1 and note that I=W r(z2n)−W r(zn+12 ).

We distinguish four cases:

(i) ξ=zn2< zn+12 =η: In this case we haveI=W r(ξ)−W r(η) = 0.

(ii) ξ = zn2 < z2n+1 < η: In this case we have W r(ξ) = 0, φ1(zn+12 ) ≥ 0, φ2(zn+12 ) = 0,φ[p]2 (zn+12 )<0, leading to

I=−W r(z2n+1) =−φ1(z2n+1[p]2 (zn+12 )≥0.

Clearly, ifI= 0 then φ1(zn+12 ) = 0 and z1m+1=zn+12 .

(iii) ξ < zn2< zn+12 =η: In this case we haveW r(η) = 0, φ1(zn2)≥0,φ2(zn2) = 0, φ[p]2 (zn2) > 0, leading toI =W r(zn2) = φ1(zn2[p]2 (zn2)≥ 0. Clearly, if I= 0 thenφ1(z2n) = 0, proving thatzm1 =z2n.

(iv) ξ < zn2 < z2n+1 < η: In this case we have φ1(zn2) ≥ 0, φ1(z2n+1) ≥ 0, φ2(zn2) = 0, φ2(zn+12 ) = 0, φ[p]2 (zn2) > 0, φ[p]2 (zn+12 ) < 0 (see Figure 1), leading toI =φ1(zn2[p]2 (zn2)−φ1(zn+12[p]2 (zn+12 )≥0. Clearly, if I = 0 thenφ1(zn2) =φ1(z2n+1) = 0, proving thatz1m=z2n andzm+11 =zn+12 .

@

@ R

@

@

@ R

zm1 z1m+1 -

zm1 zm+11

zn2 z2n+1

zn2 z2n+1

φ1

φ2

φ1 φ2

Figure 1. Bumps

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Lemma 2.9. Let ρ = (ξ, η, p, q, a, b, c, d) ∈ ∆ and let φ1, φ2 be respectively two functions in Sρk,κ∩W˜ρ. Then, there exist two intervals (ξ1, η1) and (ξ2, η2) such that φ1φ2>0 in(ξ1, η1)and in(ξ2, η2). Moreover,

Z η1 ξ1

φ1£ρφ2−φ2£ρφ1≥0, Z η2

ξ2

φ1£ρφ2−φ2£ρφ1≤0.

Proof. Without loss of generality, suppose thatκ= + and let fori= 1,2, (zij)j=kj=0 the sequence of zeros ofφi. Since the casek= 1 is obvious, we suppose thatk≥2.

We distinguish two cases

(i) z11=z12: In this case letθ= inf(z12, z22). From Lemma 2.8, we have Z z11

ξ

φ1£ρφ2−φ2£ρφ1= 0, Z θ

z11

φ1£ρφ2−φ2£ρφ1{

(≥0 ifθ=z22

≤0 ifθ=z21. Thus, ifθ=z21, we take (ξ1, η1) = (ξ, z11), (ξ2, η2) = (z11, z21) and if θ=z22, we take (ξ1, η1) = (ξ, z11), (ξ2, η2) = (z21, z22).

(ii) z12 < z11, (the case z11 < z21 is checked similarly): In this case Lemma 2.7 guarantees existence of two integers m, n≥ 1 having the same parity such that zm2 < zn1 < z1n+1 ≤z2m+1. Thus, we take (ξ1, η1) = (ξ, z21) and (ξ2, η2) = (zn1, zn+11 ) and we have from Lemma 2.8,

Z η1

ξ1

φ1£ρφ2−φ2£ρφ1≥0, Z η2

ξ2

φ1£ρφ2−φ2£ρφ1≤0.

This completes the proof.

Lemma 2.10([6]). Let ρ∈∆and letw1, w2 be two functions inW˜ρand assume that w2 does not vanish identically and £ρw1 =m1w1 and £ρw2 = m2w2 where m1, m2∈L1ρ are such that(m1−m2)∈Kρ. Suppose that either

(1) w2(ξ) =w2(η) = 0, or

(2) fori= 1,2 Bρlwi= 0 andw2(η) = 0, or (3) fori= 1,2 Bρrwi= 0 andw2(ξ) = 0 ,or (4) fori= 1,2 Bρlwi= 0 andBρrwi= 0.

Then there existsτ∈(ξ, η) such thatw1(τ) = 0.

2.4. Green’s function. Forρ= (ξ, η, p, q, a, b, c, d)∈∆ let Φρand Ψρ be respec- tively the solutions obtained from Theorem 2.3 to the equations

£ρu= 0 u(ξ) =b, u[p](ξ) =−a,

£ρu= 0 u(η) =d, u[p](η) =−c,

and W rρ = W r(Φρρ). Note that because W0rρ = 0, we have W rρ(t) = W r(Φρρ)(ξ) for allt∈(ξ, η).

Theorem 2.11. Let ρ= (ξ, η, p, q, a, b, c, d)∈∆ and assume that the trivial func- tion0 is the unique solution to the BVP

£ρu= 0 a.e. in(ξ, η),

Bρlu=Brρu= 0. (2.8)

Then, there exists a unique function Gρ: (ξ, η)×(ξ, η)→R such that

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(1) Gρ is uniformly continuous, bounded and symmetric.

(2) For s0 ∈(ξ, η) fixed, the function H0(t) = Gρ(t, s0) satisfies the differen- tial equation (2.8)in each of intervals (ξ, s0) and(s0, η)and the boundary conditions in (2.8).

(3) Fors0∈(ξ, η) fixed,G[p]ρ (s+0, s0),G[p]ρ (s0, s0)exist and we have G[p]ρ (s+0, s0)−G[p]ρ (s0, s0) = 1.

(4) Moreover, for all f ∈L1ρ, u∈W˜ρ is a solution to

£ρu=f a.e. in(ξ, η), Blρu=Bρru= 0, if and only ifu(t) =Rη

ξ Gρ(t, s)f(s)ds=Lρf(t).

(5) The operatorLρ:L1ρ→Cρ is compact.

Proof. The function

Gρ(t, s) = 1 W rρ

ρ(s)Ψρ(t) ifs≤t Φρ(t)Ψρ(s) ift≤s

is what we are seeking, whereW rρ=W r(Φρρ) =W r(Φρρ)(ξ).

Sinceq,1/p∈L1ρ, from [41, Theorem 2.3.1] we have that the functions, Φρ, Ψρ, Φ[p]ρ , Ψ[p]ρ are bounded by a constantM >0. Therefore, fort1, t2∈(ξ, η) we have

ρ(t2)−Φρ(t1)| ≤M

Z t2

t1

ds p(s)

, |Ψρ(t2)−Ψρ(t1)| ≤M| Z t2

t1

ds p(s)|, proving that Φρ, Ψρ are uniformly continuous. Then Gρ is uniformly continuous on (ξ, η)×(ξ, η). Clearly, the functionGρ satisfies Properties 1, 2, 3, and Property 4 is proved by the method of variation of constants.

At the end, note that Lρ =iρ◦Leρ, where Leρ : L1ρ →Wρ with Leρu=Lρufor allu∈L1ρ, is continuous andiρ is the continuous embedding ofWρinCρ. Because the estimate

|u(t2)−u(t1)| ≤

Z t2 t1

ds p(s)

kuk1

holds for allu∈Wρ and t1, t2 withξ≤t1< t2≤η, the embeddingiρ is compact,

and thenLρ is compact.

Lemma 2.12. Assume thatW rρ 6= 0, for some ρ= (ξ, η, p, q, a, b, c, d)∈∆, and let forθ∈(ξ, η),ρl(θ) = (ξ, θ, p, q, a, b,1,0)andρr(θ) = (θ, η, p, q, a, b,1,0).

(i) If Φρ(θ)6= 0for all θ∈(ξ, η), then for all θ∈(ξ, η),Gρl(θ) exists and we haveGρl(θ)(t, s) =Gρ(t, s)−(Ψρ(θ)/W rρΦρ(θ))Φρ(t)Φρ(s).

(ii) If Ψρ(θ)6= 0for all θ∈(ξ, η), then for all θ∈(ξ, η),Gρr(θ) exists and we haveGρr(θ)(t, s) =Gρ(t, s)−(Φρ(θ)/W rρΨρ(θ))Ψρ(t)Ψρ(s).

Proof. We need to prove that Φρ(θ)6= 0 for allθ∈(ξ, η).

(i) Let Φρl(θ)(t) = Φρ(t) and Ψρl(θ)(t) = −(Ψρ(θ)/Φρ(θ))Φρ(t) + Ψρ(t). Then Φθθ are respectively the unique solutions to

£ρl(θ)u= 0, u(ξ) =b, u[p](ξ) =−a,

£ρl(θ)u= 0, u(θ) = 0, u[p](θ) =W rρρ(θ),

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and for allθ∈(ξ, η), we haveW rρl(θ)=W rρ6= 0 and Gρl(θ)(t, s) = 1

W rρl(θ) ×

ρl(θ)(s)Ψρl(θ)(t) ifs≤t Φρl(θ)(t)Ψρl(θ)(s) ift≤s

=Gρ(t, s)−(Ψρ(θ)/W rρΦρ(θ))Φρ(t)Φρ(s).

(ii) Let Φρr(θ)and Ψρr(θ) be defined by Φρl(θ)(t) = Φρ(t)−(Ψρ(θ)/Φρ(θ))Φρ(t) and Ψρr(θ)(t) = Ψρ(t). Then, Φρr(θ)ρr(θ)are respectively the unique solutions of

£ρr(θ)u= 0, u(θ) = 0, u[p](θ) =W rρρ(θ),

£ρr(θ)u= 0, u(η) =d, u[p](η) =−c, and we have for allθ∈(ξ, η),W rρr(θ)=W rρ6= 0 and

Gρr(θ)(t, s) = 1 W rρr(θ)

×

ρr(θ)(s)Ψρr(θ)(t) ifs≤t Φρr(θ)(t)Ψρr(θ)(s) ift≤s

=Gρ(t, s)−(Φρ(θ)/W rρΨρ(θ))Ψρ(t)Ψρ(s).

2.5. Linear eigenvalue problem. Forρ= (ξ, η, p, q, a, b, c, d)∈∆ andm∈Kρ, consider the eigenvalue problem

£ρu=µmu in (ξ, η) a.e.,

Bρlu=Bρr= 0. (2.9)

Theorem 2.13 ([41, Theorem 4.9.1]). For ρ= (ξ, η, p, q, a, b, c, d)∈ ∆ and m ∈ Kρ+, BVP (2.9)admits an increasing sequences of eigenvalues (µk(ρ, m))k≥1 such that

(1) limµk(ρ, m) = +∞, (2) µk(ρ, m)is simple,

(3) If φk is an eigenvalue associated withµk(ρ, m), thenφk∈Skρ.

In what follows, we present some important properties of eigenvalues needed for the proofs of the main results of this paper.

Lemma 2.14. Let ρ = (ξ, η, p, q, a, b, c, d) ∈ ∆, m1, m2 ∈ Kρ+ and assume that m1 ≤m2 a.e. in (ξ, η) andm1 < m2 in a subset of positive measure. If for some integerk≥1, eitherµk(ρ, m1)≥0orµk(ρ, m2)≥0,thenµk(ρ, m1)> µk(ρ, m2)≥ 0.

Proof. Fori = 1,2, set µik(ρ, mi) and letφi be the eigenfunction associated with µi having a sequence of zeros (zij)j=kj=0. First, we claim that there exists j0 such that zj1

0 6= z2j

0. Indeed, assume that φ1(zj2) = 0 for all j ∈ {1, . . . , k−1}

andµ1 < µ2 and note that there existsj1 ∈ {1, . . . , k−1} such that meas({m2>

m1} ∩(zj21, zj21+1))>0 andφ1φ2>0 in (zj21, zj21+1). Applying Lemma 2.10, we get that there existsτ∈(zj21, zj21+1) such thatφ1(τ) = 0 and this contradictsφ1∈Sρk,κ. Now, letk1= max{l ≤k:z1j =z2j for allj≤l}and (ξj)j=k−kj=0 1 and (ηj)j=k−kj=0 1 be the families defined by ξj = zk1

1+j and ηj = zk2

1+j. Then we distinguish two cases.

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(i)ξ1=z1k

1+1< η1=z2k

1+1: In this case we have 0<

Z η1

η0

φ2£ρφ1−φ1£ρφ2= Z η1

η0

1m1−µ2m21φ2

= Z η1

η0

1−µ2)m1φ1φ2+ Z η1

η0

µ2(m1−m21φ2

= Z η1

η0

µ1(m1−m21φ2+ Z η1

η0

1−µ2)m2φ1φ2

and this proves that in both the casesµ1≥0 orµ2≥0, we haveµ1> µ2. (ii) ξ1 =zk1

1+1 > η1 =zk2

1+1: In this case Lemma 2.7 guarantees existence of two integersm, nhaving the same parity such that

ξm=zk11+m< ηn=zk21+n< ηn+1=zk21+n+1≤ξm+1=z1k1+m+1. As above, we have

0<

Z ηn+1

ηn

φ2£ρφ1−φ1£ρφ2= Z ηn+1

ηn

1m1−µ2m21φ2

= Z ηn+1

ηn

1−µ2)m1φ1φ2+ Z ηn+1

ηn

µ2(m1−m21φ2

= Z ηn+1

ηn

µ1(m1−m21φ2+ Z ηn+1

ηn

1−µ2)m2φ1φ2

and this proves that in both the cases µ1 ≥0 or µ2 ≥0, we haveµ1 > µ2. This

completes the proof.

Lemma 2.15. Let ρ = (ξ, η, p, q, a, b, c, d) ∈ ∆, m ∈ Kρ+ and γ, δ ∈ R with ξ < γ < δ < η. Then for all integers k ≥ 1, µk(ρ, m) ≤ µk(ρ, m) where ρ= (γ, δ, p, q,1,0,1,0).

Proof. Fix k ≥1 and set µ1k(ρ, m) and µ2 = µk(ρ, m). For i = 1,2, let φi

be an eigenfunction associated with µi, having a sequence of zeros (zij)j=kj=0, and without loss of generality, suppose thatφ1φ2>0 in a right neighborhood ofγ. We distinguish two cases.

(i)φ1>0 in (γ, δ): In this case we have 0≤ −φ1(δ)φ[p]2 (δ) +φ1(γ)φ[p]2 (γ) =

Z δ γ

φ1£ρφ2−φ2£ρφ1

= (µ2−µ1) Z δ

γ

1φ2

leading toµ2≥µ1.

(ii) φ1(t0) = 0 for some t0 ∈ (γ, δ): In this case consider the family (ξj)j=kj=00 defined by ξ0 =γ, ξk0 = δ and φ1j) = 0 for j ∈ {1, . . . , k0−1} and note that k0≤k. Thus, from Lemma 2.7 there exist two integersm, nhaving the same parity, such thatξm< zn2< zn+12 ≤ξm+1. Therefore, we haveφ1, φ2>0 in (z2n, zn+12 ) and

0≤ −φ1(zn+12[p]2 (zn+12 ) +φ1(zn2[p]2 (zn2)

= Z z2n+1

z2n

φ1£ρφ2−φ2£ρφ1

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= (µ2−µ1) Z zn+12

z2n

1φ2

leading toµ2≥µ1. This completes the proof.

Lemma 2.16. Let ρ = (ξ, η, p, q, a, b, c, d) ∈ ∆ and m ∈ Kρ+ and set for all θ ∈ (ξ, η), ρr(θ) = (θ, η, p, q,1,0, c, d) (resp. ρl(θ) = (ξ, θ, p, q, a, b,1,0)). Then, the mapping θ → µ1r(θ), m) is continuous increasing on (ξ, η) (resp. θ → µ1l(θ), m)is continuous decreasing on(ξ, η)), and we havelimθ→ηµ1r(θ), m) = +∞(resp. limθ→ξµ1l(θ), m) = +∞).

Proof. The continuity of the mapping θ → µ1r(θ), m) follows from [41, Theo- rem 4.4.1]. Let θ1, θ2 be such that ξ ≤ θ1 < θ2 < η and let for i = 1,2, φi be the eigenvector corresponding to the eigenvalue µi = µ1ri), m). Taking into considerationφ22) = 0 andW r(φ1, φ2)(η) = 0, from simple computations,

2−µ1) Z η

θ2

1φ2= Z η

θ2

φ1£ρr2)φ2−φ2£ρr1)φ112[p]22)>0, thus proving thatµ2> µ1.

Now, we understand from Theorem 2.13 that there existsµ >0 such thatµ(ρ) = µ1(ρ, m) +µ >0 and this, together withθ→µ1r(θ), m) is increasing, leads to

µ(θ) =µ1r(θ), m) +µ=µ1(ρer(θ), m)≥µ(ρ) =µ1(ρ, m)e >0 whereρer(θ) = (θ, η, p, q+µm,1,0, c, d) andρe= (ξ, η, p, q+µm,1,0, c, d).

To prove limθ→ηµ1r(θ), m) = +∞, we need to prove the existence of a positive constantM(d) such that supt∈(θ,η)

ρe(t)/Ψ

ρe(θ))≤M(d). Note that Ψ

ρe(t)6= 0 for allt∈(ξ, η); indeed, if Ψ

eρ(t0) = 0 for some t0∈(ξ, η), then there exists an integer k0 ≥ 1 such that Ψ

ρewill be an eigenfunction associated with µk0(ρer(t0), m) = 0 and yields the contradiction

0 =µk0(ρer(t0), m)≥µ1(ρer(t0), m) =µ(t0)>0.

Without loss of generality, suppose that Ψ

ρe>0 in (ξ, η) and note then thatd≥0.

We distinguish two cases:

(i)d >0: In this case we have inft∈(ξ,η)Ψ

ρe(t)>0 and sup

t∈(θ,η)

ρe(t)/Ψ

ρe(t))≤ kΨ

eρk/ inf

t∈(ξ,η)Ψ

ρe(t) =M(d).

(ii)d= 0: In this case we havec >0 and there existsδ >0 such that Ψ[p]

ρe (t)<0 for all t ∈ (δ, η). We have then supt∈(θ,η)

eρ(t)/Ψ

eρ(t)) = 1 if θ ∈ (δ, η) and supt∈(θ,η)

eρ(t)/Ψ

eρ(t))≤ kΨ

eρk/inft∈(ξ,δ)Ψ

ρe(t). Thus, sup

t∈(θ,η)

ρe(t)/Ψ

ρe(t))≤M(d) = sup(1,kΨ

ρek/ inf

t∈(ξ,δ)Ψ

ρe(t)).

Sinceµ(θ)>0,G

ρer(θ)exists and we have for all θ∈(ξ, η) and allt∈(θ, η)

|G

ρer(θ)(t, s)|=|G

ρe(t, s)−(Φ

eρ(θ)/W r

ρeΨ

eρ(θ))Ψ

ρe(t)Ψ

eρ(s)|

≤ kGeρk+W r−1

ρe M(d)kΦ

ρekkΨ

eρk.

Therefore,

0<1/µ(θ)≤ sup

t∈(θ,η)

Z η θ

|G

eρ(θ)(t, s)|m(s)ds

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