The Distance between Points of a Solution of a Second Order Linear Differential Equation Satisfying General Boundary Conditions

Abstract

This paper presents a method to obtain lower and upper bounds for the minimum distance between points 𝑎 and 𝑏 of the solution of the second order linear differential equation 𝑦^′′+𝑞(𝑥)𝑦=0 satisfying general separated boundary conditions of the type 𝑎₁₁𝑦(𝑎)+𝑎₁₂𝑦^′(𝑎)=0 and 𝑎₂₁𝑦(𝑏)+𝑎₂₂𝑦^′(𝑏)=0. The method is based on the recursive application of a linear operator to certain functions, a recursive application that makes these bounds converge to the exact distance between 𝑎 and 𝑏 as the recursivity index grows. The method covers conjugacy and disfocality as particular cases.

1. Introduction

In a recent paper of the authors (see [1]) it was shown that the recursive application of the operator 𝑃∶𝐶[𝑎,𝑏]→𝐶²[𝑎,𝑏] defined by 𝑃𝑓=^𝑥_𝑎^𝑏_𝑡𝑞(𝑠)𝑓(𝑠)𝑑𝑠𝑑𝑡, (1) where 𝑞(𝑥) is continuous on [𝑎,𝑏] and strictly positive almost everywhere on that same interval, provided a method to determine if the second order linear differential equation 𝑦^′′+𝑞(𝑥)𝑦=0 (2) was either left disfocal or left nondisfocal in the interval [𝑎,𝑏], the concept of left disfocal alluding to the nonexistence of a nontrivial solution 𝑦(𝑥) of (2) with zeroes in [𝑎,𝑏] such that 𝑦^′(𝑏)=0 (a similar definition exists for right disfocal). The method was based on two features of the operator 𝑃, namely, (i)the fact that 𝑃 is positive and monotonic, so that for 𝑓>𝑦>𝑔>0 on ]𝑎,𝑏[ one has 𝑃^𝑘𝑓(𝑥)>𝑃^𝑘𝑦(𝑥)>𝑃^𝑘𝑔(𝑥) on ]𝑎,𝑏];(ii)the fact that lim_𝑘→∞𝑃^𝑘𝑓(𝑏)=∞ if (2) is left nondisfocal in an interval interior to [𝑎,𝑏] and lim_𝑘→∞𝑃^𝑘𝑓(𝑥)=0 for any value of 𝑥∈[𝑎,𝑏] as long as (2) is left disfocal in [𝑎,𝑏].

The purpose of this paper is to extend the results of [1] to (2) with the more general boundary conditions: 𝑎₁₁𝑦(𝑎)+𝑎₁₂𝑦^′(𝑎)=0,𝑎₂₁𝑦(𝑏)+𝑎₂₂𝑦^′(𝑏)=0, (3) which contain conjugacy (𝑦(𝑎)=𝑦(𝑏)=0) and nondisfocality as particular cases, by means of the recursive application of the operator 𝐿∶𝐶[𝑎,𝑏]→𝐶²[𝑎,𝑏] defined by 𝐿𝑓=^𝑏_𝑎𝐺(𝑥,𝑡)𝑞(𝑡)𝑓(𝑡)𝑑𝑡, (4) where 𝐺(𝑥,𝑡) (or 𝐺_𝑎𝑏(𝑥,𝑡), as we will name it in some sections) is the Green function of the problem 𝑦^′′=−𝑔(𝑥),𝑎₁₁𝑦(𝑎)+𝑎₁₂𝑦^′(𝑎)𝑎=0,₂₁𝑦(𝑏)+𝑎₂₂𝑦^′(𝑏)=0. (5) In consequence this paper will yield criteria (in fact infinitely many) to obtain lower and upper bounds for the minimum distance between points 𝑎 and 𝑏 satisfying (3) when 𝑦(𝑥) is a solution of (2), bounds of which will be shown to converge to the exact values of such a minimum distance as the index 𝑘 grows. Besides extending the results of [1] to the problem (2)-(3), other results based on the same strategies will also be introduced and it will be shown that they improve the results of [1] in many cases.

Operators of the type (4) have often been used in the determination of Lyapunov inequalities for different types of equations and boundary conditions since Nehari [2] first noted that a solution 𝑦 of (2) such that 𝑦(𝑎)=𝑦(𝑏)=0 satisfied 𝑦(𝑥)=^𝑏_𝑎𝐺(𝑥,𝑡)𝑞(𝑡)𝑦(𝑡)𝑑𝑡, (6) with 𝐺()=()()𝑥,𝑡min𝑥−𝑎,𝑡−𝑎min𝑏−𝑥,𝑏−𝑡,𝑏−𝑎 (7) which implied 𝑦(𝑥)≤^𝑏_𝑎|𝐺(𝑞(𝑡𝑦(𝑡<𝑦𝑥,𝑡)|))𝑑𝑡_∞𝑏𝑎|𝐺(𝑡,𝑡)|𝑞(𝑡)𝑑𝑡; (8) that is, 1<^𝑏_𝑎|𝐺(𝑡,𝑡)|𝑞(𝑡)𝑑𝑡. (9) Beesack [3], Hinton [4], Levin [5, 6], and Reid [7–9] are other remarkable examples of such a use, as the excellent monography on Lyapunov inequalities of [10, Chapter 1] shows. Although formulae like (6) can be applied recursively if 𝑞(𝑥)≥0 and 𝐺(𝑥,𝑡)≥0 to obtain more complex versions of (9), the fact is that the iterative application of 𝐿 has rarely been proposed in any papers, with the exception of Harris [11], who suggested its application for the disfocal case—𝑦(𝑎)=𝑦^′(𝑏)=0 or 𝑦^′(𝑎)=𝑦(𝑏)=0—without getting to prove that it guaranteed any improvement. We will show in this paper that under certain conditions (3) the function 𝐺(𝑥,𝑡) is positive and the recursive application is possible and provides lower and upper bounds which improve all existing results as of today. We will also show that even in the case that 𝐺(𝑥,𝑡) gets to be negative it is still sometimes possible to obtain upper and lower bounds for the distance between 𝑎 and 𝑏, which are as close to the real distance between 𝑎 and 𝑏 as desired.

It is also worth remarking the significant interest that the calculation of lower bounds (i.e., Lyapunov kind-of inequalities) has enjoyed in comparison with the problem of the determination of upper bounds, regardless of the type of boundary conditions (3). This fact was already noted by Došlý in [12] for the conjugate case (𝑦(𝑎)=𝑦(𝑏)=0) and by the authors in [1, 13, 14] for the nondisfocal case. References [12, 15–20] are notable exceptions to this trend.

As indicated in the first paragraph, throughout the paper we will assume that 𝑞(𝑥) is continuous on an interval 𝐼⊂𝑅 such that [𝑎,𝑏]⊂𝐼 and that 𝑞(𝑥) is strictly positive almost everywhere on [𝑎,𝑏]. This allows defining the internal product 𝑓,𝑔=^𝑏_𝑎𝑞(𝑥)𝑓(𝑥)𝑔(𝑥)𝑑𝑥, (10) for 𝑓, 𝑔 continuous on [𝑎,𝑏] (it is easy to prove that (10) satisfies all the conditions required by an internal product) and the associated norm ‖⋅‖₂ defined by 𝑓₂=^𝑏_𝑎𝑞(𝑥)𝑓²(𝑥)𝑑𝑥^1/2. (11)

Likewise, we will use the notation 𝐿 to name the operator defined in (4), 𝐿𝑓 or 𝐿{𝑓} to name the function with domain [𝑎,𝑏] resulting from the application of 𝐿 to 𝑓(𝑥)∈𝐶[𝑎,𝑏], and 𝐿𝑓(𝑥) or 𝐿{𝑓}(𝑥) to name the value of the function 𝐿𝑓 at the point 𝑥 and 𝐿_𝛼,𝛽 when other extremes of integration 𝛼, 𝛽 potentially different from 𝑎, 𝑏 are used in (4). In that latter case, any ‖⋅‖₂ norm appearing in the same formula will be assumed to be calculated with 𝛼 and 𝛽 as integration extremes in (11).

We will say that the points 𝑎^′, 𝑏^′ are interior to an interval [𝑎,𝑏] if 𝑎^′,𝑏^′∈[𝑎,𝑏] and [𝑎^′,𝑏^′]≠[𝑎,𝑏].

The organization of the paper is as follows. Section 2 will present the main properties of the operator 𝐿. Sections 3 and 4 will apply these properties in different ways to find upper and lower bounds for the minimum distance between points 𝑎 and 𝑏 satisfying (3). Section 5 will introduce some formulae which simplify the calculations required in Section 3. Section 6 will apply the method to several examples. Finally Section 7 will draw several conclusions.

2. The Operator 𝐿𝑓=∫^𝑏_𝑎𝐺(𝑥,𝑡)𝑞(𝑡)𝑓(𝑡)𝑑𝑡

The purpose of this section will be to present the main properties of the operator 𝐿 defined in (4) for 𝑞(𝑥) as specified in the Introduction. As was done in [1], for the sake of clarity such properties will be presented in several lemmas which will lead to Theorem 5, which can be regarded as the key result of this section.

Lemma 1. The operator 𝐿𝑓 is linear. In addition, if 𝐺(𝑥,𝑡) is positive almost everywhere for 𝑡∈[𝑎,𝑏], then 𝐿 is positive and monotonic.

Proof. The linearity of 𝐿 is quite evident. If 𝐺(𝑥,𝑡) is positive almost everywhere for 𝑡∈[𝑎,𝑏], the positiveness (and in consequence monotonicity) of 𝐿 is a consequence of 𝑞(𝑥) being positive almost everywhere on [𝑎,𝑏] by hypothesis.

Lemma 2. The operator 𝐿𝑓 is compact.

Proof. From [21, Theorem 7.2.6] and the fact that 𝐺(𝑥,𝑡) is continuous on [𝑎,𝑏]×[𝑎,𝑏] it is straightforward to show that 𝐿 is compact with the ‖⋅‖_∞ norm and therefore with the ‖⋅‖₂ norm defined in (11).

Lemma 3. If any of the following conditions 𝑎₁₁=0,𝑎₂₁𝑎=0,₁₁𝑎=0,²²𝑎₂₁(𝑏−𝑎)+1≠0,−𝑎₁₂𝑎₁₁(𝑏−𝑎)+1≠0,𝑎₂₁𝑎=0,₁₂𝑎₁₁𝑎₂₂𝑎₂₁()−𝑎𝑏−𝑎+1²²𝑎₂₁≠0 (12) are met, then the operator 𝐿𝑓 is self-adjoint.

Proof. Let us define 𝑦₁(𝑥) as the solution of 𝑦^′′=0,𝑎₁₁𝑦(𝑎)+𝑎₁₂𝑦^′(𝑎)=0, (13) and 𝑦₂(𝑥) as the solution of 𝑦^′′=0,𝑎₂₁𝑦(𝑏)+𝑎₂₂𝑦^′(𝑏)=0. (14) A straightforward calculation shows that if any of the conditions (12) are met, then the Wronskian of 𝑦₁ and 𝑦₂, namely, 𝑊=𝑦₁(𝑥)𝑦^′₂(𝑥)−𝑦^′₁(𝑥)𝑦₂(𝑥), (15) is not zero, which implies that 𝑦₁ and 𝑦₂ are not linearly dependent and therefore no nontrivial solutions of (2)-(3) exist. In consequence from [22, Theorem IX.3] one has 𝐺(𝑥,𝑡)=−𝑦₁(𝑥)𝑦₂(𝑡)𝑦₁(𝑎)𝑦^′₂(𝑎)−𝑦^′₁(𝑎)𝑦₂(𝑎),𝑥<𝑡,−𝑦₁(𝑡)𝑦₂(𝑥)𝑦₁(𝑎)𝑦^′₂(𝑎)−𝑦^′₁(𝑎)𝑦₂(𝑎),𝑥≥𝑡. (16) Now, in order to prove self-adjointness, we need to prove that, given 𝑓,𝑔∈𝐶[𝑎,𝑏], ⟨𝐿𝑓,𝑔⟩=⟨𝑓,𝐿𝑔⟩. Thus, from (10) we have 𝐿𝑓,𝑔=^𝑏_𝑎𝑞(𝑥)(𝑥)𝑔(𝑥)=𝐿𝑓𝑑𝑥_𝑏𝑎𝑞(𝑥)𝑔(𝑥)^𝑏_𝑎𝐺(𝑥,𝑡)𝑞(𝑡)𝑓(𝑡)𝑑𝑡𝑑𝑥. (17) Combining (17) and (16) one gets to 𝐿𝑓,𝑔=−1𝑦₁(𝑎)𝑦^′₂(𝑎)−𝑦^′₁(𝑎)𝑦₂(𝑎)×^𝑏_𝑎𝑞(𝑥)𝑔(𝑥)𝑦₂(𝑥)^𝑥_𝑎𝑞(𝑡)𝑦₁(𝑡)𝑓(𝑡)+𝑑𝑡𝑑𝑥−1𝑦₁(𝑎)𝑦^′₂(𝑎)−𝑦^′₁(𝑎)𝑦₂(𝑎)×^𝑏_𝑎𝑞(𝑥)𝑔(𝑥)𝑦₁(𝑥)^𝑏_𝑥𝑞(𝑡)𝑦₂(𝑡)𝑓(𝑡)𝑑𝑡𝑑𝑥. (18) And integrating by parts the right-hand side of (18) one finally has 𝐿𝑓,𝑔=−1𝑦₁(𝑎)𝑦^′₂(𝑎)−𝑦^′₁(𝑎)𝑦₂(𝑎)×^𝑏_𝑎^𝑏_𝑥𝑞(𝑡)𝑔(𝑡)𝑦₂(𝑡)𝑑𝑡𝑞(𝑥)𝑦₁(𝑥)𝑓(𝑥)+𝑑𝑥−1𝑦₁(𝑎)𝑦^′₂(𝑎)−𝑦^′₁(𝑎)𝑦₂(𝑎)×^𝑏_𝑎^𝑥_𝑎𝑞(𝑡)𝑔(𝑡)𝑦₁(𝑡)𝑑𝑡𝑞(𝑥)𝑦₂(𝑥)𝑓(𝑥)𝑑𝑥=^𝑏_𝑎𝑞(𝑡)𝑔(𝑡)𝐺()(𝑥)𝑓(𝑥)=.𝑥,𝑡𝑑𝑡𝑞𝑑𝑥𝑓,𝐿𝑔 (19)

Lemma 4. The operator 𝐿𝑓 is bounded with the ‖⋅‖_∞ norm and verifies 𝐿𝑓(𝑥)≤max{𝐺(𝑥,𝑡),𝑡∈[𝑎,𝑏]}𝑓_∞^𝑏_𝑎𝑞(𝑡)[],𝑑𝑡,𝑥∈𝑎,𝑏𝐿𝑓(𝑥)≤max{𝐺(𝑥,𝑡),𝑡∈[𝑎,𝑏]}𝑓₂^𝑏_𝑎𝑞(𝑡)[],𝑑𝑡,𝑥∈𝑎,𝑏 (20)(21) where the ‖⋅‖₂ norm is defined as in (11).

Proof. Equation (20) is an obvious consequence of (4), and (21) follows from (4) by application of Cauchy-Schwarz inequality.

Theorem 5. If any of conditions (12) are met, then the operator 𝐿𝑓 has a countably infinite number of eigenvalues 1/𝜆_𝑛 and associated orthonormal eigenfunctions Φ_𝑛(𝑥), which allow expressing 𝐿^𝑘𝑓, 𝑘≥1, as 𝐿^𝑘𝑓=^∞_𝑛=1𝑓,Φ_𝑛𝜆^𝑘_𝑛Φ_𝑛. (22) Moreover, (i)if no nontrivial solution of (2) satisfies (3) either at 𝑎, 𝑏 or at any 𝑎^′, 𝑏^′ interior to [𝑎,𝑏], then 𝐿^𝑘𝑓²₂<𝑓²₂,𝑘≥1;lim_𝑘→∞𝐿^𝑘𝑓₂=0,lim_𝑘→∞𝐿^𝑘𝑓(𝑥)[];=0,𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑓𝑜𝑟𝑥∈𝑎,𝑏 (23)(24)(ii)if there is a nontrivial solution of (2) that satisfies (3) at some 𝑎^′, 𝑏^′ interior to [𝑎,𝑏], then 𝐿^𝑘𝑓²₂>𝑓,Φ₁²+^∞_𝑛=2𝑓,Φ_𝑛²𝜆_𝑛^2𝑘,𝑘≥1. (25) In addition, if ⟨𝑓,Φ₁⟩≠0 and 𝑐∈[𝑎,𝑏] is such that Φ₁(𝑐)≠0, then lim_𝑘→∞𝐿^𝑘𝑓₂=∞,lim_𝑘→∞𝐿^𝑘𝑓(𝑐)=±∞, (26)(27) the sign corresponding to that of ⟨𝑓,Φ₁⟩Φ₁(𝑐).

Proof. Let us consider the eigenvalue problem 𝑦^′′(𝑥)][,𝑎+𝜆𝑞𝑦=0,𝑥∈𝑎,𝑏₁₁𝑦(𝑎)+𝑎₁₂𝑦^′(𝑎)=0,𝑎₂₁𝑦(𝑏)+𝑎₂₂𝑦^′(𝑏)=0. (28) From the theory of ordinary differential equations (see [22, Theorems V.8 and V.9]) it is known that there exist a countably infinite number of eigenvalues 𝜆_𝑛 which form an increasing sequence with lim_𝑛→∞𝜆_𝑛=∞, each of which has its corresponding orthonormal (with the norm (11)) eigenfunction Φ_𝑛(𝑥), and that the set of eigenfunctions Φ_𝑛 forms an orthonormal basis of 𝐶[𝑎,𝑏]. Applying the operator 𝐿 to these eigenfunctions Φ_𝑛(𝑥) and integrating by parts it is easy to show that 𝐿Φ_𝑛=Φ_𝑛𝜆_𝑛,𝑛≥1, (29) which implies that Φ_𝑛 are also the eigenfunctions of the operator 𝐿 with corresponding eigenvalues 1/𝜆_𝑛. Since from Lemmas 1, 2 and 3  𝐿 is linear, compact, and self-adjoint, we can apply [21, Theorem 7.5.2] and represent 𝐿 in the canonical form 𝐿𝑓=^∞_𝑛=1𝑓,Φ_𝑛𝜆_𝑛Φ_𝑛. (30) Applying 𝐿 again to (30) it yields 𝐿²𝑓=^∞_𝑛=1𝐿𝑓,Φ_𝑛𝜆_𝑛Φ_𝑛=^∞_𝑛=1∑^∞_𝑗=1⟨𝑓,Φ_𝑗⟩/𝜆_𝑗Φ_𝑗,Φ_𝑛𝜆_𝑛Φ_𝑛=^∞_𝑛=1𝑓,Φ_𝑛𝜆²_𝑛Φ_𝑛, (31) given that ⟨Φ_𝑖,Φ_𝑖⟩=1 and ⟨Φ_𝑖,Φ_𝑗⟩=0 for 𝑖≠𝑗. Applying 𝐿 recursively to (31) one gets to 𝐿^𝑘𝑓=^∞_𝑛=1𝐿^𝑘−1𝑓,Φ_𝑛𝜆_𝑛Φ_𝑛=^∞_𝑛=1∑^∞_𝑗=1⟨𝑓,Φ_𝑗⟩/𝜆_𝑗^𝑘−1Φ_𝑗,Φ_𝑛𝜆_𝑛Φ_𝑛=^∞_𝑛=1𝑓,Φ_𝑛𝜆^𝑘_𝑛Φ_𝑛, (32) which is in fact (22). And the application of Parseval’s identity (see [21, Lemmas 1.5.14]) to (32) leads to 𝐿^𝑘𝑓₂=^∞_𝑛=1𝑓,Φ_𝑛²𝜆_𝑛^2𝑘. (33) Since {𝜆_𝑛} form an increasing sequence, from (33) it is clear that 𝐿^𝑘𝑓₂≤^∞_𝑛=1𝑓,Φ_𝑛²𝜆₁^2𝑘=𝑓₂𝜆^𝑘₁. (34) Now, let us note that if (2) does not have any nontrivial solution that satisfies (3) either at 𝑎, 𝑏 or at any 𝑎^′, 𝑏^′ interior to [𝑎,𝑏], the first eigenvalue 𝜆₁ (and therefore all the others) must be strictly greater than 1. In that case, from (34) one has (23) and lim_𝑘→∞𝐿^𝑘𝑓₂=0. (35) From (35) and Lemma 4 one gets (24).
Likewise, if there is a nontrivial solution of (2) that satisfies (3) at some 𝑎^′, 𝑏^′ interior to [𝑎,𝑏], then 𝜆₁ must be such that 𝜆₁<1. From this and (33) one gets (25). If, in addition, ⟨𝑓,Φ₁⟩≠0, from (33) we get to lim_𝑘→∞𝐿^𝑘𝑓₂=∞, (36) which is (26). On the other hand, we can write 𝐿^𝑘𝑓−𝑓,Φ₁𝜆^𝑘₁Φ₁=^∞_𝑛=2𝑓,Φ_𝑛𝜆^𝑘_𝑛Φ_𝑛. (37) We can divide both sides of (37) by ⟨𝑓,Φ₁⟩/𝜆^𝑘₁ to yield 𝐿^𝑘𝑓−⟨𝑓,Φ₁⟩/𝜆^𝑘₁Φ₁𝑓,Φ₁/𝜆^𝑘₁=∑^∞_𝑛=2⟨𝑓,Φ_𝑛⟩/𝜆^𝑘_𝑛Φ_𝑛⟨𝑓,Φ₁⟩/𝜆^𝑘₁. (38) Applying Parseval’s identity to (38) one gets 𝐿^𝑘𝑓−⟨𝑓,Φ₁⟩/𝜆^𝑘₁Φ₁⟨𝑓,Φ₁⟩/𝜆^𝑘₁₂=∑^∞_𝑛=2𝑓,Φ_𝑛²/𝜆_𝑛^2𝑘𝑓,Φ₁²/𝜆₁^2𝑘≤𝑓₂𝑓,Φ₁𝜆^𝑘₁𝜆^𝑘₂. (39) Since 𝜆₁<𝜆₂, from (39) one yields lim_𝑘→∞𝐿^𝑘𝑓−⟨𝑓,Φ₁⟩/𝜆^𝑘₁Φ₁⟨𝑓,Φ₁⟩/𝜆^𝑘₁₂=0, (40) which implies that there exists an index 𝑘₀ such that 𝐿^𝑘𝑓−⟨𝑓,Φ₁⟩/𝜆^𝑘₁Φ₁⟨𝑓,Φ₁⟩/𝜆^𝑘₁₂<𝜖,𝑘>𝑘₀. (41) From Lemma 4 and (41) one has 𝐿^𝑘𝑓(𝑐)−⟨𝑓,Φ₁⟩/𝜆^𝑘₁Φ₁(𝑐)⟨𝑓,Φ₁⟩/𝜆^𝑘₁<𝜖max{𝐺(𝑐,𝑡),𝑡∈[𝑎,𝑏]}^𝑏_𝑎𝑞(𝑥)𝑑𝑥, (42) for 𝑘>𝑘₀+1; that is, 𝑓,Φ₁𝜆^𝑘₁×sign𝑓,Φ₁Φ₁(𝑐){𝐺()[×−𝜖max𝑐,𝑡,𝑡∈𝑎,𝑏]}_𝑏𝑎𝑞(𝑥)𝑑𝑥<𝐿^𝑘𝑓(𝑐)<𝑓,Φ₁𝜆^𝑘₁×sign𝑓,Φ₁Φ₁(𝑐){𝐺()[×+𝜖max𝑐,𝑡,𝑡∈𝑎,𝑏]}_𝑏𝑎𝑞(𝑥)𝑑𝑥, (43) for 𝑘>𝑘₀+1. Since Φ₁(𝑐)≠0 by hypothesis and 𝜆₁<1, (43) leads to (27).

Remark 6. Theorem 5 provides two types of methods to obtain upper and lower bounds for the minimum distance between points 𝑎,𝑏 satisfying (3), one based on comparing the norm of 𝐿^𝑘𝑓 with some other constants and another based on comparing the value of 𝐿^𝑘𝑓 at concrete points of [𝑎,𝑏] for different functions 𝑓. These will be addressed separately in the next two sections.

3. Bounds for the Distance between 𝑎 and 𝑏 Based on ‖𝐿^𝑘𝑓‖₂

As stated before, Theorem 5 provides methods to get upper and lower bounds for the minimum distance between points 𝑎, 𝑏 satisfying (3), which are based on the comparison of ‖𝐿^𝑘𝑓‖₂ with some thresholds for different values of the extremes of integration 𝑎 and 𝑏 of (4) and (11).

Thus, on the one hand, if 𝑎, 𝑏 are such that there is a nontrivial solution of (2) which satisfies (3) at some 𝑎^′, 𝑏^′ interior to [𝑎,𝑏], from (26) (and as long as ⟨𝑓,Φ₁⟩≠0, i.e., 𝑓 is “close” to Φ₁) it is clear that ‖𝐿^𝑘𝑓‖₂ will grow with the index 𝑘 regardless of the choice of 𝑓 and accordingly there will be an index 𝑘₀ such that (23) is violated. This allows us to define an algorithm to find progressively better “outer” bounds of the values satisfying (3) by fixing one of them, say 𝑎, and calculating the minimum values of the extremes 𝑏_𝑘 which give 𝐿^𝑘_{𝑎,𝑏𝑘}𝑓₂=𝑓₂, (44) for different values of 𝑘, as the following theorem shows.

Theorem 7. Assume that there exists a nontrivial solution 𝑦(𝑥) of (2) satisfying (3) at 𝑎, 𝑏. Let the sequence {𝑏_𝑘} be defined by (44), where 𝑓 is a continuous function on [𝑎,𝑏_𝑘] such that ⟨𝑓,Φ₁⟩≠0 and 𝑎, 𝑏^′ fulfill any of conditions (12) for each 𝑏^′∈[𝑏,𝑏_𝑘]. Then 𝑏_𝑘≥𝑏 and {𝑏_𝑘} tends to 𝑏 as 𝑘→∞.

Proof. The fact that 𝑏_𝑘≥𝑏 is obvious from (23). Now, let us pick a 𝛿>0. From (26) lim_𝑘→∞‖𝐿^𝑘_{𝑎,𝑏+𝛿}𝑓‖₂=∞, which means that there exists an index 𝑘₁ such that 𝐿^𝑘_{𝑎,𝑏+𝛿}𝑓₂−𝑓₂>0,𝑘>𝑘₁. (45) Given that ‖𝐿^𝑘_{𝑎,𝑏−𝛿}𝑓‖₂−‖𝑓‖₂<0 for 𝑘≥1, from (23) and the continuity of ‖𝐿^𝑘_𝑎,𝑥𝑓‖₂−‖𝑓‖₂ on 𝑥 (continuity guaranteed by the hypothesis), there must exist a value 𝑏_𝑘∈[𝑏,𝑏+𝛿[ such that ‖𝐿^𝑘_{𝑎,𝑏𝑘}𝑓‖₂−‖𝑓‖₂=0 for each 𝑘>𝑘₁. This proves the second assertion of the theorem.

The application of the method based on Theorem 7 is quite straightforward given that the right-hand side of (44), that is, ‖𝑓‖²₂, is easy to calculate once fixed 𝑓. However, it is worth remarking that, from (34), the closer the selected function 𝑓 is to Φ₁, the smaller the terms ⟨𝑓,Φ_𝑛⟩ for 𝑛>1 will be and the faster the sequence 𝑏_𝑘 will converge to 𝑏. Therefore, although the method can work with any continuous function 𝑓 such that ⟨𝑓,Φ₁⟩≠0, it is desirable to select one that may be as close as possible to the expected Φ₁ associated to the problem (28).

On the other hand, if 𝑎, 𝑏 are such that no nontrivial solution of (2) satisfies (3) either at 𝑎, 𝑏 or at some 𝑎^′, 𝑏^′ interior to [𝑎,𝑏], from (24) it is clear that ‖𝐿^𝑘𝑓‖₂ will shrink with the index 𝑘 regardless of the choice of 𝑓 and accordingly there will be an index 𝑘₀ such that (25) is violated. As happened before, this allows us to define an algorithm to find progressively better “inner” bounds of the values satisfying (3) by fixing one of them, say 𝑎, and calculating the values of the extremes 𝑏_𝑘 which give 𝐿^𝑘_{𝑎,𝑏𝑘}𝑓₂=𝐵_{𝑎,𝑏𝑘},𝑘≥1, (46) where 𝐵_{𝑎,𝑏𝑘} is any lower bound of the right-hand side of (25), which has (in turn) a positive lower bound that does not depend on 𝑘. This is shown in the following theorem.

Theorem 8. Assume that there exists a nontrivial solution 𝑦(𝑥) of (2) satisfying (3) at 𝑎, 𝑏. Let the sequence {𝑏_𝑘} be defined by (46), where 𝑓 is a continuous function on [𝑎,𝑏_𝑘] and 𝑎, 𝑏^′ fulfill any of conditions (12) for each 𝑏^′∈[𝑏_𝑘,𝑏]. Then 𝑏_𝑘≤𝑏 and {𝑏_𝑘} tends to 𝑏 as 𝑘→∞.

Proof. Again, the fact that 𝑏_𝑘≤𝑏 is obvious from (25) and (46). Now, let us pick a 𝛿>0. From (24), lim_𝑘→∞‖𝐿^𝑘_{𝑎,𝑏−𝛿}𝑓‖₂=0. This and the fact that 𝐵_{𝑎,𝑏𝑘} is bounded below by a positive amount which does not vary with 𝑘 imply that there exists an index 𝑘₂ such that 𝐿^𝑘_{𝑎,𝑏−𝛿}𝑓₂−𝐵_{𝑎,𝑏−𝛿}<0,𝑘>𝑘₂. (47) Given that ‖𝐿^𝑘_{𝑎,𝑏+𝛿}𝑓‖₂−𝐵_{𝑎,𝑏+𝛿}>0 for 𝑘≥1, from (25) and the continuity of ‖𝐿^𝑘_𝑎,𝑥𝑓‖₂−𝐵_𝑎,𝑥 on 𝑥, there must exist a value 𝑏_𝑘∈]𝑏−𝛿,𝑏] such that ‖𝐿^𝑘_{𝑎,𝑏𝑘}𝑓‖₂−𝐵_{𝑎,𝑏𝑘}=0 for each 𝑘>𝑘₂. This proves the second assertion of the theorem.

Unlike what happens with the method based on Theorem 7, the method based on Theorem 8 presents some difficulties in its application due to the fact that neither ⟨𝑓,Φ_𝑛⟩ nor the eigenvalues 𝜆_𝑛 are known. We can overcome them partially by discarding, in the right-hand side of (25), the term of the series of the eigenfunctions beyond the first one, that is, by converting (25) into 𝐿^𝑘𝑓²₂≥𝑓,Φ₁²,𝑘≥1, (48) given that the discarded terms are positive (in fact ⟨𝑓,Φ₁⟩² may be a possible positive lower bound for some types of terms 𝐵_{𝑎,𝑏𝑘} used in (46), which does not depend on 𝑘). The resulting method to obtain lower bounds for the distance between 𝑎 and 𝑏 will work in the same way as the one described in the previous paragraphs, at the expense of requiring greater values of the index 𝑘 to violate (48). But even with such a simplification there is still a need to obtain a lower bound for ⟨𝑓,Φ₁⟩ which is not evident at all.

The following lemmas aim at finding cases (fundamentally choices of 𝑓 and boundary conditions (3)) where the values of ⟨𝑓,Φ₁⟩, ⟨𝑓,Φ_𝑛⟩, and 𝜆_𝑛 are bounded in a way that allows getting a lower bound for the right-hand side of (25) (or for the right-hand side of (48) if there is no way to do it with (25)), overcoming the problem.

Lemma 9. Let 𝜆_𝑛 and Φ_𝑛 be defined as in Theorem 5. Assume that 𝑎, 𝑏 satisfy any of conditions (12) of Lemma 3. Fixed 𝑥∈[𝑎,𝑏] and choosing 𝑓(𝑡)=𝐺(𝑥,𝑡) and 𝑔(𝑡)=(𝛿𝐺(𝑥,𝑡)/𝛿𝑥) one has 𝑓,Φ_𝑛=Φ_𝑛(𝑥)𝜆_𝑛,𝑛≥1,𝑔,Φ_𝑛=Φ^′_𝑛(𝑥)𝜆_𝑛,𝑛≥1. (49)(50)

Proof. From (10) one has 𝑓,Φ_𝑛=^𝑏_𝑎𝑞(𝑡)𝐺(𝑥,𝑡)Φ_𝑛(𝑡)𝑑𝑡,𝑛≥1. (51) And since Φ_𝑛 satisfies (28) one yields 𝑓,Φ_𝑛=−^𝑏_𝑎𝐺()Φ𝑥,𝑡^𝑛′′(𝑡)𝜆_𝑛𝑑𝑡,𝑛≥1. (52) As 𝐺(𝑥,𝑡) is the Green function of the problem (5), −∫^𝑏_𝑎𝐺(𝑥,𝑡)(Φ_𝑛^′′(𝑡)/𝜆_𝑛)𝑑𝑡 is the value at 𝑥 of the function satisfying (3) whose second derivative is Φ_𝑛^′′/𝜆_𝑛, that is, Φ_𝑛(𝑥)/𝜆_𝑛. This proves (49). As for (50), note simply that 𝑔,Φ_𝑛=^𝑏_𝑎𝑞(𝑡)𝛿𝐺(𝑥,𝑡)Φ𝛿𝑥^𝑛(𝑡)=𝑑𝑡𝛿∫^𝑏_𝑎𝑞(𝑡)𝐺(𝑥,𝑡)Φ_𝑛(𝑡)𝑑𝑡=1𝛿𝑥𝜆_𝑛𝛿Φ_𝑛(𝑥)𝛿𝑥,𝑛≥1. (53)

Lemma 10. Assume that either (𝑎₁₁/𝑎₁₂)≥0 or 𝑎₁₂=0 and that either −(𝑎₂₁/𝑎₂₂)≥0 or 𝑎₂₂=0. Let 𝜆_𝑛 and Φ_𝑛 be defined as in Theorem 5. Let one suppose that 𝑞(𝑥)>0 on [𝑎,𝑏] and that it can be decomposed as 𝑞(𝑥)=𝑐(𝑥)/𝑑(𝑥) with 𝑐(𝑥) and 𝑑(𝑥) being nondecreasing functions. Depending on the values of 𝑎₁₁, 𝑎₁₂, 𝑎₂₁, and 𝑎₂₂, one has (i)if 𝑎₁₂=0, then Φ_𝑛^′2(𝑎)≥2𝜆_𝑛𝑐(𝑎)∫^𝑏_𝑎𝑑(𝑡)𝑞(𝑡)𝑑𝑡,𝑛≥1; (54)(ii)if 𝑎₁₂≠0, then Φ²_𝑛(𝑎)≥2𝑑(𝑎)∫^𝑏_𝑎𝑑(𝑡)𝑞(𝑡)𝑑𝑡1+𝑎²₁₁/𝑎²₁₂𝑞(𝑎)𝜆_𝑛,𝑛≥1; (55)(iii)if 𝑎₂₂=0, then Φ_𝑛^′2(𝑏)≥2𝜆_𝑛𝑑(𝑏)∫^𝑏_𝑎𝑞(𝑡)/𝑐(𝑡)𝑑𝑡,𝑛≥1; (56)(iv)if 𝑎₂₂≠0, then Φ²_𝑛(𝑏)≥2𝑐(𝑏)∫^𝑏_𝑎𝑞(𝑡)/𝑐(𝑡)𝑑𝑡1+𝑎²₂₁/𝑎²₂₂𝑞(𝑏)𝜆_𝑛,𝑛≥1. (57)

Proof. First of all, let us note that 𝑞(𝑥) can always be decomposed in the mentioned manner, given that 𝑞(𝑥)>0 and therefore ln𝑞(𝑥) exists in [𝑎,𝑏] and can be expressed as the difference between two increasing functions 𝐴(𝑥) and 𝐵(𝑥); that is, ln𝑞(𝑥)=𝐴(𝑥)−𝐵(𝑥). (58) which, taking exponentials, becomes 𝑞(𝑥)(𝑥)(𝐴(𝑥)(𝑥=(𝑥)=expln𝑞=exp−𝐵))exp𝐴(𝑥)=𝑐(𝑥)exp𝐵𝑑(𝑥), (59) with 𝑐(𝑥) and 𝑑(𝑥) nondecreasing. Now let us consider the functional 𝐹 defined by 𝐹Φ_𝑛,𝑥=Φ²_𝑛(𝑥)+Φ^′_𝑛(𝑥)²𝜆_𝑛𝑞(𝑥),𝑥∈[𝑎,𝑏]. (60) Using integration by parts and the fact that Φ_𝑛 is orthonormal with respect to the norm (11) it is easy to prove that ^𝑏_𝑎𝑞(𝑥)𝐹Φ_𝑛,𝑥𝑑𝑥=^𝑏_𝑎𝑞(𝑥)Φ²_𝑛(𝑥)𝑑𝑥+^𝑏_𝑎Φ^′_𝑛(𝑥)²𝜆_𝑛𝑑𝑥=2^𝑏_𝑎𝑞(𝑥)Φ²_𝑛(𝑥)+1𝑑𝑥𝜆_𝑛Φ^′_𝑛(𝑏)Φ_𝑛(𝑏)−Φ^′_𝑛(𝑎)Φ_𝑛(𝑎)1=2+𝜆_𝑛Φ^′_𝑛(𝑏)Φ_𝑛(𝑏)−Φ^′_𝑛(𝑎)Φ_𝑛(𝑎). (61) From (3), (61), and the hypothesis one gets ^𝑏_𝑎𝑞(𝑥)𝐹Φ_𝑛,𝑥𝑑𝑥≥2. (62) And given that 𝑐(𝑥) and 𝑑(𝑥) are increasing on [𝑎,𝑏] and that Φ_𝑛(𝑥) verifies (28) for 𝜆=𝜆_𝑛, one has 𝑐(𝑥)𝐹Φ_𝑛,𝑥^′=𝑐(𝑥)Φ²_𝑛(𝑥)(𝑥)Φ+𝑑^′𝑛(𝑥)²𝜆_𝑛^′=𝑐^′(𝑥)Φ²_𝑛(𝑥)+𝑑^′(𝑥)Φ^′_𝑛(𝑥)²𝜆_𝑛][;≥0,𝑥∈𝑎,𝑏 (63) that is, 𝑐(𝑥)𝐹(Φ_𝑛,𝑥) is increasing on ]𝑎,𝑏[, and 𝐹Φ_𝑛,𝑥𝑑(𝑥)^′=Φ²_𝑛(𝑥)𝑑(𝑥)+Φ^′_𝑛(𝑥)²𝜆_𝑛𝑐(𝑥)^′=−𝑑^′(𝑥)Φ²_𝑛(𝑥)𝑑²(𝑥)−𝑐^′(𝑥)Φ^′_𝑛(𝑥)²𝜆_𝑛𝑐²(𝑥)][;≤0,𝑥∈𝑎,𝑏 (64) that is, 𝐹(Φ_𝑛,𝑥)/𝑑(𝑥) is decreasing on ]𝑎,𝑏[. The application of (60), (63), and (64) to (62) leads to 𝐹Φ_𝑛,𝑥𝑐(𝑥)^𝑥_𝑎𝑞(𝑡)𝑐(𝑡)1𝑑𝑡+𝑑(𝑥)^𝑏_𝑥𝑑(𝑡)𝑞(𝑡)𝑑𝑡≥^𝑥_𝑎𝑐(𝑡)𝐹Φ_𝑛𝑞(𝑡),𝑡𝑐(𝑡)+𝑑𝑡_𝑏𝑥𝐹Φ_𝑛,𝑡𝑑(𝑡)𝑑(𝑡)𝑞(𝑡)≥𝑑𝑡_𝑏𝑎𝑞(𝑥)𝐹Φ_𝑛,𝑥𝑑𝑥≥2,𝑛≥1. (65) From (3), (60), and (65) it is straightforward to obtain (54)–(57).

Lemma 11. Let 𝜆_𝑛 and Φ_𝑛 be defined as in Theorem 5. Then one has 𝜆_𝑛≤(𝑛+𝐶)²𝜋²∫^𝑏_𝑎min1,𝑞(𝑥)𝑑𝑥²=𝑀(𝑛+𝐶)²,𝑛≥1, (66) with 𝑎𝐶=0,𝑖𝑓¹¹𝑎₁₂=𝑎₂₁𝑎₂₂𝑜𝑟𝑎₂₂=𝑎₁₂−1=0;2,𝑖𝑓𝑎₁₂=0,𝑎₂₂≠0,−𝑎₂₁𝑎₂₂−1≤0;2,𝑖𝑓𝑎₂₂=0,𝑎₁₂≠0,−𝑎₁₁𝑎₁₂≥0;−1,𝑒𝑙𝑠𝑒. (67)

Proof. From [23, Theorem 8], if we define the function angle 𝜃_𝑛(𝑥) by tan𝜃_𝑛(𝑥)=sin𝜃_𝑛(𝑥)cos𝜃_𝑛(𝑥)=Φ^′_𝑛(𝑥)𝐴Φ_𝑛(𝑥), (68) where 𝐴>0 is a real constant, then 𝜃_𝑛(𝑥) satisfies the equation 𝜃^′_𝑛(𝑥)=−𝐴sin²𝜃(𝑥)−𝜆_𝑛𝑞(𝑥)𝐴cos²𝜃(𝑥), (69) being, therefore, a decreasing function. Let us fix ]−(𝜋/2),(𝜋/2)[as the range of the arctan function. In that case, (i)if 𝑎₁₂=0 we will set 𝜃_𝑛(𝑎)=𝜋/2. Else, we will set 𝜃_𝑛(𝑎)=arctan(−𝑎₁₁/𝑎₁₂𝐴);(ii)if 𝑎₂₂=0 we will define 𝜗_𝑛(𝑏)=−(𝜋/2). Else,(a)if −𝑎₁₁/𝑎₁₂≥−𝑎₂₁/𝑎₂₂ we will define 𝜗_𝑛(𝑏)=arctan(−𝑎₂₁/𝑎₂₂𝐴);(b)we will define 𝜗_𝑛(𝑏)=−𝜋+arctan(−𝑎₂₁/𝑎₂₂𝐴). With this in mind, we will also define 𝐾(𝐴)=𝜃_𝑛(𝑎)−𝜗_𝑛(𝑏)𝜋−1. (70)𝐾(𝐴) is a constant which depends only on 𝐴 and the boundary conditions (3) and is related to the angle distance between 𝑎 and 𝑏 due to (3). Since Φ_𝑛(𝑥) is a function satisfying (3) with one more zero in ]𝑎,𝑏[ than Φ_𝑛−1(𝑥), integrating (69) and taking (70) into account we obtain (𝑛+𝐾(𝐴))𝜋=−^𝑏_𝑎𝜃^′_𝑛(𝑥)𝑑𝑥=𝐴^𝑏_𝑎sin²𝜃(𝑥)+𝑑𝑥_𝑏𝑎𝜆_𝑛𝑞(𝑥)𝐴cos²𝜃(𝑥)𝑑𝑥. (71) Taking 𝐴=𝜆_𝑛 in (71) one has 𝑛+𝐾𝜆_𝑛𝜋=𝜆_𝑛^𝑏_𝑎sin²𝜃(𝑥)𝑑𝑥+^𝑏_𝑎𝑞(𝑥)cos²𝜃(𝑥)≥𝑑𝑥𝜆_𝑛^𝑏_𝑎min1,𝑞(𝑥)𝑑𝑥. (72) The problem for (72) to be used to get upper bounds for 𝜆_𝑛 is the dependence of 𝐾(𝜆_𝑛) on 𝜆_𝑛. We can eliminate it by obtaining an upper bound for 𝐾(𝜆_𝑛). Accordingly we will define 𝑎𝐶=0,if¹¹𝑎₁₂=𝑎₂₁𝑎₂₂or𝑎₂₂=𝑎₁₂−1=0;2,if𝑎₁₂=0,𝑎₂₂≠0,−𝑎₂₁𝑎₂₂−1≤0;2,if𝑎₂₂=0,𝑎₁₂≠0,−𝑎₁₁𝑎₁₂≥0;−1,else. (73) From (70), (72), and (73) one yields (𝑛+𝐶)𝜋≥𝜆_𝑛^𝑏_𝑎min1,𝑞(𝑥)𝑑𝑥. (74) Taking squares in (74) one finally gets to (66).

Finally, by (25) and Lemmas 9–11 we are in a position to prove the next theorem, which allows a straightforward application of the method based on Theorem 8.

Theorem 12. Let 𝜆_𝑛 and Φ_𝑛 be defined as in Theorem 5. Let 𝑐(𝑥) and 𝑑(𝑥) be defined as in Lemma 10. Let 𝑀 and 𝐶 be defined as in Lemma 11. Assume that there is a nontrivial solution 𝑦(𝑥) of (2) that satisfies (3) at some 𝑎^′, 𝑏^′ interior to [𝑎,𝑏]. Assume also that 𝑎, 𝑏 satisfy any of conditions (12) of Lemma 3, that either (𝑎₁₁/𝑎₁₂)≥0 or 𝑎₁₂=0, and that either −(𝑎₂₁/𝑎₂₂)≥0 or 𝑎₂₂=0. Then the following inequalities hold. (i)If 𝑎₁₂=0 then 𝐿^𝑘𝛿𝐺(𝑎,𝑡)𝛿𝑥²₂>2𝑐(𝑎)∫^𝑏_𝑎𝑑(𝑡)𝑞(𝑡)1𝑑𝑡1+(4𝑘+1)𝑀^2𝑘+1(2+𝐶)^4𝑘+1,𝑘≥1. (75) Otherwise, if 𝑁_𝑎 is the first integer greater than 1 such that 𝑀(𝑁_𝑎+𝐶)²≥𝑎²₁₁/𝑎²₁₂𝑞(𝑎), then 𝐿^𝑘{𝐺(𝑎,𝑡)}²₂>2𝑑(𝑎)∫^𝑏_𝑎𝑑(𝑡)𝑞(𝑡)×1𝑑𝑡1+𝑎²₁₁/𝑎²₁₂𝑞(𝑎)+^𝑁𝑎−1_𝑛=2𝑀^2𝑘+1(𝑛+𝐶)^4𝑘+2×𝑀(𝑛+𝐶)²+𝑎²₁₁/𝑎²₁₂𝑞(𝑎)⁻¹+12𝑀^2𝑘+2(4𝑘+3)𝑁_𝑎+𝐶^4𝑘+3,𝑘≥1. (76)(ii)If 𝑎₂₂=0 then 𝐿^𝑘𝛿𝐺(𝑏,𝑡)𝛿𝑥²₂>2𝑑(𝑏)∫^𝑏_𝑎𝑞(𝑡)(𝑡)×1/𝑐𝑑𝑡1+(4𝑘+1)𝑀^2𝑘+1(2+𝐶)^4𝑘+1,𝑘≥1. (77) Otherwise, if 𝑁_𝑏 is the first integer greater than 1 such that 𝑀(𝑁_𝑏+𝐶)²≥𝑎²₂₁/𝑎²₂₂ 𝑞(𝑏), then 𝐿^𝑘{𝐺(𝑏,𝑡)}²₂>2𝑐(𝑏)∫^𝑏_𝑎𝑞(𝑡)(𝑡)×1/𝑐𝑑𝑡1+𝑎²₂₁/𝑎²₂₂𝑞(𝑏+_{𝑁𝑏−1𝑛=2})𝑀^2𝑘+1(𝑛+𝐶)^4𝑘+2×𝑀(𝑛+𝐶)²+𝑎²₂₁/𝑎²₂₂𝑞(𝑏)⁻¹+12𝑀^2𝑘+2(4𝑘+3)𝑁_𝑏+𝐶^4𝑘+3,𝑘≥1. (78)

Proof. Let us focus first on (75). From (25), the fact that 𝜆₁<1 (from the hypothesis) and Lemmas 9-10 it is straightforward to show that 𝐿^𝑘𝛿𝐺(𝑎,𝑡)𝛿𝑥²₂>2𝑐(𝑎)∫^𝑏_𝑎𝑑(𝑡)𝑞(𝑡)𝑑𝑡1+^∞_𝑛=21𝜆_𝑛^2𝑘+1,𝑘≥1. (79) On the other hand, from (66) one has ^∞_𝑛=21𝜆_𝑛^2𝑘+1≥1𝑀^∞2𝑘+1_𝑛=21(𝑛+𝐶)^4𝑘+2,𝑘≥1. (80) But given that ^∞_𝑛=21(𝑛+𝐶)^4𝑘+2≥^∞₂𝑑𝑥(𝑥+𝐶)^4𝑘+2=1(4𝑘+1)(2+𝐶)^4𝑘+1, (81) from (79)–(81) one gets (75).
As for (76), again from (25) and Lemmas 9-10 it is straightforward to show that 𝐿^𝑘{𝐺(𝑎,𝑡)}²₂>2𝑑(𝑎)∫^𝑏_𝑎𝑑(𝑡)𝑞(𝑡)×1𝑑𝑡𝜆²₁+𝜆₁𝑎²₁₁/𝑎²₁₂𝑞(𝑎)+^∞_𝑛=21𝜆_𝑛^2𝑘𝜆²_𝑛+𝜆_𝑛𝑎²₁₁/𝑎²₁₂𝑞(𝑎),𝑘≥1. (82) On the other hand, from (66) one has ^∞_𝑛=21𝜆_𝑛^2𝑘𝜆²_𝑛+𝜆_𝑛𝑎²₁₁/𝑎²₁₂𝑞(𝑎)≥^∞_𝑛=2𝑀^2𝑘+1(𝑛+𝐶)^4𝑘+2×𝑀(𝑛+𝐶)²+𝑎²₁₁𝑎²₁₂𝑞(𝑎)⁻¹,𝑘≥1. (83) From the definition of 𝑁_𝑎, it is clear that 1𝑀^2𝑘+1(𝑛+𝐶)^4𝑘+2𝑀(𝑛+𝐶)²+𝑎²₁₁/𝑎²₁₂𝑞(𝑎)≥12𝑀^2𝑘+2(𝑛+𝐶)^4𝑘+4,𝑛≥𝑁_𝑎,𝑘≥1. (84) And given that ^∞_𝑁𝑎1(𝑛+𝐶)^4𝑘+4≥^∞_𝑁𝑎𝑑𝑥(𝑥+𝐶)^4𝑘+4=1(4𝑘+3)𝑁_𝑎+𝐶^4𝑘+3, (85) from (82)–(85) and the fact that 𝜆₁<1 by the hypothesis, one obtains (76). The proofs of (77) and (78) are very similar mutatis mutandi.

Remark 13. It is possible to improve the results of Theorem 12 by using better lower bounds for Φ²_𝑛(𝑎), Φ²_𝑛(𝑏), Φ_𝑛^′2(𝑎), and Φ_𝑛^′2(𝑏) than the ones displayed in Lemma 10 and better upper bounds for 𝜆_𝑛 than the one presented in Lemma 11 or calculating more terms in the sums of (75)–(78) before using the integrals to get lower bounds for the remainders of the series.

4. Bounds for the Distance between 𝑎 and 𝑏 Based on Concrete Values of 𝐿^𝑘𝑓(𝑥)

This section will elaborate on the results of Section 2 in order to obtain upper and lower bounds for the distance between the extremes 𝑎, 𝑏 in a similar way as it was done in [1]. A critical condition for that is the positivity of the Green function 𝐺(𝑥,𝑡), which is ensured under certain boundary conditions by the next lemma.

Lemma 14. Let 𝑎, 𝑏 be any real numbers such that 𝑎≠𝑏 and 𝑎,𝑏∈[𝑎^′,𝑏^′]. If the boundary conditions (3) verify any of the following hypotheses for 𝑎^′, 𝑏^′, 𝑎₁₂=𝑎₂₂=0,𝑎₁₂𝑎=0,1+²¹𝑎₂₂𝑏^′−𝑎^′>0,𝑎1−¹¹𝑎₁₂𝑏^′−𝑎^′>0,𝑎₂₂=0,𝑎₂₁𝑎₂₂>𝑎₁₁𝑎₁₂,𝑎₂₁𝑎₂₂−𝑎₁₁𝑎₁₂−𝑎₂₁𝑎₂₂𝑎₁₁𝑎₁₂𝑏^′−𝑎^′𝑎>0,𝑒𝑖𝑡ℎ𝑒𝑟1+²¹𝑎₂₂𝑏^′−𝑎^′𝑎>0𝑜𝑟1−¹¹𝑎₁₂𝑏^′−𝑎^′>0, (86)(87)(88)(89) then 𝐺_𝑎𝑏(𝑥,𝑡) is positive almost everywhere for 𝑡∈[𝑎,𝑏] except if 𝑥=𝑎 or 𝑥=𝑏 in case (86), 𝑥=𝑎 in case (87), and 𝑥=𝑏 in case (88).

Proof. As was mentioned in the proof of Lemma 3, if we define 𝑦₁(𝑥) and 𝑦₂(𝑥) as the solutions of 𝑦^′′=0,𝑎₁₁𝑦𝑎^′+𝑎₁₂𝑦^′𝑎^′𝑦=0,_′′=0,𝑎₂₁𝑦𝑏^′+𝑎₂₂𝑦^′𝑏^′=0, (90) respectively, then we can represent 𝐺_{𝑎′𝑏′}(𝑥,𝑡) as follows: 𝐺_{𝑎′𝑏′}(𝑥,𝑡)=−𝑦₁(𝑥)𝑦₂(𝑡)𝑦₁𝑎^′𝑦^′₂𝑎^′−𝑦^′₁𝑎^′𝑦₂𝑎^′,𝑥<𝑡,−𝑦₁(𝑡)𝑦₂(𝑥)𝑦₁𝑎^′𝑦^′₂𝑎^′−𝑦^′₁𝑎^′𝑦₂𝑎^′,𝑥≥𝑡. (91) We can divide the analysis of (91) in four cases. (i)𝑎₁₂=0 and 𝑎₂₂=0. In this case one has 𝐺_{𝑎′𝑏′}(𝑥,𝑡)=𝑥−𝑎^′𝑡−𝑏^′𝑎^′−𝑏^′,𝑥<𝑡,𝑡−𝑎^′𝑥−𝑏^′𝑎^′−𝑏^′,𝑥≥𝑡, (92) which is clearly positive almost everywhere for 𝑡∈[𝑎^′,𝑏^′] as long as 𝑥≠𝑎^′ and 𝑥≠𝑏^′. This yields condition (86).(ii)𝑎₁₂=0 and 𝑎₂₂≠0. In this case, one has 𝐺_{𝑎′𝑏′}(𝑥,𝑡)=𝑥−𝑎^′−𝑎₂₁/𝑎₂₂𝑡−𝑏^′+1−𝑎₂₁/𝑎₂₂𝑎^′−𝑏^′+1,𝑥<𝑡,𝑡−𝑎^′−𝑎₂₁/𝑎₂₂𝑥−𝑏^′+1−𝑎₂₁/𝑎₂₂𝑎^′−𝑏^′+1,𝑥≥𝑡. (93) From (93) it is clear that 𝐺_{𝑎′𝑏′}(𝑥,𝑡) will be positive almost everywhere for 𝑡∈[𝑎^′,𝑏^′] as long as 𝑥≠𝑎^′ and 𝑎1+²¹𝑎₂₂𝑏^′−𝑎^′>0, (94) which gives condition (87).(iii)𝑎₁₂≠0 and 𝑎₂₂=0. In this case, one has 𝐺_{𝑎′𝑏′}()=−𝑥,𝑡𝑡−𝑏^′−𝑎₁₁/𝑎₁₂𝑥−𝑎^′+1−𝑎₁₁/𝑎₁₂𝑏^′−𝑎^′−+1,𝑥<𝑡,𝑥−𝑏^′−𝑎₁₁/𝑎₁₂𝑡−𝑎^′+1−𝑎₁₁/𝑎₁₂𝑏^′−𝑎^′+1,𝑥≥𝑡. (95) From (95) it is clear that 𝐺_{𝑎′𝑏′}(𝑥,𝑡) will be positive almost everywhere for 𝑡∈[𝑎^′,𝑏^′] as long as 𝑥≠𝑏^′ and 𝑎1−¹¹𝑎₁₂𝑏^′−𝑎^′>0, (96) which gives condition (88).(iv)𝑎₁₂≠0 and 𝑎₂₂≠0. In this case, one has 𝐺_{𝑎′𝑏′}()=−𝑎𝑥,𝑡₂₁/𝑎₂₂𝑡−𝑏^′+1−𝑎₁₁/𝑎₁₂𝑥−𝑎^′+1−𝑎₁₁/𝑎₁₂𝑏^′−𝑎^′+1𝑎₂₁/𝑎₂₂−𝑎₁₁/𝑎₁₂,−𝑎𝑥<𝑡,₂₁/𝑎₂₂𝑥−𝑏^′+1−𝑎₁₁/𝑎₁₂𝑡−𝑎^′+1−𝑎₁₁/𝑎₁₂𝑏^′−𝑎^′+1𝑎₂₁/𝑎₂₂−𝑎₁₁/𝑎₁₂,𝑥≥𝑡. (97) From (97) it is clear that 𝐺_{𝑎′𝑏′}(𝑥,𝑡) will be positive almost everywhere for 𝑡∈[𝑎^′,𝑏^′] if 1−(𝑎₁₁/𝑎₁₂)(𝑏^′−𝑎^′)>0, 1−(𝑎₂₁/𝑎₂₂)(𝑎^′−𝑏^′)>0, and (−(𝑎₁₁/𝑎₁₂)(𝑏^′−𝑎^′)+1)(𝑎₂₁/𝑎₂₂)−(𝑎₁₁/𝑎₁₂)>0. Condition (89) is a consequence of this and the fact that (−(𝑎₁₁/𝑎₁₂)(𝑏^′−𝑎^′)+1)(𝑎₂₁/𝑎₂₂)−(𝑎₁₁/𝑎₁₂)>0 and 1−(𝑎₁₁/𝑎₁₂)(𝑏^′−𝑎^′)>0 imply 1−(𝑎₂₁/𝑎₂₂)(𝑎^′−𝑏^′)>0 and that (−(𝑎₁₁/𝑎₁₂)(𝑏^′−𝑎^′)+1)(𝑎₂₁/𝑎₂₂)−(𝑎₁₁/𝑎₁₂)>0 and 1−(𝑎₂₁/𝑎₂₂)(𝑎^′−𝑏^′)>0 imply 1−(𝑎₁₁/𝑎₁₂)(𝑏^′−𝑎^′)>0 (both proofs are straightforward and are left to the reader).To complete the theorem it suffices to prove that if (86)–(89) are met by 𝑎^′, 𝑏^′, they will also be met by any other extremes 𝑎, 𝑏 such that [𝑎,𝑏]⊂[𝑎^′,𝑏^′]. This is straightforward in the case of conditions 1−(𝑎₁₁/𝑎₁₂)(𝑏^′−𝑎^′)>0 and 1+(𝑎₂₁/𝑎₂₂)(𝑏^′−𝑎^′)>0. In the case of the condition (𝑎₂₁/𝑎₂₂)−(𝑎₁₁/𝑎₁₂)−(𝑎₂₁/𝑎₂₂)(𝑎₁₁/𝑎₁₂)(𝑏^′−𝑎^′)>0 simply note that ℎ(𝑥)=𝑎₂₁𝑎₂₂−𝑎₁₁𝑎₁₂−𝑎₂₁𝑎₂₂𝑎₁₁𝑎₁₂𝑥,𝑥∈0,𝑏^′−𝑎^′, (98) is a linear function positive at the extremes 0 and 𝑏^′−𝑎^′ by the hypothesis, and therefore it must be positive for any 𝑏−𝑎∈[0,𝑏^′−𝑎^′].

Remark 15. Conditions (86)–(89) imply conditions (12) of Lemma 3. In consequence they will replace the latter in the rest of results of this section.

Once the conditions for the positivity of the Green function have been determined, we can proceed with the key results of this method.

Lemma 16. Assume that [𝑎^′,𝑏^′]⊂[𝑎^′′,𝑏^′′] are enclosed intervals such that [𝑎^′,𝑏^′]≠[𝑎^′′,𝑏^′′] and that any of conditions (86)–(89) are met in [𝑎^′′,𝑏^′′]. Assume also that 𝑓(𝑥) is positive almost everywhere on [𝑎^′′,𝑏^′′] and that 𝑥∈[𝑎^′,𝑏^′] is such that 𝐺(𝑥,𝑡) is not identically zero. Then the operator 𝐿 satisfies 𝐿_{𝑎′,𝑏′}𝑓(𝑥)<𝐿_{𝑎′′,𝑏′′}𝑓(𝑥)<max𝐿_{𝑎′′,𝑏′′}𝑓(𝑡),𝑡∈𝑎^′′,𝑏^′′. (99)

Proof. Since 𝑞(𝑥) and 𝑓(𝑥) are positive almost everywhere on [𝑎^′′,𝑏^′′] by hypothesis, and 𝐺_{𝑎′𝑏′} and 𝐺_{𝑎′′𝑏′′} are also positive almost everywhere on [𝑎^′,𝑏^′] and [𝑎^′′,𝑏^′′], respectively, by the hypothesis and Lemma 14, it suffices to prove that 𝐺_{𝑎′𝑏′}(𝑥,𝑡)≤𝐺_{𝑎′′𝑏′′}(𝑥,𝑡) (100) for (𝑥,𝑡)∈[𝑎^′,𝑏^′]×[𝑎^′,𝑏^′]. We will do it only for the case 𝑎₁₂,𝑎₂₂≠0 (the proof of the rest of cases is very similar), for which the Green function is given by (97). If we denote by 𝑊 the expression (−(𝑎₁₁/𝑎₁₂)(𝑏^′−𝑎^′)+1)(𝑎₂₁/𝑎₂₂)−(𝑎₁₁/𝑎₁₂), a routine calculation yields 𝛿𝐺_{𝑎′𝑏′}(𝑥,𝑡)𝛿𝑎^′=−𝑎₁₁𝑎₁₂𝑊²−𝑎₂₁𝑎₂₂𝑥−𝑏^′×−𝑎+1₂₁𝑎₂₂𝑡−𝑏^′−𝑎+1,𝑥≥𝑡,₁₁𝑎₁₂𝑊²−𝑎₂₁𝑎₂₂𝑡−𝑏^′×−𝑎+1₂₁𝑎₂₂𝑥−𝑏^′+1,𝑥<𝑡. (101) From (89), which is the hypothesis applicable if 𝑎₁₂,𝑎₂₂≠0, one has 𝛿𝐺_{𝑎′𝑏′}(𝑥,𝑡)/𝛿𝑎^′<0 if 𝑎₁₁≠0 and 𝛿𝐺_{𝑎′𝑏′}(𝑥,𝑡)/𝛿𝑎^′≡0 if 𝑎₁₁=0. In other words, fixing 𝑥, 𝑡, and 𝑏^′, either 𝐺_{𝑎′𝑏′}(𝑥,𝑡) increases (𝑎₁₁≠0) or 𝐺_{𝑎′𝑏′}(𝑥,𝑡) stays constant (𝑎₁₁=0) as 𝑎^′ decreases.
In a similar manner we can show that 𝛿𝐺_{𝑎′𝑏′}(𝑥,𝑡)/𝛿𝑏^′≥0; that is, fixing 𝑥, 𝑡, and 𝑎^′, either 𝐺_{𝑎′𝑏′}(𝑥,𝑡) increases (𝑎₂₁≠0) or 𝐺_{𝑎′𝑏′}(𝑥,𝑡) stays constant (𝑎₂₁=0) as 𝑏^′ increases. This proves (100) and the lemma.

Lemma 17. Assume that any of conditions (86)–(89) holds in [𝑎,𝑏]. If (i)either 𝑎₁₂=𝑎₂₂=0,(ii)or 𝑎₁₂=0 and (𝑎₂₁/𝑎₂₂)>0,(iii)or 𝑎₂₂=0 and (𝑎₁₁/𝑎₁₂)<0,(iv)or (𝑎₁₁/𝑎₁₂)<0<(𝑎₂₁/𝑎₂₂), then one has max{𝐺(𝑥,𝑡),𝑥∈[𝑎,𝑏]}=𝐺(𝑡,𝑡),𝑡∈[𝑎,𝑏]. (102)

Proof. We will only prove the last case (the proof of the other cases is very similar). Thus, from (97) it follows that, fixing 𝑡, 𝐺(𝑥,𝑡) is a function increasing with 𝑥 for 𝑥≤𝑡 and decreasing with 𝑥 for 𝑡<𝑥. Therefore it will have a maximum at 𝑥=𝑡, which is exactly (102).

Theorem 18. Let one suppose that there exists a nontrivial solution 𝑦(𝑥) of (2) that satisfies (3) at 𝑎, 𝑏. Let 𝑦_max be defined by 𝑦_max=max{𝑦(𝑥),𝑥∈[𝑎,𝑏]}. Assume that any of conditions (86)–(89) holds in extremes 𝑎^′, 𝑏^′ such that [𝑎,𝑏]⊂[𝑎^′,𝑏^′]. Assume also that 𝑓>𝑦 almost everywhere on [𝑎,𝑏] and 𝑓>0 almost everywhere on [𝑎^′,𝑏^′]. Then 𝐿^𝑘_{𝑎′,𝑏′}𝑓(𝑥)>𝑦(𝑥),𝑘≥1,𝑥∈[𝑎,𝑏]. (103) Moreover, (i)if (𝑎₁₁/𝑎₁₂)≥0 and either 𝑎₂₂=0 or (𝑎₂₁/𝑎₂₂)>0, then 𝐿^𝑘_{𝑎′,𝑏′}𝑓𝑎^′>𝑦_max=𝑦(𝑎),𝑘≥1; (104)(ii)if (𝑎₂₁/𝑎₂₂)≤0 and either 𝑎₁₂=0 or (𝑎₁₁/𝑎₁₂)<0, then 𝐿^𝑘_{𝑎′,𝑏′}𝑓𝑏^′>𝑦_max=𝑦(𝑏),𝑘≥1; (105)(iii)if any of the conditions of Lemma 17 are met, then ^𝑏′_𝑎′𝑞(𝑡)𝐺_{𝑎′𝑏′}(𝑡,𝑡)𝐿_𝑎^𝑘−1_′𝑏′𝑓(𝑡)𝑑𝑡>𝑦_max,𝑘≥1. (106)

Proof. From the hypothesis and Lemma 14, 𝐺_𝑎𝑏(𝑥,𝑡) is either positive almost everywhere or identically zero on 𝑡∈[𝑎,𝑏], depending on the value of 𝑥. In consequence we can apply Lemma 1 recursively to yield 𝐿^𝑘_𝑎,𝑏𝑓(𝑥)≥𝐿^𝑘_𝑎,𝑏𝑦(𝑥)=𝑦(𝑥),𝑘≥1. (107) And given that 𝑓(𝑥)>0 almost everywhere on [𝑎^′,𝑏^′], from Lemma 16 and (107) one gets (103). Equations (104) and (105) are consequences of 𝑦 and 𝐿^𝑘_{𝑎′,𝑏′}𝑓 having maxima at 𝑎 and 𝑎^′, in the first case, and 𝑏 and 𝑏^′, in the second case, respectively. As for (106), it results from (103) and Lemma 17.

Theorem 19. Let one suppose that there exists a nontrivial solution 𝑦(𝑥) of (2) that satisfies (3) at 𝑎, 𝑏. Let 𝑦_max be defined by 𝑦_max=max{𝑦(𝑥),𝑥∈[𝑎,𝑏]}. Assume that any of conditions (86)–(89) holds at 𝑎, 𝑏 and that the extremes 𝑎^′, 𝑏^′ are such that [𝑎^′,𝑏^′]⊂[𝑎,𝑏]. Assume also that 0<𝑓(𝑥)<𝑦(𝑥) almost everywhere on [𝑎^′,𝑏^′]. Then 𝐿^𝑘_{𝑎′,𝑏′}𝑓(𝑥)<𝑦(𝑥),𝑘≥1,𝑥∈𝑎^′,𝑏^′. (108) Moreover, (i)if (𝑎₁₁/𝑎₁₂)≥0 and either 𝑎₂₂=0 or (𝑎₂₁/𝑎₂₂)>0, then 𝐿^𝑘_{𝑎′,𝑏′}𝑓𝑎^′<𝑦_max=𝑦(𝑎),𝑘≥1; (109)(ii)if (𝑎₂₁/𝑎₂₂)≤0 and either 𝑎₁₂=0 or (𝑎₁₁/𝑎₁₂)<0, then 𝐿^𝑘_{𝑎′,𝑏′}𝑓𝑏^′<𝑦_max=𝑦(𝑏),𝑘≥1; (110)(iii)otherwise max𝐿^𝑘_{𝑎′,𝑏′}𝑓(𝑥),𝑥∈𝑎^′,𝑏^′<𝑦_max,𝑘≥1. (111)

Proof. Applying Lemmas 1 and 16 recursively one has 𝑦_max≥𝑦(𝑥)=𝐿^𝑘_𝑎,𝑏𝑦(𝑥)>𝐿^𝑘_{𝑎′,𝑏′}𝑦(𝑥)≥𝐿^𝑘_{𝑎′,𝑏′}𝑓(𝑥),𝑘≥1,𝑥∈𝑎^′,𝑏^′. (112) This gives (108) and also (111). As before, (109) and (110) are consequences of 𝑦 and 𝐿^𝑘_{𝑎′,𝑏′}𝑓 having maxima at 𝑎 and 𝑎^′, in the first case, and 𝑏 and 𝑏^′, in the second case, respectively.

Corollary 20. Let one suppose that there exists a nontrivial solution 𝑦(𝑥) of (2) that satisfies (3) at 𝑎, 𝑏. Assume that any of conditions (86)–(89) holds at the extremes 𝑎^′, 𝑏^′. (i)If (𝑎₁₁/𝑎₁₂)≥0 and either 𝑎₂₂=0 or (𝑎₂₁/𝑎₂₂)>0, then 𝐿^𝑘_{𝑎′,𝑏′}{1}𝑎^′>1,[𝑎,𝑏]⊂𝑎^′,𝑏^′,𝑘≥1,𝐿^𝑘_{𝑎′,𝑏′}𝑏^′−𝑥𝑏^′−𝑎^′𝑎^′<1,𝑎^′,𝑏^′⊂[𝑎,𝑏],𝑘≥1. (113)(114)(ii)If (𝑎₂₁/𝑎₂₂)≤0 and either 𝑎₁₂=0 or (𝑎₁₁/𝑎₁₂)<0, then 𝐿^𝑘_{𝑎′,𝑏′}{1}𝑏^′>1,[𝑎,𝑏]⊂𝑎^′,𝑏^′,𝑘≥1,𝐿^𝑘_{𝑎′,𝑏′}𝑥−𝑎^′𝑏^′−𝑎^′𝑏^′<1,𝑎^′,𝑏^′⊂[𝑎,𝑏],𝑘≥1. (115)(116)(iii)If any of the conditions of Lemma 17 are met and 𝑐 is any real number in ]𝑎^′,𝑏^′[, then ^𝑏′_𝑎′𝑞(𝑡)𝐺_{𝑎′𝑏′}(𝑡,𝑡)𝐿_𝑎^𝑘−1_′,𝑏′{1}(𝑡)𝑑𝑡>1,[𝑎,𝑏]⊂𝑎^′,𝑏^′,𝑘≥1,𝐿^𝑘_{𝑎′,𝑏′}𝑔(𝑐)<1,𝑎^′,𝑏^′⊂[𝑎,𝑏],𝑘≥1, (117)(118) where 𝑔 is defined by 𝑔(𝑥)=𝑥−𝑎^′𝑐−𝑎^′𝑥∈𝑎^′,𝑏,𝑐_′−𝑥𝑏^′−𝑐𝑥∈𝑐,𝑏^′. (119)

Proof. Equations (113), (115), and (117) can be easily obtained by setting 𝑓(𝑥)≡𝑦_max in (104)–(106). In turn, (114) can be obtained by setting 𝑓(𝑥)≡𝑦(𝑎^′)((𝑏^′−𝑥)/(𝑏^′−𝑎^′)) (with 𝑦(𝑎^′)>0) in (108) and taking into account that 𝑦 is concave (its second derivative is negative almost everywhere on [𝑎,𝑏] since 𝑞(𝑥) and 𝑦(𝑥) are positive almost everywhere on [𝑎,𝑏] due to (86)–(89)). Equations (116) and (118) can also be obtained in a similar manner.

Remark 21. Corollary 1 of [1] is a particular case (disfocality) of Corollary 20, (115)-(116).
Theorems 18 and 19 and Corollary 20 provide conditions to assess if there exists a solution 𝑦(𝑥) satisfying (3) at extremes inner or outer to [𝑎,𝑏]. However, at least in the way they have been presented, they do not allow determining if there exists a solution satisfying (3) at exactly the extremes 𝑎, 𝑏. That will be the purpose of the next theorems.

Theorem 22. Let one suppose that there exists a nontrivial solution 𝑦(𝑥) of (2) that satisfies (3) at 𝑎, 𝑏. Let 𝑦_max be defined by 𝑦_max=max{𝑦(𝑥),𝑥∈[𝑎,𝑏]}. Assume that any of conditions (86)–(89) holds at 𝑎, 𝑏. Let 𝑓(𝑥) be such that 𝑓(𝑥)>𝑦(𝑥) almost everywhere on [𝑎,𝑏] and let 𝑏_𝑘 be a sequence of real values whose definition depends on the boundary conditions (3) as follows: (i)if (𝑎₁₁/𝑎₁₂)≥0 and either 𝑎₂₂=0 or (𝑎₂₁/𝑎₂₂)>0, then 𝐿^𝑘_{𝑎,𝑏𝑘}𝑓(𝑎)=𝑦(𝑎),𝑘≥1; (120)(ii)if (𝑎₂₁/𝑎₂₂)≤0 and either 𝑎₁₂=0 or (𝑎₁₁/𝑎₁₂)<0, then 𝐿^𝑘_{𝑎,𝑏𝑘}𝑓𝑏_𝑘=𝑦(𝑏),𝑘≥1; (121)(iii)if any of the conditions of Lemma 17 are met, then ^𝑏𝑘_𝑎𝑞(𝑡)𝐺_𝑎𝑏𝑘(𝑡,𝑡)𝐿^𝑘−1_{𝑎,𝑏𝑘}𝑓(𝑡)𝑑𝑡=𝑦_max,𝑘≥1; (122)with 𝑓>0 almost everywhere on [𝑎,𝑏_𝑘]. Then 𝑏_𝑘<𝑏 for 𝑘≥1 and {𝑏_𝑘} tends to 𝑏 as 𝑘→∞.

Proof. From the fact that 𝑓(𝑥)>𝑦(𝑥) almost everywhere on [𝑎,𝑏]and 𝑓(𝑥)>0 almost everywhere on [𝑎,𝑏_𝑘] and from (104)–(106) and (120)–(122), it is clear that [𝑎,𝑏_𝑘]⊆[𝑎,𝑏] (i.e., 𝑎<𝑏_𝑘≤𝑏) for 𝑘≥1. Now, let us assume that {𝑏_𝑘} does not have a limit in 𝑏. In that case there exist a 𝛿>0 and a subsequence {𝑏_𝑘𝑗} of {𝑏_𝑘} such that 𝑏_𝑘𝑗<𝑏−𝛿. But then, from Theorem 5 and Lemma 16 one has lim_{𝑘𝑗→∞}𝐿^𝑘𝑗_{𝑎,𝑏𝑘𝑗}𝑓(𝑥)≤lim_{𝑘𝑗→∞}𝐿^𝑘𝑗_{𝑎,𝑏−𝛿}𝑓(𝑥)=0,𝑥∈[𝑎,𝑏−𝛿],lim_{𝑘𝑗→∞}𝐿^𝑘𝑗−1_{𝑎,𝑏𝑘𝑗}𝑓(𝑥)≤lim_{𝑘𝑗→∞}𝐿^𝑘𝑗−1_{𝑎,𝑏−𝛿}𝑓(𝑥)=0,𝑥∈[𝑎,𝑏−𝛿]. (123) In both cases the convergence is uniform for 𝑥∈[𝑎,𝑏−𝛿]. Therefore, for every 𝜖>0 there will exist infinitely many 𝑏_𝑘𝑗 such that 𝐿^𝑘𝑗−1_{𝑎,𝑏𝑘𝑗}𝑓(𝑥),𝐿^𝑘𝑗_{𝑎,𝑏𝑘𝑗}𝑓(𝑥)<𝜖,𝑥∈𝑎,𝑏_𝑘𝑗 (124) which contradicts (120) and (121) and, taking into account that ^𝑏𝑘𝑗_𝑎𝑞(𝑡)𝐺(𝑡,𝑡)𝐿^𝑘𝑗−1_{𝑎,𝑏𝑘𝑗}𝑓(𝑡)𝑑𝑡<𝜖^𝑏𝑘𝑗_𝑎𝑞(𝑡)𝐺(𝑡,𝑡)𝑑𝑡, (125) also contradicts (122).

Theorem 23. Let one suppose that there exists a nontrivial solution 𝑦(𝑥) of (2) that satisfies (3) at 𝑎, 𝑏. Let 𝑦_max be defined by 𝑦_max=max{𝑦(𝑥),𝑥∈[𝑎,𝑏]}. Let 𝑏_𝑘 be a sequence of real values whose definition depends on the boundary conditions (3) as follows: (i)if (𝑎₁₁/𝑎₁₂)≥0 and either 𝑎₂₂=0 or (𝑎₂₁/𝑎₂₂)>0, then 𝐿^𝑘_{𝑎,𝑏𝑘}𝑓_{𝑎,𝑏𝑘}(𝑎)=𝑦_max,𝑘≥1; (126)(ii)if (𝑎₂₁/𝑎₂₂)≤0 and either 𝑎₁₂=0 or (𝑎₁₁/𝑎₁₂)<0, then 𝐿^𝑘_{𝑎,𝑏𝑘}𝑓_{𝑎,𝑏𝑘}𝑏_𝑘=𝑦_max,𝑘≥1; (127)(iii)if any of the conditions of Lemma 17 are met, let one pick any 𝑐∈[𝑎,𝑏_𝑘] and set 𝐿^𝑘_{𝑎,𝑏𝑘}𝑓_{𝑎,𝑏𝑘}(𝑐)=𝑦_max,𝑘≥1, (128) where 𝑓_{𝑎,𝑏𝑘} is a family of continuous functions on [𝑎,𝑏_𝑘] such that 0<𝑓_{𝑎,𝑏𝑘}(𝑥)<𝑦(𝑥) almost everywhere in [𝑎,𝑏_𝑘]. Let one suppose that there exists a continuous function 𝑓 such that 0<𝑓(𝑥)<𝑓_{𝑎,𝑏𝑘}(𝑥) almost everywhere in [𝑎,𝑏_𝑘]. Assume that any of conditions (86)–(89) holds at 𝑎, 𝑏_𝑘. Then 𝑏_𝑘>𝑏 for 𝑘≥1 and {𝑏_𝑘} tends to 𝑏 as 𝑘→∞.

Proof. From the fact that 𝑦(𝑥)>𝑓_{𝑎,𝑏𝑘}(𝑥)>0 almost everywhere on [𝑎,𝑏_𝑘], (109)–(111), and (126)–(128), it is clear that [𝑎,𝑏_𝑘]⊇[𝑎,𝑏] (i.e., 𝑎<𝑏≤𝑏_𝑘) for 𝑘≥1. Now, let us assume that {𝑏_𝑘} does not have a limit in 𝑏. In that case there exist a 𝛿>0 and a subsequence {𝑏_𝑘𝑗} of {𝑏_𝑘} such that 𝑏_𝑘𝑗>𝑏+𝛿. But then, from Theorem 5 and Lemmas 1 and 16, for those 𝑥∈[𝑎,𝑏+𝛿] such that Φ₁(𝑥)≠0, one has lim_{𝑘𝑗→∞}𝐿^𝑘𝑗_{𝑎,𝑏𝑘𝑗}𝑓_{𝑎,𝑏𝑘𝑗}(𝑥)≥lim_{𝑘𝑗→∞}𝐿^𝑘𝑗_{𝑎,𝑏𝑘𝑗}𝑓(𝑥)≥lim_{𝑘𝑗→∞}𝐿^𝑘𝑗_{𝑎,𝑏+𝛿}𝑓(𝑥)=±∞, (129) the sign depending on that of ⟨𝑓,Φ₁⟩Φ₁(𝑥). Let us suppose that such a sign is positive (the negative case can be treated in the same manner). In that case for every 𝑀>𝑦_max>0 there will exist infinitely many 𝑏_𝑘𝑗 such that 𝐿^𝑘𝑗_{𝑎,𝑏𝑘𝑗}𝑓_{𝑎,𝑏𝑘𝑗}(𝑥)>𝑀. (130) Since the values at the left-hand side of (126)-(127) are upper bounds of 𝐿^𝑘𝑗_{𝑎,𝑏𝑘𝑗}𝑓_{𝑎,𝑏𝑘𝑗}(𝑥), (130) leads to a contradiction and proves the assertion.

Remark 24. Theorems 22 and 23 guarantee that, given [𝑎,𝑏], one can determine if there exists a nontrivial solution 𝑦(𝑥) of (2) satisfying (3) at 𝑎, 𝑏 by calculating the sequences defined by (120)–(122) and (126)–(128) and checking if they converge to 𝑏.

5. The Calculation of 𝐿^𝑘𝑓(𝑥)

A common key point of Sections 3 and 4 is the importance of the calculation of 𝐿^𝑘𝑓(𝑥) for different functions 𝑓. Bearing this in mind, our aim for this section is to find manners to facilitate such a calculation. That will be done with the following theorems, in all of which the internal product ⟨⋅,⋅⟩ is assumed to be calculated on the variable 𝑡.

Theorem 25. The operator 𝐿 defined in (4) verifies 𝐿𝑓(𝑥)=𝐺(𝑥,𝑡),𝑓(𝑡),𝑥∈[𝑎,𝑏]. (131)

Proof. It is straightforward from the definition of 𝐿 given in (4) and the definition of the internal product (10).

Theorem 26. If 𝑎, 𝑏 satisfy any of conditions (12), then the operator 𝐿 defined in (4) verifies 𝐿^𝑛+𝑚+1𝑓(𝑥)=𝐿^𝑛𝐺(𝑥,𝑡),𝐿^𝑚𝑓(𝑡,[])𝑥∈𝑎,𝑏,𝑛,𝑚≥0. (132) And in particular 𝐿^𝑘+1𝑓(𝑥)=𝐿^𝑘𝐺(𝑥,𝑡),𝑓(𝑡),𝑥∈[𝑎,𝑏],𝑘≥0. (133)

Proof. From (131) one has 𝐿^𝑛+𝑚+1𝑓(𝑥)=𝐺(𝑥,𝑡),𝐿^𝑛+𝑚𝑓(𝑡,[])𝑥∈𝑎,𝑏,𝑛,𝑚≥0. (134) From the hypothesis and Lemma 3, 𝐿 is self-adjoint. We can apply that property recursively to (134) to obtain 𝐿^𝑛+𝑚+1𝑓(𝑥)=𝐿𝐺(𝑥,𝑡),𝐿^{𝑛+𝑚−1}𝑓(𝑡=𝐿_𝑛)𝐺(𝑥,𝑡),𝐿^𝑚𝑓(𝑡),𝑥∈[𝑎,𝑏],𝑛,𝑚≥0. (135) Equation (133) is a particular case of (132) for 𝑚=0.

Remark 27. The advantage of (133) is that, given (2) and fixing the value 𝑘 we want to apply to 𝐿^𝑘, it allows testing easily different functions 𝑓, 𝑔 (as many as we want) in Theorems 7 and 18–23 in a simple way, just leaving the complication of the method to the calculation of 𝐿^𝑘−1𝐺(𝑥,𝑡).

6. Some Examples

Throughout this section we will introduce examples where Theorems 7 and 12 and Corollary 20 will be used to determine upper and lower bounds for the distance between extremes 𝑎 and 𝑏 of a solution of (2) for different functions 𝑞(𝑥) and boundary conditions (3). For the sake of simplicity, the analysis will fix the value of the starting point 𝑎 (in all cases zero) and will search for upper and lower bounds of the adjacent right extreme 𝑏. The boundary conditions addressed will be disfocality, in the first three examples, and disconjugacy, in the two last ones, and the examples used to illustrate the disfocal case will be the same as those that were introduced in [1], in order to compare the bounds obtained via Theorems 7 and 12 with those coming from Corollary 20, which was already obtained in [1] for the disfocal case. A comparison between these bounds and the bounds obtained via other existing methods, like Brown and Hinton’s one (see [24]) for lower bounds, will also be done.

In all examples the calculation of 𝐿^𝑘 will be done numerically and the number of iterations will be restricted case by case.

Example 1. Let us consider the following boundary value problem 𝑦^′′+𝐴𝑒^𝑥=0,𝑥≥0,𝑦(0)=0,𝑦^′(𝑏)=0, (136) for different values of the constant 𝐴.
The application of Theorem 7 to the function 𝑓(𝑥)=𝑥, together with Theorem 12 and Corollary 20 (already applied to this problem in [1] with 𝑓(𝑥)=(𝑥/𝑏) and 𝑓(𝑥) equal to the function Φ(𝑥) of [1, Equation (47)]) gives Table 1.

Table 1 

Comparison of bounds for 𝑏 in Example 1.

As can be seen from Table 1, Theorem 7 (i.e., ‖𝐿^𝑘{𝑥}‖²₂<‖𝑥‖²₂) provides better upper bounds of 𝑏 than Corollary 20 (that is, 𝐿^𝑘{𝑥/𝑏}(𝑏)<1) with a lesser number of iterations for all values of the constant 𝐴. On the other hand, Theorem 12 yields lower bounds of 𝑏 which are either worse than Corollary 20 and Brown and Hinton’s method [24] or, when equal, require a higher number of iterations than [1]. And as was mentioned in [1], it is worth remarking the good accuracy that Brown and Hinton’s method gives for lower bounds with relatively low calculation effort.

Example 2. Let us consider the following boundary value problem: 𝑦^′′+𝐵𝑥𝑦=0,𝑥≥0,𝑦(0)=0,𝑦^′(𝑏)=0, (137) for different values of the constant 𝐵.
The application of Theorem 7 to the function 𝑓(𝑥)=𝑥, together with Theorem 12 and Corollary 20 (already applied to this problem in [1]), gives Table 2, where one can observe results and trends similar to those commented in the previous example.

Table 2 

Comparison of bounds for 𝑏 in Example 2.

Example 3. Let us consider the following boundary value problem: 𝑦^′′+𝐶𝑒^−𝑥𝑦=0,𝑥≥0,𝑦(0)=0,𝑦^′(𝑏)=0, (138) for different values of the constant 𝐶.
The application of Theorem 7 to the function 𝑓(𝑥)=𝑥, together with Theorem 12 and Corollary 20 (already applied to this problem in [1]), gives Table 3.

Table 3 

Comparison of bounds for 𝑏 in Example 3.

In this third example one can observe that Theorem 12 finally gets to yield greater (i.e., better) lower bounds of 𝑏 than those provided by Corollary 20 and Brown and Hinton’s method [24] with a lesser number of iterations 𝑘.

Example 4. Let us consider the following boundary value problem: 𝑦^′′+𝐴𝑒^𝑥=0,𝑥≥0,𝑦(0)=0,𝑦(𝑏)=0, (139) for different values of the constant 𝐴.
The application of Theorem 7 and Corollary 20—(118)—to the function 𝑓(𝑥) defined by 𝑓(𝑥)=𝑣(𝑥)=2𝑥𝑏𝑏𝑥∈0,2,2(𝑏−𝑥)𝑏𝑏𝑥∈2,𝑏; (140) together with Theorem 12 and Corollary 20—(117)—gives Table 4.

Table 4 

Comparison of bounds for 𝑏 in Example 4.

In this example the methods based on comparison of norms yield better results than those based on comparison of values of 𝐿^𝑘. The reason for this may be that the boundary conditions of this case do not allow knowing in advance the value of the maximum of 𝐿^𝑘 to be used in the comparison (note that in the three previous examples the maximum was always placed at 𝑏), causing the selection of ((𝑎+𝑏)/2) as comparison value in the case of Corollary 20—(118)—and the use of 𝐺(𝑡,𝑡) in the integral in the case of Corollary 20 —(117)—both of which require further iterations to surpass the threshold value 1.

Example 5. Let us consider the following boundary value problem 𝑦^′′+𝐵𝑥𝑦=0,𝑥≥0,𝑦(0)=0,𝑦(𝑏)=0, (141) for different values of the constant 𝐵.
The application of Theorem 7 and Corollary 20—(118)—to the function 𝑣(𝑥) defined by (140) together with Theorem 12 and Corollary 20—(117)—gives Table 5, where one can observe results and trends similar to those commented in the previous example.

Table 5 

Comparison of bounds for 𝑏 in Example 5.

7. Conclusions

Throughout this paper two different types of methods have been provided to assess whether there are nontrivial solutions of (2) satisfying the boundary conditions (3) at extremes 𝑎^′, 𝑏^′ inner, equal, or outer to other fixed extremes 𝑎, 𝑏. Both rely on an iterative application of the operator 𝐿 defined in (4) and allow obtaining sequences of extremes 𝑎_𝑘, 𝑏_𝑘 that converge to the exact values 𝑎, 𝑏 for which 𝑦(𝑥) satisfies (3). The second characteristic is by far the most relevant one when one compares them with other methods present in the literature, since these latter only provide bounds (in fact very often only lower bounds or Lyapunov inequalities since the number of methods dealing with upper bounds is quite low) which cannot be improved following the same approach.

In addition it is worth remarking that the set of methods presented over almost all possible boundary conditions (3), despite the fact that some of them (namely, the method based on Theorem 12 and those based on Theorems 18 and 19) only apply under a limited subset of boundary conditions (3). The reason for this is, on the one hand, that the only conditions common for all these methods are those of Lemma 3 (which indeed are relatively easy to satisfy) and, on the other hand, that Theorem 7 does not require any additional conditions at all and Theorem 12 requires additional conditions on (3) which complement those required by Theorems 18 and 19.

As for what method is better in what case, it is difficult to decide, but the examples show that in the quest for outer bounds the method based on Theorem 7 converges normally much faster than the one based on Theorem 18. In the search of inner bounds, as indicated before, Theorems 12 and 19 cannot be applied simultaneously to most of the boundary conditions (3), but in the cases where that is possible, the speed of convergence depends on the concrete boundary conditions and the concrete function 𝑞(𝑥). In general it seems that boundary conditions aiming for a maximum in 𝐿^𝑘 different from 𝑎 or 𝑏 tend to favour the method based on Theorem 12 versus the method based on Theorem 19, the opposite occurring (with exceptions, as Example 3 shows) when the boundary conditions force maxima of 𝐿^𝑘 at either 𝑎 or 𝑏.

As for the main drawbacks, we can comment on two. On the one hand, the fact that the speed of convergence of the method depends vastly on the selected starting function 𝑓 and can be low in some cases, many iterations being necessary to approximate the values of the extremes 𝑎 and 𝑏 that satisfy (3). On the other hand, as they have been presented, the methods cannot deal with functions 𝑞(𝑥) which are zero or negative in a subset of positive measure. However it can be shown, as was done in [1], that these methods can be extended to the case 𝑞(𝑥) zero or negative in a way similar to most of the criteria published so far for the same problem.

All in all and taking the mentioned constraints into consideration, we the authors believe that the advantages of the methods presented in this paper surpass their drawbacks largely and that they can become a very powerful tool to assess problems of conjugacy/disconjugacy, nondisfocality/disfocality, and so forth, of (2), whether directly or by means of other methods based on them which in turn improve them.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work has been supported by the Spanish Ministry of Science and Innovation Project DPI2010-C02-01.