 Research
 Open access
 Published:
The principal eigenvalue of some nth order linear boundary value problems
Boundary Value Problems volume 2021, Article number: 84 (2021)
Abstract
The purpose of this paper is to present a procedure for the estimation of the smallest eigenvalues and their associated eigenfunctions of nth order linear boundary value problems with homogeneous boundary conditions defined in terms of quasiderivatives. The procedure is based on the iterative application of the equivalent integral operator to functions of a cone and the calculation of the Collatz–Wielandt numbers of such functions. Some results on the sign of the Green functions of the boundary value problems are also provided.
1 Introduction
Let L be a disconjugate linear differential operator of nth order on an interval \([a,b]\) which, according to a wellknown theorem of Pólya [1], can be factored as a product of operators of first order as
where \(\rho _{i} > 0\), \(\rho _{0} \rho _{1} \cdots \rho _{n}=1\), and \(\rho _{i} \in C^{ni} [a,b]\). Following Elias notation we will call \(L_{0}y, L_{1}y, \ldots , L_{n}y\) the quasiderivatives of y.
In this paper we will tackle the eigenvalue problem
where \(0 \leq m \leq n1\), α is the ordered set \(\{ \alpha _{1}, \alpha _{2}, \ldots , \alpha _{k}\}\), β is the ordered set \(\{ \beta _{1}, \beta _{2}, \ldots , \beta _{nk} \}\), \(1 \leq k \leq n1\), both \(\alpha _{k}, \beta _{nk} < n\) and \(p_{l}\), \(q \in C[a,b]\) such that \((1)^{nk} q \leq 0\) a.e. on \([a,b]\) and \(p_{l}\) have signs to be determined.
A special case of these homogeneous boundary conditions, which will be very relevant in what follows, is that in which the number of boundary conditions at a and b set on quasiderivatives of order lower than t is greater or equal than t for \(t=1, \ldots , n\). Elias denoted these conditions by poised in [2], term that we will use in the rest of the manuscript. Poised boundary conditions imply that \(\lambda =0\) is not an eigenvalue of (2) as per [2, Lemma 10.3].
The purpose of this paper is to provide an iterative procedure to:

1.
Calculate the smallest or principal eigenvalue of problem (2) when the boundary conditions are poised (note that the principal label refers to the positivity of the associated eigenfunction).

2.
Calculate the second smallest eigenvalue of problem (2) when the boundary conditions are not poised and \(q(x) \equiv 0\).

3.
Estimate the eigenfunction y of problem (2) associated with such eigenvalues.
We will work on the equivalent integral eigenvalue problem
where M is the operator \(C[a,b] \rightarrow C^{n}[a,b]\) defined by
and \(G(x,t)\) is the Green function of the problem
which exists as long as the boundary conditions are poised and problem (5) does not have an extremal point in \([a,b]\) (see [2, Theorem 4.16]).
The eigenvalue problem (2) has been studied thoroughly in the literature, references [3–6] being excellent examples. A good summary can also be found in [2, Chap. 10]. However, there do not seem to exist many algorithms for the calculation of the eigenvalues, as far as the authors are aware.
As for practical applications, in [7, Appendix D] one can find multiple examples of differential problems in the theory of fluid dynamics which lead to representations like (2). In particular, the pluriharmonic equation \(\Delta ^{N} y + \lambda p y=0\), \(x \in \mathbb{R}^{q}\), \(\Delta = \sum_{i=1}^{q} \frac{\partial ^{2} }{\partial x^{2}}\), if \(p(x)= p(t)\) with \(t= (\sum_{i=1}^{q} x_{i}^{2} )^{\frac{1}{2}}\), can be represented as \(( t^{1q} \frac{d}{dt} t^{q1} \frac{d}{dt} )^{N} y + \lambda p(t) y=0\).
The calculation of the smallest eigenvalue of (2) is also relevant to proving the existence of solutions of nonlinear boundary value problems of the type \(Ly + p(x) g(y)=0\), in particular by comparing that eigenvalue with the quotient \(\frac{g(y)}{y}\) for different values of y, especially when \(y \rightarrow 0^{+}\) and when \(y \rightarrow + \infty \). This approach was started by Erbe [8] for symmetric kernels and extended by Webb and Lan [9–11] and many others, [12] being a recent example.
The procedure is based on the iterative calculation of \(M^{j}\) (that is, the composition of M with itself \(j1\) times, \(\underbrace{M \circ M \circ \cdots \circ M}_{j \text{ times } M} \)) on functions u of a cone and the determination of the socalled Collatz–Wielandt numbers of the resulting functions.
For selfcompleteness, let us recall that, given a Banach space B, a cone \(P \subset B\) is a nonempty closed set defined by the conditions:

1.
If \(u \in P\) and \(u \in P\), then \(u=0\).

2.
If \(u,v \in P\), then \(cu+dv \in P\) for any real numbers \(c,d \geq 0\).
A cone in a Banach space B allows defining a partial ordering in the Banach space by setting \(u \leq v\) if and only if \(vu \in P\). We will say that the operator M is \(u_{0}\)positive if there exists \(u_{0} \in P\) such that for any \(u \in P \backslash \{0 \}\) one can find positive constants \(\epsilon _{1}\), \(\epsilon _{2}\) such that \(\epsilon _{1} u_{0} \leq M u \leq \epsilon _{2} u_{0}\). A cone P is reproducing if \(B= P  P\) and total if \(B= \overline{ P  P}\). We will denote by \(\operatorname{int}\{P \}\) the interior of the cone P, if it exists.
Following Forster–Nagy definition [13], if \(u \in P \backslash \{0\}\), the upper and lower Collatz–Wielandt numbers are defined, respectively, as
They are called upper and lower Collatz–Wielandt numbers as they extend the estimates for the spectral radius of a nonnegative matrix given by L. Collatz [14] and H. Wielandt [15].
The properties of \(\underline{r} (M,u)\) and \(\overline{r} (M,u)\) and their relationship with the spectral radius of the operator M have been studied by several authors, starting with Marek [16, 17], Forster and Nagy [13], who corrected some previous mistakes from Marek, and Marek again [18]. The concept has been extended to multiple types of operators, Banach spaces and cones. The references [19–22] include a good account of recent results.
Although Chang [20] already proposed the use of iterative Collatz–Wielandt numbers (what he called the power method) to obtain converging upper and lower bounds of the principal eigenvalue of some boundary value problems, it was Webb [23, Sect. 6] who first applied to a boundary value problem of type (3)–(4) by taking into consideration the following result by Marek [18, Theorem 5.2].
Theorem 1
Let P be a normal cone in a Banach space. Let M be a compact linear operator \(u_{0}\)positive in P. Then, for any \(u \in P \backslash \{0 \}\), the sequences of Collatz–Wielandt numbers \(\underline{r} (M,M^{j} u)\) and \(\overline{r} (M,M^{j} u)\) converge to \(r(M)\).
Webb’s paper [23] left open the question of how to determine the \(u_{0}\)positivity of M in the general case. His article included another iterative method to bound and calculate the principal eigenvalue which had a slower convergence rate and was based on the existence of certain bounds for the Green function used as a kernel in (4).
In [24] the authors proposed an iterative approach to estimate the principal eigenvalue and associated eigenfunction of an nth order linear boundary value problem, which in essence coincides with that of Webb. The \(u_{0}\)positivity of the associated operator M was proved by using sign results of the derivatives of the corresponding Green function. This paper follows a similar modus operandi.
The organization of the paper is as follows. Section 2 elaborates on the sign properties of the quasiderivatives of the Green functions of (5). Section 3 uses them to show that, when the boundary conditions are poised, there is a cone P for which M fulfils the conditions for the Collatz–Wielandt numbers to converge, yielding a procedure for the estimation of the principal eigenvalue and eigenfunction of (2). It also shows how to adapt the process to cope with nonpoised boundary conditions. In Sect. 4 some practical considerations for the calculation of the Collatz–Wielandt numbers of this problem are presented. Section 5 gives an example of how to apply the previous theory to calculate the principal eigenvalue of a boundary value problem. Finally, Sect. 6 discusses some conclusions.
2 The sign of the quasiderivatives of the Green function
In this section we study the signs of the quasiderivatives of the Green function of the problem
with \(q \in C[a,b]\) such that either \(q \equiv 0\) or \((1)^{nk} q <0\) a.e. on \([a,b]\), provided that the boundary conditions are poised and such a problem does not have an extremal point in \([a,b]\). From nomenclature perspective we will assume that the quasiderivative \(L_{i} G(x,t)\) applies only to the variable x of \(G(x,t)\).
We will also need the following definitions.
Definition 1
Fixed \(t \in [a,b]\),

A zero component is a closed subinterval \(I \subset [a,b]\) where a quasiderivative of \(G(x,t)\) is identically zero. If a quasiderivative has several zero components, there must be subintervals of \([a,b]\) of positive measure separating them. Otherwise they will be considered the same zero component.

\(z_{i}[a,b]\) is the number of isolated zeroes or zero components of the ith quasiderivative of \(G(x,t)\) on \([a,b]\) for \(i=0, \ldots , n2\).

\(z_{i}(a,b)\) is the number of isolated zeroes or zero components of the ith quasiderivative of \(G(x,t)\) entirely lying on \((a,b)\) for \(i=0, \ldots , n2\).

\(Z_{i} \{ \alpha , \beta \}\) is the number of homogeneous boundary conditions defined in \(\{ \alpha , \beta \}\) which are lower than or equal to i.

\(E_{i}[a,b]\) is the excess of isolated zeroes or zero components of the ith quasiderivative of \(G(x,t)\) on \([a,b]\) not due to the boundary conditions and Rolle’s theorem which, for reasons that will become clear later, we will define as
$$ E_{i}[a,b] = z_{i}[a,b]  Z_{i} \{ \alpha , \beta \} +i,\quad i=0, \ldots , n2. $$(8) 
\(m(\alpha ,i)\) is the number of quasiderivatives of order equal to or higher than i which the boundary conditions α do not specify to vanish at a.

\(n(\beta ,i)\) is the number of quasiderivatives of order higher than i which the boundary conditions β do specify to vanish at b.

\(i_{\max }\) is the lowest quasiderivative such that either \(\{i_{\max }, \ldots , n1 \} \subset \alpha \) or \(\{i_{\max }, \ldots , n1 \} \subset \beta \), with \(i_{\max }=n\) if there are no boundary conditions on the \((n1)\)th quasiderivative.

\(i(\alpha ,j)\) is the lowest quasiderivative higher than or equal to j such that in all quasiderivatives between the jth and the \((i(\alpha ,j)1)\)th one there are boundary conditions set at a. If there is no boundary condition set on the jth quasiderivative at a, we will say \(i(\alpha ,j)= j\).

\(i(\beta ,j)\) is the lowest quasiderivative higher than or equal to j such that in all quasiderivatives between the jth and the \((i(\beta ,j)1)\)th one there are boundary conditions set at b. If there is no boundary condition set on the jth quasiderivative at b, we will say \(i(\beta ,j)= j\).
Lemma 1
\(E_{i}[a,b]\) satisfies \(E_{i}[a,b] \geq E_{i1}[a,b] \geq 0\) for \(i=1, \ldots , n2\).
Proof
From the definition of \(E_{i}[a,b]\) it is clear that
\(G(x,t)\) cannot be identically zero on \([a,b]\) as otherwise \(L_{n1} G(x,t)\) cannot have a discontinuity jump at \(x=t\). Therefore, it cannot have a single zero component covering \([a,b]\). This implies \(z_{0}[a,b] = E_{0}[a,b] + Z_{0} \{ \alpha , \beta \} \geq Z_{0} \{ \alpha , \beta \}\), so \(E_{0}[a,b] \geq 0\). Next, by Rolle’s theorem
and
From here and (9) one has
and
which together with (9) prove the statement. □
Remark 1
Note that Lemma 1 refers to quasiderivatives and boundary conditions, not requiring any particular condition on \(q(x)\) of (7). This could take any values as long as the associated Green function exists.
Theorem 2
Fixed \(t \in [a,b]\), if the boundary conditions \(\{ \alpha ,\beta \}\) are poised, then \(E_{i}[a,b]=0\) and \(z_{i}[a,b] \geq 1\) for \(i=0, \ldots , n2\). In addition,
and if \(Z_{i1} \{ \alpha , \beta \} = i\), then
Proof
If the boundary conditions are poised, then \(Z_{i} \{ \alpha , \beta \} \geq i+1\) for \(i=0, \ldots , n1\). This and (9) lead to \(z_{i} [a,b] \geq E_{i} [a,b] +1 \geq 1\), \(i=0, \ldots , n2\).
Next, from [2, Theorem 4.16] one knows that
which, given that \((1)^{nk} q(x) <0\) a.e. on \([a,b]\) by hypothesis, implies
This means that \(L_{n1} G(x,t)\) is increasing on \([a,b]\), including at the discontinuity point \(x=t\).
Let us assume that any \(E_{i}[a,b] \neq 0\) with \(0 \leq i \leq n2\). By Lemma 1, \(E_{n2} [a,b] \geq 1\) and there can be two cases:

Either there is no boundary condition at \(L_{n1} G(x,t)\), which, given that the total number of boundary conditions is n, implies that \(Z_{n2}\{ \alpha , \beta \} =n\) and therefore \(z_{n2} [a,b] = E_{n2} [a,b] + 2 \geq 3\). This is impossible since by Rolle’s theorem \(L_{n1} G(x,t)\) must have at least two change of signs, which is not compatible with \(L_{n1} G(x,t)\) being increasing on \([a,b]\).

Or there is one boundary condition at \(L_{n1} G(x,t)\). This implies in turn that \(Z_{n2}\{ \alpha , \beta \} =n1\) and therefore \(z_{n2} [a,b] = E_{n2} [a,b] + 1 \geq 2\). By Rolle’s theorem \(L_{n1} G(x,t)\) must have at least one change of sign, but this is also impossible since \(L_{n1} G(x,t)\) is increasing on \([a,b]\) and is set to 0 at one of the extremes.
The previous monotonicity argument can be applied to show that \(L_{n1} G(x,t)\) can only have either one zero at a or b, set by the boundary conditions, or a single zero on \((a,b)\), or one change of sign at \(x=t\), the last two cases as a result of Rolle’s theorem.
As for the sign of \(L_{i} G(x,t)\), if \(L_{i} G(a,t)=0\), then obviously \(L_{i} G(x,t) L_{i+1} G(x,t) >0\) for \(x \in (a, a+ \delta )\). Else, since \(z_{i}[a,b] \geq 1\) and \(E_{i+1} [a,b]=0\), there cannot be a zero of \(L_{i+1} G(x,t)\) on \((a,x_{i})\), where \(x_{i}\) is the smallest zero of \(L_{i} G(x,t)\). Accordingly,
so \(L_{i} G(x,t) L_{i+1} G(x,t) < 0\) for \(x \in (a, a+ \delta )\). From here and (14) one gets
In a similar way it is possible to prove that
and
Inequalities (12) are a consequence of (16) and (17).
Last, if \(Z_{i1} \{ \alpha , \beta \} = i\), then \(z_{i} [a,b] = Z_{i} \{ \alpha , \beta \}  Z_{i1} \{ \alpha , \beta \}\), that is, the only zeroes of \(L_{i} G(x,t)\) happen at a or b and \(L_{i} G(x,t)\) does not change sign on \((a,b)\). This and (16) yield (13). □
Let us turn our attention to the Green function of (7) for the case \(q \equiv 0\), that is,
For this problem we can obtain a result similar to Theorem 2.
Theorem 3
Fixed \(t \in [a,b]\), if the boundary conditions \(\{ \alpha ,\beta \}\) are poised, then \(z_{i}[a,b] \geq 1\) and \(E_{i}[a,b]=0\) for \(i=0, \ldots , n2\).
In addition
and if \(Z_{i1} \{ \alpha , \beta \} = i\) with \(i< i_{\max }\), then
Proof
If the boundary conditions \(\{ \alpha ,\beta \}\) are poised, then the Green function exists and one can reason as in Theorem 2 to obtain \(z_{i}[a,b] \geq 1\) for \(i=0, \ldots , n2\).
Next, let us assume that \(i_{\max } < n\) and \(i_{\max } \in \alpha \). From the boundary conditions and (19) one has \(L_{i} G(x,t)=0\) for \(x \in [a,t)\), \(i= i_{\max },\ldots , n1\). It cannot happen that there is \(x_{i} \in (t,b]\) such that \(L_{i} G(x_{i},t)=0\) for any \(i \geq i_{\max }\) since otherwise, by Rolle’s theorem, there should be a zero of \(L_{n1} G(x,t)\) on \((t, b)\), which is impossible since \(L_{n} G=0\) on that subinterval and \(L_{n1} G(x,t)\) has a discontinuity at \(x=t\). That implies that \((1)^{n(\beta ,i)} L_{i} G(x,t) = L_{i} G(x,t) >0\) for \(i=i_{\max },\ldots , n1\) and \(x \in (t,b]\), \(E_{n2}[a,b]=0\) and, according to Lemma 1, \(E_{i}[a,b]=0\) for \(i=0, \ldots , n2\). In addition, \(L_{i_{\max }1} G(x,t)\) must be constant on \([a,t]\) and monotonic increasing on \((t,b]\).
If the zero of \(L_{i_{\max }1} G(x,t)\) (there must be at least one!) is at b and is due to the boundary conditions, then \(L_{i_{\max }1} G(x,t) <0\) for \(x\in [a,b)\) or, in other words, \((1)^{n(\beta ,i_{\max }1)} L_{i_{\max }1} G(x,t) = (1)^{m(\alpha ,i_{ \max }1)} L_{i_{\max }1} G(x,t) >0\) for \(x\in [a,b)\). Otherwise, as the definition of \(i_{\max }\) prevents that a boundary condition on \(L_{i_{\max }1} G(x,t)\) is set at a, and the boundary conditions are poised, from Lemma 1, in particular (10), the number of isolated zeroes or zero components of \(L_{i_{\max }1} G(x,t)\) entirely lying on \((a,b)\) is
The subinterval \([a,t]\) cannot be that one, since it is not entirely within \((a,b)\), so that zero must be in \((t,b)\) and therefore \(L_{i_{\max }1} G(x,t) \neq 0\) for \(x \in [a,t]\). In particular \((1)^{n(\beta ,i_{\max }1)} L_{i_{\max }1} G(b,t) = L_{i_{\max }1} G(b,t)>0 \) and \((1)^{m(\alpha ,i_{\max }1)}\) \(L_{i_{\max }1} G(x,t) =  L_{i_{\max }1} G(x,t)>0\) for \(x \in [a, a+ \delta ]\). In both cases, since \(z_{i} [a,b] \geq 1\) for \(i = 0, \ldots , n2\), one can reason as in Theorem 2 to get to (20) and (21).
A similar result is obtained if \(i_{\max } < n\) and \(i_{\max } \in \beta \).
Last, let us assume \(i_{\max }=n\), that is, no boundary condition set at a or b for \(L_{n1} G\). From (9), \(z_{n2} [a,b] = E_{n2}[a,b] + n  (n2) \geq 2\). From Rolle’s theorem there must be at least a change of sign of \(L_{n1} G(x,t)\) in \((a,b)\). As \(L_{n} G= LG=0\) in each subinterval, that change of sign is only possible if \((1)^{m(\alpha ,n1)} L_{n1} G (x,t) =  L_{n1} G (x,t) > 0\) on \([a,t)\) and \((1)^{n(\beta ,n1)} L_{n1} G (x,t) = L_{n1} G (x,t) > 0\) on \((t,b]\). There cannot be any other zero or change of sign, so \(E_{n2}[a,b]=0\) and, from Lemma 1, \(E_{i}[a,b]=0\) for \(i=0, \ldots , n2\). As before, one can reason as in Theorem 2 to get to (20) and (21). □
3 The calculation of the principal eigenvalue
Let us first consider the eigenvalue problem
with \(\{\alpha , \beta \}\) being poised boundary conditions, \((1)^{nk} q <0\) a.e. on \([a,b]\) and \(q, p_{l} \in C[a,b]\).
Problem (22) is equivalent to the integral eigenvalue problem \(\lambda M y = y\), with M defined as
and \(G(x,t)\) being the Green function of problem (7). We will show that the problem is compliant with the conditions presented in [24] for certain Banach spaces and cones. Thus, let S be the set of the indices l such that \(Z_{l1} \{ \alpha , \beta \} = l\), \(0 \leq l \leq m\), where we assume \(Z_{1} \{ \alpha , \beta \} = 0\). If \(m=0\), we will define the Banach space B as
and if \(m > 0\) as
in all cases the associated norm being
We will define the cone P by
In a similar manner, we will define the Banach space \(\mathcal{B}\) as
with the associated norm
and the cone \(\mathcal{P}\) as
The cone \(\mathcal{P}\) is solid and its interior is defined by
These cones are the tools for proving the next theorem.
Theorem 4
Let us suppose that both \(i(\alpha ,m), i(\beta ,m) < n \). Let us also suppose that \(\{\alpha , \beta \}\) are poised,
and that there exists an index \(s \in S\) such that \((1)^{m(\alpha ,s)} p_{s}(x) > 0\) a.e. on \([a,b]\).
Then there exists a principal eigenvalue \(\lambda _{0}\) of problem (22) which satisfies
and
and its associated eigenfunction v satisfies \(v \in P \backslash \{ 0 \}\) and
for any \(u\in P \backslash \{ 0 \}\), where \(f(u)\) is a nonzero linear functional dependent on u.
Proof
The assertions are a consequence of [24, Theorems 6 and 7], which in turn require that \(\mathcal{P}\) is reproducing (this is a consequence of the cone being solid and [25, Lemma 1.1]), M is compact (this can be easily proved using Arzelà–Ascoli theorem), and that \(M \{ P \backslash \{ 0 \} \} \subset \operatorname{int} \{\mathcal{P} \}\). To prove the last assertion, let us assume that \(f \in P \backslash \{ 0 \} \) and let us check if Mf satisfies (31). From the definition of M one has
The definition of P implies, using Rolle’s theorem as in Lemma 1 and Theorem 2, that all quasiderivatives of f of order \(l \in S\) have at least a zero in \([a,b]\), but perhaps the quasiderivative of mth order. Therefore, if any of them was identically zero, all of them would vanish identically. From here, the fact that \(Z_{l1} \{ \alpha , \beta \} = l\) for \(l \in S\), (13), (30), (32), and (36) it follows that
In a similar manner, according to (12) and (13), \(n(\beta ,l)=m(\alpha ,l)\) for \(l \in S \backslash (S \cap \beta )\). From this, (12), (30), (32), and (36), one has
and
Next, from the definitions of \(m(\alpha ,l)\), \(i(\alpha ,l)\), \(n(\beta ,l)\), and \(i(\beta ,l)\) it follows
which in combination with the hypothesis \(i(\alpha ,m), i(\beta ,m) < n\), (12), (30), (32), (36), and \((1)^{n(\beta ,j)+1}=(1)^{m(\alpha ,j)}\) for \(j \in \beta \), yields
for \(l \in S\). This completes the proof. □
Remark 2
Theorem 4 shows that, selecting any \(u \in P \backslash \{ 0 \}\), the calculation of \(\underline{r} (M, M^{j} u)\) and \(\overline{r} (M, M^{j} u)\) yields lower and upper bounds for the eigenvalue \(\frac{1}{\lambda _{0}}\) which converge to this as the iteration index j grows, and that \(M^{j} u\) also converges in norm to the corresponding eigenfunction v. Such a calculation requires the comparison of \(\frac{\partial ^{l} M^{j} u}{\partial x^{l}} \) and \(\frac{\partial ^{l} M^{j+1} u}{\partial x^{l}}\) for all \(l \in S\). This is an aspect different from [24], where the comparison was restricted to the partial derivative of a certain order, but it is the price to pay for allowing a wider amount of different quasiderivatives in the righthand side of (2).
For the case \(q(x) \equiv 0\), namely for the problem
with \(\{\alpha , \beta \}\) being poised boundary conditions and \(p_{j} \in C[a,b]\), it is possible to obtain a result like Theorem 4.
Theorem 5
Let us suppose that both \(i(\alpha ,m), i(\beta ,m) \leq i_{\max }1\). Let us also suppose that \(\{\alpha , \beta \}\) are poised,
and there exists an index \(s \in S\) such that \((1)^{m(\alpha ,s)} p_{s}(x) > 0\) a.e. on \([a,b]\). Then there exists a principal eigenvalue \(\lambda _{0}\) of problem (41) which satisfies
and
and its associated eigenfunction v satisfies \(v \in P \backslash \{ 0 \}\) and
for any \(u\in P \backslash \{ 0 \}\), where \(f(u)\) is a nonzero linear functional dependent on u.
Proof
The proof is exactly the same as that of Theorem 4 by considering inequalities (20)–(21) instead of (12)–(13). □
When \(q(x) \equiv 0\) and \(m=0\), problem (22) becomes
with \(p \in C[a,b]\) such that \((1)^{nk} p >0\) on \([a,b]\). If, in addition, the boundary conditions \(\{ \alpha , \beta \}\) are not poised, the principal eigenvalue is exactly 0 due to [2, Lemma 10.3]). However, it is possible to bound and estimate the next smallest eigenvalue (and associated eigenfunction) following a similar procedure as that of Theorem 5 and taking into consideration the next theorem.
Theorem 6
Any problem of type (46) with nonpoised boundary conditions and a nonzero eigenvalue is equivalent to another problem of type (46) with poised boundary conditions.
Proof
According to [2, Lemma 4.13], for any boundary conditions \(\{\alpha , \beta \}\) there exists an index s such that the boundary conditions are spoised, that is, there are at least i conditions imposed on the first i quasiderivatives in the sequence
for \(i=1, \ldots , n\). Such an index s, according to [2, Equation (4.10)] and the definition of \(Z_{i} \{ \alpha , \beta \}\), satisfies
for \(j=0, \ldots , n1\).
Let us assume that the eigenvalue λ is not zero. As in [2, Theorem 5.5] we can permute the quasiderivatives of y in a cyclic order. Thus let \(z=L_{s} y\) and
with
As \(p=(1)^{nk} p\), this means in practice
This implies that problem (46) can be transformed into the problem
whose boundary conditions are poised, thus being compliant with the hypotheses of Theorem 5. We can apply Theorem 5, calculate z and λ, and obtain y from \(N_{ns} z\), \(\rho _{0}\) and λ. □
Remark 3
It is worth highlighting that [24] did not focus on \(\underline{r} (M, M^{j} u)\) and \(\overline{r} (M, M^{j} u)\) as the only lower and upper bounds of the principal eigenvalue, but it extended those properties to \(\underline{r} (M^{j}, M^{i} u)\) and \(\overline{r} (M^{j}, M^{i} u)\) for different values of i and j, including \(i=0\).
4 Practical considerations for calculating Collatz–Wielandt numbers
The determination of the Collatz–Wielandt numbers \(\underline{r} (M, M^{j} u)\) and \(\overline{r} (M, M^{j} u)\) requires the comparison of several quasiderivatives of \(M^{j} u\) and \(M^{j+1} u\) (those of order \(l \in S\)) across the interval \([a,b]\). When \(M^{j} u\) is calculated numerically, this is not really possible. However, in [24, Sect. 2.4] several mechanisms were shared which allow reducing the comparison of functions to one point (in some cases) or to a finite set of points. These mechanisms [24, Theorems 14 and 15] are also applicable to this case.
For the concrete case \(m=0\), if the number of boundary conditions set on the quasiderivative \(L_{0}\) is one and, in the case \(q \equiv 0\), \(i_{\max } >1\), it is straightforward to show that one can reduce the comparison of \(M^{j} u\) and u to a single point (a or b, since \(M^{j} u\) is monotonic and u can be picked up so that it is monotonic), as proposed in [24, Theorem 14]. However, this is not possible if there are two boundary conditions on the quasiderivative \(L_{0}\).
For this latter case, nevertheless, it is worth considering the following theorem.
Theorem 7
Let u be a solution of \(L u =f\) with L as in (1), poised homogeneous boundary conditions \(\{\alpha , \beta \}\) and \(f \in PC[a,b]\) such that \(f \geq 0\) on \([a,b]\) and \(f > 0\) on a subset of \([a,b]\) of positive measure. Then the number of zeroes of \(L_{i} u\) on \((a,b)\) is exactly \(Z_{i1} \{\alpha , \beta \} i\) for \(i < i_{\max }\).
Proof
Extending the definitions of \(E_{i}[a,b]\), \(Z_{i} \{\alpha , \beta \}\), and \(z_{i}(a,b)\) from \(G(x,t)\) to u, and following the same steps of Lemma 1, one gets to \(E_{n1} [a,b] \geq E_{i}[a,b] \geq 0\) for \(i=0, \ldots , n2\). If \(E_{n1} [a,b] > 0\), by Rolle’s theorem there must be a change of sign of \(L u = L_{n} u =f\) in \((a,b)\), which contradicts the hypothesis \(f \geq 0\). Therefore \(E_{n1} [a,b] = E_{i} [a,b] = 0\) and \(z_{i}(a,b)= Z_{i1} \{\alpha , \beta \} i\). □
As a result, if \((1)^{nk} z(x) \geq 0\), \(x \in [a,b]\) with \((1)^{nk} z(x) > 0\) in a subset of \([a,b]\) of positive measure, and \(Z_{0} \{\alpha , \beta \} =2\), given that \((1)^{nk} p(x) >0\) a.e. on \([a,b]\) and \(i_{\max } > 1\) (the total number of boundary conditions is n and two of them are set on \(L_{0} z\)), the previous theorem shows that the number of zeroes of \(L_{1} M z\) on \((a,b)\) is exactly one, so that \((1)^{nk} L_{0} Mz\) has only one maximum on that interval. This allows extending [24, Theorem 14] to the case \(Z_{0} \{\alpha , \beta \} =2\) by means of the following theorems.
Theorem 8
Let \(z(x)= (1)^{nk}\), \(x \in [a,b]\). If \(m=0\) and \(Z_{0} \{\alpha , \beta \} =2\), then the calculation of \(\overline{r} (M^{j}, z)\) for \(j \geq 1\), can be restricted to a single point, and these numbers satisfy \(\overline{r} (M^{j}, z) \geq \frac{1}{\lambda _{0}^{j}}\) and \(\lim_{j \rightarrow \infty } (\overline{r} (M^{j}, z) )^{\frac{1}{j}}= \frac{1}{\lambda _{0}}\).
Proof
Since z is constant, in order to calculate \(\overline{r} (M^{j}, z)\) it suffices to determine \(\sup \{ M^{j} z(x), x \in [a,b] \}\). The latter assertions are a result of [24, Theorems 7 and 8] and Theorem 5. □
Theorem 9
Let \(c_{1}\), \(c_{2}\) be such that \(a < c_{1} < c_{2} < b\), and let z be defined by
If \(m=0\) and \(Z_{0} \{\alpha , \beta \} =2\), then the calculation of \(\underline{r} (M^{j}, z)\) for \(j \geq 1\) can be restricted to the points \(c_{1}\) and \(c_{2}\). In addition, \(\underline{r} (M^{j}, z) \leq \frac{1}{\lambda _{0}^{j}}\) and \(\lim_{j \rightarrow \infty } (\underline{r} (M^{j}, z) )^{\frac{1}{j}}= \frac{1}{\lambda _{0}}\).
Proof
According to Theorem 7, \((1)^{nk} M^{j} z(x)\) must have a single maximum on \((a,b)\). Therefore, if \((1)^{nk} M^{j} z(c_{1}) \geq r^{j}\) and \((1)^{nk} M^{j} z(c_{2}) \geq r^{j}\), then \(M^{j} z \geq r^{j} z\). The latter assertions are also a result of [24, Theorems 7 and 8] and Theorem 5. □
5 Example
Let us consider the problem
with \(n=4\) and
The boundary conditions of (48) are not poised, so the smallest eigenvalue is \(\lambda =0\). We can apply Theorem 6 to transform the problem into a poised one and use the method of Theorem 5 to estimate the second smallest eigenvalue \(\lambda _{0}\). To do so we must first identify the index s for which the problem is spoised by means of equation (47).
Table 1 shows that the problem is 1, 2, and 3poised, and we can select any of these indexes for our purposes. A close examination shows that \(s=3\) provides a problem whose Green function is easy to calculate, namely
with
Therefore we have to determine the Green function of the problem
which is
Accordingly, the operator M can be calculated as
Note that the representation of Mz in (52), with the variable x outside of the integrals, reduces the memory usage of the algorithm in the case that Mz has to be calculated numerically, e.g., with Simpson’s rule for the integrals, as it allows dealing only with vectors instead of matrices.
Starting with the function \(z(x) =1\), \(x \in [1,2]\), it is possible to calculate \(M^{j} z\) for different indexes j, in this case from \(j=1\) to \(j=5\), and determine the corresponding Collatz–Wielandt numbers, which are bounds of the inverse of the nonzero smallest eigenvalue of (48). These bounds can be seen in Table 2 and yield the estimation \(\lambda _{0}=163.364\).
6 Conclusions
The procedure presented in this paper allows estimating the principal eigenvalue and associated eigenfunction of problems of type (2) as long as p and q satisfy some sign conditions and the boundary conditions are poised. It also allows estimating the second smallest eigenvalue and associated eigenfunction of that type of problems when \(m=0\) and \(q \equiv 0\), regardless of the boundary conditions.
The main limitations it presents are related to the sign requirements of p and q, and also to the possible difficulties of finding the Green function \(G(x,t)\). This mainly depends on the complexity of the differential operator L, but it is worth highlighting that in the case \(q \equiv 0\) this calculation can be done numerically in a quite straightforward manner as the Green function is just a solution of \(L_{n} G(x,t)=0\) subject to the boundary conditions \(\{\alpha , \beta \}\) and with a discontinuity jump at \(x=t\), and such a differential equation can be solved recursively by taking (1) into account.
The calculation of the Collatz–Wielandt numbers can be complex in some cases as they require comparing several quasiderivatives of two functions throughout the interval \([a,b]\). Nevertheless, as Sect. 4 points out, there exist mechanisms to reduce the comparison to a finite set of points.
Availability of data and materials
Not applicable.
References
Pólya, G.: On the mean value theorem corresponding to a given linear homogeneous differential operator. Trans. Am. Math. Soc. 24, 312–324 (1924)
Elias, U.: Oscillation Theory of TwoTerm Differential Equations. Kluwer Academic, Dordrecht (1997)
Ahmad, S., Lazer, A.C.: On nth order Sturmian theory. J. Differ. Equ. 35, 87–112 (1980)
Butler, G.J., Erbe, L.H.: Integral comparison theorems and extremal points for linear differential equations. J. Differ. Equ. 47, 214–226 (1983)
Elias, U.: Eigenvalue problems for the equation \(ly + \lambda p(x) y =0\). J. Differ. Equ. 29, 28–57 (1978)
Gaudenzi, M.: On an eigenvalue problem of Ahmad and Lazer for ordinary differential equations. Proc. Am. Math. Soc. 99(2), 237–243 (1987)
Joseph, D.D.: Stability of Fluid Motions I. Springer, Berlin (1976)
Erbe, L.H.: Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems. Math. Comput. Model. 32(5–6), 529–539 (2000)
Webb, J.R.L., Lan, K.Q.: Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topol. Methods Nonlinear Anal. 27, 91–116 (2006)
Lan, K.Q.: Eigenvalues of semipositone Hammerstein integral equations and applications to boundary value problems. Nonlinear Anal. 71(12), 5979–5993 (2009)
Webb, J.R.L.: A class of positive linear operators and applications to nonlinear boundary value problems. Topol. Methods Nonlinear Anal. 39, 221–242 (2012)
Ciancaruso, F.: Existence of solutions of semilinear systems with gradient dependence via eigenvalue criteria. J. Math. Anal. Appl. 482(1), 123547 (2020). https://doi.org/10.1016/j.jmaa.2019.123547
Forster, K.H., Nagy, B.: On the Collatz–Wielandt numbers and the local spectral radius of a nonnegative operator. Linear Algebra Appl. 120, 193–205 (1989)
Collatz, L.: Einschliessungssatze fur charakteristische zhalen von matrizen. Math. Z. 48, 221–226 (1942)
Wielandt, H.: Unzerlegbare, nicht negative matrizen. Math. Z. 52, 642–648 (1950)
Marek, I., Varga, R.S.: Nested bounds for the spectral radius. Numer. Math. 14, 49–70 (1969)
Marek, I.: Frobenius theory of positive operators: comparison theorems and applications. SIAM J. Appl. Math. 19, 607–628 (1970)
Marek, I.: Collatz–Wielandt numbers in general partially ordered spaces. Linear Algebra Appl. 173, 165–180 (1992)
Akian, M., Gaubert, S., Nussbaum, R.: A Collatz–Wielandt characterization of the spectral radius of orderpreserving homogeneous maps on cones, 1–24 (2014). arXiv:1112.5968v1 [math.FA]
Chang, K.C.: Nonlinear extensions of the Perron–Frobenius theorem and the Krein–Rutman theorem. J. Fixed Point Theory Appl. 15, 433–457 (2014)
Thieme, H.R.: Spectral radii and Collatz–Wielandt numbers for homogeneous orderpreserving maps and the monotone companion norm. In: de Jeu, M., de Pagter, O.V.G.B., Veraar, M. (eds.) Ordered Structures and Applications. Trends in Mathematics, pp. 415–467 (2016)
Chang, K.C., Wang, X., Wu, X.: On the spectral theory of positive operators and PDE applications. Discrete Contin. Dyn. Syst. 40(6), 3171–3200 (2020). https://doi.org/10.3934/dcds.2020054
Webb, J.R.L.: Estimates of eigenvalues of linear operators associated with nonlinear boundary value problems. Dyn. Syst. Appl. 23, 415–430 (2014)
Almenar, P., Jódar, L.: Estimation of the smallest eigenvalue of an nth order linear boundary value problem. Math. Methods Appl. Sci. 44, 4491–4514 (2021). https://doi.org/10.1002/mma.7047
Diaz, G.: Applications of cone theory to boundary value problems. PhD thesis, University of NebraskaLincoln (1989)
Acknowledgements
The authors wish to thank Professor Jeffrey R. L. Webb for bringing into their attention existing research about the Collatz–Wielandt numbers.
Funding
No funding was needed for this article.
Author information
Authors and Affiliations
Contributions
PA proposed the problem. Both authors discussed and agreed on the mechanisms and tools to find a solution. PA focused on the properties of the Green functions and LJ on the nature of the cones to use. PA wrote the first version of the manuscript and LJ reviewed and corrected several points. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Almenar, P., Jódar, L. The principal eigenvalue of some nth order linear boundary value problems. Bound Value Probl 2021, 84 (2021). https://doi.org/10.1186/s13661021015612
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661021015612