Estimates on the Lower Bound of the Eigenvalue of the Smallest Modulus Associated with a General Weighted Sturm-Liouville Problem

Abstract

We obtain a lower bound on the eigenvalue of smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. The main motivation for this study is the result obtained by Mingarelli (1988).

1. Introduction

In this paper, we derive some lower bound for an eigenvalue of the smallest modulus (not necessarily unique) corresponding to the problemconsolidating the results obtained by Mingarelli [1]. In the paper a lower bound for the eigenvalue of the smallest modulus was obtained under the assumptions that , a.e. on , and that and take on both positive and negative values on . The parameter is called an eigenvalue and corresponding function not identically zero on is called an eigenfunction. The eigenfunctions corresponding to real eigenvalues can be labeled in such a way that and has zeros in , . Therefore the eigenvalue of the smallest modulus will be labeled and the corresponding eigenfunction has no zero in . In this paper we assume in general that is not an eigenvalue. Our studies here are an extension of the results in [1] to the case where the assumption on the coefficient function is replaced by the more general assumption that . We use the Fredholm integral operator associated with (1)-(2) in the Hilbert space and use the estimates on its norm and the -norms of solutions of a Cauchy problem related to (1)-(2). We consider the general weighted Sturm-Liouville problem which is the case in which the coefficient functions and have no sign restrictions imposed on them. The weight function plays a critical role in the nature of In the next subsection we give a brief outline of the three different cases that arise as one varies the signs of the weight function and the coefficient function

(1) Preliminary Results. When and takes on both positive and negative values on the interval , problem (1)-(2) is right definite and only one sequence of real eigenvalues exists with as . For more information on this case, we refer the interested reader to [2–4] and so forth and the references within. Particularly, Everitt et al. [2, theorem  0] outline oscillation properties of eigenfunctions of the right definite problem. In the right definite case is the smallest eigenvalue with the corresponding eigenfunction having no zero in the interval .

When a.e. and takes on both signs on sets of positive Lebesgue measure, the problem is left definite. For more details see, for example, [4–6] and the references therein. In this case two sequences of real eigenvalues exist where , one positive and the other negative, and is then the first positive (and or negative) eigenvalue (whose eigenfunction is positive on ) labeled according to a Sturm oscillation theorem.

When a.e and changes sign, problem (1)-(2) is nondefinite (or indefinite). In this case, nonreal eigenvalues may exist and so may be either real or nonreal. In the nondefinite case, if is real, then the corresponding eigenfunction can have any number of zeros on the interval in contrast with the other two cases. There is a lot of literature covering this case (see, e.g., [4, 7, 8] and references therein). More studies are carried out on existence and estimation of nonreal eigenvalues in the recent papers [9–13] and so forth. In some of these papers upper and lower bounds on the nonreal eigenvalues of indefinite Sturm-Liouville problems are estimated and the existence of nonreal eigenvalues is discussed in others.

The discreteness of the spectrum for problem (1)-(2) and the assumption that is not an eigenvalue of the problem guarantees the existence of such an eigenvalue . Moreover, this eigenvalue is not unique, since there exist problems where the real spectrum is symmetric about zero in the left definite case. In fact even when nonreal eigenvalues exist, they appear in conjugate pairs; hence the nonreal spectrum is symmetric about the -axis.

The assumption that is not an eigenvalue of problem (1)-(2) is equivalent to the fact that the problem consisting of the equationand the boundary conditions (2) admits a unique Green’s function, , defined and continuous over . We define the inner product on byand the associated norm is given by

Green’s function for problem (3)-(2) takes the formwhere are (real) linearly independent solutions of (3) satisfying the initial conditionsrespectively. In a lot of literature (see. e.g.. [14]) it is shown that the spectral problem (1)-(2) can be reduced to a single eigenvalue equationwhere the operator is defined byon Here is Green’s function associated with problem (1)-(2), , and . That is, is an eigenfunction of problem (1)-(2) if and only if is an eigenfunction of associated with the eigenvalue .

2. The Main Results

We start by stating two important lemmas from [1] whose results we will use in proving the results in this section. We present the lemmas without proof and refer the interested reader to the cited paper.

Lemma 1. The linear operator defined by (10) maps into and is a bounded (compact but not necessarily self-adjoint) operator on whose operator norm, , is given by

In the same paper it is also shown that an eigenvalue of problem (1)-(2) having the smallest modulus admits the lower bound

Lemma 2. An eigenvalue of (1)-(2) of the smallest modulus satisfieswhere are given in (6), (7), and (8), and are their respective -norms.

We now give a variant of lemma  4 in [1].

Lemma 3. Let denote the solution of the Cauchy problemThen for fixed in ,Similarly, if denotes the solution of (14) satisfyingthen for fixed in ,

Proof. To prove (17), we use the integral representation of (14)-(15) which is given byFrom this we get the Neumann series expansion of which is given by the terms below. Generally, whereIf similar working as above yields the upper bound Therefore we have that and the series converges, so the Neumann series converges too. This yields that and the bound in (17) is established.
Similarly, the integral representation of (14)–(18) is given byThe corresponding Neumann series is given by the terms whereHence, which yields that and the bound in (19) is established.

The results in Lemma 3 lead to the estimation of the -norm of the solutions , which leads to the following result.

Lemma 4. Let denote the two linearly independent solutions of (14) satisfying (15) and (18), respectively. Then

Proof. We prove the results by calculating the -norms of and , and use the bounds in Lemma 3 as shown below. Taking the supremum on in the inequality yields the bound in (33). A similar procedure yields (34).

Applying the results in Lemma 4 to the lower bound in (13), we get the main result which is a variant of theorem  1 in [1].

Theorem 5. Let . Then for problem (1)-(2), an eigenvalue of the smallest modulus may be estimated bywhere is the solution of the Cauchy problem (14)-(15) evaluated at .

2.1. Examples

Here we give examples to verify if the inequality in (36) really holds. Without loss of generality, we consider the case where on the interval In this case, (36) becomeswhere . The eigenvalues are found using the package RootFinding(Analytic).

Example 1 (the case and changes sign). We consider the problemswhereThe solution to the problem in (38) is and so Substituting in (37) yields that , and solving problem (39) gives the spectrum to be From the spectrum we see that is not unique in this particular case since and all give the smallest modulus which is . Therefore, and (37) is satisfied.

Example 2 (the case and changes sign). We consider the problems where is as given in Example 1.
The solution to the problem in (43) is and so Substituting in (37) yields that , and solving problem (44) gives the spectrum to be From the spectrum we see that and (37) is satisfied.

Example 3 (the case and ). Here we consider problem (43) and the problemSolving problem (46) gives the spectrum to be and clearly, (37) is satisfied since .

3. Conclusion

In this paper, we undertook a study that consolidates results obtained in [1] where a lower bound for an eigenvalue of smallest modulus is obtained under the assumption that . In this paper, we have considered the general case where in which we obtain different bounds for the eigenvalue . As can be seen from Examples 1, 2, and 3 the result holds.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this article.

Acknowledgments

The authors wish to thank Professor L-E Persson (Luleå University of Technology) for reading through the manuscript. They also wish to thank the Division of Mathematical Sciences and Statistics, Luleå University of Technology, Sweden, for the financial support during the research visit of the first author. Furthermore, they wish to thank the International Science Program based at Uppsala University, Sweden, for funding the Ph.D. studies and financial assistance towards the research visits by the first author. This research is supported by International Science Programme in Mathematical Sciences, Uppsala University, Sweden, and the Department of Engineering Sciences and Mathematics, Luleå University of Technology.