×

Discontinuous Sturm-Liouville problems containing eigenparameter in the boundary conditions. (English) Zbl 1112.34070

The authors consider the following discontinuous Sturm-Liouville eigenvalue problems with eigenvalue parameters both in the equation and in one of the boundary conditions: \[ \tau u := -a(x)u^{\prime \prime} +q(x)u = \lambda u, x \in [-1, 0) \cup (0, 1] \]
with the boundary conditions at \(x = \pm 1\) \[ L_1u := \alpha_1 u(-1) + \alpha_2 u'(-1) = 0; L_4 (\lambda)u := \lambda(\beta_1'u(1) - \beta_2'u'(1)) + ((\beta_1u(1) - \beta_2u'(1)) = 0, \] and the transition conditions
\[ L_2u := \gamma_1u(-0) - \delta_1u(+0) = 0, L_3u := \gamma_2u'(-0) - \delta_2u'(+0) = 0, \]
where \(a(x) = a_1^2 > 0\) for \(x \in [-1, 0)\), \(a(x) = a_2^2 > 0\) for \(x \in (0, 1]\); \(q(x)\) is a given real-valued function which is continuous in \([-1, 0]\) and in \([0, 1]\); \(\alpha_i, \beta_i, \beta_i', \gamma_i, \delta_i, i = 1, 2\), are real numbers with \(| \alpha_1 | + | \alpha_2 | \not = 0, | \beta_1' | + | \beta_2' | + | \beta_1 | + | \beta_2 | \not = 0, | \gamma_1 | + | \delta_1 | \not = 0, | \gamma_2 | + | \delta_2 | \not = 0\), and \(\beta_1' \beta_2 - \beta_1 \beta_2' > 0\). With an operator theoretic approach, some classical properties and asymptotic approximate formulae for eigenvalues and normalized eigenfunctions are obtained.

MSC:

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34B24 Sturm-Liouville theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fulton, C. T.: Two–point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edin., 77(A), 293–308 (1977) · Zbl 0376.34008
[2] Birkhoff, G. D.: On the asymptotic character of the solution of the certain linear differential equations containing parameter. Trans. Amer. Soc., 9, 219–231 (1908) · JFM 39.0386.01 · doi:10.1090/S0002-9947-1908-1500810-1
[3] Hinton, D. B.: An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition. Quart. J. Math. Oxford., 30, 33–42 (1979) · Zbl 0427.34023 · doi:10.1093/qmath/30.1.33
[4] Schneider, A.: A note on eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z., 136, 163–167 (1974) · Zbl 0308.34023 · doi:10.1007/BF01214350
[5] Shkalikov, A. A.: Boundary value problems for ordinary differential equations with a parameter in boundary condition. Trudy Sem. Imeny I. G. Petrowsgo, 9, 190–229 (1983) · Zbl 0553.34014
[6] Walter, J.: Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z., 133, 301–312(1973) · Zbl 0259.47046 · doi:10.1007/BF01177870
[7] Yakubov, S.: Completeness of Root Functions of Regular Differential Operators, Longman, Scientific Technical, New York, 1994 · Zbl 0833.34081
[8] Yakubov, S., Yakubov, Y.: Abel basis of root functions of regular boundary value problems. Math. Nachr., 197, 157–187 (1999) · Zbl 0922.34069 · doi:10.1002/mana.19991970110
[9] Yakubov, S., Yakubov, Y.: Differential–Operator Equations, Ordinary and Partial Differential Equations, Chapman and Hall/CRC, Boca Raton, 568, 2000 · Zbl 0936.35002
[10] Titchmarsh, E. C.: Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Press, London, 1962 · Zbl 0099.05201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.