Proceedings of the American Mathematical Society

Author(s): Paul Binding; Lyonell Boulton; Jan Cepicka; Pavel Drábek; Petr Girg
Journal: Proc. Amer. Math. Soc. 134 (2006), 3487-3494.
MSC (2000): Primary 34L30; Secondary 34L10, 42A65
Posted: June 27, 2006
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Abstract: For $p\geqslant \frac{12}{11}$ , the eigenfunctions of the non-linear eigenvalue problem for the

-Laplacian on the interval

are shown to form a Riesz basis of

and a Schauder basis of

whenever $1<q<\infty$ .

1.: C. BENNEWITZ, Y. SAIT¯O. ``An embedding norm and the Lindqvist trigonometric functions'', Electronic Journal of Differential Equations 86 (2002), 1-6. MR 1938382 (2003j:46028)
2.: P. BINDING, P. DRÁBEK. ``Sturm-Liouville theory for the -Laplacian'', Studia Scientiarium Mathematicarum Hungarica 40 (2003), 375 - 396. MR 2037324 (2004j:34068)
3.: M. DEL PINO, P. DRÁBEK AND R. F. MANÁSEVICH, The Fredholm alternative at the first eigenvalue for the one-dimensional -Laplacian, J. Differential Equations, 151 (1999), 386-419. MR 1669705 (99m:34042)
4.: P. DRÁBEK, P. GIRG AND R. F. MANÁSEVICH, Generic Fredholm alternative for the one dimensional -Laplacian, Nonlin. Diff. Equations and Applications 8 (2001), 285-298. MR 1841260 (2002f:34027)
5.: W. EBERHART, ´A. ELBERT, ``On the eigenvalues of a half-linear boundary value problem'', Math. Nachr. 213 (2000), 57 - 76. MR 1755246 (2001b:34035)
6.: Á. ELBERT. ``A half-linear second order differential equation''. Coll. Math. Soc. J. Bolyai 30 (1979) 153 - 179. MR 0680591 (84g:34008)
7.: Á. ELBERT, K. TAKÂSI, T. TINAGAWA, ``An Oscillatory half-linear differential equation'', Arch. Math. 33 (1997), 355 - 361. MR 1601353 (98m:34071)
8.: C. FABRY, D. FAYYAD, ``Periodic solutions of second order differential equations with a -Laplacian and asymmetric nonlinearities'', Rend. Ist. Mat. Univ. Trieste 24 (1992), 207 - 227. MR 1310080 (96b:34027)
9.: I. GOHBERG, M. KRE ${\u{\i\/}}\kern.15em$ N, Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs, Vol. 18 American Mathematical Society, 1969. MR 0246142 (39:7447)
10.: J.R. HIGGINS, Completeness and the Basis Properties of Sets of Special Functions, Cambridge Tracts in Mathematics, Vol. 72. Cambridge University Press, 1977. MR 0499341 (58:17240)
11.: T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, 1976. MR 0407617 (53:11389)
12.: P. LINDQVIST, ``Note on a nonlinear eigenvalue problem'', Rocky Mount. J. Math. 23 (1993), 281 - 288. MR 1212743 (94d:34031)
13.: P. LINDQVIST, ``Some remarkable sine and cosine functions'', Ricerche di Matematica 44 (1995), 269 - 290. MR 1469702 (99g:33001)
14.: Y. NAITO, ``Uniqueness of positive solutions of quasilinear differential equations'', Diff. Int. Equations 8 (1995), 1813 - 1822. MR 1347982 (96g:34039)
15.: M. ÔTANI, ``A Remark on certain nonlinear elliptic equations'', Proc. Fac. Sci. Tokai Univ. 19 (1984), 23 - 28. MR 0753635 (86c:35009)
16.: W. REICHEL, W. WALTER, ``Sturm-Liouville type problems for the -Laplacian under asymptotic nonresonance conditions'', J. Differential Equations 156 (1999), 50 - 70. MR 1701814 (2000e:34036)
17.: I. SINGER, Bases in Banach Spaces I, Springer-Verlag, 1970.MR 0298399 (45:7451)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34L30, 34L10, 42A65

Paul Binding
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Lyonell Boulton
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Jan Cepicka
Affiliation: Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic

Pavel Drábek
Affiliation: Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic

Petr Girg
Affiliation: Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic

DOI: 10.1090/S0002-9939-06-08001-4
PII: S 0002-9939(06)08001-4
Keywords: $p$-Laplacian eigenvalues, eigenfunction completeness.
Received by editor(s): May 5, 2004
Received by editor(s) in revised form: October 19, 2004
Posted: June 27, 2006
Additional Notes: The research of the first author was supported by I. W. Killam Foundation and NSERC of Canada
The second author was supported by a PIMS Postdoctoral Fellowship at the University of Calgary
The research of the third, fourth, and fifth authors was supported by GACR, no. 201/03/0671
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2006, American Mathematical Society

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