Two notions which affected nonlinear analysis (Bernard Bolzano lecture)

Pavel Drábek

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 4, page 699-711
  • ISSN: 0862-7959

Abstract

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General mathematical theories usually originate from the investigation of particular problems and notions which could not be handled by available tools and methods. The Fučík spectrum and the p -Laplacian are typical examples in the field of nonlinear analysis. The systematic study of these notions during the last four decades led to several interesting and surprising results and revealed deep relationship between the linear and the nonlinear structures. This paper does not provide a complete survey. We focus on some pioneering works and present some contributions of the author. From this point of view the list of references is by no means exhaustive.

How to cite

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Drábek, Pavel. "Two notions which affected nonlinear analysis (Bernard Bolzano lecture)." Mathematica Bohemica 139.4 (2014): 699-711. <http://eudml.org/doc/269858>.

@article{Drábek2014,
abstract = {General mathematical theories usually originate from the investigation of particular problems and notions which could not be handled by available tools and methods. The Fučík spectrum and the $p$-Laplacian are typical examples in the field of nonlinear analysis. The systematic study of these notions during the last four decades led to several interesting and surprising results and revealed deep relationship between the linear and the nonlinear structures. This paper does not provide a complete survey. We focus on some pioneering works and present some contributions of the author. From this point of view the list of references is by no means exhaustive.},
author = {Drábek, Pavel},
journal = {Mathematica Bohemica},
keywords = {Fučík spectrum; $p$-Laplacian; Fučík spectrum; -Laplacian},
language = {eng},
number = {4},
pages = {699-711},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Two notions which affected nonlinear analysis (Bernard Bolzano lecture)},
url = {http://eudml.org/doc/269858},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Drábek, Pavel
TI - Two notions which affected nonlinear analysis (Bernard Bolzano lecture)
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 4
SP - 699
EP - 711
AB - General mathematical theories usually originate from the investigation of particular problems and notions which could not be handled by available tools and methods. The Fučík spectrum and the $p$-Laplacian are typical examples in the field of nonlinear analysis. The systematic study of these notions during the last four decades led to several interesting and surprising results and revealed deep relationship between the linear and the nonlinear structures. This paper does not provide a complete survey. We focus on some pioneering works and present some contributions of the author. From this point of view the list of references is by no means exhaustive.
LA - eng
KW - Fučík spectrum; $p$-Laplacian; Fučík spectrum; -Laplacian
UR - http://eudml.org/doc/269858
ER -

References

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  1. Anane, A., Simplicity and isolation of first eigenvalue of the p -Laplacian with weight, C. R. Acad. Sci., Paris, Sér. I 305 French (1987), 725-728. (1987) Zbl0633.35061MR0920052
  2. Anane, A., Tsouli, N., On the second eigenvalue of the p -Laplacian, Nonlinear Partial Differential Equations. Based on the International Conference on Nonlinear Analysis, Fés, Morocco, 1994 Pitman Res. Notes Math. Ser. 343 Longman, Harlow (1996), 1-9 A. Benkirane et al. (1996) Zbl0854.35081MR1417265
  3. Berkovits, J., Drábek, P., Leinfelder, H., Mustonen, V., Tajčová, G., Time-periodic oscillations in suspension bridges: Existence of unique solutions, Nonlinear Anal., Real World Appl. 1 (2000), 345-362. (2000) Zbl0989.74031MR1791531
  4. Binding, P. A., Drábek, P., Huang, Y. X., 10.1090/S0002-9939-97-03992-0, Proc. Am. Math. Soc. 125 (1997), 3555-3559. (1997) Zbl0882.35049MR1416077DOI10.1090/S0002-9939-97-03992-0
  5. Boccardo, L., Drábek, P., Giachetti, D., Kučera, M., 10.1016/0362-546X(86)90091-X, Nonlinear Anal., Theory Methods Appl. 10 (1986), 1083-1103. (1986) Zbl0623.34031MR0857742DOI10.1016/0362-546X(86)90091-X
  6. Dancer, E. N., 10.1017/S0004972700022747, Bull. Aust. Math. Soc. 15 (1976), 321-328. (1976) Zbl0342.34007MR0430384DOI10.1017/S0004972700022747
  7. Pino, M. del, Drábek, P., Manásevich, R., 10.1006/jdeq.1998.3506, J. Differ. Equations 151 (1999), 386-419. (1999) MR1669705DOI10.1006/jdeq.1998.3506
  8. Pino, M. A. del, Manásevich, R. F., 10.1016/0022-0396(91)90048-E, J. Differ. Equations 92 (1991), 226-251. (1991) MR1120904DOI10.1016/0022-0396(91)90048-E
  9. Drábek, P., Geometry of the energy functional and the Fredholm alternative for the p -Laplacian in higher dimensions, Proceedings of the 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, Electron. J. Differ. Equ. (electronic only) 8 (2002), 103-120. (2002) Zbl1114.35318MR1990298
  10. Drábek, P., Solvability and Bifurcations of Nonlinear Equations, Pitman Research Notes in Mathematics Series 264 Longman Scientific, Harlow; John Wiley, New York (1992). (1992) Zbl0753.34002MR1175397
  11. Drábek, P., On the global bifurcation for a class of degenerate equations, Ann. Mat. Pura Appl. (4) 159 (1991), 1-16. (1991) Zbl0814.34018MR1145086
  12. Drábek, P., 10.1016/0022-247X(87)90121-1, J. Math. Anal. Appl. 127 (1987), 435-442. (1987) Zbl0642.34009MR0915069DOI10.1016/0022-247X(87)90121-1
  13. Drábek, P., Ranges of homogeneous operators and their perturbations, Čas. Pěst. Mat. 105 (1980), 167-183. (1980) Zbl0427.47048MR0573109
  14. Drábek, P., Girg, P., Takáč, P., Ulm, M., 10.1512/iumj.2004.53.2396, Indiana Univ. Math. J. 53 (2004), 433-482. (2004) Zbl1081.35031MR2060041DOI10.1512/iumj.2004.53.2396
  15. Drábek, P., Holubová, G., 10.12775/TMNA.1999.021, Topol. Methods Nonlinear Anal. 14 (1999), 39-58. (1999) Zbl0967.35013MR1758879DOI10.12775/TMNA.1999.021
  16. Drábek, P., Holubová, G., Matas, A., Nečesal, P., 10.1023/B:APOM.0000024489.96314.7f, Appl. Math., Praha 48 (2003), 497-514. (2003) MR2025959DOI10.1023/B:APOM.0000024489.96314.7f
  17. Drábek, P., Kufner, A., 10.1090/S0002-9939-05-07958-X, Proc. Am. Math. Soc. 134 (2006), 235-242. (2006) Zbl1090.34066MR2170563DOI10.1090/S0002-9939-05-07958-X
  18. Drábek, P., Kufner, A., Nicolosi, F., Quasilinear Elliptic Equations with Degenerations and Singularities, De Gruyter Series in Nonlinear Analysis and Applications 5 Walter de Gruyter, Berlin (1997). (1997) Zbl0894.35002MR1460729
  19. Drábek, P., Kuliev, K., 10.36045/bbms/1331153412, Bull. Belg. Math. Soc. -- Simon Stevin 19 (2012), 107-119. (2012) Zbl1252.34034MR2952799DOI10.36045/bbms/1331153412
  20. Drábek, P., Robinson, S. B., 10.1016/j.jmaa.2014.04.019, J. Math. Anal. Appl. 418 (2014), 884-905. (2014) MR3206687DOI10.1016/j.jmaa.2014.04.019
  21. Drábek, P., Robinson, S. B., On the Fredholm alternative for the Fučík spectrum, Abstr. Appl. Anal. 2010 (2010), Article ID 125464, 20 pages. (2010) Zbl1214.47011MR2754193
  22. Drábek, P., Robinson, S. B., 10.1006/jdeq.2001.4070, J. Differ. Equations 181 (2002), 58-71. (2002) Zbl1163.35449MR1900460DOI10.1006/jdeq.2001.4070
  23. Drábek, P., Robinson, S. B., 10.1006/jfan.1999.3501, J. Funct. Anal. 169 (1999), Article No. jfan.1999.3501, 189-200. (1999) Zbl0940.35087MR1726752DOI10.1006/jfan.1999.3501
  24. Elbert, A., A half-linear second order differential equation, Qualitative Theory of Differential Equations, Vol. I, Szeged, 1979 Colloq. Math. Soc. János Bolyai 30 North-Holland, Amsterdam (1981), 153-180 M. Farkas. (1981) Zbl0511.34006MR0680591
  25. Fučík, S., Solvability of Nonlinear Equations and Boundary Value Problems, Mathematics and Its Applications 4 D. Reidel Publishing, Dordrecht (1980). (1980) MR0620638
  26. Fučík, S., Boundary value problems with jumping nonlinearities, Čas. Pěst. Mat. 101 (1976), 69-87. (1976) Zbl0332.34016MR0447688
  27. Fučík, S., Nečas, J., Souček, J., Souček, V., 10.1007/BFb0059360, Lecture Notes in Mathematics 346 Springer, Berlin (1973). (1973) Zbl0268.47056MR0467421DOI10.1007/BFb0059360
  28. Kováčik, O., Rákosník, J., On spaces L p ( x ) and W k , p ( x ) , Czech. Math. J. 41 (1991), 592-618. (1991) MR1134951
  29. Krejčí, P., On solvability of equations of the 4th order with jumping nonlinearities, Čas. Pěst. Mat. 108 (1983), 29-39. (1983) Zbl0515.35013MR0694138
  30. Landesman, E. M., Lazer, A. C., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609-623. (1970) Zbl0193.39203MR0267269
  31. Lazer, A. C., McKenna, P. J., 10.1137/1032120, SIAM Rev. 32 (1990), 537-578. (1990) Zbl0725.73057MR1084570DOI10.1137/1032120
  32. Růžička, M., 10.1007/BFb0104030, Lecture Notes in Mathematics 1748 Springer, Berlin (2000). (2000) Zbl0968.76531MR1810360DOI10.1007/BFb0104030
  33. Švarc, R., The solution of a Fučík's conjecture, Commentat. Math. Univ. Carol. 25 (1984), 483-517. (1984) Zbl0562.47049MR0775566

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