About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Appell, Jürgen
A la recherche du spectre perdu: An invitation to nonlinear spectral theory. (English). In: Opic, Bohumír and Rákosník, Jiří (eds.): Nonlinear Analysis, Function Spaces and Applications, Proceedings of the Spring School held in Prague, July 17-22, 2002. Vol. 7. Czech Academy of Sciences, Mathematical Institute, Praha, 2003. pp. 1-20
MSC: 35J60, 47A10, 47H09, 47H10, 47J05, 47J10, 47J25
Keywords:
Nonlinear spectrum; nonlinear eigenvalue problem; homogeneous operator; coincidence theorem; discreteness theorem; nonlinear Fredholm alternative; $p$-Laplace operator
Summary:
We give a survey on spectra for various classes of nonlinear operators, with a particular emphasis on a comparison of their advantages and drawbacks. Here the most useful spectra are the asymptotic spectrum by M. Furi, M. Martelli and A. Vignoli (1978), the global spectrum by W. Feng (1997), and the local spectrum (called “phantom”) by P. Santucci and M. Väth (2000). In the last part we discuss these spectra for homogeneous operators (of any degree), and derive a discreteness result and a nonlinear Fredholm alternative for such operators. This may be applied to an eigenvalue problem for the $p$-Laplace operator which arises in various fields of applied mathematics, mechanics, and physics.
References:
[1] Appell J., Pascale E. De, Vignoli A.: A comparison of different spectra for nonlinear operators. Nonlinear Anal., Theory Methods Appl. 40A (2000), 73–90. Zbl 0956.47035, MR 2001g:47117. Zbl 0956.47035
[2] Appell J., Dörfner M.: Some spectral theory for nonlinear operators. Nonlinear Anal., Theory Methods Appl. 28 (1997), 1955–1976. Zbl 0876.47042, MR 98e:47098. MR 1436365 | Zbl 0876.47042
[3] Appell J., Giorgieri E., Väth M.: Nonlinear spectral theory for homogeneous operators. Nonlinear Funct. Anal. Appl. 7 (2002), 589–618. MR 1959638 | Zbl 1045.47053
[4] Binding P. A., Drábek P., Huang Y. X.: On the Fredholm alternative for the $p$-Laplacian. Proc. Amer. Math. Soc. 125 (1997), 3555–3559. Zbl 0882.35049, MR 98b:35058. MR 1416077 | Zbl 0882.35049
[5] Pino M. del, Drábek P., Manásievich R.: The Fredholm alternative at the first eigenvalue for the one dimensional $p$-Laplacian. J. Differ. Equations 151 (1999), 386–419. Zbl 0931.34065, MR 99m:34042. MR 1669705
[6] Drábek P.: On the Fredholm alternative for nonlinear homogeneous operators. In: Applied nonlinear analysis (A. Sequeira et al., eds.). Kluwer Academic/Plenum Publishing, New York, 1999, 41–48. Zbl 0956.47036, MR 1 727 439. MR 1727439 | Zbl 0956.47036
[7] Drábek P.: Analogy of the Fredholm alternative for nonlinear operators. RIMS Kokyuroku 1105 (1999), 31–38. Zbl 0951.47503. MR 1747554 | Zbl 0951.47503
[8] Drábek P.: Fredholm alternative for the $p$-Laplacian: yes or no?. In: Function Spaces, Differential Operators and Nonlinear Analysis. Proceedings of the conference, Syöte, Finland, June 10–16, 1999 (V. Mustonen and J. Rákosník, eds.). Math. Inst. Acad. Sci. Czech Rep., Prague, 2000, 57–64. Zbl 0966.34012, MR 2000m:34042. MR 1755297
[9] Drábek P., Girg P., Manásievich R.: Generic Fredholm alternative-type results for the one dimensional $p$-Laplacian. Nonlinear Differential Equations Appl. 8 (2001), 285–298. Zbl pre01652489, MR 2002f:34027. MR 1841260
[10] Drábek P., Holubová G.: Fredholm alternative for the $p$-Laplacian in higher dimensions. J. Math. Anal. Appl. 263 (2001), 182–194. Zbl 1002.35046,MR 2002h:35083. MR 1864314
[11] Feng W.: A new spectral theory for nonlinear operators and its applications. Abstr. Appl. Anal. 2 (1997), 163–183. Zbl 0952.47047, MR 99d:47061. MR 1604177 | Zbl 0952.47047
[12] Fučík S.: Fredholm alternative for nonlinear operators in Banach spaces and its applications to differential and integral equations. Commentat. Math. Univ. Carol. 11 (1970), 271–284. Zbl 0995.42801, MR 42 #909. MR 0266000
[13] Furi M.: Stably solvable maps are unstable under small perturbations. Z. Anal. Anwend. 21 (2002), 203–208. Zbl pre01779543, MR 1 916 412. MR 1916412 | Zbl 1016.47042
[14] Furi M., Martelli M., Vignoli A.: Stably solvable operators in Banach spaces. Atti Accad. Naz. Lincei, VIII. Ser., Rend. Cl. Sci. Fis. Mat. Nat. 60 (1976), 21–26. Zbl 0361.47024, MR 58 #7251. MR 0487632 | Zbl 0361.47024
[15] Furi M., Martelli M., Vignoli A.: Contributions to the spectral theory for nonlinear operators in Banach spaces. Ann. Mat. Pura Appl., IV. Ser. 118 (1978), 229–294. Zbl 0409.47043, MR 80k:47070. MR 0533609 | Zbl 0409.47043
[16] Furi M., Martelli M., Vignoli A.: On the solvability of nonlinear operator equations in normed spaces. Ann. Mat. Pura Appl., IV. Ser. 128 (1980), 321–343. Zbl 0456.47051, MR 83h:47047. MR 0591562 | Zbl 0456.47051
[17] Kachurovskij R. I.: Regular points, spectrum and eigenfunctions of nonlinear operators. (Russian). Dokl. Akad. Nauk SSSR 188 (1969) 274–277. English transl. in Soviet Math. Dokl. 10 (1969), 1101–1105. Zbl 0197.40402. MR 0251599 | Zbl 0197.40402
[18] Maddox I. J., Wickstead A. W.: The spectrum of uniformly Lipschitz mappings. Proc. Royal Irish Acad., Sect. A 89 (1989), 101–114. Zbl 0661.47048, MR 90k:47120. MR 1021228 | Zbl 0661.47048
[19] Minty G.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29 (1962), 341–346. Zbl 0111.31202, MR 29 #6319. MR 0169064 | Zbl 0111.31202
[20] Nečas J.: Sur l’alternative de Fredholm pour les opérateurs non linéaires avec applications aux problèmes aux limites. Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 23 (1969), 331–345. Zbl 0187.08103, MR 42 #2332. MR 0267430 | Zbl 0187.08103
[21] Neuberger J. W.: Existence of a spectrum for nonlinear transformations. Pacific J. Math. 31 (1969), 157–159. Zbl 0182.47203, MR 41 #4329. MR 0259696 | Zbl 0182.47203
[22] Pokhozhaev S. I.: Solvability of nonlinear equations with odd operators. (Russian). Funkts. Anal. Prilozh. 1 (1967), 66–73. Zbl 0165.49502, MR 36 #4396. English transl. in Funct. Anal. Appl. 1 (1967), 227–233. Zbl 0165.49502. MR 0221344 | Zbl 0165.49502
[23] Rhodius A.: Über numerische Wertebereiche und Spektralwertabschätzungen. Acta Sci. Math. 47 (1984), 465–470. Zbl 0575.47005, MR 86i:47005. MR 0783322 | Zbl 0575.47005
[24] Santucci P., Väth M.: On the definition of eigenvalues for nonlinear operators. Nonlin. Anal., Theory Methods Appl. 40A (2000), 565–576. Zbl 0956.47038, MR 2001g:47118. MR 1768911 | Zbl 0956.47038
[25] Santucci P., Väth M.: Grasping the phantom: a new approach to nonlinear spectral theory. Ann. Mat. Pura Appl. 180 (2001), 255–284. MR 1871616 | Zbl 1150.47042
[26] Väth M.: The Furi-Martelli-Vignoli spectrum vs. the phantom. Nonlinear Anal., Theory Methods Appl. 47 (2001), 2237–2248. MR 1971633 | Zbl 1042.47533
Partner of
EuDML logo