Geodesic
On solvability of nonlinear operator equations and eigenvalues of homogeneous operators
Mathematica Bohemica, Tome 121 (1996) no. 3, pp. 301-314

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
Notions as the numerical range $W(S,T)$ and the spectrum $\s(S,T)$ of couple $(S,T)$ of homogeneous operators on a Banach space are used to derive theorems on solvability of the equation $Sx-lTx=y.$ Conditions for the existence of eigenvalues of the couple $(S,T)$ are given.
Notions as the numerical range $W(S,T)$ and the spectrum $\s(S,T)$ of couple $(S,T)$ of homogeneous operators on a Banach space are used to derive theorems on solvability of the equation $Sx-lTx=y.$ Conditions for the existence of eigenvalues of the couple $(S,T)$ are given.
DOI : 10.21136/MB.1996.125984
Classification : 47H15, 47J05
Keywords: Banach and Hilbert space; homogeneous operator; polynomial operator; symmetric operator; monotone operator; numerical range; spectrum; eigenvalue 
Burýšková, Věra; Burýšek, Slavomír. On solvability of nonlinear operator equations and eigenvalues of homogeneous operators. Mathematica Bohemica, Tome 121 (1996) no. 3, pp. 301-314. doi: 10.21136/MB.1996.125984
@article{10_21136_MB_1996_125984,
     author = {Bur\'y\v{s}kov\'a, V\v{e}ra and Bur\'y\v{s}ek, Slavom{\'\i}r},
     title = {On solvability of nonlinear operator equations and eigenvalues of homogeneous operators},
     journal = {Mathematica Bohemica},
     pages = {301--314},
     year = {1996},
     volume = {121},
     number = {3},
     doi = {10.21136/MB.1996.125984},
     mrnumber = {1419884},
     zbl = {0863.47045},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.1996.125984/}
}
TY  - JOUR
AU  - Burýšková, Věra
AU  - Burýšek, Slavomír
TI  - On solvability of nonlinear operator equations and eigenvalues of homogeneous operators
JO  - Mathematica Bohemica
PY  - 1996
SP  - 301
EP  - 314
VL  - 121
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.1996.125984/
DO  - 10.21136/MB.1996.125984
LA  - en
ID  - 10_21136_MB_1996_125984
ER  - 
%0 Journal Article
%A Burýšková, Věra
%A Burýšek, Slavomír
%T On solvability of nonlinear operator equations and eigenvalues of homogeneous operators
%J Mathematica Bohemica
%D 1996
%P 301-314
%V 121
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.1996.125984/
%R 10.21136/MB.1996.125984
%G en
%F 10_21136_MB_1996_125984

[1] F. Bonsall B. E. Cain H. Schneider: The numerical range of continuous mapping of a normed space. Aequationes Math. 2 (1968), 86-93. | MR

[2] F. E. Browder: Problemes non-lineaires. Univ. Montreal Press, 1966. | Zbl

[3] V. Burýšková: Definition und grudlegende Eigenschaften des nichtlinearen adjungierten Operators. Časopis Pěst. Mat. 103 (1978), 186-201. | MR

[4] V. Burýšková: Adjoint nonlinear opeгators. Dissertation, Praha, 1977. (In Czech.)

[5] S. Burýšek: Some remarks on polynomial operators. Comment. Math. Univ. Carolin. 10,2 (1969), 285-306. | MR

[6] S. Burýšek: On spectra of nonlinear operators. Comment. Math. Univ. Carolin. 11,4 (1970), 727-743. | MR

[7] S. Burýšek V. Burýšková: Small solutions of a nonlineaг operator equation. Acta Polytech. Práce ČVUT Praze Ser. IV Tech. Teoret. 15 (1982), No. 1, 51-54. | MR

[8] S. Burýšek V. Burýšková: Some results from theory of homogeneous operators. CTU Seminar, 1994.

[9] V. Burýšková S. Burýšek: On the convexity of the numeгical range of homogeneous operatoгs. Acta Polytech. Práce ČVUT Praze Ser. IV Tech. Teoret. 34 (1994), No. 2, 25-33.

[10] V. Burýšková: Některé výsledky z teorie nelineárních operátorů a operátorových rovnic. Habilitation Thesis, Praha, 1994. (In Czech.)

[11] S. Burýšek V. Burýšková: On the aproximative spectrum of the couple of homogeneous operators. Acta Polytech. Práce ČVUT Praze Ser. IV Tech. Teoret. 35 (1995), No. 1, 5-16.

[12] G. Conti E. DePascale: The numerical range in the nonlineaг case. Boll. Un. Mat. Ital. B(5), 15 (1978), 210-216. | MR

[13] J. A. Canavati: A theory of numerical range for nonlinear operators. J. Funct. Anal. 33 (1979), 231-258. | DOI | MR | Zbl

[14] M. Furi A. Vignoli: Spectrum of nonlinear maps and bifuгcations in the nondifferentiable case. Ann. Math. Pura Appl. (4) 113 (1977), 265-285. | MR

[15] S. K. Kyong Y. Youngoh: On the numerical range for nonlinear operators. Bull. Korean Math. Soc. 21 (1984), No. 2, 119-126. | MR

[16] J. Prüss: A characterization of uniform convexity and applications to accretive operators. Hiroshima Math. J., 11 (1981), No. 2, 229-234. | DOI | MR

[17] A. Rhodius: Deг numeгische Wertebereich für nicht netwendig lineare Abbildungen in lokalkonvexen Räumen. Math. Nachr. 72 (1976), 169-180. | DOI | MR

[18] A. E. Taylor: Úvod do funkcionální analýzy. Academia, Praha, 1973.

[19] M. M. Vajnberg: Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. Moskva, 1972. (In Russian.)

[20] Verma U. Ram: Numerical range and related nonlinear functional equations. Czechoslovak Math. J. 42 (117) (1992), No. 3, 503-513. | MR | Zbl

[21] K. Yosida: Functional Analysis. Spгinger-Verlag, Berlin, 1965. | Zbl

[22] E. H. Zarantonello: The closure of the numerical range contains the spectrum. Pacific J. Math. 22 (1967), No. 3, 575-595. | DOI | MR | Zbl

Cité par Sources :