On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces
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- by Athanassios G. Kartsatos and Igor V. Skrypnik PDF
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Abstract:
Let $X$ be a real reflexive Banach space with dual $X^{*}$ and $G\subset X$ open and bounded and such that $0\in G.$ Let $T:X\supset D(T)\to 2^{X^{*}}$ be maximal monotone with $0\in D(T)$ and $0\in T(0),$ and $C:X\supset D(C)\to X^{*}$ with $0\in D(C)$ and $C(0)\neq 0.$ A general and more unified eigenvalue theory is developed for the pair of operators $(T,C).$ Further conditions are given for the existence of a pair $(\lambda ,x) \in (0,\infty )\times (D(T+C)\cap \partial G)$ such that \[ (**)\quad \qquad \qquad \qquad \qquad \qquad \qquad Tx+\lambda Cx\owns 0.\quad \qquad \qquad \qquad \qquad \qquad \qquad \] The “implicit" eigenvalue problem, with $C(\lambda ,x)$ in place of $\lambda Cx,$ is also considered. The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators $T,~C.$ No compactness assumptions have been made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.References
- Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
- H. Brezis, M. G. Crandall, and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math. 23 (1970), 123–144. MR 257805, DOI 10.1002/cpa.3160230107
- Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
- Felix E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 1–39. MR 699315, DOI 10.1090/S0273-0979-1983-15153-4
- Felix E. Browder, The degree of mapping, and its generalizations, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982) Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 15–40. MR 729503, DOI 10.1090/conm/021/729503
- Felix E. Browder, Degree of mapping for nonlinear mappings of monotone type, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 6, i, 1771–1773. MR 699437, DOI 10.1073/pnas.80.6.1771
- Ioana Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Mathematics and its Applications, vol. 62, Kluwer Academic Publishers Group, Dordrecht, 1990. MR 1079061, DOI 10.1007/978-94-009-2121-4
- Svatopluk FuÄŤĂk, JindĹ™ich NeÄŤas, Jiřà SouÄŤek, and VladimĂr SouÄŤek, Spectral analysis of nonlinear operators, Lecture Notes in Mathematics, Vol. 346, Springer-Verlag, Berlin-New York, 1973. MR 0467421
- Zhengyuan Guan and Athanassios G. Kartsatos, On the eigenvalue problem for perturbations of nonlinear accretive and monotone operators in Banach spaces, Nonlinear Anal. 27 (1996), no. 2, 125–141. MR 1389473, DOI 10.1016/0362-546X(95)00016-O
- Z. Guan, A. G. Kartsatos, and I. V. Skrypnik, Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators, J. Differential Equations 188 (2003), no. 1, 332–351. MR 1954518, DOI 10.1016/S0022-0396(02)00066-9
- Athanassios G. Kartsatos, New results in the perturbation theory of maximal monotone and $m$-accretive operators in Banach spaces, Trans. Amer. Math. Soc. 348 (1996), no. 5, 1663–1707. MR 1357397, DOI 10.1090/S0002-9947-96-01654-6
- Athanassios G. Kartsatos and Igor V. Skrypnik, Normalized eigenvectors for nonlinear abstract and elliptic operators, J. Differential Equations 155 (1999), no. 2, 443–475. MR 1698562, DOI 10.1006/jdeq.1998.3592
- A. G. Kartsatos and I. V. Skrypnik, Topological degree theories for densely defined mappings involving operators of type $(S_+)$, Adv. Differential Equations 4 (1999), no. 3, 413–456. MR 1671257
- A. G. Kartsatos and I. V. Skrypnik, Invariance of domain for perturbations of maximal monotone operators in Banach spaces (to appear).
- A. G. Kartsatos and I. V. Skrypnik, A topological degree theory for densely defined quasibounded $(\widetilde S_{+})$-perturbations of multivalued maximal monotone operators in reflexive Banach spaces, Abstr. Appl. Anal. (to appear).
- Hong-xu Li and Fa-lun Huang, On the nonlinear eigenvalue problem for perturbations of monotone and accretive operators in Banach spaces, Sichuan Daxue Xuebao 37 (2000), no. 3, 303–309 (English, with English and Chinese summaries). MR 1784055
- N. G. Lloyd, Degree theory, Cambridge Tracts in Mathematics, No. 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978. MR 0493564
- Dan Pascali and Silviu Sburlan, Nonlinear mappings of monotone type, Martinus Nijhoff Publishers, The Hague; Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1978. MR 531036
- Wolodymyr V. Petryshyn, Approximation-solvability of nonlinear functional and differential equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 171, Marcel Dekker, Inc., New York, 1993. MR 1200455
- E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, Mathematical Surveys and Monographs, vol. 23, American Mathematical Society, Providence, RI, 1986. MR 852987, DOI 10.1090/surv/023
- Stephen Simons, Minimax and monotonicity, Lecture Notes in Mathematics, vol. 1693, Springer-Verlag, Berlin, 1998. MR 1723737, DOI 10.1007/BFb0093633
- I. V. Skrypnik and I. V. Skrypnik, NelineÄnye èllipticheskie uravneniya vysshego poryadka, Izdat. “Naukova Dumka”, Kiev, 1973 (Russian). MR 0435590
- I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, Translations of Mathematical Monographs, vol. 139, American Mathematical Society, Providence, RI, 1994. Translated from the 1990 Russian original by Dan D. Pascali. MR 1297765, DOI 10.1090/mmono/139
- Eberhard Zeidler, Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990. Nonlinear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033498, DOI 10.1007/978-1-4612-0985-0
Additional Information
- Athanassios G. Kartsatos
- Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
- Email: hermes@math.usf.edu
- Igor V. Skrypnik
- Affiliation: Institute for Applied Mathematics and Mechanics, R. Luxemburg Str. 74, Donetsk 340114, Ukraine
- Email: skrypnik@iamm.ac.donetsk.ua
- Received by editor(s): May 6, 2003
- Received by editor(s) in revised form: June 3, 2004
- Published electronically: July 26, 2005
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 3851-3881
- MSC (2000): Primary 47H14, 47H07, 47H11
- DOI: https://doi.org/10.1090/S0002-9947-05-03761-X
- MathSciNet review: 2219002