## On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces.(English)Zbl 1100.47050

Let $$X$$ be a real reflexive Banach space with dual $$X^*$$ and $$G\subset X$$ be 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 $$C(0)\neq 0$$. In the paper under review, 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 $T\,x+\lambda\,C\,x\ni 0.$ The “implicit” eigenvalue problem, with $$C(\lambda,\,x)$$ in place of $$\lambda\,C\,x$$, 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 are made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the the degree theories of the authors, involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.

### MSC:

 47H14 Perturbations of nonlinear operators 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H11 Degree theory for nonlinear operators 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems 47N20 Applications of operator theory to differential and integral equations
Full Text:

### References:

 [1] 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. · Zbl 0328.47035 [2] 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. · Zbl 0182.47501 [3] 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. [4] Felix E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 1 – 39. · Zbl 0533.47053 [5] 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. · Zbl 0531.47051 [6] 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. · Zbl 0533.47051 [7] Ioana Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Mathematics and its Applications, vol. 62, Kluwer Academic Publishers Group, Dordrecht, 1990. · Zbl 0712.47043 [8] Svatopluk Fučík, Jindřich Nečas, Jiří Souček, and Vladimír Souček, Spectral analysis of nonlinear operators, Lecture Notes in Mathematics, Vol. 346, Springer-Verlag, Berlin-New York, 1973. · Zbl 0268.47056 [9] 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. · Zbl 0864.47028 [10] 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. · Zbl 1050.47055 [11] Athanassios G. Kartsatos, New results in the perturbation theory of maximal monotone and \?-accretive operators in Banach spaces, Trans. Amer. Math. Soc. 348 (1996), no. 5, 1663 – 1707. · Zbl 0861.47029 [12] Athanassios G. Kartsatos and Igor V. Skrypnik, Normalized eigenvectors for nonlinear abstract and elliptic operators, J. Differential Equations 155 (1999), no. 2, 443 – 475. · Zbl 0931.47050 [13] A. G. Kartsatos and I. V. Skrypnik, Topological degree theories for densely defined mappings involving operators of type (\?$$_{+}$$), Adv. Differential Equations 4 (1999), no. 3, 413 – 456. · Zbl 0959.47037 [14] A. G. Kartsatos and I. V. Skrypnik, Invariance of domain for perturbations of maximal monotone operators in Banach spaces (to appear). · Zbl 1160.47044 [15] 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). · Zbl 1110.47049 [16] 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). · Zbl 0980.47056 [17] N. G. Lloyd, Degree theory, Cambridge University Press, Cambridge-New York-Melbourne, 1978. Cambridge Tracts in Mathematics, No. 73. · Zbl 0367.47001 [18] Dan Pascali and Silviu Sburlan, Nonlinear mappings of monotone type, Martinus Nijhoff Publishers, The Hague; Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1978. · Zbl 0423.47021 [19] 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. · Zbl 0772.65040 [20] 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. · Zbl 0597.47040 [21] Stephen Simons, Minimax and monotonicity, Lecture Notes in Mathematics, vol. 1693, Springer-Verlag, Berlin, 1998. · Zbl 0922.47047 [22] Нелинейные ѐллиптические уравнения высшего порядка, Издат. ”Наукова Думка”, Киев, 1973 (Руссиан). [23] 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. · Zbl 0822.35001 [24] 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. · Zbl 0684.47029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.