Note on spectral theory of nonlinear operators: Extensions of some surjectivity theorems of Fučík and Nečas. (English) Zbl 0546.47029

In this paper the existence of solutions of the equation: \[ \lambda T(x)-S(x)=f,\quad x\in X,\quad f\in Y,\quad\lambda \in R \] is studied, under the assumptions that X and Y are reflexive Banach spaces, the dual space Y’ is strictly convex, S is a bounded weakly closed operator and T is a bijective operator between X and Y. The results obtained here extend some theorems of Fučik and Nečas where S is supposed completely continuous. The method used is a topological degree theory for noncompact operators.


47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
47J05 Equations involving nonlinear operators (general)
35P05 General topics in linear spectral theory for PDEs
Full Text: EuDML


[1] S. Fučík, Nečas J., Souček J., Souček V.: Spectral Analysis of nonlinear Operators. Springer Verlag. Berlin (1973). · Zbl 0268.47056
[2] Canfora A.: La teoria del grado topologico per una classe di operatori non compatti in spazi di Hilbert. Ric. di Mat. vol. XXVIII, 109- 142 (1979). · Zbl 0428.47033
[3] Pacella F.: Il grado topologico per operatori non compatti in spazi di Banach con il duale strettamente convesso. Ric. di Mat., vol. XXIX, 211-306 (1980). · Zbl 0474.47030
[4] Nečas J.: Sur I’aternative de Fredholm pour les operateurs non-lineaires avec applications aux problèmes aux limites. Ann. Scuola Norm. Sup. Pisa, 23, 331-345 (1969). · Zbl 0187.08103
[5] Fučík S.: Note on Fredholm alternative for nonlinear operators. Comment. Math. Univ. Carolinae, 72, 213-226 (1971). · Zbl 0215.21201
[6] Nečas J.: Remark on the Fredholm alternative for nonlinear operators with application to nonlinear integral equations of generalized Hammerstein type. Comm. Math. Univ. Carolinae, 13, 109-120 (1972). · Zbl 0235.47039
[7] Petryshyn W. V.: Nonlinear equations involving noncompact operators. Proc. Symp. Pure Math. Vol. 18, Part I, Nonlinear functional Analysis, Rhode Island (1970). · Zbl 0232.47070
[8] Adams R.: Sobolev spaces. Academic Press (1975). · Zbl 0314.46030
[9] Schechter M.: Principles of functional analysis. Academic Press New York (1971). · Zbl 0211.14501
[10] Pucci C., Talenti G.: Elliptic (second-order) Partial Differential Equations with Measurable Coefficients and Approximating Integral Equations. Advances in Mathematics, 19, 48-105 (1976).
[11] Chicco M.: Solvability of the Dirichlet problem in \(H^{2},\,^{p}(\Omega )\) for a class of linear second order elliptic partial differential equations. Boll. U.M.I. (4), 374-387 (1971). · Zbl 0215.45406
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.