Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

• Autorzy: In-Sook Kim , Jung-Hyun Bae
• Języki publikacji: EN

Abstrakty

EN

Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X⊃D(T) → 2X* is a maximal monotone multi-valued operator and C: X⊃D(C) → X* is a generalized pseudomonotone quasibounded operator with L ⊂ D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T x + λ C x, with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.

Słowa kluczowe

EN

Eigenvalues; Maximal monotone operators; Pseudomonotone operators; Degree theory

Kategorie tematyczne

• 47H05: Monotone operators and generalizations
• 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems
• 47H11: Degree theory

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 5
• Strony: 851-864

Daty

wydano

2013-05-01

online

2013-03-14

Twórcy

autor

In-Sook Kim

• Sungkyunkwan University

autor

Jung-Hyun Bae

• Sungkyunkwan University

Bibliografia

• [1] Brezis H., Crandall M.G., Pazy A., Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math., 1970, 23(1), 123–144 http://dx.doi.org/10.1002/cpa.3160230107
• [2] Browder F.E., Degree of mapping for nonlinear mappings of monotone type, Proc. Nat. Acad. Sci. U.S.A., 1983, 80(6), 1771–1773 http://dx.doi.org/10.1073/pnas.80.6.1771
• [3] Browder F.E., Hess P., Nonlinear mappings of monotone type in Banach spaces, J. Funct. Anal., 1972, 11(3), 251–294 http://dx.doi.org/10.1016/0022-1236(72)90070-5
• [4] Fitzpatrick P.M., Petryshyn W.V., On the nonlinear eigenvalue problem T(u) = λC(u), involving noncompact abstract and differential operators, Boll. Un. Mat. Ital. B (5), 1978, 15(1), 80–107
• [5] Guan Z., Kartsatos A.G., On the eigenvalue problem for perturbations of nonlinear accretive and monotone operators in Banach spaces, Nonlinear Anal., 1996, 27(2), 125–141 http://dx.doi.org/10.1016/0362-546X(95)00016-O
• [6] Kartsatos A.G., New results in the perturbation theory of maximal monotone and m-accretive operators in Banach spaces, Trans. Amer. Math. Soc., 1996, 348(5), 1663–1707 http://dx.doi.org/10.1090/S0002-9947-96-01654-6
• [7] Kartsatos A.G., Skrypnik I.V., Normalized eigenvectors for nonlinear abstract and elliptic operators, J. Differential Equations, 1999, 155(2), 443–475 http://dx.doi.org/10.1006/jdeq.1998.3592
• [8] Kartsatos A.G., Skrypnik I.V., Topological degree theories for densely defined mappings involving operators of type (S +), Adv. Differential Equations, 1999, 4(3), 413–456
• [9] Kartsatos A.G., Skrypnik I.V., A new topological degree theory for densely defined quasibounded \(\tilde S_ + \) -perturbations of multivalued maximal monotone operators in reflexive Banach spaces, Abstr. Appl. Anal., 2005, 2, 121–158 http://dx.doi.org/10.1155/AAA.2005.121
• [10] Kartsatos A.G., Skrypnik I.V., On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces, Trans. Amer. Math. Soc., 2006, 358(9), 3851–3881 http://dx.doi.org/10.1090/S0002-9947-05-03761-X
• [11] Kim I.-S., Some eigenvalue results for maximal monotone operators, Nonlinear Anal., 2011, 74(17), 6041–6049 http://dx.doi.org/10.1016/j.na.2011.05.081
• [12] Krasnosel’skii M.A., Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, New York, 1964
• [13] Li H., Huang F., On the nonlinear eigenvalue problem for perturbations of monotone and accretive operators in Banach spaces, Sichuan Daxue Xuebao, 2000, 37(3), 303–309
• [14] Petryshyn W.V., Approximation-Solvability of Nonlinear Functional and Differential Equations, Monogr. Textbooks Pure Appl. Math., 171, Marcel Dekker, New York, 1993
• [15] Zeidler E., Nonlinear Functional Analysis and its Applications. II/B, Springer, New York, 1990 http://dx.doi.org/10.1007/978-1-4612-0985-0

Identyfikatory

DOI

10.2478/s11533-013-0211-2