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cone; lattice; topological degree
Using the cone theory and the lattice structure, we establish some methods of computation of the topological degree for the nonlinear operators which are not assumed to be cone mappings. As applications, existence results of nontrivial solutions for singular Sturm-Liouville problems are given. The nonlinearity in the equations can take negative values and may be unbounded from below.
[1] Deimling, K.: Nonlinear Functional Analysis. Springer Berlin (1985). MR 0787404 | Zbl 0559.47040
[2] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering 5. Academic Press Boston (1988). MR 0959889
[3] Liu, X., Sun, J.: Computation of topological degree of unilaterally asymptotically linear operators and its applications. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 96-106. DOI 10.1016/ | MR 2518016 | Zbl 1191.47076
[4] Luxemburg, W. A. J., Zaanen, A. C.: Riesz Spaces. Vol. I. North-Holland Mathematical Library. North-Holland Publishing Company Amsterdam (1971). MR 0511676
[5] Krasnosel'skij, M. A.: Positive Solutions of Operator Equations. Translated from the Russian by Richard E. Flaherty. L. F. Boron P. Noordhoff Ltd. Groningen (1964). MR 0181881 | Zbl 0121.10604
[6] Kreĭn, M. G., Rutman, M. A.: Linear operators leaving invariant a cone in a Banach space. Usp. Mat. Nauk 3 (1948), 3-95 Russian. MR 0027128 | Zbl 0030.12902
[7] Sun, J.: Nontrivial solutions of superlinear Hammerstein integral equations and applications. Chin. Ann. Math., Ser. A 7 (1986), 528-535 Chinese. MR 0886319 | Zbl 0633.45006
[8] Sun, J., Liu, X.: Computation for topological degree and its applications. J. Math. Anal. Appl. 202 (1996), 785-796. DOI 10.1006/jmaa.1996.0347 | MR 1408354 | Zbl 0866.47043
[9] Sun, J., Liu, X.: Computation of topological degree for nonlinear operators and applications. Nonlinear Anal., Theory Methods Appl. 69 (2008), 4121-4130. MR 2463359 | Zbl 1169.47043
[10] Sun, J., Liu, X.: Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations. J. Math. Anal. Appl. 348 (2008), 927-937. DOI 10.1016/j.jmaa.2008.05.023 | MR 2446045 | Zbl 1177.47065
[11] Sun, J., Zhang, G.: Nontrivial solutions of singular sublinear Sturm-Liouville problems. J. Math. Anal. Appl. 326 (2007), 242-251. DOI 10.1016/j.jmaa.2006.03.003 | MR 2277779 | Zbl 1111.34023
[12] Walter, W.: Ordinary Differential Equations. Transl. from the German by Russell Thompson. Graduate Texts in Mathematics. Readings in Mathematics 182. Springer New York (1998). MR 1629775
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