## A note on degree theory for gradient mappings.(English)Zbl 0501.58012

### MSC:

 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 47H10 Fixed-point theorems 34G20 Nonlinear differential equations in abstract spaces

Zbl 0307.47058
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### References:

 [1] Alfonso Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl. (4) 120 (1979), 113 – 137. · Zbl 0426.35038 [2] Kung Ching Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), no. 5, 693 – 712. · Zbl 0444.58008 [3] Klaus Deimling, Ordinary differential equations in Banach spaces, Lecture Notes in Mathematics, Vol. 596, Springer-Verlag, Berlin-New York, 1977. · Zbl 0361.34050 [4] M. A. Krasnosel$$^{\prime}$$skiĭ, The operator of translation along the trajectories of differential equations, Translations of Mathematical Monographs, Vol. 19. Translated from the Russian by Scripta Technica, American Mathematical Society, Providence, R.I., 1968. [5] Robert H. Martin Jr., Nonlinear operators and differential equations in Banach spaces, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. Pure and Applied Mathematics. [6] L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 267 – 302. · Zbl 0468.47040 [7] Paul H. Rabinowitz, A note on topological degree for potential operators, J. Math. Anal. Appl. 51 (1975), no. 2, 483 – 492. · Zbl 0307.47058 [8] Erich H. Rothe, A relation between the type numbers of a critical point and the index of the corresponding field of gradient vectors, Math. Nachr. 4 (1951), 12 – 17. · Zbl 0044.31903 [9] K. Thews, Untere Schranken für die Anzahl von Lösungen einer Klasse nichtlinearer Dirichlet-probleme, Doctoral Dissertation, Bochum, 1978. [10] M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. With a chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein.
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