Generalizations of Browder’s degree theory

Hu, Shou Chuan; Papageorgiou, Nikolaos S.

doi:10.1090/S0002-9947-1995-1284911-6

Generalizations of Browder’s degree theory
HTML articles powered by AMS MathViewer

by Shou Chuan Hu and Nikolaos S. Papageorgiou PDF

Trans. Amer. Math. Soc. 347 (1995), 233-259 Request permission

Abstract:

The starting point of this paper is the recent important work of F. E. Browder, who extended degree theory to operators of monotone type. The degree function of Browder is generalized to maps of the form $T + f + G$, where $T$ is maximal monotone, $f$ is of class ${(S)_ + }$ bounded, and $G( \cdot )$ is an u.s.c. compact multifunction. It is also generalized to maps of the form $f + {N_G}$, with $f$ of class ${(S)_ + }$ and ${N_G}$ the Nemitsky operator of a multifunction $G(x,r)$ satisfying various types of sign conditions. Some examples are also included to illustrate the abstract results.

References

Herbert Amann and Stanley A. Weiss, On the uniqueness of the topological degree, Math. Z. 130 (1973), 39–54. MR 346601, DOI 10.1007/BF01178975
Haïm Brézis and Felix E. Browder, Some properties of higher order Sobolev spaces, J. Math. Pures Appl. (9) 61 (1982), no. 3, 245–259 (1983). MR 690395
Felix E. Browder, Nonlinear eigenvalue problems and Galerkin approximations, Bull. Amer. Math. Soc. 74 (1968), 651–656. MR 226453, DOI 10.1090/S0002-9904-1968-11979-2
Felix E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 1–60. MR 0269962

Nonlinear operators and nonlinear equations in Banach spaces

Felix E. Browder, Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 425–500. MR 0394340
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. MR 699437, DOI 10.1073/pnas.80.6.1771
Felix E. Browder, Degree of mapping for nonlinear mappings of monotone type: densely defined mapping, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 8, i, 2405–2407. MR 700932, DOI 10.1073/pnas.80.8.2405
Felix E. Browder, Degree of mapping for nonlinear mappings of monotone type: densely defined mapping, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 8, i, 2405–2407. MR 700932, DOI 10.1073/pnas.80.8.2405
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. MR 729503, DOI 10.1090/conm/021/729503

Fixed point theory and nonlinear problems

9

Felix E. Browder, Degree theory for nonlinear mappings, Nonlinear functional analysis and its applications, Part 1 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 203–226. MR 843560
Arrigo Cellina, Approximation of set valued functions and fixed point theorems, Ann. Mat. Pura Appl. (4) 82 (1969), 17–24 (English, with Italian summary). MR 263046, DOI 10.1007/BF02410784
D. G. de Figueiredo, P.-L. Lions, and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), no. 1, 41–63. MR 664341

Linear operators

L. Führer, Ein elementarer analytischer Beweis zur Eindeutigkeit des Abbildungsgrades im $R^{n}$, Math. Nachr. 54 (1972), 259–267 (German). MR 317317, DOI 10.1002/mana.19720540117
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
Nikolaos S. Papageorgiou, On measurable multifunctions with applications to random multivalued equations, Math. Japon. 32 (1987), no. 3, 437–464. MR 914749
S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR 165 (1965), 36–39 (Russian). MR 0192184
S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1970/71), 173–180. MR 306873, DOI 10.4064/sm-37-2-173-180
Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), no. 5, 859–903. MR 486391, DOI 10.1137/0315056
Eberhard Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR 816732, DOI 10.1007/978-1-4612-4838-5