MULTIVALUED VERSIONS OF A BOLZANOS THEOREM

Title & Authors
MULTIVALUED VERSIONS OF A BOLZANOS THEOREM
Bae, Jong-Sook; Cho, Seong-Hoon;

Abstract
The intermediate value theorem for a continuous real valued function is a kind of Bolzanos theorem. Similar results also hold for compact, monotone or accretive mappings in Banach spaces. In this paper we give multivalued versions of Bolzanos theorem.
Keywords
fixed point;weakly inward set;monotone mapping;semi-monotone mapping;strongly-monotone mapping;
Language
English
Cited by
References
1.
J. S. Bae, W. K. Kim, and K. K. Tan, Another generalization of Ky Fan's minimax inequality and its applications, Bull. Inst. Math. Acad. Sinica 21 (1993), no. 3, 229-244.

2.
J. S. Bae and M. S. Park, Fixed points of ${\mu}$-condensing maps with inwardness conditions, Math. Japon. 40 (1994), no. 1, 179-183.

3.
H. Brezis, M. Crandall, and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math. 23 (1970), 123-144.

4.
F. E. Browder, Variational boundary value problems for quasi-linear elliptic equations of arbitrary order, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 31-37.

5.
F. E. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1100-1103.

6.
F. E. Browder, Remarks on nonlinear functional equations III, Illinois J. Math. 9 (1965), 617-622.

7.
J. W. Dauben, Progress of mathematics in the early 19th century: context, contents and consequences, Impact of Bolzano's epoch on the development of science (Prague, 1981), 223-260, Acta Hist. Rerum Nat. necnon Tech. Spec. Issue, 13, CSAV, Prague, 1982.

8.
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.

9.
K. Deimling, Fixed points of weakly inward multis, Nonlinear Anal. 10 (1986), no. 11, 1261-1262.

10.
Ky Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961), 305-310.

11.
Ky Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), no. 4, 519-537.

12.
R. I. Kachurovskii, Monotonic nonlinear operators in Banach spaces, Dokl. Akad. Nauk SSSR 163 (1965), 559-562.

13.
P. D. Lax, Change of variables in multiple integrals, Amer. Math. Monthly 106 (1999), no. 6, 497-501.

14.
J. Leray and J. L. Lions, Quelques resulatats de Visik sur les problemes elliptiques nonlineaires par les methodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), 97-107.

15.
G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341-346.

16.
G. J. Minty, On a "monotonicity" method for the solution of non-linear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 1038-1041.

17.
C. H. Morales, A Bolzano's theorem in the new millennium, Nonlinear Anal. 51 (2002), no. 4, Ser. A: Theory Methods, 679-691.

18.
J. A. Park, Invariance of domain theorem for demicontinuous mappings of type $(S_+)$, Bull. Korean Math. Soc. 29 (1992), no. 1, 81-87.

19.
R. Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969), 397-407.

20.
M. Shinbrot, A fixed point theorem, and some applications, Arch. Rational Mech. Anal. 17 (1964), 255-271.

21.
S. L. Trojanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1971), 173-180.

22.
M. M. Vainberg and R. I. Kachurovskii, On the variational theory of non-linear operators and equations, Dokl. Akad. Nauk SSSR 129 (1959), 1199-1202.