On the continuation principle for nonexpansive maps


In this note the continuation principle (nonlinear alternative) for nonexpansive maps on Hilbert spaces (see [5]) in extended in two directions: 1) to the case of uniformly convex Banach spaces; 2) for nonexpansive maps on a not necessarly convex set of a Hilbert space. In the proofs we use the Leray-Schauder continuation principle for condensing maps [7] , [9] (we can also  Granas’ continuation principle for contractions on complete metric spaces [6].


Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania



Paper coordinates

R. Precup, On the continuation principle for nonexpansive maps, Studia Univ. Babeş-Bolyai Math. 41 (1996) no. 3 , 85-89.


About this paper


Studia Universitatis Babes-Bolyai Mathematica

Publisher Name

Universitatis ”Babeș-Bolyai”

Print ISSN


Online ISSN


MR: 1 644 466

google scholar link

[1] Brezis, H., Analyse fonctionnelle, Masson, Paris, 1983.
[2] B rowder, F.E., Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54, 1041-1044, 1965.
[3] Browder, F.E., Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proceedings Symposium on Nonlinear Functional Analysis, Amer. Math. Soc., Chicago, 1968.
[4] Diestel, J., Geometry of Banach Spaces, Lec ture Notes in Mathematics, vol. 485, Springer, Berlin, 1975.
[5] Dugundji, J., Granas, A., Fixed Point Theory I, Monografie Mathematyczne, PWN, Warsaw, 1982.
[6] granas, A., Continuation method for contractive maps, Topol. Methods Nonlinear Anal., 3, 375-379, 1994.
[7] Krawcewicz, W., Contribution a la theorie des equations non lineaires dans les espaces de Banach, Dissertationes Math. 273, 198…
[8] Pavel, N., Ecuații diferențiale asociate unor operatori neliniari pe spații Banach, Ed. Acad. R.S.R., București, 1977.
[9] Precup, R., Nonlinear boundary value problems for infinite systems of second-order functional differential equations, Seminar on Differential Equaitons (Editor I.A.Rus), pp. 17-30, University Babes-Bolyai, Cluj, 1988

Related Posts