Fixed point theorems under combined topological and variational conditions


The new idea is to replace part of the conditions on the operator involved in the classical fixed point theorems of Schauder, Krasnoselskii, Darbo and Sadovskii, by assumptions upon the associated functional, in case that the fixed point equation has a variational form. Fixed points minimizing the associated functionals are obtained via Ekeland’s variational principle and the Palais–Smale compactness condition guaranteed by the topological properties of the nonlinear operators.


Angela Budescu
Department of Mathematics, Babes-Bolyai University,  Cluj-Napoca, Romania

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


Fixed point; Critical point; Compact nonlinear operator; Condensing operator; Ekeland’s variational principle.

Paper coordinates

A. Budescu, R. Precup, Fixed point theorems under combined topological and variational conditions, Results. Math. 70 (2016) no. 3, 487-497,



About this paper


Results in Mathematical

Publisher Name

Springer International Publishing

Print ISSN


Online ISSN


google scholar link

[1] Budescu, A.: Semilinear operator equations and systems with potential-type nonlinearities. Studia Univ. Babe¸s-Bolyai Math. 59, 199–212 (2014)
[2] Budescu, A., Precup, R.: Variational properties of the solutions of semilinear equations under nonresonance conditions. J. Nonlinear Convex Anal. (to appear)
[3] Budescu, A., Precup, R.: Variational properties of the solutions of singular second-order differential equations and systems. J. Fixed Point Theory Appl. doi:10.1007/s11784-016-0284-1
[4] Chidume, C.: Geometric Properties of Banach Spaces and Nonlinear Iterations. Springer, London (2009)
[5] Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)
[6] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
[7] Granas, A., Dugunji, J.: Fixed Point Theory. Springer, New-York (2003)
[8] Precup, R.: Nash-type equilibria and periodic solutions to nonvariational systems. Adv. Nonlinear Anal. 3, 197–207 (2014)
[9] Precup, R.: Nash-type equilibria for systems of Szulkin functionals. Set-Valued Var. Anal. (2015). doi:10.1007/s11228-015-0356-1
[10] Ricceri, B.: Uniqueness properties of functionals with Lipschitzian derivative. Port. Math. 63, 393–400 (2006)
[11] Ricceri, B.: Fixed points of nonexpansive potential operators in Hilbert spaces. Fixed Point Theory Appl. 2012(123), (2012). doi:10.1186/1687-1812-2012-123
[12] Zeidler, E.: Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems. Springer, New York (1990)


Related Posts