On the continuation method and the nonlinear alternative for Caristi-type non-self-mappings


In this note we survey some continuation fixed point theorems for Caristi-type non-self-mappings on complete metric spaces. Two new results, a continuation theorem and the nonlinear alternative are given in terms of retractions and generalized Leray–Schauder conditions.


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


Caristi’s theorem; continuation results; nonlinear alternative; contraction; metric space; retraction; fixed point.

Paper coordinates

R. Precup, On the continuation method and the nonlinear alternative for Caristi-type non-self-mappings, J. Fixed Point Theory Appl. 16 (2014), 3-10, https://doi.org/10.1007/S11784-014-0197-9


About this paper


Journal of Fixed Point Theory and Applications

Publisher Name


Print ISSN


Online ISSN


google scholar link

[1] J. Caristi, Fixed point theorems for mappings satisfying inwardness condition. Trans. Amer. Math. Soc. 215 (1976), 241–251.
[2] I. Ekeland, On the variational principle. J. Math. Anal. Appl. 47 (1974), 324–353.
[3] M. Frigon, Fixed point and continuation results for contractions in metric and gauge spaces. In: Fixed Point Theory and Its Applications, Banach Center Publications 77, Polish Acad. Sci., Warsaw, 2007, 89–114.
[4] M. Frigon, On some generalizations of Ekeland’s principle and inward contractions in gauge spaces. J. Fixed Point Theory Appl. 10 (2011), 279–298.
[5] M. Frigon, A. Granas and Z. E. A. Guennoun, Alternative non-lineaire pour les applications contractantes. Ann. Sci. Math. Qu´ebec 19 (1995), 65–68.
[6] A. Granas, Continuation method for contractive maps. Topol. Methods Nonlinear Anal. 3 (1994), 375–379.
[7] A. Granas and J. Dugundji, Fixed Point Theory. Springer, New York, 2003.
[8] J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations. In: Topological Methods in Differential Equations and Inclusions (Montreal, PQ, 1994), Mathematical and Physical Sciences 472, Kluwer, Dordrecht, 1995, 291–375.
[9] D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications. Gordon and Breach, Amsterdam, 2001.
[10] D. O’Regan and R. Precup, Continuation theory for contractions on spaces with two vector-valued metrics. Appl. Anal. 82 (2003), 131–144.
[11] R. Precup, Continuation theorems for mappings of Caristi type. Stud. Univ. Babes-Bolyai Math. 41 (1996), 101–106.
[12] R. Precup, Discrete continuation method for boundary value problems on bounded sets in Banach spaces. J. Comput. Appl. Math. 113 (2000), 267–281.
[13] H. H. Schaefer, Uber die methode der a-priori Schranken. Math. Ann. 129 (1955), 415–416.


Related Posts