Fixed point–critical point hybrid theorems and application to systems with partial variational structure


In this paper, fixed point arguments and a critical point technique are combined leading to hybrid existence results for a system of two operator equations where only one of the equations has a variational structure. An application to periodic solutions of a semi-variational system is given to illustrate the theory.


Irene Benedetti
University of Perugia, Italy

Tiziana Cardinali
University of Perugia, Italy

Radu Precup
Babeş-Bolyai University, Romania


Ekeland variational principle; Krasnosel’skii’s fixed point theorem for the sum of two operators; Perov contraction; Positive solution; Periodic problem; Second-order differential systems



Cite this paper as:

I. Benedetti, T. Cardinali, R. Precup, Fixed point–critical point hybrid theorems and application to systems with partial variational structure, Journal of Fixed Point Theory and Applications volume 23, Article number: 63 (2021),

About this paper



Publisher Name

Journal of Fixed Point Theory and Applications

Print ISSN


Online ISSN


Google Scholar Profile


[1] Avramescu, C.: On a fixed point theorem (in Romanian). St. Cerc. Mat. 22(2), 215–221 (1970)
MATH Google Scholar

[2] Basoc, I., Cardinali, T.: A hybrid nonlinear alternative theorem and some hybrid fixed point theorems for multimaps. J. Fixed Point Theory Appl. 17, 413–424 (2015)
MathSciNet Article Google Scholar

[3] Biondini, M., Cardinali, T.: Existence of solutions for a nonlinear integral equation via a hybrid fixed point theorem. Results Math. 71, 1259–1276 (2017)
MathSciNet Article Google Scholar

[4] Burton, T.A., Kirk, C.: A fixed point theorem of Krasnoselskii–Schaefer type. Math. Nachr. 189, 23–31 (1998)
MathSciNet Article Google Scholar

[5] Cabada, A.: Green’s Functions in the Theory of Ordinary Differential Equations. Springer, New York (2014)
Book Google Scholar

[6] Cardinali, T., Precup, R., Rubbioni, P.: Heterogeneous vectorial fixed point theorems. Mediterr. J. Math. 14(83), 1–12 (2017)
MathSciNet MATH Google Scholar

[7] Krasnosel’skii, M.A.: Some problems of nonlinear analysis. Am. Math. Soc. Transl. Ser. 2(10), 345–409 (1958)
MathSciNet Google Scholar

[8] Motreanu, D., Panagiotopoulos, P.D.: Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Springer, Dordrecht (1999)
Book Google Scholar

[9] O’Regan, D.: Fixed-point theory for the sum of two operators. Appl. Math. Lett. 9, 1–8 (1996)
MathSciNet Article Google Scholar

[10] Precup, R.: Methods in Nonlinear Integral Equations. Springer, Dordrecht (2002)
Book Google Scholar

[11] Precup, R.: Nash-type equilibria and periodic solutions to nonvariational systems. Adv. Nonlinear Anal. 3(4), 197–207 (2014)
MathSciNet MATH Google Scholar

[12] Precup, R.: A critical point theorem in bounded convex sets and localization of Nash-type equilibria of nonvariational systems. J. Math. Anal. Appl. 463, 412–431 (2018)
MathSciNet Article Google Scholar

[13] I.A. Rus, On the fixed points of mappings defined on a Cartesian product. III(in Romanian). Stud. Univ. Babeş Bolyai Math. 24(2), 55–56 (1979)

[14] Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 3(2), 77–109 (1986)

[15] Şerban, M.A.: Technique of fixed point structure for the mappings on product spaces, Babeş-Bolyai University of Cluj-Napoca. Semin. Fixed Point Theory, Preprint Nr. 3, 1–18 (1998)

Related Posts