A variational analogue of Krasnoselskii’s cone fixed point theory


Based on Ekeland’s principle, a variational analogue of Krasnoselskii’s cone compression-expansion fixed point theorem is presented. A general scheme of applications to semilinear equations making use of Mikhlin’s variational theory on positive linear operators is included.


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


Critical point; Fixed point; Cone; Variational principle; Semilinear operator equation; Positive solution

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R. Precup, A variational analogue of Krasnoselskii’s cone fixed point theory, in Nonlinear Analysis and Boundary Value Problems, Eds.: I. Area, A. Cabada, J. Á. Cid, D. Franco, E. Liz, R. López Pouso, R. Rodríguez López, Springer Proceedings in Mathematics & Statistics 292, Springer, 2019, 1-18. http://doi.org/10.1007/978-3-030-26987-6_1



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Springer Proceedings in Mathematics & Statistics

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Springer Cham.

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ISBN 978-3-030-26986-9

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ISBN 978-3-030-26987-6

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[1] Boulaiki, H., Moussaoui, T., Precup, R.: Multiple positive solutions for a second-order boundary value problem on the half-line. J. Nonlinear Funct. Anal. 2017, Article 17, 1–25 (2017). https://doi.org/10.23952/jnfa.2017.17
[2] Bunoiu, R., Precup, R., Varga, C.: Multiple positive standing wave solutions for Schrödinger equations with oscillating state-dependent potentials. Commun. Pure Appl. Anal. 16, 953–972 (2017). https://doi.org/10.3934/cpaa.2017046, MathSciNet CrossRef MATH Google Scholar
[3] Cabada, A., Precup, R., Saavedra, L., Tersian, S.: Multiple positive solutions to a fourth-order boundary value problem. Electron. J. Differ. Equ. 2016(254), 1–18 (2016). https://ejde.math.txstate.edu/Volumes/2016/254/abstr.html
[4] Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003). https://doi.org/10.1007/978-0-387-21593-8, CrossRef Google Scholar
[5] Krasnoselskii, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964). (OCoLC)1316344 Google Scholar
[6] Kwong, M.K.: On Krasnoselskii’s cone fixed point theorem. Fixed Point Theor. Appl. 2008, Article 164537, 1–18 (2008). https://doi.org/10.1155/2008/164537, MathSciNet CrossRef Google Scholar
[7] Lisei, H., Precup, R., Varga, C.: A Schechter type critical point result in annular conical domains of a Banach space and applications. Discret. Contin. Dyn. Syst. 36, 3775–3789 (2016). https://doi.org/10.3934/dcds.2016.36.3775 MathSciNet CrossRef MATH Google Scholar
[8] Michlin, S.G.: Partielle Differentialgleichungen in der Mathematischen Physik. Verlag Harri Deutch, Frankfurt (1978). https://katalog.ub.uni-heidelberg.de/titel/65653127
[9] Precup, R.: Critical point theorems in cones and multiple positive solutions of elliptic problems. Nonlinear Anal. 75, 834–851 (2012). https://doi.org/10.1016/j.na.2011.09.016, MathSciNet CrossRef Google Scholar
[10] Precup, R.: Abstract weak Harnack inequality, multiple fixed points and p-Laplace equations. J. Fixed Point Theor. Appl.12, 193–206 (2012). https://doi.org/10.1007/s11784-012-0091-2 MathSciNet CrossRef Google Scholar
[11] Precup, R.: On a bounded critical point theorem of Schechter. Studia Univ. Babeş–Bolyai Math. 58, 87–95 (2013). http://www.cs.ubbcluj.ro/~studia-m/2013-1/10-Precup-final2.pd
[12] Precup, R.: Critical point localization theorems via Ekeland’s variational principle. Dyn. Syst. Appl. 22, 355–370 (2013), MathSciNet MATH Google Scholar
[13] Precup, R.: Multiple periodic solutions with prescribed minimal period to second-order Hamiltonian systems. Dyn. Syst. 29, 424–438 (2014). https://doi.org/10.1080/14689367.2014.911410, MathSciNet CrossRef Google Scholar
[14] 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). https://doi.org/10.1016/j.jmaa.2018.03.035, MathSciNet CrossRef Google Scholar
[15] Precup, R., Varga, C.: Localization of positive critical points in Banach spaces and applications. Topol. Methods Nonlinear Anal. 49, 817–833 (2017). https://doi.org/10.12775/TMNA.2017.011, MathSciNet CrossRef MATH Google Scholar
[16] Precup, R., Pucci, P., Varga, C.: A three critical points result in a bounded domain of a Banach space and applications. Differ. Integral Equ. 30, 555–568 (2017), MathSciNet MATH Google Scholar
[17] Schechter, M.: Linking Methods in Critical Point Theory. Birkhäuser, Basel (1999). https://doi.org/10.1007/978-1-4612-1596-7, CrossRef Google Scholar
[18] Struwe, M.: Variational Methods. Springer, Berlin (1990). https://doi.org/10.1007/978-3-662-02624-3, CrossRef Google Scholar


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