On a bounded critical point theorem of Schechter


A new proof based on Bishop-Phelps’ variational principle is given to a critical point theorem of Schechter for extrema in a ball of a Hilbert space. The same technique is used to obtain a similar result in annular domains. Comments on the involved boundary conditions and an application to a two-point boundary value problem are included. An alternative variational approach to the compression-expansion Krasnoselskii’s fixed point method is thus provided. In addition, estimations from below are obtained here for the first time, in terms of  the energetic norm.


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


Critical point; extremum point; Palais-Smale condition; two-point boundary value problem

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R. Precup, On a bounded critical point theorem of Schechter, Stud. Univ. Babeş-Bolyai Math. 58 (2013) no. 1, 87-95.


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Studia Universitatis Babes-Bolyai Mathematica

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“Babeș-Bolyai” University

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[1] Agarwal, R., Meehan, M., O’Regan, D., Precup, R., Location of nonnegative solutions for differential equations on finite and semi-infinite intervals, Dynam. Systems Appl., 12(3-4)(2003), 323-331.
[2] Avery, R.I., Henderson, J., Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Letters, 13(2000), 1-7.
[3] Bishop, E., Phelps, R.R., The support functional of a convex set, In: Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, RI, 1963, 27-35.
[4] Erbe, L.H., Hu, S., Wang, H., Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184(1994), 640-648.
[5] Frigon, M., On some generalizations of Ekeland’s principle and inward contractions in gauge spaces, J. Fixed Point Theory Appl., 10(2011), 279-298.
[6] Granas, A., Dugundji, J., Fixed Point Theory, Springer, New York, 2003.
[7] Guo, D., Laksmikantham, V., Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces, J. Math. Anal. Appl., 129(1988), 211-222.
[8] Lian, W., Wong, F., Yeh, C., On the existence of positive solutions of nonlinear second order differential equations, Proc. Amer. Math. Soc., 124(1996), 1117-1126.
[9] Penot, J.-P., The drop theorem, the petal theorem and Ekeland’s variational principle, Nonlinear Anal., 10(1986), 813-822.
[10] Precup, R., The Leray-Schauder condition in critical point theory, Nonlinear Anal., 71(2009), 3218-3228.
[11] Precup, R., Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal., 75(2012), 834-851.
[12] Precup, R., Critical point localization theorems via Ekeland’s variational principle, Dynam. Systems Appl., to appear.
[13] Schechter, M., A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc., 331(1992), 681-703.
[14] Schechter, M., Linking Methods in Critical Point Theory, Birkhauser, Basel, 1999.
[15] Torres, P.J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190(2003), 643-662.


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