Critical point localization theorems via Ekeland’s variational principle


In this paper some existence results for critical points of extremum in conical annular regions are established by Ekeland’s variational principle. An application to two-point boundary value problems is included to illustrate the theory.


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



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R. Precup, Critical point localization theorems via Ekeland’s variational principle, Dynamic Systems and Applications, 22 (2013), pp. 355-370


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Dynamic Systems and Applications

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Academic Publications

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[1] R. Agarwal, M. Meehan, D. O’Regan and R. Precup, Location of nonnegative solutions for differential equations on finite and semi-infinite intervals, Dynam. Systems Appl., 12:323–331, 2003.
[2] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
[3] I. Ekeland, Sur les problemes variationnels, C. R. Acad. Sci. Paris Ser. A-B, 275:1057–1059, 1972.
[4] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47:324–353, 1974.
[5] L. H. Erbe, S. Hu and H. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184:640–648, 1994.
[6] L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120:743–748, 1994.
[7] A. Fonda and J. Mawhin, Quadratic forms, weighyed eigenfunctions and boundary value problems for nonlinear second order ordinary differential equations, Proc. Royal Soc. Edinburgh, 112A:145–153, 1989.
[8] D. Guo and V. Laksmikantham, Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces, J. Math. Anal. Appl., 129:211–222, 1988.
[9] J. Mawhin, Nonlinear variational two-point boundary value problems, in Variational Methods, H. Berestycki, J.-M. Coron and I. Ekeland eds., Birkh¨auser, 1990, pp 209–218.
[10] D. O’Regan and R. Precup, Compression-expansion fixed point theorem in two norms and applications, J. Math. Anal. Appl., 309:383–391, 2005.
[11] R. Precup, Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal., 75:834–851, 2012.
[12] R. Precup, Abstract weak Harnack inequality, multiple fixed points and p-Laplace equations, J. Fixed Point Theory Appl., doi 10.1007/s11784-012-0091-2.
[13] R. Precup, On a bounded critical point theorem of Schechter, Stud. Univ. Babes–Bolyai Math., 58:87–95, 2013.
[14] M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc., 331:681–703, 1992.
[15] M. Schechter, Linking Methods in Critical Point Theory, Birkh¨auser, Basel, 1999.

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