ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Ekeland's Variational Principle for Functions Unbounded from below

Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 553--558 | DOI:10.5890/DNC.2020.12.008

R. Sengupta$^{1}$ , S. Zhukovskiy$^{2}$

$^1$ Faculty of Science, Peoples' Friendship University of Russia, 117198, Moscow, Mikluho-Maklaya st., 6

$^2$ V. A. Trapeznikov Institute of Control Sciences of RAS,117997, Moscow, Profsoyuznaya st., 65

Abstract

A modification of the Ekeland variational principle for functions unbounded from below is obtained. For a wide class of differentiable functions not necessarily bounded below, it is shown that there exists a minimizing sequence satisfying the first-order necessary conditions, up to any desired approximation.

Acknowledgments

The research is supported by the Volkswagen Foundation and the Russian Foundation for Basic Research (Projects No 20-31-70013, 19-01-00080). The results in Section 3 are due to the second author who was supported by the Russian Science Foundation (Project No 20-11-20131).

References

1.  [1] Ekeland, I. (1974), On the variational principle, J. Math. Anal. Appl., 47, 324-353.
2.  [2] Aubin, J.P. and Ekeland�� I. (1984), Applied Nonlinear Analysis, J. Wiley \& Sons, N.Y.
3.  [3] Arutyunov, A.V., Gel'man, B.D., Zhukovskiy, E.S., and Zhukovskiy, S.E. (2019), Caristi-like condition. Existence of solutions to equations and minima of functions in metric spaces, Fixed Point Theory, 20(1), 31-58.
4.  [4] Vinter, R. (2000), Optimal Control, Birkhauser, Boston.
5.  [5] Arutyunov, A.V. and Tynyanskii, N.T. (1985), The maximum principle in a problem with phase constraints, Soviet Journal of Computer and System Sciences, 23, 28-35.
6.  [6] Arutyunov, A.V., de Oliveira, V.A., Pereira, F.L., Zhukovskiy, E.S., and Zhukovskiy, S.E. (2015), On the solvability of implicit differential inclusions, Applicable Analysis, 94(1), 129-143.
7.  [7] Caristi, J. (1976), Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., 215, 241-251.
8.  [8] Granas, A. and Dugundji, J. (2003), Fixed Point Theory, Springer-Verlag, N.Y.
9.  [9] Khamsi, M.A. (2009), Remarks on Caristi's fixed point theorem, Nonlinear Anal., Theory Methods Appl., 71(1-2), 227-231.
10.  [10] Arutyunov, A.V., Avakov, E.R., and Zhukovskiy, S.E. (2015), Stability theorems for estimating the distance to a set of coincidence points, SIAM Journal on Optimization, 25(2), 807-828.
11.  [11] Arutyunov, A.V., Zhukovskiy, S.E., and Pavlova, N.G. (2013), Equilibrium price as a coincidence point of two mappings, Comput. Math. Math. Phys., 53(2), 158-169.
12.  [12] Borwein, J.M. and Preiss, D. (1987), A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Am. Math. Soc., 303(2), 517-527.
13.  [13] Arutyunov, A.V., Zhukovskiy, E.S., and Zhukovskiy, S.E. (2015), Coincidence points principle for mappings in partially ordered spaces, Topology and its Applications, 179(1), 13-33.
14.  [14] Arutyunov, A.V. and Zhukovskiy, S.E. (2018), Variational principles in nonlinear analysis and their generalization, Mathematical Notes, 103(5-6), 1014�C1019.
15.  [15] Arutyunov, A.V., Zhukovskiy, E.S., and Zhukovskiy, S.E. (2019), Caristi-like condition and the existence of minima of mappings in partially ordered spaces, Journal of Optimization Theory and Applications, 180(1), 48-61.
16.  [16] Arutyunov, A.V., Greshnov, A.V., Lokoutsievskii, L.V., and Storozhuk, K.V. (2017), Topological and geometrical properties of spaces with symmetric and nonsymmetric f-quasimetrics, Topology and ita Applications, 221, 178-194.
17.  [17] Zhukovskaya, Z.T. and Zhukovskiy, S.E. (2018), On generalizations and applications of variational principles of nonlinear analysis, Tambov University Reports. Series: Natural and Technical Sciences, 23(123), 377-385. [in Russian].