Ekeland variational principle on weighted graphs

Alfuraidan, Monther; Khamsi, Mohamed

doi:10.1090/proc/14642

Ekeland variational principle on weighted graphs
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by Monther Rashed Alfuraidan and Mohamed Amine Khamsi PDF

Proc. Amer. Math. Soc. 147 (2019), 5313-5321 Request permission

Abstract:

In this work, we give a graphical version of the Ekeland variational principle which enables us to discover a new version of the Caristi fixed point theorem in weighted digraphs not necessarily generated by a partial order. Then we show that both graphical versions of the Ekeland variational principle and Caristi’s fixed point theorem are equivalent. In addition, we applied our main result on a differential structure Banach space.

References

M. R. Alfuraidan and M. A. Khamsi, Caristi fixed point theorem in metric spaces with a graph, Abstr. Appl. Anal. , posted on (2014), Art. ID 303484, 5. MR 3182273, DOI 10.1155/2014/303484
Arne Brøndsted, On a lemma of Bishop and Phelps, Pacific J. Math. 55 (1974), 335–341. MR 380343, DOI 10.2140/pjm.1974.55.335
Arne Brøndsted, Fixed points and partial orders, Proc. Amer. Math. Soc. 60 (1976), 365–366. MR 417867, DOI 10.1090/S0002-9939-1976-0417867-X
James Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241–251. MR 394329, DOI 10.1090/S0002-9947-1976-0394329-4
James Caristi, Fixed point theory and inwardness conditions, Applied nonlinear analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978) Academic Press, New York-London, 1979, pp. 479–483. MR 537557
Ivar Ekeland, Sur les problèmes variationnels, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1057–A1059 (French). MR 310670
D. G. de Figueiredo, Lectures on the Ekeland variational principle with applications and detours, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 81, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1989. MR 1019559
Jacek Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1359–1373. MR 2367109, DOI 10.1090/S0002-9939-07-09110-1
M. A. Khamsi, Remarks on Caristi’s fixed point theorem, Nonlinear Anal. 71 (2009), no. 1-2, 227–231. MR 2518029, DOI 10.1016/j.na.2008.10.042
M. J. Rossman and R. J. Twery, A solution to the travelling salesman problem by combinatorial programming [abstract], Operations Research 6 (1958) 897.
Francis Sullivan, A characterization of complete metric spaces, Proc. Amer. Math. Soc. 83 (1981), no. 2, 345–346. MR 624927, DOI 10.1090/S0002-9939-1981-0624927-9
W. D. Wallis, A beginner’s guide to graph theory, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2319463, DOI 10.1007/978-0-8176-4580-9