Browse > Article
http://dx.doi.org/10.4134/BKMS.2011.48.5.1023

AN EKELAND TYPE VARIATIONAL PRINCIPLE ON GAUGE SPACES WITH APPLICATIONS TO FIXED POINT THEORY, DROP THEORY AND COERCIVITY  

Bae, Jong-Sook (Department of Mathematics Moyngji University)
Cho, Seong-Hoon (Department of Mathematics Hanseo University)
Kim, Jeong-Jin (Department of Mathematics Moyngji University)
Publication Information
Bulletin of the Korean Mathematical Society / v.48, no.5, 2011 , pp. 1023-1032 More about this Journal
Abstract
In this paper, a new Ekeland type variational principle on gauge spaces is established. As applications, we give Caristi-Kirk type fixed point theorems on gauge spaces, and Dane$\check{s}$' drop theorem on seminormed spaces. Also, we show that the Palais-Smale condition implies coercivity on semi-normed spaces.
Keywords
variational principle; gauge space; fixed point; drop theorem Palais-Smale condition;
Citations & Related Records

Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 0
연도 인용수 순위
  • Reference
1 W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
2 M. Turinici, Mapping theorems via variable drops in Banach spaces, Istit. Lombardo Accad. Sci. Lett. Rend. A 114 (1980), 164-168.
3 X. Y. Zheng, A drop theorem in topological linear spaces, Chinese Ann. Math. Ser. A 21 (2000), no. 2, 141-148.
4 J. R. Giles and D. N. Kutzarova, Characterisation of drop and weak drop properties for closed bounded convex sets, Bull. Austral. Math. Soc. 43 (1991), no. 3, 377-385.   DOI
5 J. R. Giles, B. Sims, and A. C. Yorke, On the drop and weak drop properties for a Banach space, Bull. Austral. Math. Soc. 41 (1990), no. 3, 503-507.   DOI
6 A. Gopfert, Chr. Tammer, H. Riahi, and C. Zalinescu, Variational Method in Partially Ordered Spaces, Springer-Verlag, New York, Berlin, Heidelberg, 2003.
7 D. N. Kutzarova, On drop property of convex sets in Banach space, in "Constructive Theory of Functions '87, Sofa", 283-287.
8 R. R. Phelps, Convex Functions, Monotone operators and Differentiability, Springer- Verlag, New York, Berlin, Heidelberg-New York, 1989.
9 S. Li, An existence theorem on multiple critical points and its application in nonlinear PDE, Acta Mathematica Scientica 4 (1984).
10 J. P. Penot, The drop theorem, the petal theorem and Ekeland's variational principle, Nonlinear Anal. 10 (1986), no. 9, 813-822.   DOI   ScienceOn
11 J. Qiu, Local completeness and Drop theorem, J. Math. Anal. Appl. 266 (2002), no. 2, 288-297.   DOI   ScienceOn
12 J. Qiu, A version of Ekeland's variational principle in countable semi-normed spaces, J. Math. Res. Exposition 24 (2004), no. 1, 1-6.
13 J. H. Qiu and S. Rolewicz, Local completeness of locally pseudoconvex spaces and Borwein-Preiss variational principle, Studia Math. 183 (2007), no. 2, 99-115.   DOI
14 J. H. Qiu and S. Rolewicz, Ekeland's variational principle in locally p-convex spaces and related results, Studia Math. 186 (2008), no. 3, 219-235.   DOI
15 S. Rolewicz, On drop property, Studia Math. 85 (1987), 27-35.
16 F. Cammaroto, A. Chinni, and G. Sturiale, A remark on Ekeland's principle in locally convex topological vector spaces, Mathematical and Computer Modeling 30 (1990), 75- 79.
17 L. Cheng, Y. Zhou, and F. Zhang, Danes' drop theorem in locally convex spaces, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3699-3702.   DOI   ScienceOn
18 J. Danes, A geometric theorem useful in nonlinear functional analysis, Boll. Un. Mat. Ital. (4) 6 (1972), 369-372.
19 D. G. De Figueiredo, The Ekeland Variational Principle with Applications and Detours, Tata Institue of Fundamental esearch, Bombay, 1989.
20 J. Danes, Equivalence of some geometric and related results of nonlinear functional anal- ysis, Comment. Math. Univ. Carolin. 26 (1985), no. 3, 443-454.
21 S. Dolecki and J. P. Penot, The Clark's tangent cone and limits of tangent cones, Publ. Math. Pau 2 (1983), 1-11.
22 J. Dugundji, Topology, Ally and Bacon, Boston, 1966.
23 I. Ekeland, Sur les problemes variationnels, C. R. Acad. Sci. Paris Ser. A-B 275 (1972), 1057-1059.
24 I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.   DOI   ScienceOn
25 I. Ekeland, On convex minimization problems, Bull. Amer. Math. Soc. 1 no. 3 (1979), 445-474.
26 R. Espinola and W. A. Kirk, Set-valued contractions and fixed points, Nonlinear Anal. 54 (2003), no. 3, 485-494.   DOI   ScienceOn
27 J. X. Fang and X. Y. Lin, Fixed point theorems for set-valued $\Phi$-generalized contractions on gauge spaces, Nonlinear Anal. 69 (2008), no. 1, 201-207.   DOI   ScienceOn
28 M. Frigon, Fixed point results for generalized contractions in gauge spaces and applica- tion, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2957-2965.   DOI   ScienceOn
29 S. Al-Homidan, Q. H. Ansari, and J. C. Yao, Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Anal. 69 (2008), no. 1, 126-139.   DOI   ScienceOn
30 R. P. Agarwal and D. O.'Regan, Fixed -point theorems for multivalued maps with closed values on complete gauge spaces, Appl. Math. Lett. 14 (2001), no. 7, 831-836.   DOI   ScienceOn
31 J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, Berlin, 1990.
32 M. Bianchi, G. Kassay, and R. Pini, Existence of equilibria via Ekeland's principle, J. Math. Anal. Appl. 305 (2005), no. 2, 502-512.   DOI   ScienceOn
33 H. Brezis and F. E. Browder, A general principle on ordered sets in nonlinear functional analysis, Advances in Math. 21 (1976), no. 3, 355-364.   DOI
34 F. E. Browder, Normal solvability and the Fredholm alternative for mappings into infi- nite dimensional manifolds, J. Funct. Anal. 8 (1971), 250-274.   DOI
35 G. L. Cain Jr and M. Z. Nashed, Fixed points and stability for a sum of two operators in locally convex spaces, Pacific J. Math. 39 (1971), 581-592.   DOI
36 L. Caklovic, S. Li, and M. Willem, A note on Palais-Smale condition and coercivity, Differential Integral Equations 3 (1990), no. 4, 799-800.