Ekeland's variational principle in complete quasi-G-metric spaces

Hashemi, E. ¹ ; Ghaemi, M. B. ²

¹ Department of Mathematics, College of Basic Sciences, Karaj Branch, Islamic Azad University, Alborz, Iran
² Department of Mathematics, Iran University of Science and Technology, Tehran, Iran

Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 3, p. 184-191.

Résumé

In this paper, by concept of $\Gamma$-function which is define on q-G-m (quasi-$G$-metric) space, we establish a generalized Ekeland's variational principle in the setting of lower semicontinuous from above. As application we prove generalized flower petal theorem in q-G-m.

Détail

DOI : 10.22436/jnsa.012.03.06

Classification : 54H25, 54C60
Keywords: \( \Gamma\)-Function, q-G-m space, generalized EVP, lower semicontinuous from above function, generalized Caristi's (common) fixed point theorem, nonconvex minimax theorem, generalized flower petal theorem

Bibliographie

[1] Amemiya, M.; Takahashi, W. Fixed point theorems for fuzzy mappings in complete metric spaces, Fuzzy Sets and system, Volume 125 (2002), pp. 253-260 | DOI

[2] Aubin, J.-P.; J. Siegel Fixed points and stationary points of dissipative multivalued maps, Proc. Amer. Math. Soc., Volume 78 (1980), pp. 391-398 | Zbl | DOI

[3] Bae, J. S. Fixed point theorems for weakly contractive multivalued maps, J. Math. Anal. Appl., Volume 284 (2003), pp. 690-697 | DOI

[4] Bae, J. S.; Cho, E. W.; S. H. Yeom A generalization of the Caristi-Kirk fixed point theorem and its application to mapping theorems , J. Korean Math. Soc., Volume 31 (1994), pp. 29-48

[5] J. Caristi Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., Volume 215 (1976), pp. 241-251 | DOI

[6] Chen, Y.; Cho, Y. J.; Yang, L. Note on the results with lower semi-continuity, Bull Korean Math. Soc., Volume 39 (2002), pp. 535-541 | DOI

[7] Daffer, P. Z.; Kaneko, H.; Li, W. Variational principle and fixed points, in: Set Valued Mappings With Applications in Nonlinear Analysis, Volume 2002 (2002), pp. 129-136

[8] Dancs, S.; Hegedüs, M.; P. Medvegyev A general ordering and fixed point principle in complete metric spaces, Acta. Sci. Math. (Szeged), Volume 46 (1983), pp. 381-388

[9] Ekeland, I. Nonconvex minimization problems, Bull. Amer. Math. Soc., Volume 1 (1979), pp. 443-474 | DOI

[10] I. Ekeland On the variational principle, J. Math. Anal. Appl., Volume 47 (1974), pp. 324-353 | DOI

[11] Ekeland, I. Remarques sur les problémes variationnels , C. R. Acad. Sci. Paris Sér. A–B, Volume 275 (1972), pp. 1057-1059

[12] Gajek, L.; D. Zagrodny Geometric variational principle, Dissertationes Math. (Rozprawy Mat.), Volume 340 (1995), pp. 55-71

[13] A. Hamel Remarks to an equivalent formulation of Ekelands variational principle, Optimization, Volume 31 (1994), pp. 233-238 | Zbl | DOI

[14] Hamel, A.; A. Löhne A minimal point theorem in uniform spaces, in: Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, Volume 1, 2 (2003), pp. 557-593

[15] Hyers, D. H.; Isac, G.; Rassias, T. M. Topics in Nonlinear Analysis and Applications, World Scientific Publishing Co., River Edge, 1997 | DOI

[16] Kada, O.; Suzuki, T.; W. Takahashi Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., Volume 44 (1996), pp. 381-391

[17] Kijima, Y. On a minimization theorem, in: Nonlinear Analysis and Convex Analysis (Japanese), Volume 1995 (1995), pp. 59-62

[18] Mizoguchi, N.; Takahashi, W. Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., Volume 141 (1989), pp. 177-188 | DOI

[19] Mustafa, Z.; Sims, B. A new approach to generalized metric spaces , J. Nonlinear Convex Anal., Volume 7 (2006), pp. 289-392

[20] J.-P. Penot The drop theorem, the petal theorem and Ekelands variational principle, Nonlinear Anal., Volume 10 (1986), pp. 813-822 | DOI

[21] Saadati, R.; Vaezpoura, S. M.; P. Vetro; Rhoades, B. E. Fixed point theorems in generalized partially ordered G-metric spaces , Math. Comput. Modelling, Volume 52 (2010), pp. 797-801 | DOI

[22] Shioji, N.; Suzuki, T.; W. Takahashi Contractive mappings, Kannan mappings and metric completeness , Proc. Amer. Math. Soc., Volume 126 (1998), pp. 3117-3124 | DOI | Zbl

[23] T. Suzuki Generalized distance and existence theorems in complete metric spaces, J. Math. Anal. Appl., Volume 253 (2001), pp. 440-458 | DOI

[24] Suzuki, T. On DowningKirks theorem , J. Math. Anal. Appl., Volume 286 (2003), pp. 453-458 | DOI

[25] T. Suzuki Generalized Caristis fixed point theorems by Bae and others, J. Math. Anal. Appl., Volume 302 (2005), pp. 502-508 | DOI | Zbl

[26] Suzuki, T.; Takahashi, W. Fixed point theorems and characterizations of metric completeness, Topol. Methods Nonlinear Anal., Volume 8 (1996), pp. 371-382

[27] Takahashi, W. Existence theorems generalizing fixed point theorems for multivalued mappings , in: Fixed Point Theory and Applications, Volume 1991 (1991), pp. 397-406 | DOI

[28] Takahashi, W. Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000

[29] Tataru, D. Viscosity solutions of HamiltonJacobi equations with unbounded nonlinear terms, J. Math. Anal. Appl., Volume 163 (1992), pp. 345-392 | DOI

[30] C.-K. Zhong On Ekelands variational principle and a minimax theorem, J. Math. Anal. Appl., Volume 205 (1997), pp. 239-250 | DOI