Abstract

The aims of this paper is to give some new theorems in the field of fixed point theory. For that, we establish a generalized result of Caristi’s fixed point theorem by introducing a new type of functions that will be called the LZ-functions. And since that theorem is equivalent to Ekeland’s variational principle, we derive also an "- variational-type principle, which generalizes the latter. As application, we study the existence of solution for a system of equilibrium problem.

References

• Harjani, J., López, B., & Sadarangani, K. (2011). Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Analysis: Theory, Methods & Applications, 74(5), 1749-1760.
• Mezník, I. (2003, September). Banach fixed point theorem and the stability of the market. In Proceedings of the International Conference on Mathematics Education into the 21st Century Project.
• Cataldo, A., Lee, E., Liu, X., Matsikoudis, E., & Zheng, H. (2006, July). A constructive fixed-point theorem and the feedback semantics of timed systems. In Discrete Event Systems, 2006 8th International Workshop on (pp. 27-32). IEEE.
• Abdeljawad, T., Aydi, H., & Karapinar, E. (2012). Coupled fixed points for Meir-Keeler contractions in ordered partial metric spaces. Mathematical Problems in Engineering, 2012.
• Turinici, M. (1986). Abstract comparison principles and multivariable Gronwall-Bellman inequalities. Journal of Mathematical Analysis and Applications, 117(1), 100-127.
• Banach, Stefan, "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales", Fund. Math 3, 1 (1922), pp. 133–181.
• Nadler Jr, S. B. (1969). Multi-valued contraction mappings. Pacific J. Math, 30(2), 475-488.
• Aubin JP, Frankowska H: Set-Valued Analysis. Birkhäuser, Boston; 1990
• Aubin JP, Siegel J: Fixed points and stationary points of dissipative multivalued maps. Proc. Am. Math. Soc. 1980, 78: 391–398. 10.1090/S0002-9939-1980-0553382-1
• C´ iric´ L: Multi-valued nonlinear contraction mappings. Nonlinear Anal. 2009, 71: 2716–2723. 10.1016/j.na.2009.01.116
• Covitz H, Nadler SB: Multi-valued contraction mappings in generalized metric spaces. Isr. J. Math. 1970, 8: 5–11. 10.1007/BF02771543
• Takahashi W: Existence theorems generalizing fixed point theorems for multivalued mappings. Pitman Res. Notes Math. Ser. 252. In Fixed Point Theory and Applications. Edited by: Baillon JB, Théra M. Longman Sci. Tech., Harlow; 1991:397–406. (Marseille, 1989)
• M. Edelstein, An extension of Banach’s Contraction principle, AMS Proc. 12, No. 1 , 7 - 10 (1961)
• Lj. C´ iric´, Generalized contractions and fixed-point theorems. Publ. Inst. Math.. 12(26), 19–26 (1971)
• Brondsted, A: Fixed point and partial orders. Proc. Am. Math. Soc. 60, 365-366 (1976)
• Browder, FE: On a theorem of Caristi and Kirk. In: Fixed Point Theory and Its Applications (Proc. Sem., Dalhousie University, 1975), pp. 23-27. Academic Press, San Diego (1976)
• Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241-251 (1976)
• Kirk,WA, Caristi, J: Mapping theorems in metric and Banach spaces. Bull. Acad. Pol. Sci. 23, 891-894 (1975)
• Siegel, J: A new proof of Caristi’s fixed point theorem. Proc. Am. Math. Soc. 66, 54-56 (1977)
• C´ iric´, Lj, Fixed point theorems for mappings with a generalized contractive iterate at a point, Publ. Inst. Math. (Beograd) (N.S.) 13(27) (1972), 11–16
• L. F. Guseman, Jr., Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc, 26 (1970), 615-618
• A. Brondsted : On a lemma of Bishop and Phelps. Pac. J. Math.. 55, 335–341 (1974)
• R. Kannan, Some results on fixed points. II. American Mathematical Monthly. 76, 405–408 (1969).
• Brézis H. and Browder F. E., A general principle on ordered sets in nonlinear functional analysis, Ado. Math. 21, 355-364 (1976)
• Szàz À, An improved Altman type generalization of the Brézis-Browder ordering principle. Mathematical Communications, 2007, Vol. 12, No 2, p. 155-161.
• I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (3) (1974) 324–353.
• A. Hamel, C. Tammer, Minimal elements for product orders, Optimization 57 (2) (2008) 263–275.
• D.G. De Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Inst. Fund. Res. Lectures Math. Phys., vol. 81, Springer-Verlag, Berlin, 1989.
• A. Göpfert, H. Riahi, C. Tammer, C. Zalinescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003.
• Bianchi, M., Kassay, G., Pini, R.: Existence of equilibria via Ekeland’s principle. J. Math. Anal. Appl. 305, 502–512 (2005)
• I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.), 1 (1979), pp. 443–474.