CFP last date
20 November 2024
Reseach Article

Caristi’s Fixed Point Theorem and Ekeland’s Variational Principle for Set Valued Mapping using the LZ-functions

by Samih Lazaiz, Mohamed Aamri, Omar Zakary
International Journal of Computer Applications
Foundation of Computer Science (FCS), NY, USA
Volume 158 - Number 2
Year of Publication: 2017
Authors: Samih Lazaiz, Mohamed Aamri, Omar Zakary
10.5120/ijca2017912755

Samih Lazaiz, Mohamed Aamri, Omar Zakary . Caristi’s Fixed Point Theorem and Ekeland’s Variational Principle for Set Valued Mapping using the LZ-functions. International Journal of Computer Applications. 158, 2 ( Jan 2017), 1-6. DOI=10.5120/ijca2017912755

@article{ 10.5120/ijca2017912755,
author = { Samih Lazaiz, Mohamed Aamri, Omar Zakary },
title = { Caristi’s Fixed Point Theorem and Ekeland’s Variational Principle for Set Valued Mapping using the LZ-functions },
journal = { International Journal of Computer Applications },
issue_date = { Jan 2017 },
volume = { 158 },
number = { 2 },
month = { Jan },
year = { 2017 },
issn = { 0975-8887 },
pages = { 1-6 },
numpages = {9},
url = { https://ijcaonline.org/archives/volume158/number2/26877-2017912755/ },
doi = { 10.5120/ijca2017912755 },
publisher = {Foundation of Computer Science (FCS), NY, USA},
address = {New York, USA}
}
%0 Journal Article
%1 2024-02-07T00:03:43.449670+05:30
%A Samih Lazaiz
%A Mohamed Aamri
%A Omar Zakary
%T Caristi’s Fixed Point Theorem and Ekeland’s Variational Principle for Set Valued Mapping using the LZ-functions
%J International Journal of Computer Applications
%@ 0975-8887
%V 158
%N 2
%P 1-6
%D 2017
%I Foundation of Computer Science (FCS), NY, USA
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
  1. 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.
  2. 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.
  3. 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.
  4. Abdeljawad, T., Aydi, H., & Karapinar, E. (2012). Coupled fixed points for Meir-Keeler contractions in ordered partial metric spaces. Mathematical Problems in Engineering, 2012.
  5. Turinici, M. (1986). Abstract comparison principles and multivariable Gronwall-Bellman inequalities. Journal of Mathematical Analysis and Applications, 117(1), 100-127.
  6. Banach, Stefan, "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales", Fund. Math 3, 1 (1922), pp. 133–181.
  7. Nadler Jr, S. B. (1969). Multi-valued contraction mappings. Pacific J. Math, 30(2), 475-488.
  8. Aubin JP, Frankowska H: Set-Valued Analysis. Birkhäuser, Boston; 1990
  9. 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
  10. C´ iric´ L: Multi-valued nonlinear contraction mappings. Nonlinear Anal. 2009, 71: 2716–2723. 10.1016/j.na.2009.01.116
  11. Covitz H, Nadler SB: Multi-valued contraction mappings in generalized metric spaces. Isr. J. Math. 1970, 8: 5–11. 10.1007/BF02771543
  12. 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)
  13. M. Edelstein, An extension of Banach’s Contraction principle, AMS Proc. 12, No. 1 , 7 - 10 (1961)
  14. Lj. C´ iric´, Generalized contractions and fixed-point theorems. Publ. Inst. Math.. 12(26), 19–26 (1971)
  15. Brondsted, A: Fixed point and partial orders. Proc. Am. Math. Soc. 60, 365-366 (1976)
  16. 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)
  17. Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241-251 (1976)
  18. Kirk,WA, Caristi, J: Mapping theorems in metric and Banach spaces. Bull. Acad. Pol. Sci. 23, 891-894 (1975)
  19. Siegel, J: A new proof of Caristi’s fixed point theorem. Proc. Am. Math. Soc. 66, 54-56 (1977)
  20. 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
  21. L. F. Guseman, Jr., Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc, 26 (1970), 615-618
  22. A. Brondsted : On a lemma of Bishop and Phelps. Pac. J. Math.. 55, 335–341 (1974)
  23. R. Kannan, Some results on fixed points. II. American Mathematical Monthly. 76, 405–408 (1969).
  24. Brézis H. and Browder F. E., A general principle on ordered sets in nonlinear functional analysis, Ado. Math. 21, 355-364 (1976)
  25. Szàz À, An improved Altman type generalization of the Brézis-Browder ordering principle. Mathematical Communications, 2007, Vol. 12, No 2, p. 155-161.
  26. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (3) (1974) 324–353.
  27. A. Hamel, C. Tammer, Minimal elements for product orders, Optimization 57 (2) (2008) 263–275.
  28. 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.
  29. A. Göpfert, H. Riahi, C. Tammer, C. Zalinescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003.
  30. Bianchi, M., Kassay, G., Pini, R.: Existence of equilibria via Ekeland’s principle. J. Math. Anal. Appl. 305, 502–512 (2005)
  31. I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.), 1 (1979), pp. 443–474.
Index Terms

Computer Science
Information Sciences

Keywords

Fixed point Set valued map LZ-function Caristi Ekeland