Ordered weak $\varphi$-contractions in cone metric spaces over Banach algebras and fixed point theorems

Öz

In this  work, we introduce the class of ordered weak \varphi-contractions in cone metric spaces over Banach algebras and prove some fixed point results for the mappings belonging to this new class. Our results generalize and extend some known fixed point results in cone metric spaces to the spaces equipped with a partial order. Some examples are given which illustrate the results proved herein.

Anahtar Kelimeler

• Cone metric space,ordered weak $\varphi$-contraction,fixed point

Kaynakça

1. [2] B. Li, H. Huang, Fixed point results for weak phi-contractions in cone metric spaces over Banach algebras and applications, Journal of Function Spaces, Volume 2017(2017), Article ID 5054603, 6 pages.
2. [9] V. Berinde, Generalized contractions and applications (Romannian), Editura CubPress 22, Baia Mare (1997).
3. [1] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Sco., 132 (2004),1435-1443.
4. [11] I. A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, (2011), 198 pp, ISBN 973-8095-71-9.
5. [21] W. Rudin, Functional Analysis, 2nd edn. McGraw-Hill, New York (1991).
6. [23] Y. Feng, W. Mao, The equivalence of cone metric spaces and metric spaces, Fixed Point Theory, 11(2), (2010) 259-264.
7. [3] H. Cakall, A. Sonmez, C.Genc, On an equivalence of topological vector space valued cone metric spaces and metric spaces, Appl. Math. Lett., 25, (2012) 429-433.
8. [4] J.J. Nieto and R.R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation, Order (2005), 223-239.

9. [5] L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2) (2007), 1468--1476.
10. [6] M. Dordevic, D. Doric, Z. Kadelburg, S. Radenovic, D. Spasic, Fixed point results under c-distance in tvs-cone metric spaces, Fixed Point Theory Appl., 2011 (2011), 9 pages.
11. [7] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math., 3 (1922), 133--181.
12. [8] S. Jankovic, Z. Kadelburg, S. Radenovic, On cone metric spaces: A survey, Nonlinear Anal., 74 (7) (2011), 2591--2601.
13. [10] I. A. Rus, Generalized contractions, Seminar on Fixed Point Theory, 3 (1983),1--130.
14. [12] H. Liu, S. Xu, Cone metric spaces over Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl., 2013 (2013), 10 pages.
15. [13] H. Huang, S. Radenovic, Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications, J. Nonlinear Sci. Appl., 8 (5) (2015), 787--799.
16. [14] H. Huang, S. Radenovic, Some Fixed point results of generalized Lipschitz mappings on cone b-metric spaces over Banach algebras, J. Comput. Anal. Appl., 20 (3)(2016), 566--583.
17. [15] H. Huang, S. Radenovic, G. Deng, A sharp generalization on cone b-metric spaceover Banach algebra, J. Nonlinear Sci. Appl., 10 (2) (2017), 429--435.
18. [16] H. Huang, S. Xu, H. Liu, S. Radenovic, Fixed point theorems and T-stability of Picard iteration for generalized Lipschitz mappings in cone metric spaces over Banach algebras, J. Comput. Anal. Appl., 20 (5) (2016), 869{888.
19. [17] S.K. Malhotra, J.B. Sharma, S. Shukla, Fixed points of alpha-admissible mappings in cone metric spaces with Banach algebra, Inter. J. Anal. Appl., 9 (1) (2015), 9--18.
20. [18] S.K. Malhotra, J.B. Sharma, S. Shukla, Some fixed point theorems for G-contractions in cone b-metric spaces over Banach algebra, J. Nonlinear Funct. Anal.2017 (2017), Article ID 9, pages 17.
21. [19] S. Reich and A.J. Zaslavski, The set of non-contractive mappings is -porous in the space of all nonexpansive mappings, C. R. Acad. Sci. Paris 333 (2001), 539-541.
22. [20] S. Shukla, S. Balasubramanian, M. Pavlovic, A generalized Banach fixed point theorem, Bull. Malays. Math. Sci. Soc., 39 (4) (2016), 1529--1539.
23. [22] W.S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal., 72(5), (2010) 2259-2261.
24. [24] Z. Kadelburg, S. Radenovic, V. Rakocevic, A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett., 24, (2011) 370-374.15