Weak c-Comparison Function and a Fixed Point Theorem in Cone Metric Spaces over Banach Algebras

Abstract

In this paper, we first give the notion of weak c-comparison function, and prove some associated fixed point theorems containing classes of many well known theorems in the setting of cone metric spaces over Banach algebras with solid cones.

Keywords

• Cone Metric Space,Banach Algebras

Kaynakça

1. S. Banach, \textit{Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales}. Fundam. Math., 1922 \textbf{(3)}: 133–181.
2. R. Kannan, \textit{Some results on fixed points}. Bull.Calcultta Math.Soc., 1968 \textbf{(10)}: 71-76.
3. S.K. Chatterjea, \textit{Fixed Point Theorems}. C.R.Acad.Bulgare Sci., 1972 \textbf{(25)}: 727-730.
4. T. Zamfirescu, \textit{Fixed Point theorems in metric spaces}. Arch.Math.(Basel), 1972 \textbf{(23)}: 292-298.
5. V. Berinde, \textit{Approximating fixed points of weak contractions using the Picard iterations}. Nonlinear Anal. Forum, 2004 \textbf{(9)}: 43-53.
6. M.A Alghamdi, V. Berinde and N. Shahzad, \textit{Fixed points of non-self almost contractions}. Carp. J.Math., 2014 \textbf{(30)}: 7-14.
7. V. Berinde and M. Pacurar, \textit{Fixed points and continuity of almost contractions}. Fixed Point Theory, 2008 \textbf{(9)}: 23-34.
8. T. Suzuki, \textit{Fixed point theorems for Berinde mappings}. Bull. Kyushu Inst. Tech. Pure Appl. Math., 2011 \textbf{(58)}: 13-1

9. J. Tiammee, Y.J. Cho and S. Suanta, \textit{Fixed point theorems for nonself G-almost contractive mappings in Banach spaces endowed with graphs}. Carpathian J.Math., 2016 \textbf{(32)}: 375-382.
10. L.G. Huang and X. Zhang, \textit{Cone metric spaces and fixed point theorems of contractive mappings}. J. Math. Anal. Appl.,2007 \textbf{(332)}: 1468-1476.
11. M. Abbas, P. Vetro and S.H. Khan, \textit{On fixed points of Berinde's contractive mappings in cone metric spaces}. Carp.J.Math., 2010 \textbf{(26)}: 121-133.
12. W.S. Du, \textit{A note on cone metric fixed point theory and its equivalence}. Nonlinear Anal., 2010 \textbf{(72)}: 2259-2261.
13. S. Radenovic, S. Simic, N. Cakic and Z. Golubovic, \textit{A note on tvs-cone metric fixed point theory}. Math. Comp. Mod., 2011 \textbf{(54)}: 2418-2422.
14. H. Liu and S. Xu \textit{Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings}. Fix. P. Theory Appl., 2013, 10 pages.
15. H. Huang and S. Radenovic, \textit{Common fixed point theorems of generalized Lipschitz mappings in cone b-metricspaces over Banach algebras and applications}. J. Nonlinear Sci. Appl., 2015 \textbf{(8)}: 787-799.
16. S. Shukla, S. Balasubramanian and M. Pavlovic, \textit{A Generalized Banach Fixed Point Theorem}. Bull. Malays. Math. Sci. Soc., 2016 \textbf{(39)}: 1529-1539.
17. S. Xu and S. Radenovic, \textit{Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality}. Fix. P. Theory Appl., 2014, 12 pages.
18. M. Özavşar, \textit{Fixed point theorems for (k,l)-almost contractions in cone metric spaces over Banach algebras}. Math. Adv. Pur. Appl. Sci., 2018 \textbf{(1)}: 46-51.
19. W. Rudin, \textit{Functional Analysis}, 2nd edn. McGraw-Hill, New York 1991.
20. Sh. Rezapour and R. Hamlbarani, \textit{Some notes on the paper cone metric spaces and fixed point theorems of contractive mappings}. J. Math. Anal. Appl., 2008 \textbf{(345)}: 719–724.
21. S. Radenovic and B.E. Rhoades, \textit{Fixed point theorem for two non-self mappings in cone metric spaces}. Comput. Math.Appl., 2009 \textbf{(57)}: 1701-1707. Z.
22. Kadelburg and S. Radenovic, \textit{A note on various types of cones and fixed point results in cone metric spaces}. Asian J. Math. Appl., 2013, Article ID ama0104, 7 pages.
23. S. Jankovic, Z. Kadelburg and S. Radenovic, \textit{On cone metric spaces: a survey}. Nonlinear Anal., 2011 \textbf{(74)}: 2591-2601.
24. V. Berinde, \textit{Iterative Approximation of Fixed Points}. Editura Efemeride, Baia Mare, 2002.
25. V. Berinde, \textit{Approximating fixed points of weak $\varphi$-contractions using the Picard iterations}. Fixed Point Theory, 2003, 131-147.
26. B. Li and H. Huang, \textit{Fixed point results for weak $\varphi$-contractions in cone metric spaces over Banach algebras and applications}. Journal of Function Spaces, 2017, 6 pages.
27. H. Huang, G. Deng and S. Radenovic, \textit{Some topological properties and fixed point results in cone metric spaces over Banach algebras}. Positivity, 2019.