Fixed Point Theorems for $(k,l)$-Almost Contractions in Cone Metric Spaces over Banach Algebras

Abstract

In this paper, we first introduce the notion of $(k,l)$- weak contraction in the setting of cone metric spaces over Banach algebras. Next we prove that the class of such mappings contains those of Kannan, Chatterjea and Ciric type contractions in this new setting. Morever, by proving a fixed point theorem for such a mapping, we provide some significant extensions of the well known results in the metric fixed point theory.

Keywords

• Cone metric space,Banach algebra,Spectral Radius,Berinde Mapping,Fixed Point

References

1. \bibitem{VB} V.Berinde, \textit{Approximating fixed points of weak contractions using the Picard iterations}. Nonlinear Anal. Forum. 2004(\textbf{9}), 43-53.
2. \bibitem{RK} R.Kannan, \textit{Some results on fixed points}. Bull.Calcultta Math.Soc. 1968(\textbf{10}), 71-76.
3. \bibitem{Za}T.Zamfirescu, \textit{Fixed Point theorems in metric spaces}. Arch.Math.(Basel) 1972(\textbf{23}), 292-298.
4. \bibitem{Cir} Lj.B.Ciric, \textit{A generalization of Banach’s contraction principle}. Proc. Am.Math. Soc. 1974 (\textbf{45}), 267-273.
5. \bibitem{AL} M.A.Alghamdi, V.Berinde and N.Shahzad, \textit{Fixed points of non-self almost contractions}. Carp. J.Math. 2014 (\textbf{30}),
6. \bibitem{VBa} V.Berinde and M.Pacurar, \textit{Fixed points and continuity of almost contractions}. Fixed Point Theory. 2008 (\textbf{9})
7. \bibitem{TSU}T. Suzuki, \textit{Fixed point theorems for Berinde mappings}. Bull. Kyushu Inst. Tech. Pure Appl. Math. 2011(\textbf{58})
8. \bibitem{JT} J.Tiammee, Y.J.Cho and S.Suanta, \textit{Fixed point theorems for nonself G-almost contractive mappings in Banach spaces endowed with graphs}. Carp.J.Math. 2016(\textbf{32}), 375-382.

9. \bibitem{Hu} Huang, L.G. and Zhang, X., \textit{Cone metric spaces and fixed point theorems of contractive mappings}, J. Math. Anal. Appl. 2007(\textbf{332}), 1468-1476.
10. \bibitem{MA} M.Abbas, P.Vetro and S.H.Khan, \texit{On fixed points of Berinde's contractive mappings in cone metric spaces}. Carp.J.Math. 2010(\textbf{26}), 121-133
11. \bibitem{Du} W.S.Du, \textit{A note on cone metric fixed point theory and its equivalence}. Nonlinear Anal. 2010(\textbf{72}), 2259-2261.
12. \bibitem{Sto} 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
13. \bibitem{Liu} 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.
14. \bibitem{Cha} S.K.Chatterjea, \textit{Fixed Point Theorems}. C.R.Acad.Bulgare Sci. 1972 (\textbf{25}), 727-730.
15. \bibitem{HHu} H.Huang and S.Radenovic, \textit{Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications}. J. Nonlinear Sci. Appl. 2015 (\textbf{8}), 787-799
16. \bibitem{Sa} 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. \bibitem{Xu} 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. \bibitem{Ru} W.Rudin, \textit{Functional Analysis}, 2nd edn. McGraw-Hill, New York 1991.