Fixed Points of α-Admissible Mappings in Cone Metric Spaces with Banach Algebra

Malhotra, S.K.; Sharma, J.B.; Shukla, Satish

Research Article

Fixed Points of α-Admissible Mappings in Cone Metric Spaces with Banach Algebra

S.K. Malhotra
J.B. Sharma
Satish Shukla¹*

Shri Vaishnav Institute of Technology and Science, Gram Baroli, Indore, M.P. 453331, India.

* Corresponding author

Abstract

In this paper, we introduce the $\alpha$-admissible mappings in the setting of cone metric spaces equipped with Banach algebra and solid cones. Our results generalize and extend several known results of metric and cone metric spaces. An example is presented which illustrates and shows the significance of results proved herein.

How to Cite

S. K. Malhotra, J. B. Sharma, S. Shukla, Fixed Points of α-Admissible Mappings in Cone Metric Spaces with Banach Algebra, Int. J. Anal. Appl. 9 (2015), 9–18.

BibTeX

@article{Malhotra2015,
  author  = {Malhotra, S.K. and Sharma, J.B. and Shukla, Satish},
  title   = {Fixed Points of α-Admissible Mappings in Cone Metric Spaces with Banach Algebra},
  journal = {Int. J. Anal. Appl.},
  volume  = {9},
  year    = {2015},
  pages   = {9--18},
  doi     = {}
}

References

[1] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2003) 1435-1443.
[2] B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Analysis, 75 (2012) 2154-2165.
[3] H. C ¸ akalli, A. S ¨onmez, C ¸. Gen ¸c, On an equivalence of topological vector space valued cone metric spaces and metric spaces, Appl. Math. Lett., 25, (2012) 429-433.
[4] H. Liu and S.-Y. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl., 2013, 2013:320.
[5] H. Liu and S.-Y. Xu, Fixed point theorems of quasi-contractions on cone metric spaces with Banach algebras, Abstarct and Applied Analysis, Volume 2013, Article ID 187348, 5 pages.
[6] J.J. Nieto, R. Rodr ´iguez-L ´opez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239.
[7] L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007) 1468-1476.
[8] S. Radenovi ´c, B.E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comput. Math. Appl., 57, 1701-1707 (2009)
[9] S. Xu, S. Radenovi ´c, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl., 2014, 2014:102.
[10] Sh. Rezapour and R. Hamlbarani, Some notes on the paper Cone metric spaces and fixed point theorems of contractive mappings, Math. Anal. Appl., 345 (2008), 719-724.
[11] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, 1991.
[12] W.A. Kirk, P.S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4(1)(2003), 79-89.
[13] W.S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal., 72(5), (2010) 2259-2261.
[14] Y. Feng, W. Mao, The equivalence of cone metric spaces and metric spaces, Fixed Point Theory, 11(2), (2010) 259-264.
[15] Z. Kadelburg, M. Pavlovi ´c, S. Radenovi ´c, Common fixed point theorems for ordered contractions and quasi-contractions in ordered cone metric spaces, Comput. Math. Appl. 59, 3148-3159 (2010)
[16] Z. Kadelburg, S. Radenovi ´c, V. Rakoˇcevi ´c, A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett., 24, (2011) 370-374.