On Generalized Algebraic Cone Metric Spaces and Fixed Point Theorems

Abstract

In this paper, the author first introduces the concept of generalized algebraic cone metric spaces and some elementary results concerning generalized algebraic cone metric spaces. Next, by using these results, some new fixed point theorems on generalized (complete) algebraic cone metric spaces are proved and an example is given. As a consequence, the main results generalize the corresponding results in complete algebraic cone metric spaces and generalized complete metric spaces.

Keywords

Generalized algebraic cone metric space / Canonical decomposition / Fixed point

References

[1] Huang L G, Zhang X. Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl., 2007, 332: 1468-1476

[2] Rezapour S, Hamlbarani R. Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”. J. Math. Anal. Appl., 2008, 345: 719-724

[3] Janković S, Kadelburg Z, Radenović S. On the cone metric space: A survey. Nonlinear Anal., 2011, 74: 2591-2601

[4] Abdeljawad T, Turkoglu D, Abuloha M. Some theorems and examples of cone metric spaces. J. Comput. Anal. Appl., 2010, 12(4): 739-753

[5] Karapinar, E., Fixed point theorems in cone Banach spaces, Fixed Point Theory Appl., 2009, Article ID: 609281.

[6] Turkoglu D, Abuloha M. Cone metric spaces and fixed point theorems in diametrically contractive mappings. Acta Math. Sin., 2010, 26: 489-496

[7] Chaker W, Ghribi A, Jeribi A, Krichen B. Fixed point theorems for (p, q)-quasi-contraction mappings in cone metric spaces. Chin. Ann. Math. Ser. B, 2016, 37(2): 211-220

[8] Liu, H. and Xu, S. Y., Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl., 2013, DOI: https://doi.org/10.1186/1687-1812-2013-320

[9] Xu, S. Y. and Radenović, S., Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory Appl., 2014, DOI: https://doi.org/10.1186/1687-1812-2014-102.

[10] Tootkaboni, M. A., Salec, A. B., Algebraic cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl., 2014, DOI: https://doi.org/10.1186/1687-1812-2014-160.

[11] Diaz J B, Margolis B. A fixed point theorem of the alternative, for contractions on a generalized complete metic space. Bull. Amer. Math. Soc., 1968, 74: 305-309

[12] Jung C F K. On generalized complete metric spaces. Bull. Amer. Math. Soc., 1969, 75: 113-116