# New Conditions of The Existence of Fixed Point in Δ- Ordered Δ Banach Algebra

## DOI:

https://doi.org/10.24297/jam.v18i.8370## Keywords:

Fixed Point, Ordered Banach Algebra, Lipschitz Mapping, Contraction Mappimg## Abstract

The main idea is to construct a new algebra and find new necessary and sufficient conditions equivalent to the existence of a fixed point. In this work, an algebra is constructed, called Δ - ordered Banach algebra, we define convergent in this new space, Topological structure on Δ ordered Banach Algebra and prove this as Hausdorff space. Also, we define new conditions as Δ- *lipshtiz,, Δ- contraction contraction* , in this algebra construct, we prove this condition is the existence and uniqueness results of the fixed point. In this paper, we prove a common fixed point if the self-functions satisfy the new condition which is called *Ф-contraction.*

### Downloads

## References

Akbar Zada, Rahim Shah and Shahid Saifullah,"Some Fixed Point Theorems in Multiplicative Cone b- Matric Spaces", Journal of Analysis & Number Theory, 5, No. 1, 35-39, (2017).

B. Y.Hussein. and S. F.Hassan ,"Some properties of Characterization on ordered Banach algebra",Ms.C. thesis ,Al-Qadisiyah university,2017.

Arun Gary, Zaheer K. Ansari," Cone Metric Spaces and Fixed Point Theorems of T- Contractive Mapping", International journal of Mathematics and Statistics Invention (IJMSI), Vol.2, Iss. 2, PP. 72-76, (2014)

D. Dey and M.Saha," Partial Cone Metric Space and Some Fixed Point Theorems", TWMS J.App. Math. U_3, N.1, PP. 1-9, (2013).

Dhage, B.C.,"A study of some fixed point theorem", Ph. Thesis, Marath Wada University, Aurangabad, (1984).

Gahler, S., " 2- Metriche Raume and Ihre Topologisch Structure", Mathematiche Nachrichten, 26, 115-148, (1963).

G. S. Sulugq and B. P. Tripathi, "Some Common Fixed Point Theorems of Generalized Contractive Mappings in Cone Metric Spaces", Functional Analysis Approximation and Computation, 5:1, 33-41, (2013).

Hao Liu and Shaoman XU, " Conemetric Spaces with Banach Algebras and Fixed Point Theorems of Generalized Lipschitz Mapping, Springre, Fixed Point and Application, 320, (2013).

Huang, L., G., & Zhang X, "Cone Metric Space and Fixed Point Theorem for Contractive Mapping", Journal of Mathematical Analysis and Applications, 332, (1468- 1476).

H. K Nashine and I. Altun, "A Common Fixed Point Theorem on Ordered Metric Spaces", Bulletin of The Iranian Mathematical Society, Vol. 38 No. 4, PP. 925- 934, (2012).

H. Liu and S. Xu," Cone Metric Spaces with Banach Algebra and Fixed Point Theorems of Generalized Lipchitz Mappings," Fixed Point Theory and Applications, 320. Dio:10.118611687-1812-320 (2013).

H. Rahimi and G. Soleimani Rad, "Fixed Point Theorems Under C. Distance in Ordered Cone Metric Space", Int. J. Industrial Mathematics, Vol. 6, No. 2, Article IDISIM-00253, 9 Page, (2014).

Jerolina Fernardcz, Sompob Sale, Kalpana Saxena, Neeraj Malviya and Poom Kuman, " The A- Cone Metric Space Over Banach Algebra with Applications ", Fernardczct. at. Cogent Mathematics, 4:1828690, (2017).

K. P. R Sastry, Ch-Srinivasa Rao, A. Chandra Sekhar and M. Balaiah,"A Fixed Point Theorem in Cone Metric Spaces Under Weak Contractions", Gen. Math. Notes, Vol. 7, No. 2, PP.38-44, (2011).

K. Ha, Y. J. Choand Awhile, " Striclly Convex and 2- Convex 2- Normd Spacy." Mathematica Japonica, 33, 375-384, (1988).

L. G. Huang and X. Zhand, " Cone Metric Spaces and Fixed Point Theorems for Contractive Mappings ", Journal of Mathematics Analysis and Applications, 332, 1468-1476, (2007).

Mustafa and B. Sims, " A New Approach to Generalized Metris spaces", Journal of Nonlinear and Convex Analysis, 7, 289- 297, (2006).

Sedghi S., Shobe, N., & Aliouche, A. " A Generalization of Fixed Point Theorems in S- Metric Spaces ", Mathematichki Vesnik, 64, 258-266, (2012).

R. De Jong, "Ordered Banach Algebras", Ms.C. Thesis, Leiden University. Holland, 2010.

Sedghi S., Shobe, N., & Zhou, H., "A Common Fixed Point Theorem in D*- Metric Space", Fixed Point Theory and Applications, 29, 631-636, (2007).

Sevng Hyun Kim and Byvng Soo Lee, "Generalized Fixed Point Theorems in Cone Metric Spaces", Korean J. Math 20, No. 3, PP.353-360, (2012).

S. K. Tiwari and R. P. Dubey, "Cone Metric Spaces and Fixed Point Theorems for Generalized Y. Reich and T Rhoadg Contractive Mapping", Asin Journal of Mathematics and Applications Article Idama 0077, 9 Page, vol. (2013).

V. H. Badshah, P. Bhagatand S. Shukla, "Some Fixed Point Theorems for α-ϕ- Outmetric Mapping in 2- Metric Spaces", Scientific Publications of State University of Novipazar, Ser. A: ppl. Math. INF or M. and Mech, Vol 802, 181-199, (2016).

Zhenhua Ma, Lining Jiang, and Hongkaisun, " C^*- Algebra- Valued Metric Spaces and Related Fixed Point Theorems", Ma et al. Fixed Point and Applications Springer: 206., (2014).

Yan Han, Shaoyuan Yu, " New Common Fixed Point Results for Four Maps on Cone Metric Spaces" Applied Mathematics, 2, 1114-1118,(2011).

## Downloads

## Published

## How to Cite

*JOURNAL OF ADVANCES IN MATHEMATICS*,

*18*, 60–71. https://doi.org/10.24297/jam.v18i.8370

## Issue

## Section

## License

All articles published in *Journal of Advances in Linguistics* are licensed under a Creative Commons Attribution 4.0 International License.