Some fixed point theorems in C^* -algebra valued b -metric spaces

Subject Areas : Statistics

zahra ghorbani ¹ , javad baradaran ²

1 - Department of mathematics, Jahrom university
2 - Department of Mathematics, Jahrom University, Jahrom, Iran

Received: 2019-05-02 Accepted : 2019-09-21 Published : 2020-12-21

Keywords: نقطه تناوبی, فضای -bمتریک - C*جبر-مقدا, نقطه ثابت, - پیوسته مداری,

Abstract :

In this paper, we first define the notions of orbitally continuous and orbitally complete on a C*-algebra-valued metric space. We show that if T is an orbitally continuous mapping on a C*-algebra-valued metric space (X,A, d), where X is a nonempty set and A is a C*-algebra with the relation ⪯ and if T orbitally complete and satisfies some conditions, then for any x∈X the iterated sequence {Tn (x)} converges to a fixed point of T. Also, we prove that an orbitally continuous mapping on a C*-algebra-valued metric space (X,A, d) under conditions has a periodic point. It is prove that an orbitally continuous self-map on a C*-algebra-valued b-metric space (X,A, d) under which conditions has at least a fixed point. In additions, if (X,A, d) be a complete C*-algebra-valued metric space and T has some property. Then T has a fixed point in X provided that there exists x0∈X such that T2 (x0)≠x

References:

[1] I. A. Bakhtin. The contraction mapping principle in almost metric spaces. Journal ofFunctional Analysis 30: 26-37 (1989).

 

[2] S. Czerwik. Nonlinear set-valued contraction mappings in b−metric spaces, Atti Sem. Mat. Fis. Univ. Modena 46: 263-276 (1998).

 

[3] N. Hussain, M. H.  Shah. KKM Mappings in cone b−metric spaces. Computer and Mathematics with Applications 62: 1677-1684 (2011).

 

[4] J. B. Conway. A Course in functional analysis. Springer-Verlag, NewYork, Berlin, Heidelberg Tokyo, (1985).

 

[5] R. V. Kadison, J. R . Ringrose. Fundamentals of the theory of operator algebras; Volume I: Elementary Theory. (1997).

 

[6] T. Kamran, M. Postolache, A. Ghiura, S. Batul, R. Ali. The Banach contraction principle in C -algebra-valued b-metric spaces with application.  Journal ofFixed Point Theory and Applications 10: 1-7 (2016).

 

[7] Z. Ma, L. Jiang. C -Algebra-valued b-metric spaces and related fixed point theorems. Journal ofFixed Point Theory and Applications 222: 1-11 (2015).

 

[8] Z. Ma, L. Jiang, H. Sun. C -algebra-valued metric spaces and related fixed point theorems.  Journal ofFixed Point Theory and Applications 206 (2014)  

 

[9] G. Murphy. C -Algebras and operator theory. Academic Press, London (1990).