Solution of operator equations and proof of Kannan mapping theorem and Chatterjea-type mapping theorem in C^*-algebra valued S_b-metric spaces

Subject Areas : Statistics

S. Samira Razavi ¹ , Hashem Parvaneh Masiha ²

1 - Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
2 - Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

Received: 2019-07-08 Accepted : 2020-08-22 Published : 2021-02-19

Keywords: فضای S_b-متریک C^*-جبر مقدار, نقطه ثابت, نگاشت نوع کاترجا, نگاشت نوع کانان, معادلات عملگری,

Abstract :

In this paper, based on the results and theorems of C^*-algebra valued S_b-metric spaces, we solve one type of matrix operator equations in L(H) defined by X-∑_(n=۱)^∞▒〖A_n^* XA_n=Q〗, in which H is a Hilbert space and L(H) is the set of linear and bounded operators on H. Also, we prove that the Kannan contraction mapping has a unique fixed point in C^*-algebra valued S_b-metric spaces. Furthermore, we prove that the Chatterjea-type contraction mapping has a unique fixed point in C^*-algebra valued S_b-metric spaces. Finally, by using the Banach contraction principle in C^*-algebra valued S_b-metric spaces that has previously been studied by the authors of this article and by using the results of the above theorems, we solve the above matrix operator equation in C^*-algebra valued S_b-metric spaces. As well as, we show that this matrix operator equation has a unique solution in L(H), and this solution is a Hermitian operator.

References:

 

[1]     ناصر غفوری عدل، داود ابراهیمی بقا، محمدصادق عسگری، مهدی آژینی، پایداری معادلات تابعی مرتبه هفتمین در فضای β-گاوسی. مجله پژوهش­های نوین در ریاضی، سال چهارم، شماره پانزدهم، پاییز 1397.

 

[2]     S. Czerwik. Contraction mappings in b-metric spaces. Acta Mathematica et Informatica Universitatis Ostraviensis, 1, 5-11. (1993).

 

[3]     Q.  Xia. The geodesic problem in quasimetric spaces. Journal of Geometric Analysis, 19(2), 452-479. (2009).

 

[4]     Z.H. Ma, L.N. Jiang, H.K. Sun. C*-algebra-valued metric spaces and related fixed point theorems. Fixed Point Theory and Applications,206(2014).

 

[5]     Z. Ma, L. Jiang. C*-Algebra-valued b-metric spaces and related fixed point theorems. Fixed Point Theory and Applications, 222(2015).

 

[6]     S. Sedghi, N. Shobe, A. Aliouche. A generalization of fixed point theorem in S-metric spaces. Matematicki Vesnik, 64, 258-266(2012). 

 

[7]     N. Souayah, N. Mlaiki. A fixed point theorem in -metric space. Journal of Mathematics and Computer Science, 16, 131–139(2016).

 

[8]     M.S. Asgari, B. Mousavi, B. Solving class of nonlinear matrix equations via the coupled fixed point theorem. Appl. Math. Comput, 259, 364-373(2015).

 

[9]     N. Ghafoori, D. Ebrahimi Bagha, M.S. Asgari. Coupled fixed points of generalized Kannan contraction and its applications. Int. J. Nonlinear Anal. Appl. 9(2), 169-178(2018).

 

[10]     S.S. Razavi, H.P. Masiha. C*-algebra-valued -metric spaces and applications to Integral equations, under review

 

[11]     C. Kalaivani, G. Kalpana. Fixed point theorems in -algebra-valued S-metric spaces with some applications. U.P.B. Scientific Bulletin Series A, Vol.80, Iss. 3 (2018).