Common fixed point theorems for rational F R \(F_{\mathcal{R}}\) -contractive pairs of mappings with applications

In this paper, we study the existence of solution for the following non-linear matrix equations: X=Q+∑i=1nAi∗XAi−∑i=1nBi∗XBi,X=Q+∑i=1nAi∗ϒ(X)Ai, where Q is a Hermitian positive definite matrix, Ai\(A_{i}\), Bi\(B_{i}\) are arbitrary m×m\(m\times m\) matrices and ϒ:H(m)→P(m)\(\varUpsilon: \mathcal{H}...

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Bibliographic Details
Published in:Journal of inequalities and applications 2019-01, Vol.2019 (1), p.1-14
Main Authors: Zada, Mian Bahadur, Sarwar, Muhammad
Format: Article
Language:English
Subjects:
Fixed points (mathematics)
Mapping
Mathematical analysis
Mathematics
Matrix methods
Nonlinear equations
Theorems
Online Access:Get full text
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Summary:In this paper, we study the existence of solution for the following non-linear matrix equations: X=Q+∑i=1nAi∗XAi−∑i=1nBi∗XBi,X=Q+∑i=1nAi∗ϒ(X)Ai, where Q is a Hermitian positive definite matrix, Ai\(A_{i}\), Bi\(B_{i}\) are arbitrary m×m\(m\times m\) matrices and ϒ:H(m)→P(m)\(\varUpsilon: \mathcal{H}(m)\rightarrow \mathcal{P}(m)\) is an order preserving continuous map such that ϒ(0)=0\(\varUpsilon (0)=0\). To this aim, we establish several common fixed point theorems for two mapping satisfying a rational FR\(F_{\mathcal{R}}\)-contractive condition, where R\(\mathcal{R}\) is a binary relation.
ISSN:1025-5834
1029-242X
DOI:10.1186/s13660-018-1952-z