论文标题
关于一对非线性矩阵方程的独特常见解决方案的新存在
On new existence of a unique common solution to a pair of non-linear matrix equations
论文作者
论文摘要
本文的主要目的是研究形式\ begin {eqnarray*} x^r = q_1 + \ displayStyle \ sum_ {i = 1}^i = 1}^{m}^{m} {m_i} {a__i} {a_i}^*f（x）a_i \ mbobs A_I \ mbobs A_I \ mbobs A = s = \sum_{i=1}^{m} {A_i}^*G(X)A_i \end{eqnarray*} where $Q_1,Q_2\in P(n)$, $A_i\in M(n)$ and $F,G:P(n)\to P(n)$ are certain functions and $r,s>1$.为了实现我们的目标，我们在所有$ n \ times n $ hermitian积极确定的矩阵的集合中掌握了汤普森度量标准的优雅属性。为了进行此操作，我们首先得出了使用公制空间中的一类控制功能的一对映射的常见固定点结果。然后，我们获得了一些足够的条件，以确保对上述方程的独特积极确定的共同解决方案。最后，为了验证我们的结果，我们提供了一些数值示例，并具有迭代序列收敛行为的图表表示。
The main goal of this article is to study the existence of a unique positive definite common solution to a pair of matrix equations of the form \begin{eqnarray*} X^r=Q_1 + \displaystyle \sum_{i=1}^{m} {A_i}^*F(X)A_i \mbox{ and } X^s=Q_2 + \displaystyle \sum_{i=1}^{m} {A_i}^*G(X)A_i \end{eqnarray*} where $Q_1,Q_2\in P(n)$, $A_i\in M(n)$ and $F,G:P(n)\to P(n)$ are certain functions and $r,s>1$. In order to achieve our target, we take the help of elegant properties of Thompson metric on the set of all $n \times n$ Hermitian positive definite matrices. To proceed this, we first derive a common fixed point result for a pair of mappings utilizing a certain class of control functions in a metric space. Then, we obtain some sufficient conditions to assure a unique positive definite common solution to the said equations. Finally, to validate our results, we provide a couple of numerical examples with diagrammatic representations of the convergence behaviour of iterative sequences.
