1 Introduction
Non-linear matrix equations play an important role in several problems of engineering and applied mathematics. Various matrix equations are encountered in stability analysis [1], control theory [2, 3] and system theory [4‐6]. To test the existence of solution to non-linear matrix equations, we can have a number of advanced methods. One of these methods is to use the tools of fixed point theory. Using fixed point results, many researchers checked the existence and uniqueness of solution of non-linear matrix equations [7‐10].
An important result in fixed point theory, commonly known in the literature as the Banach principle, has been established by Banach [11]. This principle has been improved and generalized by several researchers for different kinds of contractions in various spaces. Wardowski [12] presented the concept of F-contraction and demonstrated fixed point theorems for this new type of contractions. Several authors generalized Wardowski’s theorems by extending the concept of F-contraction. Recently, Sawangsup et al. [9] introduced the concept of \(F_{\mathcal{R}}\)-contraction and established fixed point results for such type of contractions.
Anzeige
Throughout this work we use the following notation: Here \(\mathcal{P}(m)\subseteq \mathcal{H}(m)\subseteq \mathcal{M}(m)\), \(\mathcal{H}^{+}(m)\subseteq \mathcal{H}(m)\), \(\varOmega_{1}\succ 0\) and \(\varOmega_{1}\succeq 0\) means that \(\varOmega_{1}\in \mathcal{P}(m)\) and \(\varOmega_{1}\in \mathcal{H}^{+}(m)\), respectively; for \(\varOmega_{1}- \varOmega_{2}\succeq 0\) and \(\varOmega_{1}-\varOmega_{2}\succ 0\) we will use \(\varOmega_{1}\succeq \varOmega_{2}\) and \(\varOmega_{1}\succ \varOmega_{2}\), respectively. Moreover, \(\mathbb{N}= \{1,2,3,\ldots\}\), and \(\mathbb{N} _{0} = \mathbb{N}\cup \{0\}\).
-
\(\mathcal{M}(m) =\) set of \(m\times m\) complex matrices,
-
\(\mathcal{H}(m) =\) set of \(m\times m\) Hermitian matrices,
-
\(\mathcal{P}(m) =\) set of \(m\times m\) positive definite matrices,
-
\(\mathcal{H}^{+}(m) =\) set of \(m\times m\) positive semi-definite matrices.
The main concern of this paper is to study the following non-linear matrix equations:
where \(Q\in \mathcal{P}(m)\), \(A_{i}\), \(B_{i}\) are arbitrary \(m\times m\) matrices and \(\varUpsilon: \mathcal{H}(m)\rightarrow \mathcal{P}(m)\) is a continuous order preserving map such that \(\varUpsilon (0)=0\). The matrix equations (1.1) often occur in dynamic programming [13, 14], control theory [15, 16], ladder networks [17, 18], etc.
$$\begin{aligned}& X=Q+ \sum^{n}_{i=1} A^{*}_{i} X A_{i}- \sum ^{n}_{i=1} B^{*}_{i} X B_{i}, \end{aligned}$$
(1.1)
$$\begin{aligned}& X=Q+ \sum^{n}_{i=1} A^{*}_{i} \varUpsilon (X) A_{i}, \end{aligned}$$
(1.2)
Berzig [7] used coupled fixed point results to prove the existence of the unique positive definite solutions of (1.1). Recently, Sawangsup et al. [9] established fixed point theorems for \(F_{\mathcal{R}}\)-contractions and proved the existence and uniqueness of a positive definite solution of the matrix equation (1.2).
The intention of this work is to introduce the concept of rational \(F_{\mathcal{R}}\)-contractive pair of mappings, under an arbitrary binary relation \(\mathcal{R}\) and using this concept we prove fixed point results. By means of these results, we prove in the last section existence results for positive definite solutions of the two classes of non-linear matrix equations (1.1) and (1.2).
Anzeige
2 Preliminaries
In this section we recall some basic notions.
Definition 2.1
Let \(\mathbb{F}\) be the class of all functions \(\mathsf{f} : [0, \infty \mathclose{[}\rightarrow \mathbb{R}\) satisfying the following properties:
(1)
f is strictly increasing;
(2)
for every sequence \(\{s_{n}\}_{n\in \mathbb{N}}\) with \(s_{n}> 0\), we have
$$ \underset{n\rightarrow \infty }{\lim } s_{n} = 0 \quad \Longleftrightarrow\quad \underset{n\rightarrow \infty }{\lim } \mathsf{f}(s_{n})=-\infty; $$
(3)
there is \(j\in \mathopen{]}0,1[\) such that \(\underset{s\rightarrow 0^{+} }{ \lim }s^{j} \mathsf{f}(s)=0\).
Definition 2.2
([19])
Let \(\mathbb{X}\) be a non-empty set and \(\mathcal{R}\) be a binary relation on \(\mathbb{X}\). Then \(\mathcal{R}\) is transitive if \((\gamma_{2},\gamma_{1})\in \mathcal{R}\) and \((\gamma_{1},\gamma_{3})\in \mathcal{R}\) implies that \((\gamma_{2}, \gamma_{3})\in \mathcal{R}\), for all \(\gamma_{2},\gamma_{1},\gamma _{3}\in \mathbb{X}\).
Definition 2.3
Let \(\mathbb{X}\) be a non-empty set and \(\varPhi:\mathbb{X}\rightarrow \mathbb{X}\). Then a binary relation \(\mathcal{R}\) on \(\mathbb{X}\) is called Φ-closed (equivalently Φ is \(\mathcal{R}\)-non-decreasing) if for any \(\gamma_{1},\gamma _{2}\in \mathbb{X}\), we have
$$ (\gamma_{1},\gamma_{2})\in \mathcal{R}\quad \Longrightarrow \quad (\varPhi \gamma _{1},\varPhi \gamma_{2})\in \mathcal{R}. $$
Definition 2.4
([21])
Let \(\gamma_{1},\gamma_{2}\in \mathbb{X}\) and \(\mathcal{R}\) be a binary relation on a non-empty set \(\mathbb{X}\). A path (of length \(n\in \mathbb{N}\)) in \(\mathcal{R}\) from \(\gamma_{1}\) to \(\gamma_{2}\) is a sequence \(\{t_{0}, t_{1}, t_{2},\ldots,t_{n}\} \subseteq \mathbb{X}\) such that Note that \(\varGamma (\gamma_{1},\gamma_{2},\mathcal{R})\) represents the class of all paths from \(\gamma_{1}\) to \(\gamma_{2}\) in \(\mathcal{R}\).
(i)
\(t_{0}=\gamma_{1}\) and \(t_{n}=\gamma_{2}\);
(ii)
\((t_{j}, t_{j+1})\in \mathcal{R}\) for all \(j\in \{0,1,2,\ldots,n-1 \}\).
Notice that a path of length n involves \(n+1\) elements of \(\mathbb{X}\), although they are not necessarily distinct.
Definition 2.5
([22])
A metric space \((M,d)\) equipped with a binary relation \(\mathcal{R}\) is \(\mathcal{R}\)-non-decreasing-regular if for all sequences \(\{\kappa_{n}\}\) in M,
$$ \begin{aligned} &(\kappa_{n},\kappa_{n+1})\in \mathcal{R},\quad \forall n\in \mathbb{N}, \\ & \kappa_{n}\rightarrow \kappa \in M, \end{aligned} \Bigg\} \quad \Longrightarrow\quad (\kappa_{n},\kappa)\in \mathcal{R}, \forall n\in \mathbb{N}. $$
Definition 2.6
([9])
Let \((M,d)\) be a metric space, \(\mathcal{R}\) be a binary relation M and \(\varPsi: M\rightarrow M\) be a mapping. Let
Then Ψ is said to be an \(\mathsf{F}_{\mathcal{R}}\)-contraction if there exist \(\xi > 0\) and \(\mathsf{F}\in \mathbb{F}\) such that
$$ \mathcal{W} = \bigl\{ (\kappa_{1},\kappa_{2})\in \mathcal{R} : d(\varPsi \kappa _{1},\varPsi \kappa_{2}) > 0 \bigr\} . $$
$$ \xi +\mathsf{F}\bigl(d(\varPsi \kappa_{1},\varPsi \kappa_{2})\bigr) \leq \mathsf{F}\bigl(d( \kappa_{1}, \kappa_{2})\bigr),\quad \text{for all } (\kappa_{1}, \kappa_{2})\in \mathcal{W}. $$
(2.1)
3 Main results
First we modify Definition 2.6 for two maps as follows.
Definition 3.1
Let Φ, Ψ be two self-mappings and \(\mathcal{R}\) be a binary relation on a non-empty set \(\mathbb{X}\). Then \(\mathcal{R}\) is \((\varPhi,\varPsi)\)-closed if for each \(a_{1}, a_{2}\in \mathbb{X}\), we have
$$ (a_{1},a_{2})\in \mathcal{R}\quad \Longrightarrow\quad ( \varPhi a_{1},\varPsi a_{2}), ( \varPsi a_{1}, \varPhi a_{2})\in \mathcal{R}. $$
Definition 3.2
Let \((M,d)\) be a metric space, Φ, Ψ be self-mappings of M and \(\mathcal{R}\) be a binary relation on M. Let
We say that \((\varPhi, \varPsi)\) is a rational \(\mathsf{F}_{\mathcal{R}}\)-contractive pair of mappings if there exist \(\xi > 0\) and \(\mathsf{F}\in \mathbb{F}\) such that
$$ \mathcal{X} =\bigl\{ (\kappa_{1},\kappa_{2})\in \mathcal{R} : d(\varPhi \kappa _{1},\varPsi \kappa_{2})>0\bigr\} . $$
$$\begin{aligned}& \xi +\mathsf{F}\bigl(d(\varPhi \kappa_{1},\varPsi \kappa_{2})\bigr) \\& \quad \leq \mathsf{F} \biggl( d(\kappa_{1},\kappa_{2})+ \frac{d(\kappa_{2},\varPhi \kappa_{1})d( \kappa_{1},\varPsi \kappa_{2})}{1 + d(\kappa_{1},\kappa_{2})} \biggr), \quad \text{for all } (\kappa_{1}, \kappa_{2})\in \mathcal{X}. \end{aligned}$$
(3.1)
Denote by \(M((\varPhi,\varPsi);\mathcal{R})\) the set of all order pairs \((\kappa_{1},\kappa_{2})\in M\times M\) such that \((\varPhi \kappa_{1}, \varPsi \kappa_{2})\in \mathcal{R}\).
Theorem 3.3
Let
\((M,d)\)
be a complete metric space, \(\mathcal{R}\)
be a binary relation on
M
and
\(\varPhi, \varPsi: M\rightarrow M\). Suppose that the following conditions hold:
Then there is a common fixed point of
Φ
and
Ψ.
\((C_{1})\)
\(M((\varPhi,\varPsi);\mathcal{R})\)
is non-empty;
\((C_{2})\)
\(\mathcal{R}\)
is
\((\varPhi,\varPsi)\)-closed;
\((C_{3})\)
Φ
and
Ψ
are continuous;
\((C_{4})\)
the pair
\((\varPhi, \varPsi)\)
is rational
\(\mathsf{F} _{\mathcal{R}}\)-contractive.
Proof
Let \((\kappa_{0}, \kappa_{1})\) be any element of \(M((\varPhi,\varPsi); \mathcal{R})\), then \((\varPhi \kappa_{0},\varPsi \kappa_{1})\in \mathcal{R}\). Define the sequence \(\{\kappa_{n}\}\) in M by
where \(n\in \mathbb{N}_{0}\).
$$ \begin{aligned} &\kappa_{2n+1}=\varPhi \kappa_{2n}, \\ &\kappa_{2n+2}=\varPsi \kappa_{2n+1}, \end{aligned} \Bigg\} $$
(3.2)
If \(\kappa_{2n^{*}}=\kappa_{2n^{*}+1}\) for some \(n^{*}\in \mathbb{N} _{0}\), then \(\kappa_{2n^{*}}\) is a common fixed point of Φ and Ψ. If \(\kappa_{2n}\neq \kappa_{2n+1}\), for all \(n\in \mathbb{N} _{0}\). Then \(d(\varPhi \kappa_{2n} ,\varPsi \kappa_{2n+1}) > 0\), for all \(n\in \mathbb{N}_{0}\) and using assumption \((C_{2})\), we obtain
In general,
$$ \begin{aligned} &(\kappa_{1}, \kappa_{2})= ( \varPhi \kappa_{0},\varPsi \kappa_{1})\in \mathcal{R}, \\ &(\kappa_{2}, \kappa_{3})= (\varPsi \kappa_{1}, \varPhi \kappa_{2})\in \mathcal{R}, \\ &(\kappa_{3}, \kappa_{4})= (\varPhi \kappa_{2}, \varPsi \kappa_{3})\in \mathcal{R}, \\ &(\kappa_{4}, \kappa_{5})= (\varPsi \kappa_{3}, \varPhi \kappa_{4})\in \mathcal{R}, \\ \vdots \end{aligned} $$
$$ \begin{aligned} &(\kappa_{2n}, \kappa_{2n+1})=( \varPsi \kappa_{2n-1},\varPhi \kappa_{2n}) \in \mathcal{R}. \end{aligned} $$
Thus \((\kappa_{2n}, \kappa_{2n+1})\in \mathcal{X}\), for all \(n\in \mathbb{N}_{0}\). Now, taking in (3.1) \(\kappa_{1}= \kappa_{2n}\) and \(\kappa_{2}=\kappa_{2n-1}\), we have
for all \(n\in \mathbb{N}\). Similarly, setting \(\kappa_{1}=\kappa_{2n}\) and \(\kappa_{2}=\kappa_{2n+1}\) in (3.1), we can write
In general,
where \(n\in \mathbb{N}\). Now, using inequality (3.3), we can write
that is,
where \(n\in \mathbb{N}\). Thus \(\underset{n\rightarrow \infty }{\lim } \mathsf{F}(d(\kappa_{n},\kappa_{n+1}))=-\infty \), by condition (2) of Definition 2.1, we get
From condition \((3)\) of Definition 2.1, we can find \(\varepsilon \in \mathopen{]}0,1[\) such that
Using (3.4), we have
Taking the limit \(n\rightarrow \infty \) in (3.7), and using (3.5) and (3.6), we get
Hence there exists \(n_{0}\in \mathbb{N}\) such that, for all \(n\geq n_{0}\), \(n ( d(\kappa_{n},\kappa_{n+1}) ) ^{\varepsilon }\leq 1\). Consequently, we have
Now, we show that \(\{\kappa_{n}\}\) is a Cauchy sequence. For this purpose, using (3.9) and the triangular inequality, for all \(m>n\geq n_{1}\), we have
Since \(d(\kappa_{n},\kappa_{m})\leq \sum^{m-1}_{i=n} \frac{1}{i^{\frac{1}{\varepsilon }}}<\infty \), the sequence \(\{\kappa_{n}\}\) is Cauchy in M. Due to completeness of M, one can find \(t\in M\) such that \(\kappa_{n}\rightarrow t\) as \(n \rightarrow \infty \).
$$ \begin{aligned} \mathsf{F}\bigl(d(\kappa_{2n}, \kappa_{2n+1})\bigr) &=\mathsf{F}\bigl(d(\kappa_{2n+1}, \kappa_{2n})\bigr) \\ &=\mathsf{F}\bigl(d(\varPhi \kappa_{2n},\varPsi \kappa_{2n-1}) \bigr) \\ &\leq \mathsf{F} \biggl( d(\kappa_{2n},\kappa_{2n-1})+ \frac{d(\kappa _{2n-1},\varPhi \kappa_{2n})d(\kappa_{2n},\varPsi \kappa_{2n-1})}{1 + d( \kappa_{2n},\kappa_{2n-1})} \biggr) -\xi \\ &=\mathsf{F} \biggl( d(\kappa_{2n},\kappa_{2n-1})+ \frac{d(\kappa_{2n-1}, \kappa_{2n+1})d(\kappa_{2n},\kappa_{2n})}{1 + d(\kappa_{2n},\kappa _{2n-1})} \biggr) -\xi \\ &=\mathsf{F} \bigl( d(\kappa_{2n},\kappa_{2n-1})\bigr) ) -\xi, \end{aligned} $$
$$\begin{aligned} \mathsf{F}\bigl(d(\kappa_{2n+1}, \kappa_{2n+2})\bigr) =&\mathsf{F}\bigl(d(\varPhi \kappa _{2n}, \varPsi \kappa_{2n+1})\bigr) \\ \leq& \mathsf{F} \biggl( d(\kappa_{2n},\kappa_{2n+1})+ \frac{d(\kappa _{2n+1},\varPhi \kappa_{2n})d(\kappa_{2n},\varPsi \kappa_{2n+1})}{1 + d( \kappa_{2n},\kappa_{2n+1})} \biggr) -\xi \\ =&\mathsf{F} \biggl( d(\kappa_{2n},\kappa_{2n+1})+ \frac{d(\kappa_{2n+1}, \kappa_{2n+1})d(\kappa_{2n},\kappa_{2n+2})}{1 + d(\kappa_{2n}, \kappa_{2n+1})} \biggr) -\xi \\ =&\mathsf{F} \bigl( d(\kappa_{2n},\kappa_{2n+1})\bigr) ) -\xi. \end{aligned}$$
$$ \begin{aligned} \mathsf{F}\bigl(d(\kappa_{n}, \kappa_{n+1})\bigr) &\leq \mathsf{F} \bigl( d(\kappa _{n-1}, \kappa_{n})\bigr) ) -\xi, \end{aligned} $$
(3.3)
$$\begin{aligned} \mathsf{F}\bigl(d(\kappa_{n},\kappa_{n+1})\bigr) \leq & \mathsf{F}\bigl(d(\kappa_{n-1}, \kappa_{n})\bigr)-\xi \\ \leq & \mathsf{F}\bigl(d(\kappa_{n-2},\kappa_{n-1})\bigr)-2 \xi \\ \leq & \mathsf{F}\bigl(d(\kappa_{n-3},\kappa_{n-2})\bigr)-3 \xi \\ \leq & \mathsf{F}\bigl(d(\kappa_{n-4},\kappa_{n-3})\bigr)-4 \xi \\ \vdots& \\ \leq & \mathsf{F}\bigl(d(\kappa_{0},\kappa_{1})\bigr)-n \xi, \end{aligned}$$
$$ \begin{aligned} \mathsf{F}\bigl(d(\kappa_{n}, \kappa_{n+1})\bigr)\leq & \mathsf{F}\bigl(d(\kappa_{0}, \kappa_{1})\bigr)-n\xi, \end{aligned} $$
(3.4)
$$ \underset{n\rightarrow \infty }{\lim }d(\kappa_{n}, \kappa_{n+1})=0 \quad \text{or}\quad \underset{n\rightarrow \infty }{ \lim }d(\kappa_{n},\kappa_{n+1})=0^{+}. $$
(3.5)
$$ \underset{n\rightarrow \infty }{\lim } \bigl( d( \kappa_{n},\kappa_{n+1}) \bigr) ^{\varepsilon }\mathsf{F} \bigl( d(\kappa_{n},\kappa_{n+1}) \bigr) =0. $$
(3.6)
$$ \begin{aligned} \bigl( d(\kappa_{n}, \kappa_{n+1}) \bigr) ^{\varepsilon } \bigl( \mathsf{F}\bigl(d( \kappa_{n},\kappa_{n+1})\bigr)-\mathsf{F}\bigl(d( \kappa_{0},\kappa _{1})\bigr) \bigr)\leq & - \bigl( d( \kappa_{n},\kappa_{n+1}) \bigr) ^{\varepsilon }n\xi \leq 0. \end{aligned} $$
(3.7)
$$ \begin{aligned} \underset{n\rightarrow \infty }{\lim }n \bigl( d(\kappa_{n},\kappa_{n+1}) \bigr) ^{\varepsilon }=0. \end{aligned} $$
(3.8)
$$ \begin{aligned} d(\kappa_{n}, \kappa_{n+1})\leq \frac{1}{n^{\frac{1}{\varepsilon }}}, \quad \forall n\geq n_{0}. \end{aligned} $$
(3.9)
$$ \begin{aligned} d(\kappa_{n},\kappa_{m}) &\leq d( \kappa_{n},\kappa_{n+1})+d(\kappa_{n+1}, \kappa_{n+2})+d(\kappa_{n+2},\kappa_{n+3})+\cdots+d( \kappa_{m-1},\kappa _{m}) \\ &\leq \frac{1}{n^{\frac{1}{\varepsilon }}}+\frac{1}{(n+1)^{\frac{1}{ \varepsilon }}}+\frac{1}{(n+2)^{\frac{1}{\varepsilon }}}+\cdots+ \frac{1}{(m-1)^{\frac{1}{ \varepsilon }}} \\ &= \sum^{m-1}_{i=n}\frac{1}{i^{\frac{1}{\varepsilon }}}. \end{aligned} $$
Next, we show that \(\varPhi t=\varPsi t=t\). Since Φ and Ψ are continuous and \(\kappa_{2n}, \kappa_{2n-1}\rightarrow t\),
Due to the limit uniqueness, we obtain \(\varPhi t=t\) and \(\varPsi t=t\), which implies that \(\varPhi t=\varPsi t=t\) and hence there is a common fixed point of Φ and Ψ. □
$$ \begin{aligned} \kappa_{2n+1}=\varPhi \kappa_{2n} \rightarrow \varPhi t \quad \text{and}\quad \kappa_{2n}=\varPsi \kappa_{2n-1}\rightarrow \varPsi t. \end{aligned} $$
The next result ensures the uniqueness of the common fixed point in Theorem 3.3.
Theorem 3.4
Let
\(\mathcal{R}\)
be a transitive relation and
Φ, Ψ
be the self-mappings on a complete metric space
M. Assume that the following conditions hold:
Then there is a unique common fixed point of
Φ
and
Ψ.
\((C_{0})\)
for all
\((\kappa_{1},\kappa_{2})\in \mathcal{X}\), there exists
\(\mathsf{F}\in \mathbb{F}\)
such that
where
\(\xi > 0\);
$$ \xi +\mathsf{F}\bigl(d(\varPhi \kappa_{1},\varPsi \kappa_{2})\bigr) \leq \mathsf{F} \biggl( \frac{1}{2}d( \kappa_{1},\kappa_{2})+\frac{d(\kappa_{2},\varPhi \kappa_{1})d(\kappa_{1},\varPsi \kappa_{2})}{2 [ 1 + d(\kappa_{1}, \kappa_{2}) ] } \biggr), $$
(3.10)
\((C_{1})\)
\(M((\varPhi,\varPsi);\mathcal{R})\)
and
\(\varGamma (\kappa _{1},\kappa_{2},\mathcal{R})\)
are non-empty;
\((C_{2})\)
\(\mathcal{R}\)
is
\((\varPhi,\varPsi)\)-closed;
\((C_{3})\)
Φ
and
Ψ
are continuous.
Proof
Following the same steps as in the proof of Theorem 3.3, one can easily prove that there is a common fixed point of Φ and Ψ. Thus we have to show that there is a unique common fixed point of Φ and Ψ. For this purpose, assume that λ and \(\lambda^{*}\) are two distinct common fixed points of Φ and Ψ. Then since \(\varGamma (\lambda,\lambda^{*},\mathcal{R})\) is the class of paths in \(\mathcal{R}\) from λ to \(\lambda^{*}\), there is a path of finite length l, i.e. there is a sequence \(\{z_{0}, z _{1}, z_{2},\ldots, z_{l}\}\) in \(\mathcal{R}\) from λ to \(\lambda^{*}\) with
But since \(\mathcal{R}\) is transitive, we have
Now, setting \(\lambda =\lambda \) and \(\lambda^{*}=\lambda^{*}\) in contraction condition (3.10), we have
which is a contradiction. Thus \(\lambda =\lambda^{*}\) and hence λ is the unique common fixed point of Φ and Ψ. □
$$ z_{0}=\lambda,\qquad z_{l}=\lambda^{*},\qquad (z_{j},z_{j+1})\in \mathcal{R},\quad \text{for every } j=0,1,2,\ldots,(l-1). $$
$$ (\lambda,z_{1})\in \mathcal{R}, (z_{1},z_{2}) \in \mathcal{R},\ldots,\bigl(z _{k-1}, \lambda^{*}\bigr)\in \mathcal{R}\quad \Longrightarrow\quad \bigl(\lambda, \lambda^{*}\bigr) \in \mathcal{R}. $$
$$\begin{aligned}& \xi +\mathsf{F}\bigl(d\bigl(\varPhi \lambda,\varPsi \lambda^{*}\bigr)\bigr) \leq \mathsf{F} \biggl( \frac{1}{2}d \bigl(\lambda,\lambda^{*}\bigr)+\frac{d(\lambda^{*},\varPhi \lambda)d(\lambda,\varPsi \lambda^{*})}{2 [ 1 + d(\lambda,\lambda ^{*}) ] } \biggr), \\& \xi +\mathsf{F}\bigl(d\bigl(\lambda,\lambda^{*}\bigr)\bigr) \leq \mathsf{F} \biggl( \frac{1}{2}d\bigl( \lambda,\lambda^{*}\bigr)+ \frac{d(\lambda^{*},\lambda)d(\lambda,\lambda ^{*})}{2 [ 1 + d(\lambda,\lambda^{*}) ] } \biggr) \\& \hphantom{\xi +\mathsf{F}\bigl(d\bigl(\lambda,\lambda^{*}\bigr)\bigr)}< \mathsf{F} \biggl( \frac{1}{2}d\bigl(\lambda, \lambda^{*}\bigr)+\frac{1}{2}d\bigl( \lambda^{*}, \lambda\bigr) \biggr) \\& \hphantom{\xi +\mathsf{F}\bigl(d\bigl(\lambda,\lambda^{*}\bigr)\bigr)}= \mathsf{F} \bigl( d\bigl(\lambda,\lambda^{*}\bigr) \bigr), \end{aligned}$$
Corollary 3.5
Let
\(\mathcal{R}\)
be a binary relation and
Ψ
be the self-mappings on a complete metric space
M. Assume that the following conditions hold:
Then there is a fixed point of
Ψ.
\((C_{0})\)
for all
\((\kappa_{1},\kappa_{2})\in \mathcal{X}\), there exists
\(\mathsf{F}\in \mathbb{F}\)
such that
where
\(\xi > 0\);
$$ \xi +\mathsf{F}\bigl(d(\varPsi \kappa_{1},\varPsi \kappa_{2})\bigr) \leq \mathsf{F} \biggl( d(\kappa_{1}, \kappa_{2})+\frac{d(\kappa_{2},\varPsi \kappa_{1})d( \kappa_{1},\varPsi \kappa_{2})}{1 + d(\kappa_{1},\kappa_{2})} \biggr), $$
(3.11)
\((C_{1})\)
\(M(\varPsi;\mathcal{R})\)
is non-empty;
\((C_{2})\)
\(\mathcal{R}\)
is
Ψ-closed;
\((C_{3})\)
Ψ
is continuous.
Corollary 3.6
Let
\(\mathcal{R}\)
be a transitive relation and
Ψ
be the self-mappings on a complete metric space
M. Assume that the following conditions hold:
Then there is a unique fixed point of
Ψ.
\((C_{0})\)
for all
\((\kappa_{1},\kappa_{2})\in \mathcal{X}\), there exists
\(\mathsf{F}\in \mathbb{F}\)
such that
where
\(\xi > 0\);
$$ \xi +\mathsf{F}\bigl(d(\varPsi \kappa_{1},\varPsi \kappa_{2})\bigr) \leq \mathsf{F} \biggl( \frac{1}{2}d( \kappa_{1},\kappa_{2})+\frac{d(\kappa_{2},\varPsi \kappa_{1})d(\kappa_{1},\varPsi \kappa_{2})}{2[1 + d(\kappa_{1},\kappa _{2})]} \biggr), $$
(3.12)
\((C_{1})\)
\(M(\varPsi;\mathcal{R})\)
and
\(\varGamma (\kappa_{1},\kappa _{2},\mathcal{R})\)
are non-empty;
\((C_{2})\)
\(\mathcal{R}\)
is
Ψ-closed;
\((C_{3})\)
Ψ
is continuous.
Theorem 3.7
Theorem
3.3
remains true if instead of condition
\((C_{3})\), we assume that
\((M,d)\)
is
\(\mathcal{R}\)-non-decreasing regular.
Proof
In the proof of Theorem 3.3, we have seen that \((\kappa_{n}, \kappa_{n+1})\in \mathcal{R}\) and \(\kappa_{n}\rightarrow \gamma \) as \(n\rightarrow \infty \), \(\forall n\in \mathbb{N}\). Then since \((M,d)\) is \(\mathcal{R}\)-non-decreasing regular, so \((\kappa_{n}, \gamma)\in \mathcal{R}\) for every \(n\in \mathbb{N}\). Here we discuss two cases which depends on \(\mathcal{\mathcal{M}}=\{n\in \mathbb{N}: \varPhi \kappa_{2n}=\varPsi \gamma \text{ and } \varPsi \kappa_{2n+1}= \varPhi \gamma \}\).
Case (I): If \(\mathcal{M}\) finite, there exist \(n_{0}\in \mathbb{N}\) and \(\varPhi \kappa_{2n}\neq \varPsi \gamma \) and \(\varPsi \kappa_{2n+1}\neq \varPhi \gamma \), for all \(n\geq n_{0}\). Now since \(\kappa_{2n}\neq \gamma \) and \(\kappa_{2n+1}\neq \gamma \) implies that \(d(\kappa_{2n}, \gamma)>0\), \(d(\kappa_{2n+1},\gamma)>0\) and \(d(\varPhi \kappa_{2n}, \varPsi \gamma)>0\) and \(d(\varPsi \kappa_{2n+1}, \varPhi \gamma)>0\), for all \(n\geq n_{0}\).
Now, setting \(\kappa_{1}=\gamma \) and \(\kappa_{2}=\kappa_{2n+1}\) in the contractive condition (3.1), we have
But \(\{\kappa_{n}\}= \{ d(\gamma,\kappa_{2n+1})+\frac{d(\kappa _{2n+1},\varPhi \gamma)d(\gamma,\kappa_{2n+2})}{1 + d(\gamma,\kappa _{2n+1})} \} \) is a sequence of positive terms with \(\underset{n \rightarrow \infty }{\lim } \kappa_{n}=0\), so by condition (2) of Definition 2.1, \(\mathsf{F} ( \kappa_{n} ) \rightarrow -\infty \) implies that \(\mathsf{F}(d(\varPhi \gamma,\kappa_{2n+2})) \rightarrow -\infty \), again by condition (2) of Definition 2.1, \(d(\varPhi \gamma,\kappa_{2n+2})\rightarrow 0\), that is, \(\kappa_{2n+2}\rightarrow \varPhi \gamma \) as \(n \rightarrow \infty \). Also \(\kappa_{2n+2}\rightarrow \gamma \) as \(n \rightarrow \infty \), so by the uniqueness of the limit
and hence γ is the fixed point of Φ.
$$ \begin{aligned} &\xi +\mathsf{F}\bigl(d(\varPhi \gamma,\varPsi \kappa_{2n+1})\bigr) \leq \mathsf{F} \biggl( d(\gamma, \kappa_{2n+1})+\frac{d(\kappa_{2n+1},\varPhi \gamma)d( \gamma,\varPsi \kappa_{2n+1})}{1 + d(\gamma,\kappa_{2n+1})} \biggr) \\ &\quad \Longrightarrow\quad \xi +\mathsf{F}\bigl(d(\varPhi \gamma, \kappa_{2n+2})\bigr) \leq \mathsf{F} \biggl( d( \gamma, \kappa_{2n+1})+\frac{d(\kappa_{2n+1},\varPhi \gamma)d(\gamma, \kappa_{2n+2})}{1 + d(\gamma,\kappa_{2n+1})} \biggr). \end{aligned} $$
$$ \varPhi \gamma =\gamma, $$
(3.13)
Similarly, setting \(\kappa_{1}=\kappa_{2n}\) and \(\kappa_{2}=\gamma \) in contractive condition (3.1), we can easily show that \(\mathsf{F}(d(\kappa_{2n+1},\varPsi \gamma))\rightarrow -\infty \). By condition (2) of Definition 2.1, \(d(\kappa_{2n+1},\varPsi \gamma)\rightarrow 0\), that is, \(\kappa_{2n+1}\rightarrow \varPsi \gamma \) as \(n \rightarrow \infty \). Also \(\kappa_{2n+1}\rightarrow \gamma \) as \(n \rightarrow \infty \), so by the uniqueness of the limit
and hence γ is the fixed point of Ψ.
$$ \varPsi \gamma =\gamma, $$
(3.14)
From Eqs. (3.13) and (3.14), we get
Thus γ is a common fixed point of Φ and Ψ.
$$ \varPhi \gamma =\varPsi \gamma =\gamma. $$
(3.15)
Case (II): If \(\mathcal{M}\) is infinite, there exists a subsequence \(\{\kappa_{2n(j)}\}\) of \(\{\kappa_{n}\}\) with \(\kappa_{2n(j)+1}= \varPhi \kappa_{2n(j)}=\varPsi \gamma \) such that \(\kappa_{2n(j)+2}=\varPsi \kappa_{2n(j)+1}=\varPhi \gamma \) for all \(j\in \mathbb{N}\). But \(\kappa_{2n(j)+1}, \kappa_{2n(j)+2}\rightarrow \gamma \), so by the uniqueness of the limit \(\varPhi \gamma =\gamma \) and \(\varPsi \gamma = \gamma \) and hence γ is a common fixed point of Φ and Ψ.
In both cases, γ is a common fixed point of Φ and Ψ. □
Theorem 3.8
Theorem
3.4
remains true if, instead of condition
\((C_{3})\), we assume that
\((M,d)\)
is
\(\mathcal{R}\)-non-decreasing regular.
Corollary 3.9
Let
\(\mathcal{R}\)
be a binary relation and
Ψ
be the self-mappings on a complete metric space
M. Assume that the following conditions hold:
Then there is a fixed point of
Ψ.
\((C_{0})\)
for all
\((\kappa_{1},\kappa_{2})\in \mathcal{X}\), there exists
\(\mathsf{F}\in \mathbb{F}\)
such that
where
\(\xi > 0\);
$$ \xi +\mathsf{F}\bigl(d(\varPsi \kappa_{1},\varPsi \kappa_{2})\bigr) \leq \mathsf{F} \biggl( d(\kappa_{1}, \kappa_{2})+\frac{d(\kappa_{2},\varPsi \kappa_{1})d( \kappa_{1},\varPsi \kappa_{2})}{1 + d(\kappa_{1},\kappa_{2})} \biggr), $$
(3.16)
\((C_{1})\)
\(M(\varPsi;\mathcal{R})\)
is non-empty;
\((C_{2})\)
\(\mathcal{R}\)
is
Ψ-closed;
\((C_{3})\)
M
is
\(\mathcal{R}\)-non-decreasing regular.
Corollary 3.10
Let
\(\mathcal{R}\)
be a transitive relation and
Ψ
be the self-mappings on a complete metric space
M. Assume that the following conditions hold:
Then there is a unique fixed point of
Ψ.
\((C_{0})\)
for all
\((\kappa_{1},\kappa_{2})\in \mathcal{X}\), there exists
\(\mathsf{F}\in \mathbb{F}\)
such that
where
\(\xi > 0\);
$$ \xi +\mathsf{F}\bigl(d(\varPsi \kappa_{1},\varPsi \kappa_{2})\bigr) \leq \mathsf{F} \biggl( \frac{1}{2}d( \kappa_{1},\kappa_{2})+\frac{d(\kappa_{2},\varPsi \kappa_{1})d(\kappa_{1},\varPsi \kappa_{2})}{2[1 + d(\kappa_{1},\kappa _{2})]} \biggr), $$
(3.17)
\((C_{1})\)
\(M(\varPsi;\mathcal{R})\)
and
\(\varGamma (\kappa_{1},\kappa _{2},\mathcal{R})\)
are non-empty;
\((C_{2})\)
\(\mathcal{R}\)
is
Ψ-closed;
\((C_{3})\)
M
is
\(\mathcal{R}\)-non-decreasing-regular.
4 Applications
In this section, by using the previous theorems, we obtain existence results for the solutions of the matrix equations (1.1) and (1.2). We use the metric which is induced by the norm \(\|\aleph \|_{\mathrm{tr}}=\sum^{n}_{i=1} \theta_{i}( \aleph)\), where \(\theta_{i}(\aleph)\), \(i = 1,2,\ldots,n\), are the singular values of \(\aleph \in \mathcal{M}(m)\). The set \(\mathcal{H}(m)\) equipped with the trace norm \(\|\cdot\|_{\mathrm{tr}}\) is a complete metric space (see [7, 8, 23]) and partially ordered with partial ordering ⪯, where \(\aleph_{1}\preceq \aleph_{2}\Longleftrightarrow \aleph_{2}\succeq \aleph_{1}\). Also, for every \(\aleph_{1},\aleph_{2} \in \mathcal{H}(m)\) there is a glb and a lub (see [8]).
To establish the existence results we need the following lemmas.
Lemma 4.1
([8])
If
\(\aleph_{1}, \aleph_{2}\succeq O\)
are
\(m\times m\)
matrices, then
$$ 0 \leq \operatorname{tr}(\aleph_{1}\aleph_{2}) \leq \Vert \aleph_{2} \Vert \operatorname{tr}(\aleph_{1}). $$
Lemma 4.2
([24])
If
\(\aleph \in \mathcal{H}(m)\)
with
\(\aleph \prec I_{n}\), then
\(\|\aleph \| < 1\).
Define the operator \(\varPsi: \mathcal{H}(m)\rightarrow \mathcal{H}(m)\) by
where the operators \(\varPsi_{1}, \varPsi_{2}: \mathcal{H}(m)\rightarrow \mathcal{H}(m)\) are given by
and
$$ \varPsi (\mathcal{X})=\frac{1}{2} \bigl( \varPsi_{1}(\mathcal{X})+\varPsi_{2}( \mathcal{X}) \bigr), $$
$$ \varPsi_{1}(\mathcal{X})=Q + 2\sum ^{n}_{i=1} A^{*} _{i}\mathcal{X} A_{i} $$
$$ \varPsi_{2}(\mathcal{X})=Q-2\sum ^{n}_{i=1} B^{*}_{i} \mathcal{X} B_{i}. $$
Note that the solutions of the matrix equation (1.1) are the fixed points of the operator Ψ and the fixed points of the operator Ψ are the common fixed points of operators \(\varPsi_{1}\) and \(\varPsi_{2}\).
Theorem 4.3
The class of non-linear matrix equations (1.1) has a solution under the following conditions:
1.
there are two positive real numbers
\(M_{1}\)
and
\(M_{2}\)
such that
\(\sum^{n}_{i=1}A_{i} A^{*}_{i}\prec M_{1} I_{n}\)
and
\(\sum^{n}_{i=1}B_{i} B^{*}_{i}\prec M_{2}I_{n}\);
2.
for every
\(\aleph_{1},\aleph_{2}\in \mathcal{H}(m)\)
such that
\((\aleph_{1},\aleph_{2})\in \preceq \), we have
where
\(M=\max \{M_{1},M_{2}\}\)
and
ξ
is positive real number.
$$\begin{aligned}& \Vert \aleph_{1} \Vert _{\mathrm{tr}}+ \Vert \aleph_{2} \Vert _{\mathrm{tr}} \\& \quad \leq \bigl( \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}\bigl(1 + \Vert \aleph_{1}-\aleph _{2} \Vert _{\mathrm{tr}}\bigr)+ \bigl\Vert \aleph_{2}- \varPsi_{1}(\aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPsi _{2}(\aleph_{2}) \bigr\Vert _{\mathrm{tr}}\bigr) \\& \qquad {}\big/\bigl(2M\bigl( \xi \sqrt{ \Vert \aleph_{1}- \aleph_{2} \Vert _{\mathrm{tr}}\bigl(1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}\bigr)+ \bigl\Vert \aleph_{2}-\varPsi_{1}( \aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPsi_{2}( \aleph_{2}) \bigr\Vert _{\mathrm{tr}}} \\& \qquad {}+\sqrt{1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}}\bigr) ^{2}\bigr), \end{aligned}$$
Proof
Since \(\varPsi_{1}\) and \(\varPsi_{2}\) are well defined and \((\aleph_{1}, \aleph_{2})\in \preceq \) implies that \((\varPsi_{1}(\aleph_{1}),\varPsi_{2}( \aleph_{2})), (\varPsi_{2}(\aleph_{1}),\varPsi_{1}(\aleph_{2}))\in \preceq \), so that ⪯ on \(\mathcal{H}(m)\) is \((\varPsi_{1},\varPsi_{2})\)-closed.
We have to show that the operators \(\varPsi_{1}\) and \(\varPsi_{2}\) satisfy the rational type \(\mathsf{F}_{\preceq }\)-contractive conditions. For this purpose, let us consider
From conditions (1) and (2) of Theorem 4.3 it follows that
Let \(\mathsf{F}: [0, \infty)\rightarrow \mathbb{R}\) be the mapping defined by \(\mathsf{F}(\kappa_{1})=-\frac{1}{\sqrt{\kappa_{1}}}\). Then \(\mathsf{F}\in \mathbb{F}\) and the above inequality becomes
Thus
That is, the pair \((\varPsi_{1}, \varPsi_{2})\) is rational \(\mathsf{F}_{ \mathcal{R}}\)-contractive. Thus from Theorem 3.3, there is a common fixed point of \(\varPsi_{1}\) and \(\varPsi_{2}\), say \(\aleph^{*}\), i.e., \(\varPsi_{1}( \aleph^{*} )=\varPsi_{2}(\aleph^{*})= \aleph^{*}\). Consequently, Ψ has a fixed point and hence the class of non-linear matrix equation (1.1) has a solution. □
$$ \begin{aligned} \bigl\Vert \varPsi_{1}( \aleph_{1})-\varPsi_{2}(\aleph_{2}) \bigr\Vert _{\mathrm{tr}}={} &\operatorname{tr}\bigl( \varPsi_{1}( \aleph_{1})- \varPsi_{2}(\aleph_{2}) \bigr) \\ ={} &2\operatorname{tr}\Biggl( \sum^{n}_{i=1} \bigl( A^{*}_{i}\aleph _{1} A_{i}+B^{*}_{i} \aleph_{2} B_{i} \bigr) \Biggr) \\ ={} &2\sum^{n}_{i=1} \operatorname{tr}\bigl( A^{*}_{i}\aleph_{1} A _{i}+B^{*}_{i} \aleph_{2} B_{i} \bigr) \\ ={} &2 \Biggl( \sum^{n}_{i=1} \operatorname{tr}\bigl( A_{i}A^{*}_{i} \aleph_{1} \bigr) +\sum ^{n}_{i=1} \operatorname{tr}\bigl( B_{i}B ^{*}_{i}\aleph_{2} \bigr) \Biggr) \\ ={} & 2 \Biggl( \operatorname{tr}\Biggl( \sum^{n}_{i=1} A_{i}A^{*}_{i} \aleph_{1} \Biggr) +\operatorname{tr}\Biggl( \sum^{n}_{i=1}B_{i}B ^{*}_{i}\aleph_{2} \Biggr) \Biggr) \\ \leq{} &2 \Biggl( \Biggl\Vert \sum^{n}_{i=1} A_{i}A^{*} _{i} \Biggr\Vert \Vert \aleph_{1} \Vert _{\mathrm{tr}}+ \Biggl\Vert \sum ^{n}_{i=1}B _{i}B^{*}_{i} \Biggr\Vert \Vert \aleph_{2} \Vert _{\mathrm{tr}} \Biggr) \\ \leq{} &2 \bigl( M_{1} \Vert \aleph_{1} \Vert _{\mathrm{tr}}+M_{2} \Vert \aleph _{2} \Vert _{\mathrm{tr}} \bigr) \\ \leq{} &2M \bigl( \Vert \aleph_{1} \Vert _{\mathrm{tr}}+ \Vert \aleph_{2} \Vert _{\mathrm{tr}} \bigr). \end{aligned} $$
$$\begin{aligned}& \bigl\Vert \varPsi_{1}(\aleph_{1})- \varPsi_{2}(\aleph_{2}) \bigr\Vert _{\mathrm{tr}} \\& \qquad \leq \bigl( \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}} + \Vert \aleph_{1}-\aleph_{2} \Vert ^{2}_{\mathrm{tr}}+ \bigl\Vert \aleph_{2}- \varPsi_{1}(\aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPsi _{2}(\aleph_{2}) \bigr\Vert _{\mathrm{tr}}\bigr) \\& \qquad \quad \Big/\Bigl(\Bigl( \xi \sqrt{ \Vert \aleph_{1}- \aleph_{2} \Vert _{\mathrm{tr}} + \Vert \aleph_{1}- \aleph_{2} \Vert ^{2}_{\mathrm{tr}}+ \bigl\Vert \aleph_{2}-\varPsi_{1}(\aleph _{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPsi_{2}( \aleph_{2}) \bigr\Vert _{\mathrm{tr}}} \\& \qquad \quad {}+\sqrt{1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}}\Bigr) ^{2}\Bigr) \\& \quad \Longrightarrow\quad \Bigl(\xi \sqrt{ \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}} + \Vert \aleph_{1}-\aleph_{2} \Vert ^{2}_{\mathrm{tr}}+ \bigl\Vert \aleph_{2}-\varPsi_{1}(\aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}- \varPsi_{2}(\aleph_{2}) \bigr\Vert _{\mathrm{tr}}} \\& \hphantom{\quad \Longrightarrow\quad} \qquad {}+\sqrt{1 + \Vert \aleph _{1}- \aleph_{2} \Vert _{\mathrm{tr}}}\Bigr) \\& \hphantom{\quad \Longrightarrow\quad} \qquad {}\Big/\Bigl(\sqrt{ \Vert \aleph_{1}- \aleph_{2} \Vert _{\mathrm{tr}} + \Vert \aleph_{1}- \aleph_{2} \Vert ^{2}_{\mathrm{tr}}+ \bigl\Vert \aleph_{2}-\varPsi_{1}(\aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPsi_{2}( \aleph_{2}) \bigr\Vert _{\mathrm{tr}}}\Bigr) \\& \hphantom{\quad \Longrightarrow\quad}\quad \leq \frac{1}{\sqrt{ \Vert \varPsi_{1}(\aleph_{1})-\varPsi_{2}(\aleph_{2}) \Vert _{\mathrm{tr}}}} \\& \quad \Longrightarrow\quad \xi +\frac{\sqrt{1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}}}{\sqrt{ \Vert -\aleph_{2} \Vert _{\mathrm{tr}} + \Vert \aleph_{1}-\aleph_{2} \Vert ^{2}_{\mathrm{tr}}+ \Vert \aleph_{2}-\varPsi_{1}( ) \Vert \Vert \aleph_{1}-\varPsi_{2}(\aleph_{2}) \Vert _{\mathrm{tr}}}} \\& \hphantom{\hphantom{\quad \Longrightarrow\quad}}\quad \leq \frac{1}{\sqrt{ \Vert \varPsi_{1}(\aleph_{1})-\varPsi_{2}(\aleph _{2}) \Vert _{\mathrm{tr}}}} \\& \quad \Longrightarrow\quad \xi -\frac{1}{\sqrt{ \Vert \varPsi_{1}(\aleph_{1})- \varPsi_{2}(\aleph_{2}) \Vert _{\mathrm{tr}}}} \\& \hphantom{\quad \Longrightarrow\quad}\quad \leq -\frac{1}{\sqrt{\frac{ \Vert \aleph _{1}-\aleph_{2} \Vert _{\mathrm{tr}} + \Vert \aleph_{1}-\aleph_{2} \Vert ^{2}_{\mathrm{tr}}+ \Vert \aleph _{2}-\varPsi_{1}(\aleph_{1}) \Vert _{\mathrm{tr}} \Vert \aleph_{1}-\varPsi_{2}(\aleph_{2}) \Vert _{\mathrm{tr}}}{1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}}}} \\& \hphantom{\quad \Longrightarrow\quad}\quad =-\frac{1}{\sqrt{ \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}+\frac{ \Vert \aleph_{2}- \varPsi_{1}(\aleph_{1}) \Vert _{\mathrm{tr}} \Vert \aleph_{1}-\varPsi_{2}(\aleph_{2}) \Vert _{\mathrm{tr}}}{ {1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}}}}}. \end{aligned}$$
$$ \begin{aligned} \xi +\mathsf{F}\bigl( \bigl\Vert \varPsi_{1}(\aleph_{1})-\varPsi_{2}( \aleph_{2}) \bigr\Vert _{\mathrm{tr}}\bigr) \leq & \mathsf{F} \biggl( \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}+\frac{ \Vert \aleph _{2}-\varPsi_{1}(\aleph_{1}) \Vert _{\mathrm{tr}} \Vert \aleph_{1}-\varPsi_{2}(\aleph_{2}) \Vert _{\mathrm{tr}}}{1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}} \biggr). \end{aligned} $$
$$ \begin{aligned} \xi +\mathsf{F}(d\bigl(\varPsi_{1}( \aleph_{1}),\varPsi_{2}(\aleph_{2})\bigr) \leq \mathsf{F} \biggl( d(\aleph_{1},\aleph_{2})+ \frac{d(\aleph_{2},\varPsi_{1}( \aleph_{1}))d(\aleph_{1},\varPsi_{2}(\aleph_{2}))}{1 + d(\aleph_{1}, \aleph_{2})} \biggr). \end{aligned} $$
The next existence result ensures the uniqueness of solution to the non-linear matrix equation (1.1) and the proof is similar to the proof of Theorem 4.3, so we omit it.
Theorem 4.4
Under the condition (1) of Theorem
4.3, the class of non-linear matrix equations (1.1) has a unique solution if for every
\(\aleph_{1},\aleph_{2}\in \mathcal{H}(m)\)
such that
\((\aleph_{1},\aleph_{2})\in \preceq \), we have
where
\(M=\max \{M_{1},M_{2}\}\)
and
ξ
is positive real number.
$$\begin{aligned}& \Vert \aleph_{1} \Vert _{\mathrm{tr}}+ \Vert \aleph_{2} \Vert _{\mathrm{tr}} \\& \quad \leq \bigl( \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}\bigl(1 + \Vert \aleph_{1}-\aleph _{2} \Vert _{\mathrm{tr}}\bigr)+ \bigl\Vert \aleph_{2}- \varPsi_{1}(\aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPsi _{2}(\aleph_{2}) \bigr\Vert _{\mathrm{tr}}\bigr) \\& \qquad {}\big/\bigl(2M\bigl( \xi \sqrt{ \Vert \aleph_{1}- \aleph_{2} \Vert _{\mathrm{tr}}\bigl(1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}\bigr)+ \bigl\Vert \aleph_{2}-\varPsi_{1}( \aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPsi_{2}( \aleph_{2}) \bigr\Vert _{\mathrm{tr}}} \\& \qquad {}+2\sqrt{1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}}\bigr) ^{2}\bigr), \end{aligned}$$
Define the operator \(\varPhi: \mathcal{H}(m)\rightarrow \mathcal{H}(m)\) by
Note that the solutions of the matrix equation (1.2) coincide with the fixed points of the operator \(\varPhi (\aleph)\).
$$ \varPhi (\aleph)=Q + \sum^{n}_{i=1} A^{*}_{i} \varUpsilon (\aleph) A_{i}. $$
Theorem 4.5
The class of non-linear matrix equations (1.2) has a solution under the following conditions:
(1)
there is a real positive real number
M
with
\(\sum^{n}_{i=1}A_{i} A^{*}_{i}\prec M I_{n}\);
(2)
for every
\(\aleph_{1},\aleph_{2}\in \mathcal{H}(m)\)
such that
\((\aleph_{1},\aleph_{2})\in \preceq \)
and
\(\sum^{n}_{i=1} A^{*}_{i} \varUpsilon (\aleph_{1}) A _{i}\neq \sum^{n}_{i=1} A^{*}_{i} \varUpsilon (\aleph _{2}) A_{i}\), we have
where
ξ
is positive real number.
$$\begin{aligned}& \bigl\Vert \operatorname{tr}\bigl( \varUpsilon (\aleph_{1})- \varUpsilon (\aleph_{2}) \bigr) \bigr\Vert _{\mathrm{tr}} \\& \quad \leq \bigl( \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}\bigl(1 + \Vert \aleph_{1}-\aleph _{2} \Vert _{\mathrm{tr}}\bigr)+ \bigl\Vert \aleph_{2}-\varPhi ( \aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPhi ( \aleph_{2}) \bigr\Vert _{\mathrm{tr}}\bigr) \\& \qquad {}\big/\bigl(M\bigl( \xi \sqrt{ \Vert \aleph_{1}- \aleph_{2} \Vert _{\mathrm{tr}}\bigl(1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}\bigr)+ \bigl\Vert \aleph_{2}-\varPhi (\aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPhi (\aleph_{2}) \bigr\Vert _{\mathrm{tr}}} \\& \qquad {}+\sqrt{1 + \Vert \aleph_{1}-\aleph _{2} \Vert _{\mathrm{tr}}}\bigr) ^{2}\bigr), \end{aligned}$$
Proof
Since Φ is well defined and \((\aleph_{1},\aleph_{2})\in \preceq \) implies that \((\varPhi (\aleph_{1}),\varPhi (\aleph_{2}))\in \preceq \), ⪯ on \(\mathcal{H}(m)\) is Φ-closed.
We have to show that the operator \(\varPhi (\aleph_{1})\) satisfies the rational type \(\mathsf{F}_{\preceq }\)-contraction (3.16).
Let \(\aleph =\{(\aleph_{1},\aleph_{2})\in \preceq: \varUpsilon (\aleph _{1})\neq \varUpsilon (\aleph_{2})\}\). If \(\aleph_{1}, \aleph_{2}\in \aleph \), then \(\aleph_{2}\prec \aleph_{1}\). But ϒ is an order preserving mapping, so that \(\varUpsilon (\aleph_{2})\prec \varUpsilon ( \aleph_{1})\). Therefore,
using condition (2) of Theorem 4.3, we get
From condition (1) of Theorem 4.3 it follows that
Using \(\mathsf{F}(\kappa_{1})=-\frac{1}{\sqrt{\kappa_{1}}}\in \mathbb{F}\), the above inequality becomes
That is, Φ satisfies a rational type \(\mathsf{F}_{\mathcal{R}}\)-contraction (3.16). Thus from Corollary 3.9, there is a fixed point of Φ, say ℵ, i.e., \(\varPhi ({\aleph })= \aleph \). Consequently, the class of non-linear matrix equations (1.2) has a solution. □
$$\begin{aligned} \bigl\Vert \varPhi (\aleph_{1})-\varPhi ( \aleph_{2}) \bigr\Vert _{\mathrm{tr}} = &\operatorname{tr}\bigl( \varPhi (\aleph _{1})-\varPhi (\aleph_{2}) \bigr) \\ = &\operatorname{tr}\Biggl( \sum^{n}_{i=1} A^{*}_{i} \bigl( \varUpsilon (\aleph_{1})- \varUpsilon (\aleph_{2}) \bigr) A_{i} \Biggr) \\ = &\sum^{n}_{i=1} \operatorname{tr}\bigl( A^{*}_{i} \bigl( \varUpsilon (\aleph_{1})- \varUpsilon (\aleph_{2}) \bigr) A_{i} \bigr) \\ = &\sum^{n}_{i=1} \operatorname{tr}\bigl( A_{i} A^{*}_{i} \bigl( \varUpsilon ( \aleph_{1})-\varUpsilon (\aleph_{2}) \bigr) \bigr) \\ = &\operatorname{tr}\Biggl( \Biggl( \sum^{n}_{i=1}A_{i} A^{*}_{i} \Biggr) \bigl( \varUpsilon (\aleph_{1})- \varUpsilon (\aleph_{2}) \bigr) \Biggr) \\ \leq & \Biggl\Vert \sum^{n}_{i=1}A_{i} A^{*}_{i} \Biggr\Vert \bigl\Vert \varUpsilon ( \aleph_{1})-\varUpsilon (\aleph_{2}) \bigr\Vert _{\mathrm{tr}}, \end{aligned}$$
$$\begin{aligned}& \bigl\Vert \varPhi (\aleph_{1})-\varPhi ( \aleph_{2}) \bigr\Vert _{\mathrm{tr}} \\& \quad \leq \Biggl( \Biggl( \Biggl\Vert \sum^{n}_{i=1}A_{i} A ^{*}_{i} \Biggr\Vert \Biggr) \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}\bigl(1 + \Vert \aleph_{1}- \aleph_{2} \Vert _{\mathrm{tr}} \bigr)+ \bigl\Vert \aleph_{2}-\varPhi (\aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}- \varPhi ( \aleph_{2}) \bigr\Vert _{\mathrm{tr}}\Biggr) \\& \qquad {}\big/\bigl(M\bigl( \xi \sqrt{ \Vert \aleph_{1}- \aleph_{2} \Vert _{\mathrm{tr}}\bigl(1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}\bigr)+ \bigl\Vert \aleph_{2}-\varPhi (\aleph _{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPhi (\aleph_{2}) \bigr\Vert _{\mathrm{tr}}} \\& \qquad {}+\sqrt{1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}}\bigr) ^{2}\bigr). \end{aligned}$$
$$\begin{aligned}& \bigl\Vert \varPhi (\aleph_{1})-\varPhi ( \aleph_{2}) \bigr\Vert _{\mathrm{tr}} \\& \qquad \leq \bigl( \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}} + \Vert \aleph_{1}-\aleph_{2} \Vert ^{2}_{\mathrm{tr}}+ \bigl\Vert \aleph_{2}-\varPhi ( \aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPhi ( \aleph_{2}) \bigr\Vert _{\mathrm{tr}}\bigr) \\& \quad \qquad \Big/{}\Bigl(\Bigl( \xi \sqrt{ \Vert \aleph_{1}- \aleph_{2} \Vert _{\mathrm{tr}} + \Vert \aleph_{1}- \aleph_{2} \Vert ^{2}_{\mathrm{tr}}+ \bigl\Vert \aleph_{2}-\varPhi (\aleph_{1}) \bigr\Vert _{\mathrm{tr}} \bigl\Vert \aleph_{1}-\varPhi (\aleph_{2}) \bigr\Vert _{\mathrm{tr}}} \\& \quad \qquad {}+\sqrt{1 + \Vert \aleph_{1}- \aleph_{2} \Vert _{\mathrm{tr}}}\Bigr) ^{2}\Bigr) \\& \quad \Longrightarrow\quad \xi -\frac{1}{\sqrt{ \Vert \varPhi (\aleph_{1})-\varPhi (\aleph_{2}) \Vert _{\mathrm{tr}}}} \leq -\frac{1}{\sqrt{ \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}+\frac{ \Vert \aleph _{2}-\varPhi (\aleph_{1}) \Vert _{\mathrm{tr}} \Vert \aleph_{1}-\varPhi (\aleph_{2}) \Vert _{\mathrm{tr}}}{ {1 + \Vert \aleph_{1}-\aleph_{2} \Vert _{\mathrm{tr}}}}}}. \end{aligned}$$
$$ \begin{aligned} \xi +\mathsf{F}(d\bigl(\varPhi (\aleph_{1}), \varPhi (\aleph_{2})\bigr) \leq \mathsf{F} \biggl( d( \aleph_{1},\aleph_{2})+\frac{d(\aleph_{2},\varPhi ( \aleph_{1}))d(\aleph_{1},\varPhi (\aleph_{2}))}{1 + d(\aleph_{1},\aleph _{2})} \biggr). \end{aligned} $$
Theorem 4.6
5 Conclusion
Non-linear matrix equations occur in several problems of engineering and applied mathematics. Some various matrix equations are faced in stability analysis, control theory and system theory. In the current work we obtain a common fixed point theorem via rational type \(F_{\mathcal{R}}\)-contractive conditions with applications to the existence of solutions to non-linear matrix equations.
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Department of Mathematics, University of Malakand, Chakdara, Dir(L), Khyber Pakhtunkhwa, Pakistan.
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