In this section, some new upper bounds for \(\max_{d\in [0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty}\) are provided when M is a wcdd
B-matrix. Firstly, several lemmas, which will be used later, are given.
Lemma 1
[13]
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a wcdd
B-matrix with the form
\(M=B^{+}+C\), where
\(B^{+}\)
is defined as (1). Then
$$\bigl\Vert \bigl(I + \bigl(B^{+}_{D}\bigr)^{-1}C_{D} \bigr)^{-1}\bigr\Vert _{\infty} \leq n-1, $$
where
\(B^{+}_{D}=I-D+DB^{+}\)
and
\(C_{D}=DC\).
Lemma 2
[15]
Let
\(A=[a_{ij}]\in\mathbb{R}^{n\times n}\)
be a wcdd
M-matrix with
\(u_{k}(A)p_{k}(A)<1\) (\(\forall k\in\mathbb{N}\)). Then
$$\begin{aligned} \bigl\Vert A^{-1}\bigr\Vert _{\infty} &\leq \max\Biggl\{ \sum_{i=1}^{n} \Biggl( \frac{1}{ a_{ii} (1-u_{i}(A)p_{i}(A) ) } \prod_{j=1}^{i-1} \frac{u_{j}(A)}{1-u_{j}(A)p_{j}(A)} \Biggr), \\ &\quad \sum_{i=1}^{n} \Biggl( \frac{p_{i}(A)}{ a_{ii} (1-u_{i}(A)p_{i}(A) ) } \prod_{j=1}^{i-1} \frac{1}{1-u_{j}(A)p_{j}(A)} \Biggr) \Biggr\} , \end{aligned} $$
where
$$\prod_{j=1}^{i-1} \frac{u_{j}(A)}{1-u_{j}(A)p_{j}(A)} =1, \qquad \prod_{j=1}^{i-1} \frac{1}{1-u_{j}(A)p_{j}(A)}=1, \quad\textit{if } i=1. $$
Lemma 3
[14]
Let
\(\gamma>0\)
and
\(\eta\geq0\). Then, for any
\(x\in[0,1]\),
$$\frac{1}{1-x+\gamma x}\leq \frac{1}{ \min\{\gamma, 1 \}}, \qquad \frac{\eta x}{1-x+\gamma x}\leq \frac{\eta}{\gamma}. $$
Theorem 1
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a wcdd
B-matrix with the form
\(M=B^{+}+C\), where
\(B^{+}=[b_{ij}]\)
is defined as (1). If, for each
\(i\in\mathbb{N}\),
$$\hat{\beta}_{i}=b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert p_{i}\bigl(B^{+}\bigr)>0, $$
then
$$ \begin{aligned}[b] &\max_{d\in[0,1]^{n}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty } \\ &\quad\leq\max \Biggl\{ \sum_{i=1}^{n} \frac{n-1}{ \min \{\hat{\beta}_{i}, 1\} } \prod_{j=1}^{i-1} \Biggl( \frac{1}{\hat{\beta}_{j} }\sum_{k= j+1}^{n} \vert b_{jk}\vert \Biggr), \sum_{i=1}^{n} \frac{(n-1)p_{i}(B^{+})}{ \min \{\hat{\beta}_{i}, 1\} } \prod_{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} } \Biggr\} , \end{aligned} $$
(5)
where
$$\prod_{j=1}^{i-1} \Biggl(\frac{1}{\hat{\beta}_{j} } \sum_{k= j+1}^{n} \vert b_{jk} \vert \Biggr) =1, \qquad \prod_{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} }=1, \quad \textit{if } i=1. $$
Proof
Let \(M_{D}=I-D+DM\). Then
$$M_{D}=I-D+DM=I-D+D\bigl(B^{+}+C\bigr)=B_{D}^{+}+C_{D}, $$
where \(B_{D}^{+}=I-D+DB^{+}\). Similar to the proof of Theorem 2 in [13], we see that \(B_{D}^{+}\) is a wcdd
M-matrix with positive diagonal elements and \(C_{D}=DC\), and, by Lemma 1,
$$ \bigl\Vert M_{D}^{-1}\bigr\Vert _{\infty}\leq \bigl\Vert \bigl(I+\bigl(B_{D}^{+}\bigr)^{-1}C_{D} \bigr)^{-1}\bigr\Vert _{\infty} \bigl\Vert \bigl(B_{D}^{+} \bigr)^{-1}\bigr\Vert _{\infty} \leq (n-1)\bigl\Vert \bigl(B_{D}^{+}\bigr)^{-1}\bigr\Vert _{\infty}. $$
(6)
By Lemma 2, we have
$$\begin{aligned} \bigl\Vert \bigl(B_{D}^{+} \bigr)^{-1}\bigr\Vert _{\infty} &\leq \max\Biggl\{ \sum_{i=1}^{n} \frac{1}{ (1-d_{i}+b_{ii}d_{i}) (1-u_{i}(B_{D}^{+})p_{i}(B_{D}^{+}) ) } \prod_{j=1}^{i-1} \frac{u_{j}((B_{D}^{+}))}{1-u_{j}((B_{D}^{+}))p_{j}(B_{D}^{+})}, \\ &\quad \sum_{i=1}^{n} \frac{p_{i}(B_{D}^{+})}{ (1-d_{i}+b_{ii}d_{i}) (1-u_{i}((B_{D}^{+}))p_{i}(B_{D}^{+}) ) } \prod_{j=1}^{i-1} \frac{1}{1-u_{j}(B_{D}^{+})p_{j}(B_{D}^{+})} \Biggr\} . \end{aligned} $$
By Lemma 3, we can easily get the following results: for each \(i , j, k\in\mathbb{N}\),
$$\begin{aligned}& b_{k}\bigl(B_{D}^{+} \bigr)= \max_{k+1\leq i\leq n} \biggl\{ \frac{\sum_{j= k, \neq i}^{n} \vert b_{ij}\vert d_{i} }{1-d_{i}+b_{ii}d_{i}} \biggr\} \leq \max _{k+1\leq i\leq n} \biggl\{ \frac{\sum_{j= k, \neq i}^{n} \vert b_{ij}\vert }{b_{ii}} \biggr\} =b_{k} \bigl(B^{+}\bigr), \\& p_{k}\bigl(B_{D}^{+}\bigr)= \max _{k+1\leq i\leq n} \biggl\{ \frac{ \vert b_{ik}\vert d_{i}+\sum_{j= k+1, \neq i}^{n} \vert b_{ij}\vert d_{i}b_{k}(B_{D}^{+}) }{1-d_{i}+b_{ii}d_{i}} \biggr\} \\& \hphantom{p_{k}(B_{D}^{+})}\leq\max_{k+1\leq i\leq n} \biggl\{ \frac{ \vert b_{ik}\vert +\sum_{j= k+1, \neq i}^{n} \vert b_{ij}\vert b_{k}(B_{D}^{+}) }{b_{ii}} \biggr\} \\& \hphantom{p_{k}(B_{D}^{+})} \leq\max_{k+1\leq i\leq n} \biggl\{ \frac{ \vert b_{ik}\vert +\sum_{j= k+1, \neq i}^{n} \vert b_{ij}\vert b_{k}(B^{+}) }{b_{ii}} \biggr\} \\& \hphantom{p_{k}(B_{D}^{+})}=p_{k}\bigl(B^{+}\bigr), \end{aligned}$$
and
$$ \begin{aligned}[b] \frac{1}{ (1-d_{i}+b_{ii}d_{i})(1-u_{i}(B_{D}^{+})p_{i}(B_{D}^{+}))} &= \frac{1}{ 1-d_{i}+b_{ii}d_{i}-\sum_{j= i+1}^{n} \vert b_{ij}\vert d_{i} p_{i}(B_{D}^{+})} \\ &\leq\frac{1}{ \min \{ b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert p_{i}(B^{+}), 1 \} } \\ &=\frac{1}{ \min \{ \hat{\beta}_{i}, 1 \} }. \end{aligned} $$
(7)
Furthermore, by Lemma 3, we have
$$ \begin{aligned}[b] \frac{u_{i}(B_{D}^{+})}{ 1-u_{i}(B_{D}^{+})p_{i}(B_{D}^{+})} &= \frac{\sum_{j= i+1}^{n} \vert b_{ij}\vert d_{i} }{ 1-d_{i}+b_{ii}d_{i} -\sum_{j=i+1}^{n} \vert b_{ij}\vert d_{i} p_{i}(B_{D}^{+})} \\ &\leq\frac{\sum_{j= i+1}^{n} \vert b_{ij}\vert }{ b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert p_{i}(B^{+}) } \\ &=\frac{1}{\hat{\beta}_{i} }\sum_{j= i+1}^{n} \vert b_{ij}\vert \end{aligned} $$
(8)
and
$$ \begin{aligned}[b] \frac{1}{ 1-u_{i}(B_{D}^{+})p_{i}(B_{D}^{+})} &= \frac{1-d_{i}+b_{ii}d_{i} }{ 1-d_{i}+b_{ii}d_{i} -\sum_{j=i+1}^{n} \vert b_{ij}\vert d_{i} p_{i}(B_{D}^{+})} \\ &\leq\frac{1-d_{i}+b_{ii}d_{i} }{ b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert p_{i}(B^{+}) } \\ &=\frac{b_{ii}}{\hat{\beta}_{i} }. \end{aligned} $$
(9)
By (7), (8), and (9), we obtain
$$ \bigl\Vert \bigl(B_{D}^{+} \bigr)^{-1}\bigr\Vert _{\infty} \leq\max \Biggl\{ \sum _{i=1}^{n} \frac{1}{ \min \{\hat{\beta}_{i}, 1\} } \prod _{j=1}^{i-1} \Biggl(\frac{1}{\hat{\beta}_{j} }\sum _{k= j+1}^{n} \vert b_{jk}\vert \Biggr), \sum_{i=1}^{n} \frac {p_{i}(B^{+})}{ \min \{\hat{\beta}_{i}, 1\} } \prod _{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} } \Biggr\} . $$
(10)
Therefore, the result in (5) holds from (6) and (10). □
Since a B-matrix is also a wcdd
B-matrix, then by Theorem 1, we find the following result.
Corollary 1
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a
B-matrix with the form
\(M=B^{+}+C\), where
\(B^{+}=[b_{ij}]\)
is defined as (1). Then
$$ \begin{aligned}[b] & \max_{d\in[0,1]^{n}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty} \\ &\quad\leq \max \Biggl\{ \sum_{i=1}^{n} \frac{n-1}{ \min \{\hat{\beta}_{i}, 1\} } \prod_{j=1}^{i-1} \Biggl( \frac{1}{\hat{\beta}_{j} }\sum_{k= j+1}^{n} \vert b_{jk}\vert \Biggr), \sum_{i=1}^{n} \frac{(n-1)p_{i}(B^{+})}{ \min \{\hat{\beta}_{i}, 1\} } \prod_{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} } \Biggr\} , \end{aligned} $$
(11)
where
\(\hat{\beta}_{i}\)
is defined as in Theorem
1.
We next give a comparison of the bounds in (4) and (5) as follows.
Theorem 2
Let
\(M=[m_{ij}]\in\mathbb{R}^{n\times n}\)
be a wcdd
B-matrix with the form
\(M=B^{+}+C\), where
\(B^{+}=[b_{ij}]\)
is defined as (1). Let
\(\bar{\beta}_{i}\), \(\tilde{\beta}_{i}\), and
\(\hat{\beta}_{i}\)
be defined as in (3), (4), and (5), respectively. Then
$$ \begin{aligned}[b] &\max \Biggl\{ \sum _{i=1}^{n} \frac{n-1}{ \min \{\hat{\beta}_{i}, 1\} } \prod _{j=1}^{i-1} \Biggl(\frac{1}{\hat{\beta}_{j} }\sum _{k= j+1}^{n} \vert b_{jk}\vert \Biggr), \sum_{i=1}^{n} \frac{(n-1)p_{i}(B^{+})}{ \min \{\hat{\beta}_{i}, 1\} } \prod _{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} } \Biggr\} \\ &\quad \leq \sum_{i=1}^{n} \Biggl( \frac{n-1}{\min\{\tilde{\beta}_{i},1\}} \prod_{j=1}^{i-1} \frac{b_{jj}}{\tilde{\beta}_{j}} \Biggr). \end{aligned} $$
(12)
Proof
Since \(B^{+}\) is a wcdd matrix with positive diagonal elements, for any \(i\in\mathbb{N}\),
$$ 0\leq p_{i}\bigl(B^{+}\bigr)\leq1, \qquad \tilde{\beta}_{i}\leq \hat{\beta}_{i}. $$
(13)
By (13), for each \(i\in\mathbb{N}\),
$$ \frac{1}{ \hat{\beta}_{i} } \leq\frac{1}{ \tilde{\beta}_{i} }, \qquad \frac{1}{ \min\{ \hat{\beta}_{i}, 1\} } \leq\frac{1}{ \min \{ \tilde{\beta}_{i}, 1\} }. $$
(14)
The result in (12) follows by (13) and (14). □
Remark 1
-
(i)
Theorem 2 shows that the bound in (5) is better than that in (4).
-
(ii)
When n is very large, one needs more computations to obtain these upper bounds by (5) than by (4).