## 3 Main Results
Thus, there must exist a constant \(\delta_{1}\) to satisfy the following inequality when \(0<\eta<2\) and \(\mu>0\):  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]4  
Substituting (11) into (10) gives  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]5 \[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]6 \[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]7 \[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]8  
which means \(\tilde{\phi}_{i}(k)\) is bounded. Since the boundedness of \(\phi_{i}(k)\) is guaranteed by Lemma 1, \(\widehat{\phi}_{i}(k)\) is also bounded. _Step 2_ Define the collective stack vectors as follows:  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]9 \[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]0 \[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]1 \[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]2  
Now, (4) is rewritten as the collective form:  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]3Using the collective stack vectors, (7) is rewritten as the following compact form:  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]4  
where  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]5  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]6  
Similarly, the CFDL model (3) is also transformed as the collective form  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]7  
where \(\Delta\mathbf{u}(k)=\mathbf{u}(k)-\mathbf{u}(k-1)\), and \(\boldsymbol{\Phi}(k)=diag\big{(}\phi_{1}(k)...\phi_{N}(k)\big{)}\). Substituting (14) into (15), one has  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]8  
Combining (13) and (16) yields  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\]9 \[\text{or}\ |\Delta u_{i}(k-1)|\leq\epsilon\]0  
Denote \(\boldsymbol{\Omega}(k)=[\mathbf{y}(k)^{T}\mathbf{w}_{s}^{T}]^{T}\) and  
\[\boldsymbol{\Gamma}(k)=\left[ \begin{array}{cc}I-\rho\boldsymbol{\Phi}(k) \mathbf{H}(k)\mathcal{L}_{1}&-\rho\boldsymbol{\Phi}(k)\mathbf{H}(k)\mathcal{L }_{2}\\ \mathbf{0}_{N\times N}&I\end{array} \right]\]  
Then, (17) can be rewritten as  
\[\boldsymbol{\Omega}(k+1)=\boldsymbol{\Gamma}(k)\boldsymbol{\Omega}(k) \tag{18}\]  
The \(i\)th (\(i\in\mathcal{F}\)) diagonal entry \(\gamma_{ii}(k)\) of \(\boldsymbol{\Gamma}(k)\) can be expressed as  
\[\gamma_{ii}(k)=1-d_{i}\cdot\rho_{i}\cdot\frac{\phi_{i}(k)\widehat{\phi}_{i}(k )}{\lambda+|\widehat{\phi}_{i}(k)|^{2}}=1-\frac{d_{i}\rho_{i}\phi_{i}(k) \widehat{\phi}_{i}(k)}{\lambda+|\widehat{\phi}_{i}(k)|^{2}} \tag{19}\]  
If the condition \(0<\rho_{i}<\frac{1}{d_{i}}\) holds, one has \(0<d_{i}\rho_{i}<1\). According to Assumption 3 and reset condition (6), \(\phi_{i}(k)\widehat{\phi}_{i}(k)>0\), which means \(\frac{\phi_{i}(k)\widehat{\phi}_{i}(k)}{\lambda+|\phi_{i}(k)|^{2}}>0\) when the parameters satisfy \(\lambda>0\).