## 5 Main Results
This implies that there exists \(\mathfrak{d}\in\mathbb{R}^{n}\) such that  
\[D_{gH}\mathcal{H}_{k}(\bar{u},\mathfrak{d})<_{LU}[0,0],\quad\ k\in\mathcal{I}_{ m}. \tag{34}\]  
Since \(\mathcal{H}\) is continuously gH-differentiable, therefore, there exist \(\epsilon>0\) and \(\delta>0\), such that  
\[D_{gH}\mathcal{H}_{k}(u,\mathfrak{d})<_{LU}[-\epsilon,-\epsilon],\ \ \text{for all}\ k\in\mathcal{I}_{m}\ \text{and for all}\ u\in B(\bar{u},\delta). \tag{35}\]  
Since \(u^{\mathfrak{d}^{\mathfrak{d}}}\to\bar{u}\) as \(j\to\infty\), therefore, using (35), there exists \(n_{0}\in\mathbb{N}\), such that  
\[D_{gH}\mathcal{H}_{k}(u^{\mathfrak{d}^{\mathfrak{d}}},\mathfrak{d})<_{LU}[- \epsilon,-\epsilon],\ \ \text{for all}\ k\in\mathcal{I}_{m}\ \text{and for all}\ j\geq n_{0}. \tag{36}\]  
Now, for every \(j\geq n_{0}\), by defining \(\alpha_{s_{j}}:=\widetilde{\Phi}(u^{\mathfrak{d}^{\mathfrak{d}}},\mathfrak{d})\), we get  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]0  
This implies that, for every \(j\geq n_{0}\),  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]1  
Using (36) and (37), for all \(j\geq n_{0}\), we obtain  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]2  
This implies that, for all \(j\geq n_{0}\), we get  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]3  
Now, for all \(j\geq n_{0}\) and for all \(k\in\mathcal{I}_{m}\), we consider  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]4 \[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]5 \[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]6  
Therefore, for \(t=\frac{1}{2^{p}}\), the inequality in (22) is not satisfied, that is, for all \(s\geq n_{p}\), we have  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]7  
Letting \(p\to\infty\) (along a suitable subsequence, if necessary) in both sides of the inequality in (40), there exists \(k\in\mathcal{I}_{m}\) such that  
\[\mathcal{H}_{k}(\mathfrak{u}^{(s+1)})\ominus_{gH}\mathcal{H}_{k}(\mathfrak{ u}^{(s)})\leq_{LU}(\gamma\eta^{(s)})\odot D_{gH}\mathcal{H}_{k}(\mathfrak{u}^{(s)},\mathfrak{w}^{(s)}),\quad\text{ for all }k\in\mathcal{I}_{m}.\]8  
which leads to a contradiction. This completes the proof. The proof of the following lemma follows from Remark 1, Proposition 1, and Definition 3. **Lemma 8**. : _Let the functions \(\mathcal{H}_{k}:\mathbb{R}^{n}\to\mathfrak{C}(\mathbb{R})\)\((k\in\mathcal{I}_{m})\) be gH-Lipschitz continuous with Lipschitz constant \(\mathcal{L}\) and twice continuously gH-differentiable on \(\mathbb{R}^{n}\).