## 4 Illustrative Examples  
**Example 1**.: _Consider the following coupled system of second-order ordinary differential equations\[\left\{\begin{array}{l}\left(\left(\frac{9t+5}{6+t}\right)^{3}u^{ \prime}(t)\right)^{\prime}=f(t,u,v)=\frac{e^{-t}}{39}+\frac{2}{t^{3}+105}u(t)+ \frac{|v(t)|^{2}}{82(1+|v(t)|)},\ t\in[0,2],\\ \left(\sqrt{2t^{2}+1}\ v^{\prime}(t)\right)^{\prime}=g(t,u,v)= \frac{\cos(t)}{16}+\frac{1}{272\pi}\sin(4\pi u)+\left(\frac{2}{t+9}\right)^{2} v(t),\ t\in[0,2],\end{array}\right. \tag{25}\]  
_supplemented with the following boundary conditions:_  
\[u^{\prime}(0)=0,\ \ u(2)=\frac{3}{8}v\Bigg{(}\frac{1}{3}\Bigg{)}+\frac{1}{2}v \Bigg{(}\frac{2}{3}\Bigg{)}+\frac{5}{8}v(1), \tag{26}\]  
Here,  
\[p(t)=\left(\frac{9t+5}{6+t}\right)^{3}\ \ \ \ \ \text{and}\ \ \ \ \ \ \ q(t)=\sqrt{2t^{2}+1},\]  
\[a=0,\ b=2,\ \eta_{1}=1/3,\ \eta_{2}=2/3,\ \eta_{3}=1,\ \xi_{1}=6/5,\ \xi_{2}=7/5,\ \xi_{3}=8/5,\ \xi_{4}=9/5,\]  
\[\alpha_{1}=3/8,\ \alpha_{2}=1/2,\ \alpha_{3}=5/8,\ \beta_{1}=2/7,\ \beta_{2}=4/7,\ \beta_{3}=6/7,\ \beta_{4}=8/7.\]  
Using the given data, it follows that \(|B|\approx 3.285715\ \
eq 0\) (\(B\) is given by (6)), \(\bar{p}\approx 0.578704\), \(\bar{q}=1\), \(L_{1}\approx 7.437340\), \(L_{2}\approx 1.048309\), \(L_{3}\approx 4.958609\), \(L_{4}\approx 2.995171\), [\(\bar{p}\), \(\bar{q}\) and \(L_{i}\ (i=1,\ldots,4)\) are defined in (18)], \(\mathcal{O}_{1}\approx 12.395949\) and \(\mathcal{O}_{2}\approx 4.043479\) [\(\mathcal{O}_{1}\) and \(\mathcal{O}_{2}\) are given by (17)]. Obviously,  
\[|f(t,u,v)|\leq\frac{1}{39}+\frac{2}{105}\|u\|+\frac{1}{82}\|v\|,\ \ |g(t,u,v)|\leq \frac{1}{16}+\frac{1}{68}\|u\|+\frac{4}{81}\|v\|,\]  
with \(\sigma_{0}=1/39\), \(\sigma_{1}=2/105\), \(\sigma_{2}=1/82\), \(\theta_{0}=1/16\), \(\theta_{1}=1/68\), \(\theta_{2}=4/81\). Moreover, \(\mathcal{O}_{1}\sigma_{1}+\mathcal{O}_{2}\theta_{1}\approx 0.295576<1\) and \(\mathcal{O}_{1}\sigma_{2}+\mathcal{O}_{2}\theta_{2}\approx 0.350848<1\), which imply that (20) is satisfied. Clearly the hypotheses of Theorem 1 are satisfied. Therefore, by the conclusion of Theorem 1, the problems (25) and (26) has at least one solution on \([0,2]\).  
**Example 2**.: _Consider the following system:_  
\[\left\{\begin{array}{l}\left(\Big{(}\frac{9t+5}{6+t}\Big{)}^{ 3}u^{\prime}(t)\right)^{\prime}=\frac{3}{5}+\left(t^{2}+\frac{7}{12}\right) \left[\left(u(t)\right)^{\frac{2}{9}}+\left(v(t)\right)^{\frac{5}{7}}\right], \ t\in[0,2],\\ \left(\sqrt{2t^{2}+1}\ v^{\prime}(t)\right)^{\prime}=\frac{9}{4}+ \left(t+\frac{2}{11}\right)^{4}\left[\left(u(t)\right)^{\frac{2}{9}}+\left(v (t)\right)^{\frac{3}{8}}\right],\ t\in[0,2],\end{array}\right. \tag{27}\]  
_subject to the coupled boundary conditions of Example 1._  
Here \(c_{1}=2/9\), \(c_{2}=5/7\), \(k_{1}=2/5\), \(k_{2}=3/8\). Evidently, the condition \((A_{2})\) is satisfied. Thus, the conclusion of Theorem 2 applies to the system (27) with boundary conditions (26). So, there exists a solution of the problem (27) with the coupled boundary conditions (26) on \([0,2]\).  
**Example 3**.: _Consider the following system:_  
\[\left\{\begin{array}{l}\left(\left(\frac{9t+5}{6+t}\right)^{3}u^{ \prime}(t)\right)^{\prime}=\frac{3}{12\sqrt{t^{3}+121}}\Bigg{(}\tan^{-1}u(t)+ \frac{|v(t)|}{1+|v(t)|}\Bigg{)},\;t\in[0,2],\\ \left(\sqrt{2t^{2}+1}\,v^{\prime}(t)\right)^{\prime}=\frac{9}{7}+ \frac{2}{180\pi}\sin(5\pi x)+\frac{1}{18(t+1)}v(t),\;t\in[0,2],\end{array}\right. \tag{28}\]  
_with the coupled boundary conditions (26)._  
Obviously, we have  
\[|f(t,u_{1},v_{1})-f(t,u_{2},v_{2})|\leq\frac{1}{44}(|u_{1}-u_{2}|+|v_{1}-v_{2}|)\]  
and  
\[|g(t,u_{1},v_{1})-g(t,u_{2},v_{2})|\leq\frac{1}{18}(|u_{1}-u_{2}|+|v_{1}-v_{2} |),\]  
with \(\ell_{1}=1/44\) and \(\ell_{2}=1/18\). From Example 1, we conclude that  
\[\mathcal{O}_{1}\approx 12.395949,\quad\mathcal{O}_{2}\approx 4.043479\qquad \text{and}\qquad\mathcal{O}_{1}\ell_{1}+\mathcal{O}_{2}\ell_{2}\approx 0.506364 <1.\]  
Thus, by Theorem 3, the problem (28) equipped with the boundary conditions (26) has a unique solution on \([0,2]\).