## 7 Application to the Existence of the Solution to Urysohn Integral Equation (UIE)  
In this section, our concern is to obtain a unique converging point for UIE:  
\[\ell(v)=b(v)+\int\limits_{\text{IR}}L_{1}(v,t,\ell(t))\,dt. \tag{14}\]  
This above Equation (14) depending on the range of integration (IR), is a generalization of many integral equations in literature (see [40; 41; 42]). In the present article, we use a fixed point method to calculate a single converging point to Urysohn Integral Equation, which also leads to convergence of different mathematical structures. Assume IR as the set of finite measure and \(\mathcal{J}_{\text{IR}}^{2}=\{\ell\,|\,\int_{\text{IR}}|\ell(t)|^{2}dt<\infty\}\). Assume \(\left\|\cdot\right\|:\mathcal{J}_{\text{IR}}^{2}\rightarrow[0,\infty)\) with  
\[\left\|\ell\right\|_{2}=\sqrt{\int_{\text{IR}}|\ell(t)|^{2}dt},\text{ for all }\ell,t\in\mathcal{J}_{\text{IR}}^{2}.\]  
Similarly, take:  
\[\left\|\ell\right\|_{2,v}=\sqrt{\sup\{e^{-v\int_{\text{IR}}\alpha(t)dt}\int_ {\text{IR}}|\ell(t)|^{2}dt\}},\text{ for all }\ell\in\mathcal{J}_{\text{IR}}^{2},v>1.\]  
One can observe \(E=(\mathcal{J}_{\text{IR}}^{2},\left\|\cdot\right\|_{2,v})\) represents a Banach space. Assume a cone \(\mathcal{Z}=\{\ell\in\mathcal{J}_{\text{IR}}^{2}:\ell(t)>0\text{ for almost every }s\}\) in \(E\). Define \(c_{v}\) as \(c_{v}(x_{1},x_{2})=x_{1}\|x_{1}-x_{2}\|_{2,v}^{2}\) for all \(x_{1},\,x_{2}\in\mathcal{Z}\). Then, \(c_{v}\) represents a cone \(\mathbf{A}\)-metric. Define partial order \(\preceq_{\epsilon}\) on \(E\), such that  
\[\alpha\preceq_{\epsilon}\text{\ \ \ \ \ \ \ \ \ \ \  
* The kernel \(L_{1}:\mathrm{IR}\times\mathrm{IR}\times\mathbb{R}\to\mathbb{R}\) fulfils the Caratheodory axiom along with \[|L_{1}(v,t,\ell(t))|\leq w(v,t)+e(v,t)|\ell(t)|;w,e\in\mathcal{J}^{2}(\mathrm{ IR}\times\mathrm{IR}),e(v,t)>0.\]
* Assume a continuous and bounded function \(b:\mathrm{IR}\to[1,\infty)\) over IR. * There exists a constant \(C>0\) such that \[\sup_{v\in\mathrm{IR}}\int_{\mathrm{IR}}|L_{1}(v,t)|dt\leq C.\]
* For any \(\ell_{0}\in\mathcal{J}^{2}_{\mathrm{IR}}\), there exists \(\ell_{1}=K(\ell_{0})\) such that \(\ell_{1}\preceq_{e}\ell_{0}\) or \(\ell_{0}\preceq_{e}\ell_{1}\). * Take a sequence \(\{\ell_{n}\}\) satisfying \(\ell_{n-1}\preceq_{e}\ell_{n}\) such that \(\ell_{n}\to p\), which further implies to \(\ell_{n}\preceq_{e}p\) for all \(n\in\mathbb{N}\). * We can find a non-negative and measurable function \(q:\mathrm{IR}\times\mathrm{IR}\to\mathbb{R}\) such that \[\alpha(v):=\int_{\mathrm{IR}}q^{2}(v,s)\,dt\leq\frac{1}{v}\] also integrable over \(IR\) such that \[|L_{1}(v,s,\ell(s))-L_{1}(v,s,l(s))|\leq q(v,s)|\ell(s)-l(s)|\] for all \(v,s\in\mathrm{IR}\) and \(\ell,j\in E\) such that \(\ell\preceq_{e}J\). Theorem 4.1: _Assume that \(L_{1}\) satisfies all axioms (C1)-(C5), and then we obtain a single converging point for \(\mathrm{UIE}\)._  
Proof. : Defining a function \(K:E\to E\), using defined notations by  
\[(K\ell)(v)=b(v)+\int_{\mathrm{IR}}L_{1}(v,t,\ell(t))dt,\]  
\(K\) is \(\preceq\)-preserving :  
Let \(\ell,j\in E\) along with \(\ell\preceq j\), so that \(\ell(s)J(s)\geq\ell(s)\). For almost every \(v\in IR\), we have  
\[(K\ell)(v)=b(v)+\int_{\mathrm{IR}}L_{1}(v,t,\ell(t))dt\geq 1,\]  
which shows \((K\ell)(v)(K_{J})(v)\geq(K\ell)(v)\). Hence, \((K\ell)\preceq(K\ell)\). Self-operator :  
The use of (C1) and (C3) leads to a continuous and compact function \(K:\mathcal{Z}\to\mathcal{Z}\)(see [40], Lemma 3). Using (C4), we have \(\ell_{1}=K(\ell_{0})\) with \(\ell_{1}\preceq\ell_{0}\) or \(\ell_{0}\preceq\ell_{1}\), for each \(\ell_{0}\in\mathcal{Z}\) and \(K\) taken as \(\preceq\)-preserving. Hence, \(\ell_{n}=R^{n}(\ell_{0}))\) with \(\ell_{n}\preceq\ell_{n+1}\) or \(\ell_{n+1}\preceq\ell_{n}\) for all \(n\geq 0\). Applying (C5) along with Holder inequality, we find a contractive condition of Theorem 2.