# On the Existence of Solutions of Nonlinear Fredholm Integral Equations from Kantorovich's Technique
## 2 [PERSON]'s Technique
### The Majorizing Sequence  
We begin by studying the function given in (7). Firstly, we notice that we have considered any \(t_{0}\geq 0\) in the last section, but we can consider \(t_{0}=0\), so function \(\varphi(t)\) is reduced to \[\phi(t)=|\varpi|S\left(\|x_{0}\|+t\right)^{p}-t+\|\mathfrak{F}(x_{0})\|-|\varpi|S \|x_{0}\|^{p}. \tag{8}\]  
This is a consequence of the fact that \(\phi(t)=\,\varphi(t+t_{0})\), which leads us to the sequence \(\{t_{n}=N_{\varphi}(t_{n-1})\}_{n\in\mathbb{N}}\), for any \(t_{0}>0\), satisfies \(t_{n}=N_{\varphi}(t_{n-1})=t_{0}+N_{\varphi}(s_{n-1})\), \(n\in\mathbb{N}\), where \(s_{n}=N_{\varphi}(s_{n-1})\) with \(s_{0}=0\), since we have, for \(t_{0}\geq 0\) and \(s_{0}=0\),  
\[t_{0}+s_{n}=t_{0}+N_{\varphi}(s_{n-1})=t_{0}+s_{n-1}-\frac{\phi(s_{n-1})}{\phi ^{\prime}(s_{n-1})}\]  
\[=t_{0}+s_{n-1}-\frac{\varphi(s_{n-1}+t_{0})}{\phi^{\prime}(s_{n-1}+t_{0})}=t_{ n-1}-\frac{\varphi(t_{n-1})}{\phi^{\prime}(t_{n-1})}=N_{\varphi}(t_{n-1})=t_{n},\]  
for all \(n\in\mathbb{N}\). Therefore, the real sequences \(\{t_{n}\}\) and \(\{s_{n}\}\) given by [PERSON]'s method when they are constructed from \(\varphi(t)\) and \(\phi(t)\), respectively, can be obtained, one from the other, by translation. Besides, \(t_{n}-t_{n-1}=s_{n}-s_{n-1}\), for all \(n\in\mathbb{N}\), and all the results obtained previously are independent of the value \(t_{0}\geq 0\), so we choose \(t_{0}=0\) because, in practice, it is the most favourable situation.  
Secondly, we denote \(\sigma=\min\{t>0:\phi^{\prime}(t)\geq 0\}\), where \(\phi(t)\) is given in (8). Note that there exists only one positive real zero \(\sigma\) of \(\phi^{\prime}(t)\) in \((0,+\infty)\) satisfying \(\sigma=\min\{t>0:\phi^{\prime}(t)\geq 0\}\), since \(\phi^{\prime}(0)=-1<0\), \(\phi^{\prime\prime}(t)>0\) and \(\phi^{\prime}(t)>0\) as \(t\to+\infty\).  
Theorem 1: _If \(\phi(\sigma)\leq 0\), then \(\phi(t)\) has two real zeros \(r\) and \(R\) such that \(0\leq r\leq\sigma\leq R\)._  
Thirdly, by taking into account the classical Fourier conditions [19] for the convergence of [PERSON]'s method in the scalar case, we establish that sequence \(\{t_{n}\}\) is increasing and converges to \(r\) in the following result.  
Theorem 2: _If \(\phi(\sigma)\leq 0\), then sequence \(\{t_{n}\}\) is increasing and converges to the positive real zero \(r\) of \(\phi(t)\)._  
Fourthly, we prove a system of recurrence relations in the next theorem that guarantees that \(\{t_{n}\}\) is a majorizing sequence of \(\{x_{n}\}\) in \(\Omega\), whose proof is similar to that given for Lemma 7 in [18].  
Theorem 3: _Suppose that \(x_{n}\in\Omega\), for all \(n\geq 0\), and \(p\geq 3\). If \(\phi(\sigma)\leq 0\), then the following four bounds are satisfied for all \(n\in\mathbb{N}\):_  
* _there exists_ \([\mathfrak{F}^{\prime}(x_{n})]^{-1}\) _and_ \(\|[\mathfrak{F}^{\prime}(x_{n})]^{-1}\|\leq-\frac{1}{\phi^{\prime}(t_{n})}\)_,_
* \(\|\mathfrak{F}^{(i)}(x_{n})\|\leq\phi^{(i)}(t_{n})\)_, for_ \(i=2,3,\ldots,k-1\)_,_
* \(\|\mathfrak{F}(x_{n})\|\leq\phi(t_{n})\)_,_
* \(\|x_{n+1}-x_{n}\|\leq t_{n+1}-t_{n}\)_._  
Note that \((i)\), \((ii)\) and \((iv)\) are obvious if \(n=0\) and \((iii)\) are not necessary to prove \((iv)\), since it follows from the initial condition \(\|[\mathfrak{F}^{\prime}(x_{0})]^{-1}\mathfrak{F}(x_{0})\|\leq\eta\).  
Finally, if \(p\in[2,3)\), we obtain a result similar to the last theorem which can be seen in [20].