On the application of some fixed-point techniques to Fredholm integral equations of the second kind

Abstract

It is known that the global convergence of the method of successive approximations is obtained by means of the Banach contraction principle. In this paper, we study the global convergence of the method by means of a technique that uses auxiliary points and, as a consequence of this study, we obtain fixed-point type results on closed balls. We apply the study to nonlinear Fredholm integral equations of the second kind.

1 Introduction

It is well known that Fixed-Point Theory plays a prominent role in different branches of Mathematics, as well as in other disciplines such as game theory, optimization, and economics. There is a great variety of problems that naturally lead to applications of the Fixed-Point Theory [2, 4]. This theory is based on the so-called Fixed-Point Theorems, which refers to equations of the form

$$\begin{aligned} J(x)=x, \end{aligned}$$

(1)

where $J:{U}\rightarrow \mathcal {B}$ is an operator with ${U}\subset \mathcal {B}$ and $\mathcal {B}$ is a Banach space, and every solution of the Eq. (1) is called a fixed point of J.

The Fixed-Point Theory studies the conditions that the set U and the operator J must satisfy to guarantee the existence of at least one fixed point of J. In addition, the methods that approximate fixed points are also studied, as well as the set of all the fixed points of J.

In Banach spaces, the well-known Banach contraction principle (BCP for short) stands out [10]:

Let $\mathcal {B}$ be a Banach space with a contraction operator $J:\mathcal {B}\rightarrow \mathcal {B}$. Then, J admits a unique fixed point $x^*$ in $\mathcal {B}$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \mathcal {B}$.

This result is constructive and uses the construction of a solution from the method of successive approximations. And its field of application is so wide that it is usually believed that the Fixed-Point Theory is reduced to this principle [4].

The existence of a fixed point in the BCP can be proved in a simple and elegant way as follows (as we can see in [9]). Let $\varepsilon =\inf \{\Vert J(x)-x\Vert :x\in \mathcal {B}\}$. If $\sigma >0$ and $\Vert J(x)-x\Vert <\varepsilon +\sigma$, then $\varepsilon \le \Vert J(J(x))-J(x)\Vert \le K\Vert J(x)-x\Vert <K\varepsilon +\sigma$, where K is the contractivity factor. Hence, $\varepsilon =0$. Now, we consider the sets $\mathcal {B}_{\sigma }=\{x\in \mathcal {B}:\Vert J(x)-x\Vert \le \sigma \}$. Each one of these sets is nonempty and closed. If $x,y\in \mathcal {B}_{\sigma }$, then

$$\begin{aligned} \Vert x-y\Vert \le \Vert x-J(x)\Vert +\Vert J(x)-J(y)\Vert +\Vert J(y)-y\Vert \le 2\sigma +K\Vert x-y\Vert . \end{aligned}$$

If we denote the diameter of $\mathcal {B}_{\sigma }$ by $\text {diam}(\mathcal {B}_{\sigma })$, then $\text {diam}(\mathcal {B}_{\sigma })\le \frac{2\sigma }{1-K}\rightarrow 0$ when $\sigma \rightarrow 0$. Since $\mathcal {B}$ is complete, the intersection of $\{\mathcal {B}_{\sigma }:\sigma >0\}$ must therefore consist of exactly one point. It is clear that this point must be the unique fixed point of J.

Obviously, the BCP requires that Eq. (1) has a unique solution. If the operator J is nonlinear, it is clear that the previous condition is not usually satisfied. In this situation, we go to a fixed-point result restricted to a certain domain ${V}\subset \mathcal {B}$ that locates a solution of Eq. (1). We consider in our study the Restricted Contraction Principle (RCP for short) [11, 15]:

Let V be a nonempty closed subset of a Banach space $\mathcal {B}$ with a contraction operator $J:{V}\rightarrow {V}$. Then, J admits a unique fixed point $x^*$ in V. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in {V}$.

The RCP presents several difficulties when applying it. The first and fundamental is the location of the domain V. To solve this difficulty, it is necessary to have some information about the solution $x^*$ of Eq. (1), such as having prelocated the possible fixed points of J. Another difficulty is the condition $J:{V}\rightarrow {V}$, which is generally not easily verifiable. Finally, the fact that J is a contraction on V is a restriction whose modification has focused the study of numerous researchers [5, 6, 12].

If we look closely at the proof of the RCP, the condition $J:{V}\rightarrow {V}$ is necessary to correctly define the sequence $x_{n+1}=J(x_{n})$, $n\ge 0$, with $x_{0}\in {V}$. On the other hand, the contractivity condition guarantees that the sequence $x_{n+1}=J(x_{n})$, $n\ge 0$, is a Cauchy sequence and, therefore, converges on a Banach space. Finally, the condition that V is a closed set guarantees that $\lim _{n}x_{n}=x^*\in {V}$, since $\{x_{n}\}\in {V}$. Besides, the continuity of the operator J leads to $x^*=J(x^*)$. For all these reasons, it is clear that if we consider the method of successive approximations, the location of a domain of global convergence for this method give us a fixed-point result. That is, if we locate a domain V such that the method of successive approximations converges to a point $x^*$ from any $x_{0}\in V$, then $x^*$ is a fixed point of J and we obtain a fixed-point result if $x^*\in V$. This is the central idea of our work: the study of the global convergence of the method of successive approximations. For this, we give conditions on an auxiliary point $\widetilde{x}\in \mathcal {B}$ and construct a closed ball $\overline{B(\widetilde{x},r)}$ where the method of successive approximations converges globally, guaranteeing that the sequence $x_{n+1}=J(x_{n})$, $n\ge 0$, is well defined and its limit $x^*\in \overline{B(\widetilde{x},r)}$. In addition, we prove the uniqueness of the fixed point in the ball $\overline{B(\widetilde{x},r)}$. The technique based on the use of auxiliary points to prove the convergence of an iterative method has been used by the authors in Ref. [8]. A fundamental point of our work is to determine the conditions that the auxiliary point must satisfy.

Finally, we illustrate all the theoretical results presented with examples involving Fredholm integral equations of the second kind of the form [13, 14]

$$\begin{aligned} x(s) = f(s)+\eta \int _a^b \mathcal {D}(s,t) \mathcal {A}(x)(t)\,dt, \quad s\in [a,b],\quad \eta \in \mathbb {R}, \end{aligned}$$

(2)

where $f \in \mathcal {C}[a,b]$, the kernel $\mathcal {D}(s,t)$ is a known function in $[a,b]\times [a,b]$, $\mathcal {A}$ is the Nemytskii operator $\mathcal {A}:\mathcal {C}[a,b]\rightarrow \mathcal {C}[a,b]$ such that $\mathcal {A}(x)(t)={E}(x(t))$, where ${E}:\mathbb {R}\rightarrow \mathbb {R}$, and x(s) is the unknown function to be determined.

Throughout the paper, we denote $B(y,\mathfrak {r})=\{z\in \mathcal {B}:\Vert z-y\Vert <\mathfrak {r}\}$, $\overline{B(y,\mathfrak {r})}=\{z\in \mathcal {B}:\Vert z-y\Vert \le \mathfrak {r}\}$ and the set of bounded linear operators from $\mathcal {B}$ to $\mathcal {B}$ by $\mathcal {L}(\mathcal {B},\mathcal {B})$, and use the infinity norm in $\mathcal {B}$.

2 Preliminaries

As we have written previously, if we consider an operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$, where V is a nonempty open domain of a Banach space $\mathcal {B}$, the existence of more than one fixed point leads to try to apply the RCP. Therefore, we have to locate a set $\Lambda \subset {V}$, closed and nonempty, such that $J:\Lambda \rightarrow \Lambda$ is a contraction operator. In addition, to consider such set $\Lambda$, we need some information about the possible fixed points. What is usually done is to locate previously the fixed points. We illustrate this idea by considering the Fredholm integral equation of second kind given by

$$\begin{aligned} x(s) = s+\dfrac{1}{5}\int _0^1 s\,e^{-t} x(t)^2 dt, \quad s\in [0,1], \end{aligned}$$

(3)

where $\mathcal {D}(s,t)=s\,e^{-t}$ is the kernel of the integral equation and $\mathcal {A}(x)(t)=x(t)^2$ is a Nemytskii operator. Thus, we consider the operator $J_1:\mathcal {C}[0,1]\rightarrow \mathcal {C}[0,1]$ with

$$\begin{aligned}{}[J_1(x)](s) = s+\dfrac{1}{5}\int _0^1 s\,e^{-t} x(t)^2 dt, \quad s\in [0,1]. \end{aligned}$$

As the operator $J_1$ is not a contraction in the full space $\mathcal {C}[0,1]$, since

$$\begin{aligned} \Vert J_1(c)-J_1(d)\Vert \le \dfrac{e-1}{5e}\left( \Vert c\Vert +\Vert d\Vert \right) \Vert c-d\Vert , \end{aligned}$$

(4)

we try to locate previously the possible fixed points. For this, from (3), we have that

$$\begin{aligned} \Vert x^*\Vert \le \dfrac{e-1}{5e}\Vert x^*\Vert ^2+1, \end{aligned}$$

if $x^*$ is a fixed point of $J_1$. In this case, $\Vert x^*\Vert \le 1.1743\ldots$ or $\Vert x^*\Vert \ge 6.7355\ldots$ Now, we consider the set $\Lambda =\overline{B(0,2)}$, closed and nonempty, and, from (4), it follows that

$$\begin{aligned} \Vert J_1(c)-J_1(d)\Vert \le \dfrac{4(e-1)}{5e} \Vert c-d\Vert , \quad c,d\in \Lambda , \end{aligned}$$

so that $J_1$ is a contraction operator in $\Lambda$ with contractivity factor $L=\frac{4(e-1)}{5e}=0.5056\ldots$ Moreover, for each $x\in \Lambda$, we obtain

$$\begin{aligned} \Vert J_1(x)\Vert \le 1+\dfrac{4(e-1)}{5e} = 1.5057\ldots < 2, \end{aligned}$$

and then $J_1:\Lambda \rightarrow \Lambda$. Therefore, by the RCP, we see that there exists a unique fixed point $x^*$ of $J_1$ in the set $\Lambda =\overline{B(0,2)}$.

As a consequence of the above-mentioned, we can propose the following known fixed-point result on closed balls [10].

Theorem 1

Let $\mathcal {B}$ be a Banach space with a contraction operator $J:B(\widetilde{x},{r})\rightarrow \mathcal {B}$, contractivity factor $L<1$ and $\widetilde{x}\in \mathcal {B}$. If $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le (1-L){r}$, then J admits a unique fixed point.

Proof

Choose $\epsilon <r$ so that $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le (1-L)\epsilon <(1-L){r}$. We show that J maps the closed ball $\overline{B(\widetilde{x},\epsilon )}$ into itself: for $x\in \overline{B(\widetilde{x},\epsilon )}$ then

$$\begin{aligned} \Vert J(x)-\widetilde{x}\Vert \le \Vert J(x)-J(\widetilde{x})\Vert +\Vert J(\widetilde{x})-\widetilde{x}\ \le L\Vert x-\widetilde{x}\Vert +(1-L)\epsilon \le \epsilon . \end{aligned}$$

Since $\overline{B(\widetilde{x},\epsilon )}$ is complete, the conclusion follows from the BCP. $\square$

Two interesting observations follows from Theorem 1. First, from the proof of Theorem 1, we see that a ball $\overline{B(\widetilde{x},\epsilon )}$, such that $\overline{B(\widetilde{x},\epsilon )}\subset B(\widetilde{x},{r})$, is considered, so that the RCP is applied to the set $\Lambda =\overline{B(\widetilde{x},\epsilon )}$, which is nonempty and closed. In addition, we cannot guarantee the uniqueness of the fixed point in the initial ball $B(\widetilde{x},{r})$. Second, Theorem 1 gives us information about how we can choose the auxiliary point $\widetilde{x}$, which is the center of the ball, the quantity $\Vert J(\widetilde{x})-\widetilde{x}\Vert$.

We then present a fixed-point result on a closed ball without loss of uniqueness of the fixed point.

Theorem 2

Let V be a nonempty open convex domain of a Banach space $\mathcal {B}$ with a contraction operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$, contractivity factor $L<1$, $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le \alpha$ and $\widetilde{x}\in \mathcal {B}$. If $\overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }\subset {V}$, then J admits a unique fixed point $x^*$ in $\overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }$.

Proof

We start seeing that $x_2=J(x_1)$ is well defined. Indeed,

$$\begin{aligned} \Vert x_1-\widetilde{x}\Vert \le \Vert J(x_0)-J(\widetilde{x})\Vert +\Vert J(\widetilde{x})-\widetilde{x}\Vert \le L\Vert x_0-\widetilde{x}\Vert +\alpha \le \frac{\alpha }{1-L}, \end{aligned}$$

so that $x_1\in \overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }\subset {V}$ and $x_2=J(x_1)$ is then well defined.

Moreover,

$$\begin{aligned} \Vert x_2-x_1\Vert = \Vert J(x_1)-J(x_0)\Vert \le L\Vert x_1-x_0\Vert . \end{aligned}$$

Next, we suppose that

$$\begin{aligned} x_{j}\in \overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }\subset {V} \qquad \text {and}\qquad \Vert x_{j+1}-x_{j}\Vert \le L^{j}\Vert x_1-x_0\Vert , \end{aligned}$$

for $j=1,2,\ldots ,n$. As a consequence,

$$\begin{aligned} \Vert x_{n+1}-\widetilde{x}\Vert&\le \Vert J(x_n)-J(\widetilde{x})\Vert + \Vert J(\widetilde{x})-\widetilde{x}\Vert \le L\Vert x_n-\widetilde{x}\Vert +\alpha \le \frac{\alpha }{1-L}, \\ \Vert x_{n+2}-x_{n+1}\Vert&\le L\Vert x_{n+1}-x_n\Vert \le L^{n+1}\Vert x_1-x_0\Vert , \end{aligned}$$

so that, by mathematical induction on n, we obtain that $\left\{ x_{n}=J(x_{n-1})\right\} _{n\in \mathbb {N}}$ is well defined and is a Cauchy sequence in the Banach space $\mathcal {B}$. Therefore, there exists $x^*\in \mathcal {B}$ such that $x^*=\lim _{n}x_{n}$. Besides, $x^*\in \overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }$, since $x_{n}\in \overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }$, for all $n\in \mathbb {N}$. Furthermore, by the continuity of the operator J, we have that $x^*=\lim _{n}x_{n}=\lim _{n}J(x_{n-1})=J(x^*)$ and $x^*$ is then a fixed point of J.

Finally, we prove the uniqueness of $x^*\in \overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }$. We suppose that $w^*\in \overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }$ is a fixed point of J such that $J(w^*)=w^*$ and $w^*\ne x^*$. Hence,

$$\begin{aligned} \Vert w^*-x^*\Vert = \Vert J(w^*)-J(x^*)\Vert \le L\Vert w^*-x^*\Vert <\Vert w^*-x^*\Vert , \end{aligned}$$

which leads us to a contradiction. Therefore, $w^*=x^*$. $\square$

After that, we try to improve the location of a fixed point of the integral equation (3) by Theorem 2. For this, we try to choose an appropriate auxiliary point $\widetilde{x}$. Therefore, if $\widetilde{x}$ is close to the fixed point $x^*$, we have

$$\begin{aligned} \Vert J(\widetilde{x})-\widetilde{x}\Vert = \Vert J(\widetilde{x})-J(x^*)+x^*-\widetilde{x}\Vert \le (1+L)\Vert x^*-\widetilde{x}\Vert , \end{aligned}$$

so that the closer the points $\widetilde{x}$ and $x^*$ are, the smaller the parameter $\alpha$. But the opposite argument does not have to be true, since the fact that $\alpha$ is small does not guarantee that $\widetilde{x}$ and $x^*$ are close. Also, the smaller the parameter $\alpha$, the better the location of the fixed point is provided by the ball $\overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }$, since the radius of the ball is smaller.

For the integral equation (3), an appropriate choice of the auxiliary point $\widetilde{x}$ seems to be $\widetilde{x}(s)=s$. Therefore,

$$\begin{aligned} \Vert J_{1}(\widetilde{x})-\widetilde{x}\Vert \le \dfrac{1}{5} \left\| \int _0^1 e^{-t} t^2 dt \right\| \le \dfrac{2e-5}{5e} = 0.0321\ldots = \alpha . \end{aligned}$$

It is known that the function f(s) of the integral equation (2) has a lot to do with the fixed points and is a good starting point to obtain the convergence of iterative methods that are usually applied to solve (2), see [1, 3].

Then, as we have seen above, we can consider ${V}=B(0,2)$ and, by Theorem 2, we can obtain a unique fixed point in the ball $\overline{B(s,0.0649\ldots )}\subset {V}$. Obviously, Theorem 2 give us a better location of the fixed point than the RCP.

Besides, Theorem 2 allows us to locate a fixed point even if a previous location of it is not possible and the application of the RCP is difficult. See the following example where an integral equation of the form (2) is considered from a small variation of (3). Consider

$$\begin{aligned} x(s) = s+\dfrac{1}{4}\int _0^1 e^{-(s+t)} x(t)^3 dt = [J_2(x)](s), \quad s\in [0,1], \end{aligned}$$

(5)

where $J_2:\mathcal {C}[0,1]\rightarrow \mathcal {C}[0,1]$. Following the same idea as for (3) to locate previously a fixed point, we obtain that a fixed point $x^*$ of (5) must satisfy

$$\begin{aligned} \dfrac{e-1}{4e}\Vert x^*\Vert ^3-\Vert x^*\Vert +1\ge 0, \end{aligned}$$

that is always satisfied, so that we do not have any information about $\Vert x^*\Vert$. Hence, we cannot locate previously the fixed point $x^*$. Moreover,

$$\begin{aligned} \Vert J_2(c)-J_2(d)\Vert \le \dfrac{e-1}{4e} \left( \Vert c\Vert ^2+\Vert c\Vert \Vert d\Vert +\Vert d\Vert ^2\right) \Vert c-d\Vert , \end{aligned}$$

so that $J_2$ is a contraction operator in $\Lambda =\overline{B(0,{r})}$ if $r<\sqrt{\frac{4e}{3(e-1)}}=1.4523\ldots$ But $\Vert J_2(x)\Vert \nless {r}$ for all $x\in \Lambda$. As a consequence, we cannot apply the RCP in $\Lambda =\overline{B(0,{r})}$.

However, to apply Theorem 2, we can consider ${V}=B(0,5/4)$ and then $J_2$ is a contraction operator in V. Besides, if $\widetilde{x}(s)=s$, it follows $\alpha =\frac{3e-8}{2e}=0.0284\ldots$, $L=0.7407\ldots$, $\frac{\alpha }{1-L}=0.1098\ldots$ and $\overline{B\left( \widetilde{x},\frac{\alpha }{1-L}\right) }=\overline{B(s,0.1098\ldots )}\subset {V}=B(0,5/4)$. Thus, there exists a unique fixed point of $J_2$ in $\overline{B(s,0.1098\ldots )}$.

In summary, we have seen that Theorem 2 allows us to locate fixed points of contraction operators in situations where the RCP does not allow it.

3 Operators with Fréchet first derivative

In this section, we consider that $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$ is an operator with Fréchet first derivative, where V is a nonempty convex domain of a Banach space $\mathcal {B}$. As a consequence of Theorem 2, we can obtain the next result.

Corollary 3

Let V be a nonempty open convex domain of a Banach space $\mathcal {B}$ with a once Fréchet differentiable operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$, $\Vert J'(x)\Vert \le M<1$, for all $x\in {V}$, $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le \alpha$ and $\widetilde{x}\in \mathcal {B}$. If $\overline{B\left( \widetilde{x},\frac{\alpha }{1-M}\right) }\subset {V}$, then J admits a unique fixed point $x^*$ in $\overline{B\left( \widetilde{x},\frac{\alpha }{1-M}\right) }$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \overline{B\left( \widetilde{x},\frac{\alpha }{1-M}\right) }$.

Proof

From the Mean Value Theorem, it follows

$$\begin{aligned} \Vert J'(c)-J'(d)\Vert \le \Vert J'(\theta )\Vert \Vert c-d\Vert \le M\Vert c-d\Vert , \end{aligned}$$

for all $c,d\in {V}$, since $\theta =\vartheta c+(1-\vartheta )d\in {V}$ with $\vartheta \in [0,1]$. $\square$

Next, we can relax the condition required previously to the operator $J'$. Therefore, we consider that $J'$ is such that

$$\begin{aligned} \Vert J'(x)\Vert \le \ell (\Vert x\Vert ), \quad x\in {V}, \end{aligned}$$

(6)

where ${\ell }:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ is a continuous nondecreasing function. In addition, we establish the following result.

Theorem 4

Let V be a nonempty open convex domain in a Banach space $\mathcal {B}$ with a once Fréchet differentiable operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$. Suppose that the condition (6) holds and $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le \alpha$ with $\widetilde{x}\in \mathcal {B}$. If the equation $\ell (\Vert \widetilde{x}\Vert +t)t+\alpha =t$ has at least one positive root and the smallest positive root, denoted by R, satisfies $\ell (\Vert \widetilde{x}\Vert +R)<1$ and $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, then J admits a unique fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \overline{B\left( \widetilde{x},R\right) }$.

Proof

From $x_{0}\in {V}$, we have that $x_{1}$ is well defined. Moreover, from the Mean Value Theorem, it follows

$$\begin{aligned} \Vert x_1-\widetilde{x}\Vert&\le \Vert J(x_0)-J(\widetilde{x})\Vert +\Vert J(\widetilde{x})-\widetilde{x}\Vert \\&\le \Vert J'(\vartheta x_0+(1-\vartheta )\widetilde{x})\Vert \Vert x_0-\widetilde{x}\Vert + \alpha \\&\le \ell (\Vert \widetilde{x}\Vert +R)R+\alpha \\&= R, \end{aligned}$$

since $\vartheta x_0+(1-\vartheta )\widetilde{x}\in {V}$ with $\vartheta \in [0,1]$. Then, $x_1\in \overline{B\left( \widetilde{x},R\right) }$ and $x_2=J(x_1)$ is well defined.

Furthermore,

$$\begin{aligned} \Vert x_2-x_1\Vert&= \Vert J(x_1)-J(x_0)\Vert \le \Vert J'(\vartheta x_0+(1-\vartheta )x_1)\Vert \Vert x_1-x_0\Vert \\&\le \ell (\Vert \widetilde{x}\Vert +R)\Vert x_1-x_0\Vert < \Vert x_1-x_0\Vert , \end{aligned}$$

where $\vartheta \in [0,1]$.

After that, it is easy to prove by mathematical induction on n that

$$\begin{aligned} x_n\in \overline{B\left( \widetilde{x},R\right) } \quad \text {and}\quad \Vert x_{n+1}-x_n\Vert \le \ell (\Vert \widetilde{x}\Vert +R)^n\Vert x_1-x_0\Vert , \quad \text {for all}\quad n\in \mathbb {N}. \end{aligned}$$

As a consequence, the sequence $\left\{ x_{n}=J(x_{n-1})\right\} _{n\in \mathbb {N}}$ is well defined, since $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, and is of Cauchy in $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, since

$$\begin{aligned} \Vert x_{n+j}-x_{n}\Vert&\le \sum _{i=0}^{j-1} \Vert x_{n+i+1}-x_{n+i}\Vert \\&\le \sum _{i=0}^{j-1} \ell (\Vert \widetilde{x}\Vert +R)^{n+i}\Vert x_1-x_0\Vert \\&\le \dfrac{\ell (\Vert \widetilde{x}\Vert +R)^{n}-\ell (\Vert \widetilde{x}\Vert +R)^{n+j}}{1-\ell (\Vert \widetilde{x}\Vert +R)}\Vert x_1-x_0\Vert \\&\le \dfrac{1-\ell (\Vert \widetilde{x}\Vert +R)^j}{1-\ell (\Vert \widetilde{x}\Vert +R)}\,\ell (\Vert \widetilde{x}\Vert +R)^{n}\Vert x_1-x_0\Vert , \end{aligned}$$

and $\ell (\Vert \widetilde{x}\Vert +R)<1$. Therefore, there exists $x^*\in \overline{B\left( \widetilde{x},R\right) }$ such that $x^*=\lim _{n}x_{n}$. Besides, by the continuity of J, we obtain $x^*=\lim _{n}x_{n}=\lim _{n}J(x_{n-1})=J(x^*)$ and $x^*$ is then a fixed point of J.

Finally, to prove the uniqueness of the fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$, we consider that $w^*$ is a fixed point of J in $\overline{B\left( \widetilde{x},R\right) }$ such that $w^*\ne x^*$. Thus,

$$\begin{aligned} \Vert x^*-w^*\Vert&= \Vert J(x^*)-J(w^*)\le \Vert J'(\vartheta x^*+(1-\vartheta )w^*)\Vert \Vert x^*-w^*\Vert \\&\le \ell (\Vert \vartheta x^*+(1-\vartheta )w^*\Vert ) \Vert x^*-w^*\Vert \le \ell (\Vert \widetilde{x}\Vert +R) \Vert x^*-w^*\Vert \\&< \Vert x^*-w^*\Vert , \end{aligned}$$

since $\vartheta x^*+(1-\vartheta )w^*\in \overline{B\left( \widetilde{x},R\right) }\subset {V}$ with $\vartheta \in [0,1]$, which leads to a contradiction. Therefore, $w^*=x^*$. $\square$

Remark 5

Note that the result obtained in Theorem 4 generalizes that obtained in Corollary 3, since the latter is a particular case of the former, without more than taking into account $\ell (z)=M$ and $R=\frac{\alpha }{1-M}$.

Now, we illustrate the previous result with the nonlinear integral equation given in (5).

Example 6

If we consider again the nonlinear integral equation (5) and $\widetilde{x}(s)=s$, we observe that

$$\begin{aligned}{}[J_2'(x)y](s) = \dfrac{3}{4}\int _0^1 e^{-(s+t)} x(t)^2y(t) dt, \end{aligned}$$

so that $\ell (t)=\frac{3(e-1)}{4e}t^2$ and the equation $\ell (\Vert \widetilde{x}\Vert +t)t+\alpha =t$ is reduced to $(0.0284\ldots ) - (0.5259\ldots )t + (0.9481\ldots )t^2 + (0.4740\ldots )t^3=0$, whose the smallest positive root $R=0.0610\ldots$ satisfies the condition $\ell (\Vert \widetilde{x}\Vert +R)<1$, since $\ell (\Vert \widetilde{x}\Vert +R)=0.5337\ldots$ Hence, by Theorem 4, $J_2$ admits a unique fixed point in $\overline{B(s,0.0610\ldots )}$. In addition, the location of the fixed point is improved by Theorem 4 with respect to Theorem 2.

Next, we remember the following definition, which is well known and used it below.

Definition 7

Let $P:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$ be an operator, where V is a nonempty domain in a Banach space $\mathcal {B}$. We say that the operator P satisfies a L-Lipschitz condition if

$$\begin{aligned} \Vert P(x)-P(y)\Vert \le L\Vert x-y\Vert , \quad L\ge 0, \quad x,y\in {V}. \end{aligned}$$

From the last definition, we can establish the following result on fixed points.

Theorem 8

Let V be a nonempty open convex domain of a Banach space $\mathcal {B}$ with a once Fréchet differentiable operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$. Suppose that $J'$ satisfies a L-Lipschitz condition in V, $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le \alpha$ and $\Vert J'(\widetilde{x})\Vert \le \widetilde{M}<1$ with $\widetilde{x}\in \mathcal {B}$. If $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, where $R=\frac{1-\widetilde{M}-\sqrt{(1-\widetilde{M})^2-2L\alpha }}{L}$ and $\alpha <\frac{3(1-\widetilde{M})^2}{8L}$, then J admits a unique fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \overline{B\left( \widetilde{x},R\right) }$.

Proof

From $x_{0}\in {V}$, we have that $x_{1}$ is well defined. Moreover, from Taylor’s formula, it follows

$$\begin{aligned} x_1-\widetilde{x}&= J(x_0)-\widetilde{x} = J(\widetilde{x})+J'(\widetilde{x})(x_0-\widetilde{x})+\int _{\widetilde{x}}^{x_0}(J'(z)-J'(\widetilde{x}))dz-\widetilde{x} \\&= J'(\widetilde{x})(x_0-\widetilde{x})+\int _{0}^{1} (J'(\widetilde{x}+\tau (x_0-\widetilde{x}))-J'(\widetilde{x}))(x_0-\widetilde{x})d\tau +J(\widetilde{x})-\widetilde{x}, \\ \Vert x_1-\widetilde{x}\Vert&\le \widetilde{M}\Vert x_0-\widetilde{x}\Vert + \int _{0}^{1} L\tau d\tau \Vert x_0-\widetilde{x}\Vert ^2+\alpha \le \widetilde{M}R+\frac{1}{2}LR^2+\alpha = R. \end{aligned}$$

Then, $x_1\in \overline{B\left( \widetilde{x},R\right) }$ and $x_2=J(x_1)$ is well defined.

Furthermore, since

$$\begin{aligned} \Vert x_2-x_1\Vert&= \Vert J(x_1)-J(x_0)\Vert = \left\| J'(x_0)(x_1-x_0) + \int _{x_0}^{x_1}(J'(z)-J'(x_0))dz\right\| \\&\le \Vert J'(x_0)(x_1-x_0)\Vert + \left\| \int _{0}^{1} J'(x_0+\tau (x_1-x_0))-J'(x_0))(x_1-x_0)d\tau \right\| \\&\le \left( \Vert J'(x_0))\Vert +\frac{1}{2}L\Vert x_1-x_0\Vert \right) \Vert x_1-x_0\Vert , \end{aligned}$$

and

$$\begin{aligned} \Vert J'(x_0)\Vert = \Vert J'(x_0)\pm J'(\widetilde{x})\Vert \le \widetilde{M}+L\Vert x_0-\widetilde{x}\Vert \le \widetilde{M}+LR, \end{aligned}$$

we have

$$\begin{aligned} \Vert x_2-x_1\Vert \le (\widetilde{M}+2LR)\Vert x_1-x_0\Vert . \end{aligned}$$

Now, we denote ${N}=\widetilde{M}+2LR$ and see that ${N}<1$. Indeed, from $\alpha <\frac{3(1-\widetilde{M})^2}{8L}$, it follows that

$$\begin{aligned} \widetilde{M}+2L\,\frac{1-\widetilde{M}-\sqrt{(1-\widetilde{M})^2-2L\alpha }}{L}<1 \qquad \text {and}\qquad \alpha <\frac{(1-\widetilde{M})^2}{2}, \end{aligned}$$

so that ${N}<1$.

After that, it is easy to prove by mathematical induction on n that

$$\begin{aligned} x_n\in \overline{B\left( \widetilde{x},R\right) } \quad \text {and}\quad \Vert x_{n+1}-x_n\Vert \le N^n\Vert x_1-x_0\Vert , \quad \text {for all}\quad n\in \mathbb {N}. \end{aligned}$$

As a consequence, the sequence $\left\{ x_{n}=J(x_{n-1})\right\} _{n\in \mathbb {N}}$ is well defined, since $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, and is of Cauchy in $\overline{B\left( \widetilde{x},R\right) }\subset {V}$. Therefore, there exists $x^*\in \overline{B\left( \widetilde{x},R\right) }$ such that $x^*=\lim _{n}x_{n}$. Besides, by the continuity of J, we obtain $x^*=\lim _{n}x_{n}=\lim _{n}J(x_{n-1})=T(x^*)$ and $x^*$ is then a fixed point of J.

Finally, to prove the uniqueness of the fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$, we consider that $w^*$ is a fixed point of J in $\overline{B\left( \widetilde{x},R\right) }$ such that $w^*\ne x^*$. We define the operator $H:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$ such that $H(x)=x-J(x)$, that is obviously Fréchet differentiable, and $[H'(x)](y) = [I-J'(x)](y)$. In addition, from $\Vert I-J'(\widetilde{x})\Vert =\Vert J'(\widetilde{x})\Vert \le \widetilde{M}<1$, it follows that the operator $[H'(\widetilde{x})]^{-1}$ exists and $\Vert [H'(\widetilde{x})]^{-1}\Vert \le \frac{1}{1-\widetilde{M}}$ by the Banach lemma on invertible operators. Now, from

$$\begin{aligned} 0 = H(w^*)-H(x^*) = \int _0^1 H'(x^*+\tau (w^*-x^*))d\tau (w^*-x^*), \end{aligned}$$

we have that $w^*=x^*$ if there exists $Q^{-1}$, where $Q=\int _0^1 H'(x^*+\tau (w^*-x^*))d\tau$. For this, we consider the operator $T=[H'(\widetilde{x})]^{-1}Q$ and prove that $T^{-1}$ exists. Indeed, by the Banach lemma on invertible operators and

$$\begin{aligned} \left\| I-[H'(\widetilde{x})]^{-1}Q\right\|&\le \Vert [H'(\widetilde{x})]^{-1}\Vert \left\| \int _0^1(J'(\widetilde{x})-J'(x^*+\tau (w^*-x^*)))d\tau \right\| \\&\le \Vert [H'(\widetilde{x})]^{-1}\Vert \int _0^1 L\Vert (1-\tau )(x^*-w^*)+\tau (x^*-w^*)\Vert d\tau \\&\le \frac{LR}{1-\widetilde{M}} < 1, \end{aligned}$$

it follows that $T^{-1}$ exists, so that $Q^{-1}$ exists and $w^*=x^*$. $\square$

In the next example, we illustrate the last result.

Example 9

Consider the nonlinear integral equation of Fredholm given by

$$\begin{aligned} x(s) = s+\dfrac{3}{10}\int _0^1 e^{\frac{st}{2}} (x(t)-\sin (x(t))) dt, \quad s\in [0,1]. \end{aligned}$$

(7)

Hence, we define $J_3:\mathcal {C}[0,1]\rightarrow \mathcal {C}[0,1]$ with

$$\begin{aligned}{}[J_3(x)](s) = s+\dfrac{3}{10}\int _0^1 e^{\frac{st}{2}} (x(t)-\sin (x(t))) dt, \quad s\in [0,1]. \end{aligned}$$

To locate a solution of (7), we look for a fixed point of $J_3$. Then, we observe that $J_3$ is a contraction, since

$$\begin{aligned} \Vert J_3(x)-J_3(y)\Vert \le \dfrac{3}{5}(\sqrt{e}-1) \left( \Vert x-y\Vert +\Vert \sin y-\sin x\Vert \right) \le \dfrac{6}{5}(\sqrt{e}-1)\Vert x-y\Vert , \end{aligned}$$

and the contractivity factor is $L=\frac{6}{5}(\sqrt{e}-1)=0.7784\ldots <1$. Then, by the BCP, it follows that $J_3$ has a unique fixed point $x^*$ in $\mathcal {C}[0,1]$. As we cannot locate the fixed point $x^*$ from this principle, we try to locate it by the RCP. For this, we locate previously $x^*$. Therefore, we have that

$$\begin{aligned} \Vert x^*\Vert -1-\dfrac{3}{5}(\sqrt{e}-1)(1+\Vert x^*\Vert )\le 0, \end{aligned}$$

so that $\Vert x^*\Vert \le 2.2745\ldots$ In addition, we can consider $J_3:\overline{B(0,2.2745\ldots )}\rightarrow \overline{B(0,2.2745\ldots )}$ and, by the RCP, it follows that the unique fixed point of $J_3$ is in $\overline{B(0,2.2745\ldots )}$.

Next, we apply Theorem 8 to locate the fixed point $x^*$. For this, we consider ${V}=B(0,5/2)$ and

$$\begin{aligned}{}[J_3'(x)y](s) = \dfrac{3}{10}\int _0^1 e^{\frac{st}{2}} (1-\cos (x(t))) y(t)\,dt. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert J_3'(c)-J_3'(d)\Vert \le \dfrac{3}{5}(\sqrt{e}-1) \Vert \cos d-\cos c\Vert \le \dfrac{3}{5}(\sqrt{e}-1) \Vert c-d\Vert , \end{aligned}$$

so that $J_3'$ satisfies a L-Lipschitz condition in V with $L=\frac{3}{5}(\sqrt{e}-1)=0.3892\ldots <1$.

If we now choose $\widetilde{x}(s)=s$, then

$$\begin{aligned} \Vert J_3(\widetilde{x})-\widetilde{x}\Vert \le 0.0180\ldots = \alpha , \qquad \Vert J_3'(\widetilde{x})\Vert \le 0.0693\ldots = \widetilde{M} < 1, \end{aligned}$$

so that $\alpha <\frac{3(1-\widetilde{M})^2}{8L}=0.8345\ldots$ Moreover, $R=0.0195\ldots$ and $\overline{B\left( \widetilde{x},R\right) }=\overline{B(s,0.0195\ldots )}\subset {V}=B(0,5/2)$. Therefore, by Theorem 8, the fixed point $x^*$ is unique in $\overline{B(s,0.0195\ldots )}$. As a consequence, we improve the location of the fixed point $x^*$ by Theorem 8 with respect to the RCP, since $\overline{B(s,0.0195\ldots )}\subset \overline{B(0,2.2745\ldots )}$.

A usual generalization of the fact that an operator satisfies a L-Lipschitz condition is given in the following definition.

Definition 10

Let $P:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$ be an operator, where V is a nonempty domain of a Banach space $\mathcal {B}$. We say that the operator P satisfies a (L, p)-Hölder condition if

$$\begin{aligned} \Vert P(x)-P(y)\Vert \le L\Vert x-y\Vert ^p, \quad L\ge 0, \quad p\in (0,1], \quad x,y\in {V}. \end{aligned}$$

Observe that P satisfies a L-Lipschitz condition if $p=1$.

A further generalization of the facts that P satisfies a L-Lipschitz condition or a (L, p)-Hölder condition can be found in [7]. In this case, P satisfies the condition

$$\begin{aligned} \Vert P(x)-P(y)\Vert \le \ell (\Vert x\Vert ,\Vert y\Vert )\Vert x-y\Vert ^p \quad p\in (0,1], \quad x,y\in {V}, \end{aligned}$$

(8)

where $\ell :\mathbb {R}_{+}\times \mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ is a nondecreasing continuous function in both arguments. Obviously, if $\ell (s,t)=L\ge 0$, the last condition generalizes the two previous conditions (Lipschitz and Hölder).

After that, we establish the following result of fixed point under the condition (8).

Theorem 11

Let V be a nonempty open convex domain of a Banach space $\mathcal {B}$ with a once Fréchet differentiable operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$. Suppose that $J'$ satisfies the condition (8), $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le \alpha$ and $\Vert J'(\widetilde{x})\Vert \le \widetilde{M}<1$ with $\widetilde{x}\in \mathcal {B}$. If the equation

$$\begin{aligned} \widetilde{M}t+\dfrac{1}{1+p}\ell (\Vert \widetilde{x}\Vert +t,\Vert \widetilde{x}\Vert +t)t^{1+p}+\alpha = t, \end{aligned}$$

(9)

has at least one positive root and the smallest positive root, denoted by R, satisfies

$$\begin{aligned} (p+2^p)\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{1+p} <(1+p)\alpha , \end{aligned}$$

(10)

and $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, then J admits a unique fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \overline{B\left( \widetilde{x},R\right) }$.

Proof

From $x_{0}\in {V}$, we have that $x_{1}$ is well defined. Moreover, from Taylor’s formula and (9), we have

$$\begin{aligned} \Vert x_1-\widetilde{x}\Vert&= \Vert J(x_0)\pm J(\widetilde{x})-\widetilde{x}\Vert \le \Vert J(x_0)-J(\widetilde{x})\Vert +\Vert J(\widetilde{x})-\widetilde{x}\Vert \\&\le \left\| J'(\widetilde{x})(x_0-\widetilde{x})+\int _{\widetilde{x}}^{x_0}(J'(z)-J'(\widetilde{x}))dz\right\| +\alpha \\&\le \Vert J'(\widetilde{x})\Vert \Vert x_0-\widetilde{x}\Vert +\int _{0}^{1} \Vert J'(\widetilde{x}+\tau (x_0-\widetilde{x}))-J'(\widetilde{x})\Vert d\tau \Vert x_0-\widetilde{x}\Vert +\alpha \\&\le \widetilde{M}R+\int _{0}^{1} \ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)\tau ^pd\tau \Vert x_0-\widetilde{x}\Vert ^{1+p}+\alpha \\&\le \widetilde{M}R+\dfrac{1}{1+p}\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{1+p}+\alpha \\&= R. \end{aligned}$$

Then, $x_1\in \overline{B\left( \widetilde{x},R\right) }$ and $x_2=J(x_1)$ is well defined.

Furthermore,

$$\begin{aligned} \Vert x_2-x_1\Vert&= \Vert J(x_1)-J(x_0)\Vert = \left\| J'(x_0)(x_1-x_0) + \int _{x_0}^{x_1}(J'(z)-J'(x_0))dz\right\| \\&\le \Vert J'(x_0)\Vert \Vert x_1-x_0\Vert + \int _{0}^{1} \Vert J'(x_0+\tau (x_1-x_0))-J'(x_0)\Vert d\tau \Vert x_1-x_0\Vert . \end{aligned}$$

Taking then into account

$$\begin{aligned} \Vert J'(x_0)\Vert&\le \Vert J'(\widetilde{x})\Vert +\Vert J'(x_0)-J'(\widetilde{x})\Vert \le \widetilde{M}+\ell (\Vert x_0\Vert ,\Vert \widetilde{x}\Vert )\Vert x_0-\widetilde{x}\Vert ^p \\&\le \widetilde{M}+\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^p, \\ \Vert x_0+\tau (x_1-x_0)\Vert&= \Vert x_0+\tau (x_1\pm \widetilde{x}-x_0)\pm \widetilde{x}\Vert = \Vert (1-\tau )(x_0-\widetilde{x})+\tau (x_1-\widetilde{x})+\widetilde{x}\Vert \\&\le (1-\tau )R+\tau R+\Vert \widetilde{x}\Vert = \Vert \widetilde{x}\Vert +R, \end{aligned}$$

we obtain

$$\begin{aligned} \Vert x_2-x_1\Vert&\le \left( \left( \widetilde{M}+\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)\right) R^p + \dfrac{2^p}{1+p}\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^p\right) \\ {}&\quad \Vert x_1-x_0\Vert \\&= N\Vert x_1-x_0\Vert , \end{aligned}$$

where $N=\left( \widetilde{M}+\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)\right) R^p + \frac{2^p}{1+p}\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^p$.

Next, from (9), it follows that

$$\begin{aligned} \widetilde{M} = \dfrac{1}{R}\left( R-\dfrac{1}{1+p}\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{1+p}-\alpha \right) , \end{aligned}$$

so that

$$\begin{aligned} N = 1-\dfrac{\alpha }{R}+\dfrac{p}{1+p}\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^p + \dfrac{2^p}{1+p}\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^p, \end{aligned}$$

and $N<1$, since $(p+2^p)\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{1+p} <(1+p)\alpha$. Therefore,

$$\begin{aligned} \Vert x_2-x_1\Vert \le N\Vert x_1-x_0\Vert < \Vert x_1-x_0\Vert . \end{aligned}$$

After that, it is easy to prove by mathematical induction on n that

$$\begin{aligned} x_n\in \overline{B\left( \widetilde{x},R\right) } \quad \text {and}\quad \Vert x_{n+1}-x_n\Vert \le N^n\Vert x_1-x_0\Vert , \quad \text {for all}\quad n\in \mathbb {N}. \end{aligned}$$

Now, the convergence of the sequence $\{x_{n}\}$ to a fixed point $x^*\in \overline{B\left( \widetilde{x},R\right) }$ follows as in the previous theorems.

Finally, to prove the uniqueness of the fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$, we consider that $w^*$ is a fixed point of J in $\overline{B\left( \widetilde{x},R\right) }$ such that $w^*\ne x^*$. We define again the operator $H:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$ such that $H(x)=x-J(x)$ and $[H'(x)](y) = [I-J'(x)](y)$. In addition, we have

$$\begin{aligned} 0 = H(w^*)-H(x^*) = \int _0^1 H'(x^*+\tau (w^*-x^*))d\tau (w^*-x^*), \end{aligned}$$

so that $w^*=x^*$ if there exists $Q^{-1}$, where $Q=\int _0^1 H'(x^*+\tau (w^*-x^*))d\tau$. For this, by the Banach lemma on invertible operators, it is enough to prove that the operator $[H'(\widetilde{x})]^{-1}Q$ has inverse. Therefore, we have

$$\begin{aligned} \left\| I-[H'(\widetilde{x})]^{-1}Q\right\|&\le \Vert [H'(\widetilde{x})]^{-1}\Vert \left\| \int _0^1(J'(x^*+\tau (w^*-x^*))-J'(\widetilde{x}))d\tau \right\| \\&\le \dfrac{1}{1-\widetilde{M}} \int _0^1 \ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R) \Vert x^*+\tau (w^*-x^*)\Vert ^pd\tau \\&\le \frac{\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^p}{1-\widetilde{M}}. \end{aligned}$$

Taking now into account $1-\widetilde{M}=\frac{1}{1+p}\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^p+\frac{\alpha }{R}$ and

$$\begin{aligned} \ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^p< 1-\widetilde{M} \quad \Leftrightarrow \quad p\,\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{1+p} < (1+p)\alpha , \end{aligned}$$

we obtain $\frac{\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^p}{1-\widetilde{M}}<1$, since

$$\begin{aligned} p\,\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{1+p}< (p+2^p)\ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{1+p} <(1+p)\alpha . \end{aligned}$$

Therefore, $[H'(\widetilde{x})]^{-1}Q$ exists and, as a consequence, $w^*=x^*$. $\square$

From Theorem 11, we have immediately the following two results. The first is for operators whose first derivative satisfies a L-Lipschitz condition and the second is for operators whose first derivative satisfies a (L, p)-Hölder condition.

Corollary 12

Let V be a nonempty open convex domain of a Banach space $\mathcal {B}$ with a once Fréchet differentiable operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$. Suppose that $J'$ satisfies a L-Lipschitz condition in V, $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le \alpha$ and $\Vert J'(\widetilde{x})\Vert \le \widetilde{M}<1$ with $\widetilde{x}\in \mathcal {B}$. If the equation $\frac{L}{2}t^2-(1-\widetilde{M})t+\alpha = 0$ has at least one positive root and the smallest positive root, denoted by R, satisfies $3LR^2<2\alpha$ and $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, then J admits a unique fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \overline{B\left( \widetilde{x},R\right) }$.

Note that $R=\frac{1-\widetilde{M}-\sqrt{(1-\widetilde{M})^2-2L\alpha }}{L}$ in the last corollary, which is the same R that is given in Theorem 8. Moreover, the condition required to R in the last corollary, $3LR^2<2\alpha$, is equivalent to that required in Theorem 8, $\alpha <\frac{3(1-\widetilde{M})^2}{8L}$, without more than taking into account that R satisfies $\frac{L}{2}R^2-(1-\widetilde{M})R+\alpha =0$.

Corollary 13

Let V be a nonempty open convex domain of a Banach space $\mathcal {B}$ with a once Fréchet differentiable operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$. Suppose that $J'$ satisfies a (L, p)-Hölder condition in V, $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le \alpha$ and $\Vert J'(\widetilde{x})\Vert \le \widetilde{M}<1$ with $\widetilde{x}\in \mathcal {B}$. If the equation $\frac{L}{1+p}t^{1+p}-(1-\widetilde{M})t+\alpha =0$ has at least one positive root and the smallest positive root, denoted by R, satisfies $(p+2^p)LR^{1+p}<(1+p)\alpha$ and $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, then J admits a unique fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \overline{B\left( \widetilde{x},R\right) }$.

Now, we illustrate Theorem 11 with the following example.

Example 14

Consider the Fredholm integral equation

$$\begin{aligned} x(s) = 1-s+\dfrac{1}{2}\int _0^1 G(s,t) x(t)^5 dt, \quad s\in [0,1], \end{aligned}$$

(11)

where G(s, t) is the Green function in $[0,1]\times [0,1]$.

We then define $J_4:\mathcal {C}[0,1]\rightarrow \mathcal {C}[0,1]$ with

$$\begin{aligned}{}[J_4(x)](s) = 1-s+\dfrac{1}{2}\int _0^1 G(s,t) x(t)^5 dt, \quad s\in [0,1]. \end{aligned}$$

Hence,

$$\begin{aligned}{}[J_4'(x)y](s) = \dfrac{5}{2}\int _0^1 G(s,t) x(t)^4 y(t) dt. \end{aligned}$$

To use Theorem 11 to locate a fixed point of $J_4$, we begin by locating previously the possible fixed points. Therefore, if $x^*$ is a fixed point of $J_4$, then $x^*$ is a solution of (11) and

$$\begin{aligned} \Vert x^*\Vert -1-\dfrac{1}{16}\Vert x^*\Vert ^5\le 0, \end{aligned}$$

which is satisfied if $\Vert x^*\Vert \le 1.1012\ldots$ or $\Vert x^*\Vert \ge 1.5382\ldots$ As a consequence, we can consider the set ${V}=B(0,3/2)$ for Theorem 11.

On the other hand,

$$\begin{aligned} \Vert J_4'(c)-J_4'(d)\Vert&\le \dfrac{5}{16} \Vert c^4-d^4\Vert \le \dfrac{5}{16} \left( \Vert c\Vert ^3+\Vert c\Vert ^2\Vert d\Vert +\Vert c\Vert \Vert d\Vert ^2+\Vert d\Vert ^3\right) \Vert c-d\Vert \\&= \ell (\Vert c\Vert ,\Vert d\Vert )\Vert c-d\Vert , \end{aligned}$$

so that $p=1$ and $\ell (u,v)=\frac{5}{16}(u^3+u^2v+uv^2+v^3)$, which is a nondecreasing continuous function in both arguments.

If we choose $\widetilde{x}(s)=1-s$, then $\Vert J_4(\widetilde{x})-\widetilde{x}\Vert \le \frac{1}{16}=\alpha$, $\Vert J_4'(\widetilde{x})\Vert \le \frac{5}{16}=\widetilde{M}<1$ and the equation (9) is reduced to

$$\begin{aligned} \dfrac{1}{16} (1 - 11 t + 10 t^2 + 30 t^3 + 30 t^4 + 10 t^5) = 0. \end{aligned}$$

The smallest positive root of the last scalar equation is $R=0.1041\ldots$, that satisfies the inequality (10), which is reduced to

$$\begin{aligned} 3\ell (1+R,1+R)R^2 = 0.0548\ldots < 2\alpha = 0.125. \end{aligned}$$

Therefore, by Theorem 11, there exists a unique fixed point $x^*$ of $J_4$ in $\overline{B(1-s,0.1041\ldots )}\subset {V}=B(0,3/2)$.

On the other hand, it does not seem clear that we can locate a domain V of the form B(0, r) such that $J_{4}$ is a contraction operator in V and such that $J_{4}:B(0,r)\rightarrow B(0,r)$, so that the application of the RCP is not easy if we can use it to locate a domain which contains a fixed point.

4 k-times Fréchet differentiable operators

The aim of this section is to generalize the results seen in the previous section on the existence and uniqueness of a fixed point for the operator J. For this, we consider that the operator J is k-times Fréchet differentiable. We begin by generalizing the condition (6) for the k-th derivative of the operator J.

Theorem 15

Let V be a nonempty open convex domain of a Banach space $\mathcal {B}$ with a k-times Fréchet differentiable operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$ where $k\ge 2$ and

$$\begin{aligned} \Vert J^{(k)}(x)\Vert \le \ell (\Vert x\Vert ), \quad x\in {V}, \end{aligned}$$

where ${\ell }:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ is a continuous nondecreasing function. Suppose that $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le \alpha$ and $\Vert J^{(i)}(\widetilde{x})\Vert \le \widetilde{M}_{i}$, for $i=1,2,\ldots ,k-1$, with $\widetilde{x}\in \mathcal {B}$. If the equation

$$\begin{aligned} f(t)+\dfrac{1}{k!}\ell (\Vert \widetilde{x}\Vert +t)t^k+\alpha = t, \quad \text {where}\quad f(t) = \sum _{i=1}^{k-1}\dfrac{\widetilde{M}_{i}}{i!}t^{i}, \end{aligned}$$

(12)

has at least one positive root and the smallest positive root, denoted by R, satisfies

$$\begin{aligned} R(f'(R)+k-1)<k(f(R)+\alpha ), \end{aligned}$$

(13)

and $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, then J admits a unique fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$ if $\widetilde{M}_1<1$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \overline{B\left( \widetilde{x},R\right) }$.

Proof

From $x_{0}\in {V}$, we have that $x_{1}$ is well defined. Moreover, from Taylor’s formula, it follows

$$\begin{aligned} \Vert x_1-\widetilde{x}\Vert&\le \Vert J(x_0)-J(\widetilde{x})\Vert +\Vert J(\widetilde{x})-\widetilde{x}\Vert \\&\le \left\| \sum _{i=1}^{k-1}\dfrac{J^{(i)}(\widetilde{x})}{i!}(x_0-\widetilde{x})^{i} +\dfrac{1}{k!}J^{(k)}(\vartheta x_0+(1-\vartheta )\widetilde{x})(x_0-\widetilde{x})^k\right\| + \alpha , \end{aligned}$$

with $\vartheta x_0+(1-\vartheta )\widetilde{x}\in {V}$ and $\vartheta \in [0,1]$, so that

$$\begin{aligned} \Vert x_1-\widetilde{x}\Vert \le f(R)+\dfrac{1}{k!}\ell (\Vert \vartheta x_0+(1-\vartheta )\widetilde{x}\Vert )R^k+\alpha \le R. \end{aligned}$$

Then, $x_1\in \overline{B\left( \widetilde{x},R\right) }$ and $x_2=J(x_1)$ is well defined.

Furthermore, from

$$\begin{aligned} \Vert x_2-x_1\Vert = \Vert J(x_1)-J(x_0)\Vert \le \Vert J'(\xi )\Vert \Vert x_1-x_0\Vert , \end{aligned}$$

where $\xi =\vartheta x_0+(1-\vartheta )x_1\in \overline{B\left( \widetilde{x},R\right) }$ with $\vartheta \in [0,1]$, and taking into account

$$\begin{aligned} \Vert J'(\xi )\Vert = \left\| \sum _{i=1}^{k-1}\dfrac{J^{(i)}(\widetilde{x})}{(i-1)!}(\xi -\widetilde{x})^{i-1} +\dfrac{1}{(k-1)!}J^{(k)}(\mu )(\xi -\widetilde{x})^{k-1}\right\| , \end{aligned}$$

where $\mu =\vartheta \xi +(1-\vartheta )\widetilde{x}$ with $\vartheta \in [0,1]$, it follows that

$$\begin{aligned} \Vert x_2-x_1\Vert&\le \left( \sum _{i=1}^{k-1}\dfrac{\widetilde{M}_{i}}{(i-1)!}R^{i-1} + \dfrac{1}{(k-1)!}\ell (\Vert \mu \Vert )R^{k-1}\right) \Vert x_1-x_0\Vert \\&\le \left( f'(R)+\dfrac{1}{(k-1)!}\ell (\Vert \widetilde{x}\Vert +R)R^{k-1} \right) \Vert x_1-x_0\Vert \\&= \left( f'(R)+\dfrac{k}{R}(R-\alpha -f(R))\right) \Vert x_1-x_0\Vert \\&= N\Vert x_1-x_0\Vert , \end{aligned}$$

where $N=f'(R)+\dfrac{k}{R}(R-\alpha -f(R))$.

After that, it is easy to prove by mathematical induction on n that

$$\begin{aligned} x_n\in \overline{B\left( \widetilde{x},R\right) } \quad \text {and}\quad \Vert x_{n+1}-x_n\Vert \le N^n\Vert x_1-x_0\Vert , \quad \text {for all}\quad n\in \mathbb {N}. \end{aligned}$$

Besides, $N<1$, since $f'(R)+k-\frac{k}{R}(\alpha +f(R))<1$ by (13).

Next, the convergence of the sequence $\{x_{n}\}$ to a fixed point $x^*$ follows as in the previous theorems.

Finally, to prove the uniqueness of the fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$, we consider that $w^*$ is a fixed point of J in $\overline{B\left( \widetilde{x},R\right) }$ such that $w^*\ne x^*$. From the same idea developed in Theorems 8 and 11, we have that $w^*=x^*$ if the operator $Q^{-1}=\left[ \int _0^1 H'(x^*+\tau (w^*-x^*))d\tau \right] ^{-1}$ exists. For this, we prove that the operator $[H'(\widetilde{x})]^{-1}Q$ is invertible by the Banach lemma on invertible operators. Indeed, for $\delta =\vartheta \widetilde{x}+(1-\vartheta )(x^*+\tau (w^*-x^*)$ with $\vartheta \in [0,1]$, we have that

$$\begin{aligned} \left\| I-[H'(\widetilde{x})]^{-1}Q\right\|&\le \Vert [H'(\widetilde{x})]^{-1}\Vert \left\| \int _0^1(J'(x^*+\tau (w^*-x^*))-J'(\widetilde{x}))d\tau \right\| \\&\le \dfrac{1}{1-\widetilde{M}_1} \int _0^1 \left\| \sum _{i=2}^{k-1}\dfrac{J^{(i)}(\widetilde{x})}{(i-1)!} (x^*+\tau (w^*-x^*)-\widetilde{x})^{i-1}\right. \\&\quad \left. +\dfrac{1}{(k-1)!}J^{(k)}(\delta ) (x^*+\tau (w^*-x^*)-\widetilde{x})^{k-1}\right\| d\tau \\&\le \dfrac{1}{1-\widetilde{M}_1} \left( \sum _{i=2}^{k-1}\dfrac{\widetilde{M}_{i}}{(i-1)!}R^{i-1} +\dfrac{1}{(k-1)!}\ell (\Vert \widetilde{x}\Vert +R)R^{k-1}\right) \\&= \dfrac{1}{1-\widetilde{M}_1} \left( f'(R)-\widetilde{M}_1+\frac{k}{R}(R-\alpha -f(R))\right) \\&< 1, \end{aligned}$$

and the uniqueness of the fixed point $x^*$ is complete. $\square$

We see below how we can use the previous theorem in the following example.

Example 16

Consider the nonlinear Fredholm integral equation

$$\begin{aligned} x(s) = \sin (\pi s)+\dfrac{1}{5}\int _0^1 \cos (\pi s)\sin (\pi t) x(t)^5 dt, \quad s\in [0,1]. \end{aligned}$$

(14)

In this case, we consider $J_5:\mathcal {C}[0,1]\rightarrow \mathcal {C}[0,1]$ with

$$\begin{aligned}{}[J_5(x)](s) = \sin (\pi s)+\dfrac{1}{5}\int _0^1 \cos (\pi s)\sin (\pi t) x(t)^5 dt, \quad s\in [0,1]. \end{aligned}$$

Taking now into account that

$$\begin{aligned}{}[J_5'(x)a](s)&= \int _0^1 \cos (\pi s)\sin (\pi t) x(t)^4 a(t) dt, \\ [J_5''(x)ab](s)&= 4\int _0^1 \cos (\pi s)\sin (\pi t) x(t)^3 b(t)a(t) dt, \\ [J_5'''(x)abc](s)&= 12\int _0^1 \cos (\pi s)\sin (\pi t) x(t)^2 c(t)b(t)a(t) dt, \\ [J_5^{(4)}(x)abcd](s)&= 24\int _0^1 \cos (\pi s)\sin (\pi t) x(t) d(t)c(t)b(t)a(t) dt, \\ [J_5^{(5)}(x)abcde](s)&= 24\int _0^1 \cos (\pi s)\sin (\pi t) e(t)d(t)c(t)b(t)a(t) dt, \end{aligned}$$

we can obtain the function $\ell$ in the following four different ways:

$$\begin{aligned} \Vert J_5''(x)\Vert&\le \dfrac{8}{\pi }\Vert x\Vert ^3=\ell (\Vert x\Vert ),&\Vert J_5'''(x)\Vert&\le \dfrac{24}{\pi }\Vert x\Vert ^2=\ell (\Vert x\Vert ), \\ \Vert J_5^{(4)}(x)\Vert&\le \dfrac{48}{\pi }\Vert x\Vert =\ell (\Vert x\Vert ),&\Vert J_5^{(5)}(x)\Vert&\le \dfrac{48}{\pi }=\ell (\Vert x\Vert ), \end{aligned}$$

so that we can use Theorem 15 in four different ways, depending on how we choose the function $\ell$.

If we choose $\widetilde{x}(s)=\sin (\pi s)$, we obtain

$$\begin{aligned} \Vert J_5(\widetilde{x})-\widetilde{x}\Vert \le \frac{1}{16}=\alpha ,\quad \Vert J_5'(\widetilde{x})\Vert \le \frac{16}{15\pi }=0.3395\ldots =\widetilde{M}_1<1, \\ \Vert J_5''(\widetilde{x})\Vert \le \frac{3}{2}=\widetilde{M}_2,\quad \Vert J_5'''(\widetilde{x})\Vert \le \frac{16}{\pi }=\widetilde{M}_3,\quad \Vert J_5^{(4)}(\widetilde{x})\Vert \le 12=\widetilde{M}_4. \end{aligned}$$

In addition, Eq. (12) can be chosen in four different ways, depending on the function $\ell$ chosen. The smallest positive root R obtained depends on the value of k considered to obtain $\ell$ and is shown in Table 1.

Table 1 Radius of the balls of existence and uniqueness of the fixed point $x^*$

The three values of R obtained for $k=3,4,5$ satisfy the condition (13) of Theorem 15, so that there exists a unique fixed point $x^*$ of $J_5$ in $\overline{B(\sin (\pi s),R)}$.

From Table 1, we first observe that Eq. (12) has not positive roots if we consider $\ell (z)=\frac{8}{\pi }z^3$ with $k=2$. Second, we observe that the most derived from the operator $J_5$ we consider to define the function $\ell$, the best domain of existence and uniqueness of the fixed point we obtain. This makes us think that the higher the value of k in the application of Theorem 15, the better.

On the other hand, as with Eq. (5), in this case, we cannot locate previously a fixed point of (14), so that the application of the RCP is not easy if we can use it to locate a domain which contains a fixed point.

We end up studying the case in which the k-th derivative of the operator J satisfies the condition (8).

Theorem 17

Let V be a nonempty open convex domain of a Banach space $\mathcal {B}$ with a k-times Fréchet differentiable operator $J:{V}\subset \mathcal {B}\rightarrow \mathcal {B}$ where $k\ge 2$ and

$$\begin{aligned} \Vert J^{(k)}(x)-J^{(k)}(y)\Vert \le \ell (\Vert x\Vert ,\Vert y\Vert )\Vert x-y\Vert ^p \quad p\in (0,1], \quad x,y\in {V}, \end{aligned}$$

where $\ell :\mathbb {R}_{+}\times \mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ is a nondecreasing continuous function in both arguments. Suppose $\Vert J(\widetilde{x})-\widetilde{x}\Vert \le \alpha$ and $\Vert J^{(i)}(\widetilde{x})\Vert \le \widetilde{M}_{i}$, for $i=1,2,\ldots ,k$, with $\widetilde{x}\in \mathcal {B}$. If the equation

$$\begin{aligned} g(t)+\dfrac{1}{\prod _{i=1}^k(i+p)} \ell (\Vert \widetilde{x}\Vert +t,\Vert \widetilde{x}\Vert +t)t^{k+p} +\alpha = t, \quad \text {where}\quad g(t) = \sum _{i=1}^{k}\dfrac{\widetilde{M}_{i}}{i!}t^{i}, \end{aligned}$$

(15)

has at least one positive root and the smallest positive root, denoted by R, satisfies

$$\begin{aligned} g'(R)+(k-1)(R-\alpha -g(R)) < 1 \end{aligned}$$

(16)

and $\overline{B\left( \widetilde{x},R\right) }\subset {V}$, then J admits a fixed point $x^*$ in $\overline{B\left( \widetilde{x},R\right) }$, which is unique in $\overline{B\left( \widetilde{x},R\right) }$ if $\widetilde{M}_1<1$ and $R(g'(R)-1)+(k+p)(R-\alpha -g(R))<0$. Besides, $x^*$ can be approximated by $x_{n}={J}(x_{n-1})$, $n\in \mathbb {N}$, with $x_0\in \overline{B\left( \widetilde{x},R\right) }$.

Proof

From $x_{0}\in {V}$, we have that $x_{1}$ is well defined. Moreover, from Taylor’s formula, it follows

$$\begin{aligned} \Vert x_1-\widetilde{x}\Vert&\le \Vert J(x_0)-J(\widetilde{x})\Vert +\Vert J(\widetilde{x})-\widetilde{x}\Vert \\&\le \left\| \sum _{i=1}^{k}\dfrac{J^{(i)}(\widetilde{x})}{i!}(x_0-\widetilde{x})^{i} +\dfrac{1}{(k-1)!}\int _{\widetilde{x}}^{x_0} \left( J^{(k)}(z)-J^{(k)}(\widetilde{x})\right) (x_0-z)^{k-1}dz \right\| + \alpha \\&\le \sum _{i=1}^{k}\dfrac{\widetilde{M}_{i}}{i!}R^{i} +\dfrac{1}{(k-1)!}\left\| \int _{0}^{1} \left( J^{(k)}(\widetilde{x}+\tau (x_0-\widetilde{x}))\right. \right. \\&\quad \left. \left. -J^{(k)}(\widetilde{x})\right) (1-\tau )^{k-1}(x_0-\widetilde{x})^{k}d\tau \right\| +\alpha \\&\le g(R)+\dfrac{1}{\prod _{j=1}^k(j+p)} \ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{k+p}+\alpha \\&= R. \end{aligned}$$

Then, $x_1\in \overline{B\left( \widetilde{x},R\right) }$ and $x_2=J(x_1)$ is well defined.

Furthermore, from

$$\begin{aligned} \Vert x_2-x_1\Vert = \Vert J(x_1)-J(x_0)\Vert \le \Vert J'(\xi )\Vert \Vert x_1-x_0\Vert , \end{aligned}$$

where $\xi =\vartheta x_0+(1-\vartheta )x_1\in \overline{B\left( \widetilde{x},R\right) }$ with $\vartheta \in [0,1]$, and Taylor’s formula, we have

$$\begin{aligned} \Vert x_2-x_1\Vert&\le \left\| \sum _{i=0}^{k-1}\dfrac{J^{(i)}(\widetilde{x})}{i!}(\xi -\widetilde{x})^{i} + \dfrac{1}{(k-2)!}\int _{\widetilde{x}}^{\xi } \left( J^{(k)}(z)-J^{(k)}(\widetilde{x})\right) (\xi -z)^{k-1}dz\right\| \\&\quad \Vert x_1-x_0\Vert \\&\le \left( \sum _{i=0}^{k-1}\dfrac{\widetilde{M}_{i+1}}{i!}R^{i} +\dfrac{k-1}{\prod _{j=1}^k(j+p)} \ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{k+p}\right) \Vert x_1-x_0\Vert \\&= N\Vert x_1-x_0\Vert , \end{aligned}$$

where

$$\begin{aligned} N&= \sum _{i=0}^{k-1}\dfrac{\widetilde{M}_{i+1}}{i!}R^{i} +\dfrac{k-1}{\prod _{j=1}^k(j+p)} \ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{k+p} \\&= g'(R) + \dfrac{k-1}{\prod _{j=1}^k(j+p)} \ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{k+p}. \end{aligned}$$

After that, it is easy to prove by mathematical induction on n that

$$\begin{aligned} x_n\in \overline{B\left( \widetilde{x},R\right) } \quad \text {and}\quad \Vert x_{n+1}-x_n\Vert \le N^n\Vert x_1-x_0\Vert , \quad \text {for all}\quad n\in \mathbb {N}. \end{aligned}$$

Besides, $N<1$, since $g'(R)+(k-1)(R-\alpha -g(R))<1$ by hypothesis.

Next, the convergence of the sequence $\{x_{n}\}$ to a fixed point $x^*$ follows as in the previous theorems.

Finally, to prove the uniqueness of the fixed point $x^*$ of J in $\overline{B\left( \widetilde{x},R\right) }$, we consider that $w^*$ is a fixed point of J in $\overline{B\left( \widetilde{x},R\right) }$ such that $w^*\ne x^*$. From the same idea developed in Theorems 8 and 11, we have that $w^*=x^*$ if the operator $Q^{-1}=\left[ \int _0^1 H'(x^*+\tau (w^*-x^*))d\tau \right] ^{-1}$ exists. For this, we prove that the operator $[H'(\widetilde{x})]^{-1}Q$ is invertible by the Banach lemma on invertible operators. Indeed, for $\delta =\vartheta \widetilde{x}+(1-\vartheta )(x^*+\tau (w^*-x^*)$ with $\vartheta \in [0,1]$, we have that

$$\begin{aligned} \left\| I-[H'(\widetilde{x})]^{-1}Q\right\|&\le \Vert [H'(\widetilde{x})]^{-1}\Vert \left\| \int _0^1(J'(x^*+\tau (w^*-x^*))-J'(\widetilde{x}))d\tau \right\| \\&\le \dfrac{1}{1-\widetilde{M}_1} \int _0^1 \left\| \sum _{i=1}^{k}\dfrac{J^{(i)}(\widetilde{x})}{(i-1)!} (x^*+\tau (w^*-x^*)-\widetilde{x})^{i-1}\right. \\&\quad \left. +\dfrac{1}{(k-2)!} \int _0^1 \left( J^{(k)}(\delta )-J^{(k)}(\widetilde{x})\right) (1-\delta )^{k-2}\right. \\&\quad \left. (x^*+\tau (w^*-x^*)-\widetilde{x})^{k-1}d\delta \right\| \\&\le \dfrac{1}{1-\widetilde{M}_1} \left( \sum _{i=2}^{k}\dfrac{\widetilde{M}_{i}}{(i-1)!}R^{i-1}\right. \\&\quad \left. +\dfrac{1}{(k-2)!}\left( \int _0^1 \delta ^p(1-\delta )^{k-2}d\delta \right) \ell (\Vert \widetilde{x}\Vert +R,\Vert \widetilde{x}\Vert +R)R^{p+k-1}\right) \\&= \dfrac{1}{1-\widetilde{M}_1} \left( g'(R)-\widetilde{M}_1+\frac{k+p}{R}(R-\alpha -g(R))\right) \\&< 1, \end{aligned}$$

since $\widetilde{M}_1<1$ and $R(g'(R)-1)+(k+p)(R-\alpha -g(R))<0$, and the uniqueness of the fixed point $x^*$ is complete. $\square$

In the next example, we illustrate the application of Theorem 17.

Example 18

Consider the nonlinear Fredholm integral equation

$$\begin{aligned} x(s) = 1-s+\int _0^1 st x(t)^{\frac{7}{2}} dt, \quad s\in [0,1]. \end{aligned}$$

(17)

Then, we consider $J_6:\mathcal {C}[0,1]\rightarrow \mathcal {C}[0,1]$ with

$$\begin{aligned}{}[J_6(x)](s) = 1-s+\int _0^1 st x(t)^{\frac{7}{2}} dt, \quad s\in [0,1]. \end{aligned}$$

In addition,

$$\begin{aligned}{}[J_6'(x)a](s)&= \dfrac{7}{2}\int _0^1 st x(t)^{\frac{5}{2}} a(t) dt, \\ [J_6''(x)ab](s)&= \dfrac{35}{4}\int _0^1 st x(t)^{\frac{3}{2}} b(t)a(t) dt, \\ [J_6'''(x)abc](s)&= \dfrac{105}{8}\int _0^1 st x(t)^{\frac{1}{2}} c(t)b(t)a(t) dt, \end{aligned}$$

and

$$\begin{aligned} \Vert J_6'''(c)-J_6'''(d)\Vert \le \frac{105}{16} \Vert c-d\Vert ^{\frac{1}{2}}, \end{aligned}$$

so that $\ell (u,v)=\frac{105}{16}$ and $p=\frac{1}{2}$.

Now, we choose $\widetilde{x}(s)=1-s$ and obtain

$$\begin{aligned} \Vert J_6(\widetilde{x})-\widetilde{x}\Vert&\le \frac{4}{99}=\alpha ,\quad \Vert J_6'(\widetilde{x})\Vert \le \frac{14}{63}=\widetilde{M}_1<1,\quad \\ \Vert J_6''(\widetilde{x})\Vert&\le 1=\widetilde{M}_2,\quad \Vert J_6'''(\widetilde{x})\Vert \le \frac{7}{2}=\widetilde{M}_3. \end{aligned}$$

Hence, the corresponding Eq. (15) of Theorem 17 has positive roots and $R=0.0539\ldots$ is the smallest positive one, that satisfies the condition (16) and the fixed-point uniqueness condition. Therefore, there exists a unique fixed point $x^*$ of $J_6$ in $\overline{B(1-s,0.0539\ldots )}$.

On the other hand, as with Eqs. (5) and (14), in this case, we cannot locate previously a fixed point of (17), so that the application of the RCP is not easy if we can use it to locate a domain which contains a fixed point.