A Novel Fixed-Point Iteration Approach for Solving Troesch’s Problem

Abstract

This paper introduces a novel F fixed-point iteration method that leverages Green’s function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical and engineering applications. The proposed method integrates fixed-point theory with Green’s function techniques to develop an iteration process that ensures convergence, stability, and accuracy. The numerical experiments demonstrate the method’s efficiency and robustness, highlighting its potential for broader applications in solving nonlinear differential equations in Banach spaces.

1. Introduction and Preliminaries

In a set S, a distance function d is defined as symmetric if it fulfills the condition $d(p,q)=d(q,p)$ for every pair of points p and q within S. A set that has such a symmetric distance function is known as a symmetric space. It is widely recognized that Banach spaces and metric spaces inherently have this symmetry. However, pseudo-metric spaces do not always exhibit this symmetric property.

The Troesch problem appears in various physical contexts, including plasma physics and reaction–diffusion processes, see [1]. It is characterized by a nonlinear differential equation with specific boundary conditions. The traditional methods for solving the Troesch problem often encounter difficulties due to its strong nonlinearity and boundary conditions. This paper introduces a novel approach that combines fixed-point iteration and Green’s function techniques to effectively address these challenges.

In the scientific literature, solving linear and nonlinear problems involving differential and integral operators often requires exact solutions, which are frequently difficult or impossible to determine. As a result, approximate numerical solutions are highly desirable. There are many numerical methods available for solving such problems; however, many of these methods require complex parameter sets for initialization and lack established theoretical convergence criteria.

Fixed-point theory provides a powerful framework for solving nonlinear problems. The Banach fixed-point theorem [2], guarantees the existence and uniqueness of fixed points for contraction mappings in Banach spaces and ensures the convergence of iterative sequences to these fixed points. Fixed-point iteration methods leverage this theory to develop iterative schemes that converge to the solution of the nonlinear problem. According to Banach’s theorem, if S is a Banach space and $U:S\to S$ is a mapping that satisfies the property

(1)$\left|\right|Up-Uq\left|\right|\le \beta \left|\right|p-q\left|\right|\mathrm{for}\mathrm{any}p,q\in S\mathrm{and}\beta \in [0,1)$

has a unique fixed point $y^{*}\in S$. This ensures that the sequence of Picard approximations $z_{m+1}=U{z}_m$ converges to $y^{*}$ for any initial guess $z_0\in S$. It is well-known that the Picard iteration method does not converge to the fixed points of nonexpansive mappings $\left(\right||Up-Uq||\le ||p-q||\mathrm{for}\mathrm{any}p,q\in S)$. To address this scenario, Krasnoselskii [3] generalized the Picard approximation method [4] as follows:

(2)$\left\{\begin{array}{cc}{z}_0\in S,z_{m+1}=(1-\lambda )z_m+\lambda U{z}_m,\hfill & m=0,1,2,3,\dots \hfill \end{array}\right.$

where $0<\lambda <1$.

On another note, when an iteration method is known to converge to a fixed point of a nonlinear mapping, important considerations include the speed of convergence (i.e., which iteration method reaches the fixed point more quickly), the stability of iterative methods, and other relevant factors. It is widely recognized that both the Picard and Krasnoselskii iteration methods exhibit relatively slow convergence rates. Therefore, the iteration method proposed by Ali and Ali [5] defines a sequence $\left\{z_m\right\}$ with an initial guess $z_0\in S$ as follows:

(3)$\left\{\begin{array}{c}{z}_{m+1}=U_{p_m},\hfill \\ p_m=U_{q_m},\hfill \\ q_m=U\left(\right(1-{\lambda }_m)z_m+{\lambda }_{m}U{z}_m),m=0,1,2,3,\dots \hfill \end{array}\right.$

where $\left\{{\lambda }_m\right\}$ is a sequence in $(0,1)$. They asserted that this method converges more rapidly than several prominent iteration methods for Zamfirescu operators.

Now, we consider the following Troesch’s boundary value problem (BVP):

(4)$z^{″}\left(t\right)=\nu sinh\left[\nu z\left(t\right)\right];(0\le t\le 1),$

where $\nu \ge 0$, and the boundary conditions (BCs) are provided by

(5)$z\left(0\right)=0,z\left(1\right)=1.$

The Troesch’s boundary value problem (BVP) (4) and (5) was first introduced by Troesch [6]. This problem has applications in various fields of applied sciences [7,8]. The initial research on its closed-form solution was conducted by Roberts and Shipman [9], while Troesch [6] investigated numerical solutions using shooting techniques, which led to the problem being named after him. Since then, numerous methods have been proposed to solve this problem [10,11,12,13,14,15,16]. Green’s function techniques offer another robust approach to solving differential equations by transforming the problem into an integral equation. This approach is beneficial in handling boundary conditions and can be integrated into iterative methods to improve convergence and accuracy. Recently, Kafri et al. [17] combined fixed-point iteration methods and Green’s function techniques for numerically solving Troesch’s BVPs (4) and (5) and developed Picard and Krasnoselskii–Green’s iteration methods, integrating Green’s functions into these methods. They demonstrated faster convergence compared to the classical approaches but did not address stability analysis. In recent years, fixed-point iteration methods incorporating Green’s functions have shown rapid convergence for solving nonlinear problems [18,19,20,21,22,23,24]. Inspired by these methods, we propose the F-Green’s iteration method, which incorporates a Green’s function associated with the linear and nonlinear terms of Troesch’s BVPs (4) and (5). We show that our method achieves faster convergence compared to the Picard and Krasnoselskii–Green’s methods.

Despite the significant progress in the numerical methods for solving nonlinear boundary value problems like Troesch’s problem, several research gaps persist that hinder the full realization of efficient and accurate computational techniques. Many of the existing numerical methods struggle with the high nonlinearity often present in Troesch’s problem. While some techniques can approximate solutions to the problem, they frequently suffer from reduced accuracy or convergence issues. There is a need for more robust methods to overcome these issues. While theoretical analyses have provided insights into the convergence and stability of various numerical methods, there are still gaps in understanding the precise conditions under which these methods perform optimally. More comprehensive analyses are required to delineate the boundaries of the method applicability and to establish rigorous proofs for a broader range of scenarios. With these aspects in mind, we developed a novel F-Green’s iterative method to approximate the solution of Troesch’s problem with a rapid convergence rate and minimum error. The proposed method is more efficient and better than the previously defined methods (see Section 3).

This paper presents a new computational approach for solving Troesch’s problem by utilizing fixed-point iteration, Green’s function theory, and the framework of Banach spaces. The following sections will cover the mathematical formulation of Troesch’s problem, the proposed fixed-point iteration method, the theoretical analysis of the convergence properties, and numerical experiments to illustrate the method’s effectiveness in practical applications. This work aims to enhance the computational techniques for tackling the nonlinear boundary value problems in mathematical physics and related disciplines.

2. Overview of Green’s Function and Iterative Method

2.1. Construction of Green’s Function for Troesch’s BVPs

In this section, we initiate the process of constructing a Green’s function designed specifically for Troesch’s problem. Let L denote the linear differential operator such that

(6)$L\left[z\right]=z^{″}\left(t\right).$

For this problem, let $z_1$ and $z_2$ be two linearly independent solutions of L, and then the Green’s function $G(t,\xi )$ is expressed as follows:

$G(t,\xi )=\left\{\begin{array}{cc}{c}_1{z}_1+c_2{z}_2\hfill & \mathrm{for}a\le t<\xi ,\hfill \\ d_1{z}_1+d_2{z}_2\hfill & \mathrm{for}\xi \le t<b.\hfill \end{array}\right.$

Here, $c_1$, $c_2$, $d_1$, and $d_2$ are constants determined using the property that $G(t,\xi )$ solves the following boundary value problem (BVP):

(7)$L\left[G\right(t,\xi \left)\right]=\delta (t-\xi ),$

subject to the BCs:

(8)$B_1\left[G(t,\xi )\right]=0=B_2\left[G(t,\xi )\right].$

These conditions are well-known to yield specific relations between the coefficients $c_1$, $c_2$, $d_1$, and $d_2$. Additionally, properties of $G(t,\xi )$ include continuity at $t=\xi$ and a jump discontinuity of size unity.

To implement the iterative method, we begin by determining the associated Green’s function. The linear operator corresponding to problem (4) is provided by $z^{″}=0$, which has two independent solutions, namely $z_1=1$ and $z_2=t$. Thus, Green’s function can be expressed in the form

$G(t,\xi )=\left\{\begin{array}{cc}{c}_1+c_{2}t,\hfill & 0\le t<\xi \hfill \\ d_1+d_{2}t,\hfill & \xi \le t<1.\hfill \end{array}\right.$

To obtain the constants $c_1,c_2,d_1$, and $d_2$, we apply the boundary conditions and the properties of Green’s function, resulting in the following equations:

$c_1=0,c_2=\xi -1,$

$d_1=-\xi ,d_2=\xi .$

Thus, the desired Green’s function is

(9)$G(t,\xi )=\left\{\begin{array}{cc}t(\xi -1),\hfill & 0\le t<\xi \hfill \\ \xi (t-1),\hfill & \xi \le t<1.\hfill \end{array}\right.$

The problem provided in (7) leads to an integration that ensures the fulfillment of the BVP conditions. Specifically, we integrate (7) over an interval around $\xi$, leading to

(10)${\int }_{{\xi }^{-}}^{{\xi }^{+}}{G}^{″}(t,\xi )dt={\int }_{{\xi }^{-}}^{{\xi }^{+}}\delta (t-\xi )dt.$

Equation (7) can be expressed equivalently as follows:

(11)$G^{\prime }(t,{\xi }^{+})-G^{\prime }(t,{\xi }^{-})=H(t-{\xi }^{+}),$

where $H(·)$ is a heaviside function, which implies

(12)$d_1{z}_0\left({\xi }^{+}\right)+d_2{z}_0\left({\xi }^{-}\right)-c_1{z}_0\left({\xi }^{+}\right)-c_2{z}_0\left({\xi }^{-}\right)=1.$

Now, let us examine (6), and, more generally, let us consider the subsequent equation

(13)$L\left[z\right]=z^{″}\left(t\right)=f(t,z,z^{\prime },z^{″}),$

with the general boundary conditions

(14)$B_1\left[z\right]=a_{0}z\left(a\right)+a_1{z}^{\prime }\left(a\right)=0,B_2\left[z\right]=b_{0}z\left(b\right)+b_1{z}^{\prime }\left(b\right)=1,$

where $t\in [a,b]$. Note that $z_h$ represents the solution of the homogeneous part satisfying the specified boundary conditions, and $z_p$ represents the particular solution satisfying (13) and the boundary conditions $B_k=0$, $k=1,2$. We establish that problems (13) and (14) possess a general solution z with $z=z_h+z_p$. Given that G represents the Green’s function that satisfies Equations (7) through (8), it can be deduced that

(15)$z_p={\int }_0^{1}G(t,\xi )f(\xi ,z\left(\xi \right),z^{\prime }\left(\xi \right),z^{″}\left(\xi \right))d\xi .$

Now, let us consider L acting on $z_h+z_p$ as follows:

(16)$\begin{array}{cc}\hfill L[z_h+z_p]& =L\left[z_h\right]+L\left[z_p\right]=L\left[z_p\right]\hfill \\ \hfill & =L\left[{\int }_0^{1}G(t,\xi )f(\xi ,z\left(\xi \right),z^{\prime }\left(\xi \right),z^{″}\left(\xi \right))d\xi \right]\hfill \\ \hfill & ={\int }_0^{1}L\left[G(t,\xi )\right]f(\xi ,z\left(\xi \right),z^{\prime }\left(\xi \right),z^{″}\left(\xi \right))d\xi \hfill \\ \hfill & ={\int }_0^1\delta (t-\xi )f(\xi ,z\left(\xi \right),z^{\prime }\left(\xi \right),z^{″}\left(\xi \right))d\xi \hfill \\ \hfill & =f(t,z\left(t\right),z^{\prime }\left(t\right),z^{″}\left(t\right)).\hfill \end{array}$

Similarly, for the boundary conditions, we have

(17)$\begin{array}{cc}\hfill B_k[z_h+z_p]& =B_k\left[z_h\right]+B_k\left[z_p\right]=B_k\left[z_p\right]\hfill \\ \hfill & ={\int }_0^1{B}_k\left[G(t,\xi )\right]f(\xi ,z\left(\xi \right),z^{\prime }\left(\xi \right),z^{″}\left(\xi \right))d\xi .\hfill \end{array}$

Given that $B_1\left[G(\xi ,t)\right]=0$ and $B_2\left[G(\xi ,t)\right]=1$, it implies that $B_1[z_h+z_p]=0$ and $B_2[z_h+z_p]=1$. Thus, we infer that $z=z_h+z_p$ constitutes a solution to Equations (13) and (14).

2.2. A Concise Description of F-Green’s Iteration Method

This section develops a novel modified iteration approach by incorporating Green’s function. To accomplish this, we modify the F iteration method (3) for an integral operator based on Green’s function associated with Troesch’s problems (4) and (5). This integral operator possesses a unique characteristic: its set of fixed points corresponds to the solution set of Troesch’s problems (4) and (5).

To start, let us consider the following integral operator:

(18)$Nz=z_h+{\int }_0^{1}G(t,\xi )z^{″}\left(\xi \right)d\xi .$

Now, by adding and subtracting the term $f(\xi ,z,z^{\prime },z^{″})$ within the integral in (18), we obtain

(19)$Nz=z_h+{\int }_0^{1}G(t,\xi )(z^{″}\left(\xi \right)-f(\xi ,z,z^{\prime },z^{″}))d\xi +{\int }_0^{1}G(t,\xi )f(\xi ,z,z^{\prime },z^{″})d\xi .$

Keeping (15) in mind, we have from (19)

(20)$Nz=z_h+{\int }_0^{1}G(t,\xi )(z^{″}\left(\xi \right)-f(\xi ,z,z^{\prime },z^{″}))d\xi +z_p.$

Since $z_h+z_p=z$, (20) provides us

(21)$Nz=z+{\int }_0^{1}G(t,\xi )(z^{″}\left(\xi \right)-f(\xi ,z,z^{\prime },z^{″}))d\xi .$

We apply the F iteration method (3) to the operator N in (21). This yields

(22)$\left\{\begin{array}{c}{z}_{m+1}=N_{p_m},\hfill \\ p_m=N_{q_m},\hfill \\ q_m=N\left(\right(1-{\lambda }_m)z_m+{\lambda }_{m}N{z}_m),m=0,1,2,3,\dots ,\hfill \end{array}\right.$

or, equivalently, the following:

(23)$\left\{\begin{array}{c}{z}_{m+1}=p_m+{\int }_0^{1}G(t,\xi )(p_m^{″}-f(\xi ,p_m,p_m^{\prime },p_m^{″}))d\xi ,\hfill \\ p_m=q_m+{\int }_0^{1}G(t,\xi )(q_m^{″}-f(\xi ,q_m,q_m^{\prime },q_m^{″}))d\xi ,\hfill \\ q_m=w_m+{\int }_0^{1}G(t,\xi )(w_m^{″}-f(\xi ,w_m,w_m^{\prime },w_m^{″}))d\xi ,\hfill \\ w_m=z_m+{\lambda }_m{\int }_0^{1}G(t,\xi )(z_m^{″}-f(\xi ,z_m,z_m^{\prime },z_m^{″}))d\xi ,m=0,1,2,3,\dots \hfill \end{array}\right.$

3. Main Results

3.1. Convergence Analysis

In this section, we explore the convergence of the proposed iteration method. We divide our analysis into two cases:

Case I: Here, we consider the scenario where $\nu \le 1$.

By interchanging the variables t and $\xi$ in (9), the resulting Green’s function $G(t,\xi )$ is provided by

$G(t,\xi )=\left\{\begin{array}{cc}\xi (1-t)\hfill & \mathrm{for}0\le \xi <t,\hfill \\ t(1-\xi )\hfill & \mathrm{for}t\le \xi <1.\hfill \end{array}\right.$

Let $S=C[0,1]$ be a Banach space equipped with the supremum norm. Now, let us define the operator $U_G:S\to S$ as follows:

(24)$U_{G}z=z+{\int }_0^{1}G(t,\xi )(z^{″}-\nu sinh\left(\nu z\right))d\xi .$

Hence, (23) takes the following form:

(25)$\left\{\begin{array}{c}{z}_{m+1}=U_G{p}_m,\hfill \\ p_m=U_G{q}_m,\hfill \\ q_m=U_G\left(\right(1-{\lambda }_m)z_m+{\lambda }_m{U}_G{z}_m),m=0,1,2,3,\dots ,\hfill \end{array}\right.$

or, equivalently,

(26)$\left\{\begin{array}{c}{z}_{m+1}=p_m+{\int }_0^{1}G(t,\xi )(p_m^{″}-\nu sinh\left(\nu p_m\right))d\xi ,\hfill \\ p_m=q_m+{\int }_0^{1}G(t,\xi )(q_m^{″}-\nu sinh\left(\nu q_m\right))d\xi ,\hfill \\ q_m=w_m+{\int }_0^{1}G(t,\xi )(w_m^{″}-\nu sinh\left(\nu w_m\right))d\xi ,\hfill \\ w_m=z_m+{\lambda }_m{\int }_0^{1}G(t,\xi )(z_m^{″}-\nu sinh\left(\nu z_m\right))d\xi ,m=0,1,2,3,\dots \hfill \end{array}\right.$

The iteration method (25) is the desired F-Green’s iteration method. Now, we prove its convergence theoretically as follows:

Theorem 1.

Let $U_G:S\to S$ be the operator defined by

(27) $U_{G}z=z+{\int }_0^{1}G(t,\xi )\left[\nu g\left(z\right)\right]d\xi ,$

where S is a Banach space, $G(t,\xi )$ is a Green’s function associated with Troesch’s problem, and $g\left(z\right)=sinh\left(\nu z\right)$ is a function whose derivative is bounded and satisfying Lipschitz condition. Suppose that the Lipschitz constant $L_z$ and the parameter ν satisfy the condition

$L_z+\frac{{\nu }^2{max}_{t\in [0,1]}|cosh\left(\nu z\left(t\right)\right)|}{8}<1.$

If $\left\{z_m\right\}$ is the sequence obtained using the F-Green’s iteration method (25), where $\nu \le 1$ and $t\in [0,1]$, then $\left\{z_m\right\}$ converges strongly to the unique solution of Troesch’s BVPs (4) and (5). Moreover, F-Green’s iterative method converges to the solution of Troesch’s problem faster than Picard–Green’s and Mann–Green’s iterative methods.

Proof. 

Putting $L_z+\frac{{\nu }^2{max}_{t\in [0,1]}|cosh\left(\nu z\left(t\right)\right)|}{8}=\beta$$(0\le \beta <1)$, it follows that $U_G$ is a Banach’s contraction mapping. Hence, by Banach’s fixed-point theorem, $U_G$ has a unique fixed point in $S=C[0,1]$, denoted as $y^{*}$, which corresponds to the unique solution of Troesch’s BVPs (4) and (5). We now proceed to demonstrate the convergence of the method (25). To achieve this, we consider the following:

$\parallel q_m-y^{*}\parallel = \parallel U_G((1-{\lambda }_m)z_m+{\lambda }_m{U}_G{z}_m)-y^{*}\parallel$

$\le \beta \parallel (1-{\lambda }_m)(z_m-y^{*})+{\lambda }_m(U_G{z}_m-y^{*})\parallel$

$\le \beta \left((1-{\lambda }_m)\parallel z_m-y^{*}\parallel +{\lambda }_m\parallel U_G{z}_m-U_G{y}^{*}\parallel \right)$

$=\beta \left((1-{\lambda }_m)\parallel z_m-y^{*}\parallel +\beta {\lambda }_m\parallel z_m-y^{*}\parallel \right)$

$=\beta [1-{\lambda }_m(1-\beta )]\parallel z_m-y^{*}\parallel .$

Consequently, we obtain

(28)$\parallel q_m-y^{*}\parallel \le [1-{\lambda }_m(1-\beta )]\parallel z_m-y^{*}\parallel .$

Using (28),

(29)$\parallel p_m-y^{*}\parallel = \parallel U_G{q}_m-U_G{y}^{*}\parallel \le \beta \parallel q_m-y^{*}\parallel \le \beta [1-{\lambda }_m(1-\beta )]\parallel z_m-y^{*}\parallel .$

And using (29), we have

$\parallel z_{m+1}-y^{*}\parallel = \parallel U_G{p}_m-y^{*}\parallel = \parallel U_G{p}_m-U_G{y}^{*}\parallel \le \beta \parallel p_m-y^{*}\parallel \le {\beta }^{2m}{[1-{\lambda }_m(1-\beta )]}^{m+1}\parallel z_0-y^{*}\parallel .$

Since $0\le \beta <1$ and $0<1-{\lambda }_m(1-\beta )<1$, we have $\parallel z_m-y^{*}\parallel \to 0$ as $m\to \infty$. Therefore, $\left\{z_m\right\}$ converges to $y^{*}$, which is the unique fixed point of $U_G$, and corresponds to the unique solution of Troesch’s boundary value problems (4) and (5). The rest of the proof can be completed with the proof of Theorem 2.4 in [5]. □

Case II: In this scenario, we consider the parameter $\nu >1$.

Due to $\nu >1$, the presence of a boundary layer might pose challenges while solving Troesch’s BVPs (4) and (5). Therefore, we need to implement the following transformation:

$u\left(t\right)=tanh\left(\frac{\nu u\left(t\right)}{4}\right).$

In this case, the transformed Troesch’s BVPs (4) and (5) take the following form:

(30)$(1-u^2)u^{″}+2u{\left(u^{\prime }\right)}^2={\nu }^{2}u(1+u^2),$

and the new boundary conditions become

(31)$u\left(0\right)=0,u\left(1\right)=tanh\left(\frac{\nu }{4}\right).$

Now, let us examine the linear term $L\left(u\right)=u^{″}-{\nu }^{2}u=0$ and the nonlinear term $N\left(u\right)=g(u,u^{\prime },u^{″})=u^2{u}^{″}-2u{\left(u^{\prime }\right)}^2+{\nu }^2{u}^3$. Since the linear term here is different from that in Case I, Theorem 1 cannot be directly applied. Therefore, we need to derive a new Green’s function tailored for this specific linear term. To construct the Green’s function for the given problem, we use a linear combination of two independent solutions of the corresponding homogeneous linear differential equation $u^{″}-{\nu }^{2}u=0$, which are $u_1\left(t\right)=sinh\left(\nu t\right)$ and $u_2\left(t\right)=sinh\left(\nu (1-t)\right)$. These solutions meet the necessary properties of Green’s function, enabling us to construct the following Green’s function:

$G(t,\xi )=\left\{\begin{array}{cc}\frac{sinh\left(\nu \xi \right)sinh\left(\nu \right(1-t\left)\right)}{\nu sinh\left(\nu \right)}\hfill & \mathrm{when}0\le \xi <t,\hfill \\ \frac{sinh\left(\nu t\right)sinh\left(\nu \right(1-\xi \left)\right)}{\nu sinh\left(\nu \right)}\hfill & \mathrm{when}t\le \xi <1.\hfill \end{array}\right.$

Again, consider a Banach space $S=C[0,1]$ equipped with the supremum norm, and $G(t,\xi )$ represents the Green’s function provided above. Define the operator $P_G:S\to S$ as follows:

(32)$P_{G}u=u+{\int }_0^{1}G(t,\xi )[(1-u^2)u^{″}+2u{\left(u^{\prime }\right)}^2-{\nu }^{2}u(1+u^2)]d\xi .$

In this case, iteration method (23) takes the following form:

(33)$\left\{\begin{array}{c}{z}_{m+1}=P_G{p}_m,\hfill \\ p_m=P_G{q}_m,\hfill \\ q_m=P_G((1-{\lambda }_m)z_m+{\lambda }_m{P}_G{z}_m),m=0,1,2,3,\dots ,\hfill \end{array}\right.$

where $P_G$ is the operator defined in Equation (32). The convergence theorem for this case is presented below.

Theorem 2.

Consider the operator $P_G:S\to S$ defined as

(34) $P_G\left(u\right)=u+{\int }_0^{1}G(t,\xi )[(1-u^2)u^{″}+2u{\left(u^{\prime }\right)}^2-{\nu }^{2}u(1+u^2)]d\xi ,$

where S is a Banach space, $G(t,\xi )$ is a Green’s function associated with Troesch’s problem, and $g(u,u^{\prime },u^{″})=(1-u^2)u^{″}+2u{\left(u^{\prime }\right)}^2-{\nu }^{2}u(1+u^2)$ is a function satisfying derivative bound condition. Assume that the Lipschitz constant $L_u$ and the parameter ν satisfy the condition

$\left(L_u+\underset{[0,1]\times \mathbb{R}\times \mathbb{R}}{max}\left|\frac{\partial g}{\partial u}\right|\frac{cosh\left(\frac{\nu }{2}\right)-1}{{\nu }^{2}cosh\left(\frac{\nu }{2}\right)}\right)<1.$

If $\left\{z_m\right\}$ is the sequence of iterates obtained using the F-Green’s iteration method (33), where $\nu >1$ and $t\in [0,1]$, then $\left\{z_m\right\}$ converges strongly to the unique solution of the transformed Troesch’s BVPs (30) and (31).

Proof. 

Putting

$\left(L_u+\underset{[0,1]\times \mathbb{R}\times \mathbb{R}}{max}\left|\frac{\partial g}{\partial u}\right|\frac{cosh\left(\frac{\nu }{2}\right)-1}{{\nu }^{2}cosh\left(\frac{\nu }{2}\right)}\right)=\beta ,$

one can easily prove that $P_G$ is a $\beta$-contraction mapping, $(0\le \beta <1)$. Hence, by Banach’s fixed-point theorem, $P_G$ has a unique fixed point in $S=C[0,1]$, denoted as $y^{*}$, which represents the unique solution for the transformed Troesch’s BVPs (30) and (31). The remaining proof follows the same logic as the proof of Theorem 1 and is omitted. □

3.2. Stability Analysis

This section provides the stability analysis of the proposed iteration method. Numerical methods can sometimes become unstable when applied to specific operators for solving particular problems [25,26,27]. The stability of a numerical iteration method refers to its ability to consistently converge to the exact solution, even in the presence of errors between successive iterations. The concept of stability in iteration methods was initially introduced by Urabe [28], and, later, Harder and Hicks [29] provided a formal definition. Our proposed method (25) demonstrates weak $w^2$-stability, which differs from traditional stability notions.

Definition 1

([29]). Suppose U is a self-map on a Banach space S, and $\left\{z_m\right\}$ is a sequence of iterates of U generated by

(35)$\left\{\begin{array}{c}{z}_0\in S,\hfill \\ z_{m+1}=f(U,z_m),\hfill \end{array}\right.$

where $z_0$ represents the initial value and f is a function. If $\left\{z_m\right\}$ converges to a point $y^{*}\in {\mathcal{F}}_{\mathcal{U}}$, where ${\mathcal{F}}_{\mathcal{U}}$ is the fixed-point set of U, then $\left\{z_m\right\}$ is called stable with respect to U if, for every approximate sequence $\left\{a_m\right\}$ in S, the following holds:

(36)$\underset{m\to \infty }{lim}\parallel a_{m+1}-f(U,a_m)\parallel =0\Rightarrow \underset{m\to \infty }{lim}{a}_m=y^{*}.$

Definition 2

([30]). Two sequences $\left\{a_m\right\}$ and $\left\{z_m\right\}$ in a Banach space are said to be equivalent if the following property holds:

(37)$\underset{m\to \infty }{lim}\parallel a_m-z_m\parallel =0.$

In [31], Timis introduced the notion of weak $w^2$-stability, which uses the concept of equivalent sequences.

Definition 3

([31]). Let U be a self-map on a Banach space S, and consider $\left\{z_m\right\}$ as a sequence of iterates of U generated by

(38)$\left\{\begin{array}{c}{z}_0\in S,\hfill \\ z_{m+1}=f(U,z_m),\hfill \end{array}\right.$

where f represents a function. We say that $\left\{z_m\right\}$ is weak $w^2$-stable if it converges to a point $y^{*}\in {\mathcal{F}}_{\mathcal{U}}$ and, for any equivalent sequence $\left\{a_m\right\}$ in S with respect to $\left\{z_m\right\}$, the following holds:

(39)$\underset{m\to \infty }{lim}\parallel a_{m+1}-f(U,a_m)\parallel =0\Rightarrow \underset{m\to \infty }{lim}{a}_m=y^{*}.$

Theorem 3.

Let S, $U_G$, and $\left\{z_m\right\}$ be as defined in Theorem 1 and $\nu \le 1$. Then, $\left\{z_m\right\}$ is weak $w^2$-stable with respect to $U_G$.

Proof. 

Presume that $\left\{a_m\right\}$ is an equivalent sequence of $\left\{z_m\right\}$, meaning that $\left\{a_m\right\}$ satisfies ${lim}_{m\to \infty }\parallel a_m-z_m\parallel =0$. To achieve this, we define

(40)${\epsilon }_m=\parallel a_{m+1}-U_G^2{b}_m\parallel ,$

where $b_m=U_G[(1-{\lambda }_m)a_m+{\lambda }_m{U}_G{a}_m]$ and $0<{\lambda }_m<1.$

Now, we suppose that ${lim}_{m\to \infty }{\epsilon }_m=0$ and show that ${lim}_{m\to \infty }\parallel a_m-y^{*}\parallel =0$. For this, we have

$\begin{array}{cc}\hfill \parallel b_m-q_m\parallel & = \parallel U_G[(1-{\lambda }_m)a_m+{\lambda }_m{U}_G{a}_m]-U_G[(1-{\lambda }_m)z_m+{\lambda }_m{U}_G{z}_m]\parallel \hfill \\ \hfill & \le \beta \parallel [(1-{\lambda }_m)(a_m-z_m)+{\lambda }_m(U_G{a}_m-U_G{z}_m)]\parallel \hfill \\ \hfill & \le \beta [(1-{\lambda }_m)\parallel a_m-z_m\parallel +{\lambda }_m\parallel U_G{a}_m-U_G{z}_m\parallel ]\hfill \\ \hfill & \le \beta [(1-{\lambda }_m)\parallel a_m-z_m\parallel +\beta {\lambda }_m\parallel a_m-z_m\parallel ]\hfill \\ \hfill & \le \beta [1-{\lambda }_m(1-\beta )]\parallel a_m-z_m\parallel ,\hfill \end{array}$

It follows that

(41)$\parallel b_m-q_m\parallel \le [1-{\lambda }_m(1-\beta )]\parallel a_m-z_m\parallel .$

Hence, using (40) and (41), we find

$\begin{array}{cc}\hfill \parallel a_{m+1}-y^{*}\parallel & \le \parallel a_{m+1}-z_{m+1}\parallel +\parallel z_{m+1}-y^{*}\parallel \hfill \\ \hfill & \le \parallel a_{m+1}-U_G^2{b}_m\parallel +\parallel U_G^2{b}_m-z_{m+1}\parallel +\parallel z_{m+1}-y^{*}\parallel \hfill \\ \hfill & \le {\epsilon }_m+\parallel U_G^2{b}_m-U_G{p}_m\parallel +\parallel z_{m+1}-y^{*}\parallel \hfill \\ \hfill & \le {\epsilon }_m+\beta \parallel U_G{b}_m-p_m\parallel +\parallel z_{m+1}-y^{*}\parallel \hfill \\ \hfill & ={\epsilon }_m+\beta \parallel U_G{b}_m-U_G{q}_m\parallel +\parallel z_{m+1}-y^{*}\parallel \hfill \\ \hfill & \le {\epsilon }_m+{\beta }^2\parallel b_m-q_m\parallel +\parallel z_{m+1}-y^{*}\parallel \hfill \\ \hfill & \le {\epsilon }_m+{\beta }^2[1-{\lambda }_m(1-\beta )]\parallel a_m-z_m\parallel +\parallel z_{m+1}-y^{*}\parallel \hfill \\ \hfill & ={\epsilon }_m+{\beta }^2[1-{\lambda }_m(1-\beta )]\parallel a_m-z_m\parallel +\parallel z_{m+1}-y^{*}\parallel .\hfill \end{array}$

Subsequently, we obtain

(42)$\parallel a_{m+1}-y^{*}\parallel \le {\epsilon }_m+[1-{\lambda }_m(1-\beta )]\parallel a_m-z_m\parallel +\parallel z_{m+1}-y^{*}\parallel .$

Since ${lim}_{m\to \infty }{\epsilon }_m=0$ and also ${lim}_{m\to \infty }\parallel a_m-z_m\parallel =0$ as $\left\{a_m\right\}$ is an equivalent sequence for $\left\{z_m\right\}$, additionally, ${lim}_{m\to \infty }\parallel z_m-y^{*}\parallel =0$ because $\left\{z_m\right\}$ is strongly convergent to $y^{*}$. Thus, from (42), we conclude that ${lim}_{m\to \infty }\parallel a_m-y^{*}\parallel =0$. Hence, $\left\{z_m\right\}$, the sequence of iterates of F-Green’s iteration method (25) is weak $w^2$-stable with respect to $U_G$. □

Theorem 4.

Let S, $P_G$, and $\left\{z_m\right\}$ be as defined in Theorem 2. Then, $\left\{z_m\right\}$ is weak $w^2$-stable with respect to $P_G$.

Proof. 

The proof of this theorem follows similar reasoning to that of Theorem 3. Hence, we omit its explicit demonstration. □

3.3. Numerical Computations

In this section, we assess the effectiveness of our proposed F-Green’s iteration approach alongside existing methods. We present tables and graphs to illustrate the accuracy and stability of our new method across different parameter settings. These results serve to validate our theoretical findings. We set the control parameters $\nu =0.5$ and ${\lambda }_m=0.9$, both falling within the range (0, 1), and provide the obtained results for $t=0.3$ and $0.6$ in Table 1, and Table 2, respectively. Figure 1 and Figure 2 depict the convergence behavior graphically.

Evidently, our F-Green’s method demonstrates superior convergence compared to the Picard–Green’s and Mann–Green’s methods for Troesch’s BVP. In each scenario, our method exhibits convergence towards the desired solution. We initiate the process with the initial guess $z_0$, chosen as the solution of the corresponding homogeneous equation $z^{″}=0$, subject to the specified boundary conditions of $z\left(0\right)=0$ and $z\left(1\right)=1$. Subsequently, we apply the Picard–Green’s, Mann–Green’s, and F-Green’s iteration methods.

The numerical results presented in Table 3 and Figure 3 are obtained with $\nu =0.9$ and ${\lambda }_m=0.9$ for various values of t in the interval $[0,1]$, considering the error $|z_{m+1}-z_m|$. Through numerical comparison, we observe that the F-Green’s iteration method demonstrates superior convergence, with minimum error compared to the Picard and Mann–Green’s iteration methods.

4. Conclusions and Future Work

This paper presents a novel F fixed-point iteration method based on Green’s function for solving the nonlinear Troesch problem in Banach space. The method effectively combines fixed-point theory and Green’s function techniques to handle the nonlinearity and boundary conditions of the Troesch problem. The theoretical analysis, supported by numerical experiments, showcased the efficacy and stability of our approach across various parameter settings. The numerical experiments validate the method’s accuracy and efficiency. Future work will explore the optimization of the relaxation parameter, extension to multidimensional problems, and applications to other nonlinear differential equations in Banach spaces. Further comparative studies will be conducted to benchmark the F-Green’s approach against alternative numerical methods, including finite difference, finite element, and spectral methods, across a broader range of problem domains. Addressing these areas of future work will contribute to advancing the understanding and applicability of the F-Green’s iteration method in solving diverse classes of nonlinear boundary value problems.

Disadvantages of the Proposed Iterative Method

The following points include the potential disadvantages of the proposed method:

• The method relies on the accurate construction of Green’s function, which can be mathematically complex and challenging, especially for more intricate or higher-dimensional problems.

• While the method is designed to converge under certain conditions, it may struggle with convergence in practice, particularly for problems with strong nonlinearity or steep gradients. Ensuring convergence requires the careful selection of the initial guesses and may necessitate additional strategies to enhance the stability and convergence rates.

• The convergence of the iterative method is often sensitive to the choice regarding the initial guess. A poor initial guess can lead to slow convergence or divergence.

Author Contributions

Conceptualization: F.A. and M.D.; methodology: F.A. and M.A.; investigation: M.A. and M.D.; writing—original draft preparation: F.A. and M.A.; writing—review and editing: F.A. and M.D.; supervision: D.F.; project administration: D.F.; funding acquisition: D.F. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article.

Acknowledgments

The first author acknowledges the Princess Nourah bint Abdulrahman University Researchers Supporting Project, project number PNURSP2024R174, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. We thank the anonymous referees and editor for their careful reading of this paper and valuable suggestions and comments that improved the quality of the manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Comparison of iterations for different methods (for t = 0.3).

Figure 2. Comparison of iterations for different methods (for t = 0.6).

Figure 3. Comparison of first iterates ([Forumla omitted. See PDF.]) for different methods with errors.

Table 1

Comparison of iterations for different methods towards the solution of the problem for $\nu =0.5$ and $t=0.3$.

Sr. No.	Iterations	Picard–Green	Mann–Green	F-Green
1	0	0.3000000	0.3000000	0.3000000
2	1	0.2884691	0.2896222	0.2887944
3	2	0.2888032	0.2888550	0.2887944
4	3	0.2887942	0.2887988	0.2887944
5	4	0.2887944	0.2887964	0.2887944
6	5	0.2887944	0.2887948	0.2887944
7	6	0.2887944	0.2887944	0.2887944
8	7	0.2887944	0.2887944	0.2887944
9	8	0.2887944	0.2887944	0.2887944
10	9	0.2887944	0.2887944	0.2887944

Table 2

Comparison of iterations for different methods towards the solution of the problem for $\nu =0.5$ and $t=0.6$.

Sr. No.	Iterations	Picard–Green	Mann–Green	F-Green
1	0	0.6000000	0.6000000	0.6000000
2	1	0.5837262	0.5853536	0.5841332
3	2	0.5841438	0.5842272	0.5841332
4	3	0.5841330	0.5841405	0.5841332
5	4	0.5841332	0.5841370	0.5841332
6	5	0.5841332	0.5841356	0.5841332
7	6	0.5841332	0.5841332	0.5841332
8	7	0.5841332	0.5841332	0.5841332
9	8	0.5841332	0.5841332	0.5841332
10	9	0.5841332	0.5841332	0.5841332

Table 3

Comparison of first iterates ($z_1$) for different methods with errors.

Sr. No.	t	Picard–Green, Err (${\mathit{z}}_1$)	Mann–Green, Err (${\mathit{z}}_1$)	F-Green, Err (${\mathit{z}}_1$)
1	0.10	0.0139	0.0125	0.0123
2	0.20	0.0270	0.0243	0.0241
3	0.30	0.0385	0.0347	0.0345
4	0.40	0.0475	0.0428	0.0425
5	0.50	0.0532	0.0479	0.0477
6	0.60	0.0548	0.0493	0.0490
7	0.70	0.0512	0.0461	0.0458
8	0.80	0.0415	0.0374	0.0373
9	0.90	0.0248	0.0223	0.0221
10	0.99	0.0029	0.0026	0.0025

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