A Green’s Function Based Iterative Approach for Solutions of BVPs in Symmetric Spaces

Abstract

We consider the Banach space $C[0,1]$, which is a symmetric Banach space, and prove the existence and approximation of numerical solutions for a broad class of third-order BVPs. Our approach is based on an integral operator that is constructed using Green’s function. The Banach contraction principle (BCP) is applied to guarantee a unique solution to our problem. Moreover, in order to find the value of the numerical solution, this new operator is embedded within the three-step Noor iterative scheme; we named this new iterative scheme the Noor–Green iterative scheme. We provide a convergence theorem for the proposed scheme by employing suitable restrictions on the parameters involved in the problem and in the scheme. The results of the stability of our scheme are also reported. It is worth mentioning that unlike the concept of stability in the classical sense, our result for stability is based on the concept of weak $w^2$ stability. In order to support our findings, we carried out various numerical experiments using different third-order BVPs. Finally, we report on the application of our iterative scheme to solve a class of fractional BVPs in the same symmetric Banach space. Our results are essentially new in the present literature and extend several of the results found in the current literature.

1. Introduction

Let $B=(B,||.|\left|\right)$ denote a Banach space and S a selfmap of B. In this case, we call S a Banach contraction of B if for any pair of elements, $a,b\in B$, it is possible to select a constant $\alpha$ in $[0,1)$ with

$$\left|\right|Sa-Sb\left|\right|\le \alpha \left|\right|a-b\left|\right|.$$

(1)

Any point, namely, $a^{*}\in B$ satisfies the condition $a^{*}=S{a}^{*}$ is called a fixed point of S. For the equation $a=Sa$, $a\in B$ is called a fixed-point problem. The selfmap S is called nonexpansive on B if (1) holds for $\alpha =1$. It is well-known that a sought solution of a linear or a nonlinear problem can be expressed as a fixed point of a certain self map in a Banach space. A distance function, d, on space S is said to be symmetric if $d(a,b)=d(b,a)$ for all $a,b\in S$. A space with a symmetric distance function is called a symmetric space. It is well-known that Banach spaces and metric spaces are symmetric, but pseudo-metric spaces are not symmetric [1]. In 1922, Banach [2] suggested the existence and uniqueness of fixed points for contractions in Banach spaces and obtained an application for a class of integral equations. Banach [2] also proved that the unique fixed point of a contraction map can be approximated by using the one-step Picard [3] approximation scheme. However, when S is nonexpansive, then the S has a fixed point under certain restrictions (see, e.g., Browder [4] and Gohde [5]), but the Picard approximations may not converge to a fixed point. Therefore, in 1953, Mann [6] suggested a new two-step iterative scheme. After this, Ishikawa [7] provided for the first time a two-step iterative scheme for a certain class of nonlinear mappings. On the other hand, the speed of convergence of one-step and two-step iterative schemes is often very slow and unstable. Thus, in 2000, Noor [8] constructed a very general three-step iterative scheme that contains Picard, Mann, and Ishikawa iterative schemes as special cases. The Noor iterative scheme reads as follows:

$$\left\{\begin{array}{c}{a}_0\in B,\hfill \\ c_k=(1-{\gamma }_k)a_k+{\gamma }_{k}S{a}_k,\hfill \\ b_k=(1-{\beta }_k)a_k+{\beta }_{k}S{c}_k,\hfill \\ a_{k+1}=(1-{\alpha }_k)a_k+{\alpha }_{k}S{b}_k,k\in \mathbb{N}\cup \left\{0\right\},\hfill \end{array}\right.$$

(2)

where ${\alpha }_k,{\beta }_k,{\gamma }_k\in (0,1)$. When ${\alpha }_k=1$ and ${\beta }_k={\gamma }_k=0$, the Noor iterative scheme (2) reduces to a Picard iterative scheme [3]. When ${\beta }_k={\gamma }_k=0$, the Noor iterative scheme (2) reduces to a Mann iterative scheme [6]. When ${\gamma }_k=0$, the Noor iterative scheme (2) reduces to an Ishikawa iterative scheme [7]. The Noor iterative scheme is extensively used for finding fixed-point problems, but it is not studied for finding the sought solutions of boundary value problems. The purpose of this paper is to modify the Noor iterative scheme to be used for finding the numerical solutions of a broad class of third-order boundary value problems.

Boundary value problems (BVPs) play a very basic role in modeling different real-world phenomena; however, calculating the exact solutions for these BVPs using analytical approaches is not always possible (see, e.g., [9,10], and others). If an exact solution is not possible for a certain class of BVPs, one always requests a numerical solution up to desirable decimals of accuracy. Often, the classes of third-order BVPs are important and arise naturally in various fields of analysis and engineering. More specifically, the application of third-order BPVs can be found in rocket motion in space, the flow of thin films, heat problems, studying flow under the effect of gravity, flow networking in the field of biology, and so on. More applications of third-order BVPs can be found in the papers of Awoyemi [11], Awoyemi and Idowu [12], and others). On the other hand, Lu and Cui [13] studied the existence of solutions for the following class of third-order BVPs:

$$a^{‴}\left(t\right)-f(t,a\left(t\right),a^{\prime }\left(t\right),a^{″}\left(t\right))=0,$$

(3)

subjected to the boundary conditions (BCs):

$$a\left(0\right)=0,a^{\prime }\left(0\right)=0,a^{\prime }\left(1\right)=0,$$

(4)

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

For more details on the existence of solutions to third-order BVPs, we refer the reader to [14,15,16,17] and others. In the recent literature, various numerical procedures are carried out by authors for approximating the numerical solutions to third-order BVPs (see, e.g., Awoyemi [11], Awoyemi and Idowu [12], Bhrawy and Abd-Elhameed [18], and others).

On the other hand, using the technique of embedding Green’s function within fixed-point iterations is studied in [19]. Abushammala et al. [20] suggested a Mann-Green scheme by embedding a Green function into the Mann iterative scheme [6] to obtain numerical solutions to a broad class of third-order BVPs and proved that this new approach is highly accurate, corresponding to the classical approaches mentioned above. Similarly, Khuri and Louhichi [21] introduced an Ishikawa-Green iterative approach by embedding a Green function into the Ishikawa [7] iterative scheme for numerical solutions to third-order BVPs and improved the main results of Abushammala et al. [20]. In this paper, we introduce a Noor–Green iterative approach for numerical solutions to third-order BVPs and compare this scheme to two other iterative schemes. We also prove that our new approach is stable and suggest highly accurate numerical solutions for almost all the values of the parameters. In this way, we improve many of the classical results in the literature.

2. Green’s Function for Third-Order BVPs

In this paper, we consider the following third-order BVP:

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

(5)

with the BCs:

$$\left\{\begin{array}{c}{B}_{\theta }\left[a\right]={\alpha }_{1}a\left(\theta \right)+{\alpha }_2{a}^{\prime }\left(\theta \right)+{\alpha }_3{a}^{″}\left(\theta \right)=\alpha ,\hfill \\ B_{\eta }\left[a\right]={\beta }_{1}a\left(\eta \right)+{\beta }_2{a}^{\prime }\left(\eta \right)+{\beta }_3{a}^{″}\left(\eta \right)=\beta ,\hfill \\ B_{\mu }\left[a\right]={\gamma }_{1}a\left(\mu \right)+{\gamma }_2{a}^{\prime }\left(\mu \right)+{\gamma }_3{a}^{″}\left(\mu \right)=\gamma .\hfill \end{array}\right.$$

(6)

Here, $0\le t\le 1$, L is linear, whereas f is nonlinear; $\alpha ,\beta ,\gamma$ are real constants, and either $\mu =\theta$ or $\mu =\eta$. The existence of a solution to problems (5) and (6) is established under various assumptions (see, e.g., [22,23,24] and others). Following [21,24], we approach the existence and approximation of the solution for this problem under the Green functions approach as follows.

The Green’s function $G(t,s)$ for equation $L\left[a\right]=0$ takes the following form:

$$G(t,s)=\left\{\begin{array}{c}{\xi }_1{a}_1+{\lambda }_1{a}_2+{\psi }_1{a}_3\mathrm{when}a\le t<s\hfill \\ {\xi }_2{a}_1+{\lambda }_2{a}_2+{\psi }_2{a}_3\mathrm{when}s\le t<b,\hfill \end{array}\right.$$

where ${\xi }_i,{\lambda }_i,{\psi }_i,(i=1,2)$ are constant and are to be found using the properties of Green’s function listed below, and $\{a_1,a_2,a_3\}$ is the set of a linearly independent set of solutions for $L\left[a\right]=0$.

Green’s function properties: We now list the following properties of the Green functions.

($p_1$)

Green’s function G satisfies the following BCs:

$$B_{\theta }\left[G(t,s)\right]=B_{\eta }\left[G(t,s)\right]=B_{\mu }\left[G(t,s)\right]=0.$$

(7)

($p_2$)

At $t=s$, the function G is always continuous, that is,

$${\xi }_1{a}_1\left(s\right)+{\lambda }_1{a}_2\left(s\right)+{\psi }_1{a}_3\left(s\right)={\xi }_2{a}_1\left(s\right)+{\lambda }_2{a}_2\left(s\right)+{\psi }_2{a}_3\left(s\right).$$

(8)

($p_3$)

At $t=s$, the function $G^{\prime }$ is also always continuous, that is,

$${\xi }_1{a}_1^{\prime }\left(s\right)+{\lambda }_1{a}_2^{\prime }\left(s\right)+{\psi }_1{a}_3^{\prime }\left(s\right)={\xi }_2{a}_1^{\prime }\left(s\right)+{\lambda }_2{a}_2^{\prime }\left(s\right)+{\psi }_2{a}_3^{\prime }\left(s\right).$$

(9)

($p_4$)

At $t=s$, the function $G^{″}$ admits a discontinuous jump, that is,

$${\xi }_1{a}_1^{″}\left(s\right)+{\lambda }_1{a}_2^{″}\left(s\right)+{\psi }_1{a}_3^{″}\left(s\right)+\frac{1}{p\left(s\right)}={\xi }_2{a}_1^{″}\left(s\right)+{\lambda }_2{a}_2^{″}\left(s\right)+{\psi }_2{a}_3^{″}\left(s\right).$$

(10)

By using (7)–(10), we can find the values of the constants involved in Green’s functions, and any particular solution $a_p$ to problems (5) and (6) can be written as

$$a_p={\int }_a^{b}G(t,s)f(s,a_p,a_p^{\prime },a_p^{″})ds.$$

(11)

3. Noor–Green Iterative Scheme

In this section, we propose the so-called Noor–Green iterative scheme for third-order BVPs. We first consider the nonlinear differential function as follows:

$$L\left[a\right]+N\left[a\right]=f(t,a,a^{\prime },a^{″}).$$

(12)

Here, L stands for the linear term, whereas N stands for the nonlinear term, and the function f may be linear or nonlinear. Now, if $a_p$ is the particular solution for (12), then we denote this here simply as a. In this case, we assume that $G(t,s)$ is a Green function of the linear term L. Eventually, we set

$$W\left[a\right]={\int }_a^{b}G(t,s)L\left(a\right)ds.$$

(13)

Accordingly, by using (11)–(13), we have

$$\begin{array}{ccc}\hfill W\left(a\right)& =& {\int }_a^{b}G(t,s)[L\left(a\right)+N\left(a\right)-f(s,a,a^{\prime },a^{″})-N\left(a\right)+f(s,a,a^{\prime },a^{″})]ds\hfill \\ & =& {\int }_a^{b}G(t,s)[L\left(a\right)+N\left(a\right)-f(s,a,a^{\prime },a^{″})]ds+{\int }_a^{b}G(t,s)[f(s,a,a^{\prime },a^{″})-N\left(a\right)]ds\hfill \\ & =& a+{\int }_a^{b}G(t,s)[L\left(a\right)+N\left(a\right)-f(s,a,a^{\prime },a^{″})]ds.\hfill \end{array}$$

Consequently, we obtain

$$W\left(a\right)=a+{\int }_a^{b}G(t,s)[L\left(a\right)+N\left(a\right)-f(s,a,a^{\prime },a^{″})]ds.$$

(14)

Hence, when using the operator W given in (14) in the Noor iteration (2), we get

$$\left\{\begin{array}{c}{c}_k=(1-{\gamma }_k)a_k+{\gamma }_{k}W{a}_k,\hfill \\ b_k=(1-{\beta }_k)a_k+{\beta }_{k}W{c}_k,\hfill \\ a_{k+1}=(1-{\alpha }_k)a_k+{\alpha }_{k}W{b}_k,k\in \mathbb{N}\cup \left\{0\right\}.\hfill \end{array}\right.$$

(15)

It follows that

$$\left\{\begin{array}{c}{c}_k=a_k+{\gamma }_k{\int }_a^{b}G(t,s)[L\left(a_k\right)+N\left(a_k\right)-f(s,a_k,a_k^{\prime },a_k^{″})]ds,\hfill \\ b_k=(1-{\beta }_k)a_k+{\beta }_k[c_k+{\int }_a^{b}G(t,s)[L\left(c_k\right)+N\left(c_k\right)-f(s,c_k,c_k^{\prime },c_k^{″})]ds],\hfill \\ a_{k+1}=(1-{\alpha }_k)a_k+{\alpha }_k[b_k+{\int }_a^{b}G(t,s)[L\left(b_k\right)+N\left(b_k\right)-f(s,b_k,b_k^{\prime },b_k^{″})]ds],k\in \mathbb{N}\cup \left\{0\right\}.\hfill \end{array}\right.$$

(16)

The formula in (16) is the general Noor iterative scheme. For the initial guess in this case, we must choose a function that satisfies $L\left(a\right)=0$ and the given BCs. In order to prove its convergence, we will impose a certain restriction on the sequence $\left\{{\alpha }_k\right\}$.

4. Convergence Result

Here, we give the convergence and stability results of our Noor–Green iterative scheme. We also suggest some examples and numerically prove the convergence of the scheme. We also compare the numerical results with the Mann-Green and Ishikawa-Green iterative schemes. It has been shown that our novel Noor–Green iterative approach provides highly accurate results when compared to the corresponding Mann-Green and Ishikawa-Green iterative schemes.

In order to do this, we consider the following BVP:

$$a^{‴}\left(t\right)=f(t,a\left(t\right),a^{\prime }\left(t\right),a^{″}\left(t\right)),$$

(17)

with the BCs:

$$a\left(0\right)=0=a^{\prime }\left(0\right)=a\left(1\right)=0.$$

(18)

In this case, Green’s function becomes:

$$G(t,s)=\left\{\begin{array}{c}(\frac{-s^2}{2}+s-\frac{1}{2})t^2\mathrm{when}0<t<s\hfill \\ \frac{s^2}{2}-st+(\frac{-s^2}{2}+s)t^2\mathrm{when}s<t<1.\hfill \end{array}\right.$$

The adjoint value of Green’s function above is denoted by $G^{*}(t,s)$ and reads as follows:

$$G^{*}(t,s)=-\left\{\begin{array}{c}\frac{s^2}{2}-st+(\frac{-s^2}{2}+s)t^2\mathrm{when}0<s<t\hfill \\ (\frac{-s^2}{2}+s-\frac{1}{2})t^2\mathrm{when}t<s<1.\hfill \end{array}\right.$$

When applying the Noor–Green iterative scheme, we get

$$\left\{\begin{array}{c}{c}_k=a_k+{\gamma }_k{\int }_a^b{G}^{*}(t,s)[L\left(a_k\right)+N\left(a_k\right)-f(s,a_k,a_k^{\prime },a_k^{″})]ds,\hfill \\ b_k=(1-{\beta }_k)a_k+{\beta }_k[c_k+{\int }_a^b{G}^{*}(t,s)[L\left(c_k\right)+N\left(c_k\right)-f(s,c_k,c_k^{\prime },c_k^{″})]ds],\hfill \\ a_{k+1}=(1-{\alpha }_k)a_k+{\alpha }_k[b_k+{\int }_a^b{G}^{*}(t,s)[L\left(b_k\right)+N\left(b_k\right)-f(s,b_k,b_k^{\prime },b_k^{″})]ds],k\in \mathbb{N}\cup \left\{0\right\}.\hfill \end{array}\right.$$

(19)

Now, set the self map $S_{G^{*}}$ in the Banach space $C[0,1]$ as follows:

$$S_{G^{*}}a=a+{\int }_0^1{G}^{*}(t,s)[a^{‴}\left(s\right)-f(s,a\left(s\right),a^{\prime }\left(s\right),a^{″}\left(s\right))]ds.$$

(20)

Hence, (19) becomes the following:

$$\left\{\begin{array}{c}{a}_0\in B,\hfill \\ c_k=(1-{\gamma }_k)a_k+{\gamma }_k{S}_{G^{*}}{a}_k,\hfill \\ b_k=(1-{\beta }_k)a_k+{\beta }_k{S}_{G^{*}}{c}_k,\hfill \\ a_{k+1}=(1-{\alpha }_k)a_k+{\alpha }_k{S}_{G^{*}}{b}_k,k\in \mathbb{N}\cup \left\{0\right\}.\hfill \end{array}\right.$$

(21)

This scheme (21) is the general Noor iterative scheme, which is based on Green’s function. From now on, we will call the modified Noor iteration (21) the Noor–Green iterative scheme. Here, we give our main convergence result. This theorem also provides a criterion for the existence of a unique solution.

Theorem 1.

Suppose the operator $S_{G^{*}}$ is as is given in (20) such that $\frac{\sqrt{3}}{60}{sup}_{[0,1]\times \mathbb{R}\times \mathbb{R}}\left|\frac{\partial f}{\partial a}\right|<1$ and $\left\{a_k\right\}$ are sequences from a Noor–Green iteration (21). In addition, if $\sum {\alpha }_k=\infty$, then $\left\{a_k\right\}$ converges to the unique solution to problems (17) and (18).

Proof. 

In order to complete the proof, we put $\frac{\sqrt{3}}{60}{sup}_{[0,1]\times \mathbb{R}\times \mathbb{R}}\left|\frac{\partial f}{\partial a}\right|=\varrho$, and it follows that $S_{G^{*}}$ is a $\varrho$-contraction. In the view of Banach [2], there exists a unique fixed point for $S_{G^{*}}$, namely, $a^{*}$, which is the unique solution to problems (17) and (18).

We want to prove that the Noor–Green iterative scheme converges to $a^{*}$. For this,

$$\begin{array}{ccc}\hfill \left|\right|c_k-a^{*}\left|\right|& =& \left|\right|(1-{\gamma }_k)a_k+{\gamma }_k{S}_{G^{*}}{a}_k-a^{*}\left|\right|\hfill \\ & =& \left|\right|(1-{\gamma }_k)(a_k-a^{*})+{\gamma }_k(S_{G^{*}}{a}_k-a^{*})\left|\right|\hfill \\ & \le & (1-{\gamma }_k)\left|\right|a_k-a^{*}\left|\right|+{\gamma }_k\left|\right|S_{G^{*}}{a}_k-a^{*}\left|\right|\hfill \\ & \le & (1-{\gamma }_k)\left|\right|a_k-a^{*}\left|\right|+{\gamma }_k\varrho \left|\right|a_k-a^{*}\left|\right|\hfill \\ & =& [1-{\gamma }_k(1-\varrho )]\left|\right|a_k-a^{*}\left|\right|.\hfill \end{array}$$

Moreover,

$$\begin{array}{ccc}\hfill \left|\right|b_k-a^{*}\left|\right|& =& \left|\right|(1-{\beta }_k)a_k+{\beta }_k{S}_{G^{*}}{c}_k-a^{*}\left|\right|\hfill \\ & =& \left|\right|(1-{\beta }_k)(a_k-a^{*})+{\beta }_k(S_{G^{*}}{c}_k-a^{*})\left|\right|\hfill \\ & \le & (1-{\beta }_k)\left|\right|a_k-a^{*}\left|\right|+{\beta }_k\left|\right|S_{G^{*}}{c}_k-a^{*}\left|\right|\hfill \\ & \le & (1-{\beta }_k)\left|\right|a_k-a^{*}\left|\right|+{\beta }_k\varrho \left|\right|c_k-a^{*}\left|\right|\hfill \\ & \le & (1-{\beta }_k)\left|\right|a_k-a^{*}\left|\right|+{\beta }_k\varrho ([1-{\gamma }_k(1-\varrho )]\left|\right|a_k-a^{*}\left|\right|)\hfill \\ & =& \left[(1-(1-\varrho ){\beta }_k(1+\varrho {\gamma }_k)]\right||a_k-a^{*}|\left|\right]\hfill \\ & \le & [1-{\beta }_k{(1-\varrho )}^2]\left|\right|a_k-a^{*}\left|\right|.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill \left|\right|a_{k+1}-a^{*}\left|\right|& =& \left|\right|(1-{\alpha }_k)a_k+{\alpha }_k{S}_{G^{*}}{b}_k-a^{*}\left|\right|\hfill \\ & =& \left|\right|(1-{\alpha }_k)(a_k-a^{*})+{\alpha }_k(S_{G^{*}}{b}_k-a^{*})\left|\right|\hfill \\ & \le & (1-{\alpha }_k)\left|\right|a_k-a^{*}\left|\right|+{\alpha }_k\left|\right|S_{G^{*}}{b}_k-a^{*}\left|\right|\hfill \\ & \le & (1-{\alpha }_k)\left|\right|a_k-a^{*}\left|\right|+{\alpha }_k\varrho \left|\right|b_k-a^{*}\left|\right|\hfill \\ & \le & (1-{\alpha }_k)\left|\right|a_k-a^{*}\left|\right|+{\alpha }_k\varrho ([1-{\beta }_k{(1-\varrho )}^2]\left|\right|a_k-a^{*}\left|\right|)\hfill \\ & \le & [1-{\alpha }_k{(1-\varrho )}^2]\left|\right|a_k-a^{*}\left|\right|.\hfill \end{array}$$

Accordingly, we obtain

$$\begin{array}{ccc}\hfill \left|\right|a_{k+1}-a^{*}\left|\right|& \le & [1-{\alpha }_k{(1-\varrho )}^2]\left|\right|a_k-a^{*}\left|\right|\hfill \\ & \le & [1-{\alpha }_k{(1-\varrho )}^2][1-{\alpha }_{k-1}{(1-\varrho )}^2]\left|\right|a_{k-1}-a^{*}\left|\right|\hfill \\ & \le & [1-{\alpha }_k{(1-\varrho )}^2][1-{\alpha }_{k-1}{(1-\varrho )}^2][1-{\alpha }_{k-2}{(1-\varrho )}^2]\left|\right|a_{k-2}-a^{*}\left|\right|.\hfill \end{array}$$

Inductively, we obtain

$$\left|\right|a_{k+1}-a^{*}\left|\right|\le {\Pi }_{m=0}^k[1-{\alpha }_m{(1-\varrho )}^2]\left|\right|a_0-a^{*}\left|\right|.$$

(22)

Finally, we know that formula $1-a\le e^{-a}$ holds for any choice of $a\in [0,1]$. Hence, when using this fact with (22), one obtains

$$\left|\right|a_{k+1}-a^{*}\left|\right|\le e^{-{(1-\varrho )}^2{\sum }_{m=0}^k{\alpha }_m\left|\right|a_0-a^{*}\left|\right|}.$$

(23)

If it is supposed that ${\sum }_{k=0}^{\infty }=\infty$ and $\varrho$ lie in $(0,1)$, we infer from (23) that

$$\underset{k\to \infty }{lim}\left|\right|a_{k+1}-a^{*}\left|\right|=0.$$

Accordingly, $\left\{a_k\right\}$ converges to the unique fixed point $a^{*}$ of $S_{G^{*}}$ which is the unique solution to problems (17) and (18). □

5. Weak ${\mathit{w}}^{\mathbf{2}}$-Stability

The concept of stability in terms of iterative schemes pertains to their capacity to consistently and dependably yield results when applied iteratively to a problem. This notion is explored in detail by Osilike [25], along with related discussions in references like [26,27]. A well-behaved iteration, characterized as stable, should not exhibit undue sensitivity to minor alterations in the input data or initial conditions. Put differently, it should generate solutions that steadily converge towards a consistent outcome as the number of iterations increases, as opposed to veering off course or displaying erratic oscillations. Stability holds significant importance in various computational procedures, including the resolution of differential equations, optimization challenges, or numerical simulations. It plays a pivotal role in guaranteeing the precision and predictability of outcomes, thereby ensuring the successful and reliable execution of computational tasks. This section intends to introduce a theorem concerning the stability of the iterative method being examined. The idea of stability in fixed-point iterations finds its origins in the pioneering research conducted by Urabe [28], which established the initial groundwork. Expanding upon Urabe’s contributions, Harder and Hicks [29] formulated a precise mathematical definition for stability. In order to fully grasp the ensuing discussion in this study, it is crucial to revisit certain fundamental concepts, which will be succinctly summarized below.

Definition 1

([29]). Assume that S is a self map in a Banach space, B, and $\left\{a_k\right\}\subseteq B$ is a sequence of iterations produced by the following iterative formula:

$$\left\{\begin{array}{c}{a}_0\in B,\hfill \\ a_{k+1}=\Lambda (S,a_k),\hfill \end{array}\right.$$

(24)

Here, Λ is a function that is connected to the mapping S and the iterative sequence $\left\{a_k\right\}$. In this case, when the sequence $\left\{a_k\right\}$ converges in a strong sense to a fixed point, namely, $a^{*}$ of S, then $\left\{a_k\right\}$ is regarded as a stable sequence if

$$\underset{k\to \infty }{lim}\left|\right|{\widehat{a}}_{k+1}-\Lambda (S,{\widehat{a}}_k)\left|\right|=0\mathit{implies}\underset{k\to \infty }{lim}{\widehat{a}}_k=a^{*},$$

where $\left\{{\widehat{a}}_k\right\}$ is any given arbitrary sequence that is obtained from some of the elements of B.

We now consider the definition of equivalent sequences in the setting of Banach spaces.

Definition 2

([30]). The iterative sequences $\left\{a_k\right\}$ and $\left\{{\widehat{a}}_k\right\}$ are said to be equivalent to each other if and only if ${lim}_{k\to \infty }\left|\right|a_k-{\widehat{a}}_k\left|\right|=0.$

Differing from the idea of arbitrary sequences, Timis [31] introduced the notion of equivalent sequences as a basis for establishing a fresh mathematical definition of stability. This novel type of stability is denoted as “$w^2$- stability," and its precise definition is outlined below.

Definition 3

([31]). For an iterative sequence, namely $\left\{a_k\right\}$, which is iteratively produced from (24), if it is convergent to a fixed point, $a^{*}$, of a selfmap, S, in a Banach space, B, it is said to be weak $w^2$-stable if for any equivalent sequence: $\left\{{\widehat{a}}_k\right\}\subseteq B$ of $\left\{a_k\right\}$, one obtains

$$\underset{k\to \infty }{lim}\left|\right|{\widehat{a}}_{k+1}-\Lambda (S,{\widehat{a}}_k)\left|\right|=0\mathit{implies}\underset{k\to \infty }{lim}{\widehat{a}}_k=a^{*}.$$

Now, we are in the position to prove that our new Noor–Green iterative scheme (21) is weak $w^2$ stable.

Theorem 2.

Suppose that B, $S_{G^{*}}$, and $\left\{a_k\right\}$ are defined the same as is given in Theorem 1. In this case, the convergence of the sequence of the Noor–Green iterative scheme $\left\{a_k\right\}$ is weak $w^2$–stable with respect to $S_{G^{*}}$.

Proof. 

Let us take an equivalent sequence: $\left\{{\widehat{a}}_k\right\}$ of $\left\{a_k\right\}$, so we have ${lim}_{k\to \infty }\left|\right|{\widehat{a}}_k-a_k\left|\right|=0$. Now, we assume that

$${ϵ}_k=\left|\right|{\widehat{a}}_{k+1}-[(1-{\alpha }_k){\widehat{a}}_k+S_{G^{*}}{\widehat{b}}_k]\left|\right|,$$

where ${\widehat{b}}_k=(1-{\beta }_k){\widehat{a}}_k+S_{G^{*}}{\widehat{c}}_k$ and ${\widehat{c}}_k=(1-{\alpha }_k){\widehat{a}}_k+{\alpha }_k{S}_{G^{*}}{\widehat{a}}_k$.

We first find $\left|\right|{\widehat{c}}_k-c_k\left|\right|$. For this,

$$\begin{array}{ccc}\hfill \left|\right|{\widehat{c}}_k-c_k\left|\right|& =& \left|\right|[(1-{\alpha }_k){\widehat{a}}_k+{\alpha }_k{S}_{G^{*}}{\widehat{a}}_k]-[(1-{\alpha }_k)a_k+{\alpha }_k{S}_{G^{*}}{a}_k]\left|\right|\hfill \\ & =& \left|\right|[(1-{\alpha }_k)({\widehat{a}}_k-a_k)+{\alpha }_k(S_{G^{*}}{\widehat{a}}_k-S_{G^{*}}{a}_k]||\hfill \\ & \le & (1-{\alpha }_k)\left|\right|{\widehat{a}}_k-a_k\left|\right|+{\alpha }_k\left|\right|S_{G^{*}}{\widehat{a}}_k-S_{G^{*}}{a}_k\left|\right|\hfill \\ & \le & (1-{\alpha }_k)\left|\right|{\widehat{a}}_k-a_k\left|\right|+{\alpha }_k\varrho \left|\right|{\widehat{a}}_k-a_k\left|\right|\hfill \\ & \le & [1-{\alpha }_k(1-\varrho )]\left|\right|{\widehat{a}}_k-a_k\left|\right|.\hfill \end{array}$$

Hence,

$$\left|\right|{\widehat{c}}_k-c_k\left|\right|\le [1-{\alpha }_k(1-\varrho )]\left|\right|{\widehat{a}}_k-a_k\left|\right|.$$

(25)

Using (25), we get

$$\left|\right|{\widehat{b}}_k-b_k\left|\right|\le [1-{\beta }_k{(1-\varrho )}^2]\left|\right|{\widehat{a}}_k-a_k\left|\right|,$$

(26)

By keeping (26) in mind, we can proceed as follows:

$$\begin{array}{ccc}\hfill \left|\right|{\widehat{a}}_{k+1}-a^{*}\left|\right|& \le & \left|\right|{\widehat{a}}_{k+1}-a_{k+1}\left|\right|+\left|\right|a_{k+1}-a^{*}\left|\right|\hfill \\ & \le & \left|\right|{\widehat{a}}_{k+1}-[(1-{\alpha }_k){\widehat{a}}_k+S_{G^{*}}{\widehat{b}}_k]\left|\right|\hfill \\ & & +\left|\right|[(1-{\alpha }_k){\widehat{a}}_k+S_{G^{*}}{\widehat{b}}_k]-a_{k+1}\left|\right|+\left|\right|a_{k+1}-a^{*}\left|\right|\hfill \\ & =& {ϵ}_k+\left|\right|[(1-{\alpha }_k){\widehat{a}}_k+S_{G^{*}}{\widehat{b}}_k]-a_{k+1}\left|\right|+\left|\right|a_{k+1}-a^{*}\left|\right|\hfill \\ & \le & {ϵ}_k+(1-{\alpha }_k)\left|\right|{\widehat{a}}_k-a_k\left|\right|+{\alpha }_k\left|\right|S_{G^{*}}{\widehat{b}}_k-S_{G^{*}}{b}_k\left|\right|+\left|\right|a_{k+1}-a^{*}\left|\right|\hfill \\ & \le & {ϵ}_k+(1-{\alpha }_k)\left|\right|{\widehat{a}}_k-a_k\left|\right|+{\alpha }_k\varrho \left|\right|{\widehat{b}}_k-b_k\left|\right|+\left|\right|a_{k+1}-a^{*}\left|\right|\hfill \\ & \le & {ϵ}_k+(1-{\alpha }_k)\left|\right|{\widehat{a}}_k-a_k\left|\right|+{\alpha }_k\varrho [1-{\beta }_k{(1-\varrho )}^2]\left|\right|{\widehat{a}}_k-a_k\left|\right|+\left|\right|a_{k+1}-a^{*}\left|\right|\hfill \\ & \le & {ϵ}_k+[1-{\alpha }_k{(1-\varrho )}^2]\left|\right|{\widehat{a}}_k-a_k\left|\right|+\left|\right|a_{k+1}-a^{*}\left|\right|.\hfill \end{array}$$

Subsequently, we obtain

$$\left|\right|{\widehat{a}}_{k+1}-a^{*}\left|\right|\le {ϵ}_k+[1-{\alpha }_k{(1-\varrho )}^2]\left|\right|{\widehat{a}}_k-a_k\left|\right|+\left|\right|a_{k+1}-a^{*}\left|\right|.$$

(27)

Now, as was assumed, ${lim}_{k\to \infty }{ϵ}_k=0$ and ${lim}_{k\to \infty }\left|\right|{\widehat{a}}_k-a_k\left|\right|=0$. Moreover, ${lim}_{k\to \infty }\left|\right|a_{k+1}-a^{*}\left|\right|=0$ due to the convergence of $\left\{a_k\right\}$ towards $a^{*}$. Subsequently, from (27), ${lim}_{k\to \infty }\left|\right|{\widehat{a}}_k-a^{*}\left|\right|=0$. It follows that $\left\{a_k\right\}$ produced by (21) is weak $w^2$-stable with respect to the mapping $S_{G^{*}}$. □

Since every stable iterative scheme is weak $w^2$-stable but the converse is not true in general. So, the following corollary is a consequence of our Theorem 2.

Corollary 1.

Suppose that B, $S_{G^{*}}$, and $\left\{a_k\right\}$ are defined the same as is given in Theorem 1. In this case, the convergence of the sequence of the Noor–Green iterative scheme $\left\{a_k\right\}$ is stable with respect to $S_{G^{*}}$.

6. Numerical Examples

The main purpose of this section is to suggest some examples of third-order BVPs. We then prove the convergence of our novel Noor–Green iterative scheme along with some other iterative schemes. We show the high accuracy of our novel scheme under various sets of parameters.

First, we consider the following BVP.

Example 1.

Assume the following third-order BVP,

$$a^{‴}\left(t\right)+a\left(t\right)a^{″}\left(t\right)-{\left(a^{\prime }\left(t\right)\right)}^2+1=0,$$

(28)

with the BCs

$$a\left(0\right)=0=a\left(1\right)=a^{\prime }\left(0\right).$$

(29)

Notice that the exact solution to problems (28) and (29) is unknown. In this case, our novel Noor–Green iterative scheme becomes the following:

$$\left\{\begin{array}{c}{c}_k=a_k-{\gamma }_k{\int }_0^t\left[(\frac{s^2}{2}-st+(\frac{-s^2}{2}+s)t^2)(a_k^{‴}\left(s\right)+a_k\left(s\right)a_k^{″}\left(s\right)-{\left(a_k^{\prime }\left(s\right)\right)}^2+1)\right]ds\hfill \\ -{\gamma }_k{\int }_t^1\left[(\frac{-s^2}{2}+s-\frac{1}{2})t^2\right)(a_k^{‴}\left(s\right)+a_k\left(s\right)a_k^{″}\left(s\right)-{\left(a_k^{\prime }\left(s\right)\right)}^2+1)]ds,\hfill \\ b_k=(1-{\beta }_k)a_k-{\beta }_k{\int }_0^t\left[(\frac{s^2}{2}-st+(\frac{-s^2}{2}+s)t^2)(c_k^{‴}\left(s\right)+c_k\left(s\right)c_k^{″}\left(s\right)-{\left(c_k^{\prime }\left(s\right)\right)}^2+1)\right]ds\hfill \\ -{\beta }_k{\int }_t^1\left[(\frac{-s^2}{2}+s-\frac{1}{2})t^2\right)(c_k^{‴}\left(s\right)+c_k\left(s\right)c_k^{″}\left(s\right)-{\left(c_k^{\prime }\left(s\right)\right)}^2+1)]ds,\hfill \\ a_{k+1}=(1-{\alpha }_k)a_k-{\alpha }_k{\int }_0^t\left[(\frac{s^2}{2}-st+(\frac{-s^2}{2}+s)t^2)(b_k^{‴}\left(s\right)+b_k\left(s\right)b_k^{″}\left(s\right)-{\left(b_k^{\prime }\left(s\right)\right)}^2+1)\right]ds\hfill \\ -{\alpha }_k{\int }_t^1\left[(\frac{-s^2}{2}+s-\frac{1}{2})t^2\right){(b_k^{‴}\left(s\right)+b_k\left(s\right)b_k^{″}\left(s\right)-b_k^{\prime }\left(s\right))}^2+1\left)\right]ds,\hfill \\ k\in \mathbb{N}\cup \left\{0\right\}.\hfill \end{array}\right.$$

(30)

Now, we choose ${\alpha }_k={\beta }_k={\gamma }_k=0.85$. The plot of Green’s function involved in our scheme (30) is given in Figure 1. The numerical results in this case are given in

Table 1. Convergence of Noor–Green iteration with the numerical solution.

k	Noor–Green, t = 0.2	Noor–Green, t = 0.3	Noor–Green, t = 0.4	Noor–Green, t = 0.5
0	0.0000000	0.0000000	0.0000000	0.0000000
1	0.00452379	0.0089042	0.0135651	0.0176585
2	0.00519929	0.0102332	0.0155887	0.0202913
3	0.00530013	0.0104314	0.0158904	0.0206836
4	0.00531518	0.0104610	0.0159354	0.0207421
5	0.00531742	0.0104654	0.0159421	0.0207508
6	0.00531776	0.0104661	0.0159431	0.0207521
7	0.00531781	0.0104662	0.0159433	0.0207523
8	0.00531781	0.0104662	0.0159433	0.0207523

and

Table 2. Convergence of Noor–Green iteration with the numerical solution.

k	Noor–Green, t = 0.6	Noor–Green, t = 0.7	Noor–Green, t = 0.8	Noor–Green, t = 0.9
0	0.0000000	0.0000000	0.0000000	0.0000000
1	0.0203371	0.0207547	0.0180661	0.0114280
2	0.0233676	0.0238454	0.0207545	0.0133797
3	0.0238189	0.0243054	0.0211543	0.0134172
4	0.0238861	0.0243739	0.0212138	0.0134227
5	0.0238961	0.0243841	0.0212226	0.0134236
6	0.0238976	0.0243856	0.0212239	0.0134237
7	0.0238978	0.0243858	0.0212241	0.0134237
8	0.0238978	0.0243858	0.0212241	0.0134237

.

Suppose Error $=|a_k-a_{k+1}|$. Now, we choose ${\alpha }_k={\beta }_k={\gamma }_k=\frac{1}{k+1}$ and obtain the behavior of the convergence of the Noor–Green iterative scheme, as shown in Figure 2.

Now, set $RelativeError=\left|\frac{a_k-a_{k+1}}{a_{k+1}}\right|$. We compared our scheme to other schemes. We choose ${\alpha }_k={\beta }_k=\gamma =1$. For only two iterations, the relative error in

Table 3. Error comparison between different iterations.

Values of t	Mann-Green	Ishikawa-Green	Noor–Green
0.1	0.0000197656	$1.14526\times {10}^{-9}$	$6.66724\times {10}^{-14}$
0.2	0.0000220353	$1.27708\times {10}^{-9}$	$7.45389\times {10}^{-14}$
0.3	0.000024591	$1.42610\times {10}^{-9}$	$8.30384\times {10}^{-14}$
0.4	0.000024591	$1.59270\times {10}^{-9}$	$9.27027\times {10}^{-14}$
0.5	0.0000305846	$1.77967\times {10}^{-9}$	$1.03654\times {10}^{-13}$
0.6	0.000034143	$1.99336\times {10}^{-9}$	$1.16143\times {10}^{-13}$
0.7	0.0000382644	$2.24538\times {10}^{-9}$	$1.30749\times {10}^{-13}$
0.8	0.0000432057	$2.55494\times {10}^{-9}$	$1.48755\times {10}^{-13}$
0.9	0.0000493456	$2.55494\times {10}^{-9}$	$1.71615\times {10}^{-13}$

shows that our Noor–Green iterative approach is more accurate in this case than the Mann-Green and Ishikawa-Green iterative approaches. The convergence behavior of this is given in Figure 3.

We now consider another example, as follows.

Example 2.

Assume the following third-order BVP,

$$a^{‴}\left(t\right)+t{a}^{″}\left(t\right)+2t^2-t+2=0,$$

(31)

with the BCs:

$$a\left(0\right)=0=a^{\prime }\left(0\right)=a^{\prime }\left(1\right).$$

(32)

Notice that the exact solution of problems (31) and (32) is not considered. In this case, our novel Noor–Green iterative scheme becomes the following:

$$\left\{\begin{array}{c}{c}_k=a_k-{\gamma }_k{\int }_0^t\left[(\frac{s{(1-t)}^2}{2}+\frac{(s^2-s)}{2})(a_k^{‴}\left(s\right)+s{a}_k^{″}\left(s\right)+2s^2-s+2)\right]ds\hfill \\ -{\gamma }_k{\int }_t^1\left[\left(\frac{(s-1)t^2}{2}\right)(a_k^{‴}\left(s\right)+s{a}_k^{″}\left(s\right)+2s^2-s+2)\right]ds,\hfill \\ b_k=(1-{\beta }_k)a_k-{\beta }_k{\int }_0^t\left[(\frac{s{(1-t)}^2}{2}+\frac{(s^2-s)}{2})(c_k^{‴}\left(s\right)+s{c}_k^{″}\left(s\right)+2s^2-s+2)\right]ds\hfill \\ -{\beta }_k{\int }_t^1\left[\left(\frac{(s-1)t^2}{2}\right)(c_k^{‴}\left(s\right)+s{c}_k^{″}\left(s\right)+2s^2-s+2)\right]ds,\hfill \\ a_{k+1}=(1-{\alpha }_k)a_k-{\alpha }_k{\int }_0^t\left[(\frac{s{(1-t)}^2}{2}+\frac{(s^2-s)}{2})(b_k^{‴}\left(s\right)+s{b}_k^{″}\left(s\right)+2s^2-s+2)\right]ds\hfill \\ -{\alpha }_k{\int }_t^1\left[\left(\frac{(s-1)t^2}{2}\right)(b_k^{‴}\left(s\right)+s{b}_k^{″}\left(s\right)+2s^2-s+2)\right]ds,\hfill \\ k\in \mathbb{N}\cup \left\{0\right\}.\hfill \end{array}\right.$$

(33)

The plot of the Green function involved in our scheme (33) is given in Figure 4.

In the previous example, we compared our scheme to the other iterative schemes in the literature by using the relative error concept. Now, we take ${\alpha }_k={\beta }_k={\gamma }_k=0.99$ and list the comparison of absolute errors between our iterative scheme and other iterative schemes (

Table 4. Absolute error between different iterations (33).

Values of t	Mann-Green	Ishikawa-Green	Noor–Green
0.1	$3.03232\times {10}^{-12}$	$1.73472\times {10}^{-18}$	$8.67362\times {10}^{-19}$
0.2	$1.17690\times {10}^{-11}$	$6.93889\times {10}^{-18}$	$3.46945\times {10}^{-18}$
0.3	$2.49605\times {10}^{-11}$	$2.08167\times {10}^{-17}$	$6.93889\times {10}^{-18}$
0.4	$4.00038\times {10}^{-11}$	$2.08167\times {10}^{-17}$	$6.93889\times {10}^{-18}$
0.5	$5.23963\times {10}^{-11}$	$5.55112\times {10}^{-17}$	$2.77556\times {10}^{-17}$
0.6	$5.57444\times {10}^{-11}$	$6.93889\times {10}^{-17}$	0
0.7	$4.35981\times {10}^{-11}$	$8.32667\times {10}^{-17}$	0
0.8	$1.44757\times {10}^{-11}$	$8.32667\times {10}^{-17}$	0
0.9	$2.09963\times {10}^{-11}$	$1.11022\times {10}^{-16}$	0

). This shows that our new scheme has a rapid rate of convergence in the class of third-order BVPs.

Based on the absolute errors in Table 4, we have shown that our new iterative scheme converges faster to the solution when compared to the other iterative schemes in the literature. Next, by using the concept of Absolute error, we compare our new iterative scheme with other iterative schemes graphically in Figure 5.

7. Application

Fractional boundary value problems (FBVPs) represent a powerful extension of classical BVPs, incorporating non-integer order derivatives and offering a more accurate description of complex phenomena with memory and long-range dependencies. Unlike the classical notion of integer-order derivatives, which provides the instantaneous changes, FBVPs employ fractional-order derivatives that capture intricate dynamics, making them particularly relevant in various fields like physics, biology, finance, and engineering. These equations have the potential to model anomalous diffusion, viscoelasticity, and other intricate behaviors that cannot be adequately captured by integer-order counterparts. The study of FDEs has witnessed significant growth, with researchers developing novel analytical and numerical techniques to solve them, making them a vital tool for understanding the intricacies of real-world processes.

We consider the following BVPs in terms of fractional order:

$$D^{\eta }a\left(t\right)+f(t,a\left(t\right))=0,$$

(34)

with the BCs

$$a\left(0\right)=a\left(1\right)=0,$$

(35)

Here, the number t is in $[0,1]$, the real number $\eta$ is in $(1,2)$, the derivative $D^{\eta }$ is defined in terms of Caputo, called the fractional derivative of order $\alpha$, and f is a function on $[0,1]\times \mathbb{R}\to \mathbb{R}$.

Now, we consider the Banach space $B=C[0,1]$. In this case, the Green function with regard to our problems (34) and (35) is given by the following formula:

$$G(t,s)=\left\{\begin{array}{cc}\frac{1}{\Gamma \left(\eta \right)}{(t(1-s)}^{(\eta -1)}-{(t-s)}^{(\eta -1)},\hfill & \mathrm{if}s\le t\le 1,\hfill \\ \frac{t{(1-s)}^{(\eta -1)}}{\Gamma \left(\eta \right)},\hfill & \mathrm{if}t\le s\le 1.\hfill \end{array}\right.$$

We now consider mapping $S_G$ in the Banach space $B=C[0,1]$ as follows:

$$S_G\left(a\left(t\right)\right)={\int }_0^{1}G(t,s)f(s,a\left(s\right))ds,\mathrm{for}\mathrm{each}a\left(t\right)\in C[0,1].$$

(36)

In this case, our Noor–Green iterative scheme becomes

$$\left\{\begin{array}{c}{a}_0\in B,\hfill \\ c_k=(1-{\gamma }_k)a_k+{\gamma }_k{\int }_0^{1}G(t,s)f(s,a_k\left(s\right))ds,\hfill \\ b_k=(1-{\beta }_k)a_k+{\beta }_k{\int }_0^{1}G(t,s)f(s,b_k\left(s\right))ds,,\hfill \\ a_{k+1}=(1-{\alpha }_k)a_k+{\alpha }_k{\int }_0^{1}G(t,s)f(s,c_k\left(s\right))ds,,k\in \mathbb{N}\cup \left\{0\right\}.\hfill \end{array}\right.$$

(37)

We now give the proof of our final result.

Theorem 3.

Suppose the operator $S_G$ is as is given in (36), and $\left\{a_k\right\}$ is a sequence of a Noor–Green iteration (37). In addition, if $\sum {\alpha }_k=\infty$, then $\left\{a_k\right\}$ converges to the unique solution of problems (34) and (35) provided that the following condition holds:

$$\left|f\right(s,a\left(s\right))-f(s,b\left(s\right)\left)\right|\le \alpha \left|a\right(s)-b(s\left)\right|,$$

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

Proof. 

In order to complete the proof, we consider any $a\left(t\right)$ and $b\left(t\right)$ in $C[0,1]$ such that

$$\begin{array}{ccc}\hfill \left|\right|S_{G}a\left(t\right)-S_{G}b\left(t\right)\left|\right|& \le & \left|\right|{\int }_0^{1}G(s,t)f(s,a\left(s\right)))ds-{\int }_0^{1}G(s,t)f(s,b\left(s\right))\left)ds\right||\hfill \\ & =& \left|\right|{\int }_0^{1}G(s,t)[f(s,a\left(s\right))-f(s,b\left(s\right))]ds\left|\right|\hfill \\ & \le & {\int }_0^{1}G(s,t)\left|f(s,a(s\left)\right)-f(s,b(s\left)\right)\right|ds\hfill \\ & \le & \alpha \left|\right|a\left(t\right))-b\left(t\right)\left|\right|{\int }_0^{1}G(t,s)ds\hfill \\ & \le & \alpha \left|\right|a\left(t\right))-b(t\left)\right||.\hfill \end{array}$$

Consequently, we obtain

$$\left|\right|S_{G}a\left(t\right)-S_{G}b\left(t\right)\left|\right|\le \alpha \left|\right|a\left(t\right))-b\left(t\right)\left|\right|.$$

Hence, $S_G$ is a Banach contraction, and so the sequence of the Noor–Green iterative scheme converges to the unique fixed point of $S_G$ and, hence, to the unique solution of problems (34) and (35). This completes our proof. □

8. Conclusions and Open Problem

The following new outcomes were obtained from this research.

($O_1$)

The elementary three-step iterative scheme due to Noor [8] was modified based on Green’s function to obtain numerical solutions to third-order BVPs.

($O_2$)

We imposed possible mild conditions on the parameters of the scalars in our scheme and obtained a strong convergence result.

($O_3$)

Since the Noor iterative scheme [8] includes Mann [6] and Ishikawa [7] iterative schemes as special cases, our results improve the classical results of the Mann-Green and Ishikawa-Green iterative schemes to the more general context of the Noor–Green iterative scheme.

($O_4$)

We checked the high accuracy of the proposed Noor–Green iterative scheme by using two different numerical examples.

($O_5$)

The graph and tables in the paper support our theoretical findings.

($O_6$)

An application was obtained for new iterations in the class of FBVPs.

Now, we include an interesting open problem.

Open problem: Can we extend the results of this paper to the setting of PDEs?