An Efficient Iterative Procedure for Proximally Quasi-Nonexpansive Mappings and a Class of Boundary Value Problems

Abstract

It is well-known in the literature that many analytical techniques are introduced in order to find a solution for problems such as functional, differential, and integral equations. These analytical techniques sometimes fail to solve such problems, thus prompting the proposal of numerical methods for approaching their approximate solutions. This paper suggests a multi-valued version of an efficient iterative procedure called the F iterative procedure in Banach space and establishes its weak and strong convergence to fixed points of certain proximally quasi-nonexpansive operators. To support these results and to suggest the high accuracy of this procedure, we develop an example of a proximally quasi-nonexpansive operator and perform a comparative numerical experiment. As an application, we solve a two-point boundary value problem (BVP) in Banach space. Our results are new and extend some results from the literature for the new setting of mappings.

1. Introduction and Preliminaries

Let E be a subset of a normed vector space X. The operator $\mathcal{T}:E\to E$ is called a contraction if:

$$\left|\right|\mathcal{T}a-\mathcal{T}b\left|\right|\le \eta \left|\right|a-b\left|\right|,$$

(1)

where $0<\eta <1$. Banach [1] was the first to introduce these mappings and proved that such mappings in a closed subset of Banach space always posses a unique fixed point, that is, $\mathcal{T}q=q$ for some q. Note that when the real constant $\theta$ is equal to 1 in (1), then $\mathcal{T}$ is called nonexpansive. When the space X is restricted to the case of uniformly convex Banach space (UCBS), and E is closed bounded and convex in X, then each nonexpansive map $\mathcal{T}:E\to E$ has a fixed point (e.g., see Browder [2] and Gohde [3]).

Note that it is sometimes possible that a given equation (functional, differential, integral, etc.) has a solution, but its value cannot be found by applying available analytical methods. In such cases, one needs the approximate value of such solutions. The natural question of how to obtain such approximate solutions arises, which is one of the difficult tasks. One of the successful methods are iterative techniques, which are the best choice for computing such approximate solutions. To achieve the objective, we first try to express the original given equation in the form of a fixed point equation, in a way that the fixed point set of this equation coincides with the solution set of the original equation. We then use an iterative method, which converges in the fixed point set of the fixed point equation, and hence in the solution set of the original equation as well.

It is normally easy to iterate a single-valued function, but for a multi-valued map, it is a difficult task. The fixed point theory connected to multi-valued maps has played a significant role in many areas of science and technology (e.g., see [4] and others). Thus, once any result for a single-valued map is available in the literature, it is always desirable to obtain its multi-valued version. Now, we can set the concepts of fixed points for multi-valued maps: a point $q\in E$ is called a fixed point for a multi-valued map $\mathcal{T}:E\to P_E$ if q is an element of the set $\mathcal{T}q$, that is, $q\in \mathcal{T}q$, where $P_E=\{G\subseteq E:G\mathrm{is}\mathrm{proximinal}\}$. If $Tq=\left\{q\right\}$ for every fixed point q of $\mathcal{T}$, then such a multi-valued map is said to be endowed with the endpoint condition (e.g., see Jung [5]). All throughout, $Fix\left(\mathcal{T}\right)$ will denote the fixed point set of $\mathcal{T}$.

To establish the fixed point and the related results on multi-valued maps, one often requires the following concept of metric. Suppose that $E_{CB}=\{G\subseteq E:G\mathrm{is}\mathrm{closed}\mathrm{and}\mathrm{bounded}\}$ and set the real-valued function $H:E_{CB}\times E_{CB}\to \mathbb{R}$ with the following formula:

$$H(G,W)=max\left\{\underset{g\in G}{sup}d(g,W),\underset{w\in W}{sup}d(w,G)\right\},\mathrm{for}\mathrm{each}G,W\in E_{CB}.$$

Here, the function H essentially holds all requirements of the metric, which is called a Hausdorff metric on the set $E_{CB}$.

Using the function H, we thus mention the following notions of multi-valued maps.

Definition 1.

If E is any nonempty subset of Banach space, $\mathcal{T}:E\to E_{CB}$ and $a,b\in E$ and $q\in Fix\left(\mathcal{T}\right)$. Then $\mathcal{T}$ is known as:

(i) 

nonexpansive if $H(\mathcal{T}a,\mathcal{T}b)\le \left|\right|a-b\left|\right|$;

(ii) 

quasi-nonexpansive if $H(\mathcal{T}a,\mathcal{T}q)\le \left|\right|a-q\left|\right|$;

(iii) 

proximally nonexpansive if $H(P_{\mathcal{T}}a,P_{\mathcal{T}}b)\le \left|\right|a-b\left|\right|$;

(iv) 

proximally quasi-nonexpansive if $H(P_{\mathcal{T}}a,P_{\mathcal{T}}q)\le \left|\right|a-q\left|\right|$.

Remark 1.

Note that a nonexpansive multi-valued map is always continuous. Moreover, every multi-valued nonexpansive map with a nonempty fixed point set is always a multi-valued quasi-nonexpansive, but the converse is not valid in general because there exists discontinuous, multi-valued maps which are multi-valued quasi-nonexpansive (e.g., see [6] and others).

When a result on the existence of a fixed point for a given operator is proved, then calculating such a fixed point by using an appropriate fixed point iteration method is essentially an active research area on its own. To compute the value of contractions, the well-known result of Banach [1] essentially suggests the simplest Picard iteration method. However, the drawback of this method is that it fails to converge in the fixed point set of nonexpansive operators; for example, $\mathcal{T}a=\frac{a-1}{2}$ is a nonexpansive mapping on the set $[0,1]$ and admits a unique fixed point $q=\frac{1}{2}$, but the Picard iteration method suggests a divergent sequence for every selection of $a_1\in [0,1]-\left\{\frac{1}{2}\right\}$. In order to find the fixed points of nonexpansive mappings and to gain a relatively better rate of convergence, there are many iterative processes in the literature (cf. Mann [7], Ishikawa [8], Picard-Ishikawa [9], Noor [10], Agarwal [11], Abbas [12], Picard-S [13], and so on). The fixed point theory of multi-valued mappings finds its roots in the papers of Nadler [14], who studied the Banach result [1] in the setting of multi-valued mappings. In [15], Lim proved a fixed point theorem for a multi-valued nonexpansive, which is in fact the multi-valued version of the Browder–Göhde fixed point theorem. In 2005, Sastry and Babu [16] essentially constructed multi-valued versions of the Mann and Ishikawa iteration schemes, respectively, for multi-valued nonexpansive mappings and proved their convergence results in a Hilbert space setting. Panyanak [17] studied these schemes in the general context of a UCBS setting. Soon, Song, and Wang [18] improved the main outcome of Panyanak [17] by introducing the Mann and Ishikawa iterative schemes in a new sense. However, the Song and Wang [18] iterative schemes use too much time consumer estimates; these schemes also require the endpoint condition to establish their convergence. After this, Shahzad and Zegeye [6] constructed new modified Mann and Ishikawa iterative schemes and proved their convergence for proximally nonexpansive mappings without an endpoint condition. Song and Cho [19] extended their results to the setting of proximally quasi-nonexpansive mappings. Using the idea of Shahzad and Zegeye [6], Khan and Yilirim [20] used the S iteration for approximating fixed points of proximally nonexpansive mappings. Khan et al. [21] used the Abbas iteration for approximating fixed points of proximally quasi-nonexpansive nonexpansive mappings. Okeke et al. [9] used the Picard–Ishikawa iterative scheme to approximate fixed points of proximally quasi-nonexpansive mappings. The aim of this paper is to propose new efficient algorithms for the class of proximally quasi-nonexpansive mappings and a class of BVPs on a Banach space setting.

Now we list the following single-valued iterative schemes along with their multi-valued versions in

Table 1. Some well-known schemes along with their multi-valued versions.

Name of Iteration	Single-Valued Version	Multi-Valued Version
Mann	$a_{n+1}=(1-{\alpha }_n)a_n+{\alpha }_n\mathcal{T}{a}_n$	$a_{n+1}=(1-{\alpha }_n)a_n+{\alpha }_n{u}_n$
Ishikawa	$b_n=(1-{\beta }_n)a_n+{\beta }_n\mathcal{T}{a}_n$	$b_n=(1-{\beta }_n)a_n+{\beta }_n{u}_n$
	$a_{n+1}=(1-{\alpha }_n)a_n+{\alpha }_n\mathcal{T}{b}_n$	$a_{n+1}=(1-{\alpha }_n)a_n+{\alpha }_n{v}_n$
Noor	$c_n=(1-{\gamma }_n)a_n+{\gamma }_n\mathcal{T}{a}_n$	$c_n=(1-{\gamma }_n)a_n+{\gamma }_n{u}_n$
	$b_n=(1-{\beta }_n)a_n+{\beta }_n\mathcal{T}{c}_n$	$b_n=(1-{\beta }_n)a_n+{\beta }_n{w}_n$
	$a_{n+1}=(1-{\alpha }_n)a_n+{\alpha }_n\mathcal{T}{b}_n$	$a_{n+1}=(1-{\alpha }_n)a_n+{\alpha }_n{v}_n$
S	$b_n=(1-{\beta }_n)a_n+{\beta }_n\mathcal{T}{a}_n$	$b_n=(1-{\beta }_n)a_n+{\beta }_n{u}_n$
	$b_{n+1}=(1-{\alpha }_n)a_n+{\alpha }_n\mathcal{T}{b}_n$	$a_{n+1}=(1-{\alpha }_n)u_n+{\alpha }_n{v}_n$
Picard–Ishikawa	$c_n=(1-{\gamma }_n)a_n+{\gamma }_n\mathcal{T}{a}_n$	$c_n=(1-{\gamma }_n)a_n+{\gamma }_n{u}_n$
	$b_n=(1-{\beta }_n)a_n+{\beta }_n\mathcal{T}{c}_n$	$b_n=(1-{\beta }_n)a_n+{\beta }_n{w}_n$
	$a_{n+1}=T{b}_n$	$a_{n+1}=v_n$
Picard–S	$c_n=(1-{\gamma }_n)a_n+{\gamma }_n\mathcal{T}{a}_n$	$c_n=(1-{\gamma }_n)a_n+{\gamma }_n{u}_n$
	$b_n=(1-{\beta }_n)\mathcal{T}{a}_n+{\beta }_n\mathcal{T}{c}_n$	$b_n=(1-{\beta }_n)u_n+{\beta }_n{w}_n$
	$a_{n+1}=\mathcal{T}{b}_n$	$a_{n+1}=v_n$

. Set $P_{\mathcal{T}}a=\{b\in \mathcal{T}a:d(a,\mathcal{T}a)=||a-b|\left|\right\}$, $u_n\in P_{\mathcal{T}}\left(a_n\right)$, $v_n\in P_{\mathcal{T}}\left(b_n\right)$, and $w_n\in P_{\mathcal{T}}\left(c_n\right)$.

Now we set and study the following F iterative scheme, the single-valued version of which was introduced and studied by Ali and Ali [22] for multi-valued mappings:

$$\left\{\begin{array}{c}{a}_1\in E,\hfill \\ d_n=(1-{\alpha }_n)a_n+{\alpha }_n{u}_n,\hfill \\ c_n=r_n,\hfill \\ b_n=w_n,\hfill \\ a_{n+1}=v_n,n\in \mathbb{N},\hfill \end{array}\right.$$

(2)

where $u_n\in P_{\mathcal{T}}\left(a_n\right)$, $v_n\in P_{\mathcal{T}}\left(b_n\right)$, $w_n\in P_{\mathcal{T}}\left(c_n\right)$, $r_n\in P_{\mathcal{T}}\left(d_n\right)$, and ${\alpha }_n\in (0,1)$. In this research paper, we approximate fixed points for the class of proximally quasi-nonexpansive mappings (which contains the class of proximally nonexpansive mappings as a special case) using the multi-valued version of the F iterative scheme (2). We establish several weak and strong convergence results for proximally quasi-nonexpansive operators regarding the iteration method (2). To validate the theoretical outcome, we suggest some numerical examples and perform a comparative experiment.

As usual, some of the following concepts will be used.

Definition 2

([23]). A normed vector space X is said to be with Opial’s property if any $\left\{a_n\right\}\subseteq X$ converges in the weak sense to $w^{*}$, then the following equation:

$$\underset{n\to \infty }{lim\; sup}\left|\right|a_n-w^{*}\left|\right|<\underset{n\to \infty }{lim\; sup}\left|\right|a_n-w^{**}\left|\right|,$$

essentially holds for any element $w^{**}\in X-\left\{w^{*}\right\}$.

Definition 3

([20]). A multi-valued operator, namely $\mathcal{T}:E\to P_E$, is said to be demiclosed at point $d\in E$ if any $\left\{a_n\right\}\subseteq E$ admitting a weak limit, namely w, and $s_n\in \mathcal{T}{a}_n$ admitting a strong limit, namely s, then one has $s\in Ta$.

Definition 4

([24]). A multi-valued operator, namely $\mathcal{T}:E\to P_E$, is said to satisfy a condition $\left(I\right)$ if there is a continuous and non-decreasing function, namely $\psi :[0,\infty [\to [0,\infty [$, with the following properties $\psi \left(0\right)=0$, $\psi \left(t\right)>0$ for every $t\in ]0,\infty [$ such that $d(a,\mathcal{T}a)\ge \psi \left(d(a,Fix(\mathcal{T})\right)$ for each choice of $a\in E$.

Song and Cho [19] noted the following facts.

Lemma 1.

The following conditions for any multi-valued mapping $\mathcal{T}:E\to P_E$ are equivalent.

($c_1$)

$q\in Fix\left(\mathcal{T}\right)$;

($c_2$)

$P_{\mathcal{T}}\left(q\right)=\left\{q\right\}$;

($c_3$)

$q\in Fix\left(P_{\mathcal{T}}\right)$.

Furthermore, $Fix\left(\mathcal{T}\right)=Fix\left(P_{\mathcal{T}}\right)$.

Every UCBS X enjoys the following property.

Lemma 2

([25]). Suppose $0<a\le {\pi }_n\le b<1$ and $t_n,s_n\in X$ for $n\in \mathbb{N}$. If X denotes a UCBS, ${lim\; sup}_{n\to \infty }\left|\right|t_n\left|\right|\le \nu$, ${lim\; sup}_{n\to \infty }\left|\right|s_n\left|\right|\le \nu$ and ${lim}_{n\to \infty }\left|\right|{\pi }_n{t}_n+(1-{\pi }_n)s_n\left|\right|=\nu$, where $\nu \ge 0$ is some real constant, then ${lim}_{n\to \infty }\left|\right|t_n-s_n\left|\right|=0$.

2. Main Results

Lemma 3.

If the operator $\mathcal{T}:E\to P_E$ is proximally quasi-nonexpansive on a convex nonempty closed subset E of a UCBS X and the set $Fix\left(\mathcal{T}\right)\ne \mathrm{\varnothing }$, then the sequence of iterates (2) has the property ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|$ that exists for each fixed point q of $\mathcal{T}$.

Proof. 

For any $q\in Fix\left(\mathcal{T}\right)$ and $n\in \mathbb{N}$, consider the following:

$$\begin{array}{ccc}\hfill \left|\right|d_n-q\left|\right|& =& \left|\right|((1-{\alpha }_n)a_n+{\alpha }_n{u}_n)-q\left|\right|\hfill \\ & \le & (1-{\alpha }_n)\left|\right|a_n-q\left|\right|+{\alpha }_n\left|\right|u_n-q\left|\right|\hfill \\ & \le & (1-{\alpha }_n)\left|\right|a_n-q\left|\right|+{\alpha }_{n}H(P_{\mathcal{T}}{a}_n,P_{\mathcal{T}}q)\hfill \\ & \le & (1-{\alpha }_n)\left|\right|a_n-q\left|\right|+{\alpha }_n\left|\right|a_n-q\left|\right|\hfill \\ & =& \left|\right|a_n-q\left|\right|.\hfill \end{array}$$

Hence, we obtain $\left|\right|d_n-q\left|\right|\le \left|\right|a_n-q\left|\right|$. Using this, we have the following:

$$\begin{array}{ccc}\hfill \left|\right|a_{n+1}-q\left|\right|& =& \left|\right|v_n-q\left|\right|\le H(P_{\mathcal{T}}{b}_n,P_{\mathcal{T}}q)\hfill \\ & \le & \left|\right|b_n-q\left|\right|=\left|\right|w_n-q\left|\right|\le H(P_{\mathcal{T}}{c}_n,P_{\mathcal{T}}q)\hfill \\ & \le & \left|\right|c_n-q\left|\right|=\left|\right|r_n-q\left|\right|\le H(P_{\mathcal{T}}{d}_n,P_{\mathcal{T}}q)\hfill \\ & \le & \left|\right|d_n-q\left|\right|\le \left|\right|a_n-q\left|\right|.\hfill \end{array}$$

Consequently, we obtain the inequality $\left|\right|a_{n+1}-q\left|\right|\le \left|\right|a_n-q\left|\right|$ for all fixed points q of $\mathcal{T}$ and $n\in \mathbb{N}$. It follows that ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|$ exists for each fixed point q of $\mathcal{T}$. □

Using the established lemma above, we provide another key lemma, which will help us in establishing the main outcome of the sequel.

Lemma 4.

If the operator $\mathcal{T}:E\to P_E$ is proximally quasi-nonexpansive on a convex nonempty closed subset E of a UCBS X and the set $Fix\left(\mathcal{T}\right)\ne \varnothing$, then the sequence of iterates (2) has the property ${lim}_{n\to \infty }d(a_n,P_{\mathcal{T}}{a}_n)=0$.

Proof. 

We proved in Lemma 3 that ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|$ exists if one selects any $q\in Fix\left(\mathcal{T}\right)$. This strong limit establishes that ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|=\nu$, where $\nu \ge 0$ denotes any real constant. Without loss of generality, we assume that $\nu >0$. Then, since $\mathcal{T}$ is proximally quasi-nonexpansive and $u_n\in P_{\mathcal{T}}$, it follows that:

$$\underset{n\to \infty }{lim\; sup}\left|\right|u_n-q\left|\right|\le \underset{n\to \infty }{lim\; sup}H(P_{\mathcal{T}}{a}_n,P_{\mathcal{T}}q)\le \underset{n\to \infty }{lim\; sup}\left|\right|a_n-q\left|\right|=\nu .$$

(3)

Looking deeply into the proof of Lemma 3, it can be seen that:

$$\left|\right|d_n-q\left|\right|\le \left|\right|a_n-q\left|\right|,$$

$$\Rightarrow \underset{n\to \infty }{lim\; sup}\left|\right|d_n-q\left|\right|\le \underset{n\to \infty }{lim\; sup}\left|\right|a_n-q\left|\right|=\nu .$$

(4)

Once again, we can observe the proof of Lemma 3 to note the following:

$$\left|\right|a_{n+1}-q\left|\right|\le \left|\right|d_n-q\left|\right|,$$

$$\Rightarrow \nu =\underset{n\to \infty }{lim\; inf}\left|\right|a_{n+1}-q\left|\right|\le \underset{n\to \infty }{lim\; inf}\left|\right|d_n-q\left|\right|.$$

(5)

We can now combine inequalities (4) and (5) to get the following equality:

$$\underset{n\to \infty }{lim}\left|\right|d_n-q\left|\right|=\nu .$$

(6)

Using (6), we get the following equation:

$$\nu =\underset{n\to \infty }{lim}\left|\right|d_n-q\left|\right|=\underset{n\to \infty }{lim}\left|\right|(1-{\alpha }_n)(a_n-q)+{\alpha }_n(u_n-q)\left|\right|.$$

Now, all the requirements for Lemma 2 are fulfilled, and hence:

$$\underset{n\to \infty }{lim}\left|\right|a_n-u_n\left|\right|=0.$$

(7)

Since $u_n\in P_{\mathcal{T}}{a}_n$, we get the equation below:

$$\underset{n\to \infty }{lim}d(a_n,P_{\mathcal{T}}{a}_n)=0.$$

□

We now suggest the weak convergence result.

Theorem 1.

If the operator $\mathcal{T}:E\to P_E$ is proximally quasi-nonexpansive on a convex nonempty closed subset E of a UCBS X and the set $Fix\left(\mathcal{T}\right)\ne \varnothing$, then the sequence of iterates (2) converges weakly in the set $Fix\left(\mathcal{T}\right)$ whenever $I-P_{\mathcal{T}}$ is assumed to be demiclosed at zero and X is endowed with the Opial’s property.

Proof. 

Suppose $q\in Fix\left(\mathcal{T}\right)$, it follows that $p\in Fix\left(P_{\mathcal{T}}\right)$. By Lemma 3, ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|$ exists. To obtain the weak convergence of $\left\{a_n\right\}$ in the set $Fix\left(\mathcal{T}\right)$, we take two points $w_1,w_2\in E$ such that the subsequences $\left\{a_{n_j}\right\}$ and $\left\{a_{n_k}\right\}$ of $\left\{a_n\right\}$ respectively converge weakly to them. According to (7), $u_n\in \mathcal{T}{a}_n$ exists such that ${lim}_{n\to \infty }\left|\right|a_n-u_n\left|\right|=0$. According to the demicloseness of $I-P_{\mathcal{T}}$ at zero, the element $w_1\in Fix\left(\mathcal{T}\right)$. Similarly, one can prove that $w_2\in Fix\left(\mathcal{T}\right)$. It now remains to be proven that $w_1=w_2$. If $w_1\ne w_2$. Now, using Opial’s property of space, one has the following:

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}\left|\right|a_n-w_1\left|\right|& =& \underset{j\to \infty }{lim}\left|\right|a_{n_j}-w_1\left|\right|<\underset{j\to \infty }{lim}\left|\right|a_{n_j}-w_2\left|\right|\hfill \\ & =& \underset{n\to \infty }{lim}\left|\right|a_n-w_2\left|\right|=\underset{k\to \infty }{lim}\left|\right|a_{n_k}-w_2\left|\right|\hfill \\ & <& \underset{k\to \infty }{lim}\left|\right|a_{n_k}-w_1\left|\right|=\underset{n\to \infty }{lim}\left|\right|a_n-w_1\left|\right|.\hfill \end{array}$$

Hence, we have reached ${lim}_{n\to \infty }\left|\right|a_n-w_1\left|\right|<{lim}_{n\to \infty }\left|\right|a_n-w_1\left|\right|$. This strict inequality suggests a contradiction. Thus, the proof is established. □

After the weak convergence, we are now interested in providing the strong convergence result.

Theorem 2.

If the operator $\mathcal{T}:E\to P_E$ is proximally quasi-nonexpansive on a convex nonempty compact subset E of a UCBS X, and the set $Fix\left(\mathcal{T}\right)\ne \varnothing$, then the sequence of iterates (2) converges strongly in the set $Fix\left(\mathcal{T}\right)$.

Proof. 

The set E is convex and compact, so the sequence $\left\{u_n\right\}$ is contained in E and has a convergent subsequence, namely $\left\{u_{n_i}\right\}$. Suppose ${lim}_{i\to \infty }\left|\right|u_{n_i}-p\left|\right|=0$ for some $p\in E$. The target is to prove that p is a fixed point of $\mathcal{T}$ and a strong limit of $\left\{u_n\right\}$. Now, $P_{\mathcal{T}}$ is proximally nonexpansive and $a_{n_i}\in P_{\mathcal{T}}\left(u_{n_i}\right)$. Additionally, according to Lemma 4, ${lim}_{i\to \infty }\left|\right|u_{n_i}-a_{n_i}\left|\right|=0$. Accordingly, we have the following:

$$\begin{array}{ccc}\hfill d(p,P_{\mathcal{T}}p)& \le & d(p,u_{n_i})+d(u_{n_i},P_{\mathcal{T}}{u}_{n_i})+d(P_{\mathcal{T}}{u}_{n_i},P_{\mathcal{T}}p)\hfill \\ & \le & \left|\right|u_{n_i}-p\left|\right|+\left|\right|u_{n_i}-a_{n_i}\left|\right|+H(P_{\mathcal{T}}{u}_{n_i},P_{\mathcal{T}}p)\hfill \\ & \le & \left|\right|u_{n_i}-p\left|\right|+\left|\right|u_{n_i}-a_{n_i}\left|\right|+\left|\right|u_{n_i}-p\left|\right|.\hfill \end{array}$$

If we apply ${lim}_{i\to \infty }$, we obtain $d(p,P_{\mathcal{T}}\left(p\right))=0$. Thus, $p\in Fix\left(P_{\mathcal{T}}\right)$, so by Lemma 1, $p\in Fix\left(\mathcal{T}\right)$; that is, p is also a fixed point of $\mathcal{T}$. By Lemma 3, ${lim}_{n\to \infty }\left|\right|u_n-p\left|\right|$ exists. Hence, p is the strong limit of $\left\{u_n\right\}$. □

The last result is established under the compactness of the set E. For the next results, the compactness of E is not essential; however, the condition $\left(I\right)$ of $\mathcal{T}$ must be assumed.

Theorem 3.

If the operator $\mathcal{T}:E\to P_E$ is proximally quasi-nonexpansive on a convex nonempty closed subset E of a UCBS X and the set $Fix\left(\mathcal{T}\right)\ne \varnothing$, then the sequence of iterates (2) converges strongly in the set $Fix\left(\mathcal{T}\right)$ whenever $\mathcal{T}$ is assumed to be endowed with the condition $\left(I\right)$.

Proof. 

By Lemma 3, ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|$ exists for every choice of $q\in Fix\left(\mathcal{T}\right)=Fix\left(P_{\mathcal{T}}\right)$. When ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|=0$, then one has nothing to show. Hence, we consider the only case ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|=\nu >0$. We have observed in Lemma 3 that $\left|\right|a_{n+1}-q\left|\right|\le \left|\right|a_n-q\left|\right|$. This gives ${inf}_{q\in Fix\left(\mathcal{T}\right)}\left|\right|a_n-q\left|\right|\le {inf}_{q\in Fix\left(\mathcal{T}\right)}\left|\right|a_n-q\left|\right|$. Hence, we get the following equation:

$$d(a_{n+1},Fix\left(\mathcal{T}\right))\le d(a_n,Fix\left(\mathcal{T}\right)).$$

Accordingly, we notice that ${lim}_{n\to \infty }d(a_n,Fix\left(\mathcal{T}\right))$ exists. The next purpose is to show ${lim}_{n\to \infty }d(a_n,Fix\left(\mathcal{T}\right))=0$. Set ${lim}_{n\to \infty }d(a_n,Fix\left(\mathcal{T}\right))=b$ for some real constant $b>0$. Thus, for every choice of $n\in \mathbb{N}$, set the following:

$$h_n=\frac{u_n-q}{\left|\right|a_n-q\left|\right|},g_n=\frac{a_n-q}{\left|\right|a_n-q\left|\right|}.$$

It can be observed that $\left|\right|g_n\left|\right|=1$ and $\left|\right|h_n\left|\right|\le 1$ because $\left|\right|u_n-q\left|\right|\le H(P_{\mathcal{T}}{a}_n,P_{\mathcal{T}}q)\le \left|\right|a_n-q\left|\right|$. Thus, using condition $\left(I\right)$ of $\mathcal{T}$, one has:

$$\begin{array}{ccc}\hfill \underset{n}{lim\; inf}\left|\right|g_n-h_n\left|\right|& =& \underset{n}{lim\; inf}\left|\left|\frac{a_n-q}{\left|\right|a_n-q\left|\right|}-\frac{u_n-q}{\left|\right|a_n-q\left|\right|}\right|\right|=\underset{n}{lim\; inf}\left(\frac{\left|\right|a_n-u_n\left|\right|}{\left|\right|a_n-q\left|\right|}\right)\hfill \\ & \ge & \underset{n}{lim\; inf}\left(\frac{d(a_n,T{a}_n)}{\left|\right|a_n-q\left|\right|}\right)\ge \underset{n}{lim\; inf}\left(\frac{\psi (d(a_n,Fix\left(\mathcal{T}\right))}{\left|\right|a_n-q\left|\right|}\right)\hfill \\ & \ge & \frac{\psi \left(b\right)}{\nu }>0.\hfill \end{array}$$

Next, since ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|=\nu$ and ${lim}_{n\to \infty }\left|\right|d_n-q\left|\right|=\nu$ (see the proof of Lemma 4). Using these two, one has the following:

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}\left|\right|(1-{\alpha }_n)g_n+{\alpha }_n{h}_n\left|\right|& =& \underset{n\to \infty }{lim}\left|\left|(1-{\alpha }_n)\left(\frac{a_n-q}{\left|\right|a_n-q\left|\right|}\right)+{\alpha }_n\left(\frac{u_n-q}{\left|\right|a_n-q\left|\right|}\right)\right|\right|\hfill \\ & =& \underset{n\to \infty }{lim}\left|\left|\frac{(1-{\alpha }_n)a_n+{\alpha }_n{u}_n-q}{\left|\right|a_n-q\left|\right|}\right|\right|\hfill \\ & =& \frac{{lim}_{n\to \infty }\left|\right|d_n-q\left|\right|}{{lim}_{n\to \infty }\left|\right|a_n-q\left|\right|}=\frac{\nu }{\nu }=1.\hfill \end{array}$$

Thus,

$$\underset{n\to \infty }{lim}\left|\right|(1-{\alpha }_n)g_n+{\alpha }_n{h}_n\left|\right|=1.$$

Now, by Lemma 2, we get ${lim}_{n\to \infty }\left|\right|g_n-h_n\left|\right|=0$, which is clearly a contradiction to ${lim\; inf}_n\left|\right|g_n-h_n\left|\right|>0$. Thus, we conclude that ${lim}_{n\to \infty }d(a_n,Fix\left(\mathcal{T}\right))=0$. Therefore:

$$\underset{n\to \infty }{lim}\left|\right|a_n-q\left|\right|=0.$$

Hence, the sequence $\left\{a_n\right\}$ converges strongly to some fixed point q of $\mathcal{T}$. □

Now we compare the speed of convergence of our iterative process with many other processes. To do this, we consider the following example.

Example 1.

If $E=[0,5]$ and $\mathcal{T}:E\to P_E$ is defined as $\mathcal{T}a=[0,\frac{a+2}{3}]$ for each $a\in E$, then the following holds:

(i) 

$Fix\left(\mathcal{T}\right)\ne \varnothing$;

(ii) 

$\mathcal{T}$ satisfies condition I;

(iii) 

$\mathcal{T}$ is proximally nonexpansive.

Proof. 

For (i), we see that $Fix\left(\mathcal{T}\right)=[0,1$, that is, $Fix\left(\mathcal{T}\right)\ne \varnothing$. For (ii), we set a nondeacreasing function $\psi :[0,\infty )\to [0,\infty )$ such that $\psi \left(t\right)=\frac{t}{2}$ for each $t\ge 0$. Now,

Case (I). When $a\in Fix\left(T\right)=[0,1]$, then $d(a,\mathcal{T}a)=0=d(a,Fix(\mathcal{T}\left)\right)$. Hence,

$$d(a,Ta)=0\ge \psi \left(d\right(a,Fix\left(\mathcal{T}\right)\left)\right).$$

Case (II). When $a\in E-Fix\left(\mathcal{T}\right)=(1,5]$. Then,

$$\begin{array}{ccc}\hfill d(a,\mathcal{T}a)& =& d\left(a,\left[0,\frac{a+2}{3}\right]\right)\hfill \\ & =& \left|a-\left(\frac{a+2}{3}\right)\right|\hfill \\ & =& \frac{2}{3}\left(a-1\right)\hfill \\ & \ge & \frac{a-1}{2}\hfill \\ & =& \psi (a-1)\hfill \\ & =& \psi \left(d(a,[0,1\left]\right)\right)\hfill \\ & =& \psi \left(d(a,Fix(\mathcal{T}\left)\right)\right).\hfill \end{array}$$

Consequently, $d(a,\mathcal{T}a)\ge \psi \left(d\right(a,Fix\left(\mathcal{T}\right)\left)\right)$ for each $a\in E$. This proves (ii). Finally, we prove (iii). Notice that $P_{\mathcal{T}}a=\left\{a\right\}$ whenever $a\in Fix\left(\mathcal{T}\right)$, and $P_{\mathcal{T}}a=\left\{\frac{a+2}{3}\right\}$ if otherwise. There are two cases to prove the proximally quasi-nonexpansiveness of $\mathcal{T}$ as provided below.

Case (I). When $a\in Fix\left(\mathcal{T}\right)$. Then, for any fixed point q of $\mathcal{T}$, we have the following equation:

$$H(P_{\mathcal{T}}a,P_{\mathcal{T}}q)=H(\left\{a\right\},\left\{q\right\})=|a-q|.$$

Case (II). When $a\in E-Fix\left(\mathcal{T}\right)=(1,5]$. Then, for any fixed point q of $\mathcal{T}$, we have the following equation:

$$H(P_{\mathcal{T}}a,P_{\mathcal{T}}q)=H(\left\{\frac{a+2}{3}\right\},\left\{q\right\})=|\left(\frac{a+2}{3}\right)-q|\le |a-q|.$$

Consequently, $H(P_{\mathcal{T}}a,P_{\mathcal{T}}q)\le |a-q|$ for each $a\in E$ and $q\in Fix\left(\mathcal{T}\right)$. This proves (iii). Hence, all the requirements of our main results now hold. Thus, $\left\{a_n\right\}$ generated by (2) converges to a point of $Fix\left(\mathcal{T}\right)$. This convergence can be seen in the

Table 2. Some iterates suggested by the F, Picard–S, Picard–Ishikawa, S, Noor, Ishikawa, and Mann methods for $\mathcal{T}$ given in Example 1.

n	F	Picard–S	Picard–Ishikawa	S	Noor	Ishikawa	Mann
1	3.5	3.5	3.5	3.5	3.5	3.5	3.5
2	1.04938	1.19352	1.36019	1.58056	2.04264	2.08056	2.33333
3	1.00098	1.01498	1.05189	1.13482	1.43484	1.46704	1.71111
4	1.00002	1.00116	1.00748	1.03131	1.18135	1.20187	1.37926
5	1	1.00009	1.00108	1.00727	1.07563	1.08725	1.20227
6	1	1.00001	1.00016	1.00169	1.03154	1.03771	1.10788
7	1	1	1.00002	1.00039	1.01316	1.01630	1.05754
8	1	1	1	1.00009	1.00549	1.00705	1.03069
9	1	1	1	1.00002	1.00229	1.00305	1.01637
10	1	1	1	1	1.00095	1.00132	1.00873
11	1	1	1	1	1.00040	1.00057	1.00466
12	1	1	1	1	1.00017	1.00025	1.00248
13	1	1	1	1	1.00007	1.00011	1.00132
14	1	1	1	1	1.00003	1.00005	1.00071
15	1	1	1	1	1.00001	1.00002	1.00038
16	1	1	1	1	1	1.00001	1.00020
17	1	1	1	1	1	1	1.00011
18	1	1	1	1	1	1	1.00006
19	1	1	1	1	1	1	1.00003
20	1	1	1	1	1	1	1.00001
21	1	1	1	1	1	1	1

. □

Assume that ${\alpha }_n=0.70$ and ${\beta }_n=0.65$ and ${\gamma }_n=0.45$, we compare the high accuracy of the proposed F iterative method with the Picard–S [13], Picard–Ishikawa [9], S [11], Noor [10], Ishikawa [8], and Mann [7] iterative methods using $\mathcal{T}$ of Example 1. The iterative values are provided in Table 2, and the convergence behaviors can be observed in Figure 1.

Remark 2.

Table 2 and Figure 1 suggest that the F iterative method is better than all of the Picard–S, Picard–Ishikawa, S, Noor, Ishikawa, and Mann iterative processes. Every proximally nonexpansive mapping is proximally quasi-nonexpansive; thus, our results are also valid for proximally nonexpansive mappings. The presented results in this section therefore improve and extend many well-known single and multi-valued results from the current literature, especially Shahzad and Zegeye [6], Khan and Yilirim [20], Okeke [26], Gursoy and Karakaya [13], Ali and Ali [22], and many others.

3. An Application to a Class of BVPs

BVPs of various orders played a very significant role, especially in mathematical modelings, physics, and engineering problems. Finding approximate solutions for different classes of BVPs with various orders is also an active and difficult field of mathematics. In the present article, we propose and study a new approach for solving a two-point BVP (second order) by applying the F iterative method of Ali and Ali [22]. We now take a real Banach space X and select any $a_1\in X$. Then, the F iterates suggest a sequence $\left\{a_n\right\}$, which is generated in the following way:

$$\left\{\begin{array}{c}{d}_n=(1-{\alpha }_n)a_n+{\alpha }_n\mathcal{T}{a}_n,\hfill \\ c_n=\mathcal{T}{d}_n,\hfill \\ b_n=\mathcal{T}{b}_n,\hfill \\ a_{n+1}=\mathcal{T}{a}_n,n\in \mathbb{N}.\hfill \end{array}\right.$$

(8)

Notice that the sequence $0<{\alpha }_n<1$ involved in method (8) essentially holds the restriction $\sum {\alpha }_n=\infty$.

Now, the well-known Green’s function is given as follows. For each $z\in [i,j]$, assume that $h\left(z\right)$ is a continuous map on the interval $[i,j]$, then functions which satisfy $a^{″}=h\left(z\right)$ can be written as follows:

$$a=K_0+K_{1}z+{\int }_i^j(z-y)h\left(y\right)dy.$$

(9)

Thus, the unique solution of the following given BVP:

$$\left\{\begin{array}{c}{a}^{″}=h\left(z\right),\hfill \\ a\left(i\right)=0,\hfill \\ a\left(j\right)=0,\hfill \end{array}\right.$$

(10)

is in the form of (9), provided that $K_0$ and $K_1$ have appropriate values. If the values of both of the $K_0$ and $K_1$ are known, then the solution of (10) can be expressed in the following form:

$$a\left(z\right)={\int }_i^{j}G(z,y)h\left(y\right)dy,$$

where the notation $G(z,y)$ is known as Green’s function of the BVP:

$$\left\{\begin{array}{c}{a}^{″}=0,\hfill \\ a\left(i\right)=0,\hfill \\ a\left(j\right)=0,\hfill \end{array}\right.$$

and it is defined in the following way:

$$G(z,y)=\left\{\begin{array}{c}\frac{(z-i)(y-j)}{j-i}\mathrm{for}i\le z\le y,\hfill \\ \\ \frac{(z-j)(y-i)}{j-i}\mathrm{for}y\le z\le j.\hfill \end{array}\right.$$

In the same way, the solution of BVP:

$$\left\{\begin{array}{c}{a}^{″}=h\left(z\right),\hfill \\ a\left(i\right)={\zeta }_0,\hfill \\ a\left(j\right)={\zeta }_1,\hfill \end{array}\right.$$

can be easily stated in the following way:

$$a\left(z\right)={\int }_i^{j}G(z,y)h\left(y\right)dy+q\left(z\right),$$

(11)

where $q\left(z\right)$ is the solution for the equation $a^{″}=0$, with $q\left(i\right)={\zeta }_0$ and $q\left(j\right)={\zeta }_1$. Suppose the function $f(z,a,a^{\prime })$ is continuous on the set $[i,j]\times {\mathbb{R}}^2$. Then (11) suggests that whenever $q\left(z\right)$ is the solution of $a^{″}=0$ with $q\left(i\right)={\zeta }_0$ and $q\left(j\right)={\zeta }_1$, then the function $a\left(z\right)\in C^{\left(1\right)}[i,j]$ is the solution for the BVP:

$$\left\{\begin{array}{c}{a}^{″}=f(z,a,a^{\prime }),\hfill \\ a\left(i\right)={\zeta }_0,\hfill \\ a\left(j\right)={\zeta }_1,\hfill \end{array}\right.$$

(12)

if and only if $a\left(z\right)$ is the element of $C^{\left(1\right)}[a,b]$, and it is always the solution of the following given integral equation:

$$a\left(z\right)={\int }_i^{j}G(z,y)f(y,a\left(y\right),a^{\prime }\left(y\right))dy+q\left(z\right),\mathrm{on}[i,j].$$

(13)

Now assume that $\mathcal{T}:C^{\left(1\right)}[i,j]\to C^{\left(1\right)}[i,j]$ is given by the following equation:

$$\mathcal{T}\left[a\left(z\right)\right]={\int }_i^{j}G(z,y)f(y,a\left(y\right),a^{\prime }\left(y\right))dy+q\left(z\right),$$

for every choice of $a\in C^{\left(1\right)}[i,j]$ such that $i\le z\le j$, then each fixed point of the mapping $\mathcal{T}$ is the solution of (13), as well as of (12).

Recently, Bello et al. [27] suggested the following two-point BVP:

$$a^{″}=f(z,a,a^{\prime })i\le z\le j,$$

(14)

$$\left\{\begin{array}{c}{\lambda }_{0}a\left(i\right)+v_0{a}^{\prime }\left(i\right)={\zeta }_0,\hfill \\ {\lambda }_{1}a\left(j\right)+v_1{a}^{\prime }\left(j\right)={\zeta }_1,\hfill \end{array}\right.$$

(15)

where both the ${\lambda }_0$, ${\lambda }_1$, $v_0$ and $v_1$ are any real numbers satisfying ${\lambda }_0^2+v_0^2>0$ and ${\lambda }_1^2+v_1^2>0$ and so on.

Now to solve (14) and (15) with process (8), we must first transform the underlying problem in the following way:

$$\left\{\begin{array}{c}{d}_n^{″}=(1-{\alpha }_n)a_n^{″}+{\alpha }_n{a}_n^{″}=(1-{\alpha }_n)a_n^{″}+{\alpha }_{n}f(z,a_n,a_n^{\prime }),\hfill \\ \\ c_n^{″}=d_n^{″}=f(z,d_n,d_n^{\prime }),\hfill \\ \\ b_n^{″}=c_n^{″}=f(z,c_n,c_n^{\prime }),\hfill \\ \\ a_{n+1}^{″}=b_{n+1}^{″}=f(z,b_n,b_n^{\prime }),\hfill \\ \\ {\lambda }_0{a}_{n+1}\left(i\right)+v_0{a}_{n+1}^{\prime }\left(i\right)={\zeta }_0,\hfill \\ \\ {\lambda }_1{a}_{n+1}\left(j\right)+v_1{a}_{n+1}^{\prime }\left(j\right)={\zeta }_1,\hfill \end{array}\right.$$

(16)

where ${\alpha }_n\in (0,1)$ and $\sum {\alpha }_n=\infty$.

Keep in mind that $a\left(z\right)$ solves (14) as well as (15) if and only if it essentially solves the following:

$$a\left(z\right)={\int }_i^{j}G(z,y)f(y,a\left(y\right),a^{\prime }\left(y\right))dy+q\left(z\right)\mathrm{on}[i,j],$$

and the notation $G(z,y)$ represents Green’s function associated with the following BVP:

$$\left\{\begin{array}{c}{a}^{″}=0,\hfill \\ {\lambda }_{0}a\left(i\right)+v_0{a}^{\prime }\left(i\right)={\zeta }_0,\hfill \\ {\lambda }_1{a}^{\prime }\left(j\right)+v_1{a}^{\prime }\left(j\right)={\zeta }_1,\hfill \end{array}\right.$$

(17)

and $q\left(z\right)$ is the solution of (17).

We now assume that $\mathcal{T}:C^{\left(1\right)}[i,j]⟶C^{\left(1\right)}[i,j]$ is set as follows:

$$\left\{\begin{array}{c}\mathcal{T}\left[d\left(z\right)\right]={\int }_i^{j}G(z,y)f(y,d\left(y\right),d^{\prime }\left(y\right))dy+q\left(z\right),\hfill \\ \\ \mathcal{T}\left[c\left(z\right)\right]={\int }_i^{j}G(z,y)f(y,c\left(y\right),c^{\prime }\left(y\right))dy+q\left(z\right),\hfill \\ \\ \mathcal{T}\left[b\left(z\right)\right]={\int }_i^{j}G(z,y)f(y,b\left(y\right),b^{\prime }\left(y\right))dy+q\left(z\right),\hfill \\ \\ \mathcal{T}\left[a\left(z\right)\right]={\int }_i^{j}G(z,y)f(y,a\left(y\right),a^{\prime }\left(y\right))dy+q\left(z\right),\hfill \end{array}\right.$$

(18)

where the mapping $\mathcal{T}$ is such that every solution $a\left(z\right)$ of (14) and (15) is the fixed point of $\mathcal{T}$ [28].

Next, we take the derivative of our proposed iteration process defined by (8) in order to solve the suggested BVP. For this purpose, we assume E to be any nonempty convex subset of real Banach space X and $\mathcal{T}:E\to P_E$ to be a multi-valued mapping. Suppose the sequence $\left\{a_n\right\}\subset E$ is generated by (8) such that $\left\{{\alpha }_n\right\}\subseteq (0,1)$ with $\sum {\alpha }_n=\infty$.

Now we are in the position to compare our procedure (16) with (8) to establish their equivalence. First, we differentiate (18) as follows:

$$\left\{\begin{array}{c}{\left(\mathcal{T}{d}_n\right)}^{\prime }={\int }_i^j\frac{\partial }{\partial t}G(z,y)f(y,d_n\left(y\right),d_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right),\hfill \\ \\ {\left(\mathcal{T}{c}_n\right)}^{\prime }={\int }_i^j\frac{\partial }{\partial t}G(z,y)f(y,c_n\left(y\right),c_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right),\hfill \\ \\ {\left(\mathcal{T}{b}_n\right)}^{\prime }={\int }_i^j\frac{\partial }{\partial t}G(z,y)f(y,b_n\left(y\right),b_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right),\hfill \\ \\ {\left(\mathcal{T}{a}_n\right)}^{\prime }={\int }_i^j\frac{\partial }{\partial t}G(z,y)f(y,a_n\left(y\right),a_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right).\hfill \end{array}\right.$$

(19)

Second, twice differentiating (8), we have the following equation:

$$\left\{\begin{array}{c}{d}_n^{″}=(1-{\alpha }_n)a_n^{″}+{\alpha }_n{\left(\mathcal{T}{a}_n\right)}^{″}\hfill \\ \\ c_n^{″}={\left(\mathcal{T}{d}_n\right)}^{″}\hfill \\ \\ b_n^{″}={\left(\mathcal{T}{c}_n\right)}^{″}\hfill \\ \\ a_{n+1}^{″}={\left(\mathcal{T}{b}_n\right)}^{″}\hfill \\ \\ n\in \mathbb{N}.\hfill \end{array}\right.$$

(20)

Now, the third step is to differentiate (19), as follows:

$$\left\{\begin{array}{c}{\left(\mathcal{T}{d}_n\right)}^{″}={\int }_i^j\frac{{\partial }^2}{\partial z^2}G(z,d_n\left(y\right),d_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right),\hfill \\ \\ {\left(\mathcal{T}{c}_n\right)}^{″}={\int }_i^j\frac{{\partial }^2}{\partial z^2}G(z,c_n\left(y\right),c_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right),\hfill \\ \\ {\left(\mathcal{T}{b}_n\right)}^{″}={\int }_i^j\frac{{\partial }^2}{\partial z^2}G(z,b_n\left(y\right),b_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right),\hfill \\ \\ {\left(\mathcal{T}{a}_n\right)}^{″}={\int }_i^j\frac{{\partial }^2}{\partial z^2}G(z,a_n\left(y\right),a_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right).\hfill \end{array}\right.$$

(21)

Lastly, by substituting (21) in (20), one has the following:

$$\left\{\begin{array}{c}{d}_n^{″}=(1-{\alpha }_n)a_n^{″}+{\alpha }_n({\int }_i^j\frac{{\partial }^2}{\partial z^2}G(z,a_n\left(y\right),a_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right)),\hfill \\ \\ c_n^{″}={\int }_i^j\frac{{\partial }^2}{\partial z^2}G(z,d_n\left(y\right),d_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right),\hfill \\ \\ b_n^{″}={\int }_i^j\frac{{\partial }^2}{\partial z^2}G(z,c_n\left(y\right),c_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right),\hfill \\ \\ a_{n+1}^{″}={\int }_i^j\frac{{\partial }^2}{\partial z^2}G(z,b_n\left(y\right),b_n^{\prime }\left(y\right))dy+q^{\prime }\left(z\right),\hfill \\ \\ n\in \mathbb{N}.\hfill \end{array}\right.$$

(22)

Now (22) can be expressed as follows, which is our proposed fixed point method:

$$\left\{\begin{array}{c}{d}_n^{″}=(1-{\alpha }_n)a_n^{″}+{\alpha }_n{a}^{″}=(1-{\alpha }_n)a_n^{″}+{\alpha }_{n}f(z,a_n,a_n^{\prime }),\hfill \\ \\ c_n^{″}=d_n^{″}=f(z,d_n,d_n^{\prime }),\hfill \\ \\ b_n^{″}=c_n^{″}=f(z,c_n,c_n^{\prime }),\hfill \\ \\ a_{n+1}^{″}=b_n^{″}=f(z,a_n,a_n^{\prime }),\hfill \\ \\ n\in \mathbb{N}.\hfill \end{array}\right.$$

(23)

We now provide the main outcome of this section.

Theorem 4.

Suppose $\mathcal{T}:C^{\left(1\right)}[i,j]\to C^{\left(1\right)}[i,j]$ is a contraction operator (18), with a contraction constant $0<\eta <1$. If f $a_1\in C^{\left(1\right)}[i,j]$ is any affine function, and $\left\{a_n\right\}$ is a sequence of iterates (8) with $a^{″}=0$ and the boundary condition in (15) with $\left\{{\alpha }_n\right\}$ in $(0,1)$ such that $\sum {\alpha }_n=\infty$, then the sequence $\left\{a_n\right\}$ converges to the solution $q\in C^{\left(1\right)}[i,j]$ of (14) and (15)

Proof. 

The Banach result [1] suggests the uniqueness and existence of $q\in C^{\left(1\right)}[i,j]$. We show that the sequence $\left\{a_n\right\}$ converges strongly to this q. Using (8), one has the following:

$$\begin{array}{ccc}\hfill \left|\right|d_n-q\left|\right|& =& \left|\right|(1-{\alpha }_n)a_n+{\alpha }_n\mathcal{T}{a}_n-\mathcal{T}q\left|\right|\hfill \\ & \le & (1-{\alpha }_n)\left|\right|a_n-q\left|\right|+{\alpha }_n\left|\right|\mathcal{T}{a}_n-\mathcal{T}q\left|\right|\hfill \\ & \le & (1-{\alpha }_n)\left|\right|a_n-q\left|\right|+{\alpha }_n\left(\eta \right||a_n-q\left|\right|)\hfill \\ & \le & (1-{\alpha }_n(1-\eta ))\left|\right|a_n-q\left|\right|.\hfill \end{array}$$

Hence:

$$\begin{array}{ccc}\hfill \left|\right|a_{n+1}-q\left|\right|& =& \left|\right|\mathcal{T}{b}_n-\mathcal{T}q\left|\right|\hfill \\ & \le & \eta \left|\right|b_n-q\left|\right|\hfill \\ & =& \eta \left|\right|\mathcal{T}{c}_n-\mathcal{T}q\left|\right|\hfill \\ & \le & {\eta }^2\left|\right|c_n-q\left|\right|\hfill \\ & =& {\eta }^2\left|\right|\mathcal{T}{d}_n-\mathcal{T}q\left|\right|\hfill \\ & \le & {\eta }^3\left|\right|d_n-q\left|\right|\hfill \\ & \le & {\eta }^3(1-{\alpha }_n(1-\eta ))\left|\right|a_n-q\left|\right|.\hfill \end{array}$$

Inductively, we obtain the following:

$$\begin{array}{ccc}\hfill \left|\right|a_{n+1}-q\left|\right|& \le & \left|\right|a_1-q\left|\right|{\eta }^{3n}{\Pi }_{k=1}^n(1-{\alpha }_k(1-\eta )).\hfill \end{array}$$

(24)

Since $\eta \in (0,1)$ and ${\alpha }_n\in (0,1)$ for $n\in \mathbb{N}$, therefore $1-{\alpha }_n(1-\eta )<1$. Now using the well-known inequality $1-z\le e^{-z}$ for $z\in [0,1]$ together with $\sum {\alpha }_n=\infty$ and then applying $n⟶\infty$, we get from (24), ${lim}_{n\to \infty }\left|\right|a_n-q\left|\right|=0$. It follows that $a_n⟶q$. Accordingly, $\left\{a_n\right\}$ converges to a solution $q\in C^{\left(1\right)}[i,j]$ of (14) and (15). □

4. Conclusions and Future Plan

We proposed the multi-valued version of the F iterative method and proved its convergence for the class of proximally nonexpansive operators on Banach space. To support the main outcome, we presented an example of proximally nonexpansive operators and proved that its proposed method provides high accuracy corresponding to the other methods. Finally, we solved a two-point BVP, which was one of the aims of this paper. Note that:

(i)

The main outcome related to the convergence extends the idea of Ali and Ali [22], whose research idea is limited to the case of single-valued operators;

(ii)

Theorem 4 extends and unifies the main results of Bello et al. [27] and Okeke et al. [26] because, as we observed in Table 2 and Figure 1, the F iterative method is better than the Mann iteration that was used by Bello et al. [27] and the Picard–Ishikawa iteration which was used by Okeke et al. [26];

(iii)

In the future, we will try to use the obtained results of the paper to solve some problems related the class of fractional differential equations.