A new fixed point algorithm for finding the solution of a delay differential equation

1.   Introduction

Numerical reckoning fixed points for nonlinear operators is nowadays an active research problem of nonlinear analysis owing to its applications to: variational inequalities, equilibrium problems, computer simulation, image encoding and much more. Mann ^[19], Ishikawa ^[17] and Halpern ^[12] are the three basic iterative algorithms to approximate fixed points of nonexpansive mappings. Following these study, several authors constructed numerous algorithms to approximate the fixed points of different classes of nonlinear mappings mainly Noor iteration ^[20], Agarwal et al. iteration ^[4], SP iteration ^[21], Normal-S iteration ^[23], Abbas and Nazir iteration ^[1], Thakur et al. iterations ^[28,29], Karakaya et al. iteration ^[18] and many others.

In 2008, Suzuki ^[26] introduced a new generalization of nonexpansive mappings and called the defining condition as Condition (C) which is also referred as Suzuki generalized nonexpansive mappings. A mapping $T: K\rightarrow K$ defined on a nonempty subset $K$ of a Banach space $E$ is said to satisfy the Condition (C) if

for all $x$ and $y \in K.$

Suzuki proved that the mappings satisfying the Condition (C) is weaker than nonexpansive and also obtained few results regarding the existence of fixed points for such mappings. In 2011, Phuengrattana ^[22] used Ishikawa iteration to obtain some convergence results for mappings satisfying Condition (C) in uniformly convex Banach spaces. In the last few years, many authors have studied this particular class of mappings in various domain and have obtained many convergence results (e.g. ^{[2,3,9,10,28,30,31,35]}).

Recently, Ullah and Arshad ^[31] introduced a new algorithm namely M-iteration algorithm as follows:

where $\{\alpha_n\}$ is a sequence in $(0, 1)$. They proved some convergence results for Suzuki generalized nonexpansive mappings and showed that M-iteration converges faster than Picard-S ^[11] and S-iteration ^[4].

To achieve a better rate of convergence, we introduce a new iterative algorithm for approximating fixed points of Suzuki generalized nonexpansive mappings as follows:

where $\{\alpha_n\}$ is a sequence in $(0, 1).$

The aim of this paper is to prove that newly defined iterative algorithm $(1.2)$ converges faster than algorithm $(1.1)$ for contractive-like mappings. Also, we prove some convergence results involving algorithm $(1.2)$ for Suzuki generalized nonexpansive mappings. Further, we provide a numerical example to show that our iteration $(1.2)$ converges faster than a number of existing iterative algorithms in respect of Suzuki generalized nonexpansive mappings. In the last section, we use our algorithm to find solution of a delay differential equation.

2.   Preliminaries

For making our paper self contained, we collect some basic definitions and needed results.

Definition 2.1. A Banach space $E$ is said to be uniformly convex if for each $\epsilon \in (0, 2]$ there is a $\delta > 0$ such that for $x, y \in E$ with $\|x\| \leq 1$, $\|y\| \leq 1$ and $\|x - y\| > \epsilon$, we have

Definition 2.2. A Banach space $E$ is said to satisfy the Opial's condition if for any sequence $\{x_n\}$ in $E$ which converges weakly to $x \in E$ i.e. $x_n \rightharpoonup x$ implies that

for all $y \in E$ with $y \neq x.$

Examples of Banach spaces satisfying this condition are Hilbert spaces and all $l^{p}$ spaces $(1 < p < \infty)$. On the other hand, $L^{p} [0, 2\pi]$ with $1 < p \neq 2$ fail to satisfy Opial's condition.

A mapping $T : K \rightarrow E$ is demiclosed at $y \in E$ if for each sequence $\{x_n\}$ in $K$ and each $x \in E$, $x_n \rightharpoonup x$ and $Tx_n \rightarrow y$ imply that $x \in K$ and $Tx = y.$

Let $K$ be a nonempty closed convex subset of a Banach $E$, and let $\{x_n\}$ be a bounded sequence in $E$. For $x\in E$ write:

The asymptotic radius of $\{x_n\}$ relative to $K$ is given by

and the asymptotic center $A(K, \{x_n\})$ of $\{x_n\}$ is defined as:

It is known that, in a uniformly convex Banach space, $A(K, \{x_n\})$ consists of exactly one point.

The following definitions about the rate of convergence were given by Berinde ^[7].

Definition 2.3. Let $\{a_n\}$ and $\{b_n\}$ be two real sequences converging to $a$ and $b$ respectively. Then, $\{a_n\}$ converges faster then $\{b_n\}$ if $\lim\limits_{n\to\infty}\frac{\|a_n - a\|}{\|b_n - b\|} = 0.$

Definition 2.4. Let $\{u_n\}$ and $\{v_n\}$ be two fixed point iteration processes converging to the same fixed point $p$. If $\{a_n\}$ and $\{b_n\}$ are two sequences of positive numbers converging to zero such that $\|u_n - p\| \leq a_n$ and $\|v_n - p\| \leq b_n$ for all $n \geq 1$, then we say that $\{u_n\}$ converges faster than $\{v_n\}$ to $p$ if $\{a_n\}$ converges faster then $\{b_n\}$.

The following lemma due to Schu ^[24] is very useful in our subsequent discussion.

Lemma 2.1. Let $E$ be a uniformly convex Banach space and $\{t_n\}$ be any sequence such that $0 < p \leq t_n \leq q < 1$ for some $p, r \in \mathbb{R}$ and for all $n \geq 1.$ Let $\{x_n\}$ and $\{y_n\}$ be any two sequences of $E$ such that $\limsup\limits_{n\to\infty}\|x_n\| \leq r$, $\limsup\limits_{n\to\infty}\|y_n\| \leq r$ and $\limsup\limits_{n\to\infty}\|t_nx_n + (1-t_n)y_n\| = r$ for some $r \geq 0.$ Then, $\lim\limits_{n\to\infty}\|x_n - y_n\| = 0.$

Now, we list few lemmas involving Suzuki generalized nonexpansive mappings.

Lemma 2.2. (^[26]) Let $K$ be a nonempty subset of a Banach space $E$ and $T: K\rightarrow K$ be any mapping. Then,

(ⅰ) If $T$ is nonexpansive then $T$ is Suzuki generalized nonexpansive mapping.

(ⅱ) If $T$ is Suzuki generalized nonexpanisve mapping such that $F(T) \neq \emptyset,$ then $T$ is a quasi-nonexpansive mapping.

(ⅲ) If $T$ is a Suzuki generalized nonexpansive mapping, then $\|x - Ty\| \leq 3\|x - Tx\| + \|x - y\|$ for all $x$ and $y \in K.$

Lemma 2.3. (^[27]) Let $T$ be a Suzuki generalized nonexpansive mapping defined on a subset $K$ of a Banach space $E$ with the Opial property. If a sequence $\{x_n\}$ converges weakly to z and $\lim\limits_{n\to\infty}\|Tx_n - x_n\| = 0,$ then $I - T$ is demiclosed at zero.

Lemma 2.4. (^[26]) If $T$ is a Suzuki generalized nonexpansive mapping defined on a compact convex subset $K$ of a uniformly convex Banach space $E$ then, $T$ has a fixed point.

In 1972, Zamfirescu ^[34] introduced Zamfirescu mappings which serves as an important generalization for Banach contraction principle ^[5]. In 2004, Berinde ^[6] gave a more general class of mappings known as quasi-contractive mappings. Following this, Imoru and Olantiwo ^[16] gave the following definition:

Definition 2.5. A mapping $T: K\rightarrow K$ is known as contractive-like mapping if there exists a strictly increasing and continuous function $\varphi : [0, \infty) \rightarrow [0, \infty)$ with $\varphi(0) = 0$ and a constant $\delta \in [0, 1)$ such that for all $x, y \in K,$ we have

Clearly, the class of contractive-like mappings is wider than the class of quasi-contractive mappings.

3.   Rate of convergence

In this section, first we show that our algorithm $(1.2)$ converges faster than the M-iteration $(1.1)$ for contractive-like mappings.

Theorem 3.1. Let $T$ be a contractive-like mapping defined on a nonempty closed convex subset $K$ of a Banach space $E$ with $F(T) \neq \emptyset.$ If $\{x_n\}$ is a sequence defined by $(1.2)$, then $\{x_n\}$ converges faster than the iterative algorithm $(1.1).$

Proof. From $(1.1)$, for any $p \in F(T)$, we have

and

As, $\{\alpha_n\}$ is a sequence in $(0, 1),$ we can find a constant $\alpha \in \mathbb{R}$ such that $\alpha_n \leq \alpha < 1$ for all $n \in \mathbb{N}.$ So,

Now, from $(1.2)$ we get

and

So,

Let $b_n = \delta^{3n}(1 - (1 - \delta)\alpha)^n\|x_1 - p\|$ and $a_n = \delta^{2n}(1 - (1 - \delta)\alpha)^n\|d_1 - p\|$, then

Hence, $\{x_n\}$ converges faster than $\{d_n\}.$

Now, we present a example of a contractive-like mapping which is not a contraction.

Example 1: Let $E = \mathbb{R}$ and $K = [0, 6]$. Let $T:K\rightarrow K$ be a mapping defined as

Proof: Clearly $x = 0$ is the fixed point of $T.$ First, we prove that $T$ is a contractive-like mapping but not a contraction. Since $T$ is not continuous at $x = 3\in \lbrack 0, 6],$ so $T$ is not a contraction. We show that $T$ is a contractive-like mapping. For this, define $\varphi : [0, \infty) \rightarrow [0, \infty)$ as $\varphi(x) = \frac{x}{8}.$ Then, $\varphi$ is a strictly increasing as well as continuous function. Also, $\varphi(0) = 0.$

We need to show that

for all $x, y \in [0, 6]$ and $\delta$ is a constant in $[0, 1).$

Before going ahead, let us note the following. When $x \in [0, 3)$, then

and

Similarly, when $x \in [3,6]$, then

and

Consider the following cases:

Case A: Let $x, y \in [0, 3),$ then using $(3.1)$ we get

So $(A)$ is satisfied with $\delta = \frac{1}{5}.$

Case B: Let $x \in [0, 3)$ and $y \in [3,6]$ then using $(3.1)$ we get

So $(A)$ is satisfied with $\delta = \frac{1}{5}.$

Case C: Let $x \in [3,6]$ and $y \in [0, 3)$ then using $(3.2)$ we get

So $(A)$ is satisfied with $\delta = \frac{1}{5}.$

Case D: Let $x, y \in [3,6]$ then using $(3.2)$ we get

So $(A)$ is satisfied with $\delta = \frac{1}{5}.$

Consequently, $(A)$ is satisfied for $\delta = \frac{1}{5}$ and $\varphi (x) = \frac{x}{8}$ in all the possible cases. Thus, $T$ is a contractive-like mapping.

Now, using $T$, we show that our iterative algorithm $(1.2)$ has a better rate of convergence. Set $\alpha _{n} = \beta _{n} = \gamma_{n} = \frac{n}{n+1}$ for each $n\in \mathbb{N}$. Then, we get the following Table 1, Table 2, Figure 1 and Figure 2 with the initial value $4.5$.

Clearly, our algorithm $(1.2)$ converges at a faster rate for contractive-like mappings.

4.   Convergence results

First, we prove few lemmas which will be useful in obtaining convergence results.

Lemma 4.1. Let $T$ be a Suzuki generalized nonexpansive mapping defined on a nonempty closed convex subset $K$ of a Banach space $E$ with $F(T) \neq \emptyset.$ Let $\{x_n\}$ be the iterative sequence defined by the algorithm $(1.2)$. Then, $\lim\limits_{n\to\infty} \|x_n - p\|$ exists for all $p \in F(T).$

Proof. Let $p \in F(T)$ and $z \in K$. Since $T$ is a Suzuki generalized nonexpansive mapping, $\frac{1}{2}\|p - Tp\| = 0 \leq \|p - z\|$ implies that $\|Tp - Tz\| \leq \|p - z\|.$

Now we have,

and

Using $(4.1)$ and $(4.2)$, we get

Thus, $\{\|x_n - p\|\}$ is bounded and decreasing sequence of reals and hence $\lim\limits_{n\to\infty} \|x_n - p\|$ exists.

Lemma 4.2. Let $T$ be a Suzuki generalized nonexpansive mapping defined on a nonempty closed convex subset $K$ of a uniformly convex Banach space $E$. Let $\{x_n\}$ be the iterative sequence defined by the algorithm $(1.2)$. Then, $F(T) \neq \emptyset$ if and only if $\{x_n\}$ is bounded and $\lim\limits_{n\to\infty} \|Tx_n - x_n\| = 0.$

Proof. Suppose $F(T) \neq \emptyset$ and let $p \in F(T).$ Then, by Lemma 4.1, $\lim\limits_{n\to\infty}\|x_n - p\|$ exists. Let $\lim\limits_{n\to\infty} \|x_n - p\| = c.$

From Eqs $(4.1)$ and $(4.2)$, we have

and

Now,

and

So,

which along with Eq $(4.3)$ implies

Since $T$ is a Suzuki generalized nonexpansive mapping, we get

From Eq $(4.4)$, we obtain

Consider,

Using Lemma 2.3, from Eqs $(4.4)$, $(4.5)$ and $(4.6)$, we get

Now, consider

which on using Eq $(4.7)$ gives

Since,

this together with Eqs $(4.7)$ and $(4.8)$ yields that

Now, using Eqs $(4.8)$ and $(4.9)$, we have

Hence,

Conversely, suppose that $\{x_n\}$ is bounded and $\lim\limits_{n\to\infty}\|x_n - Tx_n\| = 0.$ Let $p \in A(K, \{x_n\}),$ we have

This implies that $Tp \in A(K, \{x_n\}).$ Since $E$ is uniformly convex, $A(K, \{x_n\})$ is singleton, therefore we get $Tp = p.$

Theorem 4.1. Let $T$ be a Suzuki generalized nonexpansive mapping defined on a nonempty closed convex subset $K$ of a Banach space $E$ which satisfies the Opial's condition with $F(T) \neq \emptyset.$ If $\{x_n\}$ is the iterative sequence defined by the iterative algorithm $(1.2)$, then $\{x_n\}$ converges weakly to a fixed point of $T$.

Proof. Let $p \in F(T)$. Then, from Lemma 4.1 $\lim\limits_{n\to\infty}\|x_n - p\|$ exists. In order to show the weak convergence of the algorithm $(1.2)$ to a fixed point of $T$, we will prove that $\{x_n\}$ has a unique weak subsequential limit in $F(T).$ For this, let $\{x_{n_j}\}$ and $\{x_{n_k}\}$ be two subsequences of $\{x_n\}$ which converges weakly to $u$ and $v$ respectively. By Lemma 4.1, we have $\lim\limits_{n\to\infty}\|Tx_{n} - x_{n}\| = 0$ and using the Lemma 2.3, we have $I - T$ is demiclosed at zero. So $u, v \in F(T)$.

Next, we show the uniqueness. Since $u, v \in F(T)$, so $\lim\limits_{n\to\infty}\|x_n - u\|$ and $\lim\limits_{n\to\infty}\|x_n - v\|$ exists. Let $u \neq v$. Then, by Opial's condition, we obtain

which is a contradiction, so $u = v.$ Thus, $\{x_n\}$ converges weakly to a fixed point of $T$.

Next, we establish some strong convergence results for iterative algorithm $(1.2).$

Theorem 4.2. Let $T$ be a Suzuki generalized nonexpansive mapping defined on a nonempty compact convex subset $K$ of a uniformly convex Banach space $E$. If $\{x_n\}$ is the iterative sequence defined by the iterative algorithm $(1.2)$, then $\{x_n\}$ converges strongly to a fixed point of $T$.

Proof. Using Lemma 2.4, we get $F(T) \neq \emptyset.$ So, by Lemma 4.2, we have $\lim\limits_{n\to\infty}\|Tx_n - x_n\| = 0.$ Since $K$ is compact, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $\{x_{n_k}\}$ converges strongly to $p$ for some $p \in K.$ From Lemma 2.2(ⅲ), we have

for all $n \geq 1.$ Letting $k \rightarrow \infty,$ we get that $\{x_{n_k}\}$ converges to $Tp.$ This implies that $Tp = p,$ i.e., $p \in F(T).$ Further, $\lim\limits_{n\to\infty}\|x_n - p\|$ exists by Lemma 4.1. So, $p$ is the strong limit of the sequence $\{x_n\}.$

A mapping $T:K\rightarrow K$ is said to satisfy the Condition (A) (^[25]) if there exists a nondecreasing function $f:[0, \infty) \rightarrow [0, \infty)$ with $f(0) = 0$ and $f(r) > 0$ for all $r\in (0, \infty)$ such that $\|x - Tx\| \geq f(d(x, F(T)))$ for all $x \in K,$ where $d(x, F(T)) = inf\{\|x - p\| : p \in F(T)\}.$

Theorem 4.3. Let $T$ be a Suzuki generalized nonexpansive mapping defined on a nonempty closed convex subset $K$ of a uniformly convex Banach space $E$ such that $F(T) \neq \emptyset$ and $\{x_n\}$ be the sequence defined by $(1.2)$. If $T$ satisfies Condition (A), then $\{x_n\}$ converges strongly to a fixed point of $T$.

Proof. By Lemma 4.1, $\lim\limits_{n\to\infty} \|x_n - p\|$ exists and $\|x_{n+1} - p\| \leq \|x_n - p\|$ for all $p \in F(T).$

We get

which yields

This shows that the sequence $\{d(x_n, F(T))\}$ is decreasing and bounded below, so $\lim\limits_{n\to\infty} d(x_n, F(T))$ exists.

Let $\lim\limits_{n\to\infty}\|x_n - p\| = r$ for some $r \geq 0.$ If $r = 0$ then the result follows. Assume $r > 0.$ Also, by Lemma 4.2 we have $\lim\limits_{n\to\infty} \|x_n - Tx_n\| = 0.$

It follows from Condition (A) that

so that $\lim\limits_{n\to\infty} f(d(x_n, F(T))) = 0.$

Since $f$ is a non decreasing function satisfying $f(0) = 0$ and $f(r) > 0$ for all $r \in (0, \infty)$, therefore $\lim\limits_{n\to\infty} d(x_n, F(T)) = 0.$ So, we have a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ and a sequence $\{y_k \} \subset F(T)$ such that

for all $k \in \mathbb{N}.$ Using $(4.4)$, we obtain

Therefore,

This implies that $\{y_k\}$ is a cauchy sequence in $F(T)$. Since $F(T)$ is closed, so $\{y_k\}$ converges to a point $p \in F(T).$ Then, $\{x_{n_k}\}$ converges strongly to $p$. Since $\lim\limits_{n\to\infty}\|x_n - p\|$ exists, we get $x_n \rightarrow p \in F(T).$ This completes the proof.

5.   Numerical example

In this section, first we will construct an example of a Suzuki generalized nonexpansive mapping which is not a nonexpansive mapping. Then, using that example, we will show that our iteration scheme $(1.2)$ has a better speed of convergence than number of existing iteration schemes.

Example 2: Define a mapping $T : [0, 1] \rightarrow [0, 1]$ by

First we show that $T$ is not a nonexpansive map. For this, take $x = \frac{8}{100}$ and $y = \frac{1}{12}$. Then,

and

Clearly, $\|Tx - Ty\| > \|x - y\|$ which proves that $T$ is not a nonexpansive mapping.

Now, we show that $T$ satisfies the condition $K$. For this, consider the following cases:

Case-Ⅰ: Let $x \in [0, \frac{1}{12})$, then $\frac{1}{2}\|x - Tx\| = \frac{1}{2}|2x - 1| = \frac{1}{2}(1 - 2x).$ For $\frac{1}{2}\|x - Tx\| \leq \|x - y\|$, we must have $\frac{1}{2}(1 - 2x) \leq \|x - y\|$, i.e., $\frac{1}{2}(1 - 2x) \leq |x - y|$. Here note that the case $y < x$ is not possible. So, we are left with only one case when $y > x,$ which gives $\frac{1}{2}(1 - 2x) \leq y - x,$ which yields $y \geq \frac{1}{2}$. So, $y \in [\frac{1}{2}, 1].$ Now, we have $x \in [0, \frac{1}{12})$ and $y \in [\frac{1}{2}, 1].$ So,

and

Hence,

Case-Ⅱ: Let $x \in [\frac{1}{12}, 1],$ then $\frac{1}{2}\|x - Tx\| = \frac{1}{2}|x - \frac{x + 11}{12}| = \frac{11 - 11x}{24}$. For $\frac{1}{2}\|x - Tx\| \leq \|x - y\|$, we must have $\frac{11 - 11x}{24} \leq \|x - y\|$, i.e., $\frac{11 - 11x}{24} \leq |x - y|$. Here we have two possibilities.

A: When $x < y$, we get $\frac{11 - 11x}{24} \leq y - x$, i.e., $y \geq \frac{11 + 13x}{24}$. So, $y \in [\frac{145}{288}, 1] \subset [\frac{1}{12}, 1],$ which gives $\|Tx - Ty\| = \frac{1}{12}\|x - y\| \leq \|x - y\|$. Hence,

B: When $x > y,$ then $\frac{11 - 11x}{24} \leq x - y$, i.e. $y \leq \frac{35x - 11}{24}$ which gives $y \in [0, 1].$ Also, $\frac{24y + 11}{35} \leq x$ which yields $x \in [\frac{11}{35}, 1].$ Here, for $x \in [\frac{11}{35}, 1]$ and $y \in [\frac{1}{12}, 1]$ Case IIA can be used. So, we only need to verify when $x \in [\frac{11}{35}, 1]$ and $y \in [0, \frac{1}{12}).$ For this,

and

So, $\|Tx - Ty\| \leq \|x - y\|.$ Thus, mapping $T$ satisfies the Condition (C) for all the possible cases.

Now, using above example, we will show that iteration algorithm $(1.2)$ converges faster than Thakur New, M and M$^*$ iteration. Let $\alpha_n = \beta_n = \frac{n}{n+10}$ for all $n \in \mathbb{N}$ and $x_1 = 0.02,$ then we get the following Table 3 of iteration values and Figure 3.

It is evident from above table and graph that our algorithm $(1.2)$ converges at a better speed than the above mentioned schemes.

6.   Application

In this section, we show that our iterative algorithm can be used to find a solution of a delay differential equation.

Many physical problems arising in various fields can be easily modeled with the help of ordinary differential equations. Later, it was recognized that a phenomena may have a delayed effect in a differential equation, leading to the development of concept of delay differential equations. Following this, numerous methods have been obtained to solve various kinds of delay differential equations (e.g. ^{[13,14,15,32,33]}).

In this paper, we consider the following delay differential equation

with initial condition

Now, we will show that the sequence generated by our iteration scheme $(1.2)$ converges strongly to the solution of $(6.1).$

It is well known that $(C([a, b]), ||.||_{\infty})$ is a Banach space where $C([a, b])$ denotes the space of all continuous real valued functions on a closed interval $[a, b]$ and $||.||_{\infty}$ is a Chebyshev norm $|| x - y||_{\infty} = \max\limits_{t \in [a, b]}|x(t) - y(t)|.$

Assume that the following conditions are satisfied

$(A_1)$ $t_0, b \in \mathbb{R}, \tau > 0;$

$(A_2)$ $f \in C([t_0, b] \times \mathbb{R}^{2}, \mathbb{R});$

$(A_3)$ $\psi \in C([t_0- \tau, b], \mathbb{R});$

$(A_4)$ there exists $L_f > 0$ such that

$|f(t, u_1, u_2) - f(t, v_1, v_2)| \leq L_f \sum\limits_{i = 1}^{2}|u_i - v_i|, \quad \forall u_i, v_i \in \mathbb{R}, i = 1, 2, t \in [t_0, b];$

$(A_5)$ $2L_f (b - t_0) < 1.$

We notice that the solution of $(6.1)$-$(6.2)$ if it exists is of the following form

Here, $x \in C([t_0- \tau, b], \mathbb{R}) \cap C^{1}([t_0, b], \mathbb{R}).$

Coman et al. ^[8] established the following results.

Theorem 6.1. Assume that conditions $(A_1) - (A_5)$ are satisfied. Then Problem $(6.1) - (6.2)$ has a unique solution, say $p \in C([t_0- \tau, b], \mathbb{R}) \cap C^{1}([t_0, b], \mathbb{R})$ and

Now, we prove the following result using our iterative process $(1.2).$

Theorem 6.2. Suppose that conditions $(A_1) - (A_5)$ are satisfied. Then the problem $(6.1) - (6.2)$ has a unique solution say $p \in C([t_0- \tau, b], \mathbb{R}) \cap C^{1}([t_0, b], \mathbb{R})$ and sequence generated by the algorithm $(1.2)$ converges to $p.$

Proof. Let $\{x_n\}$ be a iterative sequence generated by $(1.2)$ for the following operator:

where $\alpha_n \in (0, 1)$ for all $n \in \mathbb{N}$ such that $\sum_{n = 0}^{\infty}\alpha_n = \infty$. Denote by $p$ the fixed point of $T.$ We will show that $x_n \rightarrow p$ as $n \rightarrow \infty.$

For $t \in [t_0- \tau, t_0]$, it is easy to see that $x_n \rightarrow p$ as $n \rightarrow \infty.$

For $t \in [t_0, b],$ we have

Using $(6.3),$ $(6.4),$ $(6.5)$ and $(6.6)$ we get

On using assumption $(A_5),$ we have

Therefore, inductively we get

Since $\alpha_n \in [0, 1],$ for all $n \in \mathbb{N}$, assumption $(A_5)$ yields

Using the fact that $e^{-x} \geq 1 - x$ for all $x \in [0, 1],$ we have

which gives $\lim\limits_{n\to\infty}\|x_{n} - p\|_ {\infty} = 0.$

From the above theorem, we can say that our method will definitely converge to the unique solution of (6.1) which is a main advantage over the other methods available for the same.

7.   Conclusion

In this study a new fixed iteration process (1.2) has been obtained which is utilized to approximate fixed point of Suzuki generalized nonexpansive mappings. Further, We show that our iteration process $(1.2)$ converges faster than the recent M-iteration process $(1.1)$ for contractive-like mappings. It must be noted here that Ullah and Arshad ^[31] did not give the rate of convergence of their process analytically. They claimed just by an example. However, we not only give the proof analytically but also validate with an example. Further, we performed convergence analysis and a non trivial example has been given to illustrate the convergence behaviour. In the last section, we applied our iteration process to find the solution of delay differential equation.

Acknowledgements

We wish to pay our sincere thanks to learned referees for pointing out many omission and motivating us to study deeply for numerical aspects.

Conflict of interest

The authors declare that they have no competing interests.