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Convergence analysis of a novel iteration process with application to a fractional differential equation

Abstract

The objective of this article is to study a three-step iteration process in the framework of Banach spaces and to obtain convergence results for Suzuki generalized nonexpansive mappings. We also provide numerical examples that support our main results and illustrate the convergence behavior of the proposed process. Further, we present a data-dependence result that is also supported by a nontrivial numerical example. Finally, we discuss the solution of a nonlinear fractional differential equation by utilizing our results.

Introduction

Fixed-point theory has been gaining much attention among researchers as it provides useful tools to solve many nonlinear problems that have applications in different fields, like engineering, economics, chemistry, game theory, etc. Iteration processes play a crucial role in finding fixed points of a nonlinear mapping. Due to its simplicity and significance, the class of nonexpansive mappings is one of the most utilized class of nonlinear mappings. Let K be a nonempty closed convex subset of a Banach space E. A mapping \(S: K \rightarrow K\) is said to be nonexpansive if \(\|Sg - Sf\| \leq \|g - f\|\) for all \(g, f \in K\). S is called quasinonexpansive if \(F(S)\neq \emptyset \) and \(\|Sg - q\| \leq \|g - q\|\) for all \(g \in K\), \(q \in F(S)\), where \(F(S)\) is the set of fixed points of S, i.e., \(F(S) = \{ g \in S: Sg = g\}\). It is well known that every nonexpansive mapping with a fixed point is a quasinonexpansive mapping. One can observe that the famous Banach Contraction Principle is no longer true for nonexpansive mappings, i.e., a nonexpansive mapping need not admit a fixed point on a complete metric space. Also, Picard iteration need not be convergent for a nonexpansive map in a complete metric space. This led to the beginning of a new era of fixed-point theory for nonexpansive mappings by using geometric properties. In 1965, Browder [1], Göhde [2] and Kirk [3] gave three basic existence results in respect of nonexpansive mappings. With a view to locating fixed points of nonexpansive mappings, Mann [4], Ishikawa [5] and Halpern [6] introduced three basic iteration processes. In the same area we direct the reader to the recent works [7, 8].

Following this, several authors constructed numerous iteration processes to approximate the fixed points of different classes of nonlinear mappings, mainly Noor iteration [9], Agarwal et al. iteration [10], SP iteration [11], Normal-S iteration [12], Abbas and Nazir iteration [13], Thakur et al. iterations [14, 15], Karakaya et al. iteration [16] and many others.

In 2008, Suzuki [17] introduced a new class of mappings that is larger than the class of nonexpansive mappings and called the defining condition Condition (C), which is also referred to as generalized nonexpansive mappings. A mapping \(S: K \rightarrow K\) defined on a nonempty subset K of a Banach space E is said to satisfy the Condition (C) if

$$\frac{1}{2} \Vert g - Sg \Vert \leq \Vert g - f \Vert \quad\Rightarrow\quad \Vert Sg - Sf \Vert \leq \Vert g - f \Vert $$

for all g and \(f \in K\).

Suzuki obtained few results regarding the existence of fixed points for such mappings. In 2011, Phuengrattana [18] 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 domains and have obtained many convergence results (e.g., [14, 1925]).

Motivated and inspired by such research, we introduce a new iteration process for approximating fixed points of Suzuki generalized nonexpansive mapping as follows:

$$\begin{aligned} \textstyle\begin{cases} g_{1} \in K, \\ e_{n} = S((1-\alpha _{n})g_{n} + \alpha _{n}Sg_{n}), \\ f_{n} = Se_{n}, \\ g_{n+1} = Sf_{n}, \quad n \in \mathbb{N}, \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(\{\alpha _{n}\}\) is a sequence in \((0, 1)\).

The aim of this paper is to prove some convergence results involving process (1.1) for Suzuki generalized nonexpansive mapping. Further, we provide a numerical example to show that our iteration (1.1) converges faster than a number of existing iteration processes, such as Thakur New, Vatan, M and M iterations, etc. Further, we prove a data-dependence result along with an example to validate the analytical proof. In the last section, we use our results to find a solution to a nonlinear fractional differential equation.

Preliminaries

To make our paper self-contained, we collect some basic definitions and required 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 \(g, f \in E\) with \(\|g\| \leq 1\), \(\|f\| \leq 1\) and \(\|g - f\| > \epsilon \), we have

$$\begin{aligned} \biggl\Vert \frac{g + f}{2} \biggr\Vert < 1 - \delta. \end{aligned}$$

Definition 2.2

A Banach space E is said to satisfy Opial’s condition if for any sequence \(\{g_{n}\}\) in E that converges weakly to \(g \in E\), i.e., \(g_{n} \rightharpoonup g\) implies that

$$\begin{aligned} \limsup_{n\to \infty } \Vert g_{n} - g \Vert < \limsup _{n\to \infty } \Vert g_{n} - f \Vert \end{aligned}$$

for all \(f \in E\) with \(f \neq g\).

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 \(S: K \rightarrow E\) is demiclosed at \(f \in E\) if for each sequence \(\{g_{n}\}\) in K and each \(g \in E\), \(g_{n} \rightharpoonup g\) and \(Sg_{n} \rightarrow f\) imply that \(g \in K\) and \(Sg = f\).

Let K be a nonempty closed convex subset of a Banach E, and let \(\{g_{n}\}\) be a bounded sequence in E. For \(g \in E\), we denote:

$$\begin{aligned} r\bigl(g,\{g_{n}\}\bigr)=\limsup_{n\to \infty } \Vert g - g_{n} \Vert . \end{aligned}$$

The asymptotic radius of \(\{g_{n}\}\) relative to K is given by

$$\begin{aligned} r\bigl(K, \{g_{n}\}\bigr)=\inf \bigl\{ r\bigl(g, \{g_{n}\} \bigr):g \in K\bigr\} \end{aligned}$$

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

$$\begin{aligned} A\bigl(K, \{g_{n}\}\bigr)=\bigl\{ g \in K: r\bigl(g, \{g_{n}\}\bigr)=r\bigl(K, \{g_{n}\}\bigr)\bigr\} . \end{aligned}$$

Also, in a uniformly convex Banach space \(A(K, \{g_{n}\})\) consists of exactly one point.

The following lemma due to Schu [26] will be 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 \(\{g_{n}\}\) and \(\{f_{n}\}\) be any two sequences of E such that \(\limsup_{n\to \infty }\|g_{n}\| \leq r\), \(\limsup_{n\to \infty }\|f_{n}\| \leq r\) and \(\limsup_{n\to \infty }\|t_{n}g_{n} + (1-t_{n})f_{n}\| = r\) for some \(r \geq 0\). Then, \(\lim_{n\to \infty }\|g_{n} - f_{n}\| = 0\).

Now, we list a few lemmas involving Suzuki generalized nonexpansive mapping.

Lemma 2.2

([17])

Let K be a nonempty subset of a Banach space E and \(S: K\rightarrow K\) be any mapping. Then:

  1. (i)

    If S is nonexpansive, then S is a Suzuki generalized nonexpansive mapping.

  2. (ii)

    If S is a Suzuki generalized nonexpanisve mapping such that \(F(S) \neq \emptyset \), then S is a quasinonexpansive mapping.

  3. (iii)

    If S is a Suzuki generalized nonexpansive mapping, then \(\|g - Sf\| \leq 3\|g - Sg\| + \|g - f\|\) for all g and \(f \in K\).

Lemma 2.3

([27])

Let S be a Suzuki generalized nonexpansive mapping defined on a subset K of a Banach space E with the Opial property. If a sequence \(\{g_{n}\}\) converges weakly to e and \(\lim_{n\to \infty }\|Sg_{n} - g_{n}\| = 0\), then \(I - S\) is demiclosed at zero.

Lemma 2.4

([17])

If S is a Suzuki generalized nonexpansive mapping defined on a compact convex subset K of a uniformly convex Banach space E, then S has a fixed point.

In 1972, Zamfirescu [28] introduced Zamfirescu mappings that serve as an important generalization for the Banach contraction principle [29]. Later, in 2004, Berinde [30] gave a more general class of mappings known as quasicontractive mappings. Following this, Imoru and Olantiwo [31] gave the following definition:

Definition 2.3

A mapping \(S: K\rightarrow K\) is known as a 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 \(g, f \in K\), we have

$$\begin{aligned} \Vert Sg - Sf \Vert \leq \delta \Vert g - f \Vert + \varphi \bigl( \Vert g - Sg \Vert \bigr). \end{aligned}$$

Clearly, the class of contractive-like mappings is wider than the class of quasicontractive mappings. For more comparisons see [32].

Next, we recall the following definition and lemma that will be useful in proving our data-dependence result.

Definition 2.4

Let \(S, \tilde{S}: K \rightarrow K\) be two operators, then is said to be an approximate operator of S if \(\Vert Sg - \tilde{S}g\Vert \leq \epsilon \) for all \(g \in K\) and \(\epsilon > 0\) is a fixed number.

Lemma 2.5

([33])

If \(\{a_{n}\}\) is a nonnegative real sequence and there exists an \(m \in \mathbb{N}\) such that for all \(n \geq m\) we have the following condition:

$$a_{n+1} \leq (1 - u_{n})a_{n} +u_{n}v_{n} $$

such that \(u_{n} \in (0, 1)\) for all \(n \in \mathbb{N}\), \(\sum_{n= 0}^{\infty }u_{n} = \infty \) and \(v_{n} \geq 0\) for all \(n \in \mathbb{N}\), then the following inequality holds:

$$\begin{aligned} 0 \leq \limsup_{n\to \infty }a_{n} \leq \limsup _{n \to \infty }v_{n}. \end{aligned}$$

Convergence results

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

Lemma 3.1

Let S be a Suzuki generalized nonexpansive mapping defined on a nonempty closed convex subset K of a Banach space E with \(F(S) \neq \emptyset \). Let \(\{g_{n}\}\) be the sequence defined by the iteration process (1.1). Then, \(\lim_{n\to \infty } \|g_{n} - q\|\) exists for all \(q \in F(S)\).

Proof

Let \(q \in F(S)\) and \(e \in K\). Since S is a Suzuki generalized nonexpansive mapping,

\(\frac{1}{2}\|q - Sq\| = 0 \leq \|q - e\|\) implies that \(\|Sq - Se\| \leq \|q - e\|\).

Consider,

$$\begin{aligned} \begin{aligned} \Vert e_{n} - q \Vert &= \bigl\Vert S\bigl((1-\alpha _{n})g_{n} + \alpha _{n}Sg_{n} \bigr) - q \bigr\Vert \\ &\leq \bigl\Vert (1-\alpha _{n})g_{n} + \alpha _{n}Sg_{n} - q \bigr\Vert \\ &\leq (1 - \alpha _{n}) \Vert g_{n} - q \Vert + \alpha _{n} \Vert g_{n} - q \Vert \\ &= \Vert g_{n} - q \Vert \end{aligned} \end{aligned}$$
(3.1)

and

$$\begin{aligned} \begin{aligned} \Vert f_{n} - q \Vert &= \Vert Se_{n} - q \Vert \\ &\leq \Vert e_{n} - q \Vert \\ &\leq \Vert g_{n} - q \Vert . \end{aligned} \end{aligned}$$
(3.2)

Using (3.1) and (3.2), we obtain

$$\begin{aligned} \begin{aligned} \Vert g_{n+1} - q \Vert &= \Vert Sf_{n} - q \Vert \\ &\leq \Vert f_{n} - q \Vert \\ &\leq \Vert g_{n} - q \Vert . \end{aligned} \end{aligned}$$
(3.3)

Thus, \(\{\|g_{n} - q\|\}\) is a bounded and decreasing sequence of reals and hence \(\lim_{n\to \infty } \|g_{n} - q\|\) exists. □

Lemma 3.2

Let S be a Suzuki generalized nonexpansive mapping defined on a nonempty closed convex subset K of a Banach space E. Let \(\{g_{n}\}\) be the sequence defined by the iteration process (1.1). Then, \(F(S) \neq \emptyset \) if and only if \(\{g_{n}\}\) is bounded and \(\lim_{n\to \infty } \|Sg_{n} - g_{n}\| = 0\).

Proof

Suppose \(F(S) \neq \emptyset \) and let \(q \in F(S)\). Then, by Lemma 3.1, \(\lim_{n\to \infty }\|g_{n} - q\|\) exists and \(\{g_{n}\}\) is bounded. Let

$$\begin{aligned} \lim_{n\to \infty } \Vert g_{n} - q \Vert = r. \end{aligned}$$
(3.4)

From (3.1) and (3.4), we have

$$\begin{aligned} \limsup_{n\to \infty } \Vert e_{n} - q \Vert \leq r. \end{aligned}$$
(3.5)

Also, from (3.2) and (3.4), we obtain

$$\begin{aligned} \limsup_{n\to \infty } \Vert f_{n} - q \Vert \leq r. \end{aligned}$$
(3.6)

Further,

$$\begin{aligned} \lim_{n\to \infty } \Vert g_{n+1} - q \Vert = r = \lim _{n\to \infty } \Vert Sf_{n} - q \Vert , \end{aligned}$$

which gives

$$\begin{aligned} \lim_{n\to \infty } \Vert Sf_{n} - q \Vert = r. \end{aligned}$$

Using Condition (C), we have

$$\begin{aligned} \Vert Sf_{n} - q \Vert \leq \Vert f_{n} - q \Vert , \end{aligned}$$

which yields

$$\begin{aligned} r \leq \liminf_{n\to \infty } \Vert f_{n} - q \Vert . \end{aligned}$$
(3.7)

From (3.6) and (3.7), we obtain

$$\begin{aligned} \lim_{n\to \infty } \Vert f_{n} - q \Vert = r. \end{aligned}$$
(3.8)

Now,

$$\begin{aligned} \lim_{n\to \infty } \Vert g_{n} - q \Vert = r = \lim _{n\to \infty } \Vert f_{n} - q \Vert = \lim _{n\to \infty } \Vert Se_{n} - q \Vert . \end{aligned}$$
(3.9)

Since S is a Suzuki generalized nonexpansive mapping, we have

$$\begin{aligned} \Vert Se_{n} - q \Vert \leq \Vert e_{n} - q \Vert , \end{aligned}$$

which gives

$$\begin{aligned} r \leq \liminf_{n\to \infty } \Vert e_{n} - q \Vert . \end{aligned}$$
(3.10)

Using (3.5) and (3.10), we obtain

$$\begin{aligned} \lim_{n\to \infty } \Vert e_{n} - q \Vert = r. \end{aligned}$$
(3.11)

Consider,

$$\begin{aligned} \begin{aligned} \Vert e_{n} - q \Vert &= \bigl\Vert S\bigl((1-\alpha _{n})g_{n} + \alpha _{n}Sg_{n} \bigr) - q \bigr\Vert \\ &\leq \bigl\Vert (1-\alpha _{n})g_{n} + \alpha _{n}Sg_{n} - q \bigr\Vert \\ &\leq \Vert g_{n} - q \Vert . \end{aligned} \end{aligned}$$

From (3.4) and (3.11), we obtain

$$\begin{aligned} \lim_{n\to \infty } \bigl\Vert (1-\alpha _{n})g_{n} + \alpha _{n}Sg_{n} - q \bigr\Vert = r. \end{aligned}$$
(3.12)

On using Lemma 2.1 together with (3.4) and (3.12), we obtain

$$\begin{aligned} \lim_{n\to \infty } \Vert g_{n} - Sg_{n} \Vert = 0. \end{aligned}$$

Conversely, suppose that \(\{g_{n}\}\) is bounded and \(\lim_{n\to \infty }\|g_{n} - Sg_{n}\| = 0\). Let \(q \in A(K,\{g_{n}\})\), we have

$$\begin{aligned} \begin{aligned} r\bigl(Sq, \{g_{n}\}\bigr) &= \limsup _{n\to \infty } \Vert g_{n} - Sq \Vert \\ &\leq \limsup_{n\to \infty }\bigl(3 \Vert Sg_{n} - g_{n} \Vert + \Vert g_{n} - q \Vert \bigr) \\ &= \limsup_{n\to \infty } \Vert g_{n} - q \Vert \\ &= r\bigl(q, \{g_{n}\}\bigr). \end{aligned} \end{aligned}$$

This implies that \(Sq \in A(K, \{g_{n}\})\). Since E is uniformly convex, \(A(K, \{g_{n}\})\) is singleton, therefore we obtain \(Sq = q\). □

Theorem 3.1

Let S be a Suzuki generalized nonexpansive mapping defined on a nonempty closed convex subset K of a Banach space E that satisfies Opial’s condition with \(F(S) \neq \emptyset \). If \(\{g_{n}\}\) is the sequence defined by the iteration process (1.1), then \(\{g_{n}\}\) converges weakly to a fixed point of S.

Proof

Let \(q \in F(S)\). Then, from Lemma 3.1\(\lim_{n\to \infty }\|g_{n} - q\|\) exists. In order to show the weak convergence of the iteration process (1.1) to a fixed point of S, we will prove that \(\{g_{n}\}\) has a unique weak subsequential limit in \(F(S)\). For this, let \(\{g_{n_{j}}\}\) and \(\{g_{n_{k}}\}\) be two subsequences of \(\{g_{n}\}\) that converge weakly to u and v, respectively. By Lemma 3.2, we have \(\lim_{n\to \infty }\|Sg_{n} - g_{n}\| = 0\) and using the Lemma 2.3, we have \(I - S\) is demiclosed at zero. Hence, \(u, v \in F(S)\).

Next, we show the uniqueness. Since \(u, v \in F(S)\), \(\lim_{n\to \infty }\|g_{n} - u\|\) and \(\lim_{n\to \infty }\|g_{n} - v\|\) exists. Let \(u \neq v\). Then, by Opial’s condition, we obtain

$$\begin{aligned} \begin{aligned} \lim_{n\to \infty } \Vert g_{n} - u \Vert &= \lim_{j\to \infty } \Vert g_{n_{j}} - u \Vert \\ &< \lim_{j\to \infty } \Vert g_{n_{j}} - v \Vert \\ &= \lim_{n\to \infty } \Vert g_{n} - v \Vert \\ &= \lim_{k\to \infty } \Vert g_{n_{k}} - v \Vert \\ &< \lim_{k\to \infty } \Vert g_{n_{k}} - u \Vert \\ &= \lim_{n\to \infty } \Vert g_{n} - u \Vert , \end{aligned} \end{aligned}$$

which is a contradiction, hence \(u = v\). Thus, \(\{g_{n}\}\) converges weakly to a fixed point of S. □

Next, we establish some strong convergence results for iteration process (1.1).

Theorem 3.2

Let S be a Suzuki generalized nonexpansive mapping defined on a nonempty compact convex subset K of a uniformly convex Banach space E. If \(\{g_{n}\}\) is the iterative sequence defined by the iteration process (1.1), then \(\{g_{n}\}\) converges strongly to a fixed point of S.

Proof

Using Lemma 2.4, we obtain \(F(S) \neq \emptyset \). Hence, by Lemma 3.2, we have \(\lim_{n\to \infty }\|Sg_{n} - g_{n}\| = 0\). Since K is compact, there exists a subsequence \(\{g_{n_{k}}\}\) of \(\{g_{n}\}\) such that \(\{g_{n_{k}}\}\) converges strongly to q for some \(q \in K\). From Lemma 2.2(iii), we have

$$\Vert g_{n_{k}} - Sq \Vert \leq 3 \Vert Sg_{n_{k}} - g_{n_{k}} \Vert + \Vert g_{n_{k}} - q \Vert $$

for all \(n \geq 1\). Letting \(k \rightarrow \infty \), we obtain that \(\{g_{n_{k}}\}\) converges to Sq. This implies that \(Sq = q\), i.e., \(q \in F(S)\). Further, \(\lim_{n\to \infty }\|g_{n} - q\|\) exists by Lemma 3.1. Hence, q is the strong limit of the sequence \(\{g_{n}\}\).

A mapping \(S: K \rightarrow K\) is said to satisfy the Condition (A) ([34]) if there exists a nondecreasing function \(p:[0,\infty ) \rightarrow [0,\infty )\) with \(p(0)=0\) and \(p(r)>0\) for all \(r\in (0,\infty )\) such that \(\|g - Sg\| \geq p(d(g, F(S)))\) for all \(g \in K\), where \(d(g, F(S)) = \inf \{\|g - q\|: q \in F(S)\}\). □

Theorem 3.3

Let S be a Suzuki generalized nonexpansive mapping defined on a nonempty closed convex subset K of a uniformly convex Banach space E such that \(F(S) \neq \emptyset \) and let \(\{g_{n}\}\) be the sequence defined by (1.1). If S satisfies Condition (A), then \(\{g_{n}\}\) converges strongly to a fixed point of S.

Proof

By Lemma 3.1, \(\lim_{n\to \infty } \|g_{n} - q\|\) exists and \(\|g_{n+1} - q\| \leq \|g_{n} - q\|\) for all \(q \in F(S)\).

We obtain

$$\begin{aligned} \inf_{q \in F(S)} \Vert g_{n+1} - q \Vert \leq \inf _{q \in F(S)} \Vert g_{n} - q \Vert , \end{aligned}$$

which yields

$$\begin{aligned} d\bigl(g_{n+1}, F(S)\bigr)\leq d\bigl(g_{n}, F(S) \bigr). \end{aligned}$$

This shows that the sequence \(\{d(g_{n}, F(S))\}\) is decreasing and bounded below, hence \(\lim_{n\to \infty } d(g_{n}, F(S))\) exists.

Let \(\lim_{n\to \infty }\|g_{n} - q\| = r\) for some \(r \geq 0\). If \(r = 0\), then the result follows. Assume \(r > 0\). Also, by Lemma 3.2 we have \(\lim_{n\to \infty } \|g_{n} - Sg_{n}\|=0\).

It follows from Condition (A) that

$$\begin{aligned} \lim_{n\to \infty } p\bigl(d\bigl(g_{n}, F(S)\bigr)\bigr) \leq \lim_{n \to \infty } \Vert g_{n} - Sg_{n} \Vert =0, \end{aligned}$$

hence \(\lim_{n\to \infty } p(d(g_{n},F(S)))=0\).

Since p is a nondecreasing function satisfying \(p(0)=0\) and \(p(r) > 0\) for all \(r \in (0, \infty )\), \(\lim_{n\to \infty } d(g_{n}, F(S))=0\). Hence, we have a subsequence \(\{g_{n_{k}}\}\) of \(\{g_{n}\}\) and a sequence \(\{y_{k} \} \subset F(S)\) such that

$$\Vert g_{n_{k}} - y_{k} \Vert < \frac{1}{2^{k}} $$

for all \(k \in \mathbb{N}\). Using (3.4), we obtain

$$\begin{aligned} \Vert g_{n_{k+1}} - y_{k} \Vert < \Vert g_{n_{k}} - y_{k} \Vert < \frac{1}{2^{k}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} \Vert y_{k+1} - y_{k} \Vert & \leq \Vert y_{k+1} - g_{k+1} \Vert + \Vert g_{k+1} - y_{k} \Vert \\ &\leq \frac{1}{2^{k+1}} + \frac{1}{2^{k}} \\ &< \frac{1}{2^{k-1}} \rightarrow 0\quad \text{as }n \rightarrow \infty. \end{aligned} \end{aligned}$$

This implies that \(\{y_{k}\}\) is a Cauchy sequence in \(F(S)\). Since \(F(S)\) is closed, \(\{y_{k}\}\) converges to a point \(q \in F(S)\). Then, \(\{g_{n_{k}}\}\) converges strongly to q. Since \(\lim_{n\to \infty }\|g_{n} - q\|\) exists, we obtain \(g_{n} \rightarrow q \in F(S)\). This completes the proof. □

Now, we will construct an example of a Suzuki generalized nonexpansive mapping that is not a nonexpansive mapping. Then, using that example, we will show that our iteration scheme (1.1) has a higher speed of convergence than a number of existing iteration schemes.

Example 1

Define a mapping \(S: [0, 1] \rightarrow [0, 1]\) by

$$\begin{aligned} Sg = \textstyle\begin{cases} 1-g,& g \in [0, \frac{1}{14}), \\ \frac{g+13}{14},& g \in [\frac{1}{14}, 1]. \end{cases}\displaystyle \end{aligned}$$

First, we show that S is not a nonexpansive mapping. For this, take \(g = \frac{7}{100}\) and \(f = \frac{1}{14}\). Then,

$$\begin{aligned} \Vert Sg - Sf \Vert = \biggl\Vert (1 - g) - \biggl(\frac{f+13}{14} \biggr) \biggr\Vert = \frac{72}{19{,}600} \end{aligned}$$

and

$$\begin{aligned} \Vert g - f \Vert = \vert g - f \vert = \frac{2}{1400}. \end{aligned}$$

Clearly, \(\|Sg - Sf\| > \|g - f\|\), which proves that S is not a nonexpansive mapping.

Now, we show that S satisfies Condition \((C)\). For this, consider the following cases:

Case-I: Let \(g \in [0, \frac{1}{14})\), then \(\frac{1}{2}\|g - Sg\| = \frac{1}{2}|2g - 1| = \frac{1}{2}(1 - 2g)\). For \(\frac{1}{2}\|g - Sg\| \leq \|g - f\|\), we must have \(\frac{1}{2}(1 - 2g) \leq \|g - f\|\), i.e., \(\frac{1}{2}(1 - 2g) \leq |g - f|\). Here, note that the case \(f < g\) is not possible. Hence, we are left with only one case when \(f > g\), which gives \(\frac{1}{2}(1 - 2g) \leq f - g\), which yields \(f \geq \frac{1}{2}\). Hence, \(f \in [\frac{1}{2}, 1]\). Now, we have \(g \in [0, \frac{1}{14})\) and \(f \in [\frac{1}{2}, 1]\). Hence,

$$\begin{aligned} \Vert Sg - Sf \Vert = \biggl\Vert (1 - g) - \frac{f + 13}{14} \biggr\Vert = \biggl\vert \frac{14g + f - 1}{14} \biggr\vert < \frac{1}{14} \end{aligned}$$

and

$$\begin{aligned} \Vert g - f \Vert = \vert g - f \vert > \frac{6}{14}. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{1}{2} \Vert g - Sg \Vert \leq \Vert g - f \Vert \quad\Rightarrow\quad \Vert Sg - Sf \Vert \leq \Vert g - f \Vert . \end{aligned}$$

Case-II: Let \(g \in [\frac{1}{14}, 1]\), then \(\frac{1}{2}\|g - Sg\| = \frac{1}{2}|g - \frac{g + 13}{14}| = \frac{13 - 13g}{28}\). For \(\frac{1}{2}\|g - Sg\| \leq \|g - f\|\), we must have \(\frac{13 - 13g}{28} \leq \|g - f\|\), i.e., \(\frac{13 - 13g}{28} \leq |g - f|\). Here, we have two possibilities.

A: When \(g < f\), we obtain \(\frac{13 - 13g}{28} \leq f - g\), i.e., \(f \geq \frac{13 + 15g}{28}\). Hence, \(f \in [\frac{197}{392}, 1] \subset [\frac{1}{14}, 1]\), which gives \(\|Sg - Sf\| = \frac{1}{14}\|g - f\| \leq \|g - f\|\). Hence,

$$\begin{aligned} \frac{1}{2} \Vert g - Sg \Vert \leq \Vert g - f \Vert \quad\Rightarrow\quad \Vert Sg - Sf \Vert \leq \Vert g - f \Vert . \end{aligned}$$

B: When \(g > f\), then \(\frac{13 - 13g}{28} \leq g - f\), i.e., \(f \leq \frac{41g - 13}{28}\), which gives \(f \in [0, 1]\). Also, \(\frac{28f + 13}{41} \leq g\), which yields \(g \in [\frac{13}{41}, 1]\). Here, for \(g \in [\frac{13}{41}, 1]\) and \(f \in [\frac{1}{14}, 1]\) Case IIA can be used. Hence, we only need to verify when \(g \in [\frac{13}{41}, 1]\) and \(f \in [0, \frac{1}{14})\). For this,

$$\begin{aligned} \Vert Sg - Sf \Vert = \biggl\vert \frac{g + 13}{14} - (1 - f) \biggr\vert = \frac{1}{14} \vert 14f + g -1 \vert \leq \frac{1}{14} \end{aligned}$$

and

$$\begin{aligned} \Vert g - f \Vert = \vert g - f \vert > \frac{141}{574}. \end{aligned}$$

Hence, \(\|Sg - Sf\| \leq \|g - f\|\). Thus, mapping S satisfies the Condition (C) for all the possible cases.

Now, using the above example, we will show that the iteration algorithm (1.1) converges faster than Thakur New, Vatan, M and M iterations. Let \(\alpha _{n} = \tau _{n} = \sqrt{\frac{n}{n+100}}\) for all \(n \in \mathbb{N}\) and \(g_{1} = 0.00789\), then we obtain Table 1 and Fig. 1 showing the errors.

Figure 1
figure 1

Graph corresponding to Table 1

Table 1 Values of different iterations

It is evident from Table 1 and Fig. 1 that our iteration process (1.1) converges at a higher speed than the above-mentioned schemes.

Data dependence

In this section, we prove a data-dependence result for the iteration scheme (1.1) and we verify our theoretical result with the help of a numerical example.

Theorem 4.1

Let S be a contractive-like mapping defined on a nonempty closed convex subset K of a Banach space E with \(F(S) \neq \emptyset \). If \(\{g_{n}\}\) is a sequence defined by (1.1), then \(\{g_{n}\}\) converges to the fixed point of S.

Proof

From (1.1), for any \(q \in F(S)\), by using the fact that \((1 - (1 - \delta )\alpha _{n}) < 1\) for all \(n \in \mathbb{N}\) we have

$$\begin{aligned} \begin{aligned} \Vert e_{n} - q \Vert &= \bigl\Vert S\bigl((1-\alpha _{n})g_{n} + \alpha _{n}Sg_{n} \bigr) - q \bigr\Vert \\ &\leq \delta \bigl\Vert (1-\alpha _{n})g_{n} + \alpha _{n}Sg_{n} - q \bigr\Vert \\ &\leq \delta \bigl((1 - \alpha _{n}) \Vert g_{n} - q \Vert + \alpha _{n}\delta \Vert g_{n} - q \Vert \bigr) \\ &= \delta \bigl(1 - (1 - \delta )\alpha _{n}\bigr) \Vert g_{n} - q \Vert \\ &\leq \delta \Vert g_{n} - q \Vert \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Vert f_{n} - q \Vert &= \Vert Se_{n} - q \Vert \\ &\leq \delta \Vert e_{n} - q \Vert \\ &\leq \delta ^{2} \Vert g_{n} - q \Vert . \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} \begin{aligned} \Vert g_{n+1} - q \Vert &= \Vert Sf_{n} - q \Vert \\ &\leq \delta \Vert f_{n} - q \Vert \\ &\leq \delta ^{3} \Vert g_{n} - q \Vert \\ & \vdots \\ &\leq \delta ^{3n} \Vert g_{1} - q \Vert . \end{aligned} \end{aligned}$$

Since, \(0 \leq \delta < 1\), we obtain

$$\begin{aligned} \lim_{n\to \infty }\delta ^{3n} = 0. \end{aligned}$$

Hence,

$$\begin{aligned} \lim_{n\to \infty } \Vert g_{n+1} - q \Vert =0. \end{aligned}$$

 □

Theorem 4.2

Let S be a contractive-like mapping defined on a nonempty closed convex subset K of a Banach space E with \(F(S) \neq \emptyset \) and let be an approximate operator of S. Let \(\{g_{n}\}\) be a sequence defined by (1.1) for S, we define a sequence \(\{\tilde{g_{n}}\}\) involving as follows:

$$\begin{aligned} \textstyle\begin{cases} \tilde{g_{1}} \in K, \\ \tilde{e_{n}} = \tilde{S}((1-\alpha _{n})\tilde{g_{n}} + \alpha _{n} \tilde{S}\tilde{g_{n}}), \\ \tilde{f_{n}} = \tilde{S}\tilde{e_{n}}, \\ \tilde{g_{n+1}} = \tilde{S}\tilde{f_{n}}, \quad n \in \mathbb{N,} \end{cases}\displaystyle \end{aligned}$$
(4.1)

where \(\{\alpha _{n}\}\) is a sequence in \((0, 1)\) such that \(\frac{1}{2} \leq \alpha _{n}\) for all \(n \in \mathbb{N}\) and \(\sum_{n =0}^{\infty }\alpha _{n} = \infty \). If q and are fixed points of S and , respectively, such that \(\{\tilde{g_{n}}\} \rightarrow \tilde{q}\) as \(n \rightarrow \infty \), then we have

$$\begin{aligned} \Vert q - \tilde{q} \Vert \leq \frac{8\epsilon }{1 - \delta }. \end{aligned}$$

Proof

From (1.1) and (4.1), we have

$$\begin{aligned} \begin{aligned} \Vert e_{n} - \tilde{e_{n}} \Vert ={}& \bigl\Vert S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - \tilde{S}\bigl((1-\alpha _{n})\tilde{g_{n}} + \alpha _{n}\tilde{S} \tilde{g_{n}}\bigr) \bigr\Vert \\ \leq{}& \bigl\Vert S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1-\alpha _{n}) \tilde{g_{n}} + \alpha _{n}\tilde{S}\tilde{g_{n}} \bigr) \bigr\Vert \\ & {}+ \bigl\Vert S\bigl((1-\alpha _{n})\tilde{g_{n}} + \alpha _{n}\tilde{S} \tilde{g_{n}}\bigr) - \tilde{S} \bigl((1-\alpha _{n})\tilde{g_{n}} + \alpha _{n} \tilde{S}\tilde{g_{n}}\bigr) \bigr\Vert \\ \leq {}&\delta \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - \bigl((1- \alpha _{n})\tilde{g_{n}} + \alpha _{n}\tilde{S} \tilde{g_{n}}\bigr) \bigr\Vert \bigr) \\ &{} + \varphi \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \bigr) + \epsilon \\ \leq{}& \delta \bigl((1 - \alpha _{n}) \Vert g_{n} - \tilde{g_{n}} \Vert + \alpha _{n} \Vert Sg_{n} - \tilde{S}\tilde{g_{n}} \Vert \bigr) \\ &{} + \varphi \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \bigr) + \epsilon \\ \leq{}& \delta (1 - \alpha _{n}) \Vert g_{n} - \tilde{g_{n}} \Vert + \alpha _{n} \delta \Vert Sg_{n} - S\tilde{g_{n}} \Vert + \alpha _{n} \delta \Vert S \tilde{g_{n}} - \tilde{S}\tilde{g_{n}} \Vert \\ &{} + \varphi \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \bigr) + \epsilon \\ \leq{}& \delta \bigl(1 - \alpha _{n}(1 - \delta )\bigr) \Vert g_{n} - \tilde{g_{n}} \Vert + \alpha _{n} \delta \varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr) + (1 + \delta \alpha _{n}) \epsilon \\ &{} + \varphi \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \bigr) \end{aligned} \end{aligned}$$
(4.2)

and

$$\begin{aligned} \begin{aligned} \Vert f_{n} - \tilde{f_{n}} \Vert &= \Vert Se_{n} - \tilde{S} \tilde{e_{n}} \Vert \\ & \leq \Vert Se_{n}- S\tilde{e_{n}} \Vert + \Vert S \tilde{e_{n}} - \tilde{S} \tilde{e_{n}} \Vert \\ & \leq \delta \Vert e_{n} - \tilde{e_{n}} \Vert + \varphi \bigl( \Vert e_{n} - Se_{n} \Vert \bigr) + \epsilon. \end{aligned} \end{aligned}$$
(4.3)

Now, (4.2) and (4.3) give

$$\begin{aligned} & \Vert g_{n+1} - \tilde{g_{n+1}} \Vert \\ &\quad = \Vert Sf_{n} - \tilde{S}\tilde{f_{n}} \Vert \\ &\quad \leq \delta \Vert f_{n} - \tilde{f_{n}} \Vert + \varphi \bigl( \Vert f_{n} - Sf_{n} \Vert \bigr) + \epsilon \\ &\quad\leq \delta \bigl(\delta \Vert e_{n} - \tilde{e_{n}} \Vert + \varphi \bigl( \Vert e_{n} - Se_{n} \Vert \bigr) + \epsilon \bigr)+ \varphi \bigl( \Vert f_{n} - Sf_{n} \Vert \bigr) + \epsilon \\ &\quad= \delta ^{2} \Vert e_{n} - \tilde{e_{n}} \Vert + \delta \varphi \bigl( \Vert e_{n} - Se_{n} \Vert \bigr) + \varphi \bigl( \Vert f_{n} - Sf_{n} \Vert \bigr) + (1 + \delta )\epsilon \\ &\quad \leq \delta ^{2}\bigl(\delta \bigl(1 - \alpha _{n}(1 - \delta )\bigr) \Vert g_{n} - \tilde{g_{n}} \Vert + \alpha _{n}\delta \varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr) \\ &\qquad {}+ (1 + \delta \alpha _{n})\epsilon + \varphi \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n} \bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \bigr)\bigr) \\ & \qquad{}+ \delta \varphi \bigl( \Vert e_{n} - Se_{n} \Vert \bigr) + \varphi \bigl( \Vert f_{n} - Sf_{n} \Vert \bigr) + (1 + \delta )\epsilon \\ &\quad = \delta ^{3} \bigl(1 - \alpha _{n}(1 - \delta )\bigr) \Vert g_{n} - \tilde{g_{n}} \Vert \\ &\qquad{} + \delta ^{2}\varphi \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \bigr) \\ &\qquad{} + \alpha _{n}\delta ^{3}\varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr)+\delta \varphi \bigl( \Vert e_{n} - Se_{n} \Vert \bigr) + \varphi \bigl( \Vert f_{n} - Sf_{n} \Vert \bigr) + \bigl(1 + \delta + \delta ^{2} + \delta ^{3}\alpha _{n}\bigr)\epsilon. \end{aligned}$$
(4.4)

Since S is a contractive-like operator with \(q \in F(S)\), from Theorem 4.1 it follows that \(\lim_{n\to \infty }\Vert g_{n} - q\Vert = 0\). Hence,

$$\begin{aligned} \begin{aligned} 0 &\leq \Vert g_{n} - Sg_{n} \Vert \\ &\leq \Vert g_{n} - q \Vert + \Vert Sq - Sg_{n} \Vert \\ &\leq (1+ \delta ) \Vert g_{n} - q \Vert \rightarrow 0\quad \text{as }n \rightarrow \infty. \end{aligned} \end{aligned}$$
(4.5)

Also,

$$\begin{aligned} \begin{aligned} 0 &\leq \Vert e_{n} - Se_{n} \Vert \\ &\leq \Vert e_{n} - q \Vert + \Vert Sq - Se_{n} \Vert \\ &\leq (1+ \delta ) \Vert e_{n} - q \Vert \\ &= (1 + \delta ) \bigl\Vert S\bigl((1 - \alpha _{n})g_{n} + Sg_{n}\bigr) - q \bigr\Vert \\ &\leq (1 + \delta )\delta \Vert g_{n} - q \Vert \rightarrow 0 \quad \text{as }n \rightarrow \infty \end{aligned} \end{aligned}$$
(4.6)

and

$$\begin{aligned} \begin{aligned} 0 &\leq \Vert f_{n} - Sf_{n} \Vert \\ &\leq (1+ \delta ) \Vert f_{n} - q \Vert \\ &= (1 + \delta ) \Vert Se_{n} - q \Vert \\ &\leq (1 + \delta )\delta ^{2} \Vert g_{n} - q \Vert \rightarrow 0 \quad\text{as }n \rightarrow \infty. \end{aligned} \end{aligned}$$
(4.7)

As φ is a continuous function (4.5), (4.6) and (4.7) yield

$$\begin{aligned} \lim_{n\to \infty }\varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr) = \lim_{n\to \infty }\varphi \bigl( \Vert e_{n} - Se_{n} \Vert \bigr) = \lim _{n \to \infty }\varphi \bigl( \Vert f_{n} - Sf_{n} \Vert \bigr) = 0. \end{aligned}$$
(4.8)

On using (4.5) and (4.8), we obtain

$$\begin{aligned} 0 \leq{}& \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \\ \leq{}& (1 - \alpha _{n}) \bigl\Vert g_{n} - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n} \bigr) \bigr\Vert + {\alpha _{n}}^{2} \delta \Vert g_{n} - Sg_{n} \Vert + \alpha _{n} \varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr) \\ \leq{}& (1 - \alpha _{n}) \Vert g_{n} - q \Vert + \bigl\Vert Sq - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \\ &{} + {\alpha _{n}}^{2} \delta \Vert g_{n} - Sg_{n} \Vert + \alpha _{n} \varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr) \\ \leq{}& (1 - \alpha _{n}) \Vert g_{n} - q \Vert + \delta \bigl\Vert q - \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert + {\alpha _{n}}^{2} \delta \Vert g_{n} - Sg_{n} \Vert \\ &{} + \alpha _{n} \varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr) \\ \leq {}& (1 - \alpha _{n}) \bigl(1+ \delta - \alpha _{n} \delta + {\delta }^{2} \alpha _{n}\bigr) \Vert g_{n} - q \Vert + {\alpha _{n}}^{2} \delta \Vert g_{n} - Sg_{n} \Vert + \alpha _{n} \varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr) \\ \rightarrow{}& 0 \quad\text{as }n \rightarrow \infty. \end{aligned}$$

Again, using the fact that φ is a continuous function, we have

$$\begin{aligned} \lim_{n\to \infty }\varphi \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \bigr) = 0. \end{aligned}$$
(4.9)

Now, using the fact that \(\delta \in (0, 1)\) and \(\alpha _{n} \in (0, 1)\) with \(\alpha _{n} \geq \frac{1}{2}\), (4.4) transforms into

$$\begin{aligned} &\Vert g_{n+1} - \tilde{g_{n+1}} \Vert \\ &\quad \leq \bigl(1 - \alpha _{n}(1 - \delta )\bigr) \Vert g_{n} - \tilde{g_{n}} \Vert + \varphi \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n} \bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \bigr) \\ &\qquad{} + \alpha _{n}\varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr)+ \varphi \bigl( \Vert e_{n} - Se_{n} \Vert \bigr) + \varphi \bigl( \Vert f_{n} - Sf_{n} \Vert \bigr) + \bigl(1 + \delta + \delta ^{2} + \delta ^{3} \alpha _{n}\bigr)\epsilon \\ &\quad\leq \bigl(1 - \alpha _{n}(1 - \delta )\bigr) \Vert g_{n} - \tilde{g_{n}} \Vert + 2 \alpha _{n} \varphi \bigl( \bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) \bigr\Vert \bigr) \\ &\qquad{} + \alpha _{n}\varphi \bigl( \Vert g_{n} - Sg_{n} \Vert \bigr)+ 2\alpha _{n}\varphi \bigl( \Vert e_{n} - Se_{n} \Vert \bigr) + 2\alpha _{n} \varphi \bigl( \Vert f_{n} - Sf_{n} \Vert \bigr) + 8 \alpha _{n} \epsilon. \end{aligned}$$
(4.10)

Let \(a_{n} = \Vert g_{n} - \tilde{g_{n}}\Vert \), \(u_{n} = \alpha _{n}(1 - \delta ) \in (0, 1)\) and

$$\begin{aligned} v_{n} ={}& \bigl(2\varphi \bigl(\bigl\Vert \bigl((1 - \alpha _{n})g_{n} + \alpha _{n}Sg_{n}\bigr) - S\bigl((1 - \alpha _{n})g_{n} + Sg_{n}\bigr)\bigr\Vert \bigr) + \varphi \bigl(\Vert g_{n} - Sg_{n}\Vert \bigr)\\ &{}+ 2\varphi \bigl(\Vert e_{n} - Se_{n}\Vert \bigr) + 2\varphi \bigl(\Vert f_{n} - Sf_{n}\Vert \bigr) + 8\epsilon\bigr)\\ &{}/(1 - \delta ). \end{aligned}$$

On using (4.8) and (4.9) along with Lemma 2.5, we obtain

$$\begin{aligned} 0 \leq \limsup_{n\to \infty } \Vert g_{n} - \tilde{g_{n}} \Vert \leq \limsup_{n\to \infty }v_{n} = \frac{8\epsilon }{1 - \delta } \end{aligned}$$

that along with Theorem 4.1 yields

$$\begin{aligned} \Vert q - \tilde{q} \Vert \leq \frac{8\epsilon }{1 - \delta }. \end{aligned}$$

Now, we present an example to validate Theorem 4.2 numerically. □

Example 2

Let \(E =\mathbb{R}\) and \(K =[0,6]\). Let \(S:K\rightarrow K\) be a mapping defined as

$$\begin{aligned} S(g)= \textstyle\begin{cases} \frac{g}{4},& g \in [ 0,3), \\ \frac{g}{8},& g \in [ 3,6]. \end{cases}\displaystyle \end{aligned}$$

Proof

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

We need to show that

$$\begin{aligned} \Vert Sg -Sf \Vert \leq \delta \Vert g -f \Vert +\varphi \bigl( \Vert g - Sg \Vert \bigr) \end{aligned}$$
(A)

for all \(g,f\in [ 0,6]\) and δ is a constant in \([0,1)\).

Before going ahead, let us note the following. When \(g\in [ 0,3)\),

$$\begin{aligned} \Vert g - Sg \Vert = \biggl\Vert g-\frac{g}{4} \biggr\Vert = \frac{3g}{4} \end{aligned}$$

and

$$\begin{aligned} \varphi \biggl(\frac{3g}{4}\biggr)=\frac{3g}{24}=\frac{g}{8}. \end{aligned}$$
(4.11)

Similarly, when \(g \in [ 3,6]\), then

$$\begin{aligned} \Vert g -Sg \Vert = \biggl\Vert g-\frac{g}{8} \biggr\Vert = \frac{7g}{8} \end{aligned}$$

and

$$\begin{aligned} \varphi \biggl(\frac{7g}{8}\biggr)=\frac{7g}{48}. \end{aligned}$$
(4.12)

Now consider the following cases:

Case A: Let \(g,f\in [ 0,3)\). Using (4.11) we obtain

$$\begin{aligned} \begin{aligned} \Vert Sg -Sf \Vert & = \biggl\Vert \frac{g}{4}-\frac{f}{4} \biggr\Vert \\ & \leq \frac{1}{4} \Vert g -f \Vert \\ & \leq \frac{1}{4} \Vert g -f \Vert +\frac{g}{8} \\ & = \frac{1}{4} \Vert g - f \Vert +\varphi \biggl( \frac{3g}{4}\biggr) \\ & = \frac{1}{4} \Vert g - f \Vert +\varphi \bigl( \Vert g - Sg \Vert \bigr).\end{aligned} \end{aligned}$$

Hence, (A) is satisfied with \(\delta =\frac{1}{4}\).

Case B: Let \(g\in [ 0,3)\) and \(f\in [ 3,6]\). Using (4.11), we obtain

$$\begin{aligned} \begin{aligned} \Vert Sg - Sf \Vert & = \biggl\Vert \frac{g}{4}-\frac{f}{8} \biggr\Vert \\ & = \biggl\Vert \frac{g}{8}+\frac{g}{8}-\frac{f}{8} \biggr\Vert \\ & \leq \frac{1}{8} \Vert g-f \Vert + \biggl\Vert \frac{g}{8} \biggr\Vert \\ & \leq \frac{1}{4} \Vert g - f \Vert +\varphi \biggl( \frac{3g}{4}\biggr) \\ & = \frac{1}{4} \Vert g - f \Vert +\varphi \bigl( \Vert g - Sg \Vert \bigr).\end{aligned} \end{aligned}$$

Hence, (A) is satisfied with \(\delta =\frac{1}{4}\).

Case C: Let \(g \in [ 3,6]\) and \(f \in [ 0,3)\). Using (4.12), we obtain

$$\begin{aligned} \Vert Sg - Sf \Vert & = \biggl\Vert \frac{g}{8}-\frac{f}{4} \biggr\Vert \\ & = \biggl\Vert \frac{g}{4}-\frac{g}{8}-\frac{f}{4} \biggr\Vert \\ & \leq \frac{1}{4} \Vert g - f \Vert + \biggl\Vert \frac{g}{8} \biggr\Vert \\ & \leq \frac{1}{4} \Vert g - f \Vert + \biggl\Vert \frac{7g}{48} \biggr\Vert \\ & = \frac{1}{4} \Vert g - f \Vert +\varphi \bigl( \Vert g - Sg \Vert \bigr). \end{aligned}$$

Hence, (A) is satisfied with \(\delta =\frac{1}{4}\).

Case D: Let \(g,f\in [ 3,6]\). Using (4.12), we obtain

$$\begin{aligned} \begin{aligned} \Vert Sg - Sf \Vert & = \biggl\Vert \frac{g}{8}-\frac{f}{8} \biggr\Vert \\ & \leq \frac{1}{8} \Vert g - f \Vert + \biggl\Vert \frac{7g}{48} \biggr\Vert \\ & \leq \frac{1}{4} \Vert g - f \Vert + \biggl\Vert \frac{7g}{48} \biggr\Vert \\ & = \frac{1}{4} \Vert g - f \Vert +\varphi \bigl( \Vert g - Sg \Vert \bigr).\end{aligned} \end{aligned}$$

Hence, (A) is satisfied with \(\delta =\frac{1}{4}\).

Consequently, (A) is satisfied for \(\delta =\frac{1}{4}\) and \(\varphi (g)=\frac{g}{6}\) in all the possible cases. Thus, S is a contractive-like mapping.

Next, we define another operator \(\tilde{S}:K\rightarrow K\) as

$$\begin{aligned} \tilde{S}(g)= \textstyle\begin{cases} \frac{g}{4} + \frac{1}{10{,}000},& g \in [ 0,3), \\ \frac{g}{8} +\frac{1}{10{,}000},& g \in [ 3,6], \end{cases}\displaystyle \end{aligned}$$

for all \(g \in K\). Then, it is easy to see that is an approximate operator for S with \(\epsilon = 0.0001\) as \(\Vert Sg - \tilde{S}g\Vert \leq 0.0001\) for all \(g \in K\). Also, \(q = 0\) is the fixed point of S and \(\tilde{q} = 0.00013333333\) is the fixed point of . We obtain Table 2 of iteration values with an initial approximation of 5 and \(\alpha _{n} = \frac{n+3}{n+4}\) for all \(n \in \mathbb{N}\). Also, we have \(\Vert q - \tilde{q}\Vert = \Vert 0 - 0.00013333333 \Vert = 0.00013333333 \leq 0.0010666667 = \frac{8\epsilon }{1 - \delta }\). Hence, we can say that in a situation when it is difficult to calculate the fixed point of a mapping S we can choose a mapping closer to S and the distance between the two fixed points will reduce too.

Table 2 Iteration values involving S and

 □

Application

Fractional differential equations have applications in many fields of engineering and science including diffusive transport, electrical networks, fluid flow, probability and electromagnetic theory. Numerous fixed-point results are available for finding solutions to differential/integral equations involving fractional operators (see [3543]). This section is devoted to the study of a solution of a nonlinear fractional differential equation with the help of iteration process (1.1).

Here, we consider the following fractional differential equation:

$$\begin{aligned} \textstyle\begin{cases} D^{\lambda }g(u)+f(u, g(u))=0 & (0 \leq u \leq 1, 1 < \lambda < 2), \\ g(0)=g(1)=0, \end{cases}\displaystyle \end{aligned}$$
(5.1)

where \(D^{\lambda }\) is the Caputo fractional derivative of order λ and \(f:[0, 1] \times \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function.

Let \(E=C[0, 1]\) be a Banach space of a continuous function endowed with the maximum norm and the Green’s function associated to (5.1) is defined as:

$$\begin{aligned} G(u, v)= \textstyle\begin{cases} \frac{1}{\Gamma (\lambda )}(u(1-v)^{\lambda - 1}-(u - v)^{\lambda - 1}), & 0 \leq v \leq u \leq 1, \\ \frac{u(1-v)^{\lambda -1}}{\Gamma (\lambda )}, & 0 \leq u \leq v \leq 1. \end{cases}\displaystyle \end{aligned}$$

Assume that

$$\begin{aligned} \bigl\vert f(u, a)-f(u, b) \bigr\vert \leq \vert a - b \vert , \end{aligned}$$
(5.2)

for all \(u \in [0, 1]\) and \(a, b \in \mathbb{R}\).

Theorem 5.1

Let \(E = C[0, 1]\) and the operator \(S: E \rightarrow E\) be defined as

$$\begin{aligned} S\bigl(g(u)\bigr) = \int _{0}^{1} G(u, v) f\bigl(v, g(v)\bigr) \,dv, \end{aligned}$$

for all \(u \in [0, 1]\) and \(g \in E\). If the condition (5.2) is satisfied then the iteration process (1.1) converges to the solution of (5.1).

Proof

It is easy to see that \(g \in E\) is a solution of (5.1) if and only if g is a solution of the following integral equation

$$\begin{aligned} g(u) = \int _{0}^{1}G(u, v) f\bigl(v, g(v)\bigr) \,dv. \end{aligned}$$

Let \(g, h \in E\) and \(u \in [0, 1]\). Then,

$$\begin{aligned} \begin{aligned} \bigl\vert S\bigl(g(u)\bigr)-S\bigl(h(u)\bigr) \bigr\vert &\leq \int _{0}^{1}G(u, v) \bigl\vert f\bigl(v, g(v) \bigr)-f\bigl(v, h(v)\bigr) \bigr\vert \,dv \\ &\leq \int _{0}^{1}G(u, v) \bigl\vert g(v)-h(v) \bigr\vert \,dv \\ &\leq \Vert g - h \Vert \biggl( \int _{0}^{1}G(u, v)\,dv\biggr) \\ &\leq \Vert g - h \Vert , \end{aligned} \end{aligned}$$

which yields

$$\begin{aligned} \Vert Sg - Sh \Vert \leq \Vert g - h \Vert , \end{aligned}$$

for all \(g, h \in E\) and for all \(u \in [0, 1]\). With the help of Lemma 2.2(i) and Theorem 3.2, we can say that process (1.1) converges to the solution of (5.1). □

Conclusion

A new iteration process (1.1) is obtained for approximating fixed points of Suzuki generalized nonexpansive mappings. It is proved that it has a higher rate of convergence than the \(M^{*}\)-iteration process for contractive-like mappings. Strong and weak convergence to the fixed point of Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces are proved. The results have been supported by a newly introduced Example 1. We then presented a data-dependence result that is again followed by a numerical example. We ended by providing an application to nonlinear fractional differential equations in the framework of Caputo with a power-law singular kernel. The Riemann–Liouville integral operators used in obtaining the solution representation have semigroup properties that may make them more appropriate when we apply iterative techniques. Also, there are no restrictions on the right-hand side of the considered initial-value problem, as in the case of fractional operators with nonsingular kernels [4446]. Our new iteration process can be used by engineers, computer scientists, physicists as well as mathematicians to solve different problems more efficiently and effectively.

Availability of data and materials

No data were used to support this study.

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Acknowledgements

The last two authors would like to thank Prince Sultan University for paying the APC and for the support through the research lab TAS.

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Uddin, I., Garodia, C., Abdeljawad, T. et al. Convergence analysis of a novel iteration process with application to a fractional differential equation. Adv Cont Discr Mod 2022, 16 (2022). https://doi.org/10.1186/s13662-022-03690-z

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  • DOI: https://doi.org/10.1186/s13662-022-03690-z

MSC

  • 47H09
  • 47H10
  • 54H25

Keywords

  • Suzuki generalized nonexpansive mappings
  • Fixed point
  • Strong and weak convergence
  • Data dependence
  • Nonlinear fractional differential equation