1. Introduction
There are various problems in the field of applied mathematics that can be reformulated by means of fixed point theory. Fixed point theorems provide us with sufficient conditions for the existence of a fixed point, and thus the existence of a solution for the original problem is ensured.
The first step in the direction of a fixed point theory on metric spaces was Banach contraction principle. Came out as an abstraction for Picard iteration, this principle not only ensures the existence and uniqueness of a fixed point for contraction mappings, but also provides us an iterative algorithm to approximate this point. Finding iterative ways to approximate fixed points of different kind of mappings becomes essential as many problems of nonlinear analysis can not be solved analytically. In this regard, Picard iteration was an important starting point for the development of other processes. Despite the success it had with contraction mappings, Krasnoselskii [
1] proved in 1955 that Picard iteration does not always converge to a fixed point when taking a larger class of mappings defined on Banach spaces, namely nonexpansive mappings (for
C being a nonempty closed convex subset of a Banach space
X over the real field
$\mathbb{R}$, a mapping
$T:C\to C$ is said to be nonexpansive if it satisfies the inequality
$\u2225TxTy\u2225\le \u2225xy\u2225$, for all
x,
$y\in C$; moreover, if
$F\left(T\right)\ne \varnothing $, where
$F\left(T\right)=\{x\in C:Tx=x\}$ and
$\u2225Txp\u2225\le \u2225xp\u2225$, for all
$x\in C$ and
$p\in F\left(T\right)$, then
T is called quasinonexpansive). The main reason for such a behavior is that, unlike contraction mappings, successive iteration for nonexpansive mappings needs not converge to a fixed point. From this moment onwards, many others iterative processes have been developed for numerical reckoning fixed points of nonexpansive mappings. For instance, one of the earliest would be Mann’s [
2] iteration process, defined as follows: for an arbitrary chosen
${x}_{0}\in C$, the sequence of successive iterations is defined by
where
$\left\{{\alpha}_{n}\right\}$ is a sequence of real numbers in the interval
$(0,1)$, followed by Ishikawa [
3] iteration, a two step iteration process widely applied for numerical reckoning fixed points of nonexpansive mappings; for a starting point
${x}_{0}\in C$, this iterative scheme is defined by
where
$\left\{{\alpha}_{n}\right\},\phantom{\rule{0.166667em}{0ex}}\left\{{\beta}_{n}\right\}\in (0,1)$. Important to be mentioned would also be Agarwal et al. [
4]’s two step iteration process introduced in 2007: for an arbitrary
${x}_{0}\in C$, define
with
$\left\{{\alpha}_{n}\right\}$ and
$\left\{{\beta}_{n}\right\}$ sequences in
$(0,1)$.
In 2000, Noor [
5] introduced a new threestep iteration scheme for approximation fixed points of nonexpansive mappings as follows: starting with
${x}_{0}\in C$, define
$\left\{{x}_{n}\right\}$ iteratively by
where
$\left\{{\alpha}_{n}\right\}$,
$\left\{{\beta}_{n}\right\}$,
$\left\{{\gamma}_{n}\right\}$ are sequences of real numbers in
$(0,1)$. This has pioneered a number of new threestep iteration techniques as, for example, Abbas and Nazir [
6]: for an arbitrary
${x}_{0}\in C$, the sequence
$\left\{{x}_{n}\right\}$ is defined by
where
$\left\{{\alpha}_{n}\right\}$,
$\left\{{\beta}_{n}\right\}$,
$\left\{{\gamma}_{n}\right\}$ are real number sequences in
$(0,1)$. In the sequel, we will consider the following iterative process defined by Thakur et al. in [
7] for numerical reckoning fixed points of nonexpansive mappings; see, also [
8]: for an arbitrary chosen element
${x}_{0}\in C$, the sequence
$\left\{{x}_{n}\right\}$ is generated by
for all
$n\ge 0$, where
$\left\{{\alpha}_{n}\right\}$,
$\left\{{\beta}_{n}\right\}$,
$\left\{{\gamma}_{n}\right\}$ are sequences of real numbers in
$(0,1)$. We shall refer to this iterative procedure as TTP.
As it can be seen, nonexpansive mappings are an intensely studied category of operators in terms of finding various conditions for the existence of their fixed points (see for example Browder [
9] and Kirk [
10]), in terms of defining iterative processes to approximate the fixed points whose existence has been established, or even in connection with hybrid methods in very recent research directions (see, for instance [
11]). However, in 2008, Suzuki [
12] introduced a new class of mappings on Banach spaces (herein referred as Suzukigeneralized nonexpansive mappings or Suzuki mappings), which properly includes the class of nonexpansive mappings; this came out by limiting the range of points satisfying the nonexpansive inequality. One simple example provided by Suzuki in order to emphasize the idea that the newly introduced class is larger than nonexpansiveness is the following
This new property, named condition
$\left(C\right)$, caught the attention of many authors that searched different fixed point theorems for such mappings (see for example [
13,
14,
15,
16]). In particular, a consistent analysis in connection with condition
$\left(C\right)$ was performed in [
17], in a modular vectorial setting. An interesting extension of
$\left(C\right)$property is the class of generalized nonexpansive mappings that satisfy the socalled condition
$\left(E\right)$ introduced by GarciaFalset et al. [
18]. Condition
$\left(E\right)$ is wider than Suzuki’s condition but stronger than quasinonexpansiveness. Another extension was subject to analysis in [
19]. However, these generalized properties will not be a topic to be approached in this survey.
In this paper, we will focus on extending the study of the abovementioned TTP process to the more general class of Suzukigeneralized nonexpansive mappings. In this respect, we will provide an outcome regarding the existence of fixed points for Suzuki mappings in the framework of uniformly convex Banach spaces. In addition, some convergence theorems concerning this iterative process will be stated.
3. Convergence Theorems
Inspired by the results obtained in [
7] via the iteration procedure (
1), for nonexpansive mappings, we phrase and prove similar convergence outcomes regarding mappings satisfying condition
$\left(C\right)$. Knowing that property
$\left(C\right)$ leads to a wider class of mappings than nonexpansiveness, the results provided next are expected to be more general than the outcomes in [
7].
Lemma 4. Let C be a nonempty, closed and convex subset of a Banach space X and $T:C\to C$ a mapping satisfying condition (C) with $F\left(T\right)\ne \varnothing $. For an arbitrary chosen ${x}_{0}\in C$, let $\left\{{x}_{n}\right\}$ be the sequence generated by (1). Then $\underset{n\to \infty}{lim}\u2225{x}_{n}p\u2225$ exists for any $p\in F\left(T\right)$. Proof. Let
$p\in F\left(T\right)$. Since
T satisfies condition (C) and has at least a fixed point; it follows that
T is quasinonexpansive. Thus, from (
1), one has
The same reasoning applies to
$\u2225{y}_{n}p\u2225$, and one obtains
Now, using inequality (
2), one finds
In addition, the following inequality holds
and together with (
2) and (
3) becomes
We conclude from (
4) that
$\left\{\u2225{x}_{n}p\u2225\right\}$ is bounded and nonincreasing for all
$p\in F\left(T\right)$. Hence,
$\underset{n\to \infty}{lim}\u2225{x}_{n}p\u2225$ exists. □
Theorem 2. Let C be a nonempty, closed and convex subset of a uniformly convex Banach space X, and let $T:C\to C$ be a mapping satisfying condition (C). For an arbitrary chosen ${x}_{0}\in C$, let the sequence $\left\{{x}_{n}\right\}$ be generated by (1) for all $n\ge 0$, where $\left\{{\alpha}_{n}\right\}$, $\left\{{\beta}_{n}\right\}$, $\left\{{\gamma}_{n}\right\}\in (0,1)$, $\left\{{\gamma}_{n}\right\}$ bounded away from 0 and 1. Then $F\left(T\right)\ne \varnothing $ if and only if $\left\{{x}_{n}\right\}$ is bounded and $\underset{n\to \infty}{lim}\u2225T{x}_{n}{x}_{n}\u2225=0$. Proof. Let us first prove the direct implication. Suppose
$F\left(T\right)\ne \varnothing $ and let
$p\in F\left(T\right)$. By Lemma 4 it follows that
$\underset{n\to \infty}{lim}\u2225{x}_{n}p\u2225$ exists and
$\left\{{x}_{n}\right\}$ is bounded. Let us denote
From (
2), it is known that
$\u2225{z}_{n}p\u2225\le \u2225{x}_{n}p\u2225$. Taking lim sup on both sides of the inequality and using (
5), one obtains
Again, since
T is quasinonexpansive, one has
Now, the following inequality holds true
and combined with (3) becomes
Dividing the above relation by
$(1{\alpha}_{n})$, conducts to
and it follows that
i.e.,
Applying lim sup to (
8) and using (
5) together with (
6), one obtains
which implies
Relation (
9) can be rewritten as
From (
5), (
7), (
9) and Lemma 1 one finds
$\underset{n\to \infty}{lim\; sup}\u2225T{x}_{n}{x}_{n}\u2225=0$.
Let us now prove the converse statement. Suppose
$\left\{{x}_{n}\right\}$ is bounded and
$\underset{n\to \infty}{lim}\u2225T{x}_{n}{x}_{n}\u2225=0$. Let
$p\in A(C,\left\{{x}_{n}\right\})$. By Proposition 1(iii) one has
The above relation implies that $Tp\in A(C,\left\{{x}_{n}\right\})$. As mentioned above, when dealing with closed bounded convex subsets of uniformly convex Banach spaces, the asymptotic center is a singleton. Therefore, $Tp=p$ i.e., $F\left(T\right)\ne \varnothing $, and the proof is complete. □
Theorem 3. Let C be a nonempty closed convex subset of a uniformly convex Banach space X with Opial’s property, T and $\left\{{x}_{n}\right\}$ be as in Theorem and $F\left(T\right)\ne \varnothing $. Then $\left\{{x}_{n}\right\}$ converges weakly to a fixed point of T.
Proof. The proof is identical with the proof of the Theorem 3.3. in [
15]. This is not surprising since the conclusions of Lemma 4 and Theorem 2 are the same as in Theorem 3.2. and Lemma 3.1. in [
15], via a distinct iterative process. We chose to display the proof just for the sake of making the paper selfcontained.
Since
$F\left(T\right)\ne \varnothing $, let
$p\in F\left(T\right)$. By Theorem 2,
$\left\{{x}_{n}\right\}$ is bounded and
$\underset{n\to \infty}{lim}\u2225T{x}_{n}{x}_{n}\u2225=0$ and by Lemma 4,
$\underset{n\to \infty}{lim}\u2225{x}_{n}p\u2225$ exists. Since
X is uniformly convex, according to Milman–Pettis’s Theorem, it is reflexive. Therefore, by Eberlin’s Theorem, every bounded sequence of elements of
X contains a subsequence which converges weakly to an element of
X. Let
$\left\{{x}_{{n}_{i}}\right\}$ be the subsequence of
$\left\{{x}_{n}\right\}\in X$ which converges weakly to an element
${z}_{1}\in X$. Since
C is closed and convex, according to Mazur’s Theorem,
${z}_{1}\in C$. By Lemma 2, we obtain
$T{z}_{1}={z}_{1}$, consequently
${z}_{1}\in F\left(T\right)$. Further we will show that
$\left\{{x}_{n}\right\}$ itself converges weakly to
${z}_{1}$. Let us assume the contrary and suppose that there exists a subsequence
$\left\{{x}_{{n}_{j}}\right\}$ of
$\left\{{x}_{n}\right\}$, such that
${x}_{{n}_{j}}\rightharpoonup {z}_{2}\in C$, where
${z}_{1}\ne {z}_{2}$. Again, using Lemma 2 we have
$T{z}_{2}={z}_{2}$ i.e.,
${z}_{2}\in F\left(T\right)$. Since
X is endowed with Opial’s property, we obtain
This leads to a contradiction, so ${z}_{1}={z}_{2}$ and we conclude that $\left\{{x}_{n}\right\}$ converges weakly to a fixed point of T. □
Theorem 4. Let C be a nonempty, compact and convex subset of a uniformly convex Banach space X and let T and $\left\{{x}_{n}\right\}$ be as in Theorem. Then $\left\{{x}_{n}\right\}$ converges strongly to a fixed point of T.
Proof. Again, the proof does not differ at all from the proof of Theorem 3.4 in [
15].
By Lemma 3, we have
$F\left(T\right)\ne \varnothing $. Since
C is compact, there exists a subsequence
$\left\{{x}_{{n}_{k}}\right\}$ of
$\left\{{x}_{n}\right\}$ such that
$\left\{{x}_{{n}_{k}}\right\}$ converges strongly to an element
$p\in C$. Using Proposition 1 (iii), we have
Taking the limit of the above relation, we obtain
By Theorem 2, we have $\underset{n\to \infty}{lim}\u2225T{x}_{{n}_{k}}{x}_{{n}_{k}}\u2225=0$ and ${x}_{{n}_{k}}\to p$, so the previous inequality gives that $\underset{k\to \infty}{lim}\u2225{x}_{{n}_{k}}Tp\u2225=0$ i.e., ${x}_{{n}_{k}}\to Tp$. But the limit is unique, so $Tp=p$ which implies $p\in F\left(T\right)$. Furthermore, by Lemma 4, $\underset{n\to \infty}{lim}\u2225{x}_{n}p\u2225$ exists for any $p\in F\left(T\right)$, thus p is the strong limit of the sequence $\left\{{x}_{n}\right\}$ itself. □
Theorem 5. Let C be a nonempty closed convex subset of a uniformly convex Banach space X, let T and $\left\{{x}_{n}\right\}$ be as in Theorem and $F\left(T\right)\ne \varnothing $. If T satisfies the condition (I), then $\left\{{x}_{n}\right\}$ converges strongly to a fixed point of T.
Proof. The proof runs as in [
15] (Theorem 3.5).
By Lemma 4,
$\underset{n\to \infty}{lim}\u2225{x}_{n}p\u2225$ exists for any
$p\in F\left(T\right)$, therefore
$\underset{n\to \infty}{lim}d({x}_{n},F\left(T\right))$ exists. Suppose
${lim}_{n\to \infty}\u2225{x}_{n}p\u2225=r$, for some
$r\ge 0$. If
$r=0$, then the desired result follows. Let
$r\ne 0$. By condition (I) in Definition 4, we obtain
From the hypothesis
$F\left(T\right)\ne \varnothing $ so using Theorem 2,
$\underset{n\to \infty}{lim}\u2225T{x}_{n}{x}_{n}\u2225=0$ which implies
Considering the properties of the function
f, we find
Let
$\left\{{x}_{{n}_{k}}\right\}$ be a subsequence of
$\left\{{x}_{n}\right\}$ and
$\left\{{y}_{k}\right\}\in F\left(T\right)$ such that
For
$k\to \infty $, it follows
therefore
$\left\{{y}_{k}\right\}\in F\left(T\right)$ is a Cauchy sequence. Since
$F\left(T\right)$ is a closed set,
$\left\{{y}_{k}\right\}$ converges to a fixed point
p. Letting
$k\to \infty $ in (
10) we have
$\left\{{x}_{{n}_{k}}\right\}\to p\in F\left(T\right)$. Since
$\underset{n\to \infty}{lim}\u2225{x}_{n}p\u2225$ exists, it leads to
${x}_{n}\to p$ which completes the proof. □
5. Example and Comparative Study
In order to emphasize the value of the analyzed TTP iteration procedure in connection with Suzukitype mappings, by comparing it further with other iterative processes, we consider next an example.
Example 1. We shall further prove that T is a Suzuki but not a nonexpansive mapping. To develop the desired proof we chose to work with the Taxicab norm or 1norm on ${\mathbb{R}}^{2}$, that is ${\u2225({x}_{1},{x}_{2})\u2225}_{1}={x}_{1}+{x}_{2}$.
Proof. We start by proving that there exist ${x}_{1},{y}_{1}\in [0,2]$ and ${x}_{2},{y}_{2}\in \left[{\displaystyle 0,\frac{1}{7}}\right]$, such that, T mentioned above is not nonexpansive. Let us take for example ${x}_{1}=\frac{14}{100}\in \left[0,\frac{1}{7}\right)$, ${y}_{1}=\frac{1}{7}\in \left[\frac{1}{7},2\right]$, and ${x}_{2}={y}_{2}=0\in \left[{\displaystyle 0,\frac{1}{7}}\right]$. Then $\u2225T({x}_{1},{x}_{2})T({y}_{1},{y}_{2})\u2225}_{1}=2{x}_{1}\frac{{y}_{1}+12}{7}=\frac{614}{4900}>\frac{2}{700}=\left{x}_{1}{y}_{1}\right={\u2225({x}_{1},{x}_{2})({y}_{1},{y}_{2})\u2225}_{1$, thus T is not a nonexpansive mapping.
To prove that T satisfies condition (C), the following cases need to be analyzed:
Case I: Let ${x}_{1}\in \left[0,\frac{1}{7}\right)$. If ${y}_{1}\in \left[0,\frac{1}{7}\right)$, then it can easily be seen that T is nonexpansive and condition $\left(C\right)$ is automatically satisfied. Further, if we take ${y}_{1}\in \left[{\displaystyle \frac{1}{7},2}\right]$, then $\frac{1}{2}{\u2225({x}_{1},{x}_{2})T({x}_{1},{x}_{2})\u2225}_{1}\le {\u2225({x}_{1},{x}_{2})({y}_{1},{y}_{2})\u2225}_{1}$ stands true only for ${y}_{1}\in \left[{\displaystyle \frac{6}{7},2}\right]$. Moreover, evaluating the nonexpansiveness condition ${\u2225T({x}_{1},{x}_{2})T({y}_{1},{y}_{2})\u2225}_{1}\le {\u2225({x}_{1},{x}_{2})({y}_{1},{y}_{2})\u2225}_{1}$ for ${x}_{1}\in \left[{\displaystyle 0,\frac{1}{7}}\right]$ and ${y}_{1}\in \left[{\displaystyle \frac{6}{7},2}\right]$, one finds ${\displaystyle \frac{27{x}_{1}{y}_{1}}{7}}\le {y}_{1}{x}_{1}$ which is obviously true as ${\displaystyle \frac{27{x}_{1}{y}_{1}}{7}}\in \left[{\displaystyle 0,\frac{8}{49}}\right]$, while ${y}_{1}{x}_{1}\in \left({\displaystyle \frac{5}{7},2}\right]$. Therefore, T satisfies condition C for the case considered.
Case II: Let us now consider the rest of the interval i.e., ${x}_{1}\in \left[{\displaystyle \frac{1}{7},2}\right]$. Similarly with Case I, if ${x}_{1}$ and ${y}_{1}$ belongs to the same interval, then T is a contraction and satisfies condition C since all contractions are included in the class of Suzuki mappings. On the other side, if ${y}_{1}\in \left[{\displaystyle 0,\frac{1}{7}}\right)$ then $\frac{1}{2}{\u2225({x}_{1},{x}_{2})T({x}_{1},{x}_{2})\u2225}_{1}\le {\u2225({x}_{1},{x}_{2})({y}_{1},{y}_{2})\u2225}_{1}$ becomes $\frac{1}{2}{\displaystyle {x}_{1}\frac{{x}_{1}+12}{7}}+\frac{1}{2}{x}_{2}{x}_{2}\le {x}_{1}{y}_{1}+{x}_{2}{y}_{2}$, or, even more, $\frac{63{x}_{1}}{7}\le {x}_{1}{y}_{1}+\frac{1}{7}$, as ${x}_{2}{y}_{2}\in \left[{\displaystyle 0,\frac{1}{7}}\right]$. Further, this implies $\frac{10{x}_{1}5}{7}\ge {y}_{1}$ i.e., ${x}_{1}\in \left[{\displaystyle \frac{1}{2},2}\right]$. For ${x}_{1}\in \left[{\displaystyle \frac{1}{2},2}\right]$ and ${y}_{1}\in \left[{\displaystyle 0,\frac{1}{7}}\right]$, the nonexpansive condition is ${\displaystyle \frac{{x}_{1}+7{y}_{1}2}{7}}\le {x}_{1}{y}_{1}$ which is true as ${\displaystyle \frac{{x}_{1}+7{y}_{1}2}{7}}\in \left[{\displaystyle 0,\frac{3}{14}}\right]$ and ${x}_{1}{y}_{1}\in \left[{\displaystyle \frac{5}{14},2}\right]$, so T satisfies condition C for this case also.
Considering all the situations previously analyzed, we conclude that the above defined T is indeed an example of a Suzuki mapping, although it is not a nonexpansive one. □
Using this Suzuki mapping and the TTP iteration procedure, along with other iterative schemes mentioned in the first section, let us visualize (and analyze) the convergence behaviors by performing a numerical simulation. The results are pictured in the images included in
Figure 1,
Figure 2,
Figure 3,
Figure 4,
Figure 5,
Figure 6 and
Figure 7. The maximum number of iterations to be performed until the algorithm stops is set to
$K=30$ and the exit parameter to
$\epsilon ={10}^{15}$. The
$\left[0,2\right]\times \left[{\displaystyle 0,\frac{1}{7}}\right]$ rectangle is represented by an open window having the values of
${x}_{1}$ on the horizontal axis and those of
${x}_{2}$ on the vertical one. As a general feature of the obtained images, the first color (black) of the rightsided colorbar, corresponds to those pairs of points having long orbits, nonconvergent for the maximum number of iterations imposed. Apart from black, each color in the range corresponds to a value between 1 and 30, in ascending order, signifying the number of iterations needed to reach the fixed point of
T with the error
$\epsilon $. On what concerns the values of the involved parameters, we chose (purely arbitrary) the sequences
$\alpha}_{n}=\frac{1}{\sqrt{9n+1}$,
$\beta}_{n}=\frac{{(2n+1)}^{\frac{1}{3}}}{10n+11$ and
$\gamma}_{n}=\sqrt{\frac{n}{3n+4}$. The Algorithm 1 is used to generate these images and goes through the following steps: first, take a pair of starting points from the area
$\left[0,2\right]\times \left[{\displaystyle 0,\frac{1}{7}}\right]$, then choose an iteration procedure and perform it until the maximum number of iteration
K is reached or the exit criterion is satisfied (for this case, as we have worked with the Taxicab norm, the exit criterion is
${\u2225({x}_{{n}_{1}},{x}_{{n}_{2}})({x}_{{n+1}_{1}},{x}_{{n+1}_{2}})\u2225}_{1}={x}_{{n}_{1}}{x}_{{n+1}_{1}}+{x}_{{n}_{2}}{x}_{{n+1}_{2}}\left\right)$. When the loop terminates, the program will assign to every starting point from the specified area a pixel and a corresponding color, based on the number of iterations performed.
Algorithm 1: Convergence visualization 

Let us take a closer look to the first image; namely,
Figure 1, the one corresponding to the TTP iterative process. One can see that, as the set of fixed points is approached, the color becomes darker, meaning that the number of iterations performed decreases. Moreover, the last line of the image takes a dark blue color. This is actually to be expected since that particular line is corresponding to the set of fixed points of
T and just one iteration is needed until the exit criterion is satisfied. Similar analyses regarding convergence speed can be realized for the images provided by others iterative processes, i.e., Picard, Mann, Agarwal and AbbasNazir. It is interesting to point out that the corresponding images for iterations like Ishikawa and Noor are entirely black, meaning that the procedures are very slowly convergent (they need more than 30 iterative steps) or they do not converge at all. The explanation for such a behavior is that
T defined above is a Suzukigeneralized nonexpansive mapping, but it is not nonexpansive. Nevertheless, it is clear that, among all iterative processes, TTP remains one of the fastest; it is only surpassed by he AbbasNazir procedure. This last statement is also emphasized on the
Table 1, by taking a random point from the domain of
T (
$({x}_{1},{x}_{2})=(1,0.1)$, also marked on each image with a red ’x’) and listing the number of iterations needed to approximate the fixed point
$({x}_{1}^{*},{x}_{2}^{*})=(1,0.1)$ for each iteration procedure.
In the following, we provide a second example of a Suzuki mapping which is not nonexpansive, on a function space. This is meant to strengthen the assertion that mappings satisfying condition
C is indeed a wide class of operators, and examples for it can be provided both on
$\mathbb{R}$ (see [
15]) and
${\mathbb{R}}^{2}$, as well as on infinite dimensional spaces.
Example 2. Consider the Banach space $X={L}^{\infty}\left(\mathbb{R}\right)$ of all essentially bounded Lebesgue measurable functions, endowed with the essential supremum norm Let $C=\{f:\mathbb{R}\to [0,7]\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}f(x)=f(0),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall x\le 0\}$ and We shall further prove that T mentioned above is an example of a Suzukigeneralized nonexpansive mapping.
Proof. Suppose the inequality $\frac{1}{2}{\u2225fTf\u2225}_{\infty}\le {\u2225gf\u2225}_{\infty}$ is satisfied. This is further equivalent with $\frac{1}{2}\leftTf\left(0\right)f\left(0\right)\right\le max\left\{\leftf\left(0\right)g\left(0\right)\right,{\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right\right\}$. Thus, two cases arise:
Case 1: Let us presume that $max\left\{\leftf\left(0\right)g\left(0\right)\right,{\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right\right\}=\leftf\left(0\right)g\left(0\right)\right$. For T to satisfy condition $\left(C\right)$, this must imply $max\left\{\leftTf\left(0\right)Tg\left(0\right)\right,{\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftTf\left(x\right)Tg\left(x\right)\right\right\}\le \phantom{\rule{3.33333pt}{0ex}}\leftf\left(0\right)g\left(0\right)\right$. Because of this last inequality, it is expected the problem to be divided again into two subcases. We will analyze just the nontrivial one i.e., $\frac{1}{2}\leftTf\left(0\right)f\left(0\right)\right\le \leftf\left(0\right)g\left(0\right)\right$ implies $\leftTf\left(0\right)Tg\left(0\right)\right\le \leftf\left(0\right)g\left(0\right)\right$, as the desired result follows easly from the other one. If $f\left(0\right)\ne 7$ and $g\left(0\right)\ne 7$, or $f\left(0\right)=7$ and $g\left(0\right)=7$, it can be easly noticed that T is nonexpansive, and therefore condition $\left(C\right)$ is automatically fulfilled. For $f\left(0\right)\ne 7$ and $g\left(0\right)=7$, T is nonexpansive just for $f\left(0\right)\in \left[{\displaystyle 0,\frac{28}{5}}\right]$, and again, condition $\left(C\right)$ is satisfied. For $f\left(0\right)\in \left({\displaystyle \frac{28}{5},7}\right)$ and $g\left(0\right)=7$, $\frac{1}{2}\leftTf\left(0\right)f\left(0\right)\right\le \leftf\left(0\right)g\left(0\right)\right$ becomes $\frac{5f\left(0\right)}{14}\le 7f\left(0\right)$ which is not true as $\frac{5f\left(0\right)}{14}\in \left({\displaystyle 2,\frac{5}{2}}\right)$ and $7f\left(0\right)\in \left({\displaystyle 0,\frac{7}{5}}\right)$. The same result is obtained if we take $f\left(0\right)=7$ and $g\left(0\right)\ne 7$. Considering all the situations analyzed, we conclude that T is a Suzukimapping for the current case.
Case 2: Suppose $max\left\{\leftf\left(0\right)g\left(0\right)\right,{\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right\right\}={\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right$. The inequality $\frac{1}{2}\leftTf\left(0\right)f\left(0\right)\right\le {\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right$ must imply $max\left\{\leftTf\left(0\right)Tg\left(0\right)\right,{\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftTf\left(x\right)Tg\left(x\right)\right\right\}\le {\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right$. If we consider that $\leftTf\left(0\right)Tg\left(0\right)\right\le {\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right$, it follows ${\u2225TfTg\u2225}_{\infty}={\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right$; but as $\leftf\left(0\right)g\left(0\right)\right\le {\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right$, it follows that ${\u2225fg\u2225}_{\infty}={\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right$ too, so T is nonexpansive. If we suppose $\leftTf\left(0\right)Tg\left(0\right)\right>{\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right$, kipping in mind that ${\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right\ge \leftf\left(0\right)g\left(0\right)\right$ on one side, ${\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right\ge \frac{1}{2}\leftTf\left(0\right)f\left(0\right)\right$ on the other side and considering all combinations that T could involve, we find that the assumption is absurd and $\leftTf\left(0\right)Tg\left(0\right)\right$ could not be grater than ${\mathrm{ess}\phantom{\rule{0.166667em}{0ex}}\mathrm{sup}\phantom{\rule{0.166667em}{0ex}}}_{(0,\infty )}\leftf\left(x\right)g\left(x\right)\right$. So, overall, T is a Suzuki mapping in Case 2 also. □
6. Conclusions
This paper analyzes a threesteps Thakur iterative procedure in connection with mappings satisfying Suzuki’s generalized nonexpansiveness condition, known as property $\left(C\right)$. A necessary and sufficient condition regarding the existence of fixed points for Suzuki mappings is stated and proved via the TTP iterative process. Furthermore, convergence results are obtained when additional hypotheses related to Opial’s property, compactness or condition $\left(I\right)$ are assumed. Fresh examples of Suzuki mappings which are not nonexpansive are further provided; the settings for these examples are ${\mathbb{R}}^{2}$, with the Taxicab norm, and ${L}^{\infty}\left(\mathbb{R}\right)$ endowed with the essential supremum norm. But, the most interesting feature about the example in ${\mathbb{R}}^{2}$ is a numerical simulation, resulting in a visual comparative analysis of the convergence behaviors of several iteration procedures. This numerical modeling uses similar techniques as the rootfinding procedures for complex polynomials, which ultimately led to polynomiographic visualization.
Overall, the novelty of this paper is twofold. First, a new perspective on the TTP iteration procedure is provided; this iterative scheme was originally conceived as a tool in connection with nonexpansive mappings; now, it is proved to be an instrument as good as before for reaching the fixed points of Suzuki mappings too. Moreover, having in mind the computational dimension of an iteration procedure, a data dependency analysis is convenient, since errors can occur when using computer programs. Usually, this constrains us to actually use a perturbed mapping $\tilde{T}$, instead of the theoretical one. We managed to prove that a small perturbation of the initial data does not significantly affect the computational process of the fixed point of a contractive operator.
Secondly, more complex examples of Suzuki mappings are provided. Picking ${\mathbb{R}}^{2}$ as the setting, an interesting visual procedure is suggested as a possible new approach related to convergence analysis. In addition, another example proves that one could easily exceed the framework of finite dimensional normed spaces.