Abstract
We introduce a new iterative method in this article, called the iterative approach for fixed point approximation. Analytically, and also numerically, we demonstrate that our established I.P is faster than the well-known I.P of the prior art. Finally, in a uniformly convex Banach space environment, we present weak as well as strong convergence theorems for Suzuki’s generalized nonexpansive maps. Our findings are an extension, refinement, and induction of several existing iterative literatures.
1. Introduction and Preliminaries
In many branches of mathematics and various sciences, the existence of a fixed point is crucial. A completely fine synthesis of analysis, geometry, and topology is the fixed point theory. In particular, fixed point techniques are used in economics, biology, engineering, biochemistry, game theory, and physics. So, once the presence of a fixed point is found, it is difficult to find its value; that is why we use an I.P to calculate them. A variety of I.Ps have been introduced and it is not possible to discuss all I.Ps. The famous Banach contraction theorem uses the Picard I.P to approximate fixed points. Other well-known I.Ps can be found in references [1–8, 15–18]. The approximation speed of the I.P has a major role, and an I.P tends to be chosen in another iterative process. In [2], the writer believes that the approximation rate of the Agarwal I.P is like that of the Picard I.P and is faster than the contraction mapping of the Mann I.P. In [9], the writers help by numerical examples to prove for nonexpansive mapping, that the approximation rate of the Picard- I.P is better than existing literature. They proved that the convergence rate was good. In [6], the creators illustrate that the approximation rate of the I.P is better than existing literature. Recently, in [7], another I.P, namely, I.P, was developed, and it is shown that its approximation rate is better than existing literature. In [10], one more I.P called I.P was developed, in which they have shown that the approximation rate is better than previous literature. By hypothesis, we introduce a new I.P, namely, the iterative process.
Numerically, the approximation rate for the latest I.P is contrasted with the Agarwal I.P, Picard- I.P, I.P, I.P, and I.P. We present the weak as well as strong convergence theorems of Suzuki generalized nonexpansive maps and contraction map for our newly developed I.P. We first remember those definitions, ideas, and lemmas which we have to use in the upcoming two sections. A Banach space is referred to as uniformly convex [11] if s.t for ,
is referred to obey the Opial property [4] if , approaching weakly to , we have
If , denotes the set of all fixed points of ; a point is referred to as the fixed point of mapping .
Let , and is referred as contraction if s.t ,
is referred to as nonexpansive if , and quasinonexpansive if , so . In [12], is referred to satisfy condition if , we have
In [12], the author approved that the mapping holding condition is weaker than the nonexpansive mapping and stronger than the quasinonexpansive mapping. The mapping which holds condition is called the Suzuki generalized nonexpansive mapping. Reference [12] includes the fixed point theorem and convergence theorem for Suzuki’s generalized nonexpansive mapping. Recently, the fixed point theorem of the Suzuki generalized nonexpansive mapping has been studied by many authors [5–7]. Now, we define properties of Suzuki generalized nonexpansive mappings.
Proposition 1. Let and . Then, (i)([12], Proposition 1) If is nonexpansive, so must be a Suzuki generalized nonexpansive(ii)([12], Proposition 2) If is a Suzuki generalized nonexpansive having a fixed point, then must be a quasinonexpansive(iii)([12], Lemma 7) If is a Suzuki generalized nonexpansive, then .
Lemma 2 ([12], Proposition 3). Let which satisfies the Opial property. Suppose that is a Suzuki generalized nonexpansive. If approaches weakly to and , then . So, - is demiclosed at zero.
Lemma 3 ([12], Theorem 5). Let and is a Suzuki generalized nonexpansive. Then must have a fixed point.
Lemma 4 ([13], Lemma 1.3). Let be any real sequence in s.t . Let and be any two sequences of s.t , , and hold for some Then, .
Let , and let be a bounded sequence . For , we set
The asymptotic radius of relative to is given by and the asymptotic center of relative to is defined as
Definition 5 (see [2]). Let and be two fixed point I.P sequences that approach the single fixed point and and If the sequence and approaches to and ,respectively, and , then approaches faster as to .
2. The Iteration Process
In this section, let and and are real sequences and . For detail on I.P, please see [14].
Following is two step Agrawal I.P or I.P as
In [9], the authors develop a new I.P called Picard- I.P as given below:
They have demonstrated that with Picard- I.P, we approximate the fixed point of contraction mapping. Also, by a numerical example, they proved that the Picard- I.P have a better approximation rate than all previously developed I.P.
In [7], the authors introduced a new I.P, namely, I.P, which is given below:
In [6], the authors introduced one new I.P, namely, I.P, which is defined as
They proved that each of (10) and (11) is moving faster as compared to (8) and (9).
In [10], the authors introduced a three-step I.P, called I.P, which is given below:
They insisted that the new I.P converged quickly. By providing an example, they showed that the I.P has a better approximating rate than the I.P., Picard- I.P, I.P, and I.P.
In this competition, we developed following (new) three-step I.P, namely, I.P, defined by
In order to prove that our new I.P (13) have a better approximation rate as compared to (8),(9), (10), (11), and (12), first, we generally prove that our I.P strongly converges to unique fixed point, and then it is supported with a numerical example.
Theorem 6. Let the contraction map , where . Assume to be an iterative sequence generated by I.P, where and are real sequences satisfying or Then, approaches strongly to a unique fixed point of .
Proof. As is a contraction in , so must have a unique fixed point in . Assume is a particular fixed point of . From I.P, we get Now, Then, After repetition, we get
Therefore, we obtain Now, so and . Thus, we get and . As , . So we have
Taking the limits both sides, we get .
Theorem 7. Let , and let contraction map holding condition , having a fixed point . For a given , let and be iterative sequences developed by I.P and Picard- I.P as in [11], respectively, where , , and are real sequences satisfying and , for some and . Then, approaches to firstly rather than .
Proof. By Theorem 2.5 in [11], we have
Since and for all , we obtain . Let =.
Now, from Theorem 6, we get
Again for all gives
Let Then,
Thus, taking the limit as . Hence, the result follows.
Next, we present a result which defines the better approximation rate between the and I.P [7].
Theorem 8. Let , also let contraction map holding condition , having a fixed point . For a given , let and be iterative sequences developed by I.P and I.P as in [7], respectively, where , , and are real sequences satisfying and , for some and . Then, approaches to firstly rather than .
Proof. By Theorem 6, we have
Since and for all , we obtain
Let Then, for the iteration, we have
Now,
Therefore, we get
After repetition,
Thus, we have Now, since and for all , we obtain . Let .
Then,
Thus, taking the limit as . Hence, the result follows.
Next, we prove that I.P is faster than that of the I.P [6].
Theorem 9. Let ; also, let a contraction mapping holding condition , having a particular fixed point . For a given = , let and be iterative sequences developed by I.P and I.P as in [6], respectively, where , , and are real sequences satisfying and , for some and . Then, approaches to firstly rather than .
Proof. By result 2.1, we have Since and for all , we obtain Let Then, for the iteration, we have
Now, After repetition,
Thus, Now, since and for all , we obtain Let
Then,
Thus, taking the limit as . Hence, the result follows.
Now, we prove by numerical example that our I.P has a better approximation rate than existing I.Ps in literature.
Example 10. Let a contraction mapping be defined by Let and The iterative values for are defined in Table 1. Figures 1(a) and 1(b) present the convergence graph.
(a)
(b)
By Figures 1(a) and 1(b) and Table 1, it is clear that the new I.P has a better approximation rate then the , , , Picard-, and I.P.
Example 11. Let a contraction mapping by . Let and ; the iterative values for the initial guess are given in Table 2. Figures 2(a) and 2(b) represent the convergence graph. We can see that our new I.P (13) has a better approximation rate as compared to , Picard-, , , and I.P.
(a)
(b)
By Figures 2(a) and 2(b) and Table 2, it is proven that the new iteration has a better approximation rate than the , , , Picard-, and I.P.
3. Convergence Results of a Sequence Generated by I.P
Here, we prove some strong and weak convergence result of sequence generated by I.P for the Suzuki generalized nonexpansive mapping in uniformly convex Banach spaces.
Lemma 12. Let and let contraction mapping be a holding condition and also . For randomly chosen , let the sequence be generated by (13), then exists for any .
Proof. Let and . As holds the condition, so So from 1.1(ii), we have (a)So by using (a), we get (b)So using (b), we have (c)which means that is nonincreasing and bounded, . So exists, hence proven.
Theorem 13. Let , and let contraction mapping holding condition . For randomly chosen , let be the sequence generated by (13) where and are sequences of real numbers . Then, iff is bounded and
Proof. Assume and consider . So by result 3.1, exists and is bounded put (d)From (a) and (d), we have (e)By result 1.1(ii), we have (f)On the other hand,
This implies that
So implies that
Therefore, (g)
From (e) and (g), we get (h)
From (d), (f), and (h) together with Lemma 4, we have
On the other hand, assume that is a bounded sequence and . Also, . So by Proposition 1,
Hence, . As is uniformly convex, and is a singleton set, therefore . Hence, .
3.1. Weak and Strong Convergence Theorem
Theorem 14. Let , where satisfies Opial condition, and let holding condition , where and are real sequences , such that . For randomly selected , consider to be the sequence generated by (13) . Then, the sequence generated by the iterative process converges weakly to .
Proof. As , and from Theorem 13, is a bounded sequence and Being uniformly convex must be reflexive, so by Eberlin’s theorem , a subsequence of which approaches weakly to . As is closed and convex, by Mazur’s theorem, . By Lemma 2, . Next, we prove that approaches weakly to . Actually, if it is not true, so a subsequence for s.t converges weakly to and . By Lemma 2, . Since defined . Thus, from Theorem 13 and Opial’s property,
which is a contradiction to fact. So converges weakly to a single fixed point of .
Next, we prove the strong convergence theorem.
Theorem 15. Let , and let contraction mapping holding condition , where and are real sequences, s.t . Then, approaches strongly to
Proof. By Lemma 3, , and by Theorem 13, we have As is compact, so a subsequence of which approaches strongly to where . By Proposition 1 (iii), we have , Letting , we attain . As, by Lemma 12, has a defined value, for every , so converge strongly to .
4. Conclusion
In this article, we present the new fastest iteration method to approximating fixed point of contraction mapping. First, we present the iteration process and prove weak and strong convergence results. Also, we analytically and numerically proved that the iteration process has a better approximation rate than existing iteration processes as defined in [1–4, 6, 7, 9].
Data Availability
No data use for this study.
Conflicts of Interest
The writers announce that they do not have any competing interests.
Authors’ Contributions
All authors contributed fairly and significantly in writing this article. All authors have read and agreed to the published version of the manuscript.
Acknowledgments
The authors give thanks to their universities.