#### Abstract

We introduce a new iterative method called D-iteration to approximate a fixed point of continuous nondecreasing functions on arbitrary closed intervals. The purpose is to improve the rate of convergence compared to previous work. Specifically, our main result shows that D-iteration converges faster than P-iteration and SP-iteration to the fixed point. Consequently, we have that D-iteration converges faster than the others under the same computational cost. Moreover, the analogue of their convergence theorem holds for D-iteration.

#### 1. Introduction

Let be a closed interval. Define to be a continuous mapping. A point is said to be a fixed point of if . The set of all fixed points of is denoted by . It is a well-known fact that has a fixed point if the interval is bounded. A popular way of finding a fixed point of is an iterative method.

In 1953, Mann [1] proposed an iteration, Mann iteration, defined by andwhere and . Then, a two-step iteration, Ishikawa iteration [2], was introduced in 1974 and defined by and and . Two years later, Rhoades [3] showed that Mann and Ishikawa iterations converge for the class of continuous nondecreasing functions on a unit closed interval. Next, Borwein and Borwein [4] proved that Mann iteration converges for the class of continuous mappings on a bounded closed interval in 1991. In 2000, Noor [5] introduced a new three-step iterative method, Noor iteration, defined by andwhere and . Then, Qing and Qihou [6] extended the results of Rhoades [3] and Borwein and Borwein [4] to the class of continuous functions on an arbitrary interval in 2006. On top of that, a necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval was provided.

In 2011, Phuengrattana and Suantai [7] introduced an iteration, called SP-iteration, defined by andwhere and . In addition, the convergence of this three-step iteration holds for continuous functions on an arbitrary interval. Moreover, they showed that SP-iteration converges faster than Mann, Ishikawa, and Noor iterations for the class of continuous nondecreasing functions.

Two years later, Kosol [8] studied the convergence of S-iteration [9] for the class of continuous nondecreasing functions on a closed interval. S-iteration was first introduced by Agarwal et al. [9] and defined by andwhere and . In 2015, Sainuan [10] constructed a new iteration, called P-iteration, and showed that this iteration converges faster than S-iteration for the class of continuous nondecreasing functions. P-iteration is defined by andwhere and .

Motivated by the above results, we define D-iteration by andwhere and .

In this work, we give a necessary and sufficient condition for the convergence of D-iteration. Then, we show that D-iteration converges faster than other iterations for the class of continuous nondecreasing functions. Also, numerical examples are provided to support our result.

#### 2. Convergence Theorem

In this section, we provide the convergence theorem of D-iteration for the class of continuous nondecreasing functions on an arbitrary closed interval. First, we begin with the following lemma.

Lemma 1. *Let be a continuous nondecreasing function, and let be a sequence defined by (7).*(i)*If , then for all and is nonincreasing.*(ii)*If , then for all and is nondecreasing.*

*Proof. *(i) Assume that . We will show that for all by induction on . Clearly, this is true for . Assume that for some . From (7), we have that . Since is nondecreasing, . By the definition of , . Then, since is nondecreasing. Similarly, , and finally, we obtain that . Thus, for all . Moreover, by the proof above, we have that for all . Therefore, is nonincreasing.

(ii) The proof can be done similarly as in (i).

Theorem 2. *Let be a continuous nondecreasing function, and let be a sequence defined by (7), where and . Then, is bounded if and only if it converges to a fixed point of .*

*Proof. *Assume that is bounded. First, we will show that it is convergent. If , by (7), we obtain that for all . Therefore, is convergent. Suppose that . From Lemma 1, we have that is either nonincreasing or nondecreasing. Since is bounded, it follows that is convergent. Assume that converges to for some . Next, we will show that is a fixed point of . Since is continuous and is bounded, we have that is bounded and so are and . Note that since .

From (7), we obtain thatSince , , and is continuous, we obtain that Therefore, .

Conversely, if is convergent, then it is obvious that is bounded.

Consequently, one can see from Theorem 2 that D-iteration always converges to a fixed point of , where is a continuous nondecreasing function defined on a bounded closed interval.

Corollary 3. *Let be a continuous nondecreasing function, and let be a sequence defined by (7), where and . Then, converges to a fixed point of .*

#### 3. Rate of Convergence

To prove our main theorem, we first define how to compare the rate of convergence between two iteration methods and then give some useful lemmas to accomplish our result.

*Definition 4. *Let be a continuous function, and let and be two iterations which converge to the same point . Then is said to converge faster than if for all .

Lemma 5. *Let be a continuous nondecreasing function, and let be a sequence defined by (7). Assume that there exists a point .*(i)*If , then for all .*(ii)*If , then for all .*

*Proof. *(i) Let . We will show by induction that for all . It is clear that this is true for the case . Assume that for some . Since is nondecreasing, . By the definition of , we have that Thus, . Similarly, Therefore, . From (7), we obtain that Hence, for all .

(ii) By using the same proof as in (i), we are done.

Lemma 6. *Let be a continuous nondecreasing function, and let , , and be sequences defined by (4), (6), and (7), respectively, where .*(i)*If , then for all .*(ii)*If , then for all .*

*Proof. *(i) Let . First, we show that for all by induction. It is obvious that this inequality holds for the case . Assume that for some . Since is nondecreasing, . Since , by Lemma 1(i), . It follows that by the definition of . From iterations (6) and (7), we have that Thus, . Therefore, . Then, That is, which implies . Consider We obtain that . By induction, we can conclude that for all . Next, we show that for all by induction. It is clear that this is true for the case . Assume that for some . Then . Since , we have that (see [7] Lemma 3.2 (vii)). From (4), . Since is nondecreasing, . By the definition of , . By (4) and (6), we have that Thus, . Since is nondecreasing, . Then, That is, . Therefore, . Consider We have that . By induction, we can conclude that for all .

(ii) By using similar arguments as in (i) together with Lemma 1(ii) and Lemma 3.2 (viii) in [7], we are done.

Proposition 7. *Let be a continuous nondecreasing function such that is nonempty and bounded. If and , then defined by (7) does not converge to a fixed point of .*

*Proof. *Assume that and . Then, by Lemma 1(ii), is nondecreasing. Since , it follows that does not converge to a fixed point of .

Proposition 8. *Let be a continuous nondecreasing function such that is nonempty and bounded. If and , then defined by (7) does not converge to a fixed point of .*

*Proof. *Assume that and . Then, by Lemma 1(i), is nonincreasing. Since , does not converge to a fixed point of .

In 2011, Phuengrattana and Suantai [7] compared the rate of convergence of Mann, Ishikawa, and Noor iterations with SP-iteration. Four years later, Sainuan [10] studied the rate of convergence between P-iteration and S-iteration. Their results are concluded as the following.

Theorem 9 (see [7, 10]). *Let be a continuous nondecreasing function such that is nonempty and bounded. For the same initial point and , the following are satisfied.*(i)*Ishikawa iteration converges to if and only if Mann iteration converges to . Moreover, Ishikawa iteration converges faster than Mann iteration.*(ii)*Noor iteration converges to if and only if Ishikawa iteration converges to . Moreover, Noor iteration converges faster than Ishikawa iteration.*(iii)*SP-iteration converges to if and only if Noor iteration converges to . Moreover, SP-iteration converges faster than Noor iteration.*(iv)*If S-iteration converges to , then P-iteration converges to . Moreover, P-iteration converges faster than S-iteration.*

*Remark 10. *From Theorem 9, one can conclude that SP-iteration is better than Noor, Ishikawa, and Mann iterations. However, one can come to the different conclusion if we take the computational cost into consideration. As mentioned in [11] Remark 3.3, SP-iteration is exactly three-step Mann iteration. Thus, Mann iteration converges faster than Noor iteration and also Ishikawa iteration under the same computational cost because Ishikawa iteration is a special case of Noor iteration.

Next, we compare the rate of convergence of D-iteration with SP-iteration and P-iteration.

Theorem 11. *Let be a continuous nondecreasing function such that is nonempty and bounded, and let . Let , , and be sequences defined by (4), (6), and (7), respectively, where . Then, the following are satisfied.*(i)*If P-iteration converges to , then D-iteration converges to . Moreover, D-iteration converges faster than P-iteration.*(ii)*If SP-iteration converges to , then P-iteration converges to . Moreover, P-iteration converges faster than SP-iteration.*

*Proof. *(i) Assume that P-iteration converges to . Note that if , then we are done. Assume that . Consider the following two cases.*Case** 1* (). If , then, by the proof of Proposition 3.6 in [10], it follows that does not converge to which leads to a contradiction. Thus, . Using Lemma 5(i) and Lemma 6(i), we obtain that for all . This implies for all . By the assumption, we have that converges to . Furthermore, we also have that D-iteration converges faster than P-iteration . *Case** 2* (). Similarly, since if , then does not converge to by the proof of Proposition 3.5 in [10]. Then, by Lemmas 5(ii) and 5(ii), we obtain that for all . This implies that for all . Therefore, D-iteration converges faster than P-iteration () to .

(ii) Assume that SP-iteration converges to . By using the same proof as in (i) together with Proposition 3.5, in [7], Lemma 5, and Lemma 6, we obtain the desired result.

It follows from Theorems 9 and 11 that D-iteration converges faster than Mann, Ishikawa, Noor, SP-, S-, and P-iterations for the class of continuous nondecreasing functions.

*Remark 12. *As a result from Theorem 11, we can also conclude that D-iteration converges faster than Mann iteration and P-iteration under the same computational cost. Since S-iteration is a special case of P-iteration, Mann iteration converges faster than S-iteration under the same computational cost as well. From Remark 10, we have that D-iteration is better than other iterations despite whether computational costs being considered or not.

Next, we give numerical examples of SP-, P-, and D-iterations, where and for all .

*Example 13. *Let be defined by . We have that is a nondecreasing continuous function. Given the initial point . Then, SP-, P-, and D-iterations are presented in Table 1, where the fixed point . It can be seen that D-iteration converges faster than other iterations as a result from Theorem 11. In addition, Table 2 shows the rate of convergence for each iteration. Notice that at least 24 steps of D-iteration must be computed to obtain an error less than , at least 30 steps for P-iteration and more than 32 steps for SP-iteration. In fact, at least 119 steps are needed for SP-iteration.

*Example 14. *Let be defined by . Then, is a nondecreasing continuous function. Given the initial point . Then, Table 3 provides SP-, P-, and D-iterations, where the fixed point . Then, D-iteration converges faster than other iterations satisfying Theorem 11. Moreover, the rate of convergence of each iteration is shown in Table 4. Note that at least 87 steps of D-iteration must be computed to obtain an error less than , at least 113 steps for P-iteration, and more than 123 steps for SP-iteration. Precisely, at least 455 steps are needed for SP-iteration.

*Example 15. *Let be defined by . Then, is a nonincreasing continuous function. Given the initial point, . Then, we have SP-, P-, and D-iterations as shown in Table 5, where the fixed point which is calculated by NSolve command in Mathematica to accuracy of 40 decimal places. One can see that D-iteration converges faster than other iterations. Further, the rate of convergence for each iteration is given in Table 6. As a result, at least 14 steps of D-iteration must be computed to obtain an error less than , at least 28 steps for P-iteration, and at least 27 steps for SP-iteration.

Consequently, Example 15 suggests that this may be true for the class of nonincreasing continuous functions. This remains open. Furthermore, one can consider these iterations for larger classes of continuous functions.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research is supported by Chiang Mai University, Thailand.