- Research
- Open Access
Strong convergence theorems and rate of convergence of multi-step iterative methods for continuous mappings on an arbitrary interval
- Withun Phuengrattana^{1, 2} and
- Suthep Suantai^{1, 2}Email author
https://doi.org/10.1186/1687-1812-2012-9
© Phuengrattana and Suantai; licensee Springer. 2012
- Received: 6 October 2011
- Accepted: 31 January 2012
- Published: 31 January 2012
Abstract
In this article, by using the concept of W-mapping introduced by Atsushiba and Takahashi and K-mapping introduced by Kangtunyakarn and Suantai, we define W^{(T,N)}-iteration and K^{(T,N)}-iteration for finding a fixed point of continuous mappings on an arbitrary interval. Then, a necessary and sufficient condition for the strong convergence of the proposed iterative methods for continuous mappings on an arbitrary interval is given. We also compare the rate of convergence of those iterations. It is proved that the W^{(T,N)}-iteration and K^{(T,N)}-iteration are equivalent and the K^{(T,N)}-iteration converges faster than the W^{(T,N)}-iteration. Moreover, we also present numerical examples for comparing the rate of convergence between W^{(T,N)}-iteration and K^{(T,N)}-iteration.
MSC: 26A18; 47H10; 54C05.
Keywords
- fixed point
- continuous mapping
- W-mapping
- K-mapping
- rate of convergence
1 Introduction
where g : E → E is a contraction. Then, Picard's iteration can be applied for finding a solution of (1.2).
Question: If g : E → E is continuous but not contraction, what iteration methods can be used for finding a solution of (1.2) (that is a fixed point of g) and how about the rate of convergence of those methods.
for all n ≥ 1, where ${\left\{{\alpha}_{n}\right\}}_{n=1}^{\infty}$ is a sequence in [0,1]. The Ishikawa iteration (see [2]) is
for all n ≥ 1, where ${\left\{{\alpha}_{n}\right\}}_{n=1}^{\infty}$, ${\left\{{\beta}_{n}\right\}}_{n=1}^{\infty}$, and ${\left\{{\gamma}_{n}\right\}}_{n=1}^{\infty}$ are sequences in [0,1]. Clearly Mann iteration is special cases of SP-iteration.
In 1976, Rhoades [5] proved the convergence of the Mann and Ishikawa iterations to a solution of (1.2) when E = [0,1]. He also proved the Ishikawa iteration converges faster than the Mann iteration for the class of continuous and nondecreasing functions. Later in 1991, Borwein and Borwein [6] proved the convergence of the Mann iteration of continuous functions on a bounded closed interval. In 2006, Qing and Qihou [7] extended their results to an arbitrary interval and to the Ishikawa iteration and gave some control conditions for the convergence of Ishikawa iteration on an arbitrary interval. Recently, Phuengrattana and Suantai [4] obtained a similar result for the new iteration, called the SP-iteration, and they proved the Mann, Ishikawa, Noor and SP-iterations are equivalent and the SP-iteration converges faster than the others for the class of continuous and nondecreasing functions.
where I is the identity mapping of C and λ_{n,i}∈ [0,1] for all i = 1, 2,..., N. Such a mapping W_{ n }is called the W-mapping generated by T_{1}, T_{2},..., T_{ n }and λ_{n,1}, λ_{n,2},..., λ_{n,N}. Many researchers have studied and applied this mapping for finding a common fixed point of nonexpansive mappings, for instance, see [8–23].
where I is the identity mapping of C and λ_{n,i}∈ [0,1] for all i = 1, 2,..., N. Such a mapping K_{ n }is called the K-mapping generated by T_{1},T_{2},..., T_{ n }and λ_{n,1}, λ_{n,2},..., λ_{n,N}. They showed that if C is a nonempty closed convex subset of a strictly convex Banach space X and ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ is a finite family of nonexpansive mappings of C into itself, then $F\left({K}_{n}\right)={\bigcap}_{i=1}^{N}F\left({T}_{i}\right)$ and they also introduced an iterative method by using the concept of K-mapping for finding a common fixed point of a finite family of nonexpansive mappings and a solution of an equilibrium problem. Applications of K-mappings for fixed point problems and equilibrium problems can be found in [23–26].
By using the concept of W-mappings and K-mappings, we introduce two new iterations for finding a fixed point of a mapping T : E → E on an arbitrary interval E as follows.
where I is the identity mapping of E and λ_{n,i}∈ [0,1] for all i = 1, 2,..., N. We call a mapping ${W}_{n}^{\left(T,N\right)}$ as the W-mapping generated by T and λ_{n,1}, λ_{n,2},..., λ_{n,N}. Clearly W^{(T,1)}-iteration is Mann iteration, W^{(T,2)}-iteration is Ishikawa iteration and W^{(T,3)}-iteration is Noor iteration.
where I is the identity mapping of E and λ_{n,i}∈ [0,1] for all i = 1, 2,..., N. We call a mapping ${K}_{n}^{\left(T,N\right)}$ as the K-mapping generated by T and λ_{n,1},λ_{n,2}, ..., λ_{n,N}. Clearly K^{(T,1)}-iteration is Mann iteration and K^{(T,3)}-iteration is SP-iteration.
Obviously the mappings (1.10) and (1.12) are special cases of the W-mapping and K-mapping, respectively.
The purpose of this article is to give a necessary and sufficient condition for the strong convergence of the W^{(T,N)}-iteration and K^{(T,N)}-iteration of continuous mappings on an arbitrary interval. We also prove that the K^{(T,N)}-iteration and W^{(T,N)}-iteration are equivalent and the K^{(T,N)}-iteration converges faster than the W^{(T,N)}-iteration for the class of continuous and nondecreasing mappings. Moreover, we present numerical examples for the K^{(T,N)}-iteration to compare with the W^{(T,N)}-iteration. Our results extend and improve the corresponding results of Rhoades [5], Borwein and Borwein [6], Qing and Qihou [7], Phuengrattana and Suantai [4], and many others.
2 Convergence theorems
We first give a convergence theorem for the K^{(T,N)}-iteration for continuous mappings on an arbitrary interval.
Theorem 2.1 Let E be a closed interval on the real line and T : E → E be a continuous mapping. For x_{1} ∈ E, let the K^{(T,N)}-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$defined by (1.11), where${\left\{{\lambda}_{n,i}\right\}}_{n=1}^{\infty}$ (i = 1, 2, ..., N) are sequences in [0,1] satisfying the following conditions:
(C1)${\sum}_{n=1}^{\infty}{\lambda}_{n,i}<\infty $for all i = 1, 2,..., N - 1;
(C2) lim_{n→∞}λ_{n,N}= 0 and${\sum}_{n=1}^{\infty}{\lambda}_{n,N}=\infty $.
Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$converges to a fixed point of T.
Proof. It is obvious that if ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ converges to a fixed point of T, then it is bounded. Now, assume that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ is bounded. We will show that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ converges to a fixed point of T. First, we show that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ is convergent. To show this, we suppose not. Then there exist a, b ∈ ℝ, a = lim inf_{n→∞}x_{ n }, b = lim sup_{n→∞}x_{ n }and a < b.
By boundedness of ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$, we have ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ belongs to a bounded closed interval. Continuity of T implies that ${\left\{T{x}_{n}\right\}}_{n=1}^{\infty}$ belongs to another bounded closed interval, so ${\left\{T{x}_{n}\right\}}_{n=1}^{\infty}$ is bounded. Since U_{n,1}x_{ n }= λ_{n,1}Tx_{ n }+ (1- λ_{n,1})x_{ n }, we get ${\left\{{U}_{n,1}{x}_{n}\right\}}_{n=1}^{\infty}$ is bounded, and thus ${\left\{T{U}_{n,1}{x}_{n}\right\}}_{n=1}^{\infty}$ is bounded. Similarly, by using (1.11), we have ${\left\{{U}_{n,i}{x}_{n}\right\}}_{n=1}^{\infty}$ and ${\left\{T{U}_{n,i}{x}_{n}\right\}}_{n=1}^{\infty}$ are bounded for all i = 2, 3,..., N - 1. It follows by (1.11) that U_{n,i}x_{ n }- U_{n,i-1}x_{ n }= λ_{n,i}(TU_{n,i-1}x_{ n }- U_{n,i-1}x_{ n }) for all i = 1,2,..., N. By condition (C 1) and (C 2), we get lim_{n→∞}|U_{n,i}x_{ n }- U_{n,i-1}x_{ n }|=0 for all i = 1, 2,..., N.
By (2.4), we have x_{k+1}> x_{ k }. Thus, x_{k+1}> m.
By using the above argument, we obtain x_{k+j}> m for all j ≥ 2. Thus we get x_{ n }> m for all n > k. So a = lim inf_{n→∞}x_{ n }≥ m, which is a contradiction with a < m. Thus Tm = m. Therefore, we obtain (2.1).
For the sequence ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$, we consider the following two cases:
Case 1: There exists ${x}_{\stackrel{\u0304}{M}}$ such that $a<{x}_{\stackrel{\u0304}{M}}<b$. Then $T{x}_{\stackrel{\u0304}{M}}={x}_{\stackrel{\u0304}{M}}$. By using (1.11), we obtain that ${U}_{\stackrel{\u0304}{M},i}{x}_{\stackrel{\u0304}{M}}={x}_{\stackrel{\u0304}{M}}$ for all i = 1, 2,..., N. Thus, we have ${x}_{\stackrel{\u0304}{M}+1}={x}_{\stackrel{\u0304}{M}}$. By induction, we obtain ${x}_{\stackrel{\u0304}{M}}={x}_{\stackrel{\u0304}{M}+1}={x}_{\stackrel{\u0304}{M}+2}=...$, so ${x}_{n}\to {x}_{\stackrel{\u0304}{M}}$. This implies that ${x}_{\stackrel{\u0304}{M}}=a$ and x_{ n }→ a, which contradicts with our assumption.
Case 2: For all n, x_{ n }≤ a or x_{ n }≥ b. Because b - a > 0 and lim_{n→∞}|x_{n+1}- x_{ n }| = 0, there exists M_{1} such that $\left|{x}_{n+1}-{x}_{n}\right|<\frac{b-a}{N}$ for all n > M_{1}. It implies that either x_{ n }≤ a for all n > M_{1} or x_{ n }≥ b for all n > M_{1}. If x_{ n }≤ a for n > M_{1}, then b = lim sup_{n→∞}x_{ n }≤ a, which is a contradiction with a < b. If x_{ n }≥ b for n > M_{1}, so we have a = lim inf_{n→∞}x_{ n }≥ b, which is a contradiction with a < b.
Hence, we have ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ is convergent.
By condition (C 1), (C 2), and lim_{k→∞}h_{k,i}= w ≠ 0 for all i = 1, 2,..., N, we get that ${\sum}_{k=1}^{\infty}{\lambda}_{k,i}{h}_{k,i}$ is convergent for all i = 1, 2,..., N - 1 and ${\sum}_{k=1}^{\infty}{\lambda}_{k,N}{h}_{k,N}$ is divergent. It follows by (2.6) that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ is divergent, which is a contradiction. Hence, ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ converges to a fixed point of T.
We now obtain the convergence theorem of W^{(T,N)}-iteration. The proof is omitted because it is similar as above theorem and Theorem 2.2 of [4].
Theorem 2.2 Let E be a closed interval on the real line and T : E → E be a continuous mapping. For x_{1} ∈ E, let the W^{(T,N)}-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$defined by (1.9), where${\left\{{\lambda}_{n,1}\right\}}_{n=1}^{\infty},{\left\{{\lambda}_{n,2}\right\}}_{n=1}^{\infty}$ (i = 1,2,...,N) are sequences in [0,1] satisfying the following conditions:
(C1) lim_{n→∞}λ_{n,i}= 0 for all i = 1,2,..., N;
(C2)${\sum}_{n=1}^{\infty}{\lambda}_{n,N}=\infty $.
Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$converges to a fixed point of T.
The following results are obtained direclty from Theorem 2.1.
Corollary 2.3 ([4, Theorem 2.1]) Let E be a closed interval on the real line and T : E → E be a continuous mapping. For x_{1} ∈ E, let the SP-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$defined by (1.6), where${\left\{{\lambda}_{n,1}\right\}}_{n=1}^{\infty}$, ${\left\{{\lambda}_{n,2}\right\}}_{n=1}^{\infty}$, and${\left\{{\lambda}_{n,3}\right\}}_{n=1}^{\infty}$are sequences in [0,1] satisfying the following conditions:
(C1)${\sum}_{n=1}^{\infty}{\lambda}_{n,1}<\infty $and${\sum}_{n=1}^{\infty}{\lambda}_{n,2}<\infty $;
(C2) lim_{n→∞}λ_{n,3}= 0 and${\sum}_{n=1}^{\infty}{\lambda}_{n,3}=\infty $.
Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$converges to a fixed point of T.
Corollary 2.4 ([7, Theorem 3]) Let E be a closed interval on the real line and T : E → E be a continuous mapping. For x_{1} ∈ E, let the Mann iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$defined by (1.3), where${\left\{{\lambda}_{n,1}\right\}}_{n=1}^{\infty}$is a sequence in [0,1] satisfying lim_{n→∞}, λ_{n,1}= 0 and${\sum}_{n=1}^{\infty}{\lambda}_{n,1}=\infty $. Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$converges to a fixed point of T.
The following results are obtained directly from Theorem 2.2.
Corollary 2.5 ([4, Theorem 2.2]) Let E be a closed interval on the real line and T : E → E be a continuous mapping. For x_{1} ∈ E, let the Noor iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$defined by (1.5), where${\left\{{\lambda}_{n,1}\right\}}_{n=1}^{\infty}$, ${\left\{{\lambda}_{n,2}\right\}}_{n=1}^{\infty}$, ${\left\{{\lambda}_{n,3}\right\}}_{n=1}^{\infty}$are sequences in [0,1] satisfying the following conditions:
(C1) lim_{n→∞}λ_{n,1}= 0, lim_{n→∞}λ_{n,2}= 0 and lim_{n→∞}λ_{n,3}= 0;
(C2)${\sum}_{n=1}^{\infty}{\lambda}_{n,3}=\infty $.
Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$converges to a fixed point of T.
Corollary 2.6 ([7]) Let E be a closed interval on the real line and T : E → E be a continuous mapping. For x_{1} ∈ E, let the Ishikawa iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$defined by (1.4), where${\left\{{\lambda}_{n,1}\right\}}_{n=1}^{\infty}$are sequences in [0,1] satisfying the following conditions:
(C1) lim_{n→∞}λ_{n,1}= 0 and lim_{n→∞}λ_{n,2}= 0;
(C2)${\sum}_{n=1}^{\infty}{\lambda}_{n,2}=\infty $.
Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$converges to a fixed point of T.
3 Rate of convergence and numerical examples
There are many articles have been published on the iterative methods using for approximation of fixed points of nonlinear mappings, see for instance [1–7]. However, there are only a few articles concerning comparison of those iterative methods in order to establish which one converges faster. As far as we know, there are two ways for comparison of the rate of convergence. The first one was introduced by Berinde [27]. He used this idea to compare the rate of convergence of Picard and Mann iterations for a class of Zamfirescu operators in arbitrary Banach spaces. Popescu [28] also used this concept to compare the rate of convergence of Picard and Mann iterations for a class of quasi-contractive operators. It was shown in [29] that the Mann and Ishikawa iterations are equivalent for the class of Zamfirescu operators. In 2006, Babu and Prasad [30] showed that the Mann iteration converges faster than the Ishikawa iteration for this class of operators. Two years later, Qing and Rhoades [31] provided an example to show that the claim of Babu and Prasad [30] is false.
However, this concept is not suitable or cannot be applied to a class of continuous self-mappings defined on a closed interval. In order to compare the rate of convergence of continuous self-mappings defined on a closed interval, Rhoades [5] introduced the other concept which is slightly different from that of Berinde to compare iterative methods which one converges faster as follows.
In this section, we study the rate of convergence of W^{(T,N)}-iteration and K^{(T,N)}-iteration for continuous and nondecreasing mappings on an arbitrary interval in the sense of Rhoades. The following lemmas are useful and crucial for our following results.
Lemma 3.2 Let E be a closed interval on the real line and T : E → E be a continuous and nondecreasing mapping such that F(T) is nonempty and bounded with x_{1} > sup{p ∈ E : p = Tp}. Let${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$be defined by W^{(T,N)}-iteration or K^{(T,N)}-iteration. If Tx_{1} > x_{1}, then${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$does not converge to a fixed point of T.
Proof. We prove only the case that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ is defined by K^{(T,N)}-iteration because the other case can be proved similarly.
This implies that TU_{n,2}x_{ n }≥ Tx_{ n }≥ x_{ n }. By continuity in this way, we can show that ${x}_{n+1}={K}_{n}^{\left(T,N\right)}{x}_{n}={U}_{n,N}{x}_{n}\ge {x}_{n}$ for all n ≥ 1. Thus ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ is nondecreasing. But x_{1} > sup{p ∈ E : p = Tp}, it implies that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ does not converges to a fixed point of T.
By using the same argument of proof as in above lemma, we get the following result.
Lemma 3.3 Let E be a closed interval on the real line and T : E → E be a continuous and nondecreasing mapping such that F(T) is nonempty and bounded with x_{1} < inf{p ∈ E : p = Tp}. Let${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$be defined by W^{(T,N)}-iteration or K^{(T,N)}-iteration. If Tx_{1} < x_{1}, then${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$does not converge to a fixed point of T.
We now get the following theorem for compare rate of convergence between W^{(T,N)}-iteration and K^{(T,N)}-iteration.
Theorem 3.4 Let E be a closed interval on the real line and T : E → E be a continuous and nondecreasing mapping such that F(T) is nonempty and bounded. For u_{1} = x_{1} ∈ E, let${\left\{{u}_{n}\right\}}_{n=1}^{\infty}$and${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$are the sequences defined by (1.9) and (1.11), respectively. Let${\left\{{\lambda}_{n,i}\right\}}_{n=1}^{\infty}$be sequences in [0,1) for all i = 1,2,..., N. Then, the W^{(T,N)}-iteration${\left\{{u}_{n}\right\}}_{n=1}^{\infty}$converges to the fixed point p of T if and only if the K^{(T,N)}-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$converges to p . Moreover, the K^{(T,N)}-iteration converges faster than the W^{(T,N)}-iteration.
Proof. Put L = inf{p ∈ E : p = Tp} and U = sup{p ∈E : p = Tp}.
(⇒) Suppose that the W^{(T,N)}-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty}$ converges to the fixed point p of T.
We divide our proof into the following three cases:
Since lim_{n→∞}, u_{ n }= p, it implies that lim_{n→∞}x_{ n }= p. That is, the K^{(T,N)}-iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ converges to the same fixed point p. Moreover, by (3.2), we see that the K^{(T,N)}-iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ converges faster than the W^{(T,N)}-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty}$.
Case 2: u_{1} = x_{1} < L. By Lemma 3.3, we have Tu_{1} > u_{1} and Tx_{1} > x_{1}. By using (1.9), (1.11) and the same argument as in Case 1, we can show that x_{ n }≥ u_{ n }for all n ≥ 1. We note that x_{1} < L and by using (1.11) and mathematical induction, we can show that x_{ n }≤ L for all n ≥ 1. Thus, we have |x_{ n }- p| ≤ |u_{ n }- p| for all n ≥ 1. It follows that lim_{n→∞}x_{ n }= p and the K^{(T,N)}-iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ converges faster than the W^{(T,N)}-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty}$.
Case 3: L ≤ u_{1} = x_{1} ≤ U. Suppose that Tu_{1} ≠ u_{1}. Without loss of generality, we suppose Tu_{1} < u_{1}. It follows by (1.9) that u_{ n }≤ u_{1} for all n ≥ 1. Since lim_{n→∞}u_{ n }= p, we must get p < u_{1} = x_{1}. By the same argument as in Case 1, we have p ≤ x_{ n }≤ u_{ n }for all n ≥ 1. It follows that |x_{ n }- p| ≤ |u_{ n }- p| for all n ≥ 1. Hence, lim_{n→∞}x_{ n }= p and the K^{(T,N)}-iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ converges faster than the W^{(T,N)}-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty}$.
By mathematical induction, we have u_{ n }≤ x_{ n }for all n ≥ 1. We note that x_{1} > U and by using (3.3) and mathematical induction, we can show that x_{ n }≥ U for all n ≥ 1. Thus, we have 0 ≤ u_{ n }- p ≤ x_{ n }- p for all n ≥ 1. Since lim_{n→∞}x_{ n }= p, it follows that lim_{n→∞}u_{ n }= p That is, the W^{(T,N)}-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty}$ converges to the same fixed point p.
We also consider the speed of convergence of the K^{(T,N)}-iteration which depends on the choice of control sequences ${\left\{{\lambda}_{n,i}\right\}}_{n=1}^{\infty}$ (i = 1,2,..., N) as the following theorem.
where${\stackrel{\u0304}{K}}_{n}^{\left(T,N\right)}$is the K-mapping generated by T and${\lambda}_{n,1}^{*},{\lambda}_{n,2}^{*},...,{\lambda}_{n,N}^{*}$.
If${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$converges to the fixed point p of T, then${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty}$converges to p. Moreover, ${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty}$converges faster than${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$.
Proof. Put L = inf{p ∈ E : p = Tp} and U = sup{p ∈ E : p = Tp}. Suppose that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ converges to a fixed point p of T. We divide our proof into the following three cases:
That is, ${x}_{k+1}^{*}\le {x}_{k+1}$. By mathematical induction, we obtain ${x}_{n}^{*}\le {x}_{n}$ for all n ≥ 1. Since ${x}_{n}^{*}\ge U$ for all n ≥ 1, we get $0\le {x}_{n}^{*}-p\le {x}_{n}-p$, so $\left|{x}_{n}^{*}-p\right|\le \left|{x}_{n}-p\right|$ for all n ≥ 1. It follows that ${\text{lim}}_{n\to \infty}{x}_{n}^{*}=p$ and ${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty}$ converges faster than ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$.
Case 2: ${x}_{1}^{*}={x}_{1}<L$. By Lemma 3.3, we have $T{x}_{1}^{*}>{x}_{1}^{*}$ and Tx_{1} > x_{1}. By using (3.4), (3.5) and the same argument as in Case 1, we can show that ${x}_{n}^{*}\ge {x}_{n}$ for all n ≥ 1. We note that ${x}_{1}^{*}<L$ and by using (3.5) and mathematical induction, we can show that ${x}_{n}^{*}\le L$ for all n ≥ 1. Thus, we have $\left|{x}_{n}^{*}-p\right|\le \left|{x}_{n}-p\right|$ for all n ≥ 1. It follows that ${\text{lim}}_{n\to \infty}{x}_{n}^{*}=p$ and ${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty}$ converges faster than ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$.
Case 3: $L\le {x}_{1}^{*}={x}_{1}\le U$. Suppose that $T{x}_{1}^{*}\ne {x}_{1}^{*}$. Without loss of generality, we suppose $T{x}_{1}^{*}<{x}_{1}^{*}$. It follows by (3.5) that x_{n+1}≤ x_{ n }for all n ≥ 1. Since lim_{n→∞}x_{ n }= p, we must get $p<{x}_{1}^{*}={x}_{1}$. By the same argument as in Case 1, we have $p\le {x}_{n}^{*}\le {x}_{n}$ for all n ≥ 1. It follows that $\left|{x}_{n}^{*}-p\right|\le \left|{x}_{n}-p\right|$ for all n ≥ 1. Hence, ${\text{lim}}_{n\to \infty}{x}_{n}^{*}=p$ and ${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty}$ converges faster than ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$.
Finally, we present two numerical examples for comparing rate of convergence between W^{(T,N)}-iteration and K^{(T,N)}-iteration.
Comparison of the rate of convergence between W^{(T,10)}-iteration and the K^{(T,10)}-iteration for the mapping given in Example 3.6, for ${\lambda}_{n,i}=\frac{1}{{n}^{1.5}+1}$ (i = 1,2,...,9) and ${\lambda}_{n,10}=\frac{1}{{n}^{0.5}+1}$
W^{(T,10)}-iteration | K^{(T,10)}-iteration | |||||
---|---|---|---|---|---|---|
n | u _{ n } | | Tu _{ n } - u _{ n } | | $\left|\frac{{u}_{n}-{u}_{n-1}}{{u}_{n}}\right|$ | x _{ n } | | Tx _{ n } - x _{ n } | | $\left|\frac{{x}_{n}-{x}_{n-1}}{{x}_{n}}\right|$ |
5 | 3.148579041 | 4.2578E-01 | 5.1938E-02 | 4.038406568 | 3.8114E-03 | 1.3069E-03 |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
41 | 4.007635152 | 1.7228E-02 | 6.1858E-04 | 4.047140965 | 2.4503E-05 | 1.0900E-06 |
42 | 4.009942844 | 1.6218E-02 | 5.7549E-04 | 4.047144996 | 2.2757E-05 | 9.9604E-07 |
43 | 4.012093026 | 1.5277E-02 | 5.3593E-04 | 4.047148686 | 2.1159E-05 | 9.1160E-07 |
44 | 4.014098247 | 1.4400E-02 | 4.9954E-04 | 4.047152068 | 1.9695E-05 | 8.3559E-07 |
45 | 4.015969896 | 1.3582E-02 | 4.6605E-04 | 4.047155172 | 1.8351E-05 | 7.6704E-07 |
From Table 1, we see that the K^{(T,10)}-iteration converges faster than the W^{(T,10)}-iteration under the same control conditions. We also observe that x_{45} = 4.047155172 is an approximation of the fixed point of T with accuracy at 6 significant digits.
Comparison of the rate of convergence between the W^{(T,12)}-iteration and K^{(T,12)}-iteration for the mapping given in Example 3.7, for ${\lambda}_{n,i}=\frac{1}{{n}^{2}+1}$ (i = 1, 2,..., 11) and ${\lambda}_{n,12}=\frac{1}{{n}^{0.5}+1}$, with the initial point u_{1} = x_{1} = 5
W^{(T,12)}-iteration | K^{(T,12)}-iteration | |||||
---|---|---|---|---|---|---|
n | u _{ n } | | Tu _{ n } - u _{ n } | | $\left|\frac{{u}_{n}-{u}_{n-1}}{{u}_{n}}\right|$ | x _{ n } | | Tx _{ n } - x _{ n } | | $\left|\frac{{x}_{n}-{x}_{n-1}}{{x}_{n}}\right|$ |
5 | -0.343578991 | 7.2283E-01 | 8.7516E-01 | -1.215808886 | 4.6467E-05 | 4.2767E-05 |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
27 | -1.202983546 | 1.0858E-02 | 1.6864E-03 | -1.215863785 | 1.7805E-07 | 2.9928E-08 |
28 | -1.204709686 | 9.4031E-03 | 1.4328E-03 | -1.215863815 | 1.5237E-07 | 2.5043E-08 |
29 | -1.206182603 | 8.1616E-03 | 1.2211E-03 | -1.215863841 | 1.3080E-07 | 2.1041E-08 |
30 | -1.207442871 | 7.0994E-03 | 1.0437E-03 | -1.215863862 | 1.1261E-07 | 1.7746E-08 |
In Example 3.7, the mapping T is continuous on [-7,7] but it not differentiable at x = -4 and x = 5. In Table 2, we observe that the K^{(T,12)}-iteration and W^{(T,12)}-iteration with the initial point is x = 5 converge to a fixed point p ≈ -1.215863862 of T. Moreover, the K^{(T,12)}-iteration converges faster than the W^{(T,12)}-iteration.
Open Problem: Is it possible to prove the convergence theorem of a finite family of continuous mappings on an arbitrary interval by using W-mappings and K-mappings and how about the rate of convergence of those methods?
Declarations
Acknowledgements
The authors would like to thank the Centre of Excellence in Mathematics, the Commission on Higher Education for financial support. WP is supported by the Office of the Higher Education Commission and the Graduate School of Chiang Mai University, Thailand.
Authors’ Affiliations
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