New Picard S hybrid iteration

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Introduction of new Picard

–S

hybrid iteration with application

and some results for

nonexpansive mappings

Julee Srivastava

Department of Mathematics and Statistics,

Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur, India Abstract

Purpose– In this paper, Picard–S hybrid iterative process is defined, which is a hybrid of Picard and S-iterative process. This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard_{–Mann hybrid, Picard–Krasnoselskii hybrid and Picard–Ishikawa hybrid iterative processes} for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.

Design/methodology/approach_{– This new iteration converges faster than all of Picard, Krasnoselskii,} Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings.

Findings_{– Showed the fastest convergence of this new iteration and then other iteration defined in this paper.} The author finds the solution of delay differential equation using this hybrid iteration. For new iteration, the author also proved a theorem for nonexpansive mapping.

Originality/value_{– This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa,} S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard_{–S hybrid iterative process for nonexpansive mappings.}

Keywords Fixed point, Iteration, Contraction, Nonexpansive mappings Paper type Research paper

1. Introduction

Let E be a normed linear space and C be a non-empty convex subset of E. A mapping T : C → C is called contraction if

kTx Tyk ≤ δkx yk (1.1)

for all x; y ∈ C and δ ∈ ð0; 1Þ.

Let C be a non-empty subset of a normed linear space E and T : C → E a mapping. Then T is said to be nonexpansive if

kTx Tyk ≤ kx yk; for all x; y ∈ C

A sequence xn⊂ C is an approximating fixed point sequence of T if limn→∞kxn− Txnk ¼ 0 . We say that x ∈ C is a fixed point of T if TðxÞ ¼ x and denote FðTÞ the set of all fixed points of T.

New Picard

–S

hybrid

iteration

JEL Classification— 47H09, 47H10, 54H25, 47J25

© Julee Srivastava. Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen athttp://creativecommons.org/licences/by/4.0/legalcode The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/1319-5166.htm

Received 24 August 2020 Revised 5 November 2020 29 November 2020 Accepted 1 December 2020

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In this paper, N denotes the set of all positive integers.

The Picard iterative process [1] is defined by the sequencefu_ng as follows: u1¼ u ∈ C

unþ1¼ Tun; n ∈ N

(1.2) The Krasnoselskii iterative process [2] is defined by the sequencefv_ng:

v1¼ v ∈ C

vnþ1¼ ð1 λÞvnþ λTvn; n ∈ N

(1.3) whereλ ∈ ð0; 1Þ.

The Mann iteration [3] is defined by the sequencefw_ng: w1¼ w ∈ C

wnþ1¼ ð1 αnÞwnþαnTwn; n ∈ N

(1.4) wherefα_ng ⊂ ð0; 1Þ satisfies certain appropriate conditions.

The Ishikawa iterative process [4] is defined by the sequencefz_ng: z1¼ z ∈ C

znþ1¼ ð1 αnÞznþαnTyn

yn¼ ð1 βnÞznþ βnTzn; n ∈ N

(1.5)

wherefα_ng; fβ_ng ⊂ ð0; 1Þ satisfies certain appropriate conditions. S-iterative process [5] is defined by the sequencefq_ng:

q1¼ q ∈ C

qnþ1¼ ð1 αnÞTqnþαnTyn yn¼ ð1 βnÞqnþ βnTqn; n ∈ N

(1.6)

wherefα_ng; fβ_ng ⊂ ð0; 1Þ satisfies certain appropriate conditions.

Many important non-linear problems of applied mathematics are usually constructed in the form of fixed point equation. These problems are related with physical problem of applied sciences and engineering.

The Picard iteration is the simple iteration for approximate solution of fixed point equation for non-linear contraction mapping. Some results based on Picard iteration are introduced by Chidume and Olaleru [6].

Khan [7] introduced the Picard–Mann hybrid iterative process defined by the sequence fsng:

s1¼ s ∈ C snþ1¼ Tyn

yn¼ ð1 αnÞsnþαnTsn; n ∈ N

(1.7)

wherefα_ng is a real sequence in ð0; 1Þ.

Okeke and Abbas [8] introduced the Picard–Krasnoselskii hybrid iterative process defined by the sequencefm_ng:

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Okeke [9] introduced the Picard–Ishikawa hybrid iterative process defined by the sequenceft_ng: t1¼ t ∈ C tnþ1¼ Tvnv vn¼ ð1 αnÞtnþαnTun un¼ ð1 βnÞtnþ βnTtn; n ∈ N (1.9)

wherefα_ng; fβ_ngare real sequences in (0,1). Using hybridization with Picard, now I introduce Picard–S hybrid iterative process defined by the sequence fx_ng:

x1¼ x ∈ C xnþ1¼ Tzn

zn¼ ð1 αnÞTxnþαnTyn yn¼ ð1 βnÞxnþ βnTxn; n ∈ N

(1.10)

wherefαng and fβng are real sequences in ð0; 1Þ satisfying condition: X∞

n¼1

αnβnð1 βnÞ ¼ ∞ (1.11) Letfu_ng and fv_ng be two fixed point iteration processes that converge to a certain fixed point p of a given operator T. The sequence fung is better than fvng if

kun pk ≤ kvn pk for all n ∈ N (given by Rhodes [10]).

2. Preliminaries

Definition 2.1. Letfa_ng and fb_ng be two sequences of real numbers converging to a and b, respectively. If lim n→∞ jan aj jbn bj ¼ 0 thenfang converges faster than fbng.

Definition 2.2. Letfu_ng and fv_ng be two fixed point iterative processes, both converge to fixed point p of a given operator T. Suppose that the error estimates.

jjun pjj ≤ an; for all n ∈ N jjvn pjj ≤ bn; for all n ∈ N

are available, wherefa_ng and fb_ng are two sequences of positive numbers converging to 0. If fang converges faster than fbng, then fung converges faster than fvng to p.

Definition 2.3. LetX be a Banach space. Then a function δX : ½0; 2 → ½0; 1 is said to be the modulus of convexity of X if

δXðeÞ ¼ inf n 1x þ y 2 : kxk ≤ 1; kyk ≤ 1; kx yk ≥ eo

It is easy to see thatδ_Xð0Þ ¼ 0 and δ_XðtÞ ≥ 0 for all t ≥ 0. References [10–19] dealing with rate of convergence of iterative process. Some authors analyse its stability. We need following lemma to prove result.

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Lemma 2.1. Letfs_ng be a sequence of positive real numbers which satisfies snþ1 ≤ ð1 μnÞsn If fμng⊂ ð0; 1Þ and X∞ n¼1 μn¼ ∞; then lim_n→∞sn¼ 0:

The aim of this paper is to introduce the Picard–S hybrid iterative process and to show that this new iterative process is faster than all of Picard, Krasnoselskii, Mann, Ishikawa in sense of Berinde [20], S-iteration in sense of Agarwal [5], Picard–Mann hybrid in sense of Khan [7], Picard–Krasnoselskii hybrid in sense of Okeke [8] and Picard–Ishikawa hybrid in the sense of Okeke [9].

Okeke already proved that Picard–Krasnoselskii hybrid iterative process converges faster than Picard, Krasnoselskii, Mann and Ishikawa. Khan [7] proved that Picard–Mann hybrid iterative process converges faster than Picard, Mann, Ishikawa iterative processes. Therefore, I show that my new Picard–S hybrid iterative process converges faster than S-iteration, Picard– Mann hybrid iteration, Picard–Krasnoselskii hybrid iteration and Picard–Ishikawa hybrid iterative process in the topic Rate of Convergence. In 2020, Zhao [21] proved existence and uniqueness of pseudo almost periodic solution for a class of iterative functional differential equations with delays depending on state. In next section, I find the solution of delay differential equation using Picard–S hybrid iterative process. Aynur Sahin [22] proved some strong convergence results of Picard–Krasnoselskii hybrid iterative process for a general class of contractive-like operator in hyperbolic space. In next section, I prove some results of Picard–S hybrid iterative process for nonexpansive mappings in uniformly convex Banach space. 3. Rate of convergence

Proposition 3.1. LetC be a non-empty closed convex subset of a normed space E and let T be a contraction of C into itself. Suppose that each of the iterative process 1.6, 1.7, 1.8, 1.9 and 1.10 converges to the same fixed point p of T where fαng and fβng are sequences in (0.1) such that 0< λ ≤α_n,β_n< 1for all n ∈ N and for some λ and δ ∈ ð0; 1Þis a Lipschitz constant for contraction mapping T. Then Picard–S hybrid iterative process defined by(1.10)converges faster than all the other four iterations.

Proof: Suppose that p is the fixed point of the operator T. Using(1.1)and S-iterative process (1.6), we have kqnþ1 pk ≤ ð1 αnÞδkqn pk þαnδkyn pk ≤ δ½ð1 αnÞkqn pk þαnfð1 βnÞjjqn pjj þ δβnjjqn pjjg ¼ δ½1 ð1 δÞαnβnkqn pk ¼δ δαnβnþ δ 2_α nβn kqn pk ≤ ½δ δαnβnþ δαnβnkqn pk ¼ δkqn pk ≤ δ2_kq n−1 pk ... ≤ δn_kq 1 pk Let an¼ δnkq1 pk (3.1)

Now using(1.1)and Picard–Mann hybrid iterative process(1.7), we have

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ksnþ1 Pk ¼ kTyn pk ≤ δkyn pk ¼ δ½ð1 αnÞksn pk þαnkTsn pk ≤ δ½ð1 αnÞksn pk þαnδksn pk ¼ δ½ð1 αnþαnδÞksn pk ¼ δð1 ð1 δÞαnÞksn pk ≤ δ1 ð1 δÞλ2_Þks n pk ... ≤δ1 ð1 δÞλ2Þnks1 pk Let bn¼ δ1 ð1 δÞλ2_Þn_ks 1 pk (3.2)

Using(1.1)and Picard–Krasnoselskii hybrid iteration(1.8)

kmnþ1 pk ¼ kTyn pk ≤ δkyn pk ≤ δkð1 λÞðmn pÞ þ λðTmn pÞk ≤ δ½ð1 λÞkmn pk þ λδkmn pk ¼ δ1 ð1 δÞλ2_Þ_km n pk ... ≤δ1 ð1 δÞλ2Þnkm1 pk Let cn¼ δ1 ð1 δÞλ2_Þn_km 1 pk (3.3)

Using(1.1)and Picard–Ishikawa hybrid iterative process(1.9), we have ktnþ1 pk ¼ kTvn pk ≤ δkvn pk kun pk ¼ kð1 βnÞtnþ βnTtn pk ≤ ð1 βnÞktn pk þ βnkTtn pk ≤ ð1 βnÞktn pk þ βnδktn pk ¼ ð1 βnþ βnδÞktn pk ¼ ½1 βnð1 δÞktn pk kvn pk ¼ kð1 αnÞtnþαnTun pk ≤ ð1 αnÞktn pk þαnkTun pk ≤ ð1 αnÞktn pk þαnδkun pk ≤ ð1 αnÞktn pk þαnδ½1 βnð1 δÞktn pk ¼ ½1 _αnþαnδf1 βnð1 δÞgktn pk ¼ ½1 _αnþαnδ αnβnð1 δÞktn pk ¼ ½1 αnð1 δÞ αnβnð1 δÞktn pk ≤ ½1 αnð1 δÞktn pk Now; ktnþ1 pk ≤ δ½1 αnð1 δÞktn pk ≤ δ1 λ2ð1 δÞktn pk ... ≤ δn₁_λ2_{ð1 δÞ}n_kt 1 pk Let dn¼ δn 1 λ2ð1 δÞnkt1 pk (3.4)

Using(1.1)and Picard–S hybrid iterative process(1.10), we have

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kxnþ1 pk ¼ kTzn pk ≤ δkzn pk ≤ δkð1 αnÞTxnþαnTyn pk ≤ δkð1 αnÞðTxn pÞ þαnðTyn pÞk ≤ δ2_ð1_α nÞkxn pk þ δ2αnkyn pk ¼ δ2_½ð1_α nÞkxn pk þαnfð1 βnÞkxn pk þ βnδkxnpkg ¼ δ2_½1_α nþαnð1 βnÞ þαnβnδkxn pk ¼ δ2_{½1 ð1 δÞ}_α nβnkxn pk ≤ δ2 1 ð1 δÞλ2_kx n pk ... ≤δ2 1 ð1 δÞλ2_Þn_kx 1 pk Let en¼ δ2 1 ð1 δÞλ2_Þn kx1 pk (3.5)

Now compute the rate of convergence of Picard–S iterative process(1.10)as follows:

ðiÞ en an ¼½δ 2_{ð1 ð1 δÞλ}2_Þn δn_kq 1 pk kx 1 pk ¼ δn₁_{ð1 δÞλ}2_Þnkx1 pk kq1 pk → 0 as n → ∞

Thus,fx_ng converges faster than fq_ng to p, i.e. the Picard–S hybrid iterative process(1.10)

converges faster than the S-iterative process:

ðiiÞ en bn ¼½δ 2_{ð1 ð1 δÞλ}2_Þn ½δð1 ð1 δÞλ2_Þn kx1 pk ks1 pk ¼ δnkx1 pk ks1 pk → 0 as n → ∞

Thus,fx_ng converges faster than fs_ng to p, i.e. the Picard–S hybrid iterative process(1.10)

converges faster than the Picard–Mann hybrid iterative process.

ðiiiÞ en cn ¼½δ 2_{ð1 ð1 δÞλ}2_Þn ½δð1 ð1 δÞλ2_Þn kx1 pk km1 pk ¼ δnkx1 pk km1 pk → 0 as n → ∞ (3.6)

Thus,fx_ng converges faster than fm_ng to p, i.e. the Picard–S hybrid iterative process(1.10)

converges faster than the Picard–Krasnoselskii hybrid iterative process.

ðivÞ en dn¼ ½δ2 ð1 ð1 δÞλ2_Þn δn_{½1 λ}2_{ð1 δÞ}n kx1 pk kt1 pk ¼ δnkx1 pk kt1 pk → 0 as n → ∞ (3.7)

Thus,fx_ng converges faster than ft_ng to p, i.e. Picard–S hybrid iterative process converges faster than Picard–Ishikawa hybrid iterative process. This completes the proof of the

proposition. ,

In [8], Okeke proved that the rate of convergence of Picard–Krasnoselskii hybrid iterative process is faster than Picard, Krasnoselskii, Mann and Ishikawa iterations. Agarwal et al. [5]

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proved that S-iteration converges faster than Picard, Krasnoselskii, Mann and Ishikawa iterative processes, and Okeke [9] proved that rate of convergence of Picard–Ishikawa hybrid iterative process is faster than Picard–Mann hibrid and Picard–Krasnoselskii iterations. Therefore, I give an example to show that rate of convergence of Picard–S hybrid iterative process is faster than Picard–Mann hybrid, Picard–Krasnoselskii hybrid and S-iteration. This will show that Picard–S hybrid defined by(1.10)converges to fixed point faster than all other iterations defined in this paper.

Example 3.1. LetX ¼ R and C ¼ ½1; 10 ⊂ X and T : C → C be an operator defined by Tx ¼p3ffiffiffiffiffiffiffiffiffiffiffiffiffi2x þ 4

for all x ∈ C. Chooseαn¼ βn¼ λ ¼12for each n ∈ N with initial value x1¼ 5. For δ ¼p31ﬃﬃ₄, T is a contraction mapping. All the processes converge to the same fixed

point 2. It is clear from Table 1 and graphs that our Picard–S hybrid iterative process converges faster than Picard–Ishikawa hybrid, Picard–Mann hybrid, Picard–Krasnoselskii hybrid and S-iteration.

4. Application to delay differential equation

Here, I use this new Picard–S hybrid iterative process to find the solution of delay differential equations.

Let C½a; b be a space of all continuous real valued function on a closed interval [a, b] be endowed with the Chebyshev norm:

kx yk_∞¼ max

t ∈ ½a;bjxðtÞ yðtÞj:

Space ðC½a; b; k$k_∞Þ is known as Banach Space. In this section, the following delay differential equation has been taken:

x0ðtÞ ¼ f ðt; xðtÞ; xðt τÞÞ t ∈ ½a; b (4.1) with initial condition

xðtÞ ¼

f

ðtÞ t ∈ ½toτ; to (4.2) By the solution of above problem, we mean a function x ∈ Cð½to−τ; b; RÞ∩ ðC1½to; b; RÞ satisfying(4.1)and(4.2). Assume that the following conditions are satisfied.

ðC1Þ to; b ∈ R;τ> 0;

Step Picard–S hybrid

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ðC2Þ f ∈ Cð½to; b 3 R2; RÞ; ðC3Þ

f

∈ Cð½to−τ; b; RÞ; ðC4Þ there exists L_f > 0 such that

jf ðt; u1; u2Þ f ðt; v1; v2Þj ≤ Lf X2

i¼1

jui vij ∀ui; vi ∈ R; i ¼ 1; 2; t ∈ ½to; b ðC5Þ 2Lfðb − toÞ < 1;

Now we can reformulate problems(4.1)and(4.2)by the following integral equation:

xðtÞ ¼ 8 > < > :

f

ðtÞ t ∈ ½toτ; to

f

ðtoÞ þ Z t to f ðs; xðsÞ; xðs τÞÞds t ∈ ½to; b Coman [23] et al. established the following result.

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Theorem 4.1. Assume that the conditionsðC1− C5Þ are satisfied. Then problem(4.1)with initial condition(4.2)has unique solution p (say) in Cð½to−τ; b; RÞ∩ C1ð½to; b; RÞ and

p ¼ lim n→∞T

n_{ðxÞ for any x ∈ Cð½t}

oτ; b; RÞ: Using Picard–S hybrid iterative process, I prove the following result.

Theorem 4.2. Assume that ðC1Þ − ðC5Þ are satisfied. Then problem (4.1) with initial condition(4.2)has unique solution p (say) in Cð½to−τ; b; RÞ∩ C1ð½to; b; RÞand the Picard–S hybrid iterative process (1.10) converges to p.

Proof: Let fxng be an iterative sequence generated by the Picard–S hybrid iterative process

(1.10)for an operator defined by

TxðtÞ ¼ 8 > < > :

f

ðtÞ t ∈ ½toτ; to

f

ðtoÞ þ Z t to f ðs; xðsÞ; xðs τÞÞds t ∈ ½to; b

Let p be a fixed point of T, now I prove that xn→ p as n → ∞. It is easy to see that xn→ p for each t ∈ ½to−τ; to. Now for each t ∈ ½to; b, we have

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Using(4.6)in (4.7), we get kTyn pk∞¼ kTyn Tpk∞ ≤ 2Lfðb t0Þfð1 β_nÞkxn pk∞þ βnkTxn pk∞g ¼ 2ð1 βnÞLfðb toÞkxn pk∞þ 2βnLfðb toÞkTxn pk∞ (4.8) From(4.5)we get kTyn pk∞≤ 2ð1 βnÞLfðb toÞkxn pk∞þ 2βnLfðb toÞ2Lfðb toÞkxn pk∞ ¼ 2Lfðb toÞfð1 βnÞ þ 2βnLfðb toÞgkxn pk∞ (4.9) Using(4.5)and(4.9)in(4.4), we get

kzn pk∞≤ ð1 αnÞ2Lfðb toÞkxn pk∞þαn½2Lfðb t0Þfð1 βnÞ þ 2βnLfðb toÞgkxn pk∞

¼ 2Lfðb toÞ½ð1 αnÞ þαnð1 βnÞ þ 2αnβnLfðb toÞkxn pk∞

¼ 2Lfðb toÞ½1 αnβnþ 2αnβnLfðt toÞkxn pk∞

(4.10) Note that 2Lfðb − toÞ½1 −αnβnþ 2αnβnLfðt − toÞ ¼μn< 1 and kxn−Pk∞¼ Sn. Thus, all conditions oflemma 2.1are satisfied. Hence, lim

n→∞kxn−Pk∞¼ 0. This completes the proof of

above theorem. ,

5. Picard–S hybrid iterative process for nonexpansive mappings

Lemma 5.1. LetE be a normed space, C a non-empty convex subset of E and T : C → C a nonexpansive mapping. Iffx_ng is the iterative process defined by(1.10), then lim

n→∞kxn− Txnk exists.

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From inequality(5.2)and(5.3), we have kxn znk≤ kank þαnβnkαnk ¼ ð1 þαnβnÞkαnk (5.4) Now, kyn znk ¼ kyn fð1 αnÞTxnþαnTyngk ¼ kð1 αnÞynþαnyn ð1 αnÞTxnαnTynk ¼ kð1 αnÞðyn TxnÞ þαnðyn TynÞk ≤ ð1 αnÞkyn Txnk þαnkyn Tynk (5.5) Now, kyn Txnk ¼ kð1 βnÞxnþ βnTxn Txnk ¼ kð1 βnÞxn ð1 βnÞTxnk ≤ ð1 βnÞkxn Txnk ¼ ð1 βnÞkank (5.6) kyn Tynk ¼ kð1 βnÞxnþ βnTxn Tynk ¼ kð1 βnÞxnþ βnTxn ð1 βnþ βnÞTynk ¼ kð1 βnÞðxn TynÞ þ βnðTxn TynÞk ≤ ð1 βnÞkxn Tynk þ βnkTxn Tynk ¼ ð1 βnÞkxn Tynk þβnkxn ynk (5.7) Using(5.3)in(5.7), we get kyn Tynk ≤ ð1 βnÞkxn Tynk þ βnβnkank ¼ ð1 βnÞkxn Tynk þβ2nkank (5.8) Using(5.6)and(5.8)in(5.5), we get

kyn znk≤ ð1 αnÞð1 βnÞkank þαn ð1 βnÞkxn Tynk þβ2nkank ¼ ð1 αnÞð1 βnÞkank þαn ð1 βnÞkxn Txnþ Txn Tynk þ β2nkank ≤ ð1 αnÞð1 βnÞkank þαn ð1 βnÞðkxn Txnk þ kTxn TynkÞ þβ2nkank ≤ ð1 αnÞð1 βnÞkank þαn ðð1 βnÞkxn Txnk þ ð1 βnÞkxn ynkÞ þβ2nkank ¼ ð1 αnÞð1 βnÞkank þαn ðð1 βnÞkank þ ð1 βnÞkxn ynkÞ þ β2nkank (5.9) Using(5.3)in(5.9), we get kyn znk≤ ð1 αnÞð1 βnÞkank þαn ðð1 βnÞkank þ ð1 βnÞβnkankÞ þβ2nkank ¼ð1 _αnÞð1 βnÞ þαnð1 βnÞð1 þ βnÞ þαnβ2n kank ¼ ð1 βnþαnβnÞkank (5.10)

From(5.1),(5.4)and(5.10), we get

kanþ1k≤ ½ð1 αnÞð1 þαnβnÞkank þ ½αnð1 βnþαnβnÞkank ¼ ½ð1 αnÞð1 þαnβnÞ þαnð1 βnþαnβnÞkank ¼ ½1 þαnβnαnα2nβnþαnαnβnþα 2 nβnkank ¼ kank

So thatfka_nkg is nonincreasing and hence, lim

n→∞kank exists. ,

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Theorem 5.2. [5] Let X be a Banach space with modulus of convexity δX. Then kð1 tÞx þ tyk ≤ 1 2tð1 tÞδXðkx ykÞ

for all x; y ∈ X with kxk ≤ 1; kyk ≤ 1 and all t ∈ ½0; 1.

Theorem 5.3. LetC be a non-empty closed convex (not necessary bounded) subset of a uniformly convex Banach space X and T : C → C a nonexpansive mapping. Let fxng be the sequence defined by(1.10)with the restriction:

lim

n→∞αnβnð1 αnÞ exists and limn→∞αnβnð1 βnÞ ≠ 0

Then, for arbitrary initial value x1∈ C; fkxn− Txnkg converges to some constant γCðTÞ ¼ inffkx − Txk : x ∈ Cg, which is independent of the choice of the initial value x1∈ C.

Proof:Lemma (5.1)implies that lim

n→∞kxn− Txnkexists and denoteγðx1Þ ¼ limn→∞kxn− Txnk. Let fx_ng be another iterative sequence generated by(1.10) with the same restriction on parametersfα_ng and fβ_ng of iteration as the sequence fx_ng but with the initial value x₁∈ C. It follows fromlemma (5.1)that

lim n→∞x n Tx n ¼ γx 1 (5.11) Observe that _Ty_n_Ty* n ≤ yn y*n ≤ ð1 βnÞxn x*n þ βnTxn Tx*n ≤ ð1 βnÞxn x*n þ βnxn x*n ≤xn x*n (5.12) Now _x_nþ1_x* nþ1 ¼ Tzn Tz*n ≤zn z*n ¼ð1αnÞ Txn Tx*n þαn Tyn Ty*n ≤ ð1 αnÞTxn Tx*n þαnTyn Ty*n ≤ ð1 αnÞxn x*n þαnTyn Ty*n (5.13) Using(5.12)in(5.13), we have _x_nþ1_x* nþ1 ≤ ð1 αnÞxn x*n þαnxn x*n (5.14) ¼xn xn (5.15)

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Since xn xn _x_n_x n ¼xn xn _x_n_x n ¼ 1 Txn Txn _x_n_x n ¼Txn Txn _x_n_x n ≤ _x_n_x n _x_n_x n ¼ 1 Using theorem(5.2)and(5.16), we obtain that

_y_n_y n ≤ 1 2βnð1 βnÞδX _x_n_x n ðTxn Txn _x_n_x n xn xn It follows from(5.14)that

_xnþ1_x nþ1 ≤ ð1 αnÞxn xn þαnxn xn 1 2β_nð1 β_nÞδXxn x n Txn Txn _xn_x n ¼xn xn 2αnβnð1 βnÞxn xnδX _xn_x n Txn Txn _x1_x 1 (5.17) This gives us 2α_nβ_nð1 β_nÞxn xnδX _x_n_x n Txn Txn _x_n_x n ≤xn xn xnþ1 xnþ1 Or 2α_nβ_nð1 β_nÞxn xnδX _x_n_x n Txn Txn _x_n_x n ≤x1 x1 Using restriction lim

n→∞αnβnð1 − βnÞ ≠ 0 and limn→∞xn− x n ¼d > 0. Therefore, lim n→∞δX _x_n_x n Txn Txn _x_n_x n ¼ 0 δXis strictly increasing and continuous and lim

n→∞ _x_n_{− x} n ¼d > 0. We have lim n→∞xn x n Txn Txn ¼0 Observe that kxn Txnk xn Tx n ≤ ðxn TxnÞ x_n Tx_n which implies that

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Thus,γðx1Þ ¼ γðx1Þ. Because

kxnþ1 Txnþ1k≤ kxn Txnk≤ kx1 Tx1k for all n ∈ N and x1 ∈ C

It follows that

γCðTÞ ¼ inffkx Txk : x ∈ Cg ,

References

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