Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators

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ITERATION FOR THE CLASS OF ZAMFIRESCU OPERATORS

G. V. R. BABU AND K. N. V. V. VARA PRASAD

Received 3 February 2005; Revised 31 March 2005; Accepted 19 April 2005

The purpose of this paper is to show that the Mann iteration converges faster than the Ishikawa iteration for the class of Zamfirescu operators of an arbitrary closed convex subset of a Banach space.

1. Introduction

LetEbe a normed linear space,T:E→Ea given operator. Letx0∈Ebe arbitrary and

{αn} ⊂[0, 1] a sequence of real numbers. The sequence{xn}∞n=0⊂Edefined by

xn+1=1−αnxn+αnTxn, n=0, 1, 2,..., (1.1)

is called theMann iterationorMann iterative procedure.

Lety0∈Ebe arbitrary and{αn}and{βn}be sequences of real numbers in [0, 1]. The sequence{yn}∞n=0⊂Edefined by

yn+1=

1−αn

yn+αnTzn, n=0, 1, 2,..., zn=1−βnyn+βnT yn, n=0, 1, 2,...,

(1.2)

is called theIshikawa iterationorIshikawa iteration procedure. Zamfirescu proved the following theorem.

Theorem1.1 [5]. Let(X,d)be a complete metric space, andT:X→Xa map for which there exist real numbersa,b, andcsatisfying0< a <1,0< b,c <1/2such that for each pair x,yinX, at least one of the following is true:

(z1)d(Tx,T y)≤ad(x,y);

(z2)d(Tx,T y)≤b[d(x,Tx) +d(y,T y)];

(z3)d(Tx,T y)≤c[d(x,T y) +d(y,Tx)].

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ThenThas a unique fixed pointpand the Picard iteration{xn}∞n=0defined by

xn+1=Txn, n=0, 1, 2,..., (1.3)

converges top, for anyx0∈X.

An operatorTwhich satisfies the contraction conditions (z1)–(z3) ofTheorem 1.1will

be called aZamfirescu operator[2].

Definition 1.2[3]. Let{an}∞n=0,{bn}∞n=0be two sequences of real numbers that converge

toaandb, respectively, and assume that there exists

l=lim n→∞

_a_n₋_a

_b_n₋_b. (1.4)

Ifl=0, then we say that{an}∞n=0converges fastertoathan{bn}∞n=0tob.

Definition 1.3 [3]. Suppose that for two fixed point iteration procedures{un}∞n=0 and

{vn}∞n=0both converging to the same fixed pointpwith the error estimates

_u_n₋_p_≤_a_n_, _n₌_{0, 1, 2,}_..._, _v

n−p≤bn, n=0, 1, 2,...,

(1.5)

where{an}∞n=0and{bn}∞n=0are two sequences of positive numbers (converging to zero).

If{an}∞n=0converges faster than{bn}n∞=0, then we say that{un}∞n=0 converges fasterthan

{vn}∞n=0top.

We useDefinition 1.3to prove our main results.

Based onDefinition 1.3, Berinde [3] compared the Picard and Mann iterations of the class of Zamfirescu operators defined on a closed convex subset of a uniformly convex Banach space and concluded that the Picard iteration always converges faster than the Mann iteration, and these were observed empirically on some numerical tests in [1]. In fact, the uniform convexity of the space is not necessary to prove this conclusion, and hence the following theorem [3, Theorem 4] is established in arbitrary Banach spaces.

Theorem1.4 [3]. LetEbe an arbitrary Banach space,Ka closed convex subset of E, andT: K→Kbe a Zamfirescu operator. Let{xn}∞n=0be defined by (1.1) andx0∈K, with{αn} ⊂ [0, 1]satisfying

(i)α0=1,

(ii) 0≤αn<1forn≥1, (iii)Σ∞_n=0αn= ∞.

Then{xn}∞n=0converges strongly to the fixed point ofT and, moreover, the Picard iteration

{xn}∞n=0defined by (1.3) forx0∈K, converges faster than the Mann iteration.

Some numerical tests have been performed with the aid of the software package fixed point [1] and raised the followingopen problemin [3]:for the class of Zamfirescu operators, does the Mann iteration converge faster than the Ishikawa iteration?

The aim of this paper is to answer this open problem aﬃrmatively, that is, to show that the Mann iteration converges faster than the Ishikawa iteration.

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Theorem1.5 [2]. LetEbe an arbitrary Banach space,K a closed convex subset ofE, and T:K→K be a Zamfirescu operator. Let{yn}∞n=0be the Ishikawa iteration defined by (1.2)

fory0∈K, where{αn}∞n=0and{βn}∞n=0are sequences of real numbers in[0, 1]with{αn}∞n=0

satisfying (iii).

Then{yn}∞n=0converges strongly to the unique fixed point of T.

2. Main result

Theorem2.1. LetEbe an arbitrary Banach space,Kbe an arbitrary closed convex subset of E, andT:K→Kbe a Zamfirescu operator. Let{xn}∞n=0be defined by (1.1) forx0∈K, and

{yn}∞n=0be defined by (1.2) for y0∈Kwith{αn}∞n=0and{βn}∞n=0real sequences satisfying

(a) 0≤αn,βn≤1and(b)Σαn= ∞. Then{xn}∞n=0and{yn}∞n=0 converge strongly to the

unique fixed point ofT, and moreover, the Mann iteration converges faster than the Ishikawa iteration to the fixed point of T.

Proof. By [2, Theorem 1] (established in [4]), forx0∈K, the Mann iteration defined by

(1.1) converges strongly to the unique fixed point ofT.

ByTheorem 1.5, fory0∈K, the Ishikawa iteration defined by (1.2) converges strongly

to the unique fixed point ofT. By the uniqueness of fixed point for Zamfirescu operators, the Mann and Ishikawa iterations must converge to the same unique fixed point,p(say) ofT.

SinceTis a Zamfirescu operator, it satisfies the inequalities

Tx−T y ≤δx−y+ 2δx−Tx, (2.1)

Tx−T y ≤δx−y+ 2δy−Tx (2.2)

for allx,y∈K, whereδ=max{a,b/(1−b),c/(1−c)}, and 0≤δ <1, see [3].

Suppose thatx0∈K. Let{xn}∞n=0be the Mann iteration associated withT, and{αn}∞n=0.

Now by using Mann iteration (1.1), we have

_x_n₊₁₋_p_≤

1−αnxn−p+αnTxn−p. (2.3)

On using (2.1) withx=pandy=xn, we get

_Tx_n₋_p_≤_δ_x_n₋_p_. _(2.4)

Therefore from (2.3),

_x_n₊₁₋_p_≤

1−αnxn−p+αnδxn−p=1−αn(1−δ)xn−p (2.5)

and thus

_x_n₊₁₋_p_≤n k=1

1−αk(1−δ)]·x1−p, n=0, 1, 2,.... (2.6)

Here we observe that

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Now let{yn}∞n=0be the sequence defined by Ishikawa iteration (1.2) fory0∈K. Then

we have

_y_n₊₁₋_p_≤

1−αnyn−p+αnTzn−p. (2.8)

On using (2.2) withx=pandy=zn, we have

_Tz_n₋_p_≤_δ_z_n₋_p_{+ 2}_δ_z_n₋_p₌₃_δ_z_n₋_p_. _(2.9)

Again using (2.2) withx=pandy=yn, we have

_{T y}_n₋_p_≤_δ_y_n₋_p_{+ 2}_δ_y_n₋_p₌₃_δ_y_n₋_p_. _(2.10)

Now

_z

n−p≤

1−βnyn−p+βnT yn−p (2.11)

and hence by (2.8)–(2.11), we obtain

_y

n+1−p≤

1−αnyn−p+ 3δαnzn−p ≤1−αnyn−p+ 3δαn

1−βnyn−p+βnT yn−p =1−αnyn−p+ 3δαn1−βnyn−p+ 3δαnβnT yn−p ≤1−αnyn−p+ 3δαn1−βnyn−p+ 3δαnβn3δyn−p =1−αn+ 3δαn1−βn+ 9αnβnδ2·yn−p

=1−αn1−3δ+ 3βnδ−9βnδ2·yn−p =1−αn(1−3δ)1 + 3βnδ·yn−p.

(2.12)

Here we observe that

1−αn(1−3δ)1 + 3βnδ>0 ∀k=0, 1, 2,.... (2.13)

We have the following two cases. Case (i). Letδ∈(0, 1/3]. In this case

1−αn(1−3δ)1 + 3βnδ≤1, ∀n=0, 1, 2,..., (2.14)

and hence the inequality (2.12) becomes

_y_n₊₁₋_p_≤_y_n₋_p _∀n _(2.15)

and thus,

_y

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We now compare the coeﬃcients of the inequalities (2.6) and (2.16), usingDefinition 1.3, with

an= n

k=1

1−αk(1−δ), bn=1, (2.17)

by (b) we have limn→∞(an/bn)=0. Case (ii). Letδ∈(1/3, 1). In this case

1<1−αn(1−3δ)1 + 3βnδ≤1−αn1−9δ2 (2.18)

so that the inequality (2.12) becomes

_y_n₊₁₋_p_≤

1−αn1−9δ2yn−p ∀n. (2.19)

Therefore

_y_n₊₁₋_p_≤n k=1

1−αk

1−9δ2y1−p. (2.20)

We compare (2.6) and (2.20), usingDefinition 1.3with

an= n

k=1

1−αk(1−δ)

, bn= n

k=1

1−αk

1−δ2_. _(2.21)

Herean≥0 andbn≥0 for alln; andbn≥1 for alln.

Thusan/bn≤anand since limn→∞an=0, we have limn→∞(an/bn)=0.

Hence, from Cases(i)and(ii), it follows that{an}converges faster than{bn}, so that the Mann iteration{xn}converges faster than the Ishikawa iteration to the fixed pointp

ofT.

Corollary2.2. Under the hypotheses ofTheorem 2.1, the Picard iteration defined by (1.3) converges faster than the Ishikawa iteration defined by (1.2), to the fixed point of Zamfirescu operator.

Proof. It follows from Theorems1.4and2.1.

Remark 2.3. The Ishikawa iteration (1.2) is depending upon the parameters{αn}∞n=0and

{βn}∞n=0whereas the Mann iteration (1.1) is only on{αn}∞n=0; and byTheorem 2.1, Mann

iteration converges faster than the Ishikawa iteration. Now, the Picard iteration (1.3) is free from parameters andTheorem 1.4says that the Picard iteration converges faster than the Mann iteration.

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Acknowledgments

This work is partially supported by U. G. C. Major Research Project Grant F. 8-8/2003 (SR). One of the authors (G. V. R. Babu) thanks the University Grants commission, India, for the financial support.

References

[1] V. Berinde,Iterative Approximation of Fixed Points, Editura Efemeride, Baia Mare, 2002. [2] ,On the convergence of the Ishikawa iteration in the class of quasi contractive operators,

Acta Mathematica Universitatis Comenianae. New Series73(2004), no. 1, 119–126.

[3] ,Picard iteration converges faster than Mann iteration for a class of quasi-contractive oper-ators, Fixed Point Theory and Applications (2004), no. 2, 97–105.

[4] ,On the convergence of Mann iteration for a class of quasicontractive operators, in prepa-ration, 2004.

[5] T. Zamfirescu,Fix point theorems in metric spaces, Archiv der Mathematik23(1992), 292–298.

G. V. R. Babu: Department of Mathematics, Andhra University, Visakhapatnam, Andhra Pradesh, 530 003, India

E-mail address:gvr babu@hotmail.com

K. N. V. V. Vara Prasad: Department of Mathematics, Dr. L. B. College, Andhra University, Visakhapatnam, Andhra Pradesh, 530 013, India