International Journal of Pure and Applied Mathematics————————————————————————–Volume 66 No. 1 2011, 11-24

CONVERGENCE AND STABILITY OF THE ISHIKAWA

ITERATION SCHEMES FOR QUASI-CONTRACTIVE

MAPPINGS IN UNIFORMLY CONVEX BANACH SPACES

Zeqing Liu1, Jeong Sheok Ume2 §

1Department of MathematicsLiaoning Normal University

Dalian, Liaoning, 116029, P.R. CHINAe-mail: zeqingliu@dl.cn

2Department of Applied MathematicsChangwon National UniversityChangwon, 641-773, KOREAe-mail: jsume@changwon.ac.kr

Abstract: It is proved that certain Ishikawa iteration schemes with errorsintroduced by Xu (see J. Math. Anal. Appl., 224 (1998), 91-101) can beused to approximate fixed points of quasi-contractive mappings in p-uniformlyconvex Banach spaces. A related result deals with the stability of the Ishikawaiteration scheme for quasi-contractive mappings in p-uniformly convex Banachspaces. Our results not only resolve affirmatively two open questions put forthby Rhoades [17] and Naimpally and Singh [13] and Osilike [16], respectively, inthe more general setting, but also extend, improve and unify the correspondingresults obtained by Chidume [4], Liu [14], [15], Osilike [16] and others.

AMS Subject Classification: 47H10, 40A05Key Words: quasi-contractive mapping, fixed point, stability, Ishikawa iter-ation sequence with errors, Ishikawa iteration sequence, closed convex boundedsubset, p-uniformly convex Banach space

1. Introduction

Let X be a normed linear space with norm ‖ · ‖, T be an operator on X and

Received: July 16, 2010 c© 2011 Academic Publications§Correspondence author

12 Z. Liu, J.S. Ume

F (T ) = {x : Tx = x ∈ X} 6= φ. Let x0 be a point in X and xn+1 = f(T, xn)denote an iteration procedure which yields a sequence of point {xn}

∞n=0 in

X. Assume that {xn}∞n=0 converges strongly to q ∈ F (T ). Let {yn}

∞n=0 be an

arbitrary sequence in X, and set εn = ‖yn+1 − f(T, yn)‖. If limn→∞ εn = 0implies that limn→∞ yn = q, then the iteration procedure defined by xn+1 =f(T, xn) is said to be T -stable or stable with respect to T.

Let K be a nonempty convex subset of X and T be a mapping of K intoitself. Recall that T is quasi-contractive if there exists a constant r ∈ (0, 1)such that

‖Tx − Ty‖ ≤ r max{‖x − y‖, ‖x − Tx‖, ‖y − Ty‖,

‖x − Ty‖, ‖y − Tx‖}, ∀x, y ∈ K. (1.1)

The following iteration schemes were introduced by Ishikawa [10], Mann[12] and Xu [25], respectively.

(i) For any given x0 ∈ K the sequence {xn}∞n=0 defined by

xn+1 = (1 − an)xn + anTyn, yn = (1 − bn)xn + bnTxn, ∀n ≥ 0,

is called the Ishikawa iteration sequence, where {an}∞n=0 and {bn}

∞n=0 are real

sequences in [0, 1] satisfying appropriate conditions.

(ii) In particular, if bn = 0 for all n ≥ 0, then the sequence {xn}∞n=0 defined

by

x0 ∈ K, xn+1 = (1 − an)xn + anTxn, ∀n ≥ 0,

is called the Mann iteration sequence.

(iii) For any given x0 ∈ K the sequence {xn}∞n=0 defined by

xn+1 = anxn + bnTyn + cnun, yn = a′nxn + b′nTxn + c′nvn, ∀n ≥ 0,

where {un}∞n=0, {vn}

∞n=0 are arbitrary bounded sequences in K and {an}

∞n=0,

{bn}∞n=0, {cn}

∞n=0, {a′n}

∞n=0, {b′n}

∞n=0 and {c′n}

∞n=0 are real sequences in [0, 1]

such that an + bn + cn = a′n + b′n + c′n = 1 for all n ≥ 0 is called the Ishikawaiteration sequence with errors.

(iv) If, with the same notations and definitions as in (iii), b′n = c′n = 0 forall n ≥ 0, then the sequence {xn}

∞n=0 now defined by

x0 ∈ K, xn+1 = anxn + bnTxn + cnun, ∀n ≥ 0,

is called the Mann iteration sequence with errors.

It is clear that the Ishikawa and Mann iterative processes are all special casesof the Ishikawa and Mann iterative processes with errors, respectively. Althoughthe Mann iteration scheme can be obtained from the Ishikawa iteration schemeby setting bn = 0, ∀n ≥ 0, Rhoades [17] proved that the two iteration methods

CONVERGENCE AND STABILITY OF THE ISHIKAWA... 13

can exhibit different behaviors for different classes of nonlinear mappings.

In 1974, Ciric [1] proved first both the existence of fixed point and con-vergence of Picards iteration for quasi-contractive mappings in complete metricspaces. Rhoades [18] showed that the quasi-contractive definition (1.1) is one ofthe most general contractive-type definitions for which Picards iteration yields aunique fixed point. In [17], Rhoades noted that the Mann iteration process canbe used to approximate fixed points of quasi-contractive mappings in Hilbertspaces. Naimpally and Singh [13] and Rhoades [17] posed the following.

Question 1.1. Can the Mann iteration procedure be replaced by that ofIshikawa for quasi-contraction mapping of K into itself, where K is a compactconvex subset of a Hilbert space?

This question was resolved in the affirmative by Ding [6], Chidume [2]-[4],Chidume and Osilike [5], Liu [14], [15], Xu [23] and Zhao [26] in Hilbert spaces,Lp (or lp) spaces and Banach spaces, respectively.

A few stability results for certain classes of nonlinear mappings have beenobtained by several researchers (see, e.g, [7]-[9], [15], [19]-[21]). As was shownby Harder [7] and Harder and Hicks [9], the study on the stability is bothof theoretical and of numerical interest. Recently, Osilike [16] established thestability of certain Mann iteration procedure of quasi-contractive mappings inp-uniformly convex Banach spaces and put forth the following question:

Question 1.2. Is the Ishikawa iteration procedure T -stable for quasi-contractive maps in p-uniformly convex Banach spaces?

The purpose of this paper is to study Questions 1.1 and 1.2. In Section3, we prove certain Ishikawa iteration schemes with errors converge stronglyto fixed points for quasi-contractive mappings in p-uniformly convex Banachspaces. In Section 4, we show that the Ishikawa iteration scheme is T -stable forquasi-contractive mappings in p-uniformly convex Banach spaces. Our resultsnot only give affirmative answers to the above open questions put forth byRhoades [17] and Naimpally and Singh [13] and Osilike [16] in the more generalsetting, but also extend, improve and unify the corresponding results obtainedby Chidume [4], Liu [14], [15] Osilike [16] and others.

14 Z. Liu, J.S. Ume

2. Preliminaries

Let X be a Banach space and p > 1. The modulus of convexity of X is thefunction

δX : [0, 2] → [0, 1]

defined by

δX(ε) = inf{

1 − ‖1

2(x + y)‖ : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ε

}

.

X is called uniformly convex if δX(ε) > 0 for all ε ∈ (0, 2]. X is said to bep-uniformly convex if there exists a constant c > 0 satisfying δX(ε) ≥ cεp forall ε ∈ (0, 2]. It is well known that all Hilbert spaces, Lp (or lp ) spaces and theSobolev spaces, W

pm, 1 < p ≤ 2, are 2-uniformly convex, while for p ≥ 2, Lp (or

lp ) and Wpm spaces are p-uniformly convex.

Lemma 2.1. (see [24]) Let X be a p-uniformly convex Banach space withp > 1. Then there exists a constant λp > 0 such that

‖tx + (1 − t)y‖p ≤ t‖x‖p + (1 − t)‖y‖p − ωp(t)λp‖x − y‖p, (2.1)

for all x, y ∈ X and t ∈ (0, 1), where ωp(t) = t(1 − t)p + (1 − t)tp.

Lemma 2.2. (see [15]) Suppose that {αn}∞n=0 and {βn}

∞n=0 are nonnegative

sequences satisfying the following inequality:

αn+1 ≤ γαn + βn, ∀n ≥ 0,

where γ ∈ [0, 1) and limn→∞ βn = 0. Then limn→∞ αn = 0.

Lemma 2.3. (see [11]) Let {αn}∞n=0, {βn}

∞n=0 and {γn}

∞n=0 be three non-

negative real sequences satisfying the inequality

αn+1 ≤ (1 − ωn)αn + βn + γn

for all n ≥ 0, where {ωn}∞n=0 ⊂ [0, 1],

∑∞n=0

ωn = ∞, βn = o(ωn) and∑∞

n=0γn < ∞. Then limn→∞ αn = 0.

Lemma 2.4. Suppose that x, y are in [0,+∞) and p > 1. Then

(x + y)p ≤ xp + yp(x + y)p−1. (2.2)

Proof. Set f(x) = xp, ∀x ≥ 0. It follows from Mean-Value Theorem thatthere exists ξ ∈ [x, x + y] satisfying

(x + y)p − xp = f ′(ξ)y = pξp−1y ≤ yp(x + y)p−1,

which means that (2.2) holds. This completes the proof.

For the remainder of this paper, r and λp denote the constants appearingin (1.1) and (2.1), respectively.

CONVERGENCE AND STABILITY OF THE ISHIKAWA... 15

3. Convergence of the Ishikawa Iteration Sequences with Errors for

Quasi-Contractive Mappings

Theorem 3.1. Let K be a nonempty closed, bounded and convex subsetof a p-uniformly convex Banach space X with p > 1 and T : K → K be a quasi-contractive mapping. Suppose that {un}

∞n=0, {vn}

∞n=0 are arbitrary bounded

sequences in K and {an}∞n=0, {bn}

∞n=0, {cn}

∞n=0, {a

′n}

∞n=0, {b

′n}

∞n=0 and {c′n}

∞n=0

are any sequences in [0, 1] satisfying

an + bn + cn = a′n + b′n + c′n = 1, ∀n ≥ 0; (3.1)

limn→∞

bn = limn→∞

b′n = limn→∞

c′n = 0; (3.2)

∞∑

n=0

cn < ∞,

∞∑

n=0

bn = ∞. (3.3)

rp(1 − b′n − c′n) ≤ [(1 − bn − cn)(bn + cn)p−1

+ (1 − bn − cn)p]λp, ∀n ≥ 0. (3.4)

Then for any x0 ∈ K, the sequence {xn}∞n=0 defined iteratively by

yn = a′nxn + b′nTxn + c′nvn,

xn+1 = anxn + bnTyn + cnun, ∀n ≥ 0,(3.5)

converges strongly to a unique fixed point of T.

Proof. It follows from Theorem 1 of Ciric [1] or Theorem 11 of Rhoades[18] that T has a unique fixed point q ∈ K. Put dn = bn + cn and d′n = b′n + c′nfor all n ≥ 0. Since T is quasi-contractive, it follows that for all x ∈ K,

‖Tx − q‖ ≤ r max{‖x − q‖, ‖x − Tx‖, ‖q − Tq‖, ‖x − Tq‖, ‖q − Tx‖}

= r max{‖x − q‖, ‖x − Tx‖}, (3.6)

which implies that

‖Tx − q‖p ≤ rp[‖x − q‖p + ‖x − Tx‖p] (3.7)

for any x ∈ K. It follows from the boundedness of K, Lemmas 2.1 and 2.4 and(3.5) that there exists a constant N satisfying

p(diamK)p < N,

‖yn − Tyn‖p = ‖(1 − d′n)(xn − Tyn) + d′n(Txn − Tyn) + c′n(vn − Txn)‖p

≤(

‖(1 − d′n)(xn − Tyn) + d′n(Txn − Tyn)‖ + c′n‖vn − Txn‖)p

16 Z. Liu, J.S. Ume

≤ ‖(1 − d′n)(xn − Tyn) + d′n(Txn − Tyn)‖p + Nc′n

≤ (1 − d′n)‖xn − Tyn‖p + d′n‖Txn − Tyn‖

p

− ωp(1 − d′n)λp‖xn − Txn‖p + Nc′n, (3.8)

‖yn − q‖p = ‖(1 − d′n)(xn − q) + d′n(Txn − q) + c′n(vn − Txn)‖p

≤ (1 − d′n)‖xn − q‖p + d′n‖Txn − q‖p

− ωp(1 − d′n)λp‖xn − Txn‖p + Nc′n, (3.9)

‖xn+1 − q‖p = ‖(1 − dn)(xn − q) + dn(Tyn − q) + cn(un − Tyn)‖p

≤ (1 − dn)‖xn − q‖p + dn‖Tyn − q‖p

− ωp(1 − dn)λp‖xn − Tyn‖p + Ncn, (3.10)

for all n ≥ 0. Using (3.7)-(3.9), we have

‖Tyn − q‖p ≤ rp(‖yn − q‖p + ‖yn − Tyn‖p)

≤ rp(1 − d′n)‖xn − q‖p + rpd′n‖Txn − q‖p + rp(1 − d′n)‖xn − Tyn‖p

+ rpd′n‖Txn − Tyn‖p + 2Nrpc′n − 2rpωp(1 − d′n)λp‖xn − Txn‖

p, (3.11)

for all n ≥ 0. From (3.1) (3.4), (3.10) and (3.11), we infer that

‖xn+1 − q‖p ≤(

1 − dn + dnrp(1 − d′n))

‖xn − q‖p + rpdnd′n‖Txn − q‖p

+ rpdnd′n‖Txn − Tyn‖p + (2rpdnc′n + cn)N

+(

rp(1− d′n)dn −ωp(1− dn)λp

)

‖xn −Tyn‖p − 2dnrpωp(1− d′n)λp‖xn − Txn‖

p

≤(

1 − (1 − rp(1 − d′n))dn

)

‖xn − q‖p + 2rpNdn(d′n + c′n) + Ncn

≤(

1 − (1 − rp)dn

)

‖xn − q‖p + 4rpNdnd′n + Ncn, (3.12)

for all n ≥ 0. Set αn = ‖xn−q‖p, βn = 4rpNdnd′n, γn = Ncn and ωn = (1−rp)dn

for all n ≥ 0. Lemma 2.3 and (3.2) and (3.3) yield that αn → 0 as n → ∞.That is, xn → q as n → ∞. This completes the proof.

Theorem 3.2. Let K,X, T, p, {un}∞n=0, {vn}

∞n=0, {xn}

∞n=0, {yn}

∞n=0 be as

in Theorem 3.1. Suppose that {an}∞n=0, {bn}

∞n=0, {cn}

∞n=0, {a

′n}

∞n=0, {b

′n}

∞n=0

and {c′n}∞n=0 are any sequences in [0, 1] satisfying conditions (3.1), (3.2), (3.4)

and

bn + cn 6= 0, ∀n ≥ 0; (3.13)

limn→∞

cn

bn + cn

= 0; (3.14)

CONVERGENCE AND STABILITY OF THE ISHIKAWA... 17

∞∑

n=0

bn = ∞. (3.15)

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

Proof. As in the proof of Theorem 3.1, we conclude that (3.12) holds. Set

αn = ‖xn − q‖p, βn = dn

(

4Nrpd′n + Ncn

dn

)

,

γn = 0, ωn = (1 − rp)dn, ∀n ≥ 0.

Then (3.12) can be rewritten as

αn+1 ≤ (1 − ωn)αn + βn + γn, ∀n ≥ 0.

It is easily seen that (3.1), (3.2), (3.13)-(3.15) imply that∞∑

n=0

ωn = ∞, ωn ∈ (0, 1], βn = o(ωn), ∀n ≥ 0.

It follows from from Lemma 2.3 that αn → 0 as n → ∞. Hence xn → q asn → ∞. This completes the proof.

Theorem 3.3. Let K,X, T, p, {un}∞n=0, {vn}

∞n=0, {xn}

∞n=0, {yn}

∞n=0 be as

in Theorem 3.1. Suppose that {an}∞n=0, {bn}

∞n=0, {cn}

∞n=0, {a′n}

∞n=0, {b

′n}

∞n=0

and {c′n}∞n=0 are any sequences in [0, 1] satisfying conditions (3.1), (3.4), and

limn→∞

cn = limn→∞

b′n = limn→∞

c′n = 0; (3.16)

bn + cn ≥1 − s

1 − rp, ∀n ≥ 0, (3.17)

where s is a constant in (0, 1). Then {xn}∞n=0 converges strongly to a unique

fixed point of T.

Proof. It follows from (3.12) and (3.17) that

‖xn+1 − q‖p ≤ [1 − (1 − rp)dn]‖xn − q‖p + 4rpNdnd′n + Ncn

≤ s‖xn − q‖p + 4rpNdnd′n + Ncn, (3.18)

for all n ≥ 0. In view of (3.16), (3.18) and Lemma 2.2, we conclude immediatelythat xn → q as n → ∞. This completes the proof.

Remark 3.1. Theorems 3.1-3.3 give affirmative answers to Question 1.1of Rhoades [17] and Naimpally and Singh [13] in the more general iterationschemes.

Remark 3.2. Theorems 3.1 and 3.3 extend, improve and unify Theorems1 and 2 of Chidume [4], Theorem 1 of Liu [l4] and the theorem of Liu [15] inthe following ways:

(1) The Mann iteration scheme in [4] and the Ishikawa iteration scheme in

18 Z. Liu, J.S. Ume

[4], [14], [15] are replaced by the Ishikawa iteration scheme with errors;

(2) The Lp (or lp )spaces, 1 < p ≤ 2, in [4], Hilbert spaces in [14], [15] arereplaced by p-uniformly convex Banach spaces, 1 < p;

(3) The continuity and compactness assumptions on T and K, respectively,in [4], [14], are not needed;

(4) The condition αn ≤ βn, ∀n ≥ 0, in [4], [14] are omitted.

Remark 3.3. The following examples demonstrate that Theorems 3.1and 3.2 are independent.

Example 3.1. Let

an = 1 − (4 + n)−1

2 − (2 + n)−1, bn = (4 + n)−1

2 , cn = (2 + n)−1, ∀n ≥ 0,

a′n = 1 − 2(2 + n)−1, b′n = c′n = (2 + n)−1, ∀n ≥ 0.

Then conditions (3.1), (3.2), (3.13)-(3.15) are fulfilled. But∞

∑

n=0

cn = ∞.

That is, condition (3.3) is not satisfied.

Example 3.2. Let {a′n}∞n=0, {b

′n}

∞n=0, {c

′n}

∞n=0 be as in Example 3.1. Set

a2n−1 = 1, b2n−1 = c2n−1 = 0, ∀n ≥ 1,

a2n = 1 −(

(2n + 2)−1 − (2n2 + 2)−1)

, b2n = (2n + 2)−1,

c2n = (2n2 + 2)−1, ∀n ≥ 0.

Then conditions (3.1)-(3.3) hold. However, conditions (3.13) and (3.14) are notsatisfied.

4. Stability of Ishikawa Iteration Procedure for Quasi-Contractive

Mappings

Theorem 4.1. Let X be a p-uniformly convex Banach space with p > 1and T : X → X be a quasi-contractive mapping. Suppose that {an}

∞n=0 and

{bn}∞n=0 are real sequences in [0, 1] and s is constant in (0, 1) satisfying

rp ≤ min{(1 − an)ap−1n + (1 − an)p,

(1 − bn)bp−1n + (1 − bn)p}λp, ∀n ≥ 0, (4.1)

1 − an(1 − rp) ≤ s, ∀n ≥ 0. (4.2)

CONVERGENCE AND STABILITY OF THE ISHIKAWA... 19

For any x0 ∈ X, define the sequence {xn}∞n=0 by

zn = (1 − bn)xn + bnTxn, ∀n ≥ 0,

xn+1 = (1 − an)xn + anTzn, ∀n ≥ 0.(4.3)

Let {yn}∞n=0 be any sequence in X and define {εn}

∞n=0 by

wn = (1 − bn)yn + bnTyn, ∀n ≥ 0,

εn = ‖yn+1 − (1 − an)yn − anTwn‖, ∀n ≥ 0.(4.4)

Then:

(i) The sequence {xn}∞n=0 converges strongly to the unique fixed point q of

T ;

(ii) ‖yn+1 − q‖ ≤ εn + s1

p ‖yn − q‖, ∀n ≥ 0;

(iii) limn→∞ yn = q if and only if limn→∞ εn = 0.

Proof. The existence of a unique fixed point for T follows from Ciric [1] orRhoades [18]. Let q denote the fixed point of T . Observe that (4.2) impliesthat an ≥ 1−s

1−rp > 0 for all n ≥ 0. Hence∑∞

n=0an = ∞. It follows from the

Theorem of Xu [23] that (i) holds. Note that (4.4) means that

‖yn+1 − q‖ ≤ εn + ‖(1 − an)(yn − q) + an(Twn − q)‖ (4.5)

for all n ≥ 0. In view of Lemma 2.1, we have

‖(1 − an)(yn − q) + an(Twn − q)‖p

≤ (1 − an)‖yn − q‖p + an‖Twn − q‖p − ωp(1 − an)λp‖yn − Twn‖p, (4.6)

for all n ≥ 0. Using (1.1), we get that

‖Tx − q‖ ≤ r max{‖x − q‖, ‖x − Tx‖} (4.7)

for all x ∈ X. In particular, we also have

‖Twn − q‖ ≤ r max{‖wn − q‖, ‖wn − Twn‖} (4.8)

for all n ≥ 0.

Suppose that ‖Twn − q‖ ≤ r‖wn − q‖ for some n ≥ 0. It follows fromLemma 2.1 that

‖wn − q‖p = ‖(1 − bn)(yn − q) + bn(Tyn − q)‖p

≤ (1 − bn)‖yn − q‖p + bn‖Tyn − q‖p − ωp(1 − bn)λp‖yn − Tyn‖p. (4.9)

In virtue of (4.7), we have

‖Tyn − q‖ ≤ r max{‖yn − q‖, ‖yn − Tyn‖}. (4.10)

Hence (4.10) means that either ‖Tyn − q‖ ≤ r‖yn − q‖ or ‖Tyn − q‖ ≤ r‖yn −

20 Z. Liu, J.S. Ume

Tyn‖. If ‖Tyn − q‖ ≤ r‖yn − q‖, we conclude that by (4.9)

‖wn − q‖p ≤ (1 − bn + bnrp)‖yn − q‖p − ωp(1 − bn)λp‖yn − Tyn‖p. (4.11)

Thus (4.11) and (4.6) yield that

‖(1 − an)(yn − q) + an(Twn − q)‖p

≤(

1 − an + anrp(1 − bn + bnrp))

‖yn − q‖p − anrpωp(1 − bn)λp‖yn − Tyn‖p

− ωp(1 − an)λp‖yn − Twn‖p ≤

(

1 − an + anrp(1 − bn + bnrp))

‖yn − q‖p

≤(

1 − an(1 − rp))

‖yn − q‖p. (4.12)

If ‖Tyn − q‖ ≤ r‖yn − Tyn‖, we have from (4.9), (4.6) and (4.1)

‖(1 − an)(yn − q) + an(Twn − q)‖p ≤(

1 − an + anrp(1 − bn))

‖yn − q‖p

+ anrp(

bnrp − ωp(1 − bn)λp

)

‖yn − Tyn‖p − ωp(1 − an)λp‖yn − Twn‖

p

≤(

1 − an + anrp(1 − bn))

‖yn − q‖p ≤(

1 − an(1 − rp))

‖yn − q‖p. (4.13)

Suppose that ‖Twn − q‖ ≤ r‖wn − Twn‖ for some n ≥ 0. Then (4.6) reduce to

‖(1 − an)(yn − q) + an(Twn − q)‖p

≤ (1 − an)‖yn − q‖p + anrp‖wn − Twn‖p − ωp(1 − an)λp‖yn − Twn‖

p. (4.14)

Lemma 2.1 ensures that

‖wn−Twn‖p = ‖(1−bn)(yn−Twn)+bn(Tyn−Twn)‖p ≤ (1−bn)‖yn−Twn‖

p

+ bn‖Tyn − Twn‖p − ωp(1 − bn)λp‖yn − Tyn‖

p. (4.15)

Observe that

‖Tyn − Twn‖ ≤ r max{‖yn − Tyn‖, ‖wn − Twn‖, ‖yn − Twn‖},

which implies that either ‖Tyn − Twn‖ ≤ r‖yn − Tyn‖, or ‖Tyn − Twn‖ ≤r‖wn −Twn‖, or ‖Tyn−Twn‖ ≤ r‖yn−Twn‖. If ‖Tyn−Twn‖ ≤ r‖yn−Tyn‖,by (4.14), (4.15) and (4.1) we infer that

‖(1 − an)(yn − q) + an(Twn − q)‖p ≤ (1 − an)‖yn − q‖p

+(

anrp(1 − bn) − ωp(1 − an)λp

)

‖yn − Twn‖p

+ anrp(

bnrp − ωp(1 − bn)λp

)

‖yn − Tyn‖p ≤ (1 − an)‖yn − q‖p

+(

anrp − ωp(1 − an)λp

)

‖yn − Twn‖p ≤ (1 − an)‖yn − q‖p. (4.16)

If ‖Tyn − Twn‖ ≤ r‖yn − Twn‖, then (4.15) ensures that

CONVERGENCE AND STABILITY OF THE ISHIKAWA... 21

‖wn − Twn‖p ≤

1 − bn

1 − rpbn

‖yn − Twn‖p

− ωp(1 − bn)λp

1 − rpbn

‖yn − Tyn‖p. (4.17)

Substituting (4.17) into (4.14), by (4.1) we obtain that

‖(1 − an)(yn − q) + an(Twn − q)‖p ≤ (1 − an)‖yn − q‖p+(

anrp 1 − bn

1 − rpbn

− ωp(1 − an)λp

)

‖yn−Twn‖p−anrpωp(1−bn)

λp

1 − rpbn

‖yn−Tyn‖p

≤ (1 − an)‖yn − q‖p +(

anrp − ωp(1 − an)λp

)

‖yn − Twn‖p

≤ (1 − an)‖yn − q‖p. (4.18)

If ‖Tyn−Twn‖ ≤ r‖yn−Twn‖, we conclude immediately from (4.14) and (4.15)that

‖(1 − an)(yn − q) + an(Twn − q)‖p ≤ (1 − an)‖yn − q‖p

+(

anrp(1 − bn + bnrp) − ωp(1 − an)λp

)

‖yn − Twn‖p

− anrpωp(1 − bn)λp‖yn − Tyn‖p ≤ (1 − an)‖yn − q‖p

+(

anrp − ωp(1 − an)λp

)

‖yn − Twn‖p ≤ (1 − an)‖yn − q‖p. (4.19)

It follows from (4.2), (4.8), (4.12), (4.13), (4.16), (4.18) and (4.19) that

‖(1 − an)(yn − q) + an(Twn − q)‖ ≤(

1 − an(1 − rp))

1

p

‖yn − q‖

≤ s1

p ‖yn − q‖ (4.20)

for all n ≥ 0. Therefore, (ii) follows from (4.5) and (4.20).

Suppose that limn→∞ yn = q. Using (ii), we have

εn = ‖yn+1 − (1 − an)yn − anTwn‖

≤ ‖yn+1 − q‖ + ‖(1 − an)(yn − q) + an(Twn − q)‖

≤ ‖yn+1 − q‖ + s1

p ‖yn − q‖ → 0

as n → ∞. That is, limn→∞ εn = 0.

Suppose that limn→∞ εn = 0. It follows from (ii) and Lemma 2.2 thatlimn→∞ yn = q. Hence (iii) holds. This completes the proof.

Remark 4.1. Theorem 4.1 reveals that the Ishikawa iteration scheme isT-stable for quasi-contractive mappings in p-uniformly convex Banach spaces.Thus Theorem 4.1 resolves in the affirmative Question 1.2 raised by Osilike [16].

22 Z. Liu, J.S. Ume

As a consequence of Theorem 4.1, we have the following

Corollary 4.1. Let X be a p-uniformly convex Banach space with p > 1and T : X → X be a quasi-contractive mapping. Suppose that {an}

∞n=0 is a

real sequence in [0, 1] and s is constant in (0, 1) satisfying (4.2) and

rp ≤ [(1 − an)ap−1n + (1 − an)p]λp, ∀n ≥ 0. (4.21)

For any x0 ∈ X, define the sequence {xn}∞n=0 by

xn+1 = (1 − an)xn + anTxn, ∀n ≥ 0. (4.22)

Let {yn}∞n=0 be any sequence in X and define {εn}

∞n=0 by

εn = ‖yn+1 − (1 − an)yn − anTyn‖, ∀n ≥ 0. (4.23)

Then:

(i) The sequence {xn}∞n=0 converges strongly to the unique fixed point q of

T ;

(ii) ‖yn+1 − q‖ ≤ εn + s1

p ‖yn − q‖ ≥ 0, ∀n ≥ 0;

(iii) limn→∞ yn = q if and only if limn→∞ εn = 0.

Remark 4.2. Corollary 4.1 generalizes, improves and unifies Theorems 3and 4 of Osilike [16].

Question 4.1. Are the Mann iteration sequence with errors and theIshikawa iteration sequence with errors stable for quasi-contractive mappingsin p-uniformly convex Banach spaces with p > 1?

Question 4.2. Can be Theorem 3.1, Theorem 3.2, Theorem 3.3 and The-orem 4.1 extended to uniformly convex Banach spaces?

Acknowledgments

This research is financially supported by Changwon National University in 2009-2010.

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