STRONG CONVERGENCE IN NOOR-TYPE ITERATIVE SCHEMES IN CONVEX CONE METRIC SPACES

Abstract

The author considers a Noor-type iterative scheme to approximate com- mon fixed points of an infinite family of uniformly quasi-sup(f_n)-Lipschitzian map- pings and an infinite family of g_n-expansive mappings in convex cone metric spaces. His results generalize, improve and unify some corresponding results in convex met- ric spaces [1, 3, 9, 16, 18, 19] and convex cone metric spaces [8].

1. INTRODUCTION

Recently, a class of three-step approximation schemes, which includes Mann and Ishikawa iterative schemes for solving general variational inequalities and related problems in Hilbert spaces, was considered by Noor [11]. And then, three-step methods (named as Noor methods by some authors) for solving various classes of variational inequalities and related problems were extensively studied by the same author in [12]. Since then Noor iteration schemes have been applied to study strong and weak convergences of nonexpansive mappings [2, 10, 13, 14, 16, 20, 21]. In 2002, Xu and Noor [20] considered a three-step iterative scheme with fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces. In 2007, Noor and Huang [13] analyzed three-step iteration methods for finding the common element of the set of fixed points of nonexpansive mappings and studied the convergence criteria for three-step iterative methods. In 2007, Nammanee and Suanti [8] considered the weak and strong convergences for asymptotically nonexpansive mappings for the modified Noor iteration schemes with errors in a uniformly convex Banach spaces. In 2008, Khan et al. [7] generalized the Noor-type iterative process considered in [20] to the case of a finite family of mappings.

On the other hand, there have been many researches [1, 3, 5, 6, 9, 16-19] on iterative schemes for various kinds of nonexpansive mappings in convex metric spaces with convex structure [15] in the usual metric spaces. In 2010, Khan and Ahmed [6] introduced a generalized iterative scheme due to Khan et al. [7] in convex metric spaces and established a strong convergence to a unique common fixed point of a finite family of asymptotically quasi-nonexpansive mappings under the scheme. Very recently, Tian and Yang [17] gave some suffcient and necessary conditions for a new Noor-type iteration with errors to approximate a common fixed point for a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces.

A cone metric [4] in Banach spaces, which is a cone-version of the usual metric in ℝ is very applicable in applied mathematics including nonlinear analysis by joining it with convex structures. Very recently, Lee [8] extended an Ishikawa type itera- tive scheme with errors to approximate a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings on convex cone metric spaces.

Inspired by the works mentioned above, the author recalls some generalized nonexpansive mappings on cone metric spaces and gives some sufficient and necessary conditions for a Noor-type iteration to approximate a common fixed point of an infi- nite family of uniformly quasi-sup(fn)-Lipschitzian mappings and an infinite family of gn-expansive mappings in convex cone metric spaces. His results generalize and improve many corresponding results in convex metric spaces [1, 3, 9, 16, 18, 19] and convex cone metric spaces [8].

2. PRELIMINARIES

Throughout this paper, E is a normed vector space with a normal solid cone P.

A nonempty subset P of E is called a cone if P is closed, P ≠ {θ}, for a, b ∈ and x, y ∈ P, ax + by ∈ P and P ∩ {−P} = {θ}. We define a partial ordering in E as x y if y − x ∈ P. x << y indicates that y − x ∈ intP and x ≺ y means that x y but x ≠ y. A cone P is said to be solid if its interior intP is nonempty. A cone P is said to be normal if there exists a positive number t such that for x, y ∈ P, 0 x y implies ∥x∥ ≤ t∥y∥. The least positive number t is called the normal constant of P.

Let X be a nonempty set. A mapping d : X × X → (E, P) is called a cone metric if (i) for x, y ∈ X, θ d(x, y) and d(x, y) = θ iff x = y, (ii) for x, y ∈ X, d(x, y) = d(y, x) and (iii) for x, y, z ∈ X, d(x, y) d(x, z) + d(z, y). A nonempty set X with a cone metric d : X × X → (E, P) is called a cone metric space denoted by (X, d), where P is a solid normal cone.

A sequence {xn} in a cone metric space (X, d) is said to converge to x ∈ (X, d) and denoted as or xn → x (as n → ∞) if for any c ∈ intP, there exists a natural number N such that for all n > N, c − d(xn, x) ∈ intP. A sequence {xn} in (X, d) is called a Cauchy sequence if for any c ∈ intP, there exists a natural number N such that for all n, m > N, c − d(xn, xm) ∈ intP. A cone metric space (X, d) is said to be complete if every Cauchy sequence converges.

Lemma 2.1 ([4]). Let {xn} be a sequence in a cone metric space (X, d) and P be a normal cone with a normal constant t. Then

(i) {xn} converges to x in X if and only if d(xn, x) → θ (as n → ∞) in E. (ii) {xn} is a Cauchy sequence if and only if d(xn, xm) → θ (as n, m → ∞) in E.

We recall some generalized nonexpansive mappings and convex structures on cone metric spaces.

Definition 2.1. Let T be a self-mapping on a cone metric space (X, d) and f : X → (0, ∞) a function which is bounded above.

(i) T is f-expansive, if

In particular, T is said to be nonexpansive, if and

T is said to be contractive, if

(ii) T is asymptotically f-expansive, if there exists a sequence in X such that satisfying

d(Tnx, T ny) f(xn)d(x, y) for x, y ∈ X,

in particular, T is asymptotically nonexpansive, if there exists a sequence in [1, ∞) with satisfying

d(Tnx, T ny) knd(x, y) for x, y ∈ X .

(iii) T is asymptotically quasi-f-expansive, if there exists a sequence in X such that satisfying

d(Tnx, p) f(xn)d(x, p) for x ∈ X and p ∈ F(T) a set of fixed points of T, in particular, T is asymptotically quasi-nonexpansive, if there exists a sequence in [1, ∞) with satisfying

d(Tnx, p) knd(x, p) for x ∈ X and p ∈ F(T)

(iv) T is uniformly quasi-sup(f)-Lipschitzian, if

for x ∈ X and p ∈ F(T),

in particular, T is uniformly quasi-L-Lipschitzian, if there exists a constant L > 0 such that

d(Tnx, p) L · d(x, p) for x ∈ X and p ∈ F(T) .

Definition 2.2. Let (X, d) be a cone metric space. A mapping W : X3 ×I 3 → X is a convex structure on X if d(W(x, y, z, an, bn, cn), u) an · d(x, u) + bn · d(y, u) + cn · d(z, u) for real sequences {an}, {bn} and {cn} in I = [0, 1] satisfying an +bn +cn = 1 and x, y, z and u ∈ X. A cone metric space (X, d) with a convex structure W is called a convex cone metric space and denoted as (X, d, W). A nonempty subset C of a convex cone metric space (X, d, W) is said to be convex if W(x, y, z, a, b, c) ∈ C for all x, y, z ∈ C and a, b, c ∈ I.

In this paper, we consider the following Noor-type three-step iteration in convex cone metric spaces ;

For given x0 ∈ C, define a sequence {xn} in C as follows;

where {un}, {vn} and {wn} are any sequences in X, C is a nonempty convex subset of (X, d, W), Tn : C → C is a uniformly quasi-sup(fn)-Lipschitzian mapping, Sn : C → C is a gn-expansive mapping and {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en} and {ln} are sequences in I such that an + bn + cn = 1, αn + βn+ ≥n= 1 and dn + en + ln = 1 .

Remark 2.1. (i) We have the following iteration by putting Snxn = xn in of (2.1)

(ii) By putting Snxn = xn, we obtain the following iteration from (2.2) ;

which generalizes many kinds of Ishikawa-type iterations.

(iii) The following Ishikawa-type iteration is a special case of (2.1);

which was considered in [1].

 

3. Main Results

Throughout this paper, fn, gn : X → (0, ∞) are functions, which are bounded above, and whose least upper bounds are Mn and Nn, respectively .

Lemma 3.1 ([5, 9]). Let {an}, {bn} and {cn} be nonnegative real sequences satisfying the following conditions;

(i) an+1 ≤ (1 + bn)an + cn , (ii) and are finite, then the following hold; (1) exists. (2) If, then

The following lemma considered in convex cone metric space (X, d, W) is a result on the properties of an iteration sequence {xn} defined as (2.1) for an infinite family {Tn} of uniformly quasi-sup(fn)-Lipschitzian mappings and an infinite family {Sn} of gn-expansive mappings.

Lemma 3.2. Let d : X × X → (E, P) be a cone metric, where P is a solid normal cone with the normal constant t. Let C be a nonempty convex subset of a convex cone metric space (X, d, W). Let Tn : C → C be uniformly quasi-sup(fn)-Lipschitzian mappings and Sn : C → C be gn-expansive mappings with a nonempty set

Let {un}, {vn} and {wn} be bounded sequences in C and {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en} and {ln} be sequences in I such that αn + βn + γn = an + bn + cn = dn + en + ln = 1 .

Assume that the following conditions hold;

(i) fn > 1 and gn > 1 , (ii) and are finite, (iii) L = (1 + M + M2 ) · L0, where

is finite.

Let {xn} be the iteration defined as (2.1), then the following holds.

for p ∈ D, where ηn = αn + βn and δn = βn + γn .

(2) there exists a constant K > 0 such that

Proof. For any p ∈ D, we have

and

From (3.1) to (3.3), by the fact that 1 + x ≤ ex for x ≥ 0, we have

Moreover, we obtain the following result (2) from the result (1).

where

Remark 3.1. If fn ≥ 1 and , then

Hence from (3.4)

and from (3.5)

where

Remark 3.2. If fn ≤ 1 and , then

Hence from (3.4)

and from (3.5)

where

Remark 3.3. If fn ≤ 1 and gn ≤ 1, then from (3.4) and (3.5), respectively,

and

Remark 3.4. Lemma 2 in [1] dealt with the Ishikawa-type iteration (2.4) for an infi- nite family of uniformly quasi-Ln-Lipschitzian mappings Tn with < ∞ and an infinite family of nonexpansive mappings Sn, i.e., L ≥ 1 and gn = 1 . Hence Lemma 2 in [1] is a corollary of Lemma 3.1.

Now we introduce our main result for the iteration defined as (2.1) with an infinite family {Tn} of uniformly quasi-sup(fn)-Lipschitzian mappings and an infinite family {Sn} of gn-expansive mappings in convex cone metric spaces (X, d, W).

Theorem 3.1. Let C be a nonempty closed convex subset of a convex cone metric space (X, d, W) with a solid normal cone P with a normal constant t. Let Tn : C → C be uniformly quasi-sup(fn)-Lipschitzian mappings and Sn : C → C be gn-expansive mappings with a nonempty set Let {un}, {vn} and {wn} be bounded sequences in C and {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en} and {ln} be real sequences in I such that αn + βn + γn = an +bn + cn = dn + en + ln = 1 .

Assume that the following conditions hold;

(i) fn > 1 and gn > 1 , (ii) and are finite, (iii) L = (1 + M + M2 ) · L0, where

is finite,

(iv) and) are finite.

Let {xn} be the iteration defined as (2.1).

We have the following equivalent result;

(1) {xn} converges to a common fixed point p ∈ D. (2)

Proof. We only consider the case of fn(x) > 1 and gn(x) > 1 for x ∈ X . Obviously the statement (1) implies the statement (2). Now we show that the statement (2) implies the statement (1). From (3.4), we have

where ηn = αn + βn and δn = βn + γn for .

Thus the normality of P implies that

where t is the normal constant of P.

By the condition (iv) that and are finite, exists from Lemma 3.1. Since by the hypothesis, we have Now, we show that the sequence {xn} is convergent. For any given ε > 0, take a positive integer N0 such that

where for n ≥ N0.

By using a property of infimum for ∥d(xn, D)∥, we take a positive integer N1 ≥ N0 and p0 ∈ D such that

Hence for any positive integer n > N1, from (3.5) by the normality of P we have

Hence , which shows that , i.e., by Lemma 2.1(i). Since C is closed, .

Now, we show that the set D is closed. In fact, let {pn} a sequence in D converging to p in C. Then

Letting n → ∞ in (3.6), we have

p ∈ F(Ti) for all .

Similarly,

Letting also n → ∞ in (3.7), we have

p ∈ F(Si) for all .

Hence p ∈ D, which implies that D is closed.

Moreover, we have p∗ ∈ D. In fact, since by Lemma 2.1 (i), we have p∗ ∈ D, which says that {xn} converges to a common fixed point in D.

Remark 3.5. In Theorem 3.1, the full space need not to be complete.

Remark 3.6. We obtain the same results for iterations defined as (2.2) and (2.3) with an infinite family {Tn} of uniformly quasi-sup(fn)-Lipschitzian mappings and an infinite family {Sn} of gn-expansive mappings in convex metric spaces (X, d, W).

Remark 3.7. The main result of [1] considered the Ishikawa-type iteration (2.4) for an infinite family of uniformly quasi-Ln-Lipschitzian mappings Tn with L = and an infinite family of nonexpansive mappings Sn, i.e., L ≥ 1 and gn = 1 in convex metric spaces. Hence Theorem 1 in [1] is a corollary of Theorem 3.1.

Remark 3.8. By putting fn(x) = 1 and gn(x) = 1 for x ∈ X in Theorem 3.1, we have the corresponding results in convex metric spaces and convex cone metric spaces.

Remark 3.9. We obtain the same results as Theorem 3.1 by replacing the uniform quasi-sup(f)-Lipschitzian of Tn with the asymptotical quasi-f-expansiveness of Tn .

Remark 3.10. Theorem 3.1 generalizes, improves and unifies the corresponding results in convex metric spaces [1, 3, 9, 16, 18, 19].