Georgian Mathematical Journal Volume 9 (2002), Number 3, 591–600

NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

Abstract. In this paper, most of classical and modern convergence theorems of iterative schemes for nonexpansive mappings are presented and the main results in the paper generalize and improve the corresponding results given by many authors.

2000 Mathematics Subject Classification: Primary 47H17; secondary 47H05, 47H10. Key words and phrases: Nonexpansive mapping, Mann and Ishikawa iterative schemes, convergence theorem, uniformly convex Banach space.

1. Introduction and Preliminaries

In this paper, we will present several strong and weak convergence results of successive approximations to fixed points of nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper generalize and improve various ones concerned with constructive techniques for approximations of fixed points of nonexpansive mappings (cf. [1]–[18]).

Let X be a real normed linear space and C be a nonempty subset of X. A mapping T : C → C is said to be nonexpansive if

∥T x − T y∥ ≤ ∥x − y∥

for all x, y ∈ C. Furthermore, T is said to be quasi-nonexpansive if the set F (T) of fixed points of T is not empty, for all x ∈ D(T) and, for y ∈ F (T),

∥T x − y∥ ≤ ∥x − y∥.  (1.1)

Remark 1.1. Nonexpansive mappings with the nonempty fixed point set F (T) are quasi-nonexpansive, and linear quasi-nonxpansive mappings are nonexpansive, but it is easily seen that there exist nonlinear continuous quasi-nonexpansive mappings which are not nonexpansive, for example, consider a mapping T defined by T x = (\frac{x}{2}) sin(\frac{1}{x}) with T (0) = 0 on R.

There are two important connections between the classes of nonexpansive mappings and accretive mappings which give rise to a strong connection between the fixed point theory of nonexpansive mappings and the theory of accretive mappings.

Note that

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

(1) If T is a nonexpansive mapping of D(T) into X and if we set U = I − T D(U) = D(T), then U is an accretive mapping of D(U) into X.

(2) If {U(t) : t ≥ 0} is a semigroup of (nonlinear) mappings of X into itself with infinitesimal generator T, then all the mappings U(t) are nonexpansive if and only if (−T) is accretive.

Let X be a normed linear space, C be a nonempty convex subset of X and T : C → C be a given mapping. Then, for arbitrary x_{1} ∈ C, the Ishikawa iterative scheme {x_{n}} is defined by

\begin{cases}y_{n} = (1 − s_{n})x_{n} + s_{n}T x_{n}, n ≥ 1, \\ x_{n+1} = (1 − t_{n})x_{n} + t_{n}T y_{n}, n ≥ 1, \end{cases}  (IS)

where {s_{n}} and {t_{n}} are some suitable sequences in [0, 1]. With X, C, {t_{n}} and x_{1} as above, the Mann iterative scheme {x_{n}} is defined by

\begin{cases}x_{1} ∈ C, \\ x_{n+1} = (1 − t_{n})x_{n} + t_{n}T x_{n}, n ≥ 1. \end{cases}  (M)

We now begin with a serial of lemmas which will be needed in the sequel for the proof of our main theorems:

Lemma 1.1. Let X be a real normed linear space, C be a nonempty convex subset of X and T : C → C be a nonexpansive mapping. If {x_{n}} is the iterative scheme defined by (IS), then the following holds:

∥x_{n+1} − T x_{n+1}∥ ≤ (1 + 2τ_{n})∥x_{n} − T x_{n}∥

for all n ≥ 1, where τ_{n} = min{t_{n}, 1 − t_{n}}s_{n}. In particular, ∥x_{n+1} − T x_{n+1}∥ ≤ ∥x_{n} − T x_{n}∥ if s_{n} ≡ 0 and lim_{n→∞} ∥x_{n} − T x_{n}∥ exists.

Proof. By Tan and Xu [15, Lemma 3], we have the following estimation:

∥x_{n+1} − T x_{n+1}∥ ≤ [1 + 2s_{n}(1 − t_{n})]∥x_{n} − T x_{n}∥.  (1.2)

Observe that

∥x_{n} − T y_{n}∥ ≤ ∥x_{n} − T x_{n}∥ + ∥x_{n} − y_{n}∥ ≤ ∥x_{n} − T x_{n}∥ + s_{n}∥x_{n} − T x_{n}∥
= (1 + s_{n})∥x_{n} − T x_{n}∥.  (1.3)

It follows that

∥x_{n+1} − T x_{n+1}∥ ≤ (1 − t_{n})∥x_{n} − T x_{n+1}∥ + t_{n}∥T y_{n} − T x_{n+1}∥
≤ (1 − t_{n})(∥x_{n} − T x_{n}∥ + ∥x_{n} − x_{n+1}∥) + t_{n}∥y_{n} − x_{n+1}∥
≤ (1 − t_{n})∥x_{n} − T x_{n}∥ + ∥x_{n} − x_{n+1}∥ + t_{n}s_{n}∥x_{n} − T x_{n}∥
≤ (1 + 2t_{n}s_{n})∥x_{n} − T x_{n}∥.  (1.4)

Combining (1.2) with (1.4), we reach the desired conclusion. This completes the proof.

Lemma 1.2. Let {a_{n}} and {b_{n}} be two sequences of nonnegative numbers such that

a_{n+1} ≤ (1 + b_{n})a_{n}  (1.5)

for all n ≥ 1. If ∑_{n=1}^{∞} b_{n} converges, then lim_{n→∞} a_{n} exists. In particular, a_{n} → 0 as n → ∞ whenever {a_{n}} admits a subsequence which converges to 0.

Proof. For n, m ≥ 1, we have

a_{n+m+1} ≤ (1 + b_{n+m})a_{n+m} ≤ ⋯ ≤ ∏_{j=n}^{n+m}(1 + b_{j})a_{n},  (1.6)

which infers that {a_{n}} is bounded since ∑_{n} b_{n} converges. Set M = sup{a_{n} : n ≥ 1}. Then (1.5) reduces to

a_{n+1} ≤ a_{n} + Mb_{n}

for all n ≥ 1. Now the conclusion follows from Lemma 1 of Tan and Xu [15]. This completes the proof.

The modulus of convexity of a real Banach space X is defined by

δ_{X}(ϵ) = inf \{1 − ∥\frac{1}{2}(x + y)∥ : ∥x∥, ∥y∥ ≤ 1, ∥x − y∥ ≥ ϵ\}

for all ϵ ∈ [0, 2]. X is said to be uniformly convex if δ_{X}(0) = 0 and δ_{X}(ϵ) > 0 for all 0 < ε ≤ 2.

Lemma 1.3. Let X be a uniformly convex Banach space with the modulus of uniform convexity δ_{X}. Then δ_{X} : [0, 2] → [0, 1] is a continuous and increasing function with δ_{X}(0) = 0 and δ_{X}(t) > 0 for t > 0 and, further,

∥cu + (1 − c)v∥ ≤ 1 − 2 min{c, 1 − c}δ_{X}(∥u − v∥)  (1.7)

whenever 0 ≤ c ≤ 1 and ∥u∥, ∥v∥ ≤ 1.

Proof. See Bruck [3].

Let X be a real Banach space with a norm ∥ · ∥ and S = {x ∈ X : ∥x∥ = 1} be its unit sphere. The norm of X is said to be Fréchet differentiable if, for each x ∈ S, the limit

lim_{t→0} \frac{∥x + ty∥ − ∥x∥}{t}

exists and is attained uniformly in y ∈ S. For x ≠ 0 and y ∈ X, we denote this limit by (x, y). It is known that X has a Fréchet differentiable norm if and only if X is strongly smooth. In this case, the normalized duality mapping J : X → X^{∗} is single-valued and continuous from the strong topology of X to the strong topology of X^{∗}. As a matter of facts above mentioned, it is clear that

(x, c_{1}y_{1} + c_{2}y_{2}) = c_{1}(x, y_{1}) + c_{2}(x, y_{2})

for all c_{1}, c_{2} ∈ R, x ∈ X with x ≠ 0 and y_{1}, y_{2} ∈ X.

Lemma 1.4 ([11]). Let C be a nonempty closed convex subset of a uniformly convex Banach space X with a Frechet differentiable norm and {T_{n} : n ≥ 1} be a family of nonexpansive self-mappings of C with the nonempty common fixed point set F. If x_{1} ∈ C and define a sequence {x_{n}} in C by x_{n+1} = T_{n}x_{n} for all n ≥ 1, then lim_{n→∞}(f_{1} − f_{2}, x_{n}) exists for all f_{1}, f_{2} ∈ F with f_{1} ≠ f_{2}.

2. The Main Results

Now we prove the following convergence theorems:

Theorem 2.1. Let X be a real uniformly convex Banach space, C be a nonempty closed convex (not necessarily bounded) subset of X and T : C → C be a nonexpansive mapping. Let {x_{n}} be the iterative scheme defined by (IS) with the restrictions that ∑_{n=1}^{∞} t_{n}(1 − t_{n}) = ∞ and ∑_{n=1}^{∞} τ_{n} < ∞, where τ_{n} is as in Lemma 1.1. Then, for arbitrary initial value x_{1} ∈ C, {∥x_{n}−T x_{n}∥} converges to same constant r_{C}(T), which is independent of the choice of the initial value x_{1} ∈ C. In particular, if τ_{n} ≡ 0, then

r_{C}(T) = inf{∥x − T x∥ : x ∈ C}.

Proof. It follows from Lemmas 1.1 and 1.2 that lim_{n→∞} ∥x_{n} − T x_{n}∥ exists. We denote this limit by r(x_{1}). Let {x_{n}^{∗}} be another Ishikawa iterative scheme defined by (IS) with the same restrictions on parameters {t_{n}} and {s_{n}} as the sequence {x_{n}} but with the initial value x_{1}^{∗} ∈ C. Then r(x_{1}^{∗}) = lim_{n→∞} ∥x_{n}^{∗} − T x_{n}^{∗}∥.

It suffices to prove that r(x_{1}) = r(x_{1}^{∗}). Since ∥x_{n+1} − x_{n+1}^{∗}∥ ≤ ∥x_{n} − x_{n}^{∗}∥, we have lim_{n→∞} ∥x_{n} − x_{n}^{∗}∥ exists, which denoted this limit by d. Without loss of generality, we assume that d > 0. Observe that

x_{n+1} − x_{n+1}^{∗} = t_{n}(T y_{n} − T y_{n}^{∗}) + (1 − t_{n})(x_{n} − x_{n}^{∗})

and

∥T y_{n} − T y_{n}^{∗}∥ ≤ ∥x_{n} − x_{n}^{∗}∥.

By using Lemma 1.3, we have

∥x_{n+1} − x_{n+1}^{∗}∥ ≤ [1 − 2t_{n}(1 − t_{n})δ_{X}(\frac{∥x_{n} − x_{n}^{∗} − (T y_{n} − T y_{n}^{∗})∥}{∥x_{n} − x_{n}^{∗}∥})]∥x_{n} − x_{n}^{∗}∥,

which implies that

∑_{n=1}^{∞} t_{n}(1 − t_{n})δ_{X}(\frac{∥x_{n} − x_{n}^{∗} − (T y_{n} − T y_{n}^{∗})∥}{∥x_{n} − x_{n}^{∗}∥}) < ∞.

Since t_{n}(1 − t_{n})s_{n} ≤ τ_{n} and ∑_{n=1}^{∞} τ_{n} < ∞, we have ∑_{n=1}^{∞} t_{n}(1 − t_{n})s_{n} < ∞ and hence

∑_{n=1}^{∞} t_{n}(1 − t_{n})[δ_{X}(\frac{∥x_{n} − x_{n}^{∗} − (T y_{n} − T y_{n}^{∗})∥}{∥x_{n} − x_{n}^{∗}∥}) + s_{n}] < ∞.

Since ∑_{n=1}^{∞} t_{n}(1 − t_{n}) = ∞, we conclude

lim inf_{n→∞} [δ_{X}(\frac{∥x_{n} − x_{n}^{∗} − (T y_{n} − T y_{n}^{∗})∥}{∥x_{n} − x_{n}^{∗}∥}) + s_{n}] = 0.

Since δ_{X} is strictly increasing and continuous and lim_{n→∞} ∥x_{n} − x_{n}^{∗}∥ = d > 0, we have

lim_{k→∞} ∥x_{n_{k}} − x_{n_{k}}^{∗} − (T y_{n_{k}} − T y_{n_{k}}^{∗})∥ = 0

and lim_{k→∞} s_{n_{k}} = 0. On the other hand, we have

|∥x_{n_{k}} − T x_{n_{k}}∥ − ∥x_{n_{k}}^{∗} − T x_{n_{k}}^{∗}∥| ≤ ∥(x_{n_{k}} − T x_{n_{k}}) − (x_{n_{k}}^{∗} − T x_{n_{k}}^{∗})∥

≤ ∥x_{n_{k}} − x_{n_{k}}^{∗} − (T y_{n_{k}} − T y_{n_{k}}^{∗})∥ + ∥T x_{n_{k}} − T y_{n_{k}}∥ + ∥T x_{n_{k}}^{∗} − T y_{n_{k}}^{∗}∥

≤ ∥x_{n_{k}} − x_{n_{k}}^{∗} − (T y_{n_{k}} − T y_{n_{k}}^{∗})∥
+ s_{n_{k}}(∥x_{n_{k}} − T x_{n_{k}}∥ + ∥x_{n_{k}}^{∗} − T x_{n_{k}}^{∗}∥),  (2.1)

which converges to 0 as k → ∞. This leads to

lim_{n→∞} |∥x_{n} − T x_{n}∥ − ∥x_{n}^{∗} − T x_{n}^{∗}∥| = 0.

Therefore, we have r(x_{1}) = r(x_{1}^{∗}).

If τ_{n} ≡ 0, then ∥x_{n+1} − T x_{n+1}∥ ≤ ∥x_{n} − T x_{n}∥ for all n ≥ 1 and so r(x_{1}) ≤ ∥x_{1} − T x_{1}∥ for every x_{1} ∈ C, which implies that

r_{C}(T) = inf{∥x − T x∥ : x ∈ C}.

This completes the proof.

Theorem 2.2. Let X, C and T be as in Theorem 2.1. Then the following statements are equivalent:

(i) F (T) ≠ ∅.

(ii) For any specific x_{1} ∈ C, the sequence {x_{n}} of Picard iterates defined by

x_{n} = T^{n}x_{1} starting at x_{1} is bounded in C. (iii) For every x_{1} ∈ C, the Ishikawa iterative scheme {x_{n}} defined by (IS)

with the restrictions that t_{n} → t > 0 and s_{n} → s < 1 as n → ∞ is bounded.

(iv) There is a bounded sequence {y_{n}} ⊂ C such that y_{n}−T y_{n} → 0 as n → ∞.

(v) For every x_{1} ∈ C, the Ishikawa iterative scheme {x_{n}} defined by (IS) with the restrictions that 0 ≤ t_{n} ≤ t < 1, ∑_{n=1}^{∞} t_{n} = ∞ and ∑_{n=1}^{∞} s_{n} < ∞ is bounded.

Moreover, let {x_{n}} be the sequence as in Theorem 2.1. Then ∥x_{n} − T x_{n}∥ → 0 as n → ∞ provided that one of the above conditions (i)∼(v) holds.

Proof. For (i)⟺(ii), we may see Browder and Petryshyn [1, Theorem 1, p. 571]. For (i)⟺(v) and (i)⟺(iv), we may consult with the result of Deng [6] and references there.

Now, we will prove the equivalence of (i) and (iii) only. It is sufficient to prove that {T^{m}z} is bounded for all z ∈ C. Now we choose z ∈ C. For arbitrary but fixed m ≥ 1, using (IS), we have

∥y_{n} − T^{m−1}z∥ ≤ (1 − s_{n})∥x_{n} − T^{m−1}z∥ + s_{n}∥x_{n} − T^{m−2}z∥  (2.2)

and

∥x_{n+1} − T^{m}z∥ ≤ (1 − t_{n})∥x_{n} − T^{m}z∥ + t_{n}∥y_{n} − T^{m−1}z∥
≤ (1 − t_{n})∥x_{n} − T^{m}z∥ + t_{n}(1 − s_{n})∥x_{n} − T^{m−1}z∥
+ t_{n}s_{n}∥x_{n} − T^{m−2}z∥.  (2.3)

Since {x_{n}} is bounded, by taking lim sup on the both sides of (2.3), we have

lim sup_{n→∞} ∥x_{n} − T^{m}z∥ ≤ (1 − s) lim sup_{n→∞} ∥x_{n} − T^{m−1}z∥ + s lim sup_{n→∞} ∥x_{n} − T^{m−2}z∥.

Set a_{m} = lim sup_{n→∞} ∥x_{n} − T^{m}z∥. Then it follows that

a_{m} ≤ (1 − s)a_{m−1} + sa_{m−2}.  (2.4)

By induction, we can prove that

a_{m} ≤ [ ∑_{j=0}^{m−1}(−1)^{j}s^{j}]a_{1} + [s ∑_{j=0}^{m−2}(−1)^{j}s^{j}]a_{0}  (2.5)

for all m ≥ 2. Indeed, it is true for m = 2. Now we assume that it is true for the integers that are less than m and we have to prove that it is also true for m. By induction assumption, we have

a_{m−1} ≤ [ ∑_{j=0}^{m−2}(−1)^{j}s^{j}]a_{1} + [s ∑_{j=0}^{m−3}(−1)^{j}s^{j}]a_{0}  (2.6)

and

a_{m−2} ≤ [ ∑_{j=0}^{m−3}(−1)^{j}s^{j}]a_{1} + [s ∑_{j=0}^{m−4}(−1)^{j}s^{j}]a_{0}.  (2.7)

Substituting (2.6) and (2.7) in (2.4) yields that

a_{m} ≤ (1 − s){[ ∑_{j=0}^{m−2}(−1)^{j}s^{j}]a_{1} + [s ∑_{j=0}^{m−3}(−1)^{j}s^{j}]a_{0}} + s{[ ∑_{j=0}^{m−3}(−1)^{j}s^{j}]a_{1}

+ [s ∑_{j=0}^{m−4}(−1)^{j}s^{j}]a_{0}} ≤ [ ∑_{j=0}^{m−1}(−1)^{j}s^{j}]a_{1} + [s ∑_{j=0}^{m−2}(−1)^{j}s^{j}]a_{0},  (2.8)

which proves our claim. It follows from (2.8) that

a_{m} ≤ \frac{1}{1 + s}a_{1} + \frac{s}{1 + s}a_{0} ≤ max\{a_{0}, a_{1}\}

for all m ≥ 1, which proves the boundedness of {T^{m}z}. This completes the proof.

Recall that a Banach space X is said to satisfy Opial’s condition [10] if, for each sequence {x_{n}} in X, the condition x_{n} ⇀ x implies that

lim sup_{n→∞} ∥x_{n} − x∥ < lim sup_{n→∞} ∥x_{n} − y∥

for all y ∈ X with y ≠ x. It is well known from [10] that all l^{p} spaces for 1 < p < ∞ have this property. However, the L^{p} space do not have unless p = 2.

We say also that a mapping T : C → C is said to satisfy the Condition (A) if F (T) ≠ ∅ and there is a non-decreasing function f : R^{+} → R^{+} with f(0) = 0 and f(r) > 0 for all r > 0 such that

∥x − T x∥ ≥ f(d(x, F (T)))

for all x ∈ C, where d(x, F (T)) = inf{∥x − z∥ : z ∈ F (T)}.

Theorem 2.3. Let X be a real uniformly convex Banach space with a Fréchet differentiable norm or which satisfies Opial’s condition, C be a closed convex subset of X and T : C → C be a nonexpansive mapping with a nonempty fixed point set F (T). Then, for any initial value x_{1} ∈ C, the Ishikawa iterative scheme {x_{n}} defined by (IS) with the restrictions that ∑_{n=1}^{∞} t_{n}(1 − t_{n}) = ∞ and ∑_{n=1}^{∞} τ_{n} < ∞, where τ_{n} = min{t_{n}, 1 − t_{n}}s_{n}, converges weakly to a fixed point of T.

Proof. Set T_{n}x = t_{n}T (s_{n}T x + (1 − s_{n})x) + (1 − t_{n})x for all x ∈ C and n ≥ 1. Then the mapping T_{n} : C → C is also nonexpansive and the Ishikawa iterative scheme {x_{n}} defined by (IS) can be written as

x_{n+1} = T_{n}x_{n}

for all n ≥ 1. Furthermore, we have F (T) ⊂ F (T_{n}) for all n ≥ 1. Let ω_{w}(x_{n}) denote the weak ω-lim set of the sequence {x_{n}}. Since F (T) ≠ ∅, by Theorem 2.2, we see that

x_{n} − T x_{n} → 0

as n → ∞. Without loss of generality, assume that x_{n} ⇀ p as n → ∞. By Browder’s demi-closedness principle, we assert that T p = p, which gives that ω_{w}(x_{n}) ⊂ F (T).

To show that {x_{n}} converges weakly to a fixed point of T, it suffices to show that ω_{w}(x_{n}) consists of exactly one point. To this end, we first assume that X satisfies Opial’s condition and suppose that p ≠ q are in ω_{w}(x_{n}). Then x_{n_{k}} → p and x_{m_{k}} → q respectively. Since lim_{n→∞} ∥x_{n} − z∥ exists for any z ∈ F (T), by Opial’s condition, we conclude

lim_{n→∞} ∥x_{n} − p∥ = lim_{k→∞} ∥x_{n_{k}} − p∥ < lim_{k→∞} ∥x_{n_{k}} − q∥

= lim_{j→∞} ∥x_{m_{j}} − q∥ < lim_{j→∞} ∥x_{m_{j}} − p∥ = lim_{n→∞} ∥x_{n} − p∥,

which is a contradiction.

We now assume that X has a Fréchet differentiable norm and suppose that f_{1}, f_{2} ∈ ω_{w}(x_{n}) with f_{1} ≠ f_{2}. It follows from Lemma 1.4 that (f_{1} − f_{2}, f_{1}) = (f_{1} − f_{2}, f_{2}), which means that f_{1} = f_{2}, which is a contradiction again. This completes the proof.

Theorem 2.4. Let X, C, T and {x_{n}} be as in Theorem 2.1. Suppose, in addition, that F (T) ≠ ∅ and T satisfies the Condition (A). Then the iterative scheme {x_{n}} converges strongly to a fixed point of T.

Proof. It follows from Theorem 2.2 that x_{n} − T x_{n} → 0 as n → ∞. In view of the Condition (A), we have

∥x_{n} − T x_{n}∥ ≥ f(d(x_{n}, F (T)))

for all n ≥ 1, which implies that d(x_{n}, F (T)) → 0 as n → ∞. This leads to x_{n} → p ∈ F (T) as n → ∞. This completes the proof.

Corollary 2.1. Let X, C, T and {x_{n}} be as in Theorem 2.1. Suppose, in addition, that T (C) is relatively compact. Then the iterative scheme {x_{n}} converges strongly to a fixed point of T.

Proof. It follows from Schauder’s Fixed Point Theorem that T has at least a fixed point in C. Thus F (T) ≠ ∅. Also it is easily seen that T is demicompact, i.e., whenever {x_{n}} ⊂ C is bounded and {x_{n} − T x_{n}} converges strongly then there is a subsequence {x_{n_{j}}} of {x_{n}} which converges strongly. Since T is continuous as well as demicompact, according to Opial [10], the mapping (I −T) maps closed bounded subsets of C onto closed subsets of X. In view of Senter and Dotson [14, Lemma 1], we see that T must satisfy the Condition (A) with respect to the sequence {x_{n}}. Now Theorem 2.4 can be used to deduce the conclusion of the corollary. This completes the proof.

Remark 2.1. Theorem 2.4 shows that the Condition (A) is a sufficient condition which guarantees convergence of the iterates {x_{n}} defined by (IS). This condition is also necessary. Indeed, suppose that x_{n} → p ∈ F (T). Let A = I−T. By Xu and Roach [16, Theorem 2], the mapping A satisfies the Condition (I) (cf. [16]). Thus, from Xu and Roach [16, Theorem 1], there must be a strictly increasing function f : R^{+} → R^{+} with f(0) = 0 and j(x_{n} − p) ∈ J(x_{n} − p) such that

⟨x_{n} − T x_{n}, j(x_{n} − p)⟩ ≥ f(∥x_{n} − p∥)∥x_{n} − p∥,

which implies that

∥x_{n} − T x_{n}∥ ≥ f(∥x_{n} − p∥) ≥ f(d(x_{n}, F (T))).

This establishes the necessity.

For quasi-nonexpansive mappings, we have the following:

Theorem 2.5. Let X be a real uniformly convex Banach space, C be a nonempty closed convex subset of X and T : C → C be a quasi-nonexpansive mapping. Let {x_{n}} be the Ishikawa iterative scheme defined by (IS) with the restriction that ∑_{n=1}^{∞} t_{n}(1 − t_{n}) = ∞ and T x_{n} − T y_{n} → 0 as n → ∞. Then the iterative scheme {x_{n}} converges strongly to a fixed point of T if and only if T satisfies the Condition (A) with respect to the sequence {x_{n}}.

Proof. (⇒) As shown in Remark 2.1, A = I − T satisfies the Condition (I) and T satisfies the Condition (A) with respect to the sequence {x_{n}} and so the necessity follows.

(⇐) By Tan and Xu [15, Lemma 3], we have

lim inf_{n→∞} ∥x_{n} − T y_{n}∥ = 0.

By virtue of assumption that T x_{n} − T y_{n} → 0, we conclude

lim inf_{n→∞} ∥x_{n} − T x_{n}∥ = 0.

It follows from the Condition (A) that there is a subsequence {x_{n_{j}}} of {x_{n}} such that d(x_{n_{j}}, F (T)) → 0 as j → ∞. Therefore, we have d(x_{n}, F (T)) → 0 as n → ∞ since lim_{n→∞} d(x_{n}, F (T)) exists. This leads to the desired conclusion. This completes the proof.

Corollary 2.2. Let X, C, T and {x_{n}} be as in Theorem 2.5. Suppose in addition that T is uniformly continuous and s_{n} → 0 as n → ∞. Then the conclusion of Theorem 2.5 holds.

Proof. Observe that

x_{n} − y_{n} = s_{n}(x_{n} − T x_{n}),

while {∥x_{n} − T x_{n}∥} is bounded and so x_{n} − y_{n} → 0 as n → ∞. By the uniform continuity of T, we conclude that T x_{n}−T y_{n} → 0 as n → ∞. Now the conclusion of the corollary follows from Theorem 2.5.

Remark 2.2. Theorem 2.5 improves and generalizes the corresponding results by Tan and Xu [15, Theorems 2, 3], Xu and Roach [16, Theorem 2], Zhou and Jia [18, Theorem 2] and others.

Remark 2.3. All the results in this paper can be extended to the case in which the Ishikawa iterative scheme defined by (IS) admits errors in the senses of Liu [7] and Xu [17].

Acknowledgement

The third author wishes to acknowledge the financial support of the Korea Research Foundation (KRF-99-005-D00003).

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(Received 19.10.2001)

Author’s addresses: H. Zhou Department of Mathematics Shijiazhuang Mechanical Engineering College Shijiazhuang 050003, P. R. China Ravi P. Agarwal Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901-6975, U.S.A. E-mail: agarwal@fit.edu Yeol Je Cho and Yong Soo Kim Department of Mathematics, Gyeongsang National University Chinju 660-701, Korea, E-mail: yjcho@nongae.gsnu.ac.kr