BULLETIN OF INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN 1840-4367 Vol. 2(2012), 195-204

Former

BULLETIN OF SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p)

APPROXIMATING COMMON FIXED POINTS OF FINITE FAMILY OF ASYMPTOTICALLY NONEXPANSIVE NON-SELF MAPPINGS

A. S. Saluja, R. A. Rashwan, and Pankaj Kumar Jhade

Abstract. The aim of this paper is to study the strong convergence of an implicit iteration process to a common fixed point for a finite family of asymptotically nonexpansive nonself mappings in a uniformly convex Banach spaces.

1. Introduction and Preliminaries

Let K be a nonempty closed convex subset of a Banach space E. A self mapping T : K → K is called asymptotically nonexpansive if there exists a sequence {u_{n}} ⊂ [0, ∞); u_{n} → 0 as n → ∞ such that for all x, y ∈ K, the following inequality holds:

(1.1)  ∥T^{n}x − T^{n}y∥ ⩽ (1 + u_{n})∥x − y∥ ∀ n ⩾ 1

T is called uniformly L−Lipschitzian if there exists a constant L > 0 such that for all x, y ∈ K,

(1.2)  ∥T^{n}x − T^{n}y∥ ⩽ L∥x − y∥ ∀ n ⩾ 1

The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk ([9]), as an important generalization of the class of nonexpansive maps, who proved that if K is a nonempty closed convex subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive self mapping of K, then T has a fixed point. Iterative techniques for approximating fixed points of nonexpansive mappings and asymptotically nonexpansive mappings have been studied by various authors (See [19, 2, 3, 4, 17, 13, 5, 18, 1]) using the Mann iteration method (See e.g.[19]) or the Ishikawa iteration method (See e.g. [15]). In 1978, Bose ([15]) proved that if K is a bounded closed convex nonempty subset of a uniformly convex Banach space E satisfying Opial’s ([22]) condition and T : K → K is an asymptotically nonexpansive mapping, then the sequence {T^{n}x} converges weakly to a fixed point of T provided T is asymptotically regular at x ∈ K , i.e. lim_{n→∞} ∥T^{n}x − T^{n+1}x∥ = 0. Passty ([5]) and also Xu ([7]) proved that the requirement that T satisfies Opial’s condition can be replaced by the condition that E has a Frechet differentiable norm. Furthermore, Tan and Xu ([10, 11]) later proved that the asymptotic regularity of T can be weakend to the weakly asymptotic regularity of T at x i.e. ω − lim_{n→∞}(T^{n}x − T^{n+1}x) = 0. In all the above results, the operator T remains a self mapping of a nonempty closed convex subset K of a uniformly convex Banach space E. If, however, the domain of T, D(T) is a proper subset of E, and T maps D(T) into E, then the iteration process of Mann and Ishikawa studied by these authors. The purpose of this paper is to construct a multistep iterative scheme with errors for approximating common fixed point of a finite family of asymptotically nonexpansive nonself mappings and to prove strong convergence theorems for such maps. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Then for arbitrary x_{1} ∈ K , we define the sequence {x_{n}} iteratively as follows:

2010 Mathematics Subject Classification. Primary 47H10; Secondary 54H25.

Key words and phrases. Asymptotically nonexpansive nonself mappings, common fixed points, multistep iterative sequence with errors for non-self mappings.

(1.3)  \{ \begin{matrix}x_{n}^{1} = P (α_{n}^{1}x_{n} + β_{n}^{1}T_{1}^{n}x_{n} + γ_{n}^{1}u_{n}^{1}) \\ x_{n}^{2} = P (α_{n}^{2}x_{n} + β_{n}^{2}T_{2}^{n}x_{n}^{1} + γ_{n}^{2}u_{n}^{2}) \\ ⋯ = · · · · · · · · ·  · · · · · · · · · \\ ⋯ = · · · · · · · · ·  · · · · · · · · · \\ x_{n+1} = x_{n}^{(N)} = P (α_{n}^{N}x_{n} + β_{n}^{N}T_{N}^{n}x_{n}^{N−1} + γ_{n}^{N}u_{n}^{N})  ∀ n ⩾ 1 \end{matrix}

where {α_{n}^{1}}, {α_{n}^{2}}, · · · · · · , {α_{n}^{N}},{β_{n}^{1}}, {β_{n}^{2}}, · · · · · · , {β_{n}^{N}},{γ_{n}^{1}}, {γ_{n}^{2}}, · · · · · · , {γ_{n}^{N}} are sequences in [0, 1] with α_{n}^{i} + β_{n}^{i} + γ_{n}^{i} = 1 for all i = 1, 2, 3, · · · · · · N and {u_{n}^{1}}, {u_{n}^{2}}, · · · · · · , {u_{n}^{N}} are bounded sequences in K.

Definition 1.1. Let E be a real Banach space. A subset K of E is said to be a retract of E if there exists a continuous map P : E → E such that P x = x for all x ∈ K. A map P : E → E is said to be a retraction if P^{2} = P. It follows that if a map P is a retraction, then P y = y for all y in the range of P.

Recall that the following:

(1) A mapping T : K → K with F (T) ≠ ϕ is said to satisfy condition (A) [6] on K if there exists a non decreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and f(r) > 0 for all r ∈ (0, ∞) such that for all x ∈ K, ∥x − T x∥ ⩾ f(d(x, F)), where d(x, F (T)) = inf{∥x − p∥ : p ∈ F (T)}.

(2) A family {T_{1}, T_{2}, · · · · · · T_{n}} of N self-mappings on K with F = ⋂_{i=1}^{N} F (T_{i}) ≠ ϕ is said to satisfy condition (B) on K if there exists f and d as in (i) such that

max_{1⩽i⩽N}{∥x − T_{i}x∥} ⩾ f(d(x, F)),

for all x ∈ K.

When T_{i} = T for all i = 1, 2, · · · · · · N, then condition (B) reduces to condition (A).

Lemma 1.1. ([12]) Let {a_{n}}, {β_{n}} and {r_{n}} be non-negative sequences satisfying a_{n+1} ⩽ (1 + r_{n})a_{n} + β_{n} , ∀ n ∈ N. If ∑_{n=1}^{∞} r_{n} < ∞, ∑_{n=1}^{∞} β_{n} < ∞, then lim_{n→∞} a_{n}} exist. Moreover, if lim inf_{n→∞} a_{n} = 0 then lim_{n→∞} a_{n} = 0.

Lemma 1.2. ([14])

Let p > 1 and R > 1 be two fixed numbers and E be a Banach space. Then E is uniformly convex if and only if there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that

∥λx + (1 − λ)y∥^{p} ⩽ λ∥x∥^{p} + (1 − λ)∥y∥^{p} − w_{p}(λ)g(∥x − y∥)

for all x, y ∈ B_{R}(0) = {x ∈ E : ∥x∥ ⩽ R}, and λ ∈ [0, 1], where w_{p}(λ) = λ(1 − λ)^{p} + λ^{p}(1 − λ).

2. Main Results

Before proving our main result we shall prove the following crucial lemmas.

Lemma 2.1. Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset which is also a nonexpansive retract of E. Let T_{1}, T_{2}, · · · · · · , T_{N} : K → K be N asymptotically nonexpansive nonself mappings with sequences {r_{n}^{i}} such that ∑_{n=1}^{∞} r_{n}^{i} < ∞, for all 1 ⩽ i ⩽ N and F = ⋂_{i=1}^{N} F (T_{i}) ≠ ϕ. Let {α_{n}^{i}}, {β_{n}^{i}}, {γ_{n}^{i}} are sequences in [0,1] with α_{n}^{i} + β_{n}^{i} + γ_{n}^{i} = 1 for all i = 1, 2, 3, · · · · · · N. From arbitrary x_{1} ∈ K define the sequence {x_{n}} iteratively by (1.3), where {u_{n}^{i}} are bounded sequences in K with ∑_{n=1}^{∞} u_{n}^{i} < ∞ and ∑_{n=1}^{∞} γ_{n}^{i} < ∞. Then

∥x_{n+1} − x^{∗}∥ = ∥x_{n}^{N} − x^{∗}∥ ⩽ (1 + b_{n}^{N−1})∥x_{n} − x^{∗}∥ + d_{n}^{N−1},

for all n ⩾ 1, x^{∗} ∈ F and for some sequence {d_{n}^{i}} for all i = 1, 2, 3, · · · · · · N of numbers such that ∑_{n=1}^{∞} d_{n}^{i} < ∞.

Proof. Let x^{∗} ∈ F, then from (1.3) we get

∥x_{n}^{1} − x^{∗}∥ = ∥P (α_{n}^{1}x_{n} + β_{n}^{1}T_{1}^{n}x_{n} + γ_{n}^{1}u_{n}^{1}) − P x^{∗}∥
⩽ α_{n}^{1}∥x_{n} − x^{∗}∥ + β_{n}^{1}∥T_{1}^{n}x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x^{∗}∥
⩽ α_{n}^{1}∥x_{n} − x^{∗}∥ + β_{n}^{1}(1 + r_{n}^{1})∥x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x^{∗}∥
⩽ α_{n}^{1}(1 + r_{n}^{1})∥x_{n} − x^{∗}∥ + β_{n}^{1}(1 + r_{n}^{1})∥x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x^{∗}∥
⩽ (1 − β_{n}^{1})(1 + r_{n}^{1})∥x_{n} − x^{∗}∥ + β_{n}^{1}(1 + r_{n}^{1})∥x_{n} − x^{∗}∥
+ γ_{n}^{1}∥u_{n}^{1} − x^{∗}∥
⩽ (1 + r_{n}^{1})∥x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x^{∗}∥
⩽ (1 + r_{n}^{1})∥x_{n} − x^{∗}∥ + d_{n}^{0}

where d_{n}^{0} = γ_{n}^{1}∥u_{n}^{1} − x^{∗}∥. Since ∑_{n=1}^{∞} γ_{n}^{1} < ∞, then ∑_{n=1}^{∞} d_{n}^{0} < ∞.

Next we note that,

∥x_{n}^{2} − x^{∗}∥ = ∥P (α_{n}^{2}x_{n} + β_{n}^{2}T_{2}^{n}x_{n}^{1} + γ_{n}^{2}u_{n}^{2}) − P x^{∗}∥
⩽ α_{n}^{2}∥x_{n} − x^{∗}∥ + β_{n}^{2}∥T_{2}^{n}x_{n}^{1} − x^{∗}∥ + γ_{n}^{2}∥u_{n}^{2} − x^{∗}∥
⩽ α_{n}^{2}∥x_{n} − x^{∗}∥ + β_{n}^{2}(1 + r_{n}^{2})∥x_{n}^{1} − x^{∗}∥ + γ_{n}^{2}∥u_{n}^{2} − x^{∗}∥
⩽ α_{n}^{2}∥x_{n} − x^{∗}∥ + β_{n}^{2}(1 + r_{n}^{2})[(1 + r_{n}^{1})∥x_{n} − x^{∗}∥ + d_{n}^{0}]
+ γ_{n}^{2}∥u_{n}^{2} − x^{∗}∥
⩽ [α_{n}^{2} + β_{n}^{2}(1 + r_{n}^{2})(1 + r_{n}^{1})]∥x_{n} − x^{∗}∥ + β_{n}^{2}(1 + r_{n}^{2})d_{n}^{0}
+ γ_{n}^{2}∥u_{n}^{2} − x^{∗}∥
⩽ (α_{n}^{2} + β_{n}^{2})(1 + r_{n}^{2})(1 + r_{n}^{1})∥x_{n} − x^{∗}∥ + β_{n}^{2}(1 + r_{n}^{2})d_{n}^{0}
+ γ_{n}^{2}∥u_{n}^{2} − x^{∗}∥
⩽ (1 + r_{n}^{1} + r_{n}^{2} + r_{n}^{1}r_{n}^{2})∥x_{n} − x^{∗}∥ + β_{n}^{2}(1 + r_{n}^{2})d_{n}^{0} + γ_{n}^{2}∥u_{n}^{2} − x^{∗}∥
⩽ (1 + b_{n}^{1})∥x_{n} − x^{∗}∥ + d_{n}^{1}

where, d_{n}^{1} = β_{n}^{2}(1 + r_{n}^{2})d_{n}^{0} + γ_{n}^{2}∥u_{n}^{2} − x^{∗}∥ and b_{n}^{1} = (r_{n}^{1} + r_{n}^{2} + r_{n}^{1}r_{n}^{2}) Since ∑_{n=1}^{∞} d_{n}^{0} < ∞, ∑_{n=1}^{∞} γ_{n}^{2} < ∞, ∑_{n=1}^{∞} r_{n}^{i} < ∞, for i = 1, 2 and so ∑_{n=1}^{∞} d_{n}^{1} < ∞, and ∑_{n=1}^{∞} b_{n}^{1} < ∞.

∥x_{n}^{i} − x^{∗}∥ ⩽ (1 + b_{n}^{i−1})∥x_{n} − x^{∗}∥ + d_{n}^{i−1} ∀ n ⩾ 1, ∀ i = 1, 2, ⋯ N

Thus , ∥x_{n+1} − x^{∗}∥ = ∥x_{n}^{N} − x^{∗}∥ ⩽ (1 + b_{n}^{N−1})∥x_{n} − x^{∗}∥ + d_{n}^{N−1} for all n ⩾ 1. This completes the proof of the lemma. □

Remark 2.1. If we put P = I (Identity mapping) in Lemma (2.1), then it generalizes the corresponding lemma of Schu [8] for one mapping. Further, if F = ⋂_{i=1}^{N}F (T_{i}) ≠ ϕ and lim_{n→∞} ∥x_{n} − T_{i}^{n}x_{n}∥ = 0 for all i = 1, 2, ⋯ N, then we have lim_{n→∞} ∥x_{n+1} − x_{n}∥ = 0.

Lemma 2.2. Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset which is also a nonexpansive retract of E. Let T_{1}, T_{2}, · · · · · · , T_{N} : K → K be N uniformly continuous asymptotically nonexpansive nonself mappings with sequences {r_{n}^{i}} such that ∑_{n=1}^{∞} r_{n}^{i} < ∞, for all 1 ⩽ i ⩽ N and F = ⋂_{i=1}^{N} F (T_{i}) ≠ ϕ. Let {x_{n}} be a sequence defined by (1.3) with ∑_{n=1}^{∞} γ_{n}^{i} < ∞ and {β_{n}^{i}} ⊆ [ε, 1 − ε] for all i = 1, 2, ⋯ N & for some ε ∈ (0, 1). Then ∥x_{n} − T_{i}x_{n}∥ = 0, for all i = 1, 2, ⋯ N.

Proof. Let x^{∗} ∈ F = ⋂_{i=1}^{N} F (T_{i}). Then by Lemma (2.1) and Lemma (1.1) lim_{n→∞} ∥x_{n} − x^{∗}∥ exists. Let lim_{n→∞} ∥x_{n} − x^{∗}∥ = a. If a = 0, then by the continuity of each T_{i} the conclusion follows. Now suppose that a > 0. First, we will show that lim_{n→∞} ∥T_{N}^{n}x_{n} − x_{n}∥ = 0. Since {x_{n}} and {u_{n}^{i}} bounded for all i = 1, 2, ⋯ N, there exist R > 0 such that x_{n} − x^{∗} + γ_{n}^{i}(u_{n}^{i} − x_{n}), T _{i}^{n}x_{n}^{i−1} − x^{∗} + γ_{n}^{i}(u_{n}^{i} − x_{n}) ∈ B_{R}(0) for all n ⩾ 1 and for all i = 1, 2, ⋯ N.

APPROXIMATING COMMON FIXED POINTS 199

Now using lemma (1.3), we have

∥x_{n+1} − x^{∗}∥^{2} = ∥x_{n}^{N} − x^{∗}∥^{2}
= ∥P (α_{n}^{N}x_{n} + β_{n}^{N}T_{N}^{n}x_{n}^{N−1} + γ_{n}^{N}u_{n}^{N}) − P x^{∗}∥^{2}
= ∥α_{n}^{N}x_{n} + β_{n}^{N}T_{N}^{n}x_{n}^{N−1} + γ_{n}^{N}u_{n}^{N} − x^{∗}∥^{2}
= ∥β_{n}^{N}(T_{N}^{n}x_{n}^{N−1} − x^{∗} + γ_{n}^{N}(u_{n}^{N} − x_{n})) + (1 − β_{n}^{N})(x_{n} − x^{∗}+
γ_{n}^{N}(u_{n}^{N} − x_{n}))∥^{2}
⩽ β_{n}^{N}∥T_{N}^{n}x_{n}^{N−1} − x^{∗} + γ_{n}^{N}(u_{n}^{N} − x_{n})∥^{2}
+ (1 − β_{n}^{N})∥x_{n} − x^{∗} + γ_{n}^{N}(u_{n}^{N} − x_{n})∥^{2}
− w_{2}(β_{n}^{N})g(∥T_{N}^{n}x_{n}^{N−1} − x_{n}∥)
⩽ β_{n}^{N}(∥T_{N}^{n}x_{n}^{N−1} − x^{∗}∥ + γ_{n}^{N}∥u_{n}^{N} − x_{n}∥)^{2}
+ (1 − β_{n}^{N})(∥x_{n} − x^{∗}∥ + γ_{n}^{N}∥u_{n}^{N} − x_{n}∥)^{2}
− w_{2}(β_{n}^{N})g(∥T_{N}^{n}x_{n}^{N−1} − x_{n}∥)
⩽ β_{n}^{N}(∥x_{n} − x^{∗}∥ + d_{n}^{N−2} + γ_{n}^{N}∥u_{n}^{N} − x_{n}∥)^{2}
+ (1 − β_{n}^{N})(∥x_{n} − x^{∗}∥ + d_{n}^{N−2} + γ_{n}^{N}∥u_{n}^{N} − x_{n}∥)^{2}
− w_{2}(β_{n}^{N})g(∥T_{N}^{n}x_{n}^{N−1} − x_{n}∥)
⩽ (∥x_{n} − x^{∗}∥ + d_{n}^{N−2} + γ_{n}^{N}∥u_{n}^{N} − x_{n}∥)^{2}
− w_{2}(β_{n}^{N})g(∥T_{N}^{n}x_{n}^{N−1} − x_{n}∥)

⩽ (∥x_{n} − x^{∗}∥ + λ_{n}^{N−2})^{2} − w_{2}(β_{n}^{N})g(∥T_{N}^{n}x_{n}^{N−1} − x_{n}∥) (2.1)

where λ_{n}^{N−2} = d_{n}^{N−2} + γ_{n}^{N}∥u_{n}^{N} − x_{n}∥

Observe that ε^{3} ⩽ w_{2}(β_{n}^{N}). Now (2.1) implies that

ε^{3}g(∥T_{N}^{n}x_{n}^{N−1} − x_{n}∥) ⩽ ∥x_{n} − x^{∗}∥^{2} − ∥x_{n+1} − x^{∗}∥^{2} + ρ_{n}^{N−2}

where ρ_{n}^{N−2} = 2λ_{n}^{N−2} + (λ_{n}^{N−2})^{2}

Since ∑_{n=1}^{∞} d_{n}^{N−2} < ∞ and ∑_{n=1}^{∞} γ_{n}^{N−2} < ∞, we get ∑_{n=1}^{∞} ρ_{n}^{N−2} < ∞, which implies that

lim_{n→∞} g(∥T_{N}^{n}x_{n}^{N−1} − x_{n}∥) = 0

Since g is strictly increasing and continuous at 0, it follows that

lim_{n→∞} ∥T_{N}^{n}x_{n}^{N−1} − x_{n}∥ = 0

Since for all N, T_{N} is asymptotically nonexpansive, note that

∥x_{n} − x^{∗}∥ ⩽ ∥x_{n} − T_{N}^{n}x_{n}^{N−1}∥ + ∥T_{N}^{n}x_{n}^{N−1} − x^{∗}∥
= ∥x_{n} − T_{N}^{n}x_{n}^{N−1}∥ + (1 + r_{n}^{N})∥x_{n}^{N−1} − x^{∗}∥

for all n ⩾ 1

200 SALUJA, RASHWAN, AND JHADE

Thus

a = lim_{n→∞} ∥x_{n} − x^{∗}∥ ⩽ lim inf ∥x_{n}^{N−1} − x^{∗}∥ ⩽ lim sup ∥x_{n}^{N−1} − x^{∗}∥ ⩽ a,

and therefore

lim_{n→∞} ∥x_{n}^{N−1} − x^{∗}∥ = a.

Using the same argument in the proof above, we have

∥x_{n}^{N−1} − x^{∗}∥^{2} ⩽ β_{n}^{N−1}∥T_{N−1}^{n}x_{n}^{N−2} − x^{∗} + γ_{n}^{N−1}(u_{n}^{N−1} − x_{n})∥^{2}
+ (1 − β_{n}^{N−1})∥x_{n} − x^{∗} + γ_{n}^{N−1}(u_{n}^{N−1} − x_{n})∥^{2}
− w_{2}(β_{n}^{N−1})g(∥T_{N−1}^{n}x_{n}^{N−2} − x_{n}∥)
⩽ β_{n}^{N−1}(∥x_{n} − x^{∗}∥ + d_{n}^{N−3} + γ_{n}^{N−1}∥u_{n}^{N−1} − x_{n}∥)^{2}
+ (1 − β_{n}^{N−1})(∥x_{n} − x^{∗}∥ + d_{n}^{N−3} + γ_{n}^{N−1}∥u_{n}^{N−1} − x_{n}∥)^{2}
− w_{2}(β_{n}^{N−1})g(∥T_{N−1}^{n}x_{n}^{N−2} − x_{n}∥)
⩽ (∥x_{n} − x^{∗}∥ + d_{n}^{N−3} + γ_{n}^{N−1}∥u_{n}^{N−1} − x_{n}∥)^{2}
− w_{2}(β_{n}^{N−1})g(∥T_{N−1}^{n}x_{n}^{N−2} − x_{n}∥)

⩽ (∥x_{n} − x^{∗}∥ + λ_{n}^{N−3})^{2} − w_{2}(β_{n}^{N−1})g(∥T_{N−1}^{n}x_{n}^{N−2} − x_{n}∥) (2.2)

where λ_{n}^{N−3} = d_{n}^{N−3} + γ_{n}^{N−1}∥u_{n}^{N−1} − x_{n}∥

This implies that

ε^{3}g(∥T_{N−1}^{n}x_{n}^{N−2} − x_{n}∥) ⩽ ∥x_{n} − x^{∗}∥^{2} − ∥x_{n+1} − x^{∗}∥^{2} + ρ_{n}^{N−3},

where ρ_{n}^{N−3} = 2λ_{n}^{N−3} + (λ_{n}^{N−3})^{2}.

Therefore

lim_{n→∞} ∥T_{N−1}^{n}x_{n}^{N−2} − x_{n}∥ = 0.

Thus we have

∥x_{n} − T_{N}^{n}x_{n}∥ ⩽ ∥x_{n} − T_{N}^{n}x_{n}^{N−1}∥ + ∥T_{N}^{n}x_{n}^{N−1} − T_{N}^{n}x_{n}∥
⩽ ∥x_{n} − T_{N}^{n}x_{n}^{N−1}∥ + (1 + r_{n}^{N})∥x_{n}^{N−1} − x_{n}∥
⩽ ∥x_{n} − T_{N}^{n}x_{n}^{N−1}∥
+ (1 + r_{n}^{N})∥α_{n}^{N−1}x_{n} + β_{n}^{N−1}T_{N−1}^{n}x_{n}^{N−2} + γ_{n}^{N−1}u_{n}^{N−1} − x_{n}∥
⩽ ∥x_{n} − T_{N}^{n}x_{n}^{N−1}∥
+ (1 + r_{n}^{N})[β_{n}^{N−1}∥T_{N−1}^{n}x_{n}^{N−2} − x_{n}∥ + γ_{n}^{N−1}∥u_{n}^{N−1} − x_{n}∥]
.

Since lim_{n→∞} ∥T_{N}^{n}x_{n}^{N−1} − x_{n}∥ = 0 and lim_{n→∞} ∥T_{N−1}^{n}x_{n}^{N−2} − x_{n}∥ = 0 ,also ∑_{n=1}^{∞} γ_{n}^{N−1} < ∞ and ∑_{n=1}^{∞} r_{n}^{N} < ∞, it follows that lim_{n→∞} ∥x_{n} − T_{N}^{n}x_{n}∥ = 0. Similarly

lim_{n→∞} ∥x_{n} − T_{N−2}^{n}x_{n}^{N−3}∥ = lim_{n→∞} ∥x_{n} − T_{N−3}^{n}x_{n}^{N−4}∥ = ⋯

APPROXIMATING COMMON FIXED POINTS 201

· · · · · · · · · = lim_{n→∞} ∥x_{n} − T_{2}^{n}x_{n}^{1}∥ = 0

This implies that

lim_{n→∞} ∥x_{n} − T_{N−1}^{n}x_{n}∥ = lim_{n→∞} ∥x_{n} − T_{N−2}^{n}x_{n}∥ = ⋯ = lim_{n→∞} ∥x_{n} − T_{3}^{n}x_{n}∥ = 0

It remains to show that

lim_{n→∞} ∥x_{n} − T_{1}^{n}x_{n}∥ = 0, lim_{n→∞} ∥x_{n} − T_{2}^{n}x_{n}∥ = 0

Note that

∥x_{n}^{1} − x^{∗}∥^{2} ⩽ β_{n}^{1}(∥T_{1}^{n}x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x_{n}∥)^{2}
+ (1 − β_{n}^{1})(∥x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x_{n}∥)^{2}
− w_{2}(β_{n}^{1})g(∥T_{1}^{n}x_{n} − x_{n}∥)
⩽ β_{n}^{1}(∥x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x_{n}∥)^{2}
+ (1 − β_{n}^{1})(∥x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x_{n}∥)^{2}
− w_{2}(β_{n}^{1})g(∥T_{1}^{n}x_{n} − x_{n}∥)
⩽ (∥x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x_{n}t∥)^{2} − w_{2}(β_{n}^{1})g(∥T_{1}^{n}x_{n} − x_{n}∥)

Thus we have

ε^{2}g(∥T_{1}^{n}x_{n} − x_{n}∥) ⩽ (∥x_{n} − x^{∗}∥ + γ_{n}^{1}∥u_{n}^{1} − x_{n}∥)^{2} − ∥x_{n}^{1} − x^{∗}∥^{2}

and therefore lim_{n→∞} ∥x_{n} − T_{1}^{n}x_{n}∥ = 0. Since

∥x_{n} − T_{2}^{n}x_{n}∥ ⩽ ∥x_{n} − T_{2}^{n}x_{n}^{1}∥ + ∥T_{2}^{n}x_{n}^{1} − T_{2}^{n}x_{n}∥
⩽ ∥x_{n} − T_{2}^{n}x_{n}^{1}∥ + (1 + r_{n}^{2})∥x_{n}^{1} − x_{n}∥
⩽ ∥x_{n} − T_{2}^{n}x_{n}^{1}∥ + (1 + r_{n}^{2})∥α_{n}^{1}x_{n} + β_{n}^{1}T_{1}^{n}x_{n} + γ_{n}^{1}u_{n}^{1} − x_{n}∥
⩽ ∥x_{n} − T_{2}^{n}x_{n}^{1}∥ + (1 + r_{n}^{2})[β_{n}^{1}∥T_{1}^{n}x_{n} − x_{n}∥ + γ_{n}^{1}∥u_{n}^{1} − x_{n}∥],

Which implies that

lim_{n→∞} ∥x_{n} − T_{2}^{n}x_{n}∥ = 0.

Therefore

lim_{n→∞} ∥x_{n} − T_{i}^{n}x_{n}∥ = 0,

for all i = 1, 2, ⋯ N. On the other hand, by Remark (2.1), it is clear that

lim_{n→∞} ∥x_{n+1} − x_{n}∥ = 0.

Therefore by Lemma (2.1), we conclude that lim_{n→∞} ∥x_{n} − T_{i}x_{n}∥ = 0. □

Theorem 2.1. Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset which is also a nonexpansive retract of E. Let T_{1}, T_{2}, · · · · · · , T_{N} : K → K be N uniformly continuous asymptotically nonexpansive nonself mappings with sequences {r_{n}^{i}} such that ∑_{n=1}^{∞} r_{n}^{i} < ∞, for all 1 ⩽ i ⩽ N and F = ⋂_{i=1}^{N} F (T_{i}) ≠ ϕ. Suppose {T_{1}, T_{2}, · · · · · · , T_{N}} satisfies condition (B). Let {x_{n}} be a sequence defined by (1.3) with ∑_{n=1}^{∞} γ_{n}^{i} < ∞ and {β_{n}^{i}} ⊆ [ε, 1 − ε] for all i = 1, 2, ⋯ N and for some ε ∈ (0, 1). Then {x_{n}} converges strongly to a common fixed point of the mappings {T_{1}, T_{2}, · · · · · · , T_{N}}.

Proof. From Lemma (2.1) and (1.1), we see that lim_{n→∞} ∥x_{n} − x^{∗}∥ exist for all x^{∗} ∈ F = ⋂_{i=1}^{N} F (T_{i}). Let lim_{n→∞} ∥x_{n} − x^{∗}∥ = a for all a ⩾ 0. Without loss of generality, if a = 0, then there is nothing to prove. So that we assume that a > 0, as proved in Lemma (2.1), we have

∥x_{n+1} − x^{∗}∥ = ∥x_{n}^{N} − x^{∗}∥ ⩽ (1 + b_{n}^{N−1})∥x_{n} − x^{∗}∥ + d_{n}^{N−1},

for all n ⩾ 1, where {d_{n}^{i}}_{n=1}^{∞} , for all i = 1, 2, ⋯ N, is non-negative real sequences such that ∑_{n=1}^{∞} d_{n}^{i} < ∞ for all i = 1, 2, ⋯ N. This gives that

d(x_{n+1}, F) ⩽ (1 + b_{n}^{N−1})d(x_{n}, F) + d_{n}^{N−1} forall

n∈ N .

Applying Lemma (1.1) to the above inequality, we obtained that lim_{n→∞} d(x_{n}, F) exist. Also by Lemma (2.2) lim_{n→∞} ∥x_{n} − T_{i}^{n}x_{n}∥ = 0 for all i = 1, 2, ⋯ N. Since {T_{1}, T_{2}, · · · · · · , T_{N}} satisfies condition (B), we conclude that lim_{n→∞} d(x_{n}, F) = 0. Next we show that {x_{n}} is a Cauchy sequence. Since lim_{n→∞} d(x_{n}, F) = 0, then given any ε > 0 there exist a natural number n_{0} such that d(x_{n}, F) < \frac{ε}{3} for all n ⩾ n_{0}. So we can find p^{∗} ∈ F such that ∥x_{n_{0}} − p^{∗}∥ < \frac{ε}{2} For all n ⩾ n_{0} and m ⩾ 1, we have

∥x_{n+m} − x_{n}∥ ⩽ ∥x_{n+m} − p^{∗}∥ + ∥p^{∗} − x_{n}∥

⩽ ∥x_{n_{0}} − p^{∗}∥ + ∥x_{n_{0}} − p^{∗}∥

< \frac{ε}{2} + \frac{ε}{2} = ε

This shows that {x_{n}} is a Cauchy sequence and so is convergent ,since E is complete. Let lim x_{n} = q^{∗}. Then q^{∗} ∈ K. It remains to show that q^{∗} ∈ F. Let ε_{1} > 0 be given, then there exists a natural number n_{1} such that ∥x_{n} − x^{∗}∥ < \frac{ε_{1}}{4} for all n ⩾ n_{1}. Since lim_{n→∞} d(x_{n}, F) = 0, there exists a natural number n_{2} ⩾ n_{1} such that , for all n ⩾ n_{2}, we have d(x_{n}, F) < \frac{ε_{1}}{5} and in particular, we have d(x_{n_{2}}, F) < \frac{ε_{1}}{5} . Therefore, there exists w^{∗} ∈ K such that ∥x_{n_{2}} − w^{∗}∥ < \frac{ε_{1}}{4} For any i ∈ I and n ⩾ n_{2}, we have

∥T_{i}q^{∗} − q^{∗}∥ ⩽ ∥T_{i}q^{∗} − w^{∗}∥ + ∥w^{∗} − q^{∗}∥

⩽ 2∥q^{∗} − w^{∗}∥

⩽ 2(∥q^{∗} − x_{n_{2}}∥ + ∥x_{n_{2}} − w^{∗}∥)

< 2(\frac{ε_{1}}{4} + \frac{ε_{1}}{4} ) < ε_{1}

This implies that T_{i}q^{∗} = q^{∗}. Hence q^{∗} ∈ F (T_{i}) for all i ∈ I and so q^{∗} ∈ F = ⋂_{i=1}^{N} F (T_{i}). Thus {x_{n}} converges strongly to a common fixed point of the mappings {T_{1}, T_{2}, · · · · · · , T_{N}}. □

Remark 2.2. Theorem (2.1) extend the corresponding result of Su and Qin [21] to the case of multistep iterative sequences with errors for a finite family of asymptotically nonexpansive nonself mappings.

Remark 2.3. Our result also extend the corresponding result of Shahzad [14] to the case of multistep iterative sequences with errors for a finite family of more general class of nonexpansive mappings.

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Received 23.02.2012; available online 03.09.2012

Department of Mathematics, J H Government Post Graduate College, Betul, INDIA-460001

    E-mail address: dssaluja@rediffmail.com

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt, E-mail address: rr_{r}ashwan54@yahoo.com

Department of Mathematics, NRI Institute of Information Science and Technology, Bhopal, INDIA-462021

    E-mail address: pmathsjhade@gmail.com