PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97(111) (2015), 239–251 DOI: 10.2298/PIM130818001T

CONVERGENCE THEOREMS OF A SCHEME FOR I -ASYMPTOTICALLY QUASI-NONEXPANSIVE TYPE MAPPING IN BANACH SPACE

Seyit Temir

Abstract. Let X be a Banach space. Let K be a nonempty subset of X. Let T : K → K be an I-asymptotically quasi-nonexpansive type mapping and I : K → K be an asymptotically quasi-nonexpansive type mappings in the Banach space. Our aim is to establish the necessary and sufficient conditions for the convergence of the Ishikawa iterative sequences with errors of an I-asymptotically quasi-nonexpansive type mappping in Banach spaces to a common fixed point of T and I. Also, we study the convergence of the Ishikawa iterative sequences to common fixed point for nonself I-asymptotically quasinonexpansive type mapping in Banach spaces.

The results presented in this paper extend and generalize some recent work of Chang and Zhou [1], Wang [19 ], Yao and Wang [20 ] and many others.

1. Introduction

Let X be a real Banach space, K be a nonempty subset of Banach space and T, I : K → K. Let F (T) = {x ∈ K : T x = x} and F (I) = {x ∈ K : Ix = x} denote the set of fixed points of mappings T and I, respectively. Recall some definitions and notations. T is called nonexpansive if ∥T x − T y∥ ⩽ ∥x − y∥ for all x, y ∈ K. The quasi-nonexpansive mappings defined as the following were studied by Diaz and Metcalf [4] and Dotson [5] in Banach spaces. T is called a quasi-nonexpansive mapping if F (T) ≠ ∅ and ∥T x − p∥ ⩽ ∥x − p∥ for all x ∈ K and p ∈ F (T). The concept of asymptotically nonexpansiveness defined as the following was introduced by Goebel and Kirk [7]. T is called asymptotically quasi-nonexpansive mapping if F (T) ≠ ∅ and there exists a sequence {k_{n}} ⊂ [1, ∞) with lim_{n→∞} k_{n} = 1 such that ∥T^{n}x − p∥ ⩽ k_{n}∥x − p∥ for all x ∈ K and p ∈ F (T) and n ⩾ 1. Let X be a Banach space and K be a nonempty subset of the Banach space. Let T, I : K → K be two mappings. T is called I-nonexpansive if ∥T x − T y∥ ⩽ ∥Ix − Iy∥ for all x, y ∈ K. T is called I-quasi-nonexpansive if F (T) ∩ F (I) ≠ ∅ and ∥T x − p∥ ⩽ ∥Ix − p∥ for all x ∈ K and p ∈ F (T) ∩ F (I).

2010 Mathematics Subject Classification: 47H09; 47H10.

Key words and phrases: I-asymptotically quasi-nonexpansive type mapping, nonself Iasymptotically quasi-nonexpansive type mapping, Ishikawa iterative schemes.

Communicated by Stevan Pilipović.

From the above definitions, it follows that if F (T) ∩ F (I) is nonempty, an Inonexpansive mapping must be I quasi-nonexpansive, and linear I quasi-nonexpan-sive mappings are I-nonexpansive mappings. But it is easily seen that there exist nonlinear continuous I quasi-nonexpansive mappings which are not I-nonexpansive. T is called I-asymptotically quasi-nonexpansive if there exists a sequence {k_{n}} ⊂ [1, ∞) with lim_{n→∞} k_{n} = 1 such that ∥T^{n}x − p∥ ⩽ k_{n}∥I^{n}x − p∥ for all x ∈ K and p ∈ F (T) ∩ F (I) and n ⩾ 1. T is called I-asymptotically nonexpansive type mapping if lim sup_{n→∞} {sup{∥T^{n}x − T^{n}y∥ − ∥I^{n}x − I^{n}y∥}} ⩽ 0 for all x, y ∈ K .

T is called I-asymptotically quasi-nonexpansive type if F (T) ∩ F (I) ≠ ∅ and

(1.1)  lim sup_{n→∞}{sup{∥T^{n}x − p∥ − ∥I^{n}x − p∥}} ⩽ 0

for all x ∈ K and p ∈ F (T) ∩ F (I).

I is called asymptotically quasi-nonexpansive type if F (I) ≠ ∅ and

(1.2)  lim sup_{n→∞}{sup{∥I^{n}x − p∥ − ∥x − p∥}} ⩽ 0

for all x ∈ K and p ∈ F (I).

From the above definitions, it follows that if F (I) is nonempty, quasi-nonexpan-sive mappings, asymptotically nonexpansive mappings, asymptotically quasi-non-expansive mappings and asymptotically nonexpansive type mappings all are special cases of asymptotically quasi-nonexpansive type mappings.

Let {x_{n}} be of the Ishikawa iterative scheme [8] associated with T, x_{0} ∈ K,

y_{n} = (1 − β_{n})x_{n} + β_{n}T x_{n}
x_{n+1} = (1 − α_{n})x_{n} + α_{n}T y_{n}

for every n ∈ ℕ, where 0 ⩽ α_{n} , β_{n} ⩽ 1.

Let S, T : K → K be two mappings. In 2006, Lan [9] introduced the following iterative scheme with errors. The sequence x_{n} in K defined by

(1.3)  \begin{aligned}y_{n} & = (1 − β_{n})x_{n} + β_{n}T^{n}x_{n} + ψ_{n} \\ x_{n+1} & = (1 − α_{n})x_{n} + α_{n}S^{n}y_{n} + φ_{n} \end{aligned}

for every n ∈ ℕ, where 0 ⩽ {α_{n} }, {β_{n} } ⩽ 1 and {φ_{n} }, {ψ_{n} } are two sequences in K.

I-ASYMPTOTICALLY QUASI-NONEXPANSIVE TYPE MAPPING 241

The iterative approximation problems for nonexpansive mapping, asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mapping were studied Ghosh and Debnath [6], Goebel and Kirk [7], Liu [10, 11], Petryshyn and Williamson [13] in the settings of Hilbert spaces and uniformly convex Banach spaces. The strong and weak convergences of the sequence of Mann iterates to a fixed point of quasi-nonexpansive maps were studied by Petryshyn and Williamson [13]. Subsequently, the convergence of Ishikawa iterates of quasi-nonexpansive mappings in Banach spaces were discussed by Ghosh and Debnath [6]. The above results and some necessary and sufficient conditions for Ishikawa iterative sequences obtained to converge to a fixed point for asymptotically quasi-nonexpansive mappings were extended by Liu [10]. In [11], the results of Liu [10] were extended and some sufficient and necessary conditions for Ishikawa iterative sequences of asymptotically quasi-nonexpansive mappings with error member to converge to fixed points were proved. Recently, Temir and Gul [17] obtained the weakly almost convergence theorems for I-asymptotically quasi-nonexpansive mapping in a Hilbert space. In [20], Yao and Wang established the strong convergence of an iterative scheme with errors involving I-asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space. Temir [18], studied the convergence to common fixed point of Ishikawa iterative process of generalized I-asymptotically quasi-nonexpansive mappings to common fixed point in Banach space. In [1], the convergence theorems for Ishikawa iterative sequences with mixed errors of asymptotically quasi-nonexpansive type mappings in Banach spaces were studied.

2. Preliminaries and notations

We first recall the following definitions. A Banach space X is said to satisfy Opial’s condition [12] if, for each sequence {x_{n}} in X, the condition x_{n} ⇀ x implies

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

for all y ∈ X with y ≠ x. It is well known from [12] that all l_{r} spaces for 1 < r < ∞ have this property. However, the L_{r} space do not have unless r = 2.

In order to prove the main results of this paper, we need the following lemmas.

Lemma 2.1. [16] Let {a_{n}}, {b_{n}} be sequences of nonnegative real numbers satisfying the following conditions: ∀n ⩾ 1, a_{n+1} ⩽ a_{n} + b_{n}, where ∑ _{n=1}^{∞} b_{n} < ∞. Then lim_{n→∞} a_{n} exists.

Lemma 2.2. [15] Let K be a nonempty closed bounded convex subset of a uniformly convex Banach space X and {α_{n}} ⊆ [ϵ, 1 − ϵ] ⊂ (0, 1). Let {x_{n}} and {y_{n}} be two sequences in K such that lim sup_{n→∞} ∥x_{n}∥ ⩽ c, lim sup_{n→∞} ∥y_{n}∥ ⩽ c, and lim sup_{n→∞} ∥α_{n}x_{n} +(1−α_{n})y_{n}∥ = c for some c ⩾ 0. Then lim_{n→∞} ∥x_{n} − y_{n}∥ = 0 .

Lemma 2.3. [2] Let X be a uniformly convex Banach space, K a nonempty closed convex subset of X and T : K → K an asymptotically nonexpansive mapping with a sequence {k_{n}} ⊂ [1, ∞) and lim_{n→∞} k_{n} = 1. Then E − T is semi-closed (demi-closed) at zero, i.e., for each sequence {x_{n}} in K, if {x_{n}} converges weakly to q ∈ K and (E − T){x_{n}} converges strongly to 0, then (E − T)q = 0.

3. Convergence theorems for I-asymptotically quasi-nonexpansive type mapping

In this section, X is a Banach space and K is its nonempty subset. Let T, I : K → K be two mappings, where T is an I-asymptotically quasi-nonexpansive type mapping and I : K → K is an asymptotically quasi-nonexpansive type mapping. We study the strong and weak convergences of the sequence of Ishikawa iterates with mixed errors to a common fixed point of T and I.

Theorem 3.1. Let X be a Banach space, K its nonempty subset, and T, I : K → K two mappings. Let T be an I-asymptotically quasi-nonexpansive type and I be an asymptotically quasi-nonexpansive type in the Banach space satisfying

(3.1)  ∥T x − p∥ ⩽ L∥Ix − p∥

for all x ∈ K and p ∈ F (T) ∩ F (I), where L > 0 is a constant and

(3.2)  ∥Ix − p∥ ⩽ Γ∥x − p∥

for all x ∈ K and p ∈ F (I), where Γ > 0 is a constant. Write I : K → K instead of S : K → K in (1.3) and get

(3.3)  \begin{aligned}y_{n} & = (1 − β_{n})x_{n} + β_{n}T^{n}x_{n} + ψ_{n} \\ x_{n+1} & = (1 − α_{n})x_{n} + α_{n}I^{n}y_{n} + φ_{n} \end{aligned}

for every n ∈ ℕ, where 0 ⩽ {α_{n} }, {β_{n} } ⩽ 1 and {φ_{n} }, {ψ_{n} } be two sequences in K satisfying: (i) ∑ _{n=1}^{∞} α_{n} < ∞; (ii) {ψ_{n} } is bounded, φ_{n} = φ′_{n} + φn^{′′}, n ∈ ℕ and ∑ _{n=1}^{∞} ∥φ′_{n}∥ < ∞, ∥φ”_{n}∥ = o(α_{n}).

Then {x_{n}} converges strongly to a common fixed point of T and I in K iff

(3.4)  lim inf_{n→∞} d(x_{n}, F (T) ∩ F (I)) = 0.

Lemma 3.1. Suppose all conditions in Theorem 3.1 are satisfied; then for ε > 0, there exists a positive integer n_{0} and M > 0 such that

∥x_{n+1} − p∥ ⩽ ∥x_{n} − p∥ + α_{n}M + ∥φ′_{n}∥

for all p ∈ F (T) ∩ F (I), n ⩾ n_{0} and

∥x_{n+m} − p∥ ⩽ ∥x_{n} − p∥ + M ∑_{i=n}^{n+m−1} α_{i} + ∑_{i=n}^{n+m−1} ∥φ′_{i}∥,

for all p ∈ F (T)∩F (I), n ⩾ n_{0}, ∀m ⩾ 1, where M = sup_{n⩾0} {ε_{n} +∥ψ_{n}∥}+3ε ⩽ ∞, and ε_{n} is a sequence with ε_{n} > 0 and ε_{n} → 0 such that ∥φn^{′′}∥ = ε_{n}α_{n}.

Proof. For p ∈ F (T) ∩ F (I), from (3.3), we have

(3.5)  ∥x_{n+1} − p∥ = ∥(1 − α_{n})(x_{n} − p) + α_{n}(I^{n}y_{n} − p) + φ_{n}∥
⩽ (1 − α_{n})∥x_{n} − p∥ + α_{n}∥I^{n}y_{n} − p∥ + ∥φ_{n}∥
= (1 − α_{n})∥x_{n} − p∥ + α_{n}{∥I^{n}y_{n} − p∥ − ∥y_{n} − p∥}
+ α_{n}∥y_{n} − p∥ + ∥φ_{n}∥

Now we consider the second term 0n the right-hand side of (3.5). From (1.1) and (1.2), for any given ε > 0, there exists a positive integer n_{0} such that n ⩾ n_{0}, so we have

sup_{x∈K,p∈F (T)∩F (I)} {∥T^{n}x − p∥ − ∥I^{n}x − p∥} < ε,
sup_{x∈K,p∈F (I)} {∥I^{n}x − p∥ − ∥x − p∥} < ε.

Therefore, in particular, we have

(3.6)  {∥T^{n}x_{n} − p∥ − ∥I^{n}x_{n} − p∥} < ε,

for all p ∈ F (T) ∩ F (I) and ∀n ⩾ n_{0}.

(3.7)  {∥I^{n}y_{n} − p∥ − ∥y_{n} − p∥} < ε,

for all p ∈ F (I) and ∀n ⩾ n_{0}. From (3.7), we have

(3.8)  ∥x_{n+1} − p∥ ⩽ (1 − α_{n})∥x_{n} − p∥ + α_{n}ε + α_{n}∥y_{n} − p∥ + ∥φ_{n}∥

Consider the third term on the right-hand side of (3.8). From (3.6) and (3.7), we get

(3.9)  ∥y_{n} − p∥ = ∥(1 − β_{n})(x_{n} − p) + β_{n}(T^{n}x_{n} − p) + ψ_{n}∥
⩽ (1 − β_{n})∥x_{n} − p∥ + β_{n}{∥T^{n}x_{n} − p∥ − ∥I^{n}x_{n} − p∥}
+ β_{n}{∥I^{n}x_{n} − p∥ − ∥x_{n} − p∥} + β_{n}∥x_{n} − p∥ + ∥ψ_{n}∥
⩽ (1 − β_{n})∥x_{n} − p∥ + 2β_{n}ε + β_{n}∥x_{n} − p∥ + ∥ψ_{n}∥
= ∥x_{n} − p∥ + 2β_{n}ε + ∥ψ_{n}∥

Now consider the fourth term on the right side of (3.5); we have ∥φ_{n}∥ ⩽ ∥φ′_{n}∥ + ∥φ′′_{n}∥, ∀n ⩾ 0. Substituting (3.9) into (3.8), we have

∥x_{n+1} − p∥ ⩽ (1 − α_{n})∥x_{n} − p∥ + α_{n}ε + α_{n}{∥x_{n} − p∥ + 2β_{n}ε + ∥ψ_{n}∥} + ∥φ_{n}∥

⩽ (1 − α_{n})∥x_{n} − p∥ + α_{n}ε + α_{n}∥x_{n} − p∥

+ 2α_{n}β_{n}ε + α_{n}∥ψ_{n}∥ + ∥φ′_{n}∥ + ∥φn^{′′}∥

= ∥x_{n} − p∥ + α_{n}ε(1 + 2β_{n}) + α_{n}ε_{n} + α_{n}∥ψ_{n}∥ + ∥φ′_{n}∥

Taking M = sup_{n⩾0} {ε_{n} + ∥ψ_{n}∥} + 3ε we obtain
(3.10)  ∥x_{n+1} − p∥ ⩽ ∥x_{n} − p∥ + α_{n}M + ∥φ′_{n}∥

for all p ∈ F (T)∩F (I), n ⩾ n_{0}. Writing n+m−1 instead of n in inequality (3.10), for m ⩾ 1, we get

∥x_{n+m} − p∥ ⩽ ∥x_{n+m−1} − p∥ + α_{n+m−1} M + ∥φ′_{n+m−1} ∥

⩽ ∥x_{n+m−2} − p∥ + (α_{n+m−1} + α_{n+m−2} )M + ∥φ′_{n+m−2} ∥ + ∥φ′_{n+m−1} ∥

⋮

⩽ ∥x_{n} − p∥ + M ∑_{i=n}^{n+m−1} α_{i} + ∑_{i=n}^{n+m−1} ∥φ′_{i}∥

for all p ∈ F (T) ∩ F (I), n ⩾ n_{0}. Thus Lemma 3.1 is proved. □

Since {ψ_{n} } is bounded, φ_{n} = φ′_{n} + φn^{′′}, n ∈ ℕ and ∑ _{n=0}^{∞} ∥φ′_{n}∥ < ∞, ∥φn^{′′}∥ =

o(α_{n}), then we have ∑ _{n=0}^{∞} (Mα_{n} + ∥φ′_{n}∥) < ∞. From Lemma 2.1, we take {a_{n}} = {x_{n} − p} and {b_{n}} = Mα_{n} + ∥φ′_{n}∥. This implies that lim_{n→∞} ∥x_{n} − p∥ exists.

Proof of Theorem 3.1. We only prove the sufficiency of Theorem 3.1. Suppose that (3.4) is satisfied; then lim_{n→∞} d(x_{n}, F (T) ∩ F (I)) = 0.

First we show that {x_{n}} is a Cauchy sequence in K. For ε > 0 and n ⩾ n_{1} there exists n_{1} ⩾ n_{0} such that d(x_{n}, F (T)∩F (I)) < ε, ∑ _{n=n_{1}}^{∞} α_{n} < \frac{ε}{M} , ∑ _{n=n_{1}}^{∞} ∥φ′_{n}∥ < ε. By the definition of infimum and d(x_{n}, F (T) ∩ F (I)) < ε there exists p_{0} ∈ F (T) ∩ F (I) such that d(x_{n_{1}} , p) < 2ε. Furthermore, for n ⩾ n_{1} ⩾ n_{0} and ∀m ⩾ 1

∥x_{n+m} − x_{n}∥ ⩽ ∥x_{n+m} − p_{0}∥ + ∥x_{n} − p_{0}∥

⩽ ∥x_{n_{1}} − p_{0}∥ + M ∑_{i=n_{1}}^{n+m−1} α_{i} + ∑_{i=n_{1}}^{n+m−1} ∥φ′_{i}∥

+ ∥x_{n_{1}} − p_{0}∥ + M ∑_{i=n_{1}}^{n−1} α_{i} + ∑_{i=n_{1}}^{n−1} ∥φ′_{i}∥.

Then for n ⩾ n_{1} ⩾ n_{0} and ∀m ⩾ 1 we have ∥x_{n+m} − x_{n}∥ ⩽ 8ε. Since ε is arbitrary, then {x_{n}} is a Cauchy sequence in K. Since X is a Banach space, let {x_{n}} → p^{∗} as n → ∞. We prove that p^{∗} ∈ F (T) ∩ F (I). We have {x_{n}} → p^{∗} as n → ∞ and lim_{n→∞} d(x_{n}, F (T) ∩ F (I)) = 0, for ε > 0, there exists a positive integer n_{2} ⩾ n_{1} ⩾ n_{0} and n ⩾ n_{2} we have ∥x_{n} − p^{∗}∥ < ε, d(x_{n}, F (T) ∩ F (I)) < ε. Then there exists q ∈ F (T) ∩ F (I) such that d(x_{n_{2}} , q) < 2ε. Furthermore, for n ⩾ n_{2}

∥T^{n}p^{∗} − p^{∗}∥ ⩽ {∥T^{n}p^{∗} − q∥ − ∥p^{∗} − q∥} + 2∥p^{∗} − q∥

⩽ {∥T^{n}p^{∗} − q∥ − ∥I^{n}p^{∗} − q∥} + {∥I^{n}p^{∗} − q∥ − ∥p^{∗} − q∥} + 3∥p^{∗} − q∥

< 2ε + 3{3ε} = 11ε

Since T is I-asymptotically quasi nonexpansive type and I is asymptotically quasi nonexpansive type, this implies that {T^{n}p^{∗}} → p^{∗} as n → ∞. Furthermore,

∥T^{n}p^{∗} − T p^{∗}∥ ⩽ {∥T^{n}p^{∗} − q∥ − ∥p^{∗} − q∥} + ∥p^{∗} − q∥ + ∥T p^{∗} − q∥.

Then for n ⩾ n_{2} by (3.1), (3.2), (3.6) and (3.7) we have

∥T^{n}p^{∗} − T p^{∗}∥⩽ {∥T^{n}p^{∗} − q∥ − ∥I^{n}p^{∗} − q∥} + {∥I^{n}p^{∗} − q∥ − ∥p^{∗} − q∥}
+ 2∥p^{∗} − q∥ + L∥Ip^{∗} − q∥
⩽ 2ε + 2∥p^{∗} − q∥ + LΓ∥p^{∗} − q∥
⩽ 2ε + (2 + LΓ){∥x_{n_{2}} − p^{∗}∥ + ∥x_{n_{2}} − q∥}
< 2ε + (2 + LΓ)3ε < ε(8 + 3LΓ)

Since ε is arbitrary, {T^{n}p^{∗}} → T p^{∗} as n → ∞, implying T p^{∗} = p^{∗} ∈ F (T) ∩ F (I).

Further we apply for I : K → K asymptotically quasi nonexpansive type mapping. Then for n ⩾ n_{2} we have

∥I^{n}p^{∗} − p^{∗}∥ ⩽ {∥I^{n}p^{∗} − q∥ − ∥q − p^{∗}∥} + 2∥p^{∗} − q∥
⩽ ε + 2{∥x_{n_{2}} − p^{∗}∥ + ∥x_{n_{2}} − q∥} < ε + 2{ε + 2ε} = 7ε

This implies that {I^{n}p^{∗}} → p^{∗} as n → ∞. Furthermore,

∥I^{n}p^{∗} − Ip^{∗}∥ ⩽ {∥I^{n}p^{∗} − q∥ − ∥p^{∗} − q∥} + ∥p^{∗} − q∥ + ∥Ip^{∗} − q∥.

Then for n ⩾ n_{2} by (3.2) and (3.7) we have

∥I^{n}p^{∗} − Ip^{∗}∥ ⩽ {∥I^{n}p^{∗} − q∥ − ∥p^{∗} − q∥} + ∥p^{∗} − q∥ + Γ∥Ip^{∗} − q∥
⩽ ε + ∥p^{∗} − q∥ + Γ∥p^{∗} − q∥
⩽ ε + (1 + Γ){∥x_{n_{2}} − p^{∗}∥ + ∥x_{n_{2}} − q∥}
< ε + (1 + Γ)3ε < ε(4 + 3Γ).

Since ε is arbitrary, {I^{n}p^{∗}} → p^{∗} as n → ∞. Also

∥I^{n}p^{∗} − Ip^{∗}∥ ⩽ ∥I^{n}p^{∗} − q∥ + ∥Ip^{∗} − q∥ < 2ε

Since ε is arbitrary, {I^{n}p^{∗}} → Ip^{∗} as n → ∞. This shows that Ip^{∗} = p^{∗} ∈ F (I). From this we obtain p^{∗} ∈ F (T) ∩ F (I).

Thus {x_{n}} converges strongly to a common fixed point of T and I in K, subset of X Banach space. □

Now we establish the weak convergence theorem for Ishikawa iterates of Iasymptotically quasi-nonexpansive type mappings in Banach spaces. First, we prove the following lemma.

Lemma 3.2. Let X be a uniformly convex Banach space and K be a nonempty closed convex subset of X. Let T, I and {x_{n}} be the same as in Lemma 3.1. If F = F (T) ∩ F (I) ≠ ∅, then lim_{n→∞} ∥T x_{n} − x_{n}∥ = lim_{n→∞} ∥Ix_{n} − x_{n}∥ = 0.

Proof. By Lemma 3.1, for any p ∈ F (T) ∩ F (I), lim_{n→∞} ∥x_{n} − p∥ exists. Let lim_{n→∞} ∥x_{n} − p∥ = c. If c = 0, then the proof is completed.

Now suppose c > 0. From (3.9), we have ∥y_{n} − p∥ ⩽ ∥x_{n} − p∥ + 2β_{n}ε + ∥ψ_{n}∥. Taking lim sup on both sides in the above inequality,

(3.11)  lim sup_{n→∞}∥y_{n} − p∥ ⩽ c.

Since I is asymptotically nonexpansive type self-mappings on K, from (3.7), which is on taking lim sup_{n→∞} and using (3.11), then we get lim sup_{n→∞} ∥I^{n}y_{n} − p∥ ⩽ c. Further, lim_{n→∞} ∥x_{n+1} − p∥ = c means that

lim_{n→∞} ∥α_{n}I^{n}y_{n} + (1 − α_{n})x_{n} − p∥ = c
lim_{n→∞} (1 − α_{n})∥x_{n} − p∥ + α_{n}∥I^{n}y_{n} − p∥ = c.

It follows from Lemma 2.2

(3.12)  lim_{n→∞} ∥I^{n}y_{n} − x_{n}∥ = 0.

Further,

lim_{n→∞} ∥α_{n}(T^{n}x_{n} − p) + (1 − α_{n})(x_{n} − p)∥ = lim_{n→∞} ∥y_{n} − p∥ = c.

By Lemma 2.2, we have

(3.13)  lim_{n→∞} ∥T^{n}x_{n} − x_{n}∥ = 0.

From (3.12) and (3.13), we have

(3.14)  lim_{n→∞} ∥I^{n}x_{n} − x_{n}∥ = 0.

Using (3.1), (3.2), (3.3), (3.13) and (3.14), it is easy to show that

(3.15)  lim_{n→∞} ∥T x_{n} − x_{n}∥ = 0
(3.16)  lim_{n→∞} ∥Ix_{n} − x_{n}∥ = 0.

Then the proof is completed. □

Theorem 3.2. Let X be a uniformly convex Banach space which satisfies Opial’s condition, K be a nonempty closed convex subset of X. Let T, I and {x_{n}} be the same as in Lemma 3.1. If F (T) ∩ F (I) ≠ ∅, the mappings E − T and E − I are semi-closed at zero, then {x_{n}} converges weakly to a common fixed point of T and I.

Proof. By assumption, F (T) ∩ F (I) is nonempty. Take p ∈ F (T) ∩ F (I). It follows from Lemma 3.1 that the limit lim_{n→∞} ∥x_{n} −p∥ exists. Therefore, {x_{n} −p} is a bounded sequence in X. Since X is a uniformly convex Banach space and K is a nonempty closed convex subset of X, then K is weakly compact. This implies that there exists a subsequence {x_{n_{k}} } of {x_{n}} such that {x_{n_{k}} } converges to a point p ∈ w({x_{n}}), where w({x_{n}}) denotes the weak limit set of {x_{n}}, which shows that w({x_{n}}) is nonempty. For any p ∈ w({x_{n}}), there exists a subsequence {x_{n_{k}} } of {x_{n}} such that {x_{n_{k}} } → p weakly. Hence, it follows from (3.15) and (3.16) in Lemma 3.2 that T p = p and Ip = p. By Opial’s condition, {x_{n}} has only one weak limit point, i.e., {x_{n}} converges weakly to a common fixed point of T and I. □

4. Convergence for nonself I-asymptotically quasi-nonexpansive type mappings

In this section, the convergence of the Ishikawa iterative sequences to common fixed point for nonself I-asymptotically quasi-nonexpansive type mapppings is obtained in Banach spaces.

A subset K of X is called a retract of X if there exists a continuous map P : X → K such that P x = x for all x ∈ K. A map P : X → K is called a retraction if P^{2} = P. In particular, a subset K is called a nonexpansive retract of X if there exists a nonexpansive retraction P : X → K such that P x = x for all x ∈ K.

Next, we introduce the following concepts for nonself mappings. Let X be a real Banach space. A subset K of X be nonempty nonexpansive retraction of X and P be nonexpansive retraction from X onto K. A nonself mapping T : K → X is called asymptotically nonexpansive if there exists a sequence {υ_{n}} ⊂ [1, ∞) with lim_{n→∞} υ_{n} = 1 such that

∥T (P T)^{n−1} x − T (P T)^{n−1} y∥ ⩽ υ_{n}∥x − y∥

for all x, y ∈ K and n ⩾ 1. T is called uniformly L-Lipschitzian if there exists a constant L > 0 such that

∥T (P T)^{n−1} x − T (P T)^{n−1} y∥ ⩽ L∥x − y∥

for all x, y ∈ K and n ⩾ 1. From the above definition, it is obvious that nonself asymptotically nonexpansive mappings is uniformly L-Lipschitzian.

Let I : K → X be a nonself asymptotically quasi-nonexpansive type mappings and T : K → X be a nonself I-asymptotically quasi-nonexpansive type mappings with F (T) ∩ F (I) = {x ∈ K : T x = x = Ix} ≠ ∅. A mapping T : K → X is called Λ-Lipschitzian if there exists constant Λ > 0 such that

∥T (P T)^{n−1} x − T (P T)^{n−1} y∥ ⩽ Λ∥I(P I)^{n−1} x − I(P I)^{n−1} y∥

for all x, y ∈ K and n ⩾ 1.

Iterative techniques for converging fixed points of nonexpansive non-self mappings have been studied by many authors (see, for example, [3, 19, 14]). The concept of nonself asymptotically nonexpansive mappings was introduced in [3] as a generalization of asymptotically nonexpansive self-mappings and some strong and weak convergence theorems for such mappings were obtained. The sequence {x_{n}}_{n⩾1} generated as follows: x_{1} ∈ K,

y_{n} = P (α_{n}T (P T)^{n−1} x_{n} + β_{n}x_{n}),
x_{n+1} = P (α′_{n}T (P T)^{n−1} y_{n} + β′_{n}x_{n}), ∀n ⩾ 1,

where {α_{n}}, {β_{n}}, {α′_{n}}, {β′_{n}} ∈ (0, 1).

Let T : K → X be a nonself I-asymptotically quasi-nonexpansive type mapping and I : K → X be a nonself asymptotically quasi-nonexpansive type mapping.

Now we define an {x_{n}}_{n⩾1} sequence as follows:

(4.1)  \begin{aligned}y_{n} & = P (α_{n}T (P T)^{n−1} x_{n} + β_{n}x_{n} + γ_{n}ψ_{n}), \\ x_{n+1} & = P (α′_{n}I(P I)^{n−1} y_{n} + β′_{n}x_{n} + γ′_{n}φ_{n}), ∀n ⩾ 1, \end{aligned}

where {α_{n}}, {β_{n}}, {γ_{n}}, {α′_{n}}, {β′_{n}}, {γ′_{n}} are sequences in (0, 1) with α_{n} +β_{n} +γ_{n} = 1 = α′_{n} + β′_{n} + γ′_{n} and {ψ_{n}}, {φ_{n}} are bounded sequences in K.

lim sup_{n→∞}( sup_{x∈X,p∈F (T)∩F (I)} {∥T (P T)^{n−1} x − p∥ − ∥I(P I)^{n−1} x − p∥}) ⩽ 0.

Observe that

lim sup_{n→∞}( sup_{x∈X,p∈F (T)∩F (I)} {∥T (P T)^{n−1} x − p∥ − ∥I(P I)^{n−1} x − p∥})

× lim sup_{n→∞}( sup_{x∈X,p∈F (T)∩F (I)} {∥T (P T)^{n−1} x − p∥ + ∥I(P I)^{n−1} x − p∥})

= lim sup_{n→∞}( sup_{x∈X,p∈F (T)∩F (I)} {∥T (P T)^{n−1} x − p∥^{2} − ∥I(P I)^{n−1} x − p∥^{2}}) ⩽ 0.

Therefore we have

lim sup_{n→∞}( sup_{x∈X,p∈F (T)∩F (I)} {∥T (P T)^{n−1} x − p∥ − ∥I(P I)^{n−1} x − p∥}) ⩽ 0.

This implies that for any given ε > 0, there exists a positive integer n_{0} such that for n ⩾ n_{0} we have

( sup_{x∈X,p∈F (T)∩F (I)} {∥T (P T)^{n−1} x − p∥ − ∥I(P I)^{n−1} x − p∥}) ⩽ 0.

Theorem 4.1. Let X be a Banach space and K be a nonempty subset of the Banach space. Let T, I : K → X be two nonself mappings. Let T be a nonself I-asymptotically quasi-nonexpansive type and I be a nonself asymptotically quasinonexpansive type in Banach space with F (T) ∩ F (I) ≠ ∅. Let the sequence {x_{n}} be defined by (4.1) and for every n ∈ ℕ, where {α_{n}}, {β_{n}}, {γ_{n}}{α′_{n}}, {β′_{n}}, {γ′_{n}} are sequences in (0, 1) with α_{n} + β_{n} + γ_{n} = 1 = α′_{n} + β′_{n} + γ′_{n}, ∑ _{i=1}^{∞} γ_{n} < ∞, ∑ _{i=1}^{∞} γ′_{n} < ∞, and {ψ_{n}}, {φ_{n}} are bounded sequences in K.

Then {x_{n}} converges strongly to a common fixed point of T and I in K iff

(4.2)  lim inf_{n→∞} d(x_{n}, F (T) ∩ F (I)) = 0.

Proof. The necessity of condition (4.2) is obvious. Next we prove the sufficiency of condition (4.2). Let the sequence {x_{n}} be defined by (4.1). Let p ∈ F (T) ∩ F (I), by boundedness of the sequences {ψ_{n}}, {φ_{n}}, so we can put

M = max{sup_{n⩾1}∥ψ_{n} − p∥, sup_{n⩾1}∥φ_{n} − p∥}.

For any given ε > 0, there exists a positive integer n_{0} such that n ⩾ n_{0}

sup_{x∈K,p∈F (T)∩F (I)} {∥T (P T)^{n−1} x − p∥ − ∥I(P I)^{n−1} x − p∥} < ε.

sup_{x∈K,p∈F (I)} {∥I(P I)^{n−1} x − p∥ − ∥x − p∥} < ε.

Therefore, in particular, we have

(4.3)  {∥T (P T)^{n−1} x_{n} − p∥ − ∥I(P I)^{n−1} x_{n} − p∥} < ε,

for all p ∈ F (T) ∩ F (I) and ∀n ⩾ n_{0}.

(4.4)  {∥I(P I)^{n−1} y_{n} − p∥ − ∥y_{n} − p∥} < ε,

for all p ∈ F (I) and ∀n ⩾ n_{0}. Thus for each n ⩾ 1 and for any p ∈ F (T) ∩ F (I), using (4.1), (4.3) and (4.4), we have

(4.5)  ∥x_{n+1} − p∥ = ∥P (α′_{n}x_{n} + β′_{n}I(P I)^{n−1} y_{n} + γ′_{n}φ_{n} − p)∥
⩽ α′_{n}∥x_{n} − p∥ + β′_{n}∥I(P I)^{n−1} y_{n} − p∥ + γ′_{n}∥φ_{n} − p∥
= α′_{n}∥x_{n} − p∥ + β′_{n}{∥I(P I)^{n−1} y_{n} − p∥ − ∥y_{n} − p∥}
+ β′_{n}∥y_{n} − p∥ + γ′_{n}∥φ_{n} − p∥
⩽ α′_{n}∥x_{n} − p∥ + β′_{n}{ε} + β′_{n}∥y_{n} − p∥ + γ′_{n}M

and

(4.6) ∥y_{n} − p∥ = ∥P (α_{n}x_{n} + β_{n}T (P T)^{n−1} x_{n} + γ_{n}ψ_{n} − p)∥
⩽ α_{n}∥x_{n} − p∥ + β_{n}∥T (P T)^{n−1} x_{n} − p∥ + γ_{n}∥ψ_{n} − p∥
⩽ α_{n}∥x_{n} − p∥ + β_{n}{∥T (P T)^{n−1} x_{n} − p∥ − ∥I(P I)^{n−1} x_{n} − p∥}
+ β_{n}{∥I(P I)^{n−1} x_{n} − p∥ − ∥x_{n} − p∥} + β_{n}∥x_{n} − p∥ + γ_{n}M
⩽ α_{n}∥x_{n} − p∥ + 2β_{n}{ε} + β_{n}∥x_{n} − p∥ + γ_{n}M
⩽ (α_{n} + β_{n})∥x_{n} − p∥ + 2β_{n}ε + γ_{n}M
⩽ (1 − γ_{n})∥x_{n} − p∥ + 2β_{n}ε + γ_{n}M
⩽ ∥x_{n} − p∥ + D_{n}

where D_{n} = 2β_{n}ε + γ_{n}M. Then ∑ _{n=1}^{∞} D_{n} < ∞ since ∑ _{n=1}^{∞} γ_{n} < ∞.

Substituting (4.6) into (4.5), we have

(4.7)  ∥x_{n+1} − p∥ ⩽ α′_{n}∥x_{n} − p∥ + β′_{n}ε + β′_{n}(∥x_{n} − p∥ + D_{n}) + γ′_{n}M
⩽ (α′_{n} + β′_{n})∥x_{n} − p∥ + β′_{n}(ε + D_{n}) + γ′_{n}M
⩽ (1 − γ′_{n})∥x_{n} − p∥ + G_{n}
⩽ ∥x_{n} − p∥ + G_{n}

where G_{n} = β′_{n}(ε + D_{n}) + γ′_{n}M. Then ∑ _{n=1}^{∞} G_{n} < ∞ since ∑ _{n=1}^{∞} γ′_{n} < ∞ and ∑ _{n=1}^{∞} D_{n} < ∞ .

It follows from (4.7) that d(x_{n+1} , F (T) ∩ F (I)) ⩽ d(x_{n}, F (T) ∩ F (I)) + G_{n}. By Lemma 2.1, we can get that lim_{n→∞} d(x_{n}, F (T)∩F (I)) exists. By condition

lim inf_{n→∞} d(x_{n}, F (T) ∩ F (I)) = 0, we have

(4.8)  lim_{n→∞} d(x_{n}, F (T) ∩ F (I)) = 0.

Next we prove that {x_{n}} is a Cauchy sequence in X. In fact, for any n ⩾ n_{0}, any m ⩾ n_{1} and any p ∈ F (T) ∩ F (I) we have

(4.9)  ∥x_{n+m} − p∥ ⩽ ∥x_{n+m−1} − p∥ + G_{n+m−1}

⩽ ∥x_{n+m−2} − p∥ + G_{n+m−1} + G_{n+m−2}

⩽ … ⩽ ∥x_{n} − p∥ + ∑_{k=n}^{∞} G_{k}.

So by (4.9), we have

(4.10)  ∥x_{n+m} − x_{n}∥ ⩽ ∥x_{n+m} − p∥ + ∥x_{n} − p∥ ⩽ 2∥x_{n} − p∥ + ∑_{k=n}^{∞} G_{k}.

By the arbitrariness of p ∈ F (T) ∩ F (I) and (4.10), we have

∥x_{n+m} − p∥ ⩽ 2d(x_{n}, F (T) ∩ F (I)) + ∑_{k=n}^{∞} G_{k} ∀n ⩾ n_{0}.

For any given ε > 0, there exists a positive integer n_{1} ⩾ n_{0}, such that for any n ⩾ n_{1}, d(x_{n}, F (T) ∩ F (I)) < \frac{ε}{4} and ∑ _{k=n}^{∞} G_{k} < \frac{ε}{2}, we have ∥x_{n+m} − x_{n}∥ < ε, and so for any m ⩾ 1

lim_{n→∞} ∥x_{n+m} − x_{n}∥ = 0.

This shows that {x_{n}} is a Cauchy sequence in X. Since X is complete, there exists a p^{∗} ∈ X such that x_{n} → p^{∗} as n → ∞.

Finally, by the routine method, we have to prove that p^{∗} ∈ F (T) ∩ F (I). By contradiction, we assume that p^{∗} is not in F (T) ∩ F (I). Since F (T) ∩ F (I) is a closed set, d(p^{∗}, F (T) ∩ F (I)) > 0. Hence for all p ∈ F (T) ∩ F (I), we have

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

This implies that

(4.11)  d(p^{∗}, F (T) ∩ F (I)) ⩽ ∥x_{n} − p^{∗}∥ + d(x_{n}, F (T) ∩ F (I)).

Letting n → ∞ in (4.11) and noting (4.8), we have d(p^{∗}, F (T) ∩ F (I)) ⩽ 0. This is a contradiction. Hence p^{∗} ∈ F (T) ∩ F (I). This completes the proof of Theorem 4.1. □

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Department of Mathematics (Received 14 04 2012) Art and Science Faculty (Revised 16 11 2012) Harran University Sanliurfa Turkey

temirseyit@harran.edu.tr