Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 2(2011), Pages 246-251.
SOME CONVERGENCE RESULTS FOR SEQUENCES OF OPERATORS IN BANACH SPACES
(COMMUNICATED BY EBERHARD MALKOWSKY)
MEMUDU OLAPOSI OLATINWO
Abstract. In this paper, we establish some fixed point theorems in connection with sequences of operators in the Banach space setting for Mann and Ishikawa iterative processes. Our results extend some of the results of Berinde, Bonsall, Nadler and Rus from complete metric space to the Banach space setting.
1. Introduction
In this paper, we establish some fixed point theorems in connection with se-quences of operators in the Banach space settings for Mann and Ishikawa iterative processes. Our results extend some of the results of Berinde [1, 2], Bonsall [3], Nadler [6] and Rus [8, 9] from complete metric space to the Banach space setting. Let (𝐸,∣∣.∣∣) be a Banach space and𝑇 :𝐸 →𝐸 a selfmap of𝐸. Suppose that 𝐹𝑇 ={𝑝∈𝐸 ∣𝑇 𝑝=𝑝}is the set of fixed points of 𝑇.
In the Banach space setting, we state the following well-known iterative processes which have been employed to approximate the fixed points of various operators over the years:
Define the sequence{𝑥𝑛}∞𝑛=0 by
𝑥𝑛+1= (1−𝜆)𝑥𝑛+𝜆𝑇 𝑥𝑛, 𝑛= 0,1,⋅ ⋅ ⋅, 𝑥0∈𝐸, (1)
where𝜆∈[0,1].Then, Eqn, (1) is called the Schaefer iterative process (see Schaefer [10]).
For𝑥0∈𝐸,the sequence{𝑥𝑛}∞𝑛=0 defined by
𝑥𝑛+1= (1−𝛼𝑛)𝑥𝑛+𝛼𝑛𝑇 𝑥𝑛, 𝑛= 0,1,⋅ ⋅ ⋅, (2)
where{𝛼𝑛} ∞
𝑛=0⊂[0,1],is called the Mann iteration process (see Mann [5]). For𝑥0∈𝐸,the sequence{𝑥𝑛}
∞
𝑛=0 defined by 𝑥𝑛+1= (1−𝛼𝑛)𝑥𝑛+𝛼𝑛𝑇 𝑧𝑛
𝑧𝑛= (1−𝛽𝑛)𝑥𝑛+𝛽𝑛𝑇 𝑥𝑛
}
, 𝑛= 0,1,⋅ ⋅ ⋅ , (3)
2000Mathematics Subject Classification. 47H06, 54H25.
Key words and phrases. some fixed point theorems; sequences of operators; Banach space setting.
c
⃝2011 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted February 10, 2011. Published March 24, 2011.
where{𝛼𝑛}∞𝑛=0 and {𝛽n} ∞
n=0are sequences in [0,1],is called the Ishikawa iteration process (see Ishikawa [4]).
Remark 1.1: If𝜆= 1 in (1), or,𝛼𝑛 = 1 in (2), then we obtain
𝑥𝑛+1=𝑇 𝑥𝑛, 𝑛= 0,1,⋅ ⋅ ⋅, 𝑥0∈𝐸, (4)
which is called the Picard iteration. See [1, 2, 8, 9] for Picard iteration.
Definition 1.1 [1, 2, 8]: (a) A function 𝜓: IR+ → IR+ is called a comparison function if it satisfies the following conditions:
(i)𝜓is monotone increasing; (ii) lim
𝑛→∞𝜓
𝑛(𝑡))→0, ∀𝑡≥0.
(b) A comparison function satisfying 𝑡−𝜓(𝑡) → ∞ as t → ∞ is called a strict comparison function.
Remark 1.2: Every comparison function satisfies𝜓(0) = 0 and𝜓(𝑡)< 𝑡, ∀𝑡∈IR+. We shall employ the following contractive conditons:
(i) For a selfmapping 𝑇: 𝐸 → 𝐸, there exist real numbers 𝐿 ≥0 and 𝑎 ∈[0,1), such that
∣∣𝑇 𝑥−𝑇 𝑦∣∣ ≤ 𝜙(∣∣𝑥−𝑇 𝑥∣∣) +𝑎∣∣𝑥−𝑦∣∣
1 +𝐿∣∣𝑥−𝑇 𝑥∣∣ , ∀𝑥, 𝑦 ∈𝐸, (5)
where𝜙: IR+→IR+ is a monotone increasing function such that𝜙(0) = 0.
(ii) For a selfmapping𝑇:𝐸→𝐸,there exist a strict comparison function𝜓: IR+→ IR+ and a monotone increasing function𝜙: IR+→IR+,with𝜙(0) = 0,such that
∣∣𝑇 𝑥−𝑇 𝑦∣∣ ≤𝜙(∣∣𝑥−𝑇 𝑥∣∣) +𝜓(∣∣𝑥−𝑦∣∣), ∀ 𝑥, 𝑦∈𝐸. (6)
2. Main Results
Theorem 2.1. Let(𝐸,∣∣.∣∣)be a Banach space and{𝑇𝑛}∞𝑛=0a sequence of operators 𝑇𝑛 :𝐸 → 𝐸 such that 𝐹𝑇𝑛 ={𝑥∗𝑛} for each 𝑛 ∈IN. For 𝑥0 ∈ 𝐸, let {𝑥𝑛}∞𝑛=0 be
the Mann iterative process defined by (2), 𝛼𝑛 ∈ [0,1]. Suppose that the sequence
{𝑇𝑛}∞𝑛=0 converges uniformly to a mapping 𝑇 : 𝐸 →𝐸 satisfying (5), with 𝐹𝑇 =
{𝑥∗}, where𝜙:IR+→IR+ is a monotone increasing function such that
𝜙(0) = 0.
Then, 𝑥∗
𝑛→𝑥∗ as n→ ∞.
Proof. Let 𝜖 >0 and choose a natural number 𝑁 such that for 𝑛 ≥ 𝑁,we have ∣∣𝑇𝑛𝑥−𝑇 𝑥∣∣<(1−𝑎)𝜖, for all x∈E. Then, for𝑛≥𝑁 we have
∣∣𝑥∗
𝑛−𝑥∗∣∣ ≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛∣∣𝑇 𝑥∗−𝑇𝑛𝑥∗𝑛∣∣
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛[∣∣𝑇 𝑥∗−𝑇 𝑥∗𝑛∣∣+∣∣𝑇 𝑥𝑛∗−𝑇𝑛𝑥∗𝑛∣∣]
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛[𝜑
(∣∣𝑥∗−𝑇 𝑥∗∣∣)+𝑎∣∣𝑥∗−𝑥∗
𝑛∣∣
1+𝐿∣∣𝑥∗−𝑇 𝑥∗∣∣ +∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣]
<(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝑎𝛼𝑛∣∣𝑥∗−𝑥∗𝑛∣∣+𝛼𝑛(1−𝑎)𝜖,
from which we have∣∣𝑥∗
𝑛−𝑥∗∣∣< 𝜖,
Also, since𝜖 >0 is arbitrary, then 𝑥∗𝑛→𝑥∗ as n→ ∞.
□
Theorem 2.2. Let(𝐸,∣∣.∣∣)be a Banach space and{𝑇𝑛}∞𝑛=0a sequence of operators 𝑇𝑛 :𝐸 → 𝐸 such that 𝐹𝑇𝑛 ={𝑥
∗
𝑛} for each 𝑛 ∈IN. For 𝑥0 ∈ 𝐸, let {𝑥𝑛}∞𝑛=0 be
the Mann iterative process defined by (2), 𝛼𝑛 ∈ [0,1]. Suppose that the sequence
{𝑇𝑛}∞𝑛=0 converges uniformly to a mapping 𝑇 : 𝐸 →𝐸 satisfying (6), with 𝐹𝑇 =
{𝑥∗}, where𝜙:IR+
→IR+ is a monotone increasing function such that 𝜙(0) = 0
and𝜓:IR+→IR+ is a strict comparison function. Then, 𝑥∗𝑛 →𝑥∗ as n→ ∞.
Proof. Let 𝜖 >0 and choose a natural number 𝑁 such that for 𝑛 ≥ 𝑁,we have ∣∣𝑇𝑛𝑥−𝑇 𝑥∣∣< 𝜖, for all x∈E.Then, for𝑛≥𝑁 we have
∣∣𝑥∗
𝑛−𝑥∗∣∣ ≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛∣∣𝑇 𝑥∗−𝑇𝑛𝑥∗𝑛∣∣
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛[∣∣𝑇 𝑥∗−𝑇 𝑥∗𝑛∣∣+∣∣𝑇 𝑥𝑛∗−𝑇𝑛𝑥∗𝑛∣∣]
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛[𝜑(∣∣𝑥∗−𝑇 𝑥∗∣∣) +𝜓(∣∣𝑥∗−𝑥∗𝑛∣∣) +∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣]
<(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛𝜓(∣∣𝑥∗−𝑥∗𝑛∣∣) +𝛼𝑛𝜖,
from which we have∣∣𝑥∗
𝑛−𝑥∗∣∣ −𝜓(∣∣𝑥∗−𝑥∗𝑛∣∣)< 𝜖,
leading to∣∣𝑥∗
𝑛−𝑥∗∣∣< 𝜖, ∀ 𝑛≥𝑁,
since𝜓 is a strict comparison function. Also, since 𝜖 >0 is arbitrary, then𝑥∗ 𝑛 →
𝑥∗as n→ ∞.
□
Theorem 2.3. Let (𝐸, ∣∣.∣∣)be a Banach space and{𝑇𝑛} ∞
𝑛=0 a sequence of
opera-tors 𝑇𝑛 :𝐸→𝐸 such that 𝐹𝑇𝑛={𝑥 ∗
𝑛},for each 𝑛∈IN. For𝑥0 ∈𝐸, let{𝑥𝑛} ∞ 𝑛=0
be the Ishikawa iterative process defined by (3), 𝛼𝑛, 𝛽𝑛 ∈ [0,1]. Suppose that the
sequence {𝑇𝑛} ∞
𝑛=0 converges uniformly to a mapping 𝑇 : 𝐸 → 𝐸 satisfying (6),
with 𝐹𝑇 ={𝑥∗}, where 𝜓 : IR+ →IR+ is a sublinear, strict comparison function
and 𝜙: IR+ → IR+ is a monotone increasing function such that 𝜙(0) = 0. Then,
𝑥∗𝑛→𝑥∗ as n→ ∞.
Proof. Let 𝜖 >0 and choose a natural number 𝑁 such that for 𝑛 ≥ 𝑁,we have ∣∣𝑇𝑛𝑥−𝑇 𝑥∣∣<2𝛽𝑛𝜖 , 𝛽𝑛 >0, for all x∈E and∣∣𝑇𝑛𝑧−𝑇 𝑧∣∣< 𝜖2, for all z∈E,
𝑧∗= (1−𝛽𝑛)𝑥∗+𝛽𝑛𝑇 𝑥∗, 𝑧𝑛∗ = (1−𝛽𝑛)𝑥∗𝑛+𝛽𝑛𝑇𝑛𝑥∗𝑛.
Therefore, we have by using (6) and the fact that𝜓(𝑡)< 𝑡 ∀𝑡∈IR+ that
∣∣𝑥∗
𝑛−𝑥∗∣∣ ≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛[∣∣𝑇 𝑧∗−𝑇 𝑧𝑛∗∣∣+∣∣𝑇 𝑧∗𝑛−𝑇𝑛𝑧𝑛∗∣∣]
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛[𝜑(∣∣𝑧∗−𝑇 𝑧∗∣∣) +𝜓(∣∣𝑧∗−𝑧∗𝑛∣∣) +∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣]
= (1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛𝜓(∣∣𝑧𝑛∗−𝑧∗∣∣) +𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧∗𝑛∣∣
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛(1−𝛽𝑛)𝜓(∣∣𝑥∗−𝑥∗𝑛∣∣)
+𝛼𝑛𝛽𝑛𝜓2(∣∣𝑥∗−𝑥∗𝑛∣∣) +𝛼𝑛𝛽𝑛𝜓(∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣) +𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧𝑛∗∣∣
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛(1−𝛽𝑛)𝜓(∣∣𝑥∗−𝑥∗𝑛∣∣)
+𝛼𝑛𝛽𝑛𝜓(∣∣𝑥∗−𝑥∗𝑛∣∣) +𝛼𝑛𝛽𝑛𝜓(∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣ ) +𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧∗𝑛∣∣
= (1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛𝜓(∣∣𝑥∗𝑛−𝑥∗∣∣) +𝛼𝑛𝛽𝑛𝜓(∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣)
+𝛼𝑛∣∣𝑇 𝑧∗𝑛−𝑇𝑛𝑧𝑛∗∣∣
from which we have
𝛼𝑛[∣∣𝑥∗𝑛−𝑥∗∣∣ −𝜓(∣∣𝑥∗𝑛−𝑥∗∣∣) ]≤𝛼𝑛𝛽𝑛𝜓(∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣) +𝛼𝑛∣∣𝑇 𝑧∗𝑛−𝑇𝑛𝑧∗𝑛∣∣,
so that
∣∣𝑥∗𝑛−𝑥∗∣∣ −𝜓(∣∣𝑥∗𝑛−𝑥∗∣∣) ≤𝛽𝑛𝜓(∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣) +∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧∗𝑛∣∣
< 𝛽𝑛∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣+∣∣𝑇 𝑧∗𝑛−𝑇𝑛𝑧∗𝑛∣∣
𝜓being a strict comparison function leads to∣∣𝑥∗
𝑛−𝑥∗∣∣< 𝜖, ∀𝑛≥𝑁.Since𝜖 >0
is arbitrary, then𝑥∗
𝑛→𝑥∗ as n→ ∞.
□
Theorem 2.4. Let (𝐸, ∣∣.∣∣)be a Banach space and{𝑇𝑛}∞𝑛=0 a sequence of
opera-tors 𝑇𝑛 :𝐸→𝐸 such that 𝐹𝑇𝑛={𝑥∗𝑛},for each 𝑛∈IN. For𝑥0 ∈𝐸, let{𝑥𝑛}∞𝑛=0
be the Ishikawa iterative process defined by (3), 𝛼𝑛, 𝛽𝑛 ∈ [0,1]. Suppose that the
sequence{𝑇𝑛}∞𝑛=0converges uniformly to a mapping 𝑇 :𝐸→𝐸 satisfying (5) with 𝐹𝑇 = {𝑥∗}, where 𝜙 : IR+ → IR+ is a monotone increasing function such that
𝜙(0) = 0. Then,𝑥∗𝑛→𝑥∗ as n→ ∞.
Proof. Let 𝜖 >0 and choose a natural number 𝑁 such that for 𝑛 ≥ 𝑁,we have ∣∣𝑇𝑛𝑥−𝑇 𝑥∣∣ < (1−𝑎2)(1+𝑎𝛽𝑛𝑎𝛽𝑛)𝜖, 𝑎 > 0, 𝛽𝑛 > 0, for all x ∈ E and ∣∣𝑇𝑛𝑧−𝑇 𝑧∣∣ <
(1−𝑎)(1+𝑎𝛽𝑛)
2 𝜖, for all z∈E, 𝑧∗= (1−𝛽
𝑛)𝑥∗+𝛽𝑛𝑇 𝑥∗, 𝑧𝑛∗ = (1−𝛽𝑛)𝑥∗𝑛+𝛽𝑛𝑇𝑛𝑥∗𝑛.
Therefore, we have by using (5) that
∣∣𝑥∗
𝑛−𝑥∗∣∣ ≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛∣∣𝑇 𝑧∗−𝑇𝑛𝑧∗𝑛∣∣
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛[∣∣𝑇 𝑧∗−𝑇 𝑧𝑛∗∣∣+∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧𝑛∗∣∣]
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛[𝜑(∣∣𝑧∗−𝑇 𝑧∗∣∣) +𝑎∣∣𝑧∗−𝑧𝑛∗∣∣+∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣]
= (1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝑎𝛼𝑛∣∣𝑧𝑛∗−𝑧∗∣∣+𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧∗𝑛∣∣
≤(1−𝛼𝑛+𝑎𝛼𝑛−𝑎𝛼𝑛𝛽𝑛)∣∣𝑥∗−𝑥∗𝑛∣∣+𝑎𝛼𝑛𝛽𝑛[𝜑(∣∣𝑥∗−𝑇 𝑥∗∣∣)
+𝑎∣∣𝑥∗−𝑥∗
𝑛∣∣+∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣] +𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧𝑛∗∣∣
= (1−𝛼𝑛+𝑎𝛼𝑛−𝑎𝛼𝑛𝛽𝑛+𝑎2𝛼𝑛𝛽𝑛)∣∣𝑥∗−𝑥∗𝑛∣∣+𝑎𝛼𝑛𝛽𝑛∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣
+𝛼𝑛∣∣𝑇 𝑧∗𝑛−𝑇𝑛𝑧𝑛∗∣∣,
from which we have
𝛼𝑛(1−𝑎)(1 +𝑎𝛽𝑛)∣∣𝑥∗𝑛−𝑥 ∗∣∣ ≤
𝑎𝛼𝑛𝛽𝑛∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣+𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧∗𝑛∣∣,
leading to
∣∣𝑥∗
𝑛−𝑥∗∣∣ ≤
𝑎𝛽𝑛
(1−𝑎)(1+𝑎𝛽𝑛)∣∣𝑇 𝑥 ∗
𝑛−𝑇𝑛𝑥∗𝑛∣∣+
1
(1−𝑎)(1+𝑎𝛽𝑛)∣∣𝑇 𝑧 ∗
𝑛−𝑇𝑛𝑧∗𝑛∣∣,
<(1−𝑎)(1+𝑎𝛽𝑛𝑎𝛽𝑛)
(1−𝑎)(1+𝑎𝛽𝑛) 2𝑎𝛽𝑛 𝜖+
1 (1−𝑎)(1+𝑎𝛽𝑛)
(1−𝑎)(1+𝑎𝛽𝑛)
2 𝜖
= 𝜖
2+
𝜖
2 =𝜖,
leading to∣∣𝑥∗
𝑛−𝑥∗∣∣< 𝜖, ∀𝑛≥𝑁.Since𝜖 >0 is arbitrary, then𝑥∗𝑛→𝑥∗as n→ ∞. □
Remark 2.1: Theorem 2.1 - Theorem 2.4 are extensions of both Theorem 3.2 and Theorem 3.6 of Rus [9]. Theorem 3.2 of Rus [9] is itself Theorem 1 of Nadler [6] and Theorem 7.8 of Berinde [1, 2].
We state the following results for pointwise convergence cases for the iterative processes:
Theorem 2.5. Let (𝐸, ∣∣.∣∣)be a Banach space and{𝑇𝑛}∞𝑛=0 a sequence of
opera-tors 𝑇𝑛 :𝐸→𝐸 such that 𝐹𝑇𝑛={𝑥∗𝑛},for each 𝑛∈IN. For𝑥0 ∈𝐸, let{𝑥𝑛}∞𝑛=0
be the Ishikawa iterative process defined by (3), 𝛼𝑛, 𝛽𝑛 ∈ [0,1]. Suppose that the
sequence {𝑇𝑛}∞𝑛=0 converges pointwise to a mapping𝑇 :𝐸→𝐸 satisfying (5) with 𝐹𝑇 = {𝑥∗}, where 𝜙 : IR+ → IR+ is a monotone increasing function such that
Proof. We shall use the contractive condition (5) and the pointwise convergence of 𝑇𝑛 to𝑇,where𝑧∗= (1−𝛽𝑛)𝑥∗+𝛽𝑛𝑇 𝑥∗, 𝑧𝑛∗= (1−𝛽𝑛)𝑥𝑛∗ +𝛽𝑛𝑇𝑛𝑥∗𝑛.
Therefore we have ∣∣𝑥∗
𝑛−𝑥∗∣∣ ≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛∣∣𝑇 𝑧∗−𝑇𝑛𝑧∗𝑛∣∣
≤(1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝛼𝑛[𝜑(∣∣𝑧∗−𝑇 𝑧∗∣∣) +𝑎∣∣𝑧∗−𝑧𝑛∗∣∣+∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣]
= (1−𝛼𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣+𝑎𝛼𝑛∣∣𝑧𝑛∗−𝑧∗∣∣+𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧∗𝑛∣∣
≤(1−𝛼𝑛+𝑎𝛼𝑛−𝑎𝛼𝑛𝛽𝑛)∣∣𝑥∗−𝑥∗𝑛∣∣+𝑎𝛼𝑛𝛽𝑛[∣∣𝑇 𝑥∗−𝑇 𝑥∗𝑛∣∣
+∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣] +𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧𝑛∗∣∣
≤(1−𝛼𝑛+𝑎𝛼𝑛−𝑎𝛼𝑛𝛽𝑛)∣∣𝑥∗−𝑥∗𝑛∣∣+𝑎𝛼𝑛𝛽𝑛[𝜑(∣∣𝑥∗−𝑇 𝑥∗∣∣)
+𝑎∣∣𝑥∗−𝑥∗
𝑛∣∣+∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣] +𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧𝑛∗∣∣
= (1−𝛼𝑛+𝑎𝛼𝑛−𝑎𝛼𝑛𝛽𝑛+𝑎2𝛼𝑛𝛽𝑛)∣∣𝑥∗−𝑥∗𝑛∣∣+𝑎𝛼𝑛𝛽𝑛∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣
+𝛼𝑛∣∣𝑇 𝑧∗𝑛−𝑇𝑛𝑧𝑛∗∣∣,
from which we have
𝛼𝑛(1−𝑎)(1 +𝑎𝛽𝑛)∣∣𝑥∗𝑛−𝑥∗∣∣ ≤𝑎𝛼𝑛𝛽𝑛∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣+𝛼𝑛∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧∗𝑛∣∣,
leading to
∣∣𝑥∗
𝑛−𝑥∗∣∣ ≤[(1−𝑎)(1 +𝑎𝛽𝑛)] −1
[𝑎𝛽𝑛∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣+∣∣𝑇 𝑧𝑛∗−𝑇𝑛𝑧∗𝑛∣∣]→0 as n→ ∞,
since𝑇𝑛 converges pointwise to 𝑇.Hence, we have that∣∣𝑥∗𝑛−𝑥∗∣∣ →0 as n→ ∞. □
Theorem 2.6. Let(𝐸,∣∣.∣∣)be a Banach space and{𝑇𝑛}∞𝑛=0a sequence of operators 𝑇𝑛 : 𝐸 → 𝐸 such that 𝐹𝑇𝑛 ={𝑥∗𝑛} for each 𝑛 ∈ 𝑁. For 𝑥0 ∈ 𝐸, let {𝑥𝑛}∞𝑛=0 be
the Mann iterative process defined by (2), 𝛼𝑛 ∈ [0,1]. Suppose that the sequence
{𝑇𝑛}∞𝑛=0 converges pointwise to a mapping 𝑇 :𝐸 →𝐸 satisfying (5), with 𝐹𝑇 =
{𝑥∗}, where𝜙:IR+→IR+ is a monotone increasing function such that 𝜙(0) = 0.
Then, 𝑥∗
𝑛→𝑥∗ as n→ ∞.
Proof. We shall use (2), the contractive condition (5) and the pointwise convergence of𝑇𝑛 to𝑇.Therefore we have that
∣∣𝑥∗𝑛−𝑥∗∣∣ ≤(1−𝑎)−1∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣ →0𝑎𝑠 𝑛→ ∞,
since𝑇𝑛 converges pointwise to 𝑇.
□
Theorem 2.7. Let (𝐸,∣∣.∣∣) be a Banach space and {𝑇𝑛}∞𝑛=0 a sequence of
op-erators 𝑇𝑛 : 𝐸 → 𝐸 such that 𝐹𝑇𝑛 = {𝑥∗𝑛} for each 𝑛 ∈ 𝑁. For 𝑥0 ∈ 𝐸, let {𝑥𝑛}∞𝑛=0 be the Mann iterative process defined by (2),𝛼𝑛∈[0,1]. Suppose that the
sequence{𝑇𝑛}∞𝑛=0 converges pointwise to a mapping𝑇:𝐸→𝐸 satisfying (6), with 𝐹𝑇 ={𝑥∗},where𝜓:IR+→IR+is a strict comparison function and𝜙:IR+→IR+
is a monotone increasing function such that𝜙(0) = 0. Then,𝑥∗𝑛→𝑥∗ as n→ ∞.
Proof. By using (2), the contractive condition (6) and the pointwise convergence of𝑇𝑛 to𝑇,we have that
∣∣𝑥∗𝑛−𝑥∗∣∣ −𝜓(∣∣𝑥∗𝑛−𝑥∗∣∣)≤ ∣∣𝑇 𝑥∗𝑛−𝑇𝑛𝑥∗𝑛∣∣ →0 as n→ ∞,
since𝑇𝑛 converges pointwise to𝑇.It follows that∣∣𝑥∗𝑛−𝑥∗∣∣ →0 as n→ ∞,since
𝜓is a strict comparison function.
Theorem 2.8. Let(𝐸,∣∣.∣∣)be a Banach space and{𝑇𝑛}∞𝑛=0a sequence of operators 𝑇𝑛:𝐸 →𝐸 such that 𝐹𝑇𝑛 ={𝑥
∗
𝑛} for each𝑛∈𝑁. For𝑥0∈𝐸, let{𝑥𝑛}∞𝑛=0 be the
Mann iterative process defined by (2), where𝛼𝑛∈[0,1].Suppose that the sequence
{𝑇𝑛}∞𝑛=0 converges pointwise to a mapping 𝑇 :𝐸→𝐸 satisfying
∣∣𝑇 𝑥−𝑇 𝑦∣∣ ≤𝐿∣∣𝑥−𝑇 𝑥∣∣+𝜓(∣∣𝑥−𝑦∣∣), ∀ 𝑥, 𝑦∈𝐸, 𝐿≥0, (7)
with 𝐹𝑇 = {𝑥∗}, where 𝜓 : IR+ → IR+ is a strict comparison function. Then,
𝑥∗𝑛→𝑥∗ as n→ ∞.
Proof. The proof of this result is similar to that of Theorem 2.7.
□
Theorem 2.9. Let(𝐸,∣∣.∣∣)be a Banach space and{𝑇𝑛}∞𝑛=0a sequence of operators 𝑇𝑛 : 𝐸 → 𝐸 such that 𝐹𝑇𝑛 ={𝑥∗𝑛} for each 𝑛 ∈ 𝑁. For 𝑥0 ∈ 𝐸, let {𝑥𝑛}∞𝑛=0 be
the Ishikawa iterative process defined by (3), where 𝛼𝑛, 𝛽𝑛 ∈[0,1]. Suppose that
the sequence {𝑇𝑛}∞𝑛=0 converges pointwise to a mapping 𝑇 :𝐸→𝐸 satisfying (6),
with 𝐹𝑇 ={𝑥∗}, where 𝜓 : IR+ →IR+ is a sublinear, strict comparison function
and 𝜙: IR+ → IR+ is a monotone increasing function such that 𝜙(0) = 0. Then,
𝑥∗𝑛→𝑥∗ as n→ ∞.
Proof. The proof of this result is similar to that of Theorem 2.7.
□
Remark 2.2: Theorem 2.5 - Theorem 2.9 extend a result of Bonsall [3] (which is Theorem 3.1 of Rus [9]). Theorem 2.7 - Theorem 2.9 are also extensions of Theorem 7.2.1 of Rus [8] (which is Theorem 7.9 of Berinde [1, 2]).
Remark 2.3: Corresponding results can also be deduced from our results for the Schaefer’s iterative process defined in (1). See [7].
References
[1] V. Berinde;Iterative Approximation of Fixed Points,Editura Efemeride (2002).
[2] V. Berinde; Iterative Approximation of Fixed Points, Springer-Verlag Berlin Heidelberg (2007).
[3] F. F. Bonsall;Lectures on Some Fixed Point Theorems of Functional Analysis,Tata Inst. Fund. Res., Bombay (1962).
[4] S. Ishikawa;Fixed Point by a New Iteration Method,Proc. Amer. Math. Soc. 44 (1) (1974), 147-150. 41 (1973), 430-438.
[5] W. R. Mann;Mean Value Methods in Iteration,Proc. Amer. Math. Soc. 44 (1953), 506-510. [6] S. B. Nadler (JR.); Sequences of Contractions and Fixed points, Pacific J. Math. 27 (3)
(1968), 579-585.
[7] M. O. Olatinwo;Some Results on Stability and Sequences of Operators for Schaefer’s Fixed Point Iterative Algorithm in Hilbert and Normed Linear Spaces,Proc. Jangeon Math. Soc. (To appear in 2011).
[8] I. A. Rus;Generalized Contractions and Applications,Cluj Univ. Press, Cluj Napoca, (2001). [9] I. A. Rus;Sequences of Operators and Fixed Points, Fixed Point Theory,5 (2) (2004),
349-368.
[10] H. Schaefer; Uber die Methode Sukzessiver Approximationen, Jahresber. Deutsch. Math. Verein. 59 (1957), 131-140.
Memudu Olaposi Olatinwo
Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria
