Strong Convergence for Hybrid 𝑆-Iteration Scheme

Abstract

We establish a strong convergence for the hybrid 𝑆-iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces.

1. Introduction and Preliminaries

Let 𝐸 be a real Banach space and let 𝐾 be a nonempty convex subset of 𝐸. Let 𝐽 denote the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽(𝑥)=𝑓^∗∈𝐸^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖²,𝑓^∗=‖𝑥‖,∀𝑥,𝑦∈𝐸, (1) where 𝐸^∗ denotes the dual space of 𝐸 and ⟨⋅,⋅⟩ denotes the generalized duality pairing. We will denote the single-valued duality map by 𝑗.

Let 𝑇∶𝐾→𝐾 be a mapping.

Definition 1. The mapping 𝑇 is said to be Lipschitzian if there exists a constant 𝐿>1 such that 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐾. (2)

Definition 2. The mapping 𝑇 is said to be nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐾. (3)

Definition 3. The mapping 𝑇 is said to be pseudocontractive if for all 𝑥,𝑦∈𝐾, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦². (4)

Definition 4. The mapping 𝑇 is said to be strongly pseudocontractive if for all 𝑥,𝑦∈𝐾, there exists 𝑘∈(0,1) such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑘𝑥−𝑦². (5)
Let 𝐾 be a nonempty convex subset 𝐶 of a normed space 𝐸.(a)The sequence {𝑥_𝑛} defined by, for arbitrary 𝑥₁∈𝐾, 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥1, (6) where {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1], is known as the Ishikawa iteration process [1]. If 𝛽_𝑛=0 for 𝑛≥1, then the Ishikawa iteration process becomes the Mann iteration process [2].(b)The sequence {𝑥_𝑛} defined by, for arbitrary 𝑥₁∈𝐾, 𝑥_𝑛+1=𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥1, (7) where {𝛽_𝑛} is a sequence in [0,1], is known as the 𝑆-iteration process [3, 4].
In the last few years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudocontractive mappings using the Ishikawa iteration scheme (see, e.g., [1]). Results which had been known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see, e.g., [5–10] and the references cited therein).
In 1974, Ishikawa [1] proved the following result.

Theorem 5. Let 𝐾 be a compact convex subset of a Hilbert space 𝐻 and let 𝑇∶𝐾→𝐾 be a Lipschitzian pseudocontractive mapping. For arbitrary 𝑥₁∈𝐾, let {𝑥_𝑛} be a sequence defined iteratively by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥1, (8) where {𝛼_𝑛} and {𝛽_𝑛} are sequences satisfying(i)0≤𝛼_𝑛≤𝛽_𝑛≤1, (ii)lim_𝑛→∞𝛽_𝑛=0,(iii)∑_𝑛≥1𝛼_𝑛𝛽_𝑛=∞.
Then the sequence {𝑥_𝑛} converges strongly at a fixed point of 𝑇.

In [6], Chidume extended the results of Schu [9] from Hilbert spaces to the much more general class of real Banach spaces and approximated the fixed points of (strongly) pseudocontractive mappings.

In [11], Zhou and Jia gave the more general answer of the question raised by Chidume [5] and proved the following.

If 𝑋 is a real Banach space with a uniformly convex dual 𝑋^∗, 𝐾 is a nonempty bounded closed convex subset of 𝑋, and 𝑇∶𝐾→𝐾 is a continuous strongly pseudocontractive mapping, then the Ishikawa iteration scheme converges strongly at the unique fixed point of 𝑇.

In this paper, we establish the strong convergence for the hybrid 𝑆-iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces. We also improve the result of Zhou and Jia [11].

2. Main Results

We will need the following lemmas.

Lemma 6 (see [12]). Let 𝐽∶𝐸→2^𝐸 be the normalized duality mapping. Then for any 𝑥,𝑦∈𝐸, one has 𝑥+𝑦²≤‖𝑥‖²,.+2𝑦,𝑗𝑥+𝑦∀𝑗𝑥+𝑦∈𝐽𝑥+𝑦 (9)

Lemma 7 (see [10]). Let {𝜌_𝑛} be nonnegative sequence satisfying 𝜌_𝑛+1≤1−𝜃_𝑛𝜌_𝑛+𝜔_𝑛, (10) where 𝜃_𝑛∈[0,1],∑_𝑛≥1𝜃_𝑛=∞, and 𝜔_𝑛=o(𝜃_𝑛). Then lim_𝑛→∞𝜌_𝑛=0. (11)

The following is our main result.

Theorem 8. Let 𝐾 be a nonempty closed convex subset of a real Banach space 𝐸, let 𝑆∶𝐾→𝐾 be nonexpansive, and let 𝑇∶𝐾→𝐾 be Lipschitz strongly pseudocontractive mappings such that 𝑝∈𝐹(𝑆)∩𝐹(𝑇)={𝑥∈𝐾∶𝑆𝑥=𝑇𝑥=𝑥} and ≤≤𝑥−𝑆𝑦𝑆𝑥−𝑆𝑦,∀𝑥,𝑦∈𝐾,𝑥−𝑇𝑦𝑇𝑥−𝑇𝑦,∀𝑥,𝑦∈𝐾. (C) Let {𝛽_𝑛} be a sequence in [0,1] satisfying(iv) ∑_𝑛≥1𝛽_𝑛=∞,(v) lim_𝑛→∞𝛽_𝑛=0.
For arbitrary 𝑥₁∈𝐾, let {𝑥_𝑛} be a sequence iteratively defined by 𝑥_𝑛+1=𝑆𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥1. (12)
Then the sequence {𝑥_𝑛} converges strongly at the common fixed point 𝑝 of 𝑆 and 𝑇.

Proof. For strongly pseudocontractive mappings, the existence of a fixed point follows from Delmling [13]. It is shown in [11] that the set of fixed points for strongly pseudocontractions is a singleton.
By (v), since lim_𝑛→∞𝛽_𝑛=0, there exists 𝑛₀∈ℕ such that for all 𝑛≥𝑛₀, 𝛽_𝑛1≤min,4𝑘1−𝑘(),1+𝐿)(1+3𝐿 (13) where 𝑘<1/2. Consider 𝑥_𝑛+1−𝑝²=𝑥_𝑛+1−𝑝,𝑗𝑥_𝑛+1=−𝑝𝑆𝑦_𝑛−𝑝,𝑗𝑥_𝑛+1=−𝑝𝑇𝑥_𝑛+1−𝑝,𝑗𝑥_𝑛+1+−𝑝𝑆𝑦_𝑛−𝑇𝑥_𝑛+1,𝑗𝑥_𝑛+1𝑥−𝑝≤𝑘_𝑛+1−𝑝²+𝑆𝑦_𝑛−𝑇𝑥_𝑛+1𝑥_𝑛+1−𝑝, (14) which implies that 𝑥_𝑛+1≤1−𝑝1−𝑘𝑆𝑦_𝑛−𝑇𝑥_𝑛+1, (15) where 𝑆𝑦_𝑛−𝑇𝑥_𝑛+1≤𝑆𝑦_𝑛−𝑇𝑦_𝑛+𝑇𝑦_𝑛−𝑇𝑥_𝑛+1≤𝑥_𝑛−𝑆𝑦_𝑛+𝑥_𝑛−𝑇𝑦_𝑛+𝑇𝑦_𝑛−𝑇𝑥_𝑛+1≤𝑆𝑥_𝑛−𝑆𝑦_𝑛+𝑇𝑥_𝑛−𝑇𝑦_𝑛+𝑇𝑦_𝑛−𝑇𝑥_𝑛+1≤𝑆𝑥_𝑛−𝑆𝑦_𝑛+𝐿𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛+1,𝑦_𝑛−𝑥_𝑛+1≤𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1=𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑆𝑦_𝑛≤𝑦_𝑛−𝑥_𝑛+𝑆𝑥_𝑛−𝑆𝑦_𝑛, (16)(17) and consequently from (16), we obtain 𝑆𝑦_𝑛−𝑇𝑥_𝑛+1≤(1+𝐿)𝑆𝑥_𝑛−𝑆𝑦_𝑛+2𝐿𝑥_𝑛−𝑦_𝑛≤(1+3𝐿)𝑥_𝑛−𝑦_𝑛=(1+3𝐿)𝛽_𝑛𝑥_𝑛−𝑇𝑥_𝑛≤(1+𝐿)(1+3𝐿)𝛽_𝑛𝑥_𝑛−𝑝. (18)
Substituting (18) in (15) and using (13), we get 𝑥_𝑛+1≤()−𝑝1+𝐿)(1+3𝐿𝛽1−𝑘^𝑛𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝. (19)
So, from the above discussion, we can conclude that the sequence {𝑥_𝑛−𝑝} is bounded. Since 𝑇 is Lipschitzian, so {𝑇𝑥_𝑛−𝑝} is also bounded. Let 𝑀₁=sup_𝑛≥1‖𝑥_𝑛−𝑝‖+sup_𝑛≥1‖𝑇𝑥_𝑛−𝑝‖. Also by (ii), we have 𝑥_𝑛−𝑦_𝑛=𝛽_𝑛𝑥_𝑛−𝑇𝑥_𝑛≤𝑀₁𝛽_𝑛→0 (20) as 𝑛→∞, implying that {𝑥_𝑛−𝑦_𝑛} is bounded, so let 𝑀₂=sup_𝑛≥1‖𝑥_𝑛−𝑦_𝑛‖+𝑀₁. Further, 𝑦_𝑛−𝑝≤𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑝≤𝑀₂, (21) which implies that {𝑦_𝑛−𝑝} is bounded. Therefore, {𝑇𝑦_𝑛−𝑝} is also bounded.
Set 𝑀₃=sup_𝑛≥1𝑦_𝑛−𝑝+sup_𝑛≥1𝑇𝑦_𝑛−𝑝. (22)
Denote 𝑀=𝑀₁+𝑀₂+𝑀₃. Obviously, 𝑀<∞.
Now from (12) for all 𝑛≥1, we obtain 𝑥_𝑛+1−𝑝²=𝑆𝑦_𝑛−𝑝²≤𝑦_𝑛−𝑝², (23) and by Lemma 6, we get 𝑦_𝑛−𝑝²=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛−𝑝²=1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑇𝑥_𝑛−𝑝²≤1−𝛽_𝑛²𝑥_𝑛−𝑝²+2𝛽_𝑛𝑇𝑥_𝑛−𝑝,𝑗𝑦_𝑛=−𝑝1−𝛽_𝑛²𝑥_𝑛−𝑝²+2𝛽_𝑛⟨𝑇𝑦_𝑛−𝑝,𝑗𝑦_𝑛−𝑝⟩+2𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛,𝑗𝑦_𝑛≤−𝑝1−𝛽_𝑛²𝑥_𝑛−𝑝²+2𝑘𝛽_𝑛𝑦_𝑛−𝑝²+2𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛𝑦_𝑛≤−𝑝1−𝛽_𝑛²𝑥_𝑛−𝑝²+2𝑘𝛽_𝑛𝑦_𝑛−𝑝²+2𝑀𝐿𝛽_𝑛𝑥_𝑛−𝑦_𝑛, (24) which implies that 𝑦_𝑛−𝑝²≤1−𝛽_𝑛²1−2𝑘𝛽_𝑛𝑥_𝑛−𝑝²+2𝑀𝐿𝛽_𝑛1−2𝑘𝛽_𝑛𝑥_𝑛−𝑦_𝑛≤1−𝛽_𝑛𝑥_𝑛−𝑝²+4𝑀𝐿𝛽_𝑛𝑥_𝑛−𝑦_𝑛 (25) because by (13), we have ((1−𝛽_𝑛)/(1−2𝑘𝛽_𝑛))≤1 and (1/(1−2𝑘𝛽_𝑛))≤2. Hence, (23) gives us 𝑥_𝑛+1−𝑝²≤1−𝛽_𝑛𝑥_𝑛−𝑝²+4𝑀𝐿𝛽_𝑛𝑥_𝑛−𝑦_𝑛. (26)
For all 𝑛≥1, put 𝜌_𝑛=𝑥_𝑛,𝜃−𝑝_𝑛=𝛽_𝑛,𝜔_𝑛=4𝑀𝐿𝛽_𝑛𝑥_𝑛−𝑦_𝑛, (27) then according to Lemma 7, we obtain from (26) that lim_𝑛→∞𝑥_𝑛−𝑝=0. (28)
This completes the proof.

Corollary 9. Let 𝐾 be a nonempty closed convex subset of a real Hilbert space 𝐻, let 𝑆∶𝐾→𝐾 be nonexpansive, and let 𝑇∶𝐾→𝐾 be Lipschitz strongly pseudocontractive mappings such that 𝑝∈𝐹(𝑆)∩𝐹(𝑇) and the condition (C). Let {𝛽_𝑛} be a sequence in [0,1] satisfying the conditions (iv) and (v).
For arbitrary 𝑥₁∈𝐾, let {𝑥_𝑛} be a sequence iteratively defined by (12). Then the sequence {𝑥_𝑛} converges strongly at the common fixed point 𝑝 of 𝑆 and 𝑇.

Example 10. As a particular case, we may choose, for instance, 𝛽_𝑛=1/𝑛.

Remark 11. (1) The condition (C) is not new and it is due to Liu et al. [14].
(2) We prove our results for a hybrid iteration scheme, which is simple in comparison to the previously known iteration schemes.

Acknowledgment

This study was supported by research funds from Dong-A University.