Strong Convergence of an Implicit 𝑆-Iterative Process for Lipschitzian Hemicontractive Mappings

Abstract

We establish the strong convergence for the implicit 𝑆-iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

1. Introduction

Let 𝐻 be a Hilbert space and let 𝑇∶𝐻→𝐻 be a mapping.

The mapping 𝑇 is called Lipshitzian if there exists 𝐿>0 such that 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐻.(1.1)

If 𝐿=1, then 𝑇 is called nonexpansive and if 0≤𝐿<1, then 𝑇 is called contractive.

The mapping 𝑇 is said to be pseudocontractive ([1, 2]) if 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐻,(1.2) and the mapping 𝑇 is said to be strongly pseudocontractive if there exists 𝑘∈(0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐻.(1.3)

Let 𝐹(𝑇)∶={𝑥∈𝐻∶𝑇𝑥=𝑥} and the mapping 𝑇 is called hemicontractive if 𝐹(𝑇)≠∅ and 𝑇𝑥−𝑥^∗2≤𝑥−𝑥^∗2+‖𝑥−𝑇𝑥‖²,∀𝑥∈𝐻,𝑥^∗∈𝐹(𝑇).(1.4)

It is easy to see the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractive mappings. For the importance of fixed points of pseudocontractions the reader may consult [1].

In 1974, Ishikawa [3] proved the following result.

Theorem 1.1. 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 by𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥1,(1.5) where {𝛼_𝑛} and {𝛽_𝑛} are sequences satisfying the conditions:(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 to a fixed point of 𝑇.

Another iteration scheme which has been studied extensively in connection with fixed points of pseudocontractive mappings.

In 2011, Sahu [4] and Sahu and Petruşel [5] introduced the 𝑆-iterative process as follows.

Let 𝐾 be a nonempty convex subset of a normed space 𝑋 and let 𝑇∶𝐾→𝐾 be a mapping. Then, for arbitrary 𝑥₁∈𝐾, the 𝑆-iterative process is defined by 𝑥_𝑛+1=𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥1,(1.6) where {𝛽_𝑛} is a real sequence in [0,1].

In this paper, we establish the strong convergence for the implicit 𝑆-iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

2. Main Results

We need the follwing lemma.

Lemma 2.1 (see [6]). For all 𝑥,𝑦∈𝐻 and 𝜆∈[0,1], the following well-known identity holds (1−𝜆)𝑥+𝜆𝑦²=(1−𝜆)‖𝑥‖²+𝜆𝑦²−𝜆(1−𝜆)𝑥−𝑦².(2.1)

Now we prove our main results.

Theorem 2.2. Let 𝐾 be a compact convex subset of a real Hilbert space 𝐻 and let 𝑇∶𝐾→𝐾 be a Lipschitzian hemicontractive mapping satisfying 𝑥−𝑇𝑦≤𝑇𝑥−𝑇𝑦,∀𝑥,𝑦∈𝐾.(𝐶)
Let {𝛽_𝑛} be a sequence in [0,1] satisfying(iv) ∑^∞_𝑛=1𝛽_𝑛=∞,(v) ∑^∞_𝑛=1𝛽²_𝑛<∞.
For arbitrary 𝑥₀∈𝐾, let {𝑥_𝑛} be a sequence defined iteratively by 𝑥_𝑛=𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥1.(2.2)
Then the sequence {𝑥_𝑛} converges strongly to the fixed point 𝑥^∗ of 𝑇.

Proof. From Schauder’s fixed point theorem, 𝐹(𝑇) is nonempty since 𝐾 is a convex compact set and 𝑇 is continuous, let 𝑥^∗∈𝐹(𝑇). Using the fact that 𝑇 is hemicontractive we obtain 𝑇𝑥_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑇𝑥_𝑛²,(2.3)𝑇𝑦_𝑛−𝑥^∗2≤𝑦_𝑛−𝑥^∗2+𝑦_𝑛−𝑇𝑦_𝑛².(2.4)
Now by (v), there exists 𝑛₀∈ℕ such that for all 𝑛≥𝑛₀, 𝛽_𝑛1≤min3,1𝐿²,(2.5) which implies that 2𝛽_𝑛1−𝛽_𝑛≤1.(2.6)
With the help of (2.2), (2.3), and Lemma 2.1, we obtain the following estimates: 𝑦_𝑛−𝑥^∗2=1−𝛽_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛−𝑥^∗2=1−𝛽_𝑛𝑥_𝑛−1−𝑥^∗+𝛽_𝑛𝑇𝑥_𝑛−𝑥^∗2=1−𝛽_𝑛𝑥_𝑛−1−𝑥^∗2+𝛽_𝑛𝑇𝑥_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²≤1−𝛽_𝑛𝑥_𝑛−1−𝑥^∗2+𝛽_𝑛𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑇𝑥_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²,𝑦_𝑛−𝑇𝑦_𝑛²=1−𝛽_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛²=1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑦_𝑛+𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛²=1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑦_𝑛²+𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛².(2.7)
Substituting (2.7) in (2.4) we obtain 𝑇𝑦_𝑛−𝑥^∗2≤1−𝛽_𝑛𝑥_𝑛−1−𝑥^∗2+𝛽_𝑛𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑇𝑥_𝑛²+1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑦_𝑛²+𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛²−2𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛².(2.8)
Also with the help of condition (𝐶) and (2.8), we have 𝑥_𝑛+1−𝑥^∗2=𝑇𝑦_𝑛−𝑥^∗2≤1−𝛽_𝑛𝑥_𝑛−1−𝑥^∗2+𝛽_𝑛𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑇𝑥_𝑛²+1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑦_𝑛²+𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛²−2𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²≤1−𝛽_𝑛𝑥_𝑛−1−𝑥^∗2+𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑦_𝑛²+2𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛²−2𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²,(2.9) which implies that 𝑥_𝑛+1−𝑥^∗2≤𝑥_𝑛−1−𝑥^∗2+𝑥_𝑛−1−𝑇𝑦_𝑛²+2𝛽_𝑛1−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛²−2𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²≤𝑥_𝑛−1−𝑥^∗2+𝑥_𝑛−1−𝑇𝑦_𝑛²+𝑇𝑥_𝑛−𝑇𝑦_𝑛²−2𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²,(2.10) where 𝑥_𝑛−1−𝑇𝑦_𝑛²≤𝑇𝑥_𝑛−1−𝑇𝑦_𝑛²≤𝐿²𝑥_𝑛−1−𝑦_𝑛²=𝐿²𝛽²_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²,(2.11)𝑇𝑥_𝑛−𝑇𝑦_𝑛²≤𝐿²𝑥_𝑛−𝑦_𝑛²≤𝐿²𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝑦_𝑛²≤𝐿²𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²≤𝐿²𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛𝑀²,𝑥_𝑛−𝑥_𝑛−1=𝑥_𝑛−1−𝑇𝑦_𝑛≤𝑇𝑥_𝑛−1−𝑇𝑦_𝑛≤𝐿𝑥_𝑛−1−𝑦_𝑛=𝐿𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛≤𝐿𝛽_𝑛𝑀(2.12) and consequently from (2.12), we obtain 𝑇𝑥_𝑛−𝑇𝑦_𝑛²≤𝐿²(1+𝐿)²𝑀²𝛽²_𝑛.(2.13)
Hence by (2.5), (2.10), (2.11), and (2.13), we have 𝑥_𝑛−𝑥^∗2≤𝑥_𝑛−1−𝑥^∗2+𝐿²𝛽²_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²+𝐿²(1+𝐿)²𝑀²𝛽²_𝑛−2𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²=𝑥_𝑛−1−𝑥^∗2+𝐿²(1+𝐿)²𝑀²𝛽²_𝑛−𝛽_𝑛2−𝐿²𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²≤𝑥_𝑛−1−𝑥^∗2+𝐿²(1+𝐿)²𝑀²𝛽²_𝑛−𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²,(2.14) which implies that 𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛²≤𝑥_𝑛−1−𝑥^∗2−𝑥_𝑛−𝑥^∗2+𝐿²(1+𝐿)²𝑀²𝛽²_𝑛,(2.15) so that 12^𝑛_𝑗=𝑁𝛽_𝑗𝑥_𝑗−1−𝑇𝑥_𝑗²≤𝑥_𝑁−𝑥^∗2−𝑥_𝑛−𝑥^∗2+𝐿²(1+𝐿)²𝑀^2𝑛_𝑗=𝑁𝛽²_𝑗.(2.16)
Hence by conditions (iv) and (v), we get ^∞_𝑗=0𝑥_𝑗−1−𝑇𝑥_𝑗²<∞.(2.17)
It implies that lim_𝑛→∞𝑥_𝑛−1−𝑇𝑥_𝑛=0.(2.18)
Consider 𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝑇𝑥_𝑛,(2.19) which implies that lim_𝑛→∞𝑥_𝑛−𝑇𝑥_𝑛=0.(2.20)
The rest of the argument follows exactly as in the proof of Theorem of [3]. This completes the proof.

Theorem 2.3. Let 𝐾 be a compact convex subset of a real Hilbert space 𝐻 and let 𝑇∶𝐾→𝐾 be a Lipschitzian hemicontractive mapping satisfying the condition (𝐶). Let {𝛽_𝑛} be a sequence in [0,1] satisfying the conditions (iv) and (v).
Assume that 𝑃_𝐾∶𝐻→𝐾 be the projection operator of 𝐻 onto 𝐾. Let {𝑥_𝑛} be a sequence defined iteratively by 𝑥_𝑛=𝑃_𝐾𝑇𝑦_𝑛,𝑦_𝑛=𝑃_𝐾1−𝛽_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥1.(2.21)
Then the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. The operator 𝑃_𝐾 is nonexpansive (see, e.g., [2]). 𝐾 is a Chebyshev subset of 𝐻 so that, 𝑃_𝐾 is a single-valued mapping. Hence, we have the following estimate: 𝑥_𝑛−𝑥^∗2=𝑃_𝐾𝑇𝑦_𝑛−𝑃_𝐾𝑥^∗2≤𝑇𝑦_𝑛−𝑥^∗2≤𝑥_𝑛−1−𝑥^∗2+𝐿²(1+𝐿)²𝑀²𝛽²_𝑛−𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛².(2.22)
The set 𝐾=𝐾∪𝑇(𝐾) is compact and so the sequence {‖𝑥_𝑛−𝑇𝑥_𝑛‖} is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.2. This completes the proof.

Remark 2.4. In main results, the condition (𝐶) is not new and it is due to Liu et al. [7].

Acknowledgment

The authors would like to thank the referees for thier useful comments and suggestions.