Asymptotically Nonexpansive Mappings in Banach Spaces

Volume 2013, Article ID 202095,8pages http://dx.doi.org/10.1155/2013/202095

Research Article

Strong Convergence Theorems for Semigroups of

Asymptotically Nonexpansive Mappings in Banach Spaces

D. R. Sahu,

Ngai-Ching Wong,

and Jen-Chih Yao

1Department of Mathematics, Banaras Hindu University, Varanasi 221005, India

2Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan

3Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

4Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Ngai-Ching Wong; wong@math.nsysu.edu.tw Received 4 February 2013; Accepted 4 April 2013

Academic Editor: Qamrul Hasan Ansari

Copyright © 2013 D. R. Sahu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let𝑋be a real reflexive Banach space with a weakly continuous duality mapping𝐽_𝜑. Let𝐶be a nonempty weakly closed star-shaped (with respect to𝑢) subset of𝑋. LetF={𝑇(𝑡) : 𝑡 ∈ [0, +∞)}be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of𝐶, which is uniformly continuous at zero. We will show that the implicit iteration scheme:𝑦_𝑛= 𝛼_𝑛𝑢+(1−𝛼_𝑛)𝑇(𝑡_𝑛)𝑦_𝑛, for all𝑛 ∈N, converges strongly to a common fixed point of the semigroupFfor some suitably chosen parameters{𝛼_𝑛}and{𝑡_𝑛}.

Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).

1. Introduction

Let𝐶be a nonempty subset of a (real) Banach space𝑋and 𝑇 : 𝐶 → 𝐶a mapping. Thefixed point setof𝑇is defined by𝐹(𝑇) = {𝑥 ∈ 𝐶 : 𝑇𝑥 = 𝑥}. We say that𝑇isnonexpansive if‖𝑇𝑥 − 𝑇𝑦‖ ≤ ‖𝑥 − 𝑦‖for all𝑥, 𝑦in𝐶andasymptotically nonexpansiveif there exists a sequence{𝑘_𝑛}in[1, +∞)with lim_{𝑛 → ∞}𝑘_𝑛= 1such that‖𝑇^𝑛𝑥 − 𝑇^𝑛𝑦‖ ≤ 𝑘_𝑛‖𝑥 − 𝑦‖for all𝑥, 𝑦 in𝐶and𝑛inN.

SetR⁺ := [0, +∞). We call a one-parameter familyF:=

{𝑇(𝑡) : 𝑡 ∈ R⁺}of mappings from𝐶into𝐶astrongly con- tinuous semigroup of Lipschitzian mappingsif

(1) for each𝑡 > 0, there exists a bounded function𝑘(⋅) : (0, +∞) → (0, +∞)such that

󵄩󵄩󵄩󵄩𝑇(𝑡)𝑥 − 𝑇(𝑡)𝑦󵄩󵄩󵄩󵄩 ≤ 𝑘(𝑡)󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩 ∀𝑥,𝑦 ∈ 𝐶, (1) (2) 𝑇(0)𝑥 = 𝑥for all𝑥in𝐶,

(3) 𝑇(𝑠 + 𝑡) = 𝑇(𝑠)𝑇(𝑡)for all𝑠, 𝑡inR⁺,

(4) for each𝑥in𝐶, the mapping𝑇(⋅)𝑥fromR⁺into𝐶is continuous.

Note that lim inf_{𝑡 → 0}+𝑘(𝑡) ≥ 1. If𝑘(𝑡) = 𝐿for all𝑡 > 0in (1), thenFis called astrongly continuous semigroup of uniformly

𝐿-Lipschitzian mappings.If𝑘(𝑡) = 1for all𝑡 > 0in (1), then Fis called astrongly continuous semigroup of nonexpansive mappings. If𝑘(𝑡) ≥ 1for all𝑡 > 0and lim_{𝑡 → +∞}𝑘(𝑡) = 1 in (1), then Fis called a strongly continuous semigroup of asymptotically nonexpansive mappings.Moreover,Fis said to be(right) uniformly continuousif it also holds:

(5) For any bounded subset𝐵of𝐶, we have

𝑡 → 0lim⁺sup

𝑥∈𝐵‖𝑇 (𝑡) 𝑥 − 𝑥‖ = 0. (2) We denote by𝐹(F)the set of common fixed points ofF; that is,𝐹(F) := ⋂_𝑡∈R+𝐹(𝑇(𝑡)).

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that if𝐶is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping has a fixed point. Several authors have studied the problem of the existence of fixed points of asymp- totically nonexpansive mappings in Banach spaces having rich geometric structure; see [2] and the references therein.

Consider a nonempty closed convex subset𝐶of a (real) Banach space𝑋. A classical method to study nonexpansive

mappings is to approximate them by contractions. More precisely, for a fixed element𝑢in𝐶, define for each𝑡in(0, 1) a contraction𝐺_𝑡by

𝐺_𝑡𝑥 = 𝑡𝑢 + (1 − 𝑡) 𝑇𝑥 ∀𝑥 ∈ 𝐶. (3) Let𝑥_𝑡be the fixed point of𝐺_𝑡; that is,

𝑥_𝑡= 𝑡𝑢 + (1 − 𝑡) 𝑇𝑥_𝑡. (4) Browder [3] (Reich [4], resp.) proves that as𝑡 → 0⁺, the point𝑥_𝑡 converges strongly to a fixed point of 𝑇if 𝑋 is a Hilbert space (uniformly smooth Banach space, resp.). Many authors (see, e.g., [5–13]) have studied strong convergence of approximates{𝑥_𝑡} for asymptotically nonexpansive self- mappings𝑇in Banach spaces under the additional assump- tion𝑥_𝑡− 𝑇𝑥_𝑡 → 0as𝑛 → ∞. This additional assumption can be removed when𝑇is uniformly asymptotically regular.

Suzuki [14] initiated the following implicit iteration pro- cess for a semigroupF := {𝑇(𝑡) : 𝑡 ∈ R⁺}of nonexpansive mappings in a Hilbert space:

𝑦_𝑛 = 𝛼_𝑛𝑢 + (1 − 𝛼_𝑛) 𝑇 (𝑡_𝑛) 𝑦_𝑛, ∀𝑛 ∈N. (5) Xu [15] extended Suzuki’s result to uniformly convex Banach spaces with weakly sequentially continuous duality map- pings. Recently, Zegeye and Shahzad [16] extended results of Xu [15] and established the following strong convergence theorem.

Theorem ZS. Let 𝐶 be a nonempty closed convex bounded subset of a real uniformly convex Banach space𝑋with a weakly continuous duality mapping𝐽_𝜑with gauge𝜑. LetF:= {𝑇(𝑡) : 𝑡 ∈R⁺}be a strongly continuous asymptotically nonexpansive semigroups with net{𝑘(𝑡)} ⊂ [1, +∞). Assume that𝐹(F)is a sunny nonexpansive retract of𝐶with𝑃as the sunny non- expansive retraction. Assume that{𝑡_𝑛} ⊂ (0, +∞)and{𝛼_𝑛} ⊂ (0, 1)such that(1 − 𝛼_𝑛)𝑘(𝑡_𝑛) ≤ 1for all𝑛 ∈ N,lim_{𝑛 → ∞}𝑡_𝑛 = lim_{𝑛 → ∞}(𝛼_𝑛/𝑡_𝑛) = 0, and{𝛼_𝑛/(1 − (1 − 𝛼_𝑛)𝑘(𝑡_𝑛))}is bounded.

Let𝑢in𝐶be fixed.

(1) There exists a sequence{𝑦_𝑛}in𝐶such that

𝑦_𝑛= 𝛼_𝑛𝑢 + (1 − 𝛼_𝑛) 𝑇 (𝑡_𝑛) 𝑦_𝑛 ∀𝑛 ∈N, (6) which converges strongly to an element of𝐹(F).

(2) Every sequence{𝑥_𝑛}defined iteratively with any𝑥₁in 𝐶, and

𝑥_𝑛+1= 𝛼_𝑛𝑢 + (1 − 𝛼_𝑛) 𝑇 (𝑡_𝑛) 𝑥_𝑛 ∀𝑛 ∈N, (7) converges strongly to an element of𝐹(F), provided that

‖𝑥_𝑛+1− 𝑥_𝑛‖ ≤ 𝐷𝛼_𝑛for some𝐷 > 0.

Problem 1. Is it possible to drop the uniform convexity assumption in Theorem ZS?

Motivated by Schu [13], the purpose of this paper is to further analyze strong convergence of (6) and (7) for strongly

continuous semigroups of asymptotically nonexpansive map- pings defined on a set which is not necessarily convex. It is important and actually quite surprising that we are able to do so for the class of Banach spaces which are not necessarily uniformly convex. It should be noted that, in this generality, Theorem ZS does not apply. Our results are definitive, settle Problem1, and also improve results of Suzuki [14], Xu [15], and Zegeye and Shahzad [16].

2. Preliminaries

Let𝐶be a nonempty subset of a (real) Banach space𝑋with dual space𝑋^∗. We call a mapping𝑇 : 𝐶 → 𝐶weakly con- tractiveif

󵄩󵄩󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩 − 𝜓(󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩) ∀𝑥,𝑦 ∈ 𝐶, (8) where𝜓 : [0, +∞) → [0, +∞)is a continuous and nonde- creasing function such that𝜓(0) = 0,𝜓(𝑡) > 0for𝑡 > 0, and lim_{𝑡 → +∞}𝜓(𝑡) = +∞.

By agaugewe mean a continuous strictly increasing func- tion𝜑defined onR⁺ := [0, +∞)such that𝜑(0) = 0and lim_{𝑟 → +∞}𝜑(𝑟) = +∞. Associated with a gauge𝜑, the (gen- erally multivalued)duality mapping𝐽_𝜑: 𝑋 → 𝑋^∗is defined by

𝐽_𝜑(𝑥) := {𝑥^∗ ∈ 𝑋^∗: ⟨𝑥, 𝑥^∗⟩ = ‖𝑥‖ 𝜑 (‖𝑥‖) ,

󵄩󵄩󵄩󵄩𝑥^∗󵄩󵄩󵄩󵄩 = 𝜑(‖𝑥‖)}. (9) Clearly, the (normalized) duality mapping𝐽corresponds to the gauge𝜑(𝑡) = 𝑡. In general,

𝐽_𝜑(𝑥) = 𝜑 (‖𝑥‖)

‖𝑥‖ 𝐽 (𝑥) , 𝑥 ̸= 0. (10) Recall that𝑋is said to have aweakly(resp,sequentially) continuous duality mapping if there exists a gauge 𝜑 such that the duality mapping𝐽_𝜑 is single valued and (resp, seq- uentially) continuous from𝑋with the weak topology to𝑋^∗ with the weak^∗topology. Everyℓ^𝑝(1 < 𝑝 < +∞)space has a weakly continuous duality mapping with the gauge𝜑(𝑡) = 𝑡^𝑝−1(for more details see [17,18]). We know that if𝑋admits a weakly sequentially continuous duality mapping, then 𝑋 satisfiesOpial’s condition; that is, if{𝑥_𝑛}is a sequence weakly convergent to𝑥in𝑋, then there holds the inequality

lim sup

𝑛 → ∞ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥󵄩󵄩󵄩󵄩 <lim sup

𝑛 → ∞ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦󵄩󵄩󵄩󵄩 , 𝑦 ∈ 𝑋with𝑦 ̸= 𝑥.

(11) Browder [19] initiated the study of certain classes of non- linear operators by means of a duality mapping𝐽_𝜑. Define

Φ (𝑡) := ∫^𝑡

0𝜑 (𝑠) 𝑑𝑠 ∀𝑡 ≥ 0. (12) Then, it is known that𝐽_𝜑(𝑥)is the subdifferential of the convex functionΦ(‖ ⋅ ‖)at𝑥; that is,

𝐽_𝜑(𝑥) = 𝜕Φ (‖𝑥‖) , 𝑥 ∈ 𝑋, (13)

where 𝜕denotes the subdifferential in the sense of convex analysis. We need the subdifferential inequality

Φ (󵄩󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩󵄩) ≤ Φ(‖𝑥‖) + ⟨𝑦,𝑗(𝑥 + 𝑦)⟩

∀𝑥, 𝑦 ∈ 𝑋, 𝑗 (𝑥 + 𝑦) ∈ 𝐽_𝜑(𝑥 + 𝑦) . (14) For a smooth𝑋, we have

Φ (󵄩󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩󵄩) ≤ Φ(‖𝑥‖) + ⟨𝑦,𝐽^𝜑(𝑥 + 𝑦)⟩ ∀𝑥, 𝑦 ∈ 𝑋, (15) or considering the normalized duality mapping𝐽, we have

󵄩󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩󵄩²≤ ‖𝑥‖²+ 2 ⟨𝑦, 𝐽 (𝑥 + 𝑦)⟩ ∀𝑥, 𝑦 ∈ 𝑋. (16) Assume that a sequence{𝑥_𝑛}in𝑋converges weakly to a point𝑥in𝑋. Then the following identity holds;

lim sup

𝑛 → ∞ Φ (󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦󵄩󵄩󵄩󵄩) =lim sup

𝑛 → ∞ Φ (󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥󵄩󵄩󵄩󵄩) + Φ (󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩)

∀𝑦 ∈ 𝑋.

(17) Remark 2. For any𝑘with0 ≤ 𝑘 ≤ 1, we have

𝜑 (𝑘𝑡) ≤ 𝜑 (𝑡) ∀𝑡 > 0, Φ (𝑘𝑡) = ∫^𝑘𝑡

0 𝜑 (𝑠) 𝑑𝑠 = 𝑘 ∫^𝑡

0𝜑 (𝑘𝑢) 𝑑𝑢

≤ 𝑘 ∫^𝑡

0𝜑 (𝑢) 𝑑𝑢 = 𝑘Φ (𝑡) ∀𝑡 > 0.

(18)

We need the following demiclosedness principle for asymptotically nonexpansive mappings in a Banach space.

Lemma 3 (see [17, Corollary 5.6.4], [10]). Let𝑋be a Banach space with a weakly continuous duality mapping𝐽_𝜑 : 𝑋 → 𝑋^∗with gauge function𝜑. Let𝐶be a nonempty closed convex subset of𝑋and𝑇 : 𝐶 → 𝐶an asymptotically nonexpansive mapping. Then,𝐼 − 𝑇is demiclosed at zero; that is, if{𝑥_𝑛}is a sequence in𝐶which converges weakly to𝑥and if the sequence {𝑥_𝑛− 𝑇𝑥_𝑛}converges strongly to zero, then𝑥 − 𝑇𝑥 = 0.

Let𝐶be a convex subset of a Banach space𝑋and𝐷a nonempty subset of𝐶. Then, a continuous mapping𝑃from 𝐶onto𝐷is called aretractionif𝑃 𝑥 = 𝑥for all𝑥in𝐷; that is, 𝑃²= 𝑃. A retraction𝑃is said to besunnyif𝑃(𝑃𝑥+𝑡(𝑥−𝑃𝑥)) = 𝑃𝑥for each𝑥in𝐶and𝑡 ≥ 0with𝑃 𝑥 + 𝑡(𝑥 − 𝑃 𝑥)in𝐶. If the sunny retraction𝑃is also nonexpansive, then𝐷is said to be asunny nonexpansive retractof𝐶. The sunny nonexpansive retraction𝑄from𝐶onto𝐷is unique if𝑋is smooth.

Lemma 4 (see Goebel and Reich [20, Lemma 13.1]). Let𝐶be a convex subset of a smooth Banach space𝑋,𝐷a nonempty subset of 𝐶, and 𝑃 a retraction from𝐶 onto 𝐷. Then, the following are equivalent.

(a)𝑃is sunny and nonexpansive.

(b)⟨𝑥 − 𝑃𝑥, 𝐽(𝑧 − 𝑃𝑥)⟩ ≤ 0for all𝑥 ∈ 𝐶, 𝑧 ∈ 𝐷.

(c)⟨𝑥 − 𝑦, 𝐽(𝑃 𝑥 − 𝑃 𝑦)⟩ ≥ ‖ 𝑃 𝑥 − 𝑃 𝑦‖² for all𝑥, 𝑦 ∈ 𝐶.

Lemma 5 (see [21]). Let{𝛼_𝑛}and{𝛾_𝑛}be two real sequences such that

(i){𝛼_𝑛} ⊂ [0, 1]and∑^∞_𝑛=1𝛼_𝑛= +∞, (ii) lim sup_{𝑛 → ∞}𝛾_𝑛≤ 0.

Let{𝜆_𝑛}be a sequence of nonnegative numbers which satis- fies the inequality

𝜆_𝑛+1≤ (1 − 𝛼_𝑛) 𝜆_𝑛+ 𝛼_𝑛𝛾_𝑛, 𝑛 ∈N. (19) Then,lim_{𝑛 → ∞}𝜆_𝑛= 0.

3. Existence of Common Fixed Points

We begin with the following.

Proposition 6. Let𝐶be a nonempty closed subset of a Banach space𝑋. LetF = {𝑇(𝑡) : 𝑡 ∈R⁺}be a uniformly continuous semigroup of asymptotically nonexpansive mappings from𝐶 into itself with a net{𝑘(𝑡) : 𝑡 ∈ (0, +∞)}. Let{𝑏_𝑛}be a sequence in(0, 1)and{𝑡_𝑛}a sequence in(0, +∞)with𝑘(𝑡_𝑛) − 1 < 𝑏_𝑛for all𝑛inN. Assume that𝑢 ∈ 𝐶and𝑏_𝑛𝑢 + (1 − 𝑏_𝑛)𝑇(𝑡_𝑛)𝑥 ∈ 𝐶for all𝑥in𝐶and𝑛inN.

(a) There exists a sequence{𝑦_𝑛}in𝐶defined by

𝑦_𝑛= 𝑏_𝑛𝑢 + (1 − 𝑏_𝑛) 𝑇 (𝑡_𝑛) 𝑦_𝑛 ∀𝑛 ∈N. (20) (b) If the sequence {𝑦_𝑛} described by (20) is bounded

andlim_{𝑛 → ∞}𝑡_𝑛=lim_{𝑛 → ∞}(𝑏_𝑛/𝑡_𝑛) = 0, then

𝑦_𝑛− 𝑇 (𝑡) 𝑦_𝑛 󳨀→ 0 𝑎𝑠 𝑛 󳨀→ ∞, ∀𝑡 > 0. (21) Proof. (a) Set󰜚_𝑛 := (𝑘(𝑡_𝑛) − 1)/𝑏_𝑛. Since𝑘(𝑡_𝑛) − 1 < 𝑏_𝑛 for all𝑛inN, it follows that󰜚_𝑛 < 1 ≤ 𝑘(𝑡_𝑛), and hence0 < (1 − 𝑏_𝑛)𝑘(𝑡_𝑛) = (1 − 𝑏_𝑛)(1 + 󰜚_𝑛𝑏_𝑛) < 1for all𝑛inN. For each𝑛in N, the mapping𝐺_𝑛: 𝐶 → 𝐶defined by

𝐺_𝑛𝑦 := 𝑏_𝑛𝑢 + (1 − 𝑏_𝑛) 𝑇 (𝑡_𝑛) 𝑦, 𝑦 ∈ 𝐶 (22) is a contraction with Lipschitz constant(1 − 𝑏_𝑛)𝑘(𝑡_𝑛). There- fore, there exists a sequence{𝑦_𝑛}in𝐶described by (20).

(b) Suppose that the sequence{𝑦_𝑛}in𝐶described by (20) is bounded and lim_{𝑛 → ∞}𝑡_𝑛 = lim_{𝑛 → ∞}(𝑏_𝑛/𝑡_𝑛) = 0.

From (20), we have

󵄩󵄩󵄩󵄩𝑇(𝑡^𝑛) 𝑦_𝑛󵄩󵄩󵄩󵄩 = 11 − 𝑏_𝑛󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑏_𝑛𝑢󵄩󵄩󵄩󵄩 ≤ 1

1 − 𝑏_𝑛 (󵄩󵄩󵄩󵄩𝑦^𝑛󵄩󵄩󵄩󵄩 + 𝑏^𝑛‖𝑢‖) . (23) Without loss of generality, we may assume that {𝑏_𝑛} is bounded away from 1. Then, there exists a positive constant𝛿 such that𝑏_𝑛 ≤ 𝛿 < 1for all𝑛inN. Since𝐵 := {𝑦_𝑛 : 𝑛 ∈N}is bounded, it follows from (23) that{𝑇(𝑡_𝑛)𝑦_𝑛}is bounded.

Set𝐿 :=sup{𝑘(𝑡) : 𝑡 ∈R⁺}. For𝑡 > 0, we have

󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑇 (𝑡) 𝑦_𝑛󵄩󵄩󵄩󵄩

≤^[𝑡/𝑡∑^𝑛]−1

𝑖=0 󵄩󵄩󵄩󵄩𝑇(𝑖𝑡^𝑛) 𝑦_𝑛− 𝑇 ((𝑖 + 1) 𝑡_𝑛) 𝑦_𝑛󵄩󵄩󵄩󵄩

+󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑇([𝑡

𝑡_𝑛] 𝑡_𝑛) 𝑦_𝑛− 𝑇 (𝑡) 𝑦_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

≤ [𝑡

𝑡_𝑛] 𝐿 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑇 (𝑡_𝑛) 𝑦_𝑛󵄩󵄩󵄩󵄩 + 𝐿󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩𝑇(𝑡 − [𝑡

𝑡_𝑛] 𝑡_𝑛) 𝑦_𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩

= 𝐿𝑏_𝑛[𝑡

𝑡_𝑛] 󵄩󵄩󵄩󵄩𝑢 − 𝑇 (𝑡^𝑛) 𝑦_𝑛󵄩󵄩󵄩󵄩 + 𝐿󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑇(𝑡 − [𝑡

𝑡_𝑛] 𝑡_𝑛) 𝑦_𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

≤ 𝑡𝐿 (𝑏_𝑛

𝑡_𝑛) 󵄩󵄩󵄩󵄩𝑢 − 𝑇 (𝑡^𝑛) 𝑦_𝑛󵄩󵄩󵄩󵄩 + 𝐿sup

𝑦∈𝐵󵄩󵄩󵄩󵄩𝑇(𝑠^𝑛) 𝑦 − 𝑦󵄩󵄩󵄩󵄩 , (24) for all𝑛inN, where𝑠_𝑛 = 𝑡 − [𝑡/𝑡_𝑛]𝑡_𝑛. Note that𝑠_𝑛 = 𝑡 − [𝑡/𝑡_𝑛]𝑡_𝑛 ≤ 𝑡_𝑛 → 0as𝑛 → ∞. SinceFis uniformly con- tinuous at0, it follows that sup_𝑦∈𝐵 ‖ 𝑇(𝑠_𝑛)𝑦 − 𝑦 ‖ → 0as 𝑛 → ∞. Therefore,𝑦_𝑛− 𝑇(𝑡)𝑦_𝑛 → 0as𝑛 → ∞.

Remark 7. (a) If𝐶is star shaped with respect to𝑢in𝐶, then the assumption “𝑏_𝑛𝑢 + (1 − 𝑏_𝑛)𝑇(𝑡_𝑛)𝑥 ∈ 𝐶for all𝑥in𝐶and𝑛 inN” inProposition 6is automatically satisfied.

(b) If F = {𝑇(𝑡) : 𝑡 ∈ R⁺} is a strongly continuous semigroup of nonexpansive mappings with 𝐹(F) ̸= 0, the sequence{𝑦_𝑛} in 𝐶described by (20) is bounded (see [15, Theorem 3.3]).

Theorem 8. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping𝐽_𝜑with gauge function𝜑, and𝐶a nonempty weakly closed subset of𝑋. LetF= {𝑇(𝑡) : 𝑡 ∈R⁺}be a uniformly continuous semigroup of asymptotically nonexpansive mappings from𝐶into itself with a net{𝑘(𝑡) : 𝑡 ∈ (0, +∞)}. Let{𝑏_𝑛}be a sequence in(0, 1)and{𝑡_𝑛}a sequence in (0, +∞)with𝑘(𝑡_𝑛)−1 < 𝑏_𝑛for all𝑛inNsatisfying the condition

𝑛 → ∞lim 𝑡_𝑛= lim

𝑛 → ∞

𝑏_𝑛 𝑡_𝑛 = lim

𝑛 → ∞

𝑘 (𝑡_𝑛) − 1

𝑏_𝑛 = 0. (25)

Assume that𝑢 ∈ 𝐶such that the sequence{𝑦_𝑛}described by (20)is bounded in𝐶. Then,

(a) 𝐹(F) ̸= 0.

(b) {𝑦_𝑛}converges strongly to an element𝑦^∗ ∈ 𝐹(F)which holds the inequality

⟨𝑦^∗− 𝑢, 𝐽 (𝑦^∗−V)⟩ ≤ 0 ∀V∈ 𝐹 (F) . (26)

Proof. ByProposition 6(b), we have𝑦_𝑛−𝑇(𝑡)𝑦_𝑛 → 0as𝑛 →

∞for all𝑡 > 0. Since {𝑦_𝑛}is bounded, there exists a sub- sequence{𝑦_𝑛𝑖}of{𝑦_𝑛}such that𝑦_𝑛𝑖⇀ 𝑦^∗∈ 𝐶.

(a) For each𝑟 > 0, we have

󵄩󵄩󵄩󵄩𝑇(𝑟)^𝑚𝑥 − 𝑇(𝑟)^𝑚𝑦󵄩󵄩󵄩󵄩

= 󵄩󵄩󵄩󵄩𝑇 (𝑚𝑟) 𝑥 − 𝑇 (𝑚𝑟) 𝑦󵄩󵄩󵄩󵄩

≤ 𝑘 (𝑚𝑟) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩 ∀𝑥,𝑦 ∈ 𝐶, 𝑚 ∈N,

(27)

that is, each𝑇(𝑟)is asymptotically nonexpansive mapping.

Since lim_{𝑛 → ∞}‖𝑦_𝑛− 𝑇(𝑡)𝑦_𝑛‖ = 0for all𝑡 > 0, it follows from Lemma 3that𝑇(𝑡)𝑦^∗ = 𝑦^∗for all𝑡 > 0. Hence,𝐹(F) ̸= 0.

(b) Since{𝑦_𝑛}is bounded, there exists a constant𝑀 ≥ 0 such that‖𝑦_𝑛− 𝑝‖ ≤ 𝑀for all𝑛inNand𝑝in𝐹(F).

For anyVin𝐹(F), from (15) andRemark 2, we have Φ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑦^∗󵄩󵄩󵄩󵄩)

= Φ (󵄩󵄩󵄩󵄩𝑏^𝑛𝑢 + (1 − 𝑏_𝑛) 𝑇 (𝑡_𝑛) 𝑦_𝑛− 𝑦^∗󵄩󵄩󵄩󵄩)

≤ Φ ((1 − 𝑏_𝑛) 󵄩󵄩󵄩󵄩𝑇 (𝑡^𝑛) 𝑦_𝑛− 𝑦^∗󵄩󵄩󵄩󵄩) + 𝑏_𝑛⟨𝑢 − 𝑦^∗, 𝐽_𝜑(𝑦_𝑛− 𝑦^∗)⟩

≤ Φ ((1 − 𝑏_𝑛) 𝑘 (𝑡_𝑛) 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑦^∗󵄩󵄩󵄩󵄩) + 𝑏_𝑛⟨𝑢 − 𝑦^∗, 𝐽_𝜑(𝑦_𝑛− 𝑦^∗)⟩

≤ (1 − 𝑏_𝑛) 𝑘 (𝑡_𝑛) Φ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑦^∗󵄩󵄩󵄩󵄩) + 𝑏_𝑛⟨𝑢 − 𝑦^∗, 𝐽_𝜑(𝑦_𝑛− 𝑦^∗)⟩ .

(28)

Since𝑘(𝑡) ≥ 1for all𝑡 > 0, we have Φ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑦^∗󵄩󵄩󵄩󵄩) ≤ 𝑘(𝑡^𝑛) − 1

𝑏_𝑛 Φ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑦^∗󵄩󵄩󵄩󵄩) + ⟨𝑢 − 𝑦^∗, 𝐽_𝜑(𝑦_𝑛− 𝑦^∗)⟩

≤ 𝑘 (𝑡_𝑛) − 1

𝑏_𝑛 Φ (𝑀)+⟨𝑢 − 𝑦^∗, 𝐽_𝜑(𝑦_𝑛− 𝑦^∗)⟩ . (29) Observing that(𝑘(𝑡_𝑛𝑖) − 1)/𝑏_𝑛𝑖 → 0, 𝑦_𝑛𝑖 ⇀ 𝑦^∗, and𝐽_𝜑 is weakly continuous, we conclude from (29) that𝑦_𝑛𝑖 → 𝑦^∗ ∈ 𝐶as𝑖 → ∞because𝐶is closed.

We prove that {𝑦_𝑛} converges strongly to𝑦^∗. Suppose, for contradiction, that{𝑦_𝑛𝑗}is another subsequence of{𝑦_𝑛} such that𝑦_𝑛𝑗 → 𝑧^∗ ∈ 𝐶with𝑧^∗ ̸= 𝑦^∗. Since lim_{𝑛 → ∞}(𝑦_𝑛− 𝑇(𝑡)𝑦_𝑛) = 0for all𝑡 > 0, we have𝑧^∗ ∈ 𝐹(F). For anyVin 𝐹(F), we have

⟨𝑦_𝑛− 𝑇 (𝑡_𝑛) 𝑦_𝑛, 𝐽_𝜑(𝑦_𝑛−V)⟩

= ⟨𝑦_𝑛−V+ 𝑇 (𝑡_𝑛)V− 𝑇 (𝑡_𝑛) 𝑦_𝑛, 𝐽_𝜑(𝑦_𝑛−V)⟩

= 󵄩󵄩󵄩󵄩𝑦^𝑛−V󵄩󵄩󵄩󵄩𝜑(󵄩󵄩󵄩󵄩𝑦^𝑛−V󵄩󵄩󵄩󵄩)

− ⟨𝑇 (𝑡_𝑛) 𝑦_𝑛− 𝑇 (𝑡_𝑛)V, 𝐽_𝜑(𝑦_𝑛−V)⟩

≥ − (𝑘 (𝑡_𝑛) − 1) 󵄩󵄩󵄩󵄩𝑦^𝑛−V󵄩󵄩󵄩󵄩𝜑(󵄩󵄩󵄩󵄩𝑦^𝑛−V󵄩󵄩󵄩󵄩) ∀𝑛 ∈N.

(30)

From (20), we have

⟨𝑦_𝑛− 𝑢, 𝐽_𝜑(𝑦_𝑛−V)⟩

= (1 − 𝑏_𝑛) ⟨𝑇 (𝑡_𝑛) 𝑦_𝑛− 𝑢, 𝐽_𝜑(𝑦_𝑛−V)⟩

= (1 − 𝑏_𝑛) ⟨𝑇 (𝑡_𝑛) 𝑦_𝑛− 𝑦_𝑛+ 𝑦_𝑛− 𝑢, 𝐽_𝜑(𝑦_𝑛−V)⟩ . (31)

It follows that

⟨𝑦_𝑛− 𝑢, 𝐽_𝜑(𝑦_𝑛−V)⟩

≤ 1 − 𝑏_𝑛

𝑏_𝑛 ⟨𝑇 (𝑡_𝑛) 𝑦_𝑛− 𝑦_𝑛, 𝐽_𝜑(𝑦_𝑛−V)⟩

≤ (1 − 𝑏_𝑛)𝑘 (𝑡_𝑛) − 1

𝑏_𝑛 󵄩󵄩󵄩󵄩𝑦^𝑛−V󵄩󵄩󵄩󵄩𝜑(󵄩󵄩󵄩󵄩𝑦^𝑛−V󵄩󵄩󵄩󵄩)

≤ 𝑘 (𝑡_𝑛) − 1

𝑏_𝑛 𝑀𝜑 (𝑀)

(32)

for allVin𝐹(F)and𝑛inN. Since lim_{𝑛 → ∞}(𝑘(𝑡_𝑛) − 1)/𝑏_𝑛 = 0, we obtain from (32) that

⟨𝑦^∗− 𝑢, 𝐽_𝜑(𝑦^∗− 𝑧^∗)⟩ ≤ 0,

⟨𝑧^∗− 𝑢, 𝐽_𝜑(𝑧^∗− 𝑦^∗)⟩ ≤ 0. (33) Addition of (33) yields

⟨𝑦^∗− 𝑧^∗, 𝐽_𝜑(𝑦^∗− 𝑧^∗)⟩ ≤ 0. (34) Hence,𝑦^∗ = 𝑧^∗, a contradiction. Therefore,{𝑦_𝑛}converges strongly to𝑦^∗∈ 𝐶.

Finally, from (32), we conclude that𝑦^∗satisfies (26).

LetF = {𝑇(𝑡) : 𝑡 ∈R⁺}be a strongly continuous semi- group of asymptotically nonexpansive mappings from𝐶into itself and𝑢 ∈ 𝐶. Motivated by Morales and Jung [22, Theorem 1] and Morales [23, Theorem 2], we define

𝐸^𝑢_F(𝐶) = {𝜆𝑢 + (1 − 𝜆) 𝑇 (𝑡) 𝑥 : 𝑥 ∈ 𝐶, 𝜆 ∈ [0, 1] , 𝑡 > 0} . (35) Corollary 9. Let 𝑋be a real reflexive Banach space with a weakly continuous duality mapping𝐽_𝜑with gauge function𝜑 and𝐶a nonempty weakly closed subset of𝑋. LetF= {𝑇(𝑡) : 𝑡 ∈R⁺}be a uniformly continuous semigroup of asymptotically nonexpansive mappings from𝐶into itself with a net{𝑘(𝑡) : 𝑡 ∈ (0, +∞)}. Let{𝑏_𝑛}be a sequence in(0, 1)and{𝑡_𝑛}a sequence in (0, +∞)with𝑘(𝑡_𝑛) − 1 < 𝑏_𝑛for all𝑛inNsatisfying condition (25). Assume that𝑢 ∈ 𝐶such that𝐶is star shaped with respect to𝑢, and set𝐸^𝑢_F(𝐶)defined by(35)is bounded.

(a)For each𝑛 inN, there is exactly one point𝑦_𝑛 in 𝐶 described by(20).

(b)𝐹(F)is nonempty. Moreover,{𝑦_𝑛}converges strongly to an element𝑦^∗in𝐹(F)which holds(26).

Corollary 9 improves and generalizes several recent results of this nature. Indeed, it extends [13, Theorem 1.7]

from the class of asymptotically nonexpansive mappings to a uniformly continuous semigroup of asymptotically nonex- pansive mappings without uniformly asymptotic regularity assumption. In particular,Corollary 9improves Theorem ZS in the following ways.

(1) Convexity of𝐶is not required.

(2) The assumption “uniform convexity” of the underly- ing space is not required.

(3) For convergence of{𝑦_𝑛}, condition “𝐹(F)is a sunny nonexpansive retract of𝐶” is not assumed.

Next, we show that𝐹(F)is a nonempty sunny nonexpan- sive retract of𝐶.

Theorem 10. Let𝑋 be a real reflexive Banach space with a weakly continuous duality mapping𝐽_𝜑with gauge function𝜑.

Let𝐶be a nonempty closed convex-bounded subset of𝑋. Let F= {𝑇(𝑡) : 𝑡 ∈R⁺}be a uniformly continuous semigroup of asymptotically nonexpansive mappings from𝐶into itself with net{𝑘(𝑡) : 𝑡 ∈ (0, +∞)}. Then,𝐹(F)is a nonempty sunny nonexpansive retract of𝐶.

Proof. ByTheorem 8(a),𝐹(F) ̸= 0. As inTheorem 8(b), for each𝑢in𝐶the sequence{𝑦_𝑛}converges strongly to an ele- ment𝑦^∗in𝐹(F). Define a mapping𝑄 : 𝐶 → 𝐹(F)by

𝑄𝑢 =_{𝑛 → ∞}lim𝑦_𝑛. (36)

By (32), we have

⟨𝑦_𝑛− 𝑢, 𝐽_𝜑(𝑦_𝑛−V)⟩ ≤ 𝑘 (𝑡_𝑛) − 1

𝑏_𝑛 diam(𝐶) 𝜑 (diam(𝐶)) (37) for all𝑢in𝐶,Vin𝐹(F), and𝑛inN, where diam(𝐵)is the diameter of set𝐵. Letting𝑛 → ∞, we obtain that

⟨𝑄𝑢 − 𝑢, 𝐽_𝜑(𝑄𝑢 −V)⟩ ≤ 0 ∀V∈ 𝐹 (F) . (38) Therefore, byLemma 4, we conclude that𝑄is sunny nonex- pansive.

Remark 11. In view ofRemark 7(b), the nonemptℎness of the common fixed point set of a uniformly continuous semigroup of nonexpansive mappings implies that the sequence{𝑦_𝑛}in 𝐶described by (20) is bounded. So we can drop the bound- edness assumption of domain𝐶inTheorem 10, provided that 𝐹(F) ̸= 0.

Corollary 12. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping𝐽_𝜑 with gauge function 𝜑 and𝐶 a nonempty weakly closed subset of 𝑋. Let F = {𝑇(𝑡) : 𝑡 ∈R⁺}be a uniformly continuous semigroup of non- expansive mappings from𝐶into itself with𝐹(F) ̸= 0. Let{𝑏_𝑛} be a sequence in(0, 1)and{𝑡_𝑛}a sequence in(0, +∞)such that lim_{𝑛 → ∞}𝑡_𝑛=lim_{𝑛 → ∞}(𝑏_𝑛/𝑡_𝑛) = 0. Assume that𝑢 ∈ 𝐶and that 𝐶is star shaped with respect to𝑢.

(a)For each𝑛inN, there is exactly one point𝑦_𝑛in𝐶des- cribed by(20).

(b){𝑦_𝑛}converges strongly to an element𝑦^∗in𝐹(F)which holds(26).

Corollary 12is an improvement of [14, Theorem 3] and [15, Theorem 3.3], where strong convergence theorems were established in Hilbert and uniformly convex Banach spaces, respectively.

4. Approximation of Common Fixed Points

Theorem 13. Let 𝑋be a real reflexive Banach space with a weakly continuous duality mapping𝐽_𝜑with gauge function𝜑.

Let𝐶be a nonempty closed convex bounded subset of𝑋and F = {𝑇(𝑡) : 𝑡 ∈ R⁺}a uniformly continuous semigroup of asymptotically nonexpansive mappings from𝐶into itself with a net{𝑘(𝑡) : 𝑡 ∈ (0, +∞)}. For given𝑢, 𝑥₁in𝐶, let{𝑥_𝑛}be a sequence in𝐶generated by(7), where{𝛼_𝑛}is a sequence in (0, 1)and{𝑡_𝑛}is a decreasing sequence in(0, +∞)satisfying the following conditions.

(C1) lim_{𝑛 → ∞}𝑡_𝑛=lim_{𝑛 → ∞}(𝛼_𝑛/𝑡_𝑛) = 0and∑^∞_𝑛=1𝛼_𝑛= ∞, (C2) lim_{𝑛 → ∞}(|𝛼_𝑛−𝛼_𝑛+1|/𝛼_𝑛) = 0or∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛+1| < +∞, (C3) lim_{𝑛 → ∞}(‖𝑇(𝑡_𝑛− 𝑡_𝑛+1)𝑥_𝑛− 𝑥_𝑛‖/𝛼_𝑛+1) = 0,

(C4) (1 − 𝛼_𝑛)𝑘(𝑡_𝑛) ≤ 1for all𝑛 ∈Nandlim_{𝑛 → ∞}((𝑘(𝑡_𝑛) − 1)/𝛼_𝑛) = 0.

Then,

(a) 𝐹(F) ̸= 0.

(b) {𝑥_𝑛}converges strongly to𝑄_𝐹(F)(𝑢), where𝑄_𝐹(F)is a sunny nonexpansive retraction of𝐶onto𝐹(F).

Proof. It follows fromTheorem 10that𝐹(F) ̸= 0, and there is a sunny nonexpansive retraction𝑄_𝐹(F)of𝐶onto𝐹(F). Set 𝑦^∗ := 𝑄_𝐹(F)(𝑢),𝐵 := {𝑥_𝑛},𝐿 :=sup{𝑘(𝑡) : 𝑡 > 0}, and𝑀 :=

sup{‖𝑥_𝑛 − 𝑦^∗‖ : 𝑛 ∈ N}. Since𝐶is bounded, there exists a constant𝐾 > 0such that

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑥_𝑛󵄩󵄩󵄩󵄩 ≤ 𝐾, 󵄩󵄩󵄩󵄩𝑇(𝑡^𝑛) 𝑥_𝑛− 𝑢󵄩󵄩󵄩󵄩 ≤ 𝐾 ∀𝑛 ∈N. (39) Hence, for all𝑛 > 1, we have

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑥_𝑛󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩(𝛼^𝑛− 𝛼_𝑛−1) (𝑢 − 𝑇 (𝑡_𝑛−1) 𝑥_𝑛−1) + (1 − 𝛼_𝑛) (𝑇 (𝑡_𝑛) 𝑥_𝑛− 𝑇 (𝑡_𝑛−1) 𝑥_𝑛−1)󵄩󵄩󵄩󵄩

≤ 󵄨󵄨󵄨󵄨𝛼^𝑛− 𝛼_𝑛−1󵄨󵄨󵄨󵄨𝐾

+ (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑇 (𝑡^𝑛) 𝑥_𝑛− 𝑇 (𝑡_𝑛−1) 𝑥_𝑛−1󵄩󵄩󵄩󵄩

≤ 󵄨󵄨󵄨󵄨𝛼^𝑛− 𝛼_𝑛−1󵄨󵄨󵄨󵄨𝐾

+ (1 − 𝛼_𝑛) (󵄩󵄩󵄩󵄩𝑇 (𝑡^𝑛) 𝑥_𝑛− 𝑇 (𝑡_𝑛) 𝑥_𝑛−1󵄩󵄩󵄩󵄩

+ 󵄩󵄩󵄩󵄩𝑇 (𝑡^𝑛) 𝑥_𝑛−1− 𝑇 (𝑡_𝑛−1) 𝑥_𝑛−1󵄩󵄩󵄩󵄩)

≤ 󵄨󵄨󵄨󵄨𝛼^𝑛− 𝛼_𝑛−1󵄨󵄨󵄨󵄨𝐾 + (1 − 𝛼^𝑛) 𝑘 (𝑡_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥_𝑛−1󵄩󵄩󵄩󵄩

+ 󵄩󵄩󵄩󵄩𝑇 (𝑡^𝑛) 𝑥_𝑛−1− 𝑇 (𝑡_𝑛−1) 𝑥_𝑛−1󵄩󵄩󵄩󵄩

≤ (1 − 𝛼_𝑛) 𝑘 (𝑡_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥_𝑛−1󵄩󵄩󵄩󵄩

+ 󵄨󵄨󵄨󵄨𝛼^𝑛− 𝛼_𝑛−1󵄨󵄨󵄨󵄨𝐾 + 𝑘(𝑡^𝑛) 󵄩󵄩󵄩󵄩𝑇 (𝑡^𝑛−1− 𝑡_𝑛) 𝑥_𝑛−1− 𝑥_𝑛−1󵄩󵄩󵄩󵄩

≤ (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥_𝑛−1󵄩󵄩󵄩󵄩 + 󵄨󵄨󵄨󵄨𝛼^𝑛− 𝛼_𝑛−1󵄨󵄨󵄨󵄨𝐾 + 𝐿 󵄩󵄩󵄩󵄩𝑇 (𝑡^𝑛−1− 𝑡_𝑛) 𝑥_𝑛−1− 𝑥_𝑛−1󵄩󵄩󵄩󵄩

+ (1 − 𝛼_𝑛) (𝑘 (𝑡_𝑛) − 1) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥_𝑛−1󵄩󵄩󵄩󵄩

≤ (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥_𝑛−1󵄩󵄩󵄩󵄩

+ 󵄨󵄨󵄨󵄨𝛼^𝑛− 𝛼_𝑛−1󵄨󵄨󵄨󵄨𝐾 + 𝐿󵄩󵄩󵄩󵄩𝑇(𝑡^𝑛−1− 𝑡_𝑛) 𝑥_𝑛−1− 𝑥_𝑛−1󵄩󵄩󵄩󵄩

+ (𝑘 (𝑡_𝑛) − 1) 𝐾.

(40) Hence, byLemma 5and assumptions (C1)∼(C4), we conclude that𝑥_𝑛+1− 𝑥_𝑛 → 0as𝑛 → ∞. Fix𝑡 > 0, we have

󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇 (𝑡) 𝑥_𝑛󵄩󵄩󵄩󵄩

≤^[𝑡/𝑡∑^𝑛]−1

𝑖=0 󵄩󵄩󵄩󵄩𝑇(𝑖𝑡^𝑛) 𝑥_𝑛− 𝑇 ((𝑖 + 1) 𝑡𝑛) 𝑥_𝑛󵄩󵄩󵄩󵄩

+󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩𝑇([ 𝑡

𝑡_𝑛] 𝑡_𝑛) 𝑥_𝑛− 𝑇 (𝑡) 𝑥_𝑛󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩

≤ [𝑡

𝑡_𝑛] 𝐿 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇 (𝑡_𝑛) 𝑥_𝑛󵄩󵄩󵄩󵄩

+ 𝐿󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩𝑇(𝑡 − [𝑡

𝑡_𝑛] 𝑡_𝑛) 𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩

≤ 𝐿 [𝑡

𝑡_𝑛] (󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥_𝑛+1󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑇 (𝑡_𝑛) 𝑥_𝑛󵄩󵄩󵄩󵄩) + 𝐿󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩𝑇(𝑡 − [𝑡

𝑡_𝑛] 𝑡_𝑛) 𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩

≤ 𝐿 (𝑡

𝑡_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥_𝑛+1󵄩󵄩󵄩󵄩

+ 𝑡𝐿 (𝛼_𝑛

𝑡_𝑛) 󵄩󵄩󵄩󵄩𝑢 − 𝑇 (𝑡^𝑛) 𝑥_𝑛󵄩󵄩󵄩󵄩

+ 𝐿 max{󵄩󵄩󵄩󵄩𝑇 (𝑠) 𝑥^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩 : 0 ≤ 𝑠 ≤ 𝑡^𝑛}

(41) for all𝑛inN, which gives that𝑥_𝑛− 𝑇(𝑡)𝑥_𝑛 → 0as𝑛 → ∞.

We may assume that𝑥_𝑛𝑖 ⇀ 𝑝 ∈ 𝐶as𝑖 → ∞. In view of the assumption that the duality mapping𝐽_𝜑is weakly sequentially continuous, it follows fromLemma 3that𝑝 ∈ 𝐹(F). Thus, by the weak continuity of𝐽_𝜑andLemma 4, we have

lim sup

𝑛 → ∞ ⟨𝑢 − 𝑄_𝐹(F)(𝑢) , 𝐽_𝜑(𝑥_𝑛− 𝑄_𝐹(F)(𝑢))⟩

= lim

𝑖 → ∞⟨𝑢 − 𝑄_𝐹(F)(𝑢) , 𝐽_𝜑(𝑥_𝑛𝑖− 𝑄_𝐹(F)(𝑢))⟩

= ⟨𝑢 − 𝑄_𝐹(F)(𝑢) , 𝐽_𝜑(𝑝 − 𝑄_𝐹(F)(𝑢))⟩

≤ 0.

(42) Define𝛾_𝑛 := max{0, ⟨𝑢 − 𝑦^∗, 𝐽_𝜑(𝑥_𝑛+1− 𝑦^∗)⟩}. From (7), we obtain

Φ (󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑦^∗󵄩󵄩󵄩󵄩)

= Φ (󵄩󵄩󵄩󵄩𝛼^𝑛(𝑢 − 𝑦^∗) + (1 − 𝛼_𝑛) (𝑇 (𝑡_𝑛) 𝑥_𝑛− 𝑦^∗)󵄩󵄩󵄩󵄩)

≤ Φ ((1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑇^𝑛𝑥_𝑛− 𝑦^∗󵄩󵄩󵄩󵄩) + 𝛼_𝑛⟨𝑢 − 𝑦^∗, 𝐽_𝜑(𝑥_𝑛+1− 𝑦^∗)⟩

≤ Φ ((1 − 𝛼_𝑛) 𝑘 (𝑡_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦^∗󵄩󵄩󵄩󵄩) + 𝛼^𝑛𝛾_𝑛.

(43)

Since(1 − 𝛼_𝑛)𝑘(𝑡_𝑛) ≤ 1for all𝑛inN, it follows from Remark 2that

Φ (󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑦^∗󵄩󵄩󵄩󵄩)

≤ (1 − 𝛼_𝑛) 𝑘 (𝑡_𝑛) Φ (󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦^∗󵄩󵄩󵄩󵄩) + 𝛼^𝑛𝛾_𝑛

≤ (1 − 𝛼_𝑛) Φ (󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦^∗󵄩󵄩󵄩󵄩) + 𝛼_𝑛𝛾_𝑛+ (𝑘 (𝑡_𝑛) − 1) Φ (𝑀) .

(44)

Note that lim_{𝑛 → ∞}𝛾_𝑛 = 0and lim_{𝑛 → ∞}((𝑘(𝑡_𝑛) − 1)/𝛼_𝑛) = 0.

UsingLemma 5, we obtain that{𝑥_𝑛}converges strongly to𝑦^∗. One can carry overTheorem 13to the so-called viscosity approximation technique (see Xu [21]). We derive a more general result in this direction which is an improvement upon several convergence results in the context of viscosity approximation technique.

Theorem 14. Let 𝑋 be a real reflexive Banach space with a weakly continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let𝐶be a nonempty closed convex-bounded subset of𝑋, 𝑓 : 𝐶 → 𝐶a weakly contraction with function𝜓, andF= {𝑇(𝑡) : 𝑡 ∈R⁺}a uniformly continuous semigroup of asympto- tically nonexpansive mappings from𝐶 into itself with a net {𝑘(𝑡) : 𝑡 ∈ (0, +∞)}. For an arbitrary initial value𝑥₁ in𝐶, let{𝑥_𝑛}be a sequence in𝐶generated by

𝑥_𝑛+1= 𝛼_𝑛𝑓𝑥_𝑛+ (1 − 𝛼_𝑛) 𝑇 (𝑡_𝑛) 𝑥_𝑛 ∀𝑛 ∈N. (45) Here, {𝛼_𝑛} is a sequence in (0, 1), and {𝑡_𝑛} is a decreasing sequence in(0, +∞)satisfying conditions (C1)∼(C4). Then,

(a)𝐹(F) ̸= 0.

(b){𝑥_𝑛} converges strongly to 𝑥^∗ in𝐹(F), where 𝑥^∗ = 𝑄_𝐹(F)(𝑓𝑥^∗)and𝑄_𝐹(F)is a sunny nonexpansive retrac- tion of𝐶onto𝐹(F).

Proof. It follows fromTheorem 10that 𝐹(F) ̸= 0, and there is a sunny nonexpansive retraction𝑄_𝐹(F) of𝐶onto𝐹(F).

Since𝑄_𝐹(F)𝑓is a weakly contractive mapping from𝐶into itself, it follows from [24, Theorem 1] that there exists a unique

element𝑥^∗in𝐶such that𝑥^∗ = 𝑄_𝐹(F)(𝑓𝑥^∗). Such𝑥^∗in𝐶is an element of𝐹(F). Now, we define a sequence{𝑧_𝑛}in𝐶by

𝑧_𝑛+1= 𝛼_𝑛𝑓𝑥^∗+ (1 − 𝛼_𝑛) 𝑇 (𝑡_𝑛) 𝑧_𝑛 ∀𝑛 ∈N. (46) ByTheorem 13, we have that𝑧_𝑛 → 𝑥^∗ = 𝑄_𝐹(F)(𝑓𝑥^∗). By boundedness of{𝑥_𝑛}and{𝑧_𝑛}, there exists a constant𝑀 ≥ 0 such that‖𝑥_𝑛− 𝑧_𝑛‖ ≤ 𝑀for all𝑛 ∈N. Observe that

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑧_𝑛+1󵄩󵄩󵄩󵄩

≤ 𝛼_𝑛󵄩󵄩󵄩󵄩𝑓𝑥^𝑛− 𝑓𝑥^∗󵄩󵄩󵄩󵄩

+ (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑇 (𝑡^𝑛) 𝑥_𝑛− 𝑇 (𝑡_𝑛) 𝑧_𝑛󵄩󵄩󵄩󵄩

≤ 𝛼_𝑛(󵄩󵄩󵄩󵄩𝑓𝑥^𝑛− 𝑓𝑧_𝑛󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑓𝑧^𝑛− 𝑓𝑥^∗󵄩󵄩󵄩󵄩) + (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑇 (𝑡^𝑛) 𝑥_𝑛− 𝑇 (𝑡_𝑛) 𝑧_𝑛󵄩󵄩󵄩󵄩

≤ 𝛼_𝑛(󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧_𝑛󵄩󵄩󵄩󵄩 − 𝜓(󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧_𝑛󵄩󵄩󵄩󵄩) + 󵄩󵄩󵄩󵄩𝑧^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩) + (1 − 𝛼_𝑛) 𝑘 (𝑡_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧_𝑛󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧_𝑛󵄩󵄩󵄩󵄩 − 𝛼^𝑛𝜓 (󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧_𝑛󵄩󵄩󵄩󵄩)

+ 𝛼_𝑛󵄩󵄩󵄩󵄩𝑧^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩 + (1 − 𝛼^𝑛) (𝑘 (𝑡_𝑛) − 1) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧_𝑛󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧_𝑛󵄩󵄩󵄩󵄩 − 𝛼^𝑛𝜓 (󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧_𝑛󵄩󵄩󵄩󵄩) + 𝛼_𝑛[󵄩󵄩󵄩󵄩𝑧^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩 + (𝑘(𝑡^𝑛) − 1)

𝛼_𝑛 𝑀] .

(47)

By [25, Lemma 3.2], we obtain‖ 𝑥_𝑛− 𝑧_𝑛 ‖ → 0. Therefore, 𝑥_𝑛 → 𝑥^∗ = 𝑄_𝐹(F)(𝑓𝑥^∗).

Theorem 15. Let 𝑋be a real reflexive Banach space with a weakly continuous duality mapping𝐽_𝜑with gauge function𝜑.

Let𝐶be a nonempty closed convex subset of𝑋and𝑓 : 𝐶 → 𝐶 a weakly contraction. LetF= {𝑇(𝑡) : 𝑡 ∈ R⁺}be a uniformly continuous semigroup of nonexpansive mappings from𝐶into itself with𝐹(F) ̸= 0. For given𝑥₁in𝐶, let{𝑥_𝑛}be a sequence in 𝐶generated by(45), where{𝛼_𝑛}is a sequence in(0, 1)and{𝑡_𝑛} is a decreasing sequence in(0, +∞)satisfying conditions (C1)∼

(C3). Then,{𝑥_𝑛}converges strongly to𝑥^∗ ∈ 𝐹(F), where𝑥^∗= 𝑄_𝐹(F)(𝑓𝑥^∗)and𝑄_𝐹(F)is a sunny nonexpansive retraction of𝐶 onto𝐹(F).

Remark 16. (a) For a related result concerning the strong convergence of the explicit iteration procedure𝑧_𝑛+1 := (𝜆/

𝑘_𝑛)𝑇^𝑛(𝑧_𝑛)to some fixed point of an asymptotically nonexpan- sive mapping𝑇on star-shaped domain in a reflexive Banach space with a weakly continuous duality mapping, we refer the reader to Schu [13].

(b) We remark that condition “‖𝑥_𝑛+1−𝑥_𝑛‖ ≤ 𝐷𝛼_𝑛for some 𝐷 > 0” implies that‖𝑥_𝑛+1− 𝑥_𝑛‖ → 0as𝛼_𝑛 → 0. Thus, the assumption “‖𝑥_𝑛+1− 𝑥_𝑛‖ ≤ 𝐷𝛼_𝑛for some𝐷 > 0” imposed in Theorem ZS is very strong. In our results, such assumption is avoided. Under a mild assumption,Theorem 13shows that the sequence{𝑥_𝑛}generated by (7) converges strongly to a common fixed point of a uniformly continuous semigroup of asymptotically nonexpansive mappings in a real Banach space without uniform convexity. Therefore,Theorem 13is a

significant improvement of a number of known results (e.g., Theorem ZS and [12, Theorem 4.7]) for semigroups of asym- ptotically nonexpansive mappings.Corollary 9and Theorem 13provide an affirmative answer to Problem1.

Acknowledgments

This research is supported partially by Taiwan NSC Grants 99-2115-M-037-002-MY3, 98-2923-E-037-001-MY3, 99-2115- M-110-007-MY3. The authors would like to thank the referees for their careful reading and helpful comments.

References

[1] K. Goebel and W. A. Kirk, “A fixed point theorem for asymp- totically nonexpansive mappings,”Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.

[2] T. D. Benavides and P. L. Ram´ırez, “Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings,”Proceedings of the American Mathematical Society, vol. 129, no. 12, pp. 3549–3557, 2001.

[3] F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,”Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.

[4] S. Reich, “Strong convergence theorems for resolvents of ac- cretive operators in Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980.

[5] L. C. Ceng, A. R. Khan, Q. H. Ansari, and J. C. Yao, “Viscosity approximation methods for strongly positive and monotone operators,”Fixed Point Theory, vol. 10, no. 1, pp. 35–71, 2009.

[6] L. C. Ceng, H. K. Xu, and J. C. Yao, “The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol.

69, no. 4, pp. 1402–1412, 2008.

[7] L. C. Ceng, N. C. Wong, and J. C. Yao, “Fixed point solutions of variational inequalities for a finite family of asymptotically non- expansive mappings without common fixed point assumption,”

Computers & Mathematics with Applications, vol. 56, no. 9, pp.

2312–2322, 2008.

[8] H. Fukhar-ud-din and A. R. Khan, “Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces,”Computers & Mathematics with Appli- cations, vol. 53, no. 9, pp. 1349–1360, 2007.

[9] H. Fukhar-ud-din, A. R. Khan, D. O. Regan, and R. P. Agarwal,

“An implicit iteration scheme with errors for a finite family of uniformly continuous mappings,”Functional Differential Equa- tions, vol. 14, no. 2–4, pp. 245–256, 2007.

[10] T. C. Lim and H. K. Xu, “Fixed point theorems for asymp- totically nonexpansive mappings,”Nonlinear Analysis: Theory, Methods & Applications, vol. 22, no. 11, pp. 1345–1355, 1994.

[11] D. R. Sahu, R. P. Agarwal, and D. O’Regan, “Structure of the fixed point set of asymptotically nonexpansive mappings in Banach spaces with weak uniformly normal structure,”Journal of Applied Analysis, vol. 17, no. 1, pp. 51–68, 2011.

[12] D. R. Sahu, Z. Liu, and S. M. Kang, “Iterative approaches to com- mon fixed points of asymptotically nonexpansive mappings,”

The Rocky Mountain Journal of Mathematics, vol. 39, no. 1, pp.

281–304, 2009.

[13] J. Schu, “Approximation of fixed points of asymptotically non- expansive mappings,”Proceedings of the American Mathematical Society, vol. 112, no. 1, pp. 143–151, 1991.

[14] T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,”Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2003.

[15] H. K. Xu, “A strong convergence theorem for contraction semi- groups in Banach spaces,”Bulletin of the Australian Mathemat- ical Society, vol. 72, no. 3, pp. 371–379, 2005.

[16] H. Zegeye and N. Shahzad, “Strong convergence theorems for continuous semigroups of asymptotically nonexpansive map- pings,”Numerical Functional Analysis and Optimization, vol. 30, no. 7-8, pp. 833–848, 2009.

[17] R. P. Agarwal, D. O’Regan, and D. R. Sahu,Fixed Point Theory for Lipschitzian-Type Mappings with Applications, vol. 6 of Topological Fixed Point Theory and Its Applications, Springer, New York, NY, USA, 2009.

[18] I. Cioranescu,Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dor- drecht, The Netherlands, 1990.

[19] F. E. Browder, “Convergence theorems for sequences of nonlin- ear operators in Banach spaces,”Mathematische Zeitschrift, vol.

100, pp. 201–225, 1967.

[20] K. Goebel and S. Reich,Uniform Convexity, Hyperbolic Geome- try, and Nonexpansive Mappings, Marcel Dekker, 1984.

[21] H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,”Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.

[22] C. H. Morales and J. S. Jung, “Convergence of paths for pseudo- contractive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 128, no. 11, pp. 3411–3419, 2000.

[23] C. H. Morales, “Strong convergence of path for continuous pseudo-contractive mappings,” Proceedings of the American Mathematical Society, vol. 135, no. 9, pp. 2831–2838, 2007.

[24] B. E. Rhoades, “Some theorems on weakly contractive maps,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no.

4, pp. 2683–2693, 2001.

[25] Y. I. Alber, C. E. Chidume, and H. Zegeye, “Approximating fixed points of total asymptotically nonexpansive mappings,”Fixed Point Theory and Applications, vol. 2006, Article ID 10673, 20 pages, 2006.

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