Mixed Approximation for Nonexpansive Mappings in Banach Spaces

Abstract

The mixed viscosity approximation is proposed for finding fixed points of nonexpansive mappings, and the strong convergence of the scheme to a fixed point of the nonexpansive mapping is proved in a real Banach space with uniformly Gâteaux differentiable norm. The theorem about Halpern type approximation for nonexpansive mappings is shown also. Our theorems extend and improve the correspondingly results shown recently.

1. Introduction and Preliminaries

Let 𝐸 be a real Banach space with norm ‖⋅‖, 𝐸^∗ denote the dual space of 𝐸, and ⟨⋅,⋅⟩ denote the generalized duality pairing. Let 𝐽∶𝐸→2^𝐸∗ denote the normalized duality mapping defined by

𝐽(𝑥)={𝑓^∗∈𝑋^∗∶⟨𝑥,𝑓^∗⟩=‖𝑓^∗‖‖𝑥‖,‖𝑓^∗‖=‖𝑥‖},∀𝑥∈𝐸. (1.1) It is well known that the following results: ∀𝑥,𝑦∈𝐸, ∀𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦), ∀𝑗(𝑥)∈𝐽(𝑥),

‖𝑥‖²+2⟨𝑦,𝑗(𝑥)⟩≤‖𝑥+𝑦‖²≤‖𝑥‖²+⟨𝑦,𝑗(𝑥+𝑦)⟩. (1.2)

Let 𝑈={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of Banach space 𝐸, the norm of 𝐸 is said to be Gâteaux differentiable if the limit

lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡 (1.3) exists for each 𝑥,𝑦∈𝑈. Such an 𝐸 is called a smooth Banach space. The norm of Banach space 𝐸 is said to be uniformly Gâteaux differentiable if for each 𝑦∈𝑈, the limit lim_𝑡→0((‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡) is attained uniformly for 𝑥∈𝑈. A Banach space 𝐸 is said to be strictly convex if

‖𝑥‖=‖𝑦‖=1,𝑥≠𝑦implies‖𝑥+𝑦‖2<1, (1.4) to be uniformly convex if for all 𝜀∈[0,2], ∃𝛿_𝜀>0 such that

‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀implies‖𝑥+𝑦‖2<1−𝛿_𝜀. (1.5)

It is well known that (see [1, 2]): (1) if 𝐸 has a uniformly Gâteaux differentiable norm, then 𝐽 is norm-to-weak∗ continuous on bounded set of 𝐸. (2) If a Banach space 𝐸 admits a sequentially continuous duality mapping 𝐽 from weak topology to weak star topology, then the duality mapping 𝐽 is single-valued. (3) Each uniformly convex Banach space 𝐸 is reflexive and strictly convex and has fixed point property for nonexpansive self-mappings; every uniformly smooth Banach space 𝐸 is a reflexive Banach space with a uniformly Gâteaux differentiable norm and has fixed point property for nonexpansive self-mappings.

Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸. If 𝐷 is a nonempty subset of 𝐶, then a mapping 𝑃∶𝐶→𝐷 is said to be a retraction if 𝑃𝑥=𝑥 for all 𝑥∈𝐷. A mapping 𝑃∶𝐶→𝐷 is said to be a sunny if 𝑃(𝑃𝑥+𝑡(𝑥−𝑃𝑥))=𝑃𝑥,∀𝑥∈𝐶 where 𝑡>0. A subset 𝐷 of 𝐶 is said to be a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction of 𝐶 onto 𝐷. For more details, see [2]. Note that every closed convex subset of a Hilbert space is a nonexpansive retract. In the sequel, we always take 𝑃 to denote the sunny nonexpansive retraction of 𝐸 onto 𝐶.

Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping; for a sequence {𝛼_𝑛}⊂(0,1) and a fixed contractive mapping 𝑓∶𝐶→𝐶, the sequence {𝑥_𝑛}, iteratively defined in 𝐶 by

𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑇𝑥_𝑛,𝑛=0,1,2,…, (1.6) is said to be viscosity approximation. If 𝑓(𝑥_𝑛)=𝑦₀ for a given 𝑦₀∈𝐶, it is called Halpern approximation which was first introduced by Halpern [3] in 1967. Under the following assumption:

(i)lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞,(ii)^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞(orlim_𝑛→∞𝛼_𝑛+1𝛼_𝑛=1), (1.7) Xu [4] proved the strong convergence of {𝑥_𝑛} to a fixed point of 𝑇 in Hilbert spaces and in uniformly smooth Banach spaces in 2004. In [3], Halpern pointed out that the condition (i) is necessary for the convergence of the Halpern approximation to a fixed point of 𝑇. At the same time, he put forth the following open problem: is the condition (i) a sufficient condition for the convergence of the Halpern approximation to a fixed point of 𝑇? which was put forward by Reich in [5] also. In order to answer the open question, many authors have done extensively some works; see [6–11] and the references therein. In [7–9], the strong convergence of the Halpern approximation depends on the convergence of the path 𝑥_𝑡=𝑡𝑦₀+(1−𝑡)𝑇𝑥_𝑡(𝑡∈(0,1)). In [6], Song got rid of the dependence on the convergence of the path 𝑥_𝑡, and proved the convergence of the Halpern approximation under the assumptions for {𝛼_𝑛} and 𝑇 as follows:

(iii)^∞_𝑛=0𝛼_𝑛=∞,^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞(orlim_𝑛→∞𝛼_𝑛+1𝛼_𝑛=1),(iv)𝐾_min𝐹(𝑇)≠∅, (1.8)

where 𝐾_min={𝑧∈𝐶∶𝒰_𝑛‖𝑥_𝑛−𝑧‖²=inf_𝑦∈𝐶𝒰_𝑛‖𝑥_𝑛−𝑦‖²} and 𝒰_𝑛 is a Banach limit. Recently, many authors have studied extensively the problem of approximating a fixed point of nonexpansive nonself-mappings 𝑇∶𝐶→𝐸 in a Banach space, by using the Halpern type iteration (see [12–16]) and the viscosity type iteration (see [17–23]).

In this work, on one hand, we will prove the strong convergence of the mixed viscosity iterative scheme, which is introduced as follows: for any chosen 𝑥₀∈𝐶,

𝑦_𝑛=(1−𝜃_𝑛)𝑥_𝑛+𝜃_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛]),∀𝑛≥0, (1.9) where 𝑓∶𝐶→𝐶 is a fixed contractive mapping, {𝛼_𝑛}, {𝛽_𝑛}, and {𝜃_𝑛} are the real number sequences in (0,1), to a fixed point of the nonexpansive mapping 𝑇∶𝐶→𝐶 in a real Banach space with uniformly Gâteaux differentiable norm under the following conditions:

lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞,0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,lim_𝑛→∞𝜃_𝑛=0,((iv)^′𝐾_min𝐹(𝑇)≠∅, (𝐶1))(1.10) where 𝐾_min={𝑧∈𝐾∶𝒰_𝑛‖𝑥_𝑛−𝑧‖²=inf_𝑦∈𝐶𝒰_𝑛‖𝑥_𝑛−𝑦‖²}, 𝒰_𝑛 is a Banach limit, and {𝑥_𝑛} is defined by (1.9). On the other hand, we will show that the condition (iv)^′ is not necessary for proving the strong convergence of the mixed viscosity iterative scheme. As the applications, we will show some results about mixed viscosity type approximation and Halpern type approximation for nonexpansive nonself-mappings also. Our theorems complement and generalize the corresponding results in [8, 9, 16, 24–26].

Now, we recall the following lemmas for proving our theorems firstly.

Lemma 1.1 (see [27]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝐸, {𝑡_𝑛}⊂[0,1] a sequence satisfying 0<liminf_𝑛→∞𝑡_𝑛≤limsup_𝑛→∞𝑡_𝑛<1. (1.11) Suppose that 𝑥_𝑛+1=𝑡_𝑛𝑦_𝑛+(1−𝑡_𝑛)𝑥_𝑛, for all 𝑛=0,1,2,…, and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. (1.12) Then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 1.2 (see [28]). Let {𝑎_𝑛}, {𝑏_𝑛} and, {𝑐_𝑛} be sequences of nonnegative real numbers such that 𝑎_𝑛+1≤(1−𝛼_𝑛)𝑎_𝑛+𝛼_𝑛𝑏_𝑛+𝑐_𝑛,∀𝑛≥0. If {𝛼_𝑛}⊂[0,1], ^∞_𝑛=1𝛼_𝑛=∞,limsup_𝑛→∞𝑏_𝑛≤0,^∞_𝑛=1𝑐_𝑛<∞, (1.13) then lim_𝑛→∞𝑎_𝑛=0.

2. Main Results

In this section, the mixed viscosity iterations for a contractive self-mapping 𝑓 for approximating to a fixed point of nonexpansive mapping 𝑇∶𝐶→𝐶 are studied in a real Banach space.

Theorem 2.1. Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}≠∅ and 𝑓∶𝐶→𝐶 a fixed contractive mapping with the contractive coefficient 𝑘∈(0,1). If the sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝜃_𝑛} in (0,1) satisfy (𝐶1) and for any 𝑥₀∈𝐶, the sequence {𝑥_𝑛} is defined as follows: 𝑦_𝑛=(1−𝜃_𝑛)𝑥_𝑛+𝜃_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛]),∀𝑛≥0; (2.1) then we obtain the following:(1){𝑥_𝑛} is bounded;(2)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.

Proof. First we show that {𝑥_𝑛} is bounded. Now let 𝑧∈𝐹(𝑇), then ‖𝑦_𝑛−𝑧‖=‖(1−𝜃_𝑛)(𝑥_𝑛−𝑧)+𝜃_𝑛(𝑇𝑥_𝑛−𝑇𝑧)‖≤‖𝑥_𝑛−𝑧‖;‖𝑥_𝑛+1−𝑧‖=‖𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛]−𝑧‖,≤𝛼_𝑛‖(𝑓(𝑥_𝑛)−𝑓(𝑧))+(𝑓(𝑧)−𝑧)‖+(1−𝛼_𝑛)‖(1−𝛽_𝑛)(𝑥_𝑛−𝑧)+𝛽_𝑛(𝑇𝑦_𝑛−𝑧)‖≤𝛼_𝑛𝑘‖𝑥_𝑛−𝑧‖+𝛼_𝑛‖𝑓(𝑧)−𝑧‖+(1−𝛼_𝑛)𝛽_𝑛‖𝑇𝑦_𝑛−𝑇𝑧‖+(1−𝛼_𝑛)(1−𝛽_𝑛)‖𝑥_𝑛−𝑧‖≤𝛼_𝑛𝑘‖𝑥_𝑛−𝑧‖+𝛼_𝑛‖𝑓(𝑧)−𝑧‖+(1−𝛼_𝑛)𝛽_𝑛‖𝑦_𝑛−𝑧‖+(1−𝛼_𝑛)(1−𝛽_𝑛)‖𝑥_𝑛−𝑧‖≤𝛼_𝑛‖𝑓(𝑧)−𝑧‖+(1−𝛼_𝑛(1−𝑘))‖𝑥_𝑛1−𝑧‖≤max{1−𝑘‖𝑓(𝑧)−𝑧‖,‖𝑥_𝑛1−𝑧‖}≤max{1−𝑘‖𝑓(𝑧)−𝑧‖,‖𝑥₀−𝑧‖}. (2.2) Therefore, {𝑥_𝑛} is bounded, so are {𝑇𝑥_𝑛}, {𝑦_𝑛}, {𝑇𝑦_𝑛}, and {𝑓(𝑥_𝑛)}. Next, we show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (2.3)
For all 𝑛≥0, let 𝑥_𝑛+1=𝛾_𝑛𝑥_𝑛+(1−𝛾_𝑛)𝑢_𝑛, (2.4) where 𝛾_𝑛=(1−𝛼_𝑛)(1−𝛽_𝑛)∈(0,1), then 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1. (2.5) Note that 𝑢_𝑛+1−𝑢_𝑛=𝑥_𝑛+2−𝛾_𝑛+1𝑥_𝑛+11−𝛾_𝑛+1−𝑥_𝑛+1−𝛾_𝑛𝑥_𝑛1−𝛾_𝑛=𝛼_𝑛+1𝑓(𝑥_𝑛+1)+(1−𝛼_𝑛+1)[(1−𝛽_𝑛+1)𝑥_𝑛+1+𝛽_𝑛+1𝑇𝑦_𝑛+1]−𝛾_𝑛+1𝑥_𝑛+11−𝛾_𝑛+1−𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛]−𝛾_𝑛𝑥_𝑛1−𝛾_𝑛=𝛼_𝑛+11−𝛾_𝑛+1(𝑓(𝑥_𝑛+1)+𝑇𝑦_𝑛+1𝛼)−^𝑛1−𝛾_𝑛(𝑓(𝑥_𝑛)+𝑇𝑦_𝑛)−(𝑇𝑦_𝑛+1−𝑇𝑦_𝑛),‖𝑇𝑦_𝑛+1−𝑇𝑦_𝑛‖≤‖𝑦_𝑛+1−𝑦_𝑛‖≤‖𝑥_𝑛+1−𝑥_𝑛‖+𝜃_𝑛+1‖𝑥_𝑛+1−𝑇𝑥_𝑛+1‖+𝜃_𝑛‖𝑥_𝑛−𝑇𝑥_𝑛‖. (2.6)(2.7) Hence, we have that ‖𝑢_𝑛+1−𝑢_𝑛𝛼‖≤^𝑛+11−𝛾_𝑛+1‖𝑓(𝑥_𝑛+1)+𝑇𝑦_𝑛+1𝛼‖+^𝑛1−𝛾_𝑛‖𝑓(𝑥_𝑛)+𝑇𝑦_𝑛‖+‖𝑥_𝑛+1−𝑥_𝑛‖+𝜃_𝑛+1‖𝑥_𝑛+1−𝑇𝑥_𝑛+1‖+𝜃_𝑛‖𝑥_𝑛−𝑇𝑥_𝑛‖. (2.8) Then it follows from the boundedness of {𝑥_𝑛}, {𝑇𝑥_𝑛}, {𝑇𝑦_𝑛}, and {𝑓(𝑥_𝑛)}, and (𝐶1) that limsup_𝑛→∞(‖𝑢_𝑛+1−𝑢_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. (2.9) It follow from (2.9) and Lemma 1.1 that lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0,lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (2.10)(2.11) Since 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)+(1−𝛼_𝑛)𝛽_𝑛(𝑇𝑦_𝑛−𝑥_𝑛𝑥),_𝑛−𝑦_𝑛=𝜃_𝑛(𝑥_𝑛−𝑇𝑥_𝑛),(1−𝛼_𝑛)𝛽_𝑛‖𝑇𝑦_𝑛−𝑥_𝑛‖≤‖𝑥_𝑛+1−𝑥_𝑛‖+𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑥_𝑛‖, (2.12)(2.13) from (𝐶1), (2.11) and, the boundedness of {𝑥_𝑛}, {𝑇𝑥_𝑛}, and {𝑓(𝑥_𝑛)}, we have lim_𝑛→∞‖𝑥_𝑛−𝑇𝑦_𝑛‖=0,lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. (2.14) From ‖𝑥_𝑛+1−𝑇𝑥_𝑛‖≤‖𝑇𝑦_𝑛−𝑇𝑥_𝑛‖+‖𝑥_𝑛−𝑇𝑦_𝑛‖+‖𝑥_𝑛+1−𝑥_𝑛‖≤‖𝑦_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑇𝑦_𝑛‖+‖𝑥_𝑛+1−𝑥_𝑛‖, (2.15) we obtain lim_𝑛→∞‖𝑥_𝑛+1−𝑇𝑥_𝑛‖=0. (2.16) Therefore, lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. This completes the proof.

Proposition 2.2 (see [18]). Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸 which has uniformly Gâteaux differentiable norm. Suppose that {𝑥_𝑛} is a bounded sequence of 𝐸 such that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and 𝒰_𝑛 is a Banach limit. If 𝑧∈𝐶 such that 𝒰_𝑛‖𝑥_𝑛−𝑧‖²=inf_𝑦∈𝐶𝒰_𝑛‖𝑥_𝑛−𝑦‖², then limsup_𝑛→∞⟨𝑦−𝑧,𝑗(𝑥_𝑛−𝑧)⟩≤0,∀𝑦∈𝐶. (2.17)

Let {𝑥_𝑛} be defined by (2.1) and 𝛼_𝑛∈(0,1), it follows from Theorem 2.1 that {𝑥_𝑛} is bounded. Let 𝜑(𝑦)=𝒰_𝑛‖𝑥_𝑛−𝑦‖²∀𝑦∈𝐶, (2.18) then 𝜑(𝑦) is convex and continuous. If 𝐸 is reflexive, there exists 𝑧∈𝐶 such that 𝜑(𝑧)=inf_𝑦∈𝐶𝜑(𝑦) (see [2], Theorem 1.3.11). Let

𝐾_min={𝑧∈𝐶∶𝜑(𝑧)=inf_𝑦∈𝐶𝜑(𝑦)}, (2.19) then 𝐾_min≠∅ is a closed convex subset in a reflexive Banach space 𝐸.

Theorem 2.3. Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸 which has uniformly Gâteaux differentiable norm. Suppose that 𝑇∶𝐶→𝐶 is a nonexpansive mapping with 𝐹(𝑇)≠∅, 𝑓 is a fixed contractive mapping with the contractive coefficient 𝑘∈(0,1), and the sequences {𝛼_𝑛}, {𝛽_𝑛} and {𝜃_𝑛} in (0,1) satisfy (𝐶1). If 𝐾_min⋂𝐹(𝑇)≠∅, then the sequence {𝑥_𝑛} defined by (2.1) converges strongly to a fixed point of 𝑇 as 𝑛→∞.

Proof. Take 𝑧∈𝐾_min⋂𝐹(𝑇). It follows from Theorem 2.1 that {𝑥_𝑛} is bounded and lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. For 𝑓(𝑧)∈𝐶, by Proposition 2.2, we have that limsup_𝑛→∞⟨𝑓(𝑧)−𝑧,𝑗(𝑥_𝑛−𝑧)⟩≤0. (2.20)
Next, we show that lim_𝑛→∞𝑥_𝑛=𝑧. Since 𝑥_𝑛+1−(𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑧)=(𝑥_𝑛+1−𝑧)−𝛼_𝑛(𝑓(𝑥_𝑛)−𝑧),‖𝑥_𝑛+1−(𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑧)‖≤‖(1−𝛼_𝑛)[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛]−(1−𝛼_𝑛)𝑧‖=‖(1−𝛼_𝑛)[(1−𝛽_𝑛)(𝑥_𝑛−𝑧)+𝛽_𝑛(𝑇𝑦_𝑛−𝑇𝑧)]‖≤(1−𝛼_𝑛)‖𝑥_𝑛−𝑧‖, (2.21) then ‖𝑥_𝑛+1−𝑧‖²=⟨𝑥_𝑛+1−(𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑧),𝑗(𝑥_𝑛+1−𝑧)⟩+𝛼_𝑛⟨𝑓(𝑥_𝑛)−𝑧,𝑗(𝑥_𝑛+1−𝑧)⟩≤‖𝑥_𝑛+1−(𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑧)‖‖𝑥_𝑛+1−𝑧‖+𝛼_𝑛(⟨𝑓(𝑥_𝑛)−𝑓(𝑧),𝑗(𝑥_𝑛+1−𝑧)⟩+⟨𝑓(𝑧)−𝑧,𝑗(𝑥_𝑛+1≤1−𝑧)⟩)2(1−𝛼_𝑛)(‖𝑥_𝑛−𝑧‖²+‖𝑥_𝑛+1−𝑧‖²1)+2𝛼_𝑛(‖𝑓(𝑥_𝑛)−𝑓(𝑧)‖²+‖𝑥_𝑛+1−𝑧‖²)+𝛼_𝑛⟨𝑓(𝑧)−𝑧,𝑗(𝑥_𝑛+1≤1−𝑧)⟩2(1−𝛼_𝑛(1−𝑘²))‖𝑥_𝑛−𝑧‖²+12‖𝑥_𝑛+1−𝑧‖²+𝛼_𝑛⟨𝑓(𝑧)−𝑧,𝑗(𝑥_𝑛+1−𝑧)⟩. (2.22) Hence, ‖𝑥_𝑛+1−𝑧‖²≤(1−𝛼_𝑛(1−𝑘²))‖𝑥_𝑛−𝑧‖²+2𝛼_𝑛⟨𝑓(𝑧)−𝑧,𝑗(𝑥_𝑛+1−𝑧)⟩. (2.23) Let 𝑎_𝑛=‖𝑥_𝑛−𝑧‖² and 𝑏_𝑛=(2/(1−𝑘²))⟨𝑦−𝑧,𝑗(𝑥_𝑛+1−𝑧)⟩; it follows from Lemma 1.2 that {𝑥_𝑛} converges strongly to 𝑧∈𝐶. This completes the proof.

Proposition 2.4. Let 𝐸 be a real reflexive Banach space with uniformly Gâteaux differentiable norm, 𝐶 a nonempty closed convex subset of 𝐸, 𝑇∶𝐶→𝐶 a nonexpansive mapping with 𝐹(𝑇)≠∅, and {𝑥_𝑛} defined by (2.1). Then 𝐾_min⋂𝐹(𝑇)≠∅.

Proof. From the reflexivity of 𝐸 and the definition of 𝐾_min, it follows that 𝐾_min is a nonempty closed convex subset of 𝐸. By Theorem 2.1, we know that lim_𝑛→∞‖𝑥_𝑛+1−𝑇𝑥_𝑛‖=0.
Claim that 𝑇(𝐾_min)⊂𝐾_min. Indeed, for any 𝑥∈𝐾_min, we have 𝜑(𝑇𝑥)=𝒰_𝑛‖𝑥_𝑛−𝑇𝑥‖²=𝒰_𝑛‖𝑥_𝑛+1−𝑇𝑥‖²≤𝒰_𝑛(‖𝑥_𝑛+1−𝑇𝑥_𝑛‖+‖𝑇𝑥_𝑛−𝑇𝑥‖)²≤𝒰_𝑛‖𝑥_𝑛−𝑥‖²=𝜑(𝑥). (2.24) Therefore, 𝑇𝑥∈𝐾_min and 𝑇(𝐾_min)⊂𝐾_min.
Since 𝐹(𝑇)≠∅, there exists unique 𝑢∈𝐾_min such that ‖𝑧−𝑢‖=inf_𝑥∈𝐾_min‖𝑧−𝑥‖, for all 𝑧∈𝐹(𝑇). By 𝑇𝑧=𝑧 and 𝑇𝑢∈𝐾_min, we have ‖𝑧−𝑇𝑢‖=‖𝑇𝑧−𝑇𝑢‖≤‖𝑧−𝑢‖. (2.25) Hence 𝑢=𝑇𝑢 by the uniqueness of 𝑢∈𝐾_min. Thus 𝐾_min⋂𝐹(𝑇)≠∅.

By the above results, we can obtain the following theorem.

Theorem 2.5. Let 𝐸 be a real reflexive Banach space with uniformly Gâteaux differentiable norm, 𝐶 a nonempty closed convex subset of 𝐸, 𝑇∶𝐶→𝐶 a nonexpansive mapping with 𝐹(𝑇)≠∅, and 𝑓 a fixed contractive mapping with the contractive coefficient 𝑘∈(0,1). If the sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝜃_𝑛} in (0,1) satisfy (𝐶1), then the sequence {𝑥_𝑛} defined by (2.1) converges strongly to a fixed point of 𝑇.

Remark 2.6. Theorem 2.5 shakes off the assumption lim_𝑛→∞‖𝑇𝑦_𝑛−𝑦_𝑛‖=0 in [26] and extends Theorem 1 in [24] shown in uniformly smooth Banach spaces.

Theorem 2.7. Let 𝐸 be a real strictly convex Banach space 𝐸 with uniformly Gâteaux differentiable norm and 𝐶 a nonempty closed convex subset of 𝐸 which is a sunny nonexpansive retract of 𝐸. Let 𝑇∶𝐶→𝐸 be a nonexpansive nonself-mapping with 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}≠∅ and 𝑓∶𝐶→𝐶 a fixed contractive mapping with the contractive coefficient 𝑘∈(0,1). Suppose that the sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝜃_𝑛} in (0,1) satisfy (𝐶1), 𝐶 is a sunny nonexpansive retract of 𝐸, and the sequence {𝑥_𝑛} is defined as follows: forany𝑥₀𝑦∈𝐶,_𝑛=(1−𝜃_𝑛)𝑥_𝑛+𝜃_𝑛𝑃𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑃𝑇𝑦_𝑛]),∀𝑛≥0; (2.26) If {𝑧∈𝐶∶𝒰_𝑛‖𝑥_𝑛−𝑧‖²=inf_𝑦∈𝐶𝒰_𝑛‖𝑥_𝑛−𝑦‖²}⋂𝐹(𝑇)≠∅, then the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇 as 𝑛→∞.

Proof. It follows from [12, Lemmas 3.1 and 3.3] that 𝐹(𝑇)=𝐹(𝑃𝑇), and then 𝐹(𝑃𝑇)≠∅. By replacing 𝑇 by 𝑃𝑇 in Theorem 2.3, we can show that the conclusion holds.

Theorem 2.8. Let 𝐸 be a real reflexive and strictly convex Banach space with uniformly Gâteaux differentiable norm, 𝐶 a nonempty closed convex subset of a real Banach space 𝐸 which is a sunny nonexpansive retract of 𝐸, 𝑇∶𝐶→𝐸 a nonexpansive nonself-mapping with 𝐹(𝑇)≠∅, and 𝑓 a fixed contractive mapping with the contractive coefficient 𝑘∈(0,1). If the sequences {𝛼_𝑛}, {𝛽_𝑛} and {𝜃_𝑛} in (0,1) satisfy (𝐶1), then the sequence {𝑥_𝑛} defined by (2.26) converges strongly to a fixed point of 𝑇.

Proof. It follows from [12, Lemmas 3.1 and 3.3] that 𝐹(𝑇)=𝐹(𝑃𝑇), and then 𝐹(𝑃𝑇)≠∅. By replacing 𝑇 by 𝑃𝑇 in Theorem 2.5, we can show that the conclusion holds.

Corollary 2.9. Let 𝐸 be a real uniformly convex Banach space with uniformly Gâteaux differentiable norm, 𝐶, 𝑇, 𝑓, {𝛼_𝑛}, and {𝛽_𝑛} as Theorem 2.8. Then the sequence {𝑥_𝑛} defined by (2.1) converges strongly to a fixed point of 𝑇.

3. Some Applications

In this section, we introduce the following Halpern type approximation: for the given 𝑥₀,𝑢₀∈𝐶, the sequence {𝑥_𝑛} is defined by

𝑦_𝑛=(1−𝜃_𝑛)𝑥_𝑛+𝜃_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢₀+(1−𝛼_𝑛)[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛]),∀𝑛≥0, (3.1) and show some results about Halpern type approximation for nonexpansive mappings, which generalize and improve some known conclusions.

Define 𝐶_min={𝑧∈𝐶∶𝜑(𝑧)=inf_𝑦∈𝐶𝜑(𝑦)}, where 𝜑(𝑦)=𝒰_𝑛‖𝑥_𝑛−𝑦‖² for all 𝑦∈𝐶 and {𝑥_𝑛} is defined by (3.1).

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸 which has uniformly Gâteaux differentiable norm. Suppose that 𝑇∶𝐶→𝐶 is a nonexpansive mapping with 𝐹(𝑇)≠∅, 𝑓 is a fixed contractive mapping with the contractive coefficient 𝑘∈(0,1), and the sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝜃_𝑛} in (0,1) satisfy (𝐶1). If 𝐶_min⋂𝐹(𝑇)≠∅, then the sequence {𝑥_𝑛} defined by (3.1) converges strongly to a fixed point of 𝑇 as 𝑛→∞.

Proof. First we show that {𝑥_𝑛} is bounded. Now let 𝑧∈𝐹(𝑇), then ‖𝑦_𝑛−𝑧‖=‖(1−𝜃_𝑛)(𝑥_𝑛−𝑧)+𝜃_𝑛(𝑇𝑥_𝑛−𝑇𝑧)‖≤‖𝑥_𝑛−𝑧‖,‖𝑥_𝑛+1−𝑧‖=‖(𝛼_𝑛𝑢₀+(1−𝛼_𝑛)[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛])−𝑧‖≤𝛼_𝑛‖𝑢₀−𝑧‖+(1−𝛼_𝑛)‖(1−𝛽_𝑛)(𝑥_𝑛−𝑧)+𝛽_𝑛(𝑇𝑦_𝑛−𝑧)‖≤𝛼_𝑛‖𝑢₀−𝑧‖+(1−𝛼_𝑛)𝛽_𝑛‖𝑇𝑦_𝑛−𝑇𝑧‖+(1−𝛼_𝑛)(1−𝛽_𝑛)‖𝑥_𝑛−𝑧‖≤𝛼_𝑛‖𝑢₀−𝑧‖+(1−𝛼_𝑛)‖𝑥_𝑛−𝑧‖≤max{‖𝑢₀−𝑧‖,‖𝑥₀−𝑧‖}. (3.2) Therefore, {𝑥_𝑛} is bounded. Then 𝐶_min is well defined.
In the proofs of Theorems 2.1 and 2.3, take 𝑓(𝑥)=𝑢₀ for all 𝑥∈𝐶; we can show similarly that the conclusion holds.

Remark 3.2. If 𝜃_𝑛=0 for all 𝑛=0,1,2,…, Theorem 3.1 weakens the condition of {𝛽_𝑛} of Theorem 4.1 in [6].

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of a real reflexive Banach space 𝐸 which has uniformly Gâteaux differentiable norm. Suppose that 𝑇∶𝐶→𝐶 is a nonexpansive mapping with 𝐹(𝑇)≠∅ and the sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝜃_𝑛} in (0,1) satisfy (𝐶1). Then the sequence {𝑥_𝑛} defined by (3.1) converges strongly to a fixed point of 𝑇 as 𝑛→∞.

Proof. Take 𝑓(𝑥)=𝑢₀ for all 𝑥∈𝐶 in Theorem 2.5; it is easy to show that the conclusion holds.

Remark 3.4. If 𝜃_𝑛=0 for all 𝑛=0,1,2,…, Theorem 3.3 gets rid of the dependence on the implicit anchor-like continuous path 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝑇𝑧_𝑡 in Suzuki's Theorem 3 in [9] and Theorem 3.1 of C. E. Chidume and C. O. Chidume [8]. It also complements and generalizes [25, Theorem 1], which is proved in uniformly smooth Banach spaces.

Theorem 3.5. Let 𝐶 be a nonempty closed convex subset of a real reflexive and strictly convex Banach space 𝐸 which has uniformly Gâteaux differentiable norm. Suppose that 𝐶 is a sunny nonexpansive retract of 𝐸, 𝑇∶𝐶→𝐸 is a nonexpansive nonself-mapping with 𝐹(𝑇)≠∅, 𝑓 is a fixed contractive mapping with the contractive coefficient 𝑘∈(0,1), the sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝜃_𝑛} in (0,1) satisfy (𝐶1), and the sequence {𝑥_𝑛} is defined as follows, forthegiven𝑥₀,𝑢₀𝑦∈𝐶,_𝑛=(1−𝜃_𝑛)𝑥_𝑛+𝜃_𝑛𝑃𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢₀+(1−𝛼_𝑛)[(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑃𝑇𝑦_𝑛]),∀𝑛≥0. (3.3) Then the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇 as 𝑛→∞.

Remark 3.6. If 𝜃_𝑛=0 for all 𝑛=0,1,2,…, Theorem 3.3 improves and generalizes Theorem 3.2 in [16]: it gets rid of the restriction of 𝛿∈(0,1) and dependence on the implicit anchor-like continuous path 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝑃𝑇𝑧_𝑡.

Acknowledgments

The authors are grateful to Professor Jean Pierre Gossez and the referees for the careful reading and many valuable suggestions. This paper is supported partially by National Natural Science Foundation of China (no. 10871217), the Grant from the “project 211(Phase III)” (no. QN09-106) and the Scientific Research Fund (no. 09XG052) of the Southwestern University of Finance and Economics.