Iterative Methods for Pseudocontractive Mappings in Banach Spaces

Abstract

Let 𝐸 a reflexive Banach space having a uniformly Gâteaux differentiable norm. Let 𝐶 be a nonempty closed convex subset of 𝐸, 𝑇∶𝐶→𝐶 a continuous pseudocontractive mapping with 𝐹(𝑇)≠∅, and 𝐴∶𝐶→𝐶 a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant 𝑘∈(0,1). Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,1) satisfying suitable conditions and for arbitrary initial value 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated by 𝑥_𝑛=𝛼_𝑛𝐴𝑥_𝑛+𝛽_𝑛𝑥_𝑛−1+(1−𝛼_𝑛−𝛽_𝑛)𝑇𝑥_𝑛,𝑛≥1. If either every weakly compact convex subset of 𝐸 has the fixed point property for nonexpansive mappings or 𝐸 is strictly convex, then {𝑥_𝑛} converges strongly to a fixed point of 𝑇, which solves a certain variational inequality related to 𝐴.

1. Introduction and Preliminaries

Throughout this paper, we denote by 𝐸 the norm ‖⋅‖ and 𝐸^∗ a real Banach space and the dual space of 𝐸, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐸. For the mapping 𝑇∶𝐶→𝐶, we denote the fixed point set of 𝑇 by 𝐹(𝑇); that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}.

Let 𝐽 denote the normalized duality mapping from 𝐸 into 2^𝑋∗ defined by 𝐽(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖𝑓,𝑓=‖𝑥‖,∀𝑥∈𝐸, (1) where ⟨⋅,⋅⟩ denotes the generalized duality pair between 𝐸 and 𝐸^∗. Recall that the norm of 𝐸 is said to be Gâteaux differentiable if lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (2) exists for each 𝑥,𝑦 in its unit sphere 𝑈={𝑥∈𝐸∶‖𝑥‖=1}. Such an 𝐸 is called a smooth Banach space. The norm is said to be uniformly Gâteaux differentiable if for 𝑦∈𝑈, the limit is attained uniformly for 𝑥∈𝑈. The space 𝐸 is said to have a uniformly Fréchet differentiable norm (and 𝐸 is said to be uniformly smooth) if the limit in (2) is attained uniformly for (𝑥,𝑦)∈𝑈×𝑈. It is known that 𝐸 is smooth if and only if the normalized duality mapping 𝐽 is single valued. It is well known that if 𝐸 is uniformly smooth, then the duality mapping is norm-to-norm uniformly continuous on bounded subsets of 𝐸, and that if 𝐸 has a uniformly Gâteaux differentiable norm, then 𝐽 is norm-to-weak* uniformly continuous on each bounded subsets of 𝐸 [1, 2].

It is relevant to the results of this paper to note that while every uniformly smooth Banach space is a reflexive Banach space having a uniformly Gâteaux differentiable norm, the converse does not hold. To see this, consider 𝐸 to be the direct sum 𝑙²(𝑙^𝑝𝑛), the class of all those sequences 𝑥={𝑥_𝑛} with 𝑥_𝑛∈𝑙^𝑝𝑛 and ‖𝑥‖=(∑_𝑛<∞‖𝑥_𝑛‖²)^1/2 (see [3]). If 1<𝑝_𝑛<∞ for 𝑛∈ℕ, where either limsup_𝑛→∞𝑝_𝑛=∞ or liminf_𝑛→∞𝑝_𝑛=1, then 𝐸 is a reflexive Banach space with a uniformly Gâteaux differentiable norm but is not uniformly smooth (see [3–5]). We also observe that the spaces which enjoy the fixed point property (shortly, F.P.P) for nonexpansive mappings are not necessarily spaces having a uniformly Gâteaux differentiable norm. On the other hand, the converse of this fact appears to be unknown as well.

A Banach space 𝐸 is said to be strictly convex if ‖𝑥‖=𝑦=1,𝑥≠𝑦implies𝑥+𝑦2<1. (3) A Banach space 𝐸 is said to be uniformly convex if 𝛿_𝐸(𝜀)>0 for all 𝜀>0, where 𝛿_𝐸(𝜀) is the modulus of convexity of 𝐸 defined by 𝛿_𝐸(𝜀)=inf1−𝑥+𝑦2∶‖𝑥‖𝑦,[].≤1,≤1,𝑥−𝑦≥𝜀𝜀∈0,2 (4) It is well known that a uniformly convex Banach space 𝐸 is reflexive and strictly convex [1] and satisfies the F.P.P. for nonexpansive mappings. However, it appears to be unknown whether a reflexive and strictly convex space satisfies the F.P.P. for nonexpansive mappings.

Recall that a mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is called pseudocontractive if the inequality 𝑥−𝑦≤𝑥−𝑦+𝑟(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦 (5) holds for each, 𝑦∈𝐷(𝑇) and for all 𝑟>0. From a result of Kato [6], we know that (5) is equivalent to (6) below; there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦² (6) for all 𝑥,𝑦∈𝐷(𝑇). The mapping 𝑇 is said to be strongly pseudocontractive it there exists a constant 𝑘∈(0,1) and 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑘𝑥−𝑦² (7) for all 𝑥,𝑦∈𝐷(𝑇).

The class of pseudocontractive mappings is one of the most important classes of mappings in nonlinear analysis and it has been attracting mathematician’s interest. In addition to generalizing the nonexpansive mappings (the mappings 𝑇∶𝐷→𝐸 for which ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐷), the pseudocontractive ones are characterized by the fact that 𝑇 is pseudocontractive if and only if 𝐼−𝑇 is accretive, where a mapping 𝐴 with domain 𝐷(𝐴) and range 𝑅(𝐴) in 𝐸 is called accretive if the inequality 𝑥−𝑦≤𝑥−𝑦+𝑠𝐴𝑥−𝐴𝑦, (8) holds for every 𝑥,𝑦∈𝐷(𝐴) and for all 𝑠>0.

Within the past 40 years or so, many authors have been devoting their study to the existence of zeros of accretive mappings or fixed points of pseudocontractive mappings and iterative construction of zeros of accretive mappings and of fixed points of pseudocontractive mappings (see [5, 7–10]). Also, several iterative methods for approximating fixed points (zeros) of nonexpansive and pseudocontractive mappings (accretive mappings) in Hilbert spaces and Banach spaces have been introduced and studied by many authors. We can refer to [11–17] and the references in therein.

In 2007, Rafiq [15] introduced a Mann-type implicit iterative method (9) for a hemicontractive mapping 𝑇 as 𝑥₀∈𝐾,𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥1, (9) where 𝐾 is a compact convex subset of a real Hilbert space 𝐻 and {𝛼_𝑛}⊂[𝛿,1−𝛿] for some 𝛿∈(0,1), and proved that {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

In 2007, Yao et al. [16] introduced an iterative method (10) below for approximating fixed points of a continuous pseudocontractive mapping 𝑇 without compactness assumption on its domain in a uniformly smooth Banach space: for arbitrary initial value 𝑥₀∈𝐶 and a fixed anchor 𝑢∈𝐶, 𝑥_𝑛=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑇𝑥_𝑛,∀𝑛≥1, (10) where {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} are three sequences in (0,1) satisfying some appropriate conditions. By using the Reich inequality [9] in uniformly smooth Banach spaces 𝑥+𝑦²≤‖𝑥‖²(𝑥)⟩𝑥‖𝑦𝑏𝑦+2⟨𝑦,𝐽+max{‖,1},∀𝑥,𝑦∈𝐸, (11) where 𝑏∶[0,∞)→[0,∞) is a nondecreasing continuous function, they proved that the sequence {𝑥_𝑛} generated by (10) converges strongly to a fixed point of 𝑇. In particular, in 2007, by using the viscosity iterative method studied by [18, 19], Song and Chen [17] introduced a modified implicit iterative method (12) below for a continuous pseudocontractive mapping 𝑇 without compactness assumption on its domain in a real reflexive and strictly convex Banach space having a uniformly Gâteaux differentiable norm: for arbitrary initial value 𝑥₀∈𝐶, 𝑥_𝑛=𝛼_𝑛𝑦_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑦_𝑛=𝛽_𝑛𝑓𝑥_𝑛−1+1−𝛽_𝑛𝑥_𝑛−1,∀𝑛≥1, (12) where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) satisfying some appropriate conditions and 𝑓∶𝐶→𝐶 is a contractive mapping, and proved that the sequence {𝑥_𝑛} generated by (12) converges strongly to a fixed point of 𝑇, which is the unique solution of a certain variational inequality related to 𝑓.

In this paper, inspired and motivated by the above-mentioned results, we introduce the following the iterative method for a continuous pseudocontractive mapping 𝑇: for arbitrary initial value 𝑥₀∈𝐶, 𝑥_𝑛=𝛼_𝑛𝐴𝑥_𝑛+𝛽_𝑛𝑥_𝑛−1+1−𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛,∀𝑛≥1, (13) where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) and 𝐴∶𝐶→𝐶 is a bounded continuous strongly pseudocontractive mapping with a pseudocontractive constant 𝑘∈(0,1). Whether a reflexive Banach space has a uniformly Gâteaux differentiable norm such that every weakly compact convex subset of 𝐸 has the fixed point property for nonexpansive mappings, or a reflexive and strict convex Banach space has a uniformly Gâteaux differentiable norm, we establish the strong convergence of the sequence generated by proposed iterative method (13) to a fixed point of the mapping, which solves a certain variational inequality related to 𝐴. The main result generalizes, improves, and develops the corresponding results of Yao et al. [16] and Song and Chen [17] as well as Rafiq [15].

We need the following well-known lemmas for the proof of our main result.

Lemma 1 (see [1, 2]). Let 𝐸 be a Banach space and let 𝐽 be the normalized duality mapping on 𝐸. Then for any 𝑥,𝑦∈𝐸, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2⟨𝑦,𝑗𝑥+𝑦⟩,∀𝑗𝑥+𝑦∈𝐽𝑥+𝑦. (14)

Lemma 2 (see [20]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1≤1−𝜆_𝑛𝑠_𝑛+𝜆_𝑛𝛿_𝑛,∀𝑛≥0, (15) where {𝜆_𝑛} and {𝛿_𝑛} satisfy the following conditions: (i){𝜆_𝑛}⊂[0,1] and ∑^∞_𝑛=0𝜆_𝑛=∞ or, equivalently, ∏^∞_𝑛=0(1−𝜆_𝑛)=0,(ii)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=0𝜆_𝑛𝛿_𝑛<∞.
Then lim_𝑛→∞𝑠_𝑛=0.

2. Iterative Methods

We need the following result which was given in [10].

Proposition 3. Let 𝐶 be a closed convex subset of a Banach space 𝐸. Suppose that 𝑇,𝐴 are two continuous mappings from 𝐶 into itself, which are pseudocontractive and strongly pseudocontractive, respectively. Then there exists a unique path 𝑡↦𝑥_𝑡∈𝐶,  𝑡∈(0,1), satisfying 𝑥_𝑡=𝑡𝐴𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡. (16) Further, the following hold.(i)Suppose that there exists a bounded sequence {𝑥_𝑛} in 𝐶 such that 𝑥_𝑛−𝑇𝑥_𝑛→0, while {𝑥_𝑛−𝐴𝑥_𝑛} is bounded. Then the path {𝑥_𝑡} is bounded. (ii)In particular, if 𝑇 has a fixed point in 𝐶, then the path {𝑥_𝑡} is bounded. (iii)If 𝑝 is a fixed point of 𝑇, there exists 𝑗∈𝐽(𝑥_𝑡−𝑝) such that 𝑥_𝑡−𝐴𝑥_𝑡,𝑗≤0. (17)

We prepare the following result for the existence of a solution of the variational inequality related to 𝐴.

Theorem 4. Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸 and let 𝑇 be a continuous pseudocontractive mapping from 𝐶 into itself with 𝐹(𝑇)≠∅ and let 𝐴∶𝐶→𝐶 be a continuous bounded strongly pseudocontractive mapping with a pseudocontractive coefficient 𝑘∈(0,1). For each 𝑡∈(0,1), let 𝑥_𝑡∈𝐶 be defined by 𝑥_𝑡=𝑡𝐴𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡. (18) If one of the following assumptions holds: (H1)𝐸 is a reflexive Banach space, the norm of 𝐸 is uniformly Gâteaux differentiable, and every weakly compact convex subset of 𝐸 has the fixed point property for nonexpansive mappings; (H2)𝐸 is a reflexive and strictly convex Banach space and the norm of 𝐸 is uniformly Gâteaux differentiable,
then the path {𝑥_𝑡} converges strongly to a point 𝑢 in 𝐹(𝑇), which is the unique solution of the variational inequality ⟨(𝐼−𝐴)𝑢,𝐽(𝑢−𝑣)⟩≤0,∀𝑣∈𝐹(𝑇). (19)

Proof. In case of (H1), the result follows from Theorem 2 of [10]. So, we prove only the case of (H2). We follow the method of proof in [5, 10].
By Proposition 3, the path {𝑥_𝑡∶𝑡∈(0,1)} exists. It remains to show that it converges strongly to a fixed point of 𝑇 as 𝑡→0. As a consequence of Theorem 6 of [21], the mapping 2𝐼−𝑇 has a nonexpansive inverse, denoted by 𝑔, which maps 𝐶 into itself with 𝐹(𝑇)=𝐹(𝑔). By Proposition 3(ii), {𝑥_𝑡} is bounded. Since 𝐴 is a bounded mapping, {𝐴𝑥_𝑡∶𝑡∈(0,1)} is bounded. From (18), we have 𝑇𝑥_𝑡=1𝑥1−𝑡^𝑡−𝑡1−𝑡𝐴𝑥_𝑡≤1𝑥1−𝑡^𝑡+𝑡1−𝑡𝐴𝑥_𝑡, (20) and so {𝑇𝑥_𝑡} is bounded (as 𝑡→0). Since 𝑥_𝑡−𝑇𝑥_𝑡=𝑡𝐴𝑥_𝑡−𝑇𝑥_𝑡→0as𝑡→0, (21) we derive that 𝑥_𝑡−𝑔𝑥_𝑡→0as𝑡→0. (22) Let {𝑡_𝑛} be a sequence in (0,1) such that 𝑡_𝑛→0 as 𝑛→∞ and let 𝑥_𝑛=𝑥_𝑡𝑛. Since {𝑥_𝑛} is bounded, we may define 𝑓∶𝐶→ℝ⁺ by 𝑓(𝑥)=limsup_𝑛→∞‖𝑥_𝑛−𝑥‖. Since 𝐸 is reflexive, 𝑓(𝑥)→∞ as ‖𝑥‖→∞, and 𝑓 is continuous and convex, and the set 𝑀=𝑢∈𝐶∶𝑓(𝑢)=inf𝑓(𝑥)∶𝑥∈𝐶 (23) is a nonempty (due to Theorem 1.2 of [22]). Since 𝐹(𝑇)=𝐹(𝑔)≠∅, let 𝑣∈𝐹(𝑔). Then, since 𝐸 is strict convex, the set 𝑀₀={𝑢∈𝑀∶‖𝑢−𝑣‖=inf{‖𝑥−𝑣‖∶𝑥∈𝑀}} (24) is singleton. Let 𝑀₀={𝑢} for some 𝑢∈𝑀. We also know that 𝑔(𝑣)=𝑣 and 𝑔(𝑢)−𝑣=𝑔(𝑢)−𝑔(𝑣)≤‖𝑢−𝑣‖. (25) Therefore, 𝑔(𝑢)=𝑢. Since 𝑇 is pseudocontractive and 𝑢 is a fixed point of 𝑇, we derive from Proposition 3(iii) that 𝑥_𝑛−𝐴𝑥_𝑛,𝐽𝑥_𝑛−𝑢≤0,∀𝑛∈ℕ. (26) Now let 𝑥=𝐴𝑢−𝑢 and 𝑡∈(0,1). Then by Lemma 1, we have 𝑥_𝑛−𝑢−𝑡𝑥²≤𝑥_𝑛−𝑢²+2⟨−𝑡𝑥,𝐽𝑥_𝑛−𝑢−𝑡𝑥⟩. (27) Let 𝜀>0. Then by the assumption on 𝐸, there exists 𝛿>0 such that 𝑢+𝑡𝑥∈𝐶 and 𝑥,𝐽𝑥_𝑛−𝑢<𝜀+𝑥,𝐽𝑥_𝑛−𝑢−𝑡𝑥 (28) for all 𝑡∈(0,𝛿). Consequently, ⟨𝑥,𝐽𝑥_𝑛1−𝑢⟩<𝜀+𝑥2𝑡^𝑛−𝑢²−𝑥_𝑛−𝑢−𝑡𝑥². (29) Therefore, we may choose a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑘→∞𝐴𝑢−𝑢,𝐽𝑥_𝑛𝑘−𝑢≤0. (30) From (26) and the fact that 𝐴 is strongly pseudocontractive, we have 𝑥_𝑛𝑘−𝑢²=𝑥_𝑛𝑘−𝑢,𝐽𝑥_𝑛𝑘=𝑥−𝑢_𝑛𝑘−𝐴𝑥_𝑛𝑘,𝐽𝑥_𝑛𝑘+−𝑢𝐴𝑥_𝑛𝑘−𝐴𝑢,𝐽𝑥_𝑛𝑘+𝑥−𝑢𝐴𝑢−𝑢,𝐽_𝑛𝑘𝑥−𝑢≤𝑘_𝑛𝑘−𝑢²+𝐴𝑢−𝑢,𝐽𝑥_𝑛𝑘−𝑢, (31) which implies that (1−𝑘)‖𝑥_𝑛𝑘−𝑢‖²≤⟨𝐴𝑢−𝑢,𝐽(𝑥_𝑛𝑘−𝑢)⟩. Thus, by (30), we conclude that {𝑥_𝑛𝑘} converges strongly to 𝑢. To prove that actually the net {𝑥_𝑡} converges strongly to 𝑢, let {𝑥_𝑚𝑘} be another subsequence of {𝑥_𝑡∶𝑡∈(0,1)} such that 𝑥_𝑚𝑘=𝑥_𝑡𝑚𝑘, 𝑡_𝑚𝑘→0 as 𝑘→∞ and 𝑥_𝑚𝑘→𝑣, where 𝑇𝑣=𝑣. Then Proposition 3(iii) implies that ⟨𝑢−𝐴𝑢,𝐽(𝑢−𝑣)⟩≤0 and ⟨𝑣−𝐴𝑣,𝐽(𝑣−𝑢)⟩≤0. This implies that ⟨𝑢−𝑣−(𝐴𝑢−𝐴𝑣),𝐽(𝑢−𝑣)⟩≤0. (32) Since 𝐴 is strongly pseudocontractive, 𝑢=𝑣 and the strong lim_𝑡→0𝑥_𝑡 exists. This same argument may be used to conclude that 𝑢 is the only solution of the variational inequality (19). This completes the proof.

Using Theorem 4, we establish our second main result.

Theorem 5. Let 𝐸 be a Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a continuous pseudocontractive mapping such that 𝐹(𝑇)≠∅, and let 𝐴∶𝐶→𝐶 be a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant 𝑘∈(0,1). Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,1) satisfying the following conditions:(C1)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞𝛽_𝑛=0;(C2)∑^∞_𝑛=1(𝛼_𝑛/(𝛼_𝑛+𝛽_𝑛))=∞.
For arbitrary initial value 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be defined by 𝑥_𝑛=𝛼_𝑛𝐴𝑥_𝑛+𝛽_𝑛𝑥_𝑛−1+1−𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛,∀𝑛≥1. (33) If one of the following assumptions holds: (H1)𝐸 is a reflexive Banach space, the norm of 𝐸 is uniformly Gâteaux differentiable, and every weakly compact convex subset of 𝐸 has the fixed point property for nonexpansive mappings; (H2)𝐸 is a reflexive and strictly convex Banach space and the norm of 𝐸 is uniformly Gâteaux differentiable,
then {𝑥_𝑛} converges strongly to a fixed point 𝑝 of 𝑇, which is the unique solution of the variational inequality ⟨(𝐼−𝐴)𝑝,𝐽𝑝−𝑞⟩≤0,∀𝑞∈𝐹(𝑇). (34)

Proof. We divide the proof into several steps as follows.
Step  1. We show that {𝑥_𝑛} is bounded. To this end, let 𝑞∈𝐹(𝑇). Then, noting that 𝑥_𝑛−𝑞=𝛼_𝑛𝐴𝑥_𝑛−𝑞+𝛽_𝑛𝑥_𝑛−1+−𝑞1−𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑞,⟨𝑇𝑥_𝑛−𝑞,𝐽𝑥_𝑛−𝑞⟩≤𝑥_𝑛−𝑞²𝐴𝑥_𝑛−𝐴𝑞,𝐽𝑥_𝑛−𝑞⟩≤𝑘𝑥_𝑛−𝑞², (35) we have 𝑥_𝑛−𝑞²=𝛼_𝑛𝐴𝑥_𝑛−𝐴𝑞+𝐴𝑞−𝑞+𝛽_𝑛𝑥_𝑛−1+−𝑞1−𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑞,𝐽𝑥_𝑛−𝑞≤𝛼_𝑛𝑘𝑥_𝑛−𝑞²+𝛼_𝑛𝐴𝑞−𝑞𝑥_𝑛−𝑞+𝛽_𝑛𝑥_𝑛−1−𝑞𝑥_𝑛+−𝑞1−𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝑞², (36) which implies 𝑥_𝑛−𝑞≤1−𝛼_𝑛(1−𝑘)−𝛽_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝐴𝑞−𝑞+𝛽_𝑛𝑥_𝑛−1−𝑞. (37) So, we obtain 𝑥_𝑛≤𝛼−𝑞^𝑛(1−𝑘)𝛼_𝑛+𝛽_𝑛+𝛽𝐴𝑞−𝑞_𝑛(1−𝑘)𝛼_𝑛+𝛽_𝑛𝑥_𝑛−1=()𝛼−𝑞1−𝑘_𝑛(1−𝑘)𝛼_𝑛+𝛽_𝑛𝐴𝑞−𝑞+𝛽1−𝑘_𝑛(1−𝑘)𝛼_𝑛+𝛽_𝑛𝑥_𝑛−1𝑥−𝑝≤max_𝑛−1−𝑞,𝐴𝑞−𝑞.1−𝑘 (38) By induction, we have 𝑥_𝑛−𝑞≤max𝑥₀,1−𝑞1−𝑘𝐴𝑞−𝑞for𝑛≥1. (39) Hence, {𝑥_𝑛} is bounded. Since 𝐴 is a bounded mapping, {𝐴𝑥_𝑛} is bounded. From (33), it follows that 𝑇𝑥_𝑛=11−𝛼_𝑛−𝛽_𝑛𝑥_𝑛+𝛼_𝑛𝐴𝑥_𝑛+𝛽_𝑛𝑥_𝑛−1, (40) and so {𝑇𝑥_𝑛} is bounded (as 𝑛→∞).
Step  2. We show that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. In fact, by (33) and the condition (C1), we have 𝑥_𝑛−𝑇𝑥_𝑛≤𝛼_𝑛𝐴𝑥_𝑛−𝑇𝑥_𝑛+𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛→0. (41)
Step  3. We show that limsup_𝑛→∞⟨𝐴𝑝−𝑝,𝐽𝑥_𝑛−𝑝⟩≤0, (42) where 𝑝=lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡∈𝐶 being defined by 𝑥_𝑡=𝑡𝐴𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡. To this end, we note that 𝑥_𝑡−𝑥_𝑛=𝑡𝐴𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡−𝑥_𝑛=𝑡𝐴𝑥_𝑡−𝑥_𝑡+𝑇𝑥_𝑡−𝑥_𝑛−𝑡𝑇𝑥_𝑡−𝑥_𝑡=𝑡𝐴𝑥_𝑡−𝑥_𝑡+𝑇𝑥_𝑡−𝑇𝑥_𝑛+𝑇𝑥_𝑛−𝑥_𝑛+𝑡²𝐴𝑥_𝑡−𝑇𝑥_𝑡. (43) Then, it follows that 𝑥_𝑡−𝑥_𝑛²=𝑡⟨𝐴𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩+⟨𝑇𝑥_𝑡−𝑇𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛⟩+⟨𝑇𝑥_𝑛−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛⟩+𝑡²⟨𝐴𝑥_𝑡−𝑇𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩≤𝑡⟨𝐴𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩+𝑥_𝑡−𝑥_𝑛²+𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑡−𝑥_𝑛+𝑡²𝐴𝑥_𝑡−𝑇𝑥_𝑡𝑥_𝑡−𝑥_𝑛, (44) which implies that 𝐴𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡≤𝑇𝑥_𝑛−𝑥_𝑛𝑡𝑥_𝑡−𝑥_𝑛+𝑡𝐴𝑥_𝑡−𝑇𝑥_𝑡𝑥_𝑡−𝑥_𝑛. (45) From Proposition 3, we know that {𝑥_𝑡}, {𝐴𝑥_𝑡} and {𝑇𝑥_𝑡}, are bounded. Since {𝑥_𝑛} and {𝑇𝑥_𝑛} are also bounded and 𝑥_𝑛−𝑇𝑥_𝑛→0 by Step 2, taking the upper limit as 𝑛→∞ in (45), we get limsup_𝑛→∞⟨𝐴𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡⟩≤𝑡𝐿, (46) where 𝐿>0 is a constant such that ‖𝐴𝑥_𝑡−𝑇𝑥_𝑡‖‖𝑥_𝑡−𝑥_𝑛‖≤𝐿 for all 𝑛≥0 and 𝑡∈(0,1). Taking the limsup as 𝑡→0 in (46) and noticing the fact that the two limits are interchangeable due to the fact that 𝐽 is norm-to-weak* uniformly continuous on each bounded subsets of 𝐸, we have limsup_𝑛→∞𝐴𝑝−𝑝,𝐽𝑥_𝑛−𝑝≤0. (47)
Step  4. We show that lim_𝑛→∞‖𝑥_𝑛−𝑝‖=0, where 𝑝=lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡∈𝐶 being defined by 𝑥_𝑡=𝑡𝐴𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡 and 𝑝 is the unique solution of the variational inequality (34) by Theorem 4. First, from (33) and (35), we have 𝑥_𝑛−𝑝²=⟨𝑥_𝑛−𝑝,𝐽𝑥_𝑛−𝑝⟩=⟨𝛼_𝑛𝐴𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−1+−𝑝1−𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑝,𝐽𝑥_𝑛−𝑝⟩=⟨𝛼_𝑛𝐴𝑥_𝑛−𝐴𝑝,𝐽𝑥_𝑛−𝑝⟩+𝛽_𝑛⟨𝑥_𝑛−1−𝑝,𝐽𝑥_𝑛⟩+−𝑝1−𝛼_𝑛−𝛽_𝑛⟨𝑇𝑥_𝑛−𝑝,𝐽𝑥_𝑛−𝑝⟩+𝛼_𝑛⟨𝐴𝑝−𝑝,𝐽𝑥_𝑛−𝑝⟩≤𝛼_𝑛𝑘𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−1−𝑝𝑥_𝑛+−𝑝1−𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛⟨𝐴𝑝−𝑝,𝐽𝑥_𝑛−𝑝⟩≤𝛼_𝑛𝑘𝑥_𝑛−𝑝²+𝛽_𝑛2𝑥_𝑛−1−𝑝²+𝑥_𝑛−𝑝²+1−𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛⟨𝐴𝑝−𝑝,𝐽𝑥_𝑛−𝑝⟩. (48) This implies that 𝑥_𝑛−𝑝²≤𝛽_𝑛2(1−𝑘)𝛼_𝑛+𝛽_𝑛𝑥_𝑛−1−𝑝²+2𝛼_𝑛2(1−𝑘)𝛼_𝑛+𝛽_𝑛⟨𝐴𝑝−𝑝,𝐽𝑥_𝑛⟩=2()𝛼−𝑝1−1−𝑘_𝑛2(1−𝑘)𝛼_𝑛+𝛽_𝑛𝑥_𝑛−1−𝑝²+2(1−𝑘)𝛼_𝑛2(1−𝑘)𝛼_𝑛+𝛽_𝑛𝐴𝑝−𝑝,𝐽𝑥_𝑛−𝑝=1−𝑘1−𝜆_𝑛𝑥_𝑛−1−𝑝²+𝜆_𝑛𝛿_𝑛, (49) where 𝜆_𝑛=2(1−𝑘)𝛼_𝑛/(2(1−𝑘)𝛼_𝑛+𝛽_𝑛) and 𝛿_𝑛=(1/(1−𝑘))⟨𝐴𝑝−𝑝,𝐽(𝑥_𝑛−𝑝)⟩. We observe that 0≤2(1−𝑘)𝛼_𝑛/(2(1−𝑘)𝛼_𝑛+𝛽_𝑛)≤1 and (1−𝑘)𝛼_𝑛/(𝛼_𝑛+𝛽_𝑛)=2(1−𝑘)𝛼_𝑛/(2𝛼_𝑛+2𝛽_𝑛)<2(1−𝑘)𝛼_𝑛/(2(1−𝑘)𝛼_𝑛+𝛽_𝑛). From the condition (C2) and Step 3, it is easily seen that ∑^∞_𝑛=1𝜆_𝑛=∞ and limsup_𝑛→∞𝛿_𝑛≤0. Thus, applying Lemma 2 to (49), we conclude that lim_𝑛→∞𝑥_𝑛=𝑝. This completes the proof.

Corollary 6. Let 𝐸 be a uniformly smooth Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a continuous pseudocontractive mapping such that 𝐹(𝑇)≠∅ and let 𝐴∶𝐶→𝐶 be a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant 𝑘∈(0,1). Let {𝛼_𝑛} and {𝛽_𝑛} be two sequences in (0,1) satisfying the conditions (C1) and (C2) in Theorem 5. For arbitrary initial value 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated by (33) in Theorem 5. Then {𝑥_𝑛} converges strongly to a fixed point 𝑝 of 𝑇, which is the unique solution of the variational inequality (34).

Corollary 7 (see [16, Theorem 3.1]). Let 𝐸 be a uniformly smooth Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a continuous pseudocontractive mapping such that 𝐹(𝑇)≠∅. Let {𝛼_𝑛}, {𝛽_𝑛} and {𝛾_𝑛} be three sequences in (0,1) satisfying the conditions (C1) and (C2) in Theorem 5 and 𝛾_𝑛=1−𝛼_𝑛−𝛽_𝑛  for 𝑛≥1. For arbitrary initial value 𝑥₀∈𝐶 and a fixed anchor 𝑢∈𝐶, let the sequence {𝑥_𝑛} be generated by 𝑥_𝑛=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑇𝑥_𝑛,∀𝑛≥1. (50) Then {𝑥_𝑛} converges strongly to a fixed point 𝑝 of 𝑇, which is the unique solution of the variational inequality ⟨𝑝−𝑢,𝐽𝑝−𝑞⟩≤0,∀𝑞∈𝐹(𝑇). (51)

Proof. Taking 𝐴𝑥=𝑢,  for  all  𝑥∈𝐶 as a constant function, the result follows from Corollary 6.

Corollary 8. Let 𝐸 be a uniformly convex Banach space has a uniformly Gâteaux differentiable norm and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a continuous pseudocontractive mapping such that 𝐹(𝑇)≠∅ and let 𝐴∶𝐶→𝐶 be a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant 𝑘∈(0,1). Let {𝛼_𝑛} and {𝛽_𝑛} be two sequences in (0,1) satisfying the conditions (C1) and (C2) in Theorem 5. For arbitrary initial value 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated by (33) in Theorem 5. Then {𝑥_𝑛} converges strongly to a fixed point 𝑝 of 𝑇, which is the unique solution of the variational inequality (34).

Remark 9.   (1) Theorem 5 extends and improves Theorem 3.1 of Yao et al. [16] in the following aspects.(a) 𝑢 is replaced by a continuous bounded strongly pseudocontractive mapping 𝐴.  (b) The uniformly smooth Banach space is extended to a reflexive Banach space having a uniformly Gâteaux differentiable norm. (c) The condition 𝛼_𝑛/𝛽_𝑛→0 in [16] is weakened to 𝛼_𝑛→0 and 𝛽_𝑛→0 as 𝑛→∞. (2)It is worth pointing out that in Corollaries 6 and 7, we do not use the Reich inequality (11) in comparison with Theorem 3.1 of Yao et al. [16].(3)Theorem 5 and Corollary 8 also develop and complement Theorem 3.1 and Corollary 3.2 of Song and Chen [17] by replacing the contractive mapping with a continuous bounded strongly pseudocontractive mapping in the iterative scheme (1.7) in [17].(4)The assumption (H1) in Theorems 4 and 5 appears to be independent of the assumption (H2).(5)We point out that the results in this paper apply to all 𝐿^𝑝 spaces, 1<𝑝<∞.

Acknowledgment

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