Strong Convergence Theorems for a Common Fixed Point of a Family of Asymptotically 𝑘-Strict Pseudocontractive Mappings

Abstract

We provide an iterative process which converges strongly to a common fixed point of finite family of asymptotically 𝑘-strict pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

1. Introduction

Let 𝐸 be a real normed linear space with dual 𝐸^∗. A gauge function 𝜑∶[0,∞]∶=𝑅⁺→𝑅⁺ is a continuous and strictly increasing function satisfying 𝜑(0)=0 and 𝜑(𝑡)→∞, as 𝑡→∞. The generalized duality mapping from 𝐸 to 2^𝐸∗ associated with the gauge function 𝜑 (see, e.g., [1]) is defined by 𝐽_𝜑(𝑥)∶=𝑥^∗∈𝐸^∗∶𝑥,𝑥^∗=𝑥𝜑𝑥,𝑥||||(||||)_∗=𝜑(||𝑥||),∀𝑥∈𝐸, (1) where ⟨⋅,⋅⟩ denotes the duality pairing. In the case that 𝜑(𝑡)=𝑡, the duality mapping 𝐽_𝜑=𝐽 is called the normalized duality mapping.

Following Browder [2], we say that a Banach space 𝐸 has a weakly continuous duality mapping if there exists a gauge 𝜑 for which the duality mapping 𝐽_𝜑 is single valued and weak-to-weak* sequentially continuous (i.e., if {𝑥_𝑛} is a sequence in 𝐸 weakly convergent to a point 𝑥, then the sequence 𝐽_𝜑(𝑥_𝑛) converges weak* to 𝐽_𝜑(𝑥)). It is known that 𝑙_𝑝 has a weakly continuous duality mapping with a gauge function 𝜑(𝑡)=𝑡^𝑝−2, for all 1<𝑝<∞.

Let 𝐾 be a nonempty subset of 𝐸. A mapping 𝑇∶𝐾→𝐾 is called asymptotically 𝑘-strict pseudocontractive, with sequence {𝑙_𝑛}⊆[1,∞), lim_𝑛→∞𝑙_𝑛=1 (see, e.g., [3–6]) if for all 𝑥,𝑦∈𝐾, there exist 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) and a constant 𝑘∈[0,1) such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑗𝑥−𝑦≤𝑙_𝑛𝑥−𝑦²−𝑘𝐼−𝑇^𝑛𝑥−𝐼−𝑇^𝑛𝑦², (2) for all 𝑛≥1.

If 𝐼 denotes the identity operator, then (2) can be equivalently written as 𝐼−𝑇^𝑛𝑥−𝐼−𝑇^𝑛𝑦,𝑗𝑥−𝑦≥𝑘(𝐼−𝑇^𝑛)𝑥−(𝐼−𝑇^𝑛)𝑦²−𝑙_𝑛−1𝑥−𝑦², (3) for all 𝑛≥1.

If 𝐸=𝐻, a real Hilbert space, it is shown by Osilike et al. [4] that (2) (and hence (3)) is equivalent to the inequality 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤1+2𝑙_𝑛−1𝑥−𝑦²+𝜆(𝐼−𝑇^𝑛)𝑥−(𝐼−𝑇^𝑛)𝑦², (4) where 𝜆=(1−2𝑘). 𝑇 is called uniformly Lipschitz if there exists 𝐿≥0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐷(𝑇). It is shown in [4] that an asymptotically 𝑘-strict pseudocontractive mapping is uniformly Lipschitz.

The class of asymptotically 𝑘-strict pseudocontractive mappings was first introduced in Hilbert spaces by Liu [5]. He proved the following theorem.

Theorem Q (see [5]). Let 𝐾 be a closed convex and bounded subset of a Hilbert space 𝐻. Let 𝑇∶𝐾→𝐾 be completely continuous asymptotically 𝑘-strict pseudocontractive mapping for some 0≤𝑘<1 with sequence {𝑙_𝑛}⊂[0,∞) such that ∑(𝑙_𝑛−1)<∞ and 𝐹(𝑇)≠∅. Let {𝑥_𝑛} be a sequence generated by the modified Mann's iteration method: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1, (5) where {𝛼_𝑛} is a real sequence satisfying 𝜖≤𝛼_𝑛≤1−𝑘−𝜖 for all 𝑛≥1 and some 𝜖>0. Then, {𝑥_𝑛} converges strongly to a fixed point of  𝑇.

The iteration scheme (5) is called modified Mann’s iterative processes which was introduced by Schu [7, 8] and has been used by several authors (see, e.g., [3–5, 9–17]). We observe that Liu [5] proved strong convergence of scheme (5) to a fixed point of asymptotically 𝑘-strict pseudocontractive mapping 𝑇 with additional assumption that 𝑇 is completely continuous, where 𝑇∶𝐶→𝐶 is said to be completely continuous if for every bounded sequence {𝑥_𝑛}, there exists a subsequence, say {𝑥_𝑛𝑗} of {𝑥_𝑛} such that the sequence {𝑇𝑥_𝑛𝑗} converges strongly to some element of the range of 𝑇.

In [12], Kim and Xu studied weak convergence theorem for the class of asymptotically 𝑘-strict pseudocontractive mappings in the frame work of Hilbert spaces. In fact, they proved the following.

Theorem KX (see [12]). Let 𝐾 be a closed and convex subset of a Hilbert space 𝐻. Let 𝑇∶𝐾→𝐾 be an asymptotically 𝑘-strict pseudocontractive mapping for some 0≤𝑘<1 with sequence {𝑙_𝑛}⊂[0,∞) such that ∑(𝑙_𝑛−1)<∞ and 𝐹(𝑇)≠∅. Let {𝑥_𝑛} be a sequence generated by the modified Mann's iteration method: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1, (6) where {𝛼_𝑛} is a real sequence satisfying 𝑘+𝜆≤𝛼_𝑛≤1−𝜆, for all 𝑛≥1, and 𝜆∈(0,1). Then, {𝑥_𝑛} converges weakly to a fixed point of  𝑇.

In 2007, Osilike et al. [13] extended Theorem KX by proving weak convergence of scheme (6) to a fixed point of 𝑇 in the frame work of 𝑞 uniformly smooth Banach spaces which are also uniformly convex under suitable control conditions.

In 2011, Zhang and Xie [17] extended Theorem of Osilike et al. [13] to a more general real uniformly convex Banach space 𝐸 with Fréchet differentiable norm. In addition, they proved strong convergence of scheme (5) to a fixed point of asymptotically 𝑘-strict pseudocontractive mapping provided that liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹(𝑇))=0, where 𝑑(𝑥_𝑛,𝐹(𝑇))=inf_{𝑝∈𝐹(𝑇)}||𝑥_𝑛−𝑝||.

However, we observe that the convergence obtained above is either weak or requiring additional assumption like liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹(𝑇))=0 or 𝑇 is completely continuous. But the requirement that liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹(𝑇))=0 is not easy to verify, as 𝐹(𝑇) is in general unknown, and there is also an example of asymptotically 𝑘-strict pseudocontractive mapping which is not completely continuous as shown below.

An example of asymptotically 𝑘-strict pseudocontractive mapping which is not completely continuous.

Example 1. Let 𝐸=𝑙₂={𝑥={𝑥_𝑖}^∞_𝑖=1,𝑥_𝑖∈ℝ,∑^∞_𝑖=1|𝑥_𝑖|²<∞} and 𝐵={𝑥∈𝑙₂∶||𝑥||≤1}. Define 𝑇∶𝐵→𝐵 by 𝑇𝑥=(0,𝑥²₁,𝑎₂𝑥₂,𝑎₃𝑥₃,…), where {𝑎_𝑘}^∞_𝑘=1 is a real sequence satisfying 0<𝑎_𝑘<1, 𝑘≥2, and Π^∞_𝑘=2𝑎_𝑘=1/2. Then it is shown in [13] that 𝑇 is asymptotically 𝑘-strict pseudocontractive mapping.

Now, we show that 𝑇 is not completely continuous. Let {𝑥_𝑛} be a sequence in 𝐵 defined by 𝑥₁=(1,0,0,…), 𝑥₂=(0,1,0,0,…), 𝑥₃=(0,0,1,0,0,…),…. Then {𝑥_𝑛}⊂𝐵 and {𝑇𝑥_𝑛}={𝑦_𝑛} is given by 𝑦₁=(0,1,0,0,…), 𝑦₂=(0,0,𝑎₂,0,0,…), 𝑦₃=(0,0,0,𝑎₃,0,0,…),…. Hence, since 𝑎_𝑘→1, as 𝑘→∞, there is no subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that {𝑇𝑥_𝑛𝑖} converges strongly to a point in 𝐵, as ||𝑇𝑥_𝑛𝑖−𝑇𝑥_𝑛𝑗||=||𝑦_𝑛𝑖−𝑦_𝑛𝑗||=|𝑎_𝑛𝑖|²+|𝑎_𝑛𝑗|²↛0, as 𝑖,𝑗→∞. Therefore, 𝑇 is not completely continuous.

Thus, one question is raised naturally: can we obtain a scheme that converges strongly to a fixed point of asymptotically 𝑘-strict pseudocontractive mappings without those additional assumptions?

It is our purpose in this paper to provide an iterative scheme {𝑥_𝑛} which converges strongly to a common fixed point of finite family of asymptotically 𝑘-strict pseudocontractive mappings in Banach spaces. The assumption that liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹(𝑇))=0 or 𝑇 is completely continuous is not required.

2. Preliminaries

We need the following definitions from [18]. The Banach space 𝐸 is said to be uniformly convex if, given 𝜀>0, there exists 𝛿>0, such that, for all 𝑥,𝑦∈𝐸 with ‖𝑥‖≤1,‖𝑦‖≤1 and ‖𝑥−𝑦‖≥𝜀, ‖(1/2)(𝑥+𝑦)‖≤1−𝛿. It is well known that 𝐿_𝑝, ℓ_𝑝, and Sobolev spaces 𝑊^𝑝_𝑚, (1<𝑝<∞), are uniformly convex.

A Banach space 𝐸 is said to have a Fréchet differentiable norm if for all 𝑥∈𝐵={𝑥∈𝐸∶||𝑥||=1}lim_𝑡→0𝑥+𝑡𝛾−‖𝑥‖𝑡 (7) exists and is attained uniformly in 𝑦∈𝐵. It is well known that uniformly smooth Banach spaces has a Fréchet differentiable norm.

In order to prove our results, we need the following lemmas.

Lemma 2 (see [19]). Let 𝐶 be a nonempty close convex subset of a real Banach space 𝐸 which has the Fréchet differentiable norm. For 𝑥∈𝐸, let 𝜌 be defined for 0<𝑡<∞ by 𝜌(𝑡)=sup_𝛾∈𝐵𝑥+𝑡𝛾²−‖𝑥‖²𝑡−2𝛾,𝑗(𝑥). (8) Then, lim_𝑡→0𝜌(𝑡)=0 and ‖𝑥+ℎ‖²≤‖𝑥‖²+2ℎ,𝑗(𝑥)+||ℎ||𝜌(‖ℎ‖),∀ℎ∈𝐸⧵{0}. (9)

It is shown in [19] that if 𝐸=𝐻, a real Hilbert space, then 𝜌(𝑡)=𝑡, for 𝑡>0. In our general setting, throughout this paper we assume that 𝜌(𝑡)≤2𝑡.

Lemma 3. Let 𝐸 be a real Banach space. Then the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+,.𝑦,𝑗𝑥+𝑦∀𝑥,𝑦∈𝐻,𝑗𝑥+𝑦∈𝐽𝑥+𝑦 (10)

Lemma 4 (see [20]). Let 𝐸 be a uniformly convex Banach space and 𝐵_𝑅(0) a closed ball of 𝐸. Then, there exists a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝛼₀𝑥₀+𝛼₁𝑥₁+𝛼₂𝑥₂+⋯+𝛼_𝑘𝑥_𝑘²≤^𝑘_𝑖=0𝛼_𝑖𝑥_𝑖²−𝛼_𝑖𝛼_𝑗𝑔𝑥_𝑖−𝑥_𝑗, (11) for each 𝛼_𝑖∈(0,1) and for 𝑥_𝑖∈𝐵_𝑅(0)∶={𝑥∈𝐸∶||𝑥||≤𝑅}, 𝑖=0,1,2,…,𝑘 with ∑^𝑘_𝑖=0𝛼_𝑖=1.

Lemma 5 (see [21]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛼_𝑛𝛿_𝑛,𝑛≥𝑛₀, (12) where {𝛼_𝑛}⊂(0,1) and {𝛿_𝑛}⊂𝑅 satisfying the following conditions: lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=1𝛼_𝑛=∞, and limsup_𝑛→∞𝛿_𝑛≤0. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 6 (see [17]). Let 𝐶 be a nonempty closed convex subset of a real uniformly convex Banach space 𝐸 which has the Fréchet differentiable norm. Let 𝑇∶𝐶→𝐶 be an asymptotically 𝑘-strict pseudocontractive mapping with fixed point of 𝑇, 𝐹(𝑇)∶={𝑥∈𝐶∶𝑇𝑥=𝑥}≠∅. Then (𝐼−𝑇) is demiclosed at zero, that is, if 𝑥_𝑛⇀𝑥 and 𝑇𝑥_𝑛−𝑥_𝑛→0, as 𝑛→∞, then 𝑥=𝑇(𝑥).

Lemma 7 (see [22]). Let {𝑎_𝑛} be sequences of real numbers such that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑎_𝑛𝑖<𝑎_𝑛𝑖+1 for all 𝑖∈𝑁. Then there exists a nondecreasing sequence {𝑚_𝑘}⊂𝑁 such that 𝑚_𝑘→∞ and the following properties are satisfied by all (sufficiently large) numbers 𝑘∈𝑁: 𝑎_𝑚𝑘≤𝑎_𝑚𝑘+1,𝑎_𝑘≤𝑎_𝑚𝑘+1. (13) In fact, 𝑚_𝑘=max{𝑗≤𝑘∶𝑎_𝑗<𝑎_𝑗+1}.

3. Main Results

We now prove our main theorem.

Theorem 8. Let 𝐶 be a nonempty, closed, and convex subset of a real uniformly convex Banach space 𝐸 which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from 𝐸 into 𝐸^∗. Let 𝑇_𝑖∶𝐶→𝐶 be asymptotically 𝑘_𝑖-strict pseudocontractive mappings for 0≤𝑘_𝑖<1 with sequences {𝑙_𝑛,𝑖}⊂[1,∞), for 𝑖=1,2,…,𝑁. Assume that 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence defined by 𝑥₁=𝑢∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑆_𝑛𝑥_𝑛,𝑛≥1, (14) where 𝑆_𝑛∶=𝜃_𝑛,1𝑇^𝑛₁+𝜃_𝑛,2𝑇^𝑛₂+⋯+𝜃_𝑛,𝑁𝑇^𝑛_𝑁, such that 𝜃_𝑛,1+𝜃_𝑛,2+⋯+𝜃_𝑛,𝑁=1, for each 𝑛≥1, {𝛼_𝑛},{𝜃_𝑛,𝑖}⊂(0,𝑐)⊂(0,1), satisfying liminf_𝑛𝜃_𝑛,𝑖>0, lim_𝑛→∞𝛼_𝑛=0, ∑𝛼_𝑛=∞, lim_𝑛→∞((𝑙_𝑛,𝑖−1)/𝛼_𝑛)=0, for 𝑖=1,2,…,𝑁 and {𝛽_𝑛}⊂[𝑎,𝑏]⊂(0,𝑘) (𝑎, 𝑏, and 𝑐 constants), for 𝑘=min_{1≤𝑖≤𝑁}{𝑘_𝑖}, Then the sequence {𝑥_𝑛} generated by (14) converges strongly to a common fixed point of {𝑇_𝑖∶𝑖=1,2,…,𝑁}.

Proof. Fix 𝑥^∗∈𝐹. Let 𝑦_𝑛=(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑆_𝑛𝑥_𝑛 and 𝑙_𝑛∶=max{𝑙_𝑛,𝑖∶𝑖=1,2,…,𝑁}. Then, using Lemma 2 and (3) we have that 𝑦_𝑛−𝑥^∗2=𝑥_𝑛−𝑥^∗−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²=𝑥_𝑛−𝑥^∗−𝛽_𝑛𝑥_𝑛−𝜃_𝑛,1𝑇^𝑛₁+𝜃_𝑛,2𝑇^𝑛₂+⋯+𝜃_𝑛,𝑁𝑇^𝑛_𝑁𝑥_𝑛²≤𝑥_𝑛−𝑥^∗2−2𝛽_𝑛𝜃_𝑛,1𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛+𝜃_𝑛,2𝑥_𝑛−𝑇^𝑛₂𝑥_𝑛+⋯+𝜃_𝑛,𝑁𝑥_𝑛−𝑇^𝑛_𝑁𝑥_𝑛,𝑗𝑥_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛𝜌𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝑥_𝑛−𝑥^∗2−2𝛽_𝑛𝜃_𝑛,1𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛,𝑗𝑥_𝑛−𝑥^∗−2𝛽_𝑛𝜃_𝑛,2𝑥_𝑛−𝑇^𝑛₂𝑥_𝑛,𝑗𝑥_𝑛−𝑥^∗−⋯−2𝛽_𝑛𝜃_𝑛,𝑁𝑥_𝑛−𝑇^𝑛_𝑁𝑥_𝑛,𝑗𝑥_𝑛−𝑥^∗+2𝛽²_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²≤𝑥_𝑛−𝑥^∗2−2𝛽_𝑛𝜃_𝑛,1𝑘𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛²−𝑙_𝑛−1𝑥_𝑛−𝑥^∗2−2𝛽_𝑛𝜃_𝑛,2𝑘𝑥_𝑛−𝑇^𝑛₂𝑥_𝑛²−𝑙_𝑛−1𝑥_𝑛−𝑥^∗2−⋯−2𝛽_𝑛𝜃_𝑛,𝑁𝑘𝑥_𝑛−𝑇^𝑛_𝑁𝑥_𝑛²−𝑙_𝑛−1𝑥_𝑛−𝑥^∗2+2𝛽²_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²≤1+2𝛽_𝑛𝑙_𝑛−1𝑥_𝑛−𝑥^∗2−2𝛽_𝑛𝜃_𝑛,1𝑘𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛²−2𝛽_𝑛𝜃_𝑛,2𝑘𝑥_𝑛−𝑇^𝑛₂𝑥_𝑛²−⋯−2𝛽_𝑛𝜃_𝑛,𝑁𝑘𝑥_𝑛−𝑇^𝑛_𝑁𝑥_𝑛²+2𝛽²_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛². (15)
On the other hand using Lemma 4 we get that 𝑥_𝑛−𝑆_𝑛𝑥_𝑛²=𝑥_𝑛−𝜃_𝑛,1𝑇^𝑛₁+𝜃_𝑛,2𝑇^𝑛₂+⋯+𝜃_𝑛,𝑁𝑇^𝑛_𝑁𝑥_𝑛²=𝜃_𝑛,1𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛+𝜃_𝑛,2𝑥_𝑛−𝑇^𝑛₂𝑥_𝑛+⋯+𝜃_𝑛,𝑁(𝑥_𝑛−𝑇^𝑛_𝑁𝑥_𝑛)²≤𝜃_𝑛,1𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛²+𝜃_𝑛,2𝑥_𝑛−𝑇^𝑛₂𝑥_𝑛²+⋯+𝜃_𝑛,𝑁𝑥_𝑛−𝑇^𝑛_𝑁𝑥_𝑛². (16)
Now substituting (16) into (15) we obtain that 𝑦_𝑛−𝑥^∗2≤1+2𝛽_𝑛𝑙_𝑛−1𝑥_𝑛−𝑥^∗2−2𝛽_𝑛𝜃_𝑛,1𝑘−𝛽_𝑛𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛²−2𝛽_𝑛𝜃_𝑛,2𝑘−𝛽_𝑛𝑥_𝑛−𝑇^𝑛₂𝑥_𝑛²−⋯−2𝛽_𝑛𝜃_𝑛,𝑁𝑘−𝛽_𝑛𝑥_𝑛−𝑇^𝑛_𝑁𝑥_𝑛²≤1+2𝛽_𝑛𝑙_𝑛−1𝑥_𝑛−𝑥^∗2, (17)(18) since (𝑘−𝛽_𝑛)≥0 for each 𝑛≥1. Then now, from (14) and (18) we get that 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛(𝑢−𝑥^∗)+(1−𝛼_𝑛)(𝑦_𝑛−𝑥^∗)²≤𝛼_𝑛𝑢−𝑥^∗2+1−𝛼_𝑛𝑦_𝑛−𝑥^∗2≤𝛼_𝑛𝑢−𝑥^∗2+1−𝛼_𝑛1+2𝛽_𝑛𝑙_𝑛−1𝑥_𝑛−𝑥^∗2≤𝛼_𝑛𝑢−𝑥^∗2+1−𝛼_𝑛+𝜖𝛼_𝑛𝑥_𝑛−𝑥^∗2,∀𝑛≥𝑁₀,≤𝛼_𝑛𝑢−𝑥^∗2+1−𝛼_𝑛(1−𝜖)𝑥_𝑛−𝑥^∗2,∀𝑛≥𝑁₀, (19) where 𝑁₀ is a positive integer such that 2(1−𝛼_𝑛)𝛽_𝑛(𝑙_𝑛−1)/𝛼_𝑛<𝜖, for all 𝑛≥𝑁₀, for some 𝜖>0. Therefore, by induction, 𝑥_𝑛+1−𝑥^∗2≤max𝑥_𝑁0−𝑥^∗2,(1−𝜖)⁻¹𝑢−𝑥^∗2,∀𝑛≥𝑁₀, (20) which implies that {𝑥_𝑛} and hence {𝑦_𝑛} are bounded.
Furthermore, from (14), Lemma 3, and (17) we get that 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝑢−𝑥^∗+(1−𝛼_𝑛)(𝑦_𝑛−𝑥^∗)²≤1−𝛼_𝑛𝑦_𝑛−𝑥^∗2+2𝛼_𝑛𝑢−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛1+2𝛽_𝑛𝑙_𝑛−1𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑢−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗−2𝛽_𝑛𝜃_𝑛,1𝑘−𝛽_𝑛1−𝛼_𝑛𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛²−2𝛽_𝑛𝜃_𝑛,2𝑘−𝛽_𝑛1−𝛼_𝑛𝑥_𝑛−𝑇^𝑛₂𝑥_𝑛²−⋯−2𝛽_𝑛𝜃_𝑛,𝑁𝑘−𝛽_𝑛1−𝛼_𝑛𝑥_𝑛−𝑇^𝑛_𝑁𝑥_𝑛²≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑢−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗+2𝛽_𝑛𝑀𝑙_𝑛−1−2𝛽_𝑛𝜃_𝑛,11−𝛼_𝑛𝑘−𝛽_𝑛𝑥_𝑛−𝑇^𝑛₁𝑥_𝑛²−2𝛽_𝑛𝜃_𝑛,2𝑘−𝛽_𝑛1−𝛼_𝑛𝑥_𝑛−𝑇^𝑛₂𝑥_𝑛²−⋯−2𝛽_𝑛𝜃_𝑛,𝑁𝑘−𝛽_𝑛1−𝛼_𝑛𝑥_𝑛−𝑇^𝑛_𝑁𝑥_𝑛²≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑢−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗+2𝛽_𝑛𝑀𝑙_𝑛−1, (21)(22) for some 𝑀>0.
Now, the rest of the proof is divided into two parts.
Case 1. Suppose that there exists 𝑁₁≥0 such that {||𝑥_𝑛−𝑥^∗||} is decreasing for all 𝑛≥𝑁₁. Then we have that {||𝑥_𝑛−𝑥^∗||)} is convergent. Then from (21) and the assumptions on {𝛽_𝑛}, {𝛼_𝑛}, and {𝑙_𝑛} we have that 𝛽_𝑛𝜃_𝑛,𝑖(1−𝑐)(𝑘−𝛽_𝑛)||𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛||²→0, as 𝑛→∞, which implies that 𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛⟶0,as𝑛⟶∞, (23) for 𝑖=1,2,…,𝑁. Then from (14) we obtain that 𝑥_𝑛+1−𝑦_𝑛=𝛼_𝑛𝑢−𝑦_𝑛⟶0,as𝑛⟶∞. (24) Again, from (23) we get that 𝑦_𝑛−𝑥_𝑛=𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑥_𝑛≤𝑆_𝑛𝑥_𝑛−𝑥_𝑛≤𝜃_𝑛,1𝑇^𝑛₁𝑥_𝑛−𝑥_𝑛+𝜃_𝑛,2𝑇^𝑛₂𝑥_𝑛−𝑥_𝑛+⋯+𝜃_𝑛,𝑁𝑇^𝑛_𝑁𝑥_𝑛−𝑥_𝑛⟶0, (25) as 𝑛→∞. Thus, (24) and (25) imply that 𝑥_𝑛+1−𝑥_𝑛⟶0,as𝑛⟶∞. (26) Therefore, since each 𝑇_𝑖, for 𝑖=1,2,…,𝑁, is uniformly 𝐿-Lipschitzian and 𝑥_𝑛−𝑇_𝑖𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇_𝑖^𝑛+1𝑥_𝑛+1+𝑇_𝑖^𝑛+1𝑥_𝑛+1−𝑇_𝑖^𝑛+1𝑥_𝑛+𝑇_𝑖^𝑛+1𝑥_𝑛−𝑇_𝑖𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇_𝑖^𝑛+1𝑥_𝑛+1+𝐿𝑥_𝑛+1−𝑥_𝑛+𝑇_𝑖𝑇^𝑛_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛=(1+𝐿)𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇_𝑖^𝑛+1𝑥_𝑛+1+𝑇_𝑖𝑇^𝑛_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛, (27) we have from (23), (26), and uniform continuity of 𝑇_𝑖 that 𝑥_𝑛−𝑇_𝑖𝑥_𝑛⟶0,as𝑛⟶∞, (28) for each 𝑖=1,2,…,𝑁. Furthermore, the fact that {𝑥_𝑛} is bounded and 𝐸 is reflexive implies that we can choose a subsequence {𝑥_𝑛𝑖+1} of {𝑥_𝑛+1} such that 𝑥_𝑛𝑖+1⇀𝑧 andlimsup_𝑛→∞𝑢−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗=lim_𝑖→∞𝑢−𝑥^∗,𝐽𝑥_𝑛𝑖+1−𝑥^∗. (29) Now, from (26) we get that 𝑥_𝑛𝑖⇀𝑧 and from Lemma 6 we have that 𝑧∈𝐹(𝑇_𝑖), for each 𝑖=1,2,…,𝑁. Hence, 𝑧∈∩^𝑁_𝑖=1𝐹(𝑇_𝑖). Therefore, putting 𝑥^∗=𝑧 in (29) and using the fact that 𝐽 is weakly sequentially continuous we immediately obtain that limsup_𝑛→∞⟨𝑢−𝑧,𝐽(𝑥_𝑛+1−𝑧)⟩=lim_𝑖→∞⟨𝑢−𝑧,𝐽(𝑥_𝑛𝑖+1−𝑧)⟩=⟨𝑢−𝑧,𝐽(𝑧−𝑧)⟩=0. Again, putting 𝑥^∗=𝑧 in inequality (22), we get that 𝑥_𝑛+1−𝑧²≤1−𝛼_𝑛𝑥_𝑛−𝑧²+2𝛼_𝑛𝑢−𝑧,𝐽𝑥_𝑛+1−𝑧+2𝛽_𝑛𝑀𝑙_𝑛−1, (30) and, hence, it follows from (30) and Lemma 5 that ||𝑥_𝑛−𝑧||→0, as 𝑛→∞. Consequently, 𝑥_𝑛→𝑧.
Case 2. Suppose that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑥_𝑛𝑖−𝑥^∗<𝑥_𝑛𝑖+1−𝑥^∗, (31) for all 𝑖∈𝑁. Then, by Lemma 7, there exists a nondecreasing sequence {𝑚_𝑗}⊂𝑁 such that 𝑚_𝑗→∞, ||𝑥_𝑚𝑗−𝑥^∗||≤||𝑥_𝑚𝑗+1−𝑥^∗|| and ||𝑥_𝑗−𝑥^∗||≤||𝑥_𝑚𝑗+1−𝑥^∗|| for all 𝑗∈𝑁. Then from (21) and following the method of Case 1, we get that 𝑥_𝑚𝑗−𝑇^𝑚𝑗_𝑖𝑥_𝑚𝑗⟶0,as𝑗⟶∞, (32) for each 𝑖=1,2,…,𝑁. Thus, again following the method of Case 1, we obtain that 𝑥_𝑚𝑗+1−𝑥_𝑚𝑗→0 and 𝑥_𝑚𝑗−𝑇_𝑖𝑥_𝑚𝑗→0, as 𝑗→∞, for each 𝑖=1,2,…,𝑁 and there exists 𝑧^∗∈∩^𝑁_𝑖=1𝐹(𝑇_𝑖) such that limsup_𝑗→∞𝑢−𝑧^∗,𝐽𝑥_𝑚𝑗+1−𝑧^∗=0. (33) Then now, putting 𝑥^∗=𝑧^∗ in (22) we have that 𝑥_𝑚𝑗+1−𝑧^∗2≤1−𝛼_𝑚𝑗𝑥_𝑚𝑗−𝑧^∗2+2𝛼_𝑚𝑗𝑢−𝑧^∗,𝐽𝑥_𝑚𝑗+1−𝑧^∗+2𝛽_𝑚𝑗𝑀𝑙_𝑚𝑗−1. (34) Since ‖𝑥_𝑚𝑗−𝑧^∗‖²≤‖𝑥_𝑚𝑗+1−𝑧^∗‖², (34) implies that 𝛼_𝑚𝑗𝑥_𝑚𝑗−𝑧^∗2≤𝑥_𝑚𝑗−𝑧^∗2−𝑥_𝑚𝑗+1−𝑧^∗2+2𝛼_𝑚𝑗𝑢−𝑧^∗,𝐽𝑥_𝑚𝑗+1−𝑧^∗+2𝛽_𝑚𝑗𝑀𝑙_𝑚𝑗−1. (35) Moreover, since 𝛼_𝑚𝑗>0, inequality (35) gives that 𝑥_𝑚𝑗−𝑧^∗2≤2𝑢−𝑧^∗,𝐽𝑥_𝑚𝑗+1−𝑧^∗+2𝛽_𝑚𝑗𝑀𝑙_𝑚𝑗−1𝛼_𝑚𝑗. (36) Then, from (33) and the fact that 2𝛽_𝑚𝑗𝑀(𝑙_𝑚𝑗−1)/𝛼_𝑚𝑗→0, we obtain that ||𝑥_𝑚𝑗−𝑧^∗||→0, as 𝑗→∞. This together with (34) gives that ||𝑥_𝑚𝑗+1−𝑧^∗||→0, as 𝑗→∞. But ||𝑥_𝑗−𝑧^∗||≤||𝑥_𝑚𝑗+1−𝑧^∗||, for all 𝑗∈𝑁; thus we obtain that 𝑥_𝑗→𝑧^∗. Therefore, from the above two cases, we can conclude that {𝑥_𝑛} converges strongly to an element of 𝐹 and the proof is complete.

If, in Theorem 8, we assume a single asymptotically 𝑘-strict pseudocontractive mapping we get the following corollary.

Corollary 9. Let 𝐶 be a nonempty, closed, and convex subset of a real uniformly convex Banach space 𝐸 which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from 𝐸 into 𝐸^∗. Let 𝑇∶𝐶→𝐶 be an asymptotically 𝑘-strict pseudocontractive mapping for 0≤𝑘<1 with sequences {𝑙_𝑛}⊂[1,∞). Assume that 𝐹(𝑇) is nonempty. Let {𝑥_𝑛} be a sequence defined by 𝑥₁=𝑢∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1, (37) where {𝛼_𝑛}⊂(0,𝑐)⊂(0,1), satisfying lim_𝑛→∞𝛼_𝑛=0, ∑𝛼_𝑛=∞, lim_𝑛→∞((𝑙_𝑛−1)/𝛼_𝑛)=0, and {𝛽_𝑛}⊂[𝑎,𝑏]⊂(0,𝑘) (𝑎, 𝑏, and 𝑐 constants). Then the sequence {𝑥_𝑛} generated by (37) converges strongly to a fixed point of 𝑇.

Proof. Putting 𝑇=𝑇₁=𝑇₂=⋯=𝑇_𝑁 in (14), we get that 𝑆_𝑛=𝑇^𝑛 and the scheme reduces to scheme (37) and following the method of proof of Theorem 8 we get that (see (21) and (22)) 𝑥_𝑛+1−𝑥^∗2≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑢−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗−2𝛽_𝑛𝑘−𝛽_𝑛1−𝛼_𝑛𝑥_𝑛−𝑇^𝑛𝑥_𝑛²+2𝛽_𝑛𝑀^′𝑙_𝑛≤−11−𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑢−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗−2𝛽_𝑛𝑘−𝛽_𝑛(1−𝑐)𝑥_𝑛−𝑇^𝑛𝑥_𝑛²+2𝛽_𝑛𝑀^′𝑙_𝑛≤−11−𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑢−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗+2𝛽_𝑛𝑀^′𝑙_𝑛−1, (38) for some 𝑀^′>0. Now, considering cases, as in the proof of Theorem 8, we obtain the required result.

Corollary 10. Let 𝐾 be a nonempty, closed, and convex subset of 𝑙_𝑝, 1<𝑝<∞. Let 𝑇_𝑖∶𝐶→𝐶 be asymptotically 𝑘_𝑖-strict pseudocontractive mappings for 0≤𝑘_𝑖<1 with sequences {𝑙_𝑛,𝑖}⊂[1,∞), for 𝑖=1,2,…,𝑁. Assume that 𝐹∶=∩^𝑛_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence defined by 𝑥₁=𝑢∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑆_𝑛𝑥_𝑛,𝑛≥1, (39) where 𝑆_𝑛∶=𝜃_𝑛,1𝑇^𝑛₁+𝜃_𝑛,2𝑇^𝑛₂+⋯+𝜃_𝑛,𝑁𝑇^𝑛_𝑁, such that 𝜃_𝑛,1+𝜃_𝑛,2+⋯+𝜃_𝑛,𝑁=1, for each 𝑛≥1, {𝛼_𝑛},{𝜃_𝑛,𝑖}⊂(0,𝑐)⊂(0,1), satisfying liminf_𝑛→∞𝜃_𝑛,𝑖>0, lim_𝑛→∞𝛼_𝑛=0, ∑𝛼_𝑛=∞, lim_𝑛→∞((𝑙_𝑛,𝑖−1)/𝛼_𝑛)=0 and {𝛽_𝑛}⊂[𝑎,𝑏]⊂(0,𝑘) (for 𝑎, 𝑏, and 𝑐 constants), for 𝑘=min_{1≤𝑖≤𝑁}{𝑘_𝑖}. Then the sequence {𝑥_𝑛} generated by (39) converges strongly to a common fixed point of {𝑇_𝑖∶𝑖=1,2,…,𝑁}.

Proof. We note that 𝑙_𝑝, 1<𝑝<∞, spaces are uniformly convex which have Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from 𝐸 into 𝐸^∗ (see, e.g., [18]). Thus, the result follows from Theorem 8.

Corollary 11. Let 𝐾 be a nonempty, closed, and convex subset of 𝑙_𝑝, 1<𝑝<∞. Let 𝑇∶𝐶→𝐶 be an asymptotically 𝑘-strict pseudocontractive mapping for some 0≤𝑘<1 with sequences {𝑙_𝑛}⊂[1,∞). Assume that 𝐹(𝑇) is nonempty. Let {𝑥_𝑛} be a sequence defined by 𝑥₁=𝑢∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1, (40) where {𝛼_𝑛}⊂(0,𝑐)⊂(0,1), and {𝛽_𝑛}⊂[𝑎,𝑏]⊂(0,𝑘) (for 𝑎, 𝑏, and 𝑐 constants) satisfying lim_𝑛→∞𝛼_𝑛=0, ∑𝛼_𝑛=∞ and lim_𝑛→∞((𝑙_𝑛−1)/𝛼_𝑛)=0. Then the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

If in Theorem 8 we have that 𝐸=𝐻, a real Hilbert space, then 𝐸 is uniformly convex with Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping. Thus, we have the following corollary.

Corollary 12. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖∶𝐶→𝐶 be asymptotically 𝑘_𝑖-strict pseudocontractive mappings for 0≤𝑘_𝑖<1 with sequences {𝑙_𝑛,𝑖}⊂[1,∞), for 𝑖=1,2,…,𝑁. Assume that 𝐹∶=∩^𝑛_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence defined by 𝑥₁=𝑢∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑆_𝑛𝑥_𝑛,𝑛≥1, (41) where 𝑆_𝑛∶=𝜃_𝑛,1𝑇^𝑛₁+𝜃_𝑛,2𝑇^𝑛₂+⋯+𝜃_𝑛,𝑁𝑇^𝑛_𝑁, such that 𝜃_𝑛,1+𝜃_𝑛,2+⋯+𝜃_𝑛,𝑁=1, for each 𝑛≥1, {𝛼_𝑛},{𝜃_𝑛,𝑖}⊂(0,𝑐)⊂(0,1), satisfying liminf_𝑛𝜃_𝑛,𝑖>0, lim_𝑛→∞𝛼_𝑛=0, ∑𝛼_𝑛=∞, lim_𝑛→∞((𝑙_𝑛,𝑖−1)/𝛼_𝑛)=0 and {𝛽_𝑛}⊂[𝑎,𝑏]⊂(0,𝑘) (for 𝑎, 𝑏, and 𝑐 constants), for 𝑘=min_{1≤𝑖≤𝑁}{𝑘_𝑖}. Then the sequence {𝑥_𝑛} generated by (41) converges strongly to a common fixed point of {𝑇_𝑖∶𝑖=1,2,…,𝑁}.

Corollary 13. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be an asymptotically 𝑘-strict pseudocontractive mapping for some 0≤𝑘<1 with sequences {𝑙_𝑛}⊂[1,∞). Assume that 𝐹(𝑇) is nonempty. Let {𝑥_𝑛} be a sequence defined by 𝑥₁=𝑢∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1, (42) where {𝛼_𝑛}⊂(0,𝑐)⊂(0,1), and {𝛽_𝑛}⊂[𝑎,𝑏]⊂(0,𝑘) (for 𝑎, 𝑏, and 𝑐 constants) satisfying lim_𝑛→∞𝛼_𝑛=0, ∑𝛼_𝑛=∞ and lim_𝑛→∞((𝑙_𝑛−1)/𝛼_𝑛)=0. Then the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Remark 14. We note that Corollary 9 generalizes several recent results of this nature. Particularly, it extends Theorem KX of [12], Theorem 2 of Liu [5], and corresponding theorem of Schu [7] in the sense that our convergence is strong in more general Banach spaces possessing weakly sequentially continuous duality mappings without the requirement that 𝑇 be completely continuous.

Remark 15. Corollary 9 is an improvement of Theorem 3.2 of Osilike et al. [13] and Theorems 3.1 and 3.2 of Zhang and Xie [17] in the sense that our convergence is strong without the requirement that liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹(𝑇))=0, provided that 𝐸 possesses weakly sequentially continuous duality mappings.

Acknowledgments

N. Shahzad gratefully acknowledges research support from the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.