Approximation Analysis for a Common Fixed Point of Finite Family of Mappings Which Are Asymptotically 𝑘-Strict Pseudocontractive in the Intermediate Sense

Abstract

We introduce an iterative process which converges strongly to a common fixed point of a finite family of uniformly continuous asymptotically 𝑘_𝑖-strict pseudocontractive mappings in the intermediate sense for 𝑖=1,2,…,𝑁. The projection of 𝑥₀ onto the intersection of closed convex sets 𝐶_𝑛 and 𝑄_𝑛 for each 𝑛≥1 is not required. Moreover, the restriction that the interior of common fixed points is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

1. Introduction and Preliminaries

Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻. A mapping 𝑇∶𝐶→𝐻 is called Lipschitzian if there exists 𝐿>0 such that ‖𝑇𝑥−𝑇𝑦‖≤𝐿‖𝑥−𝑦‖, for all 𝑥, 𝑦∈𝐶. If 𝐿=1, then 𝑇 is called nonexpansive, and if 𝐿∈[0,1), 𝑇 is called contraction. 𝑇 is called uniformly 𝐿-Lipschitzian if there exists 𝐿>0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖, for all 𝑥, 𝑦∈𝐶 and all 𝑛≥1. Clearly, every contraction mapping is nonexpansive and every nonexpansive mapping is uniformly 𝐿-Lipschitzian with 𝐿=1 and hence Lipschitzian.

A mapping 𝑇∶𝐶→𝐻 is said to be asymptotically nonexpansive if there exists a sequence {𝛾_𝑛}⊂[0,∞) with 𝛾_𝑛→0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤(1+𝛾_𝑛)‖𝑥−𝑦‖ for all integers 𝑛≥1 and all 𝑥,𝑦∈𝐶. 𝑇 is said to be asymptotically nonexpansive in the intermediate sense if there exist sequences {𝛾_𝑛},{𝑐_𝑛}⊂[0,∞) with 𝛾_𝑛→0, 𝑐_𝑛→0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤(1+𝛾_𝑛)‖𝑥−𝑦‖+𝑐_𝑛 for all integers 𝑛≥1 and all 𝑥,𝑦∈𝐶.

A mapping 𝑇∶𝐶→𝐻 is said to be asymptotically 𝑘-strict pseudocontractive if there exist a constant 𝑘∈[0,1) and a sequence {𝛾_𝑛}⊂[0,∞) with 𝛾_𝑛→0, as 𝑛→∞, such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤1+𝛾_𝑛𝑥−𝑦²+𝑘(𝐼−𝑇^𝑛)𝑥−(𝐼−𝑇^𝑛)𝑦²,∀𝑥,𝑦∈𝐶. (1) The class of asymptotically 𝑘-strict pseudocontractive mappings which includes the class of asymptotically nonexpansive, and hence the class of nonexpansive mappings was introduced by Liu [1] in 1996 (see, also [2]). Kim and Xu [3] proved that the fixed point set of asymptotically 𝑘-strict pseudocontractions is closed and convex. Recall that a fixed point of a map 𝑇∶𝐶→𝐻 is a set {𝑥∈𝐶∶𝑇𝑥=𝑥}, and it is denoted by 𝐹(𝑇). In addition, it is noted in [3] that every asymptotically 𝑘-strict pseudocontractive mapping with sequence {𝛾_𝑛} is a uniformly 𝐿-Lipschitzian mapping with 𝐿∶=sup{(𝑘+1+(1−𝑘)𝛾_𝑛)/(1−𝑘)∶𝑛∈ℕ}.

A mapping 𝑇 is said to be an asymptotically 𝑘-strict pseudocontractive in the intermediate sense if limsup_𝑛→∞sup_{𝑥,𝑦∈𝐶}𝑇^𝑛𝑥−𝑇^𝑛𝑦²−1+𝛾_𝑛𝑥−𝑦²−𝑘𝐼−𝑇^𝑛𝑥−𝐼−𝑇^𝑛𝑦≤0, (2) where 𝑘∈[0,1) and {𝛾_𝑛}⊂[0,∞) such that 𝛾_𝑛→0, as 𝑛→∞. If we put 𝑐_𝑛∶=max0,sup_{𝑥,𝑦∈𝐶}𝑇^𝑛𝑥−𝑇^𝑛𝑦²−1+𝛾_𝑛𝑥−𝑦²−𝑘𝐼−𝑇^𝑛𝑥−𝐼−𝑇^𝑛𝑦². (3) It follows that 𝑐_𝑛→0, as 𝑛→∞, and (2) is reduced to the following: 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤1+𝛾_𝑛𝑥−𝑦²+𝑘𝐼−𝑇^𝑛𝑥−𝐼−𝑇^𝑛𝑦²+𝑐_𝑛,∀𝑥,𝑦∈𝐶. (4) We note that the class of asymptotically 𝑘-strict pseudocontractive mappings is properly contained in a class of asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense (see examples in [4]). The class of asymptotically 𝑘-strict pseudocontractive mappings in the intermediate sense was introduced by Sahu et al. [4]. They obtained a weak convergence theorem of modified Mann iterative processes for these class of mappings. In [4], Sahu et al. established the following classical result.

Theorem SXY1 (see [4]). Let 𝐻 be a real Hilbert space, and let 𝐶⊂𝐻 be a nonempty closed and convex set. Let 𝑇 be a uniformly continuous and asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense with sequences {𝛾_𝑛} and {𝑐_𝑛} such that 𝐹(𝑇) is nonempty and ∑^𝑛_𝑛=1𝛾_𝑛<∞. Assume that {𝛼_𝑛} is a sequence in (0,1) such that 0<𝛿≤𝛼_𝑛≤1−𝑘−𝛿<1 and ∑^𝑛_𝑛=1𝛼_𝑛𝑐_𝑛<∞. Let {𝑥_𝑛} be a sequence defined by 𝑥₁=𝑥∈𝐶 and 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝑛≥1. (5) Then, {𝑥_𝑛} converges weakly to a fixed point of 𝑇.

But it is worth mentioning that the convergence obtained is a weak convergence. Furthermore, we observe from the proof of Theorem SXY1 that if, in addition, 𝐶 or 𝑇 has some compactness assumption, we obtain that the sequence {𝑥_𝑛} given by (5) converges strongly to a fixed point of 𝑇.

Attempts to modify the Mann iteration method (5) so that strong convergence is guaranteed, without compactness assumption on 𝐶 or 𝑇, have recently been made. Sahu et al. [4] established the following hybrid Mann algorithm for approximating fixed points of asymptotically 𝑘-strict pseudocontractive mappings in the intermediate sense.

Theorem SXY2 (see [4]). Let 𝐻 be a real Hilbert space, and let 𝐶⊂𝐻 be a nonempty, closed, and convex set. Let 𝑇 be a uniformly continuous asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense with sequences {𝛾_𝑛} and {𝑐_𝑛} such that 𝐹(𝑇) is nonempty and bounded. Assume that {𝛼_𝑛} is a sequence in [0,1] such that 0<𝛿≤𝛼_𝑛≤1−𝑘, for all 𝑛∈𝑁. Let {𝑥_𝑛} be a sequence in 𝐶 defined by 𝑢=𝑥₁∈𝐶 and 𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑥_𝑛𝐶,𝑛≥1,_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑢−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}(𝑢),𝑛≥1, (6) where 𝜃_𝑛=𝑐_𝑛+𝛾_𝑛𝛿_𝑛 and 𝛿_𝑛=sup{‖𝑥_𝑛−𝑧‖∶𝑧∈𝐹(𝑇)}<∞. Then, {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)(𝑢).

Recently, Hu and Cai [5] studied the strong convergence of the modified Mann iteration process (5) for a finite family of asymptotically 𝑘-strict pseudocontractive mappings in the intermediate sense. More precisely, they obtained the following theorem.

Theorem HC (see [5]). Let 𝐻 be a real Hilbert space, let 𝐶⊂𝐻 be a nonempty, closed, and convex set. Let 𝑇_𝑖∶𝐶→𝐶 be uniformly continuous asymptotically 𝑘_𝑖-strict pseudocontractive mappings in the intermediate sense for some 0≤𝑘_𝑖<1 with sequences {𝛾_𝑛,𝑖} and {𝑐_𝑛,𝑖} such that lim_𝑛→∞𝛾_𝑛,𝑖=0 and lim_𝑛→∞𝑐_𝑛,𝑖=0 for 𝑖=1, 2, …, 𝑁. Let 𝑘=max{𝑘_𝑖∶1≤𝑖≤𝑁}, 𝛾_𝑛=max{𝛾_𝑛,𝑖∶1≤𝑖≤𝑁}, and 𝑐_𝑛=max{𝑐_𝑛,𝑖∶1≤𝑖≤𝑁}. Assume that 𝐹∶=⋂^𝑁_𝑛=1𝐹(𝑇_𝑖) is nonempty and bounded. Let {𝛽_𝑛} be a sequence in [0,1] such that 0<𝛿≤𝛽_𝑛≤1−𝑘 for all 𝑛∈𝑁. Let {𝑥_𝑛} be a sequence defined by 𝑥₀∈𝐶 and 𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑘(𝑛)_𝑖(𝑛)𝑥_𝑛𝐶,𝑛≥1,_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥1, (7) where 𝜃_𝑛=𝛾_𝑘(𝑛)𝜌²_𝑛+𝑐_𝑘(𝑛)→0, as 𝑛→∞, for 𝜌_𝑛=sup{‖𝑥_𝑛−𝑧‖∶𝑧∈𝐹}<∞, 𝑛=(ℎ−1)𝑁+𝑖, where 𝑖=𝑖(𝑛)∈{1,2,…,𝑁}, ℎ=ℎ(𝑛)≥1 a positive integer such that ℎ(𝑛)→∞, as 𝑛→∞. Then, {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑥₀).

But we observe that the iterative algorithms (6) and (7) generate a sequence {𝑥_𝑛} by projecting 𝑥₀ onto the intersection of closed convex sets 𝐶_𝑛 and 𝑄_𝑛 for each 𝑛≥1 which is not easy to compute.

Attempts to remove projection mapping onto the intersection of closed convex sets 𝐶_𝑛 and 𝑄_𝑛 for each 𝑛≥1 have recently been made. In [6], Zegeye et al. studied the strong convergence of the modified Mann iteration process for the class of asymptotically 𝑘-strict pseudocontractive mappings in the intermediate sense. More precisely, they proved the following theorem.

Theorem ZRS (see [6]). Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian and asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense with sequences {𝛾_𝑛}⊂[0,∞) and {𝑐_𝑛}⊂[0,∞). Assume that the interior of 𝐹(𝑇) is nonempty. Let {𝑥_𝑛} be a sequence defined by 𝑥₁=𝑥∈𝐶 and 𝑦_𝑛=𝛽_𝑛𝑇^𝑛𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑇^𝑛𝑦_𝑛+1−𝛼_𝑛𝑥_𝑛,𝑛≥1, (8) where {𝛼_𝑛} and {𝛽_𝑛} satisfy certain conditions. Then, {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

But it is worth to mention that the assumption interior of 𝐹(𝑇) is nonempty is severe restriction.

It is our purpose, in this paper, to construct an iteration scheme which converges strongly to a common fixed point of a finite family of uniformly continuous asymptotically 𝑘_𝑖-strict pseudocontractive mappings in the intermediate sense for 𝑖=1, 2, …, 𝑁. The projection of 𝑥₀ onto the intersection of closed convex sets 𝐶_𝑛 and 𝑄_𝑛 for each 𝑛≥1 is not required. Furthermore, the restriction that the interior of 𝐹(𝑇) is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

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

Lemma 1. Let 𝐻 be a real Hilbert space. Then, for any given 𝑥, 𝑦∈𝐻, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦. (9)

Lemma 2 (see [7]). Let H be a real Hilbert space. Then, for all 𝑥, 𝑦∈𝐻 and 𝛼_𝑖,∈[0,1] for 𝑖=1, 2, …, 𝑛 such that 𝛼₀+𝛼₁+⋯+𝛼_𝑛=1, the following equality holds: 𝛼₀𝑥₀+𝛼₁𝑥₁+⋯+𝛼_𝑛𝑥_𝑛²=^𝑛_𝑖=0𝛼_𝑖𝑥_𝑖²−_{0≤𝑖,𝑗≤𝑛}𝛼_𝑖𝛼_𝑗𝑥_𝑖−𝑥_𝑗². (10)

Lemma 3 (see [8]). 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. (11) In fact, 𝑚_𝑘=max{𝑗≤𝑘∶𝑎_𝑗<𝑎_𝑗+1}.

Lemma 4 (see [9]). 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 5 (see [4]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a continuous asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense. Then, 𝐹(𝑇) is closed and convex.

Lemma 6 (see [4]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a uniformly continuous asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense. Let {𝑥_𝑛} be a sequence in 𝐶 such that ‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖→0 and ‖𝑥_𝑛−𝑥_𝑛+1‖→0, as 𝑛→∞. Then, ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0, as 𝑛→∞.

Lemma 7 (see [4]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, and 𝑇∶𝐶→𝐶 be a continuous asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense. Then if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛 converges weakly 𝑥∈𝐶 and limsup_𝑚→∞limsup_𝑛→∞‖𝑥_𝑛−𝑇^𝑚𝑥_𝑛‖=0, then (𝐼−𝑇)𝑥=0.

Lemma 8 (see [10]). Let 𝐶 be a nonempty closed, convex subset of a Hilbert space 𝐻 and let 𝑃_𝐶 be the metric projection mapping from 𝐻 onto 𝐶. Given 𝑧=𝑃_𝐶𝑥 if and only if ⟨𝑦−𝑧,𝑥−𝑧⟩≤0, for all 𝑦∈𝐶.

2. Main Result

Theorem 9. Let 𝐶 be a nonempty, closed and convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖∶𝐶→𝐶 be uniformly continuous asymptotically 𝑘_𝑖-strict pseudocontractive mappings in the intermediate sense for some 0≤𝑘_𝑖<1 with sequences {𝛾_𝑛,𝑖} and {𝑐_𝑛,𝑖}, for 𝑖=1, 2, …, 𝑁. Assume that 𝐹∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑦=𝑤∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛽_𝑛,0𝑥_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝑇^𝑛_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝑦_𝑛, (13) where 0<𝛼_𝑛≤𝑎₀<1 such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, lim_𝑛→∞[𝛾_𝑛,𝑖/𝛼_𝑛]=0=lim_𝑛→∞[𝑐_𝑛,𝑖/𝛼_𝑛], and 0<𝛿≤𝛽_𝑛,𝑖≤1−𝑘−𝛿<1 satisfying 𝛽_𝑛,0+𝛽_𝑛,1+⋯+𝛽_𝑛,𝑁=1 for each 𝑛≥1 and 𝑘∶=max{𝑘_𝑖∶𝑖=1,2,…,𝑁}. Then, {𝑥_𝑛} converges strongly to an element of 𝐹.

Proof. Let 𝑥^∗=𝑃_𝐹𝑤. Let 𝛾_𝑛∶=max{𝛾_𝑛,𝑖∶𝑖=1,2,…,𝑁} and 𝑐_𝑛∶=max{𝑐_𝑛,𝑖∶𝑖=1,2,…,𝑁}. Then, from (13), Lemma 2, and asymptotically 𝑘_𝑖-strict pseudocontractiveness of 𝑇_𝑖, for each 𝑖∈{1,2,…,𝑁}, we get that 𝑦_𝑛−𝑥^∗2=𝛽_𝑛,0𝑥_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝑇^𝑛_𝑖𝑥_𝑛−𝑥^∗2≤𝛽_𝑛,0𝑥_𝑛−𝑥^∗2+^𝑁_𝑖=1𝛽_𝑛,𝑖𝑇^𝑛_𝑖𝑥_𝑛−𝑥^∗2−^𝑁_𝑖=1𝛽_𝑛,0𝛽_𝑛,𝑖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²≤𝛽_𝑛,0𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛,01+𝛾_𝑛𝑥_𝑛−𝑥^∗2+^𝑁_𝑖=1𝛽_𝑛,𝑖𝑘𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²+1−𝛽_𝑛,0𝑐_𝑛−^𝑁_𝑖=1𝛽_𝑛,0𝛽_𝑛,𝑖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²≤1+𝛾_𝑛𝑥_𝑛−𝑥^∗2−^𝑁_𝑖=1𝛽_𝑛,𝑖(1−𝑘−𝛿)𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²+1−𝛽_𝑛,0𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑥^∗2−𝛿^2𝑁_𝑖=1𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²+1−𝛽_𝑛,0𝑐_𝑛,𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝑤+1−𝛼_𝑛𝑦_𝑛−𝑥^∗2≤𝛼_𝑛𝑤−𝑥^∗2+1−𝛼_𝑛𝑦_𝑛−𝑥^∗2≤𝛼_𝑛𝑤−𝑥^∗2+1−𝛼_𝑛×1+𝛾_𝑛𝑥_𝑛−𝑥^∗2−𝛿²×^𝑁_𝑖=1𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²+1−𝛽_𝑛,0𝑐_𝑛≤𝛼_𝑛𝑤−𝑥^∗2+1−𝛼_𝑛1+𝛾_𝑛𝑥_𝑛−𝑥^∗2−1−𝛼_𝑛𝛿^2𝑁_𝑖=1𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²+1−𝛼_𝑛1−𝛽_𝑛,0𝑐_𝑛≤𝛼_𝑛𝑤−𝑥^∗2+1+1−𝛼_𝑛(1−𝜖)𝑥_𝑛−𝑥^∗2−1−𝛼_𝑛𝛿^2𝑁_𝑖=1𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²≤𝛼_𝑛𝑤−𝑥^∗2+1+1−𝛼_𝑛(1−𝜖)𝑥_𝑛−𝑥^∗2 (14)(15) since there exists 𝑁₀>0 such that 𝛾_𝑛/𝛼_𝑛≤𝜖 and 𝑐_𝑛/𝛼_𝑛≤1 for all 𝑛≥𝑁₀ and for some 𝜖>0 satisfying (1−𝜖)𝛼_𝑛≤1. Thus, by induction, 𝑥_𝑛+1−𝑥^∗2≤max(1−𝜖)⁻¹𝑤−𝑥^∗2+1,𝑥₀−𝑥^∗2,∀𝑛≥𝑁₀, (16) which implies that {𝑥_𝑛}, and hence {𝑇^𝑛_𝑖𝑥_𝑛} and {𝑦_𝑛} are bounded. Moreover, from (13), (14) and Lemma 1, we obtain that 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝑤−𝑥^∗+1−𝛼_𝑛𝑦_𝑛−𝑥^∗2≤1−𝛼_𝑛𝑦_𝑛−𝑥^∗2+2𝛼_𝑛𝑤−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛1+𝛾_𝑛𝑥_𝑛−𝑥^∗2−𝛿^2𝑁_𝑖=1𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²+1−𝛽_𝑛,0𝑐_𝑛+2𝛼_𝑛𝑤−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥^∗2−𝛿²1−𝛼_𝑛^𝑁_𝑖=1𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²+1−𝛼_𝑛1−𝛽_𝑛,0𝑐_𝑛+2𝛼_𝑛𝑤−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑤−𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛾_𝑛+𝑐_𝑛𝑀−𝛿²1−𝑎₀^𝑁_𝑖=1𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛²≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑤−𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛾_𝑛+𝑐_𝑛𝑀 (17)(18) for some 𝑀>0 and for all 𝑛∈𝑁.
Now, following the method of proof of Lemma 3.2 of Maingé [8], we consider two cases.
Case  1. Suppose that there exists 𝑛₀∈𝑁 such that {‖𝑥_𝑛−𝑥^∗‖} is nonincreasing for all 𝑛≥𝑛₀. In this situation, {‖𝑥_𝑛−𝑥^∗‖)} is convergent. Then, from (17), we have that 𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛⟶0,as𝑛⟶∞, (19) for 𝑖=1, 2, …, 𝑁. Moreover, from (13) and (19) and the fact that 𝛼_𝑛→0, we get that 𝑦_𝑛−𝑥_𝑛≤^𝑁_𝑖=1𝛽_𝑛,𝑖𝑇^𝑛_𝑖𝑥_𝑛−𝑥_𝑛𝑥⟶0,_𝑛+1−𝑦_𝑛=𝛼_𝑛𝑤−𝑦_𝑛⟶0, (20) as 𝑛→∞, and hence 𝑥_𝑛+1−𝑥_𝑛⟶0,as𝑛⟶∞. (21) Furthermore, from (19), (21), and Lemma 6, we obtain that 𝑥_𝑛−𝑇_𝑖𝑥_𝑛⟶0,as𝑛⟶∞. (22) Let {𝑥_𝑛𝑘+1} be a subsequence of {𝑥_𝑛+1} such that limsup_𝑛→∞⟨𝑤−𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩ = lim_𝑘→∞⟨𝑤−𝑥^∗,𝑥_𝑛𝑘+1−𝑥^∗⟩ and {𝑥_𝑛𝑘+1} converges weakly to 𝑣. Then, from (21), we also get that {𝑥_𝑛𝑘} converges weakly to 𝑣. Moreover, since 𝑇_𝑖 is uniformly continuous and ‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖→0, for all 𝑖∈{1,2,…,𝑁}, we get that ‖𝑥_𝑛−𝑇^𝑚_𝑖𝑥_𝑛‖→0, as 𝑛→∞, for all 𝑚∈𝑁. Therefore, by Lemma 7, we obtain that 𝑣∈⋂^𝑁_𝑖=1𝐹(𝑇_𝑖). Now, from Lemma 8, we have that limsup_𝑛→∞𝑤−𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝑤−𝑥^∗,𝑣−𝑥^∗≤0. (23) Then, from (18), (23), and Lemma 4, we obtain that ‖𝑥_𝑛−𝑥^∗‖→0, as 𝑛→∞. Consequently, 𝑥_𝑛→𝑥^∗.
Case  2. Suppose that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑥_𝑛𝑖−𝑥^∗<𝑥_𝑛𝑖+1−𝑥^∗ (24) for all 𝑖∈𝑁. Then, by Lemma 3, there exist a nondecreasing sequence {𝑚_𝑘}⊂𝑁 such that 𝑚_𝑘→∞, 𝑥_𝑚𝑘−𝑥^∗≤𝑥_𝑚𝑘+1−𝑥^∗ (25) and ‖𝑥_𝑘−𝑥^∗‖≤‖𝑥_𝑚𝑘+1−𝑥^∗‖, for all 𝑘∈𝑁. Then, from (17) and the fact that 𝛼_𝑛→0, we have 𝛿²1−𝑎₀^𝑁_𝑖=1𝑥_𝑚𝑘−𝑇^𝑚𝑘_𝑖𝑥_𝑚𝑘²≤𝑥_𝑚𝑘−𝑥^∗2−𝑥_𝑚𝑘+1−𝑥^∗2+𝛼_𝑚𝑘𝑥_𝑚𝑘−𝑥^∗2+2𝛼_𝑚𝑘𝑤−𝑥^∗,𝑥_𝑚𝑘+1−𝑥^∗+𝛾_𝑚𝑘+𝑐_𝑚𝑘𝑀⟶0,as𝑘⟶∞. (26) Then, we get that 𝑥_𝑚𝑘−𝑇^𝑚𝑘_𝑖𝑥_𝑚𝑘→0, as 𝑛→∞, for each 𝑖=1, 2, …, 𝑁. Thus, as in Case 1, we obtain that 𝑥_𝑚𝑘−𝑦_𝑚𝑘→0 and 𝑥_𝑚𝑘+1−𝑥_𝑚𝑘→0, as 𝑘→∞ and limsup_𝑘→∞𝑤−𝑥^∗,𝑥_𝑚𝑘+1−𝑥^∗≤0. (27) Now, from (18), we have that 𝑥_𝑚𝑘+1−𝑥^∗2≤1−𝛼_𝑚𝑘𝑥_𝑚𝑘−𝑥^∗+2𝛼_𝑚𝑘𝑤−𝑥^∗,𝑥_𝑚𝑘+1−𝑥^∗+𝛾_𝑚𝑘+𝑐_𝑚𝑘𝑀. (28) This implies that 𝛼_𝑚𝑘𝑥_𝑚𝑘−𝑥^∗2≤𝑥_𝑚𝑘−𝑥^∗2−𝑥_𝑚𝑘+1−𝑥^∗2+2𝛼_𝑚𝑘𝑤−𝑥^∗,𝑥_𝑚𝑘+1−𝑥^∗+𝛾_𝑚𝑘+𝑐_𝑚𝑘𝑀. (29) Then, using (25), inequality (29) implies that 𝛼_𝑚𝑘𝑥_𝑚𝑘−𝑥^∗2≤2𝛼_𝑚𝑘𝑤−𝑥^∗,𝑥_𝑚𝑘+1−𝑥^∗+𝛾_𝑚𝑘+𝑐_𝑚𝑘𝑀. (30) In particular, since 𝛼_𝑚𝑘>0, we get 𝑥_𝑚𝑘−𝑥^∗2≤2𝑤−𝑥^∗,𝑥_𝑚𝑘+1−𝑥^∗+𝛾_𝑚𝑘+𝑐_𝑚𝑘𝛼_𝑚𝑘. (31) Furthermore, using (27) and the fact that (𝛾_𝑚𝑘+𝑐_𝑚𝑘)/𝛼_𝑚𝑘→0, we obtain that ‖𝑥_𝑚𝑘−𝑥^∗‖→0, as 𝑘→∞. This together with (28) give ‖𝑥_𝑚𝑘+1−𝑥^∗‖→0, as 𝑘→∞. But ‖𝑥_𝑘−𝑥^∗‖≤‖𝑥_𝑚𝑘+1−𝑥^∗‖ for all 𝑘∈𝑁. Thus we obtain that 𝑥_𝑘→𝑥^∗. 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 9, we assume that 𝑁=1, then we get the following corollary.

Corollary 10. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a uniformly continuous asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense for some 0≤𝑘<1 with sequences {𝛾_𝑛} and {𝑐_𝑛}. Assume that 𝐹∶=𝐹(𝑇) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑦=𝑤∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝑦_𝑛, (32) where 0<𝛼_𝑛≤𝑎₀<1 such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, lim_𝑛→∞[𝛾_𝑛/𝛼_𝑛]=0=lim_𝑛→∞[𝑐_𝑛/𝛼_𝑛], and 0<𝛿≤𝛽_𝑛≤1−𝑘−𝛿<1 for each 𝑛≥1. Then, {𝑥_𝑛} converges strongly to an element of 𝐹.

If in Theorem 9, we assume that each 𝑇_𝑖 is asymptotically 𝑘_𝑖-strict pseudocontractive mapping, then we get the following corollary.

Corollary 11. 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 some 0≤𝑘_𝑖<1 with sequences {𝛾_𝑛,𝑖}, for 𝑖=1, 2, …, 𝑁. Assume that 𝐹∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑦=𝑤∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛽_𝑛,0𝑥_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝑇^𝑛_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝑦_𝑛, (33) where 0<𝛼_𝑛≤𝑎₀<1 such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, lim_𝑛→∞[𝛾_𝑛,𝑖/𝛼_𝑛]=0, and 0<𝛿≤𝛽_𝑛,𝑖≤1−𝑘−𝛿<1 satisfying 𝛽_𝑛,0+𝛽_𝑛,1+⋯+𝛽_𝑛,𝑁=1 for each 𝑛≥1 and 𝑘∶=max{𝑘_𝑖∶𝑖=1,2,…,𝑁}. Then, {𝑥_𝑛} converges strongly to an element of 𝐹.

Proof. Since every asymptotically 𝑘_𝑖-strict pseudocontractive mapping is uniformly continuous and asymptotically 𝑘_𝑖-strict pseudocontractive mapping in the intermediate sense with 𝑐_𝑛,𝑖≡0, for all 𝑛≥1 and each 𝑖=1,2,…,𝑁, the conclusion follows from Theorem 9.

If in Theorem 9, we assume that each 𝑇_𝑖 is asymptotically nonexpansive in the intermediate sense we obtain 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 uniformly continuous asymptotically nonexpansive mappings in the intermediate sense with sequences {𝛾_𝑛,𝑖} and {𝑐_𝑛,𝑖}, for 𝑖=1, 2, …, 𝑁. Assume that 𝐹∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑦=𝑤∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛽_𝑛,0𝑥_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝑇^𝑛_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝑦_𝑛, (34) where 0<𝛼_𝑛≤𝑎₀<1 such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, lim_𝑛→∞[𝛾_𝑛,𝑖/𝛼_𝑛] = 0 = lim_𝑛→∞[𝑐_𝑛,𝑖/𝛼_𝑛], and 0<𝛿≤𝛽_𝑛,𝑖≤1−𝛿<1 satisfying 𝛽_𝑛,0+𝛽_𝑛,1+⋯+𝛽_𝑛,𝑁=1 for each 𝑛≥1. Then, {𝑥_𝑛} converges strongly to an element of 𝐹.

Proof. Since every asymptotically nonexpansive mapping in the intermediate sense is asymptotically 𝑘_𝑖-strict pseudocontractive mapping in the intermediate sense with 𝑘_𝑖=0, for each 𝑖=1, 2, …, 𝑁, the conclusion follows from Theorem 9.

If in Theorem 9, we assume that each 𝑇_𝑖 is asymptotically nonexpansive, we obtain the following corollary.

Corollary 13. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖∶𝐶→𝐶 be asymptotically nonexpansive mappings with sequences {𝛾_𝑛,𝑖}, for 𝑖=1, 2, …, 𝑁. Assume that 𝐹∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑦=𝑤∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛽_𝑛,0𝑥_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝑇^𝑛_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝑦_𝑛, (35) where 0<𝛼_𝑛≤𝑎₀<1 such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, lim_𝑛→∞[𝛾_𝑛,𝑖/𝛼_𝑛]=0, and 0<𝛿≤𝛽_𝑛,𝑖≤1−𝛿<1 satisfying 𝛽_𝑛,0+𝛽_𝑛,1+⋯+𝛽_𝑛,𝑁=1 for each 𝑛≥1. Then, {𝑥_𝑛} converges strongly to an element of 𝐹.

Proof. Since every asymptotically nonexpansive mapping is uniformly continuous and asymptotically 𝑘_𝑖-strict pseudocontractive mapping with 𝑘_𝑖=0 and 𝑐_𝑛,𝑖=0, for all 𝑛≥1 and 𝑖=1, 2, …, 𝑁, the conclusion follows from Theorem 9.

Remark 14. Our results extend and unify most of the results that have been proved for this important class of nonlinear mappings. In particular, Theorem 9 extends Theorem SXY1, SXY2, HC, and Theorem ZRS in the sense that our convergence is either strong, does not require computation of closed convex sets 𝐶_𝑛 and 𝑄_𝑛 for each 𝑛≥1, or does not require the assumption that interior of set of fixed points is nonempty.

Remark 15. We also remark that Corollary 11 is more general than Theorem 3.1 of Kim and Xu [3] and Corollary 13 is more general than Theorem 2.2 of Kim and Xu [11] in the sense that our convergence is either strong, does not require computation of closed convex sets 𝐶_𝑛 and 𝑄_𝑛 for each 𝑛≥1, or does not require the assumption that interior of set of fixed points is nonempty.

Acknowledgment

The research of N. Shahzad was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.