Strong Convergence of Non-Implicit Iteration Process with Errors in Banach Spaces

Abstract

The purpose of this paper is to study the strong convergence of a non-implicit iteration process with errors for asymptotically I-nonexpansive mappings in the intermediate sense in the framework of Banach spaces. The results presented in this paper extend and improve the corresponding results recently announced.

1. Introduction and Preliminaries

Let 𝐾 be a nonempty, closed, and convex subset of a real Banach space 𝑋 and let 𝑇∶𝐾→𝐾 be a mapping. In this paper, we use 𝐹(𝑇) to stand for the set of fixed points of 𝑇, that is 𝐹(𝑇)={𝑥∈𝐾∶𝑇𝑥=𝑥}.

Recall that 𝑇 is said to be nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐾. (1.1)

𝑇is said to be asymptotically nonexpansive if there exists a sequence {ℎ_𝑛} with ℎ_𝑛⊂[1,+∞) with lim_𝑛→∞ℎ_𝑛=1 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤ℎ_𝑛𝑥−𝑦,∀𝑥,𝑦∈𝐾,𝑛≥1. (1.2)

𝑇 is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds: limsup_𝑛→∞sup_{𝑥,𝑦∈𝑘}𝑇^𝑛𝑥−𝑇^𝑛𝑦−𝑥−𝑦≤0. (1.3)

Observe that if we define 𝑎_𝑛=sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖),𝜎_𝑛=max{0,𝑎_𝑛}, then 𝜎_𝑛→0 as 𝑛→∞ and (1.3) reduces to 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝑥−𝑦+𝜎_𝑛,∀𝑥,𝑦∈𝐾,𝑛≥1. (1.4)

It is easy to see that every nonexpansive mapping is asymptotically nonexpansive. And every asymptotically nonexpansive mapping is asymptotically nonexpansive in the intermediated sense. In [1], Goebel and Kirk proved that, if 𝐾 is a nonempty closed convex bounded subset of a real uniformly convex Banach space 𝑋, and 𝑇 is an asymptotically nonexpansive self-mapping on 𝐾, then 𝑇 has a fixed point in 𝐾. The class of mappings which are asymptotically nonexpansive in the intermediat sense was investigated by Bruck et al. [2] and Kirk [3]. Since then, many authors have investigated the fixed point problem of these mappings based on implicit iterative methods or non-implicit iterative methods; see, for example, [4–21].

Let 𝐼∶𝐾→𝐾 be a mapping. Recall that 𝑇 is said to be asymptotically 𝐼-nonexpansive if there exists a sequence {ℎ_𝑛} with {ℎ_𝑛}⊂[1,+∞) with lim_𝑛→∞ℎ_𝑛=1 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤ℎ_𝑛𝐼^𝑛𝑥−𝐼^𝑛𝑦,∀𝑥,𝑦∈𝐾,𝑛≥1. (1.5) Recently, weak and strong convergence theorems for fixed points of 𝐼-nonexpansive mappings, and asymptotically 𝐼-nonexpansive mappings have been established by many scholar, see, for example, [22–25].

In this paper, we consider a new mapping based on asymptotically nonexpansive mappings in the intermediate sense and asymptotically 𝐼-nonexpansive mappings.

Let 𝑇∶𝐾→𝐾,𝐼∶𝐾→𝐾 be two mappings. 𝑇 is said to be asymptotically 𝐼-nonexpansive in the intermediate sense if it is continuous and the following inequality holds: limsup_𝑛→∞sup_{𝑥,𝑦∈𝑘}𝑇^𝑛𝑥−𝑇^𝑛𝑦−𝐼^𝑛𝑥−𝐼^𝑛𝑦≤0. (1.6) Observe that if we define 𝑎_𝑛=sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝐼^𝑛𝑥−𝐼^𝑛𝑦‖), 𝜎_𝑛=max{0,𝑎_𝑛}, then 𝜎_𝑛→0 as 𝑛→∞ and (1.6) reduces to 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝐼^𝑛𝑥−𝐼^𝑛𝑦+𝜎_𝑛,∀𝑥,𝑦∈𝐾,𝑛≥1. (1.7) Note that if 𝐼=Id, where Id is the identity mapping, then (1.7) reduces to (1.4).

In this paper, we investigate asymptotically 𝐼-nonexpansive mappings in the intermediate sense based on a non-implicit iterative algorithm. Strong convergence of the implicit iterative algorithm is obtained in the framework of Banach spaces.

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

Lemma 1.1 (see [21]). let 𝑋 be a uniformly convex Banach space. Let 𝑏 and 𝑐 be two constants with 0<𝑏<𝑐<1. Suppose that {𝑡_𝑛} is a sequence in [𝑏,𝑐]. Let {𝑥_𝑛} and {𝑦_𝑛} be two sequences in 𝑋 such that limsup_𝑛→∞𝑥_𝑛≤𝑑,limsup_𝑛→∞𝑦_𝑛≤𝑑,lim_𝑛→∞𝑡_𝑛𝑥_𝑛+1−𝑡_𝑛𝑦_𝑛=𝑑 (1.8) hold for some 𝑑≥0, then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 1.2 (see [26]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} be three nonnegative sequences satisfying the following condition: 𝑎_𝑛+1≤1+𝑏_𝑛𝑎_𝑛+𝑐_𝑛,∀𝑛≥𝑛₀, (1.9) where 𝑛₀ is some nonnegative integer, ∑^∞_𝑛=1𝑏_𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑛<∞. Then the limit lim_𝑛→∞𝑎_𝑛 exists.

2. Main Results

Lemma 2.1. Let 𝑋 be a real Banach space and 𝐾 a nonempty closed and convex subset of 𝑋. Let 𝑇∶𝐾→𝐾 be a asymptotically 𝐼-nonexpansive in the intermediate sense and 𝐼∶𝐾→𝐾 a asymptotically nonexpansive in the intermediate sense. Assume that 𝐹∶=𝐹(𝑇)∩𝐹(𝐼)≠∅. Let 𝜎_𝑛=max{0,sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝐼^𝑛𝑥−𝐼^𝑛𝑦‖)} and 𝜎_𝑛=max{0,sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)}. Let {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛} be six real number sequences in (0,1). Let {𝑥_𝑛} be a sequence generated in the following iterative process: 𝑥₁𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝐼^𝑛𝑥_𝑛+𝛾_𝑛𝑣_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑢_𝑛,𝑛≥1, (2.1) where {𝑢_𝑛} and {𝑣_𝑛} be two bounded sequences in 𝐾. Assume that the following restrictions are satisfied: (a)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1; (b)∑^∞_𝑛=1𝜎_𝑛<∞,∑^∞_𝑛=1𝜎_𝑛<∞; (c)∑^∞_𝑛=1𝛾_𝑛<∞,∑^∞_𝑛=1𝛾_𝑛<∞. Then lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for all 𝑝∈𝐹.

Proof. Letting 𝑝∈𝐹, we see that 𝑦_𝑛−𝑝=1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛+𝛽_𝑛𝐼^𝑛𝑥_𝑛+𝛾_𝑛𝑣_𝑛≤𝛽−𝑝1−_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝐼^𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛≤𝛽−𝑝1−_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝜎_𝑛+𝛾_𝑛𝑣_𝑛=𝛾−𝑝1−_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑝+𝛽_𝑛𝜎_𝑛≤𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑝+𝛽_𝑛𝜎_𝑛,𝑥_𝑛+1−𝑝=1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑢_𝑛≤−𝑝1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑇^𝑛𝑦_𝑛−𝑝+𝛾_𝑛𝑢_𝑛=−𝑝1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑇^𝑛𝑦_𝑛−𝑇^𝑛𝑝+𝛾_𝑛𝑢_𝑛≤−𝑝1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝐼^𝑛𝑦_𝑛−𝐼^𝑛𝑝+𝜎_𝑛+𝛾_𝑛𝑢_𝑛=−𝑝1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝐼^𝑛𝑦_𝑛−𝐼^𝑛𝑝+𝛽_𝑛𝜎_𝑛+𝛾_𝑛𝑢_𝑛≤−𝑝1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑦_𝑛−𝑝+𝜎_𝑛+𝛽_𝑛𝜎_𝑛+𝛾_𝑛𝑢_𝑛=−𝑝1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑦_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑝+𝛽_𝑛𝜎_𝑛+𝜎_𝑛≤1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑦_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑝+𝛽_𝑛𝜎_𝑛+𝜎_𝑛. (2.2)(2.3) Substituting (2.2) into (2.3),we obtain that 𝑥_𝑛+1−𝑝≤1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑝+𝛽_𝑛𝜎_𝑛+𝛾_𝑛𝑢_𝑛−𝑝+𝛽_𝑛𝜎_𝑛+𝜎_𝑛=𝑥_𝑛−𝑝+𝛽_𝑛𝛾_𝑛𝑣_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑝+𝛽_𝑛𝜎_𝑛+𝛽_𝑛𝜎_𝑛1+𝛽_𝑛. (2.4) Let 𝑎_𝑛=‖𝑥_𝑛−𝑝‖, 𝑏_𝑛=0, and 𝑐_𝑛=𝛽_𝑛𝛾_𝑛𝑣_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑝+𝛽_𝑛𝜎_𝑛+𝛽_𝑛𝜎_𝑛1+𝛽_𝑛. (2.5) It follows from (2.4) that 𝑎_𝑛+1≤𝑎_𝑛+𝑐_𝑛. (2.6) In view of the restrictions (b) and (c), we see that ∑^∞_𝑛=1𝑐_𝑛<∞. We can easily conclude the desired conclusion with the aid of Lemma 1.2. This completes the proof of Lemma 2.1.

Theorem 2.2. Let 𝑋 be a real Banach space and 𝐾 a nonempty closed and convex subset of 𝑋. Let 𝑇∶𝐾→𝐾 be a asymptotically 𝐼-nonexpansive in the intermediate sense and 𝐼∶𝐾→𝐾 a asymptotically nonexpansive in the intermediate sense. Assume that 𝐹∶=𝐹(𝑇)∩𝐹(𝐼)≠∅. Let 𝜎_𝑛=max{0,sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝐼^𝑛𝑥−𝐼^𝑛𝑦‖)} and 𝜎_𝑛=max{0,sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)}. Let {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} be six real number sequences in (0,1). Let {𝑥_𝑛} be a sequence generated in the following iterative process: 𝑥₁𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝐼^𝑛𝑥_𝑛+𝛾_𝑛𝑣_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑢_𝑛,𝑛≥1, (2.7) where {𝑢_𝑛} and {𝑣_𝑛} be two bounded sequences in 𝐾. Assume that the following restrictions are satisfied: (a)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1; (b)∑^∞_𝑛=1𝜎_𝑛<∞,∑^∞_𝑛=1𝜎_𝑛<∞; (c)∑^∞_𝑛=1𝛾_𝑛<∞,∑^∞_𝑛=1𝛾_𝑛<∞. If both 𝑇 and 𝐼 are continuous, then the sequence {𝑥_𝑛} strongly converges to a common fixed point of 𝑇 and 𝐼 if and only if liminf_𝑛→∞𝑑𝑥_𝑛,𝐹=0. (2.8)

Proof. The necessity is obvious. Next, we prove the sufficiency part of the theorem. Note that continuity of 𝑇 and 𝐼 implies that the set 𝐹(𝑇) and 𝐹(𝐼) are closed. It follows from (2.6) that 𝑥_𝑛+1−𝑝≤𝑥_𝑛−𝑝+𝑐_𝑛. (2.9) This implies in turn that 𝑑𝑥_𝑛+1,𝐹≤𝑑𝑥_𝑛,𝐹+𝑐_𝑛. (2.10) Now applying Lemma 1.2 to (2.10), we obtain the existence of the limit lim_𝑛→∞𝑑(𝑥_𝑛,𝐹). By condition (2.8), we have lim_𝑛→∞𝑑𝑥_𝑛,𝐹=liminf_𝑛→∞𝑑𝑥_𝑛,𝐹=0. (2.11)
Next we prove that the sequence {𝑥_𝑛} is a Cauchy sequence in 𝐾. For any positive integers 𝑛, 𝑚, from (2.9) it follows that 𝑥_𝑛+𝑚−𝑝≤𝑥_{𝑛+𝑚−1}−𝑝+𝑐_{𝑛+𝑚−1}≤𝑥_{𝑛+𝑚−2}−𝑝+𝑐_{𝑛+𝑚−2}+𝑐_{𝑛+𝑚−1}≤𝑥≤⋯_𝑛−𝑝+^{𝑛+𝑚−1}_𝑖=𝑛𝑐_𝑖≤𝑥_𝑛−𝑝+^∞_𝑖=𝑛𝑐_𝑖. (2.12) Since lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, and ∑^∞_𝑛=1𝑐_𝑛<∞, for any given 𝜖>0, there exists a positive integer 𝑛₀ such that 𝑑𝑥_𝑛<𝜖,𝐹8,^∞_𝑖=𝑛𝑐_𝑖<𝜖2,∀𝑛≥𝑛₀. (2.13) Therefore there exists 𝑝₁∈𝐹 such that 𝑑(𝑥_𝑛,𝑝₁)<(𝜖/4), ∀𝑛≥𝑛₀. Consequently, for any 𝑛≥𝑛₀ and for all 𝑚≥1, we have 𝑥_𝑛+𝑚−𝑥_𝑛≤𝑥_𝑛+𝑚−𝑝₁+𝑥_𝑛−𝑝₁≤2𝑥_𝑛−𝑝₁+^∞_𝑖=𝑛𝑐_𝑖≤𝜖4𝜖⋅2+2=𝜖. (2.14) This implies that {𝑥_𝑛} is a Cauchy sequence in 𝐾. Let 𝑥_𝑛→𝑥^∗∈𝐾. Since 𝐹 is closed, this implies that 𝑥^∗∈𝐹. This shows that {𝑥_𝑛} strongly converges to a common fixed of 𝑇 and 𝐼. This completes the proof of Theorem 2.2.

Lemma 2.3. Let 𝑋 be a real Banach space and 𝐾 a nonempty closed and convex subset of 𝑋. Let 𝑇∶𝐾→𝐾 be a asymptotically 𝐼-nonexpansive in the intermediate sense and 𝐼∶𝐾→𝐾 a asymptotically nonexpansive in the intermediate sense. Assume that 𝐹∶=𝐹(𝑇)∩𝐹(𝐼)≠∅. Let 𝜎_𝑛=max{0,sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝐼^𝑛𝑥−𝐼^𝑛𝑦‖)} and 𝜎_𝑛=max{0,sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)}. Let {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛} be six real number sequences in (0,1). Let {𝑥_𝑛} be a sequence generated in the following iterative process: 𝑥₁𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝐼^𝑛𝑥_𝑛+𝛾_𝑛𝑣_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑢_𝑛,𝑛≥1, (2.15) where {𝑢_𝑛} and {𝑣_𝑛} be two bounded sequences in 𝐾. Assume that the following restrictions are satisfied: (a)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, ∀𝑛≥1; (b)∑^∞_𝑛=1𝜎_𝑛<∞, ∑^∞_𝑛=1𝜎_𝑛<∞; (c)there exist constants 𝜏₁,𝜏₂∈(0,1) such that 𝜏₁≤𝛽_𝑛, 𝛽_𝑛≤𝜏₂, ∀𝑛≥1; (d)∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾_𝑛<∞. Then lim_𝑛→∞𝑥_𝑛−𝑇^𝑛𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝐼^𝑛𝑥_𝑛=0. (2.16)(2.17)

Proof. According to Lemma 2.1, for any 𝑝∈𝐹, we have lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Without loss of generality, we may assume that lim_𝑛→∞𝑥_𝑛−𝑝=𝑑, (2.18) where 𝑑>0 is some constant. It follows that lim_𝑛→∞𝑥_𝑛+1−𝑝=lim_𝑛→∞1−𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑦_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑥_𝑛=𝑑. (2.19) Notice that 𝑥_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑥_𝑛. (2.20) It follows from the restriction (d) and (2.18) that limsup_𝑛→∞𝑥_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑥_𝑛≤𝑑. (2.21) Notice that limsup_𝑛→∞𝑇^𝑛𝑦_𝑛−𝑝+𝛾_𝑛𝑢_𝑛−𝑥_𝑛≤limsup_𝑛→∞𝑇^𝑛𝑦_𝑛−𝑝+limsup_𝑛→∞𝛾_𝑛𝑢_𝑛−𝑥_𝑛=limsup_𝑛→∞𝑇^𝑛𝑦_𝑛−𝑝≤limsup_𝑛→∞𝐼^𝑛𝑦_𝑛−𝐼^𝑛𝑝+𝜎_𝑛≤limsup_𝑛→∞𝑦_𝑛−𝑝+𝜎_𝑛+𝜎_𝑛=limsup_𝑛→∞𝑦_𝑛−𝑝≤limsup_𝑛→∞𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑝+𝛽_𝑛𝜎_𝑛=𝑑. (2.22) In view of (2.19), (2.21) and (2.22), we obtain from Lemma 1.1 that lim_𝑛→∞𝑥_𝑛−𝑇^𝑛𝑦_𝑛=0. (2.23) Notice that 𝑥_𝑛+1−𝑥_𝑛=𝛽_𝑛𝑇^𝑛𝑦_𝑛−𝑥_𝑛+𝛾_𝑛𝑢_𝑛−𝑥_𝑛≤𝛽_𝑛𝑇^𝑛𝑦_𝑛−𝑥_𝑛+𝛾_𝑛𝑢_𝑛−𝑥_𝑛. (2.24) It follows from (2.23) and the restriction (d) that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (2.25) Notice that 𝐼^𝑛𝑥_𝑛−𝑝≤𝑥_𝑛−𝑝+𝜎_𝑛. (2.26) It follows that limsup_𝑛→∞𝐼^𝑛𝑥_𝑛−𝑝≤𝑑. (2.27) On the other hand, we have lim_𝑛→∞𝑦_𝑛−𝑝=lim_𝑛→∞1−𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑥_𝑛+𝛽_𝑛𝐼^𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑥_𝑛=𝑑. (2.28) Notice that 𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑥_𝑛. (2.29) It follows that limsup_𝑛→∞𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑥_𝑛≤𝑑. (2.30) Notice that 𝐼^𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑥_𝑛≤𝐼^𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑥_𝑛. (2.31) It follows from (2.27) that limsup_𝑛→∞𝐼^𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑣_𝑛−𝑥_𝑛≤𝑑. (2.32) In view of (2.28), (2.30), and (2.32), we obtain from Lemma 1.1 that lim_𝑛→∞𝑥_𝑛−𝐼^𝑛𝑥_𝑛=0. (2.33) On the other hand, we have 𝑥_𝑛−𝑇^𝑛𝑥_𝑛≤𝑥_𝑛−𝑇^𝑛𝑦_𝑛+𝑇^𝑛𝑦_𝑛−𝑇^𝑛𝑥_𝑛≤𝑥_𝑛−𝑇^𝑛𝑦_𝑛+𝑦_𝑛−𝑥_𝑛+𝜎_𝑛+𝜎_𝑛≤𝑥_𝑛−𝑇^𝑛𝑦_𝑛+𝛽_𝑛𝐼^𝑛𝑥_𝑛−𝑥_𝑛+𝛾_𝑛𝑣_𝑛−𝑥_𝑛+𝜎_𝑛+𝜎_𝑛. (2.34) In view of (2.23) and (2.33), we have lim_𝑛→∞‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖=0. This completes the proof of Lemma 2.3.

Theorem 2.4. Let 𝑋 be a real Banach space and 𝐾 a nonempty closed and convex subset of 𝑋. Let 𝑇∶𝐾→𝐾 be a asymptotically 𝐼-nonexpansive in the intermediate sense and 𝐼∶𝐾→𝐾 a asymptotically nonexpansive in the intermediate sense. Assume that 𝐹∶=𝐹(𝑇)∩𝐹(𝐼)≠∅. Let 𝜎_𝑛=max{0,sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝐼^𝑛𝑥−𝐼^𝑛𝑦‖)} and 𝜎_𝑛=max{0,sup_{𝑥,𝑦∈𝑘}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖)}. Let {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛} be six real number sequences in (0,1). Assume that both 𝑇 and 𝐼 are Lipschitz continuous. Let {𝑥_𝑛} are a sequence generated in the following iterative process: 𝑥₁𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝐼^𝑛𝑥_𝑛+𝛾_𝑛𝑣_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑦_𝑛+𝛾_𝑛𝑢_𝑛,𝑛≥1, (2.35) where {𝑢_𝑛} and {𝑣_𝑛} be two bounded sequences in 𝐾. Assume that the following restrictions are satisfied: (a)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,∀𝑛≥1; (b)∑^∞_𝑛=1𝜎_𝑛<∞,∑^∞_𝑛=1𝜎_𝑛<∞; (c)there exist constants 𝜏₁,𝜏₂∈(0,1) such that 𝜏₁≤𝛽_𝑛,𝛽_𝑛≤𝜏₂,∀𝑛≥1; (d)∑^∞_𝑛=1𝛾_𝑛<∞,∑^∞_𝑛=1𝛾_𝑛<∞. If at least one of the mappings 𝑇 and 𝐼 is compact, then the sequence convergence strongly to a common fixed point of 𝑇 and 𝐼.

Proof. Without loss of generality, we may assume that 𝑇 is compact; this means that there exists a subsequence {𝑇^𝑛𝑘𝑥_𝑛𝑘} of {𝑇^𝑛𝑥_𝑛} such that {𝑇^𝑛𝑘𝑥_𝑛𝑘} converges strongly to 𝑥^∗∈𝐾, then (2.16) implies that {𝑥_𝑛𝑘} converges strongly to 𝑥^∗. Since 𝑇 is continuous, then {𝑇^𝑛𝑘+1𝑥_𝑛𝑘} converges strongly to 𝑇𝑥^∗. On the other hand, according to (2.17) and the continuity of 𝐼, we obtain that {𝐼^𝑛𝑘𝑥_𝑛𝑘}, {𝐼^𝑛𝑘+1𝑥_𝑛𝑘} converge strongly to 𝑥^∗, 𝐼𝑥^∗, respectively. Since lim_𝑘→∞‖𝑥_𝑛𝑘+1−𝑥_𝑛𝑘‖=0, then 𝐼^𝑛𝑘+1𝑥_𝑛𝑘+1−𝐼^𝑛𝑘+1𝑥_𝑛𝑘≤𝑥_𝑛𝑘+1−𝑥_𝑛𝑘+𝜎_𝑛𝑘𝑇→0,as𝑘→∞,_𝑛𝑘+1𝑥_𝑛𝑘+1−𝑇^𝑛𝑘+1𝑥_𝑛𝑘≤𝐼^𝑛𝑘+1𝑥_𝑛𝑘+1−𝐼^𝑛𝑘+1𝑥_𝑛𝑘+𝜎_𝑛𝑘≤𝑥_𝑛𝑘+1−𝑥_𝑛𝑘+𝜎_𝑛𝑘+𝜎_𝑛𝑘→0,as𝑘→∞. (2.36) Observe that 𝑥^∗−𝑇𝑥^∗≤𝑥^∗−𝑥_𝑛𝑘+1+𝑥_𝑛𝑘+1−𝑇^𝑛𝑘+1𝑥_𝑛𝑘+1+𝑇^𝑛𝑘+1𝑥_𝑛𝑘+1−𝑇^𝑛𝑘+1𝑥_𝑛𝑘+𝑇^𝑛𝑘+1𝑥_𝑛𝑘−𝑇𝑥^∗,𝑥^∗−𝐼𝑥^∗≤𝑥^∗−𝑥_𝑛𝑘+1+𝑥_𝑛𝑘+1−𝐼^𝑛𝑘+1𝑥_𝑛𝑘+1+𝐼^𝑛𝑘+1𝑥_𝑛𝑘+1−𝐼^𝑛𝑘+1𝑥_𝑛𝑘+𝐼^𝑛𝑘+1𝑥_𝑛𝑘−𝐼𝑥^∗. (2.37) Taking limit as 𝑘→∞ in the above inequality, we find 𝑥^∗=𝑇𝑥^∗,𝑥^∗=𝐼𝑥^∗, which means 𝑥^∗∈𝐹. However, due to Lemma 2.1, the limit lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ exists, therefore lim_𝑛→∞𝑥_𝑛−𝑥^∗=lim_𝑘→∞𝑥_𝑛𝑘−𝑥^∗=0, (2.38) which means that {𝑥_𝑛} converges strongly to 𝑥^∗∈𝐹. This completes the proof of Theorem 2.4.

Acknowledgment

The work was supported by Natural Science Foundation of Zhejiang Province (Y6110270).