Implicit Mann Type Iteration Method Involving Strictly Hemicontractive Mappings in Banach Spaces

Abstract

We proved that the modified implicit Mann iteration process can be applied to approximate the fixed point of strictly hemicontractive mappings in smooth Banach spaces.

1. Introduction

Let 𝐾 be a nonempty subset of an arbitrary Banach space 𝑋 and let 𝑋^∗ be its dual space. The symbols 𝐷(𝑇) and 𝐹(𝑇) stand for the domain and the set of fixed points of 𝑇 (for a single-valued mapping 𝑇∶𝑋→𝑋,𝑥∈𝑋 is called a fixed point of 𝑇 iff 𝑇𝑥=𝑥). We denote by 𝐽 the normalized duality mapping from 𝑋 to 2^𝑋∗ defined by 𝐽(𝑥)=𝑓^∗∈𝑋^∗∶𝑥,𝑓^∗=‖𝑥‖²=𝑓^∗2,𝑥∈𝑋, (1.1) where ⟨⋅,⋅⟩ denotes the duality pairing. In a smooth Banach space, 𝐽 is singlevalued (we denoted by 𝑗).

Remark 1.1. (1) 𝑋 is called uniformly smooth if 𝑋^∗ is uniformly convex.
(2) In a uniformly smooth Banach space, 𝐽 is uniformly continuous on bounded subsets of 𝑋.

Let 𝑇∶𝐷(𝑇)⊂𝑋→𝑋 be a mapping.

Definition 1.2. The mapping 𝑇 is called Lipshitz if there exists a constant 𝐿>0 such that𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦, (1.2)
for all 𝑥,𝑦∈𝐷(𝑇). If 𝐿=1, then 𝑇 is called nonexpansive and if 0≤𝐿<1, then 𝑇 is called contractive.

Definition 1.3 (see [1, 2]). (1) The mapping 𝑇 is said to be pseudocontractive if𝑥−𝑦≤𝑥−𝑦+𝑟(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦, (1.3) for all 𝑥,𝑦∈𝐷(𝑇) and 𝑟>0.
(2) The mapping 𝑇 is said to be strongly pseudocontractive if there exists a constant 𝑡>1 such that𝑥−𝑦≤(1+𝑟)𝑥−𝑦−𝑟𝑡𝑇𝑥−𝑇𝑦, (1.4) for all 𝑥,𝑦∈𝐷(𝑇) and 𝑟>0.
(3) The mapping 𝑇 is said to be local strongly pseudocontractive if for each 𝑥∈𝐷(𝑇) there exists a constant 𝑡>1 such that𝑥−𝑦≤(1+𝑟)𝑥−𝑦−𝑟𝑡𝑇𝑥−𝑇𝑦, (1.5) for all 𝑦∈𝐷(𝑇) and 𝑟>0.
(4) The mapping 𝑇 is said to be strictly hemicontractive if 𝐹(𝑇)≠∅ and if there exists a constant 𝑡>1 such that𝑥−𝑞≤(1+𝑟)𝑥−𝑞−𝑟𝑡𝑇𝑥−𝑞, (1.6) for all 𝑥∈𝐷(𝑇),  𝑞∈𝐹(𝑇) and 𝑟>0.

Clearly, each strongly pseudocontractive mapping is local strongly pseudocontractive.

Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of 𝑇 in case 𝑇 is a Lipschitz strongly pseudocontractive mapping from a bounded closed convex subset of 𝐿_𝑝 (or 𝑙_𝑝) into itself. Schu [3] generalized the result in [1] to both uniformly continuous strongly pseudocontractive mappings and real smooth Banach spaces. Park [4] extended the result in [1] to both strongly pseudocontractive mappings and certain smooth Banach spaces. Rhoades [5] proved that the Mann and Ishikawa iteration methods may exhibit different behaviors for different classes of nonlinear mappings. Afterwards, several generalizations have been made in various directions (see, e.g., [6–13]).

In 2001, Xu and Ori [14] introduced the following implicit iteration process for a finite family of nonexpansive mappings {𝑇_𝑖∶𝑖∈𝐼} (here 𝐼={1,2,…,𝑁}) with {𝛼_𝑛} a real sequence in (0,1) and an initial point 𝑥₀∈𝐾:𝑥₁=1−𝛼₁𝑥₀+𝛼₁𝑇₁𝑥₁,𝑥₂=1−𝛼₂𝑥₁+𝛼₂𝑇₂𝑥₂,⋮𝑥_𝑁=1−𝛼_𝑁𝑥_𝑁−1+𝛼_𝑁𝑇_𝑁𝑥_𝑁,𝑥_𝑁+1=1−𝛼_𝑁+1𝑥_𝑁+𝛼_𝑁+1𝑇_𝑁+1𝑥_𝑁+1,⋮ (1.7) which can be written in the following compact form: 𝑥_𝑛=1−𝛼_𝑛𝑥_𝑛−1+𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝑛≥1, (1.8) where 𝑇_𝑛=𝑇_{𝑛(mod𝑁)} (here the mod𝑁 function takes values in 𝐼). Xu and Ori [14] proved the weak convergence of this process to a common fixed point of the finite family defined in a Hilbert space. They further remarked that it is yet unclear what assumptions on the mappings and/or the parameters {𝛼_𝑛} are sufficient to guarantee the strong convergence of the sequence {𝑥_𝑛}.

In [11], Osilike proved the following results.

Theorem 1.4. Let 𝑋 be a real Banach space and let 𝐾 be a nonempty closed convex subset of 𝑋. Let {𝑇_𝑖∶𝑖∈𝐼} be 𝑁 strictly pseudocontractive mappings from 𝐾 to 𝐾 with ℱ=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)≠∅. Let {𝛼_𝑛} be a real sequence satisfying the following conditions: (i)0<𝛼_𝑛<1, (ii)∑^∞_𝑛=1(1−𝛼_𝑛)=∞, (iii)∑^∞_𝑛=1(1−𝛼_𝑛)²<∞.
From arbitrary 𝑥₀∈𝐾, define the sequence {𝑥_𝑛} by the implicit iteration process (1.8). Then {𝑥_𝑛} converges strongly to a common fixed point of the mappings {𝑇_𝑖∶𝑖∈𝐼} if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,ℱ)=0.

Remark 1.5. One can easily see that for 𝛼_𝑛=1−1/𝑛^1/2,  ∑^∞_𝑛=1(1−𝛼_𝑛)²=∞. Hence the results of Osilike [11] are needed to be improved.

Let 𝐾 be a nonempty closed bounded convex subset of an arbitrary smooth Banach space 𝑋 and let 𝑇∶𝐾→𝐾 be a continuous strictly hemicontractive mapping. We proved that the implicit Mann type iteration method converges strongly to a unique fixed point of 𝑇.

The results presented in this paper extend and improve the corresponding results particularly in [1, 3, 4, 7, 8, 10, 11, 13, 15].

2. Preliminaries

We need the following results.

Lemma 2.1 (see [4]). Let 𝑋 be a smooth Banach space. Suppose that one of the following holds: (a)𝐽 is uniformly continuous on any bounded subsets of 𝑋,(b)⟨𝑥−𝑦,𝑗(𝑥)−𝑗(𝑦)⟩≤‖𝑥−𝑦‖² for all 𝑥,𝑦 in 𝑋,(c)for any bounded subset 𝐷 of 𝑋, there is a function 𝑐∶[0,∞)→[0,∞) such thatRe𝑥−𝑦,𝑗(𝑥)−𝑗𝑦≤𝑐𝑥−𝑦, (2.1)
for all 𝑥,𝑦∈𝐷, where 𝑐 satisfies lim_𝑡→0+(𝑐(𝑡)/𝑡)=0.
Then for any 𝜖>0 and any bounded subset 𝐾, there exists 𝛿>0 such that𝑠𝑥+(1−𝑠)𝑦²≤(1−2𝑠)𝑦²+2𝑠Re𝑥,𝑗𝑦+2𝑠𝜖, (2.2) for all 𝑥,𝑦∈𝐾 and 𝑠∈[0,𝛿].

Remark 2.2. (1) If X is uniformly smooth, then (a) in Lemma 2.1 holds.
(2) If 𝑋 is a Hilbert space, then (𝑏) in Lemma 2.1 holds.

Lemma 2.3 (see [8]). Let 𝑇∶𝐷(𝑇)⊂𝑋→𝑋 be a mapping with 𝐹(𝑇)≠∅. Then 𝑇 is strictly hemicontractive if and only if there exists a constant 𝑡>1 such that for all 𝑥∈𝐷(𝑇) and 𝑞∈𝐹(𝑇), there exists 𝑗(𝑥−𝑞)∈𝐽(𝑥−𝑞) satisfying≥1Re𝑥−𝑇𝑥,𝑗𝑥−𝑞1−𝑡𝑥−𝑞². (2.3)

Lemma 2.4 (see [10]). Let 𝑋 be an arbitrary normed linear space and let 𝑇∶𝐷(𝑇)⊂𝑋→𝑋 be a mapping. (a)If 𝑇 is a local strongly pseudocontractive mapping and 𝐹(𝑇)≠∅, then 𝐹(𝑇) is a singleton and 𝑇 is strictly hemicontractive.(b)If 𝑇 is strictly hemicontractive, then 𝐹(𝑇) is a singleton.

Lemma 2.5 (see [10]). Let {𝜃_𝑛},{𝜎_𝑛}, and {𝜔_𝑛} be nonnegative real sequences and let 𝜖^′>0 be a constant satisfying 𝜎_𝑛+1≤1−𝜃_𝑛𝜎_𝑛+𝜖^′𝜃_𝑛+𝜔_𝑛,𝑛≥1, (2.4) where ∑^∞_𝑛=1𝜃_𝑛=∞,  𝜃_𝑛≤1 for all 𝑛≥1 and ∑^∞_𝑛=1𝜔_𝑛<∞. Then limsup_𝑛→∞𝜎_𝑛≤𝜖^′.

3. Main Results

We now prove our main results.

Lemma 3.1. Let 𝑋 be a smooth Banach space. Suppose that one of the following holds:(a)𝐽 is uniformly continuous on any bounded subsets of 𝑋,(b)⟨𝑥−𝑦,𝑗(𝑥)−𝑗(𝑦)⟩≤‖𝑥−𝑦‖² for all 𝑥,𝑦 in 𝑋,(c)for any bounded subset 𝐷 of 𝑋, there is a function 𝑐∶[0,∞)→[0,∞) such thatRe𝑥−𝑦,𝑗(𝑥)−𝑗𝑦≤𝑐𝑥−𝑦 (3.1)
for all 𝑥,𝑦∈𝐷, where 𝑐 satisfies lim_𝑡→0+𝑐((𝑡)/𝑡)=0.
Then for any 𝜖>0 and any bounded subset 𝐾, there exists 𝛿>0 such that𝛼𝑥+𝛽𝑦+𝛾𝑧²≤(1−2𝛼)‖𝑥‖²+2𝛼𝛽(𝑥1−𝛼Re𝑦,𝑗)+2𝛼𝛾(𝑥1−𝛼Re𝑧,𝑗)+2𝜖𝛼 (3.2) for all 𝑥,𝑦,𝑧∈𝐾 and 𝛼,𝛽,𝛾∈[0,𝛿];𝛼+𝛽+𝛾=1.Proof. For 𝛼,𝛽,𝛾∈[0,𝛿];𝛼+𝛽+𝛾=1, by using (2.2), consider 𝛼𝑥+𝛽𝑦+𝛾𝑧²=𝛽𝛼𝑥+(1−𝛼)𝛾1−𝛼𝑦+𝑧1−𝛼²≤(1−2𝛼)‖𝑥‖²𝛽+2𝜖𝛼+2𝛼Re𝛾1−𝛼𝑦+(𝑥=(𝑥‖1−𝛼𝑧,𝑗)1−2𝛼)‖²+2𝜖𝛼+2𝛼𝛽(𝑥1−𝛼Re𝑦,𝑗)+2𝛼𝛾(𝑥.1−𝛼Re𝑧,𝑗) (3.3) This completes the proof.

Theorem 3.2. Let 𝑋be a smooth Banach space satisfying any one of the Axioms (𝑎)–(𝑐) of Lemma 3.1. Let 𝐾 be a nonempty closed bounded convex subset of 𝑋 and let 𝑇∶𝐾→𝐾 be a continuous strictly hemicontractive mapping. Let {𝛼_𝑛},{𝛽_𝑛} and {𝛾_𝑛} be real sequences in [0,1] satisfying conditions (iv)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, forall𝑛≥1, (v)∑^∞_𝑛=1𝛼_𝑛=∞, (vi)∑^∞_𝑛=1𝛽_𝑛<∞ and ∑^∞_𝑛=1𝛾_𝑛<∞.
For a sequence {𝑣_𝑛} in 𝐾, suppose that {𝑥_𝑛} is the sequence generated from an arbitrary 𝑥₀∈𝐾 by𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛+𝛾_𝑛𝑇𝑣_𝑛,𝑛≥1, (3.4) satisfying ∑^∞_𝑛=1‖𝑣_𝑛−𝑥_𝑛‖<∞.
Then the sequence {𝑥_𝑛} converges strongly to a unique fixed point 𝑞 of  𝑇.

Proof. By [2, Corollary 1], 𝑇 has a unique fixed point 𝑞 in 𝐾. It follows from Lemma 2.4 that 𝐹(𝑇) is a singleton. That is, 𝐹(𝑇)={𝑞} for some 𝑞∈𝐾.
Set 𝑀=1+diam𝐾. It is easy to verify that𝑀=sup_𝑛≥1𝑥_𝑛−𝑞+sup_𝑛≥1𝑇𝑥_𝑛−𝑞+sup_𝑛≥1𝑇𝑣_𝑛−𝑞. (3.5) Also 𝑣_𝑛−𝑞²≤𝑣_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑞²+2𝑣_𝑛−𝑥_𝑛𝑥_𝑛≤𝑣−𝑞_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑞²+2𝑀𝑣_𝑛−𝑥_𝑛. (3.6) Consider 𝑥_𝑛−𝑞²=𝛼_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛+𝛾_𝑛𝑇𝑣_𝑛−𝑞²=𝛼_𝑛(𝑥_𝑛−1−𝑞)+𝛽_𝑛(𝑇𝑥_𝑛−𝑞)+𝛾_𝑛(𝑇𝑣_𝑛−𝑞)²≤𝛼_𝑛𝑥_𝑛−1−𝑞²+𝛽_𝑛𝑇𝑥_𝑛−𝑞²+𝛾_𝑛𝑇𝑣_𝑛−𝑞²≤𝑥_𝑛−1−𝑞²+𝑀²𝛽_𝑛+𝛾_𝑛, (3.7) where the first inequality holds by the convexity of ‖⋅‖².
Now we put 𝑘=1/𝑡, where 𝑡 satisfies (2.3). Using (3.4) and Lemma 3.1, we infer that 𝑥_𝑛−𝑞²=𝛼_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛+𝛾_𝑛𝑇𝑣_𝑛−𝑞²=𝛼_𝑛(𝑥_𝑛−1−𝑞)+𝛽_𝑛(𝑇𝑥_𝑛−𝑞)+𝛾_𝑛(𝑇𝑣_𝑛−𝑞)²≤1−2𝛼_𝑛𝑥_𝑛−1−𝑞²𝛼+2^𝑛𝛽_𝑛1−𝛼_𝑛Re𝑇𝑥_𝑛−𝑞,𝑗𝑥_𝑛−1𝛼−𝑞+2^𝑛𝛾_𝑛1−𝛼_𝑛Re𝑇𝑣_𝑛−𝑞,𝑗𝑥_𝑛−1−𝑞+2𝜖𝛼_𝑛=1−2𝛼_𝑛𝑥_𝑛−1−𝑞²𝛼+2^𝑛𝛽_𝑛1−𝛼_𝑛Re𝑇𝑥_𝑛−𝑞,𝑗𝑥_𝑛𝛼−𝑞+2^𝑛𝛽_𝑛1−𝛼_𝑛Re𝑇𝑥_𝑛−𝑞,𝑗𝑥_𝑛−1−𝑞−𝑗𝑥_𝑛𝛼−𝑞+2^𝑛𝛾_𝑛1−𝛼_𝑛Re𝑇𝑣_𝑛−𝑞,𝑗𝑣_𝑛𝛼−𝑞+2^𝑛𝛾_𝑛1−𝛼_𝑛Re𝑇𝑣_𝑛−𝑞,𝑗𝑥_𝑛−1−𝑞−𝑗𝑣_𝑛−𝑞+2𝜖𝛼_𝑛≤1−2𝛼_𝑛𝑥_𝑛−1−𝑞²𝛼+2^𝑛𝛽_𝑛1−𝛼_𝑛𝑘𝑥_𝑛−𝑞²𝛼+2^𝑛𝛽_𝑛1−𝛼_𝑛𝑇𝑥_𝑛−𝑞𝑗𝑥_𝑛−1−𝑞−𝑗𝑥_𝑛𝛼−𝑞+2^𝑛𝛾_𝑛1−𝛼_𝑛𝑘𝑣_𝑛−𝑞²𝛼+2^𝑛𝛾_𝑛1−𝛼_𝑛𝑇𝑣_𝑛−𝑞𝑗𝑥_𝑛−1−𝑞−𝑗𝑣_𝑛−𝑞+2𝜖𝛼_𝑛≤1−2𝛼_𝑛𝑥_𝑛−1−𝑞²𝛼+2^𝑛𝛽_𝑛1−𝛼_𝑛𝑘𝑥_𝑛−𝑞²𝛼+2𝑀^𝑛𝛽_𝑛1−𝛼_𝑛𝛿_𝑛𝛼+2^𝑛𝛾_𝑛1−𝛼_𝑛𝑘𝑣_𝑛−𝑞²𝛼+2𝑀^𝑛𝛾_𝑛1−𝛼_𝑛𝜂_𝑛+2𝜖𝛼_𝑛≤1−2𝛼_𝑛𝑥_𝑛−1−𝑞²𝛼+2^𝑛𝛽_𝑛1−𝛼_𝑛𝑘𝑥_𝑛−𝑞²𝛼+2^𝑛𝛾_𝑛1−𝛼_𝑛𝑘𝑣_𝑛−𝑞²+2𝑀𝛼_𝑛max𝛿_𝑛,𝜂_𝑛+2𝜖𝛼_𝑛, (3.8) where 𝛿_𝑛=𝑗𝑥_𝑛−1−𝑞−𝑗𝑥_𝑛,𝜂−𝑞_𝑛=𝑗𝑥_𝑛−1−𝑞−𝑗𝑣_𝑛−𝑞. (3.9) Also, we have 𝑥_𝑛−1−𝑥_𝑛=𝑥_𝑛−1−𝛼_𝑛𝑥_𝑛−1−𝛽_𝑛𝑇𝑥_𝑛−𝛾_𝑛𝑇𝑣_𝑛=𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛+𝛾_𝑛𝑥_𝑛−1−𝑇𝑣_𝑛≤𝛽_𝑛𝑥_𝑛−1−𝑇𝑥_𝑛+𝛾_𝑛𝑥_𝑛−1−𝑇𝑣_𝑛≤2𝑀𝛽_𝑛+𝛾_𝑛<∞ (3.10) implies 𝑥_𝑛−1−𝑥_𝑛⟶0, (3.11) as 𝑛→∞, and consequently 𝑥_𝑛−1−𝑣_𝑛≤𝑥_𝑛−1−𝑥_𝑛+𝑥_𝑛−𝑣_𝑛⟶0 (3.12) as 𝑛→∞. Since 𝐽 is uniformly continuous on any bounded subsets of 𝑋, we have 𝛿_𝑛,𝜂_𝑛⟶0as𝑛⟶∞. (3.13) For any given 𝜖>0 and the bounded subset 𝐾, there exists a 𝛿>0 satisfying (2.2). Note that (3.13) and (vi) ensure that there exists an 𝑁 such that 𝛽_𝑛,𝛾_𝑛𝜖<min𝛿,8𝑀²𝑘,𝛿_𝑛,𝜂_𝑛≤𝜖4𝑀,𝑛≥𝑁. (3.14) Now substituting (3.6) in (3.8) to obtain𝑥_𝑛−𝑞²≤1−2𝛼_𝑛𝑥_𝑛−1−𝑞²+2𝑘𝛼_𝑛𝑥_𝑛−𝑞²+2𝑀𝛼_𝑛max𝛿_𝑛,𝜂_𝑛+2𝜖𝛼_𝑛𝛼+2^𝑛𝛾_𝑛1−𝛼_𝑛𝑘𝑣_𝑛−𝑥_𝑛²+2𝑀𝑣_𝑛−𝑥_𝑛, (3.15) by using (3.7), implies 𝑥_𝑛−𝑞²≤1−2(1−𝑘)𝛼_𝑛𝑥_𝑛−1−𝑞²+2𝜖𝛼_𝑛+2𝑀²𝑘𝛼_𝑛𝛽_𝑛+𝛾_𝑛+2𝑀𝛼_𝑛max𝛿_𝑛,𝜂_𝑛𝛼+2^𝑛𝛾_𝑛1−𝛼_𝑛𝑘𝑣_𝑛−𝑥_𝑛²+2𝑀𝑣_𝑛−𝑥_𝑛≤1−2(1−𝑘)𝛼_𝑛𝑥_𝑛−1−𝑞²+3𝜖𝛼_𝑛𝛼+2^𝑛𝛾_𝑛1−𝛼_𝑛𝑘𝑣_𝑛−𝑥_𝑛²+2𝑀𝑣_𝑛−𝑥_𝑛 (3.16) for all 𝑛≥𝑁.
Put𝜎_𝑛=𝑥_𝑛−1−𝑞²,𝜃_𝑛=2(1−𝑘)𝛼_𝑛,𝜖^′=3𝜖2(),𝜔1−𝑘_𝑛𝛼=2^𝑛𝛾_𝑛1−𝛼_𝑛𝑘𝑣_𝑛−𝑥_𝑛²+2𝑀𝑣_𝑛−𝑥_𝑛, (3.17) and we have from (3.16) 𝜎_𝑛+1≤1−𝜃_𝑛𝜎_𝑛+𝜖^′𝜃_𝑛+𝜔_𝑛,𝑛≥1. (3.18) For 𝑘<1/2, set 𝛿=1/2(1−𝑘)<1. Because 𝛼_𝑛≤𝛿, we imply 1−𝛼_𝑛≥1−𝛿 and 2(1−𝑘)𝛼_𝑛≤1. Now observe that ∑^∞_𝑛=1𝜃_𝑛=∞,𝜃_𝑛≤1 for all 𝑛≥1 and ∑^∞_𝑛=1𝜔_𝑛<∞. It follows from Lemma 2.5 that limsup_𝑛→∞𝑥_𝑛−𝑞²≤𝜖^′. (3.19) Letting 𝜖^′→0⁺, we obtain that limsup_𝑛→∞‖𝑥_𝑛−𝑞‖²=0, which implies that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

Corollary 3.3. Let 𝑋be a smooth Banach space satisfying any one of the Axioms (𝑎)–(𝑐) of Lemma 3.1. Let 𝐾 be a nonempty closed bounded convex subset of 𝑋 and let 𝑇∶𝐾→𝐾 be a Lipschitz strictly hemicontractive mapping. Let {𝛼_𝑛},{𝛽_𝑛} and {𝛾_𝑛} be real sequences in [0,1] satisfying the conditions (iv)–(vi).
From arbitrary 𝑥₀∈𝐾, define the sequence {𝑥_𝑛} by the implicit iteration process (3.4). Then the sequence {𝑥_𝑛} converges strongly to a unique fixed point 𝑞 of 𝑇.

Corollary 3.4. Let 𝑋be a smooth Banach space satisfying any one of the Axioms (𝑎)-(𝑐) of Lemma 3.1. Let 𝐾 be a nonempty closed bounded convex subset of 𝑋 and let 𝑇∶𝐾→𝐾 be a continuous strictly hemicontractive mapping. Suppose that {𝛼_𝑛} be a real sequence in [0,1] satisfying the conditions (v) and lim_𝑛→∞𝛼_𝑛=0.
From arbitrary 𝑥₀∈𝐾, define the sequence {𝑥_𝑛} by the implicit iteration process (1.8). Then the sequence {𝑥_𝑛} converges strongly to a unique fixed point 𝑞 of 𝑇.

Corollary 3.5. Let 𝑋 be a smooth Banach space satisfying any one of the Axioms (𝑎)–(𝑐) of Lemma 3.1. Let 𝐾 be a nonempty closed bounded convex subset of 𝑋 and let 𝑇∶𝐾→𝐾 be a Lipschitz strictly hemicontractive mapping. Suppose that {𝛼_𝑛} be a real sequence in [0,1] satisfying the conditions (v) and 𝑙𝑖𝑚_𝑛→∞𝛼_𝑛=0.
From arbitrary 𝑥₀∈𝐾, define the sequence {𝑥_𝑛} by the implicit iteration process (1.8). Then the sequence {𝑥_𝑛} converges strongly to a unique fixed point 𝑞 of 𝑇.

Remark 3.6. Similar results can be found for the iteration processes involved error terms; we omit the details.

Remark 3.7. Theorem 3.2 and Corollary 3.3 extend and improve Theorem 1.4 in the following directions.
We do not need the assumption liminf_𝑛→∞𝑑(𝑥_𝑛,ℱ)=0 as in Theorem 1.4.

4. Applications for Multistep Implicit Iterations

Let 𝐾 be a nonempty closed convex subset of a smooth Banach space 𝑋 and let 𝑇,𝑇₁,𝑇₂,…,𝑇_𝑝∶𝐾→𝐾(𝑝≥2) be a family of 𝑝+1 mappings.

Algorithm 4.1. For a given x₀∈K, compute the sequence {x_𝑛} by the implicit iteration process of arbitrary fixed order p≥2∶𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛+𝛾_𝑛𝑇₁𝑦¹_𝑛,𝑦^𝑖_𝑛=𝛽^𝑖_𝑛𝑥_𝑛−1+1−𝛽^𝑖_𝑛𝑇_𝑖+1𝑦_𝑛^𝑖+1𝑦,𝑖=1,2,…,𝑝−2,_𝑛𝑝−1=𝛽_𝑛^𝑝−1𝑥_𝑛−1+1−𝛽_𝑛^𝑝−1𝑇_𝑝𝑥_𝑛,𝑛≥1, (4.1) which is called the multistep implicit iteration process, where {𝛼_𝑛}, {𝛽_𝑛},  {𝛾_𝑛}, and {𝛽^𝑖_𝑛},𝑖=1,2,…,𝑝−1 are real sequences in [0,1] and 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, forall𝑛≥1.

For 𝑝=3, we obtain the following three-step implicit iteration process.

Algorithm 4.2. For a given 𝑥₀∈𝐾, compute the sequence {𝑥_𝑛} by the iteration process 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛+𝛾_𝑛𝑇₁𝑦¹_𝑛,𝑦¹_𝑛=𝛽¹_𝑛𝑥_𝑛−1+1−𝛽¹_𝑛𝑇₂𝑦²_𝑛,𝑦²_𝑛=𝛽²_𝑛𝑥_𝑛−1+1−𝛽²_𝑛𝑇₃𝑥_𝑛,𝑛≥1, (4.2) where {𝛼_𝑛}, {𝛽_𝑛},  {𝛾_𝑛}, {𝛽¹_𝑛} and {𝛽²_𝑛} are real sequences in [0,1] satisfying some certain conditions.
For 𝑝=2, we obtain the following two-step implicit iteration process.

Algorithm 4.3. For a given 𝑥₀∈𝐾, compute the sequence {𝑥_𝑛} by the iteration process 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+𝛽_𝑛𝑇𝑥_𝑛+𝛾_𝑛𝑇₁𝑦¹_𝑛,𝑦¹_𝑛=𝛽¹_𝑛𝑥_𝑛−1+1−𝛽¹_𝑛𝑇₂𝑥_𝑛,𝑛≥1, (4.3) where {𝛼_𝑛}, {𝛽_𝑛},  {𝛾_𝑛} and {𝛽¹_𝑛} are real sequences in [0,1] satisfying some certain conditions.
If 𝑇₁=𝑇,  𝑇₂=𝐼 and 𝛽¹_𝑛=0 in (4.3), we obtain the following implicit Mann iteration process.

Algorithm 4.4. For any given 𝑥₀∈𝐾, compute the sequence {𝑥_𝑛} by the iteration process 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥1, (4.4) where {𝛼_𝑛} is a real sequence in [0,1] satisfying some certain conditions.

Theorem 4.5. Let 𝑋be a smooth Banach space satisfying any one of the Axioms (𝑎)–(𝑐) of Lemma 3.1. Let 𝐾 be a nonempty closed bounded convex subset of 𝑋 and let 𝑇,𝑇₁,𝑇₂,…,𝑇_𝑝∶𝐾→𝐾(𝑝≥2) be 𝑝+1 mappings. Let 𝑇,𝑇₁ be continuous strictly hemicontractive mappings. Let {𝛼_𝑛}, {𝛽_𝑛},  {𝛾_𝑛} and {𝛽^𝑖_𝑛},𝑖=1,2,…,𝑝−1 be real sequences in [0,1] satisfying the conditions (iv)–(vi) and ∑^∞_𝑛=1(1−𝛽¹_𝑛)<∞. For arbitrary 𝑥₀∈𝐾, define the sequence {𝑥_𝑛} by (4.1). Then {𝑥_𝑛} converges strongly to the common fixed point of ⋂^𝑝_𝑖=1𝐹(𝑇_𝑖)∩𝐹(𝑇)≠∅.

Proof. By applying Theorem 3.2 under assumption that 𝑇 and 𝑇₁ are continuous strictly hemicontractive mappings, we obtain Theorem 4.5 which proves strong convergence of the iteration process defined by (4.1). Consider by taking 𝑇₁=𝑇 and 𝑣_𝑛=𝑦¹_𝑛, 𝑣_𝑛−𝑥_𝑛=𝑦¹_𝑛−𝑥_𝑛=𝛽¹_𝑛𝑥_𝑛−1+1−𝛽¹_𝑛𝑇₂𝑦²_𝑛−𝑥_𝑛=𝛽¹_𝑛𝑥_𝑛−1−𝑥_𝑛+1−𝛽¹_𝑛𝑇₂𝑦²_𝑛−𝑥_𝑛≤𝛽¹_𝑛𝑥_𝑛−1−𝑥_𝑛+1−𝛽¹_𝑛𝑇₂𝑦²_𝑛−𝑥_𝑛≤𝛽¹_𝑛𝑥_𝑛−1−𝑥_𝑛+𝑀^′1−𝛽¹_𝑛. (4.5) From (4.5) and the condition ∑^∞_𝑛=1(1−𝛽¹_𝑛)<∞, we obtain ^∞_𝑛=1𝑣_𝑛−𝑥_𝑛<∞. (4.6) This completes the proof.

Corollary 4.6. Let 𝑋be a smooth Banach space satisfying any one of the Axioms (𝑎)-(𝑐) of Lemma 3.1. Let 𝐾 be a nonempty closed bounded convex subset of 𝑋 and let 𝑇,𝑇₁,𝑇₂,…,𝑇_𝑝∶𝐾→𝐾(𝑝≥2) be 𝑝+1 mappings. Let 𝑇,𝑇₁ be Lipschitz strictly hemicontractive mappings. Let {𝛼_𝑛}, {𝛽_𝑛},   {𝛾_𝑛} and {𝛽^𝑖_𝑛}, 𝑖=1,2,…,𝑝−1 be real sequences in [0,1] satisfying the conditions (iv)-(vi) and ∑^∞_𝑛=1(1−𝛽¹_𝑛)<∞. For arbitrary 𝑥₀∈𝐾, define the sequence {𝑥_𝑛} by (4.1). Then {𝑥_𝑛} converges strongly to the common fixed point of ⋂^𝑝_𝑖=1𝐹(𝑇_𝑖)∩𝐹(𝑇)≠∅.