Convergence of a New Modified Ishikawa Type Iteration for Common Fixed Points of Total Asymptotically Strict Pseudocontractive Semigroups

Abstract

The purpose of this paper is to give a new modified Ishikawa type iteration algorithm for common fixed points of total asymptotically strict pseudocontractive semigroups. Under the reduction of some conditions, both strong convergence and weak convergence of the iteration algorithm are proved in Banach spaces with new methods of proofs, respectively. The main results presented in this paper extend and improve the corresponding recent results of many others.

1. Introduction

Throughout this paper, we assume that 𝐸 is a real Banach space with the norm ‖⋅‖, 𝐸^∗ the dual space of 𝐸, ⟨⋅,⋅⟩ the duality between 𝐸 and 𝐸^∗, and 𝐶 a nonempty closed convex subset of 𝐸. ℝ⁺ denotes the set of nonnegative real numbers and ℕ the natural number set. The mapping 𝐽∶𝐸→2^𝐸∗ with 𝐽(𝑥)=𝑓^∗∈𝐸^∗∶𝑥,𝑓^∗=‖𝑥‖²,𝑓^∗=‖𝑥‖,𝑥∈𝐸, (1) is called the normalized duality mapping.

Let 𝑇∶𝐶→𝐶 be a nonlinear mapping. 𝐹(𝑇) denotes the set of the fixed points of 𝑇.

As we know, a mapping 𝑇∶𝐶→𝐶 is said to be pseudocontractive, if, for all 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦), such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦². (2)

Variational inequalities introduced by Stampacchia in the early sixties have had a great impact and influence on the development of almost all branches of pure and applied sciences and have witnessed an explosive growth in theoretical advances, algorithmic development, and so forth. Recently, some authors also studied the problem of finding the solution set of variational inequalities and the common element of the fixed point set for generalized nonexpansive mappings in the framework of real Hilbert spaces and Banach spaces. As is known to all, the variational inequality problem, nonlinear optimization problem, and fixed point problem are equivalent to each other under certain conditions.

In 2012, Chang et al. [1] introduced a more general class of pseudocontractive mappings and studied the methods for approximation of the split common fixed points.

Definition 1 (see [1]). (I) A mapping 𝑇∶𝐶→𝐶 is said to be (𝛾,𝜇_𝑛,𝜉_𝑛,𝜙)-totally asymptotically strictly pseudocontractive, if there exist a constant 𝛾∈[0,1] and sequences {𝜇_𝑛},{𝜉_𝑛}⊂[0,∞) with 𝜇_𝑛→0 and 𝜉_𝑛→0, such that, for all 𝑥,𝑦∈𝐶, 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤𝑥−𝑦²+𝛾𝐼−𝑇^𝑛𝑥−𝐼−𝑇^𝑛𝑦²+𝜇_𝑛𝜙𝑥−𝑦+𝜉_𝑛,∀𝑛≥1, (3) where 𝜙∶[0,∞)→[0,∞) is continuous and a strict increasing function with 𝜙(0)=0.
(II) A mapping 𝑇∶𝐶→𝐶 is said to be (𝛾,𝑘_𝑛)-asymptotically strictly pseudocontractive, if there exist a constant 𝛾∈[0,1) and a sequence 𝑘_𝑛⊂[1,∞) with 𝑘_𝑛→1, such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤𝑘_𝑛𝑥−𝑦²+𝛾(𝐼−𝑇^𝑛)𝑥−(𝐼−𝑇^𝑛)𝑦²,∀𝑥,𝑦∈𝐶. (4)

Definition 2. (I) One-parameter family 𝒯∶={𝑇(𝑡)∶𝐶→𝐶,𝑡≥0} is said to be a pseudocontractive semigroup on 𝐶, if the following conditions are satisfied: (a)𝑇(0)𝑥=𝑥 for each 𝑥∈𝐶; (b)𝑇(𝑡+𝑠)𝑥=𝑇(𝑡)𝑇(𝑠)𝑥 for any 𝑡,𝑠∈ℝ⁺ and 𝑥∈𝐶; (c)the mapping 𝑡→𝑇(𝑡)𝑥 is continuous for any given 𝑥∈𝐶; (d)for any 𝑡≥0, 𝑇(𝑡) is pseudocontractive; that is, for any 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦), such that 𝑇^𝑛(𝑡)𝑥−𝑇^𝑛(𝑡)𝑦,𝑗𝑥−𝑦≤𝑥−𝑦². (5)
(II) One-parameter family 𝒯∶={𝑇(𝑡)∶𝐶→𝐶,𝑡≥0} is said to be strict pseudocontractive semigroup on 𝐶, if the conditions (a)–(c) and the following condition (e) are satisfied. (e) For any 𝑥,𝑦∈𝐶, there exist 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) and a bounded function 𝜂∶[0,∞)→[0,∞), such that, for any 𝑡≥0, 𝑇(𝑡)(𝑡)≤𝑥−𝑇𝑦,𝑗𝑥−𝑦𝑥−𝑦²−𝜂(𝑡)(𝑥−𝑦)−(𝑇(𝑡)𝑥−𝑇(𝑡)𝑦)². (6)
(III) 𝒯∶={𝑇(𝑡)∶𝐶→𝐶,𝑡≥0} is said to be asymptotically strict pseudocontractive semigroup, if the conditions (a)–(c) and the following condition (f) are satisfied. (f) There exist a bounded function 𝜂∶[0,∞)→[0,∞) and a sequence 𝑘_𝑛⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞. For any given 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that for, any 𝑡≥0, 𝑇^𝑛(𝑡)𝑥−𝑇^𝑛(𝑡)𝑦,𝑗𝑥−𝑦≤𝑘_𝑛𝑥−𝑦²−𝜂(𝑡)(𝑥−𝑦)−𝑇^𝑛(𝑡)𝑥−𝑇^𝑛(𝑡)𝑦². (7)

Osilike and Akuchu [2] established an iterative scheme for approximation of common fixed points of a finite family of asymptotically pseudocontractive mappings. Miao et al. [3] introduced an implicit iteration process for a finite family of total asymptotically pseudocontractive maps. And in recent years, many researchers focused on the convergence of pseudocontractive and asymptotically strict pseudocontractive semigroups; see [4–8] and their references. In [9, 10] especially, the authors gave the modified Mann type iteration algorithm and studied its convergence.

Inspired and motivated by the above works, in this paper, we give a new modified Ishikawa type iteration algorithm for total asymptotically strict pseudocontractive semigroups. Under the reducation of some conditions, we prove both strong convergence and weak convergence of the iteration algorithm by using the method of the subsequence of a subsequence of the sequence {𝑥_𝑛} in Banach spaces, respectively. The results presented in this paper extend and improve the corresponding recent results of many authors, such as [1, 7–10].

2. Preliminaries

This section contains some definitions, notations, and lemmas, which will be used in the proofs of our main results in the next section.

A Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0((‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡) exists for each 𝑥,𝑦∈{𝑥∈𝐸∶‖𝑥‖=1}. It is well known that if 𝐸 is reflexive and smooth, then the duality mapping 𝐽 is single valued.

A Banach space 𝐸 is said to have Opial condition if, for any sequence {𝑥_𝑛}⊂𝐸 weakly convergent to 𝑥₀∈𝐸, liminf_𝑛→∞𝑥_𝑛−𝑥₀<liminf_𝑛→∞𝑥_𝑛−𝑥 (8) holds for any 𝑥≠𝑥₀.

A mapping 𝑇 is said to be demiclosed, if, for any sequence {𝑥_𝑛}⊂𝐸,  𝑥_𝑛⇀𝑦 and ‖(𝐼−𝑇)𝑥_𝑛‖→0 imply that (𝐼−𝑇)𝑦=0.

Definition 3 (see [9]). One-parameter 𝒯∶={𝑇(𝑡)∶𝐶→𝐶,𝑡≥0} is said to be a (𝜂,{𝜇_𝑛},{𝜉_𝑛},𝜙)-total asymptotically strict pseudocontractive semigroup on 𝐶, if the conditions (a)–(c) in Definition 2 and the following condition (g) are satisfied.(g) There exist a bounded function 𝜂∶[0,∞)→[0,∞) and sequences {𝜇_𝑛}⊂[0,∞) and {𝜉_𝑛}⊂[0,∞) with 𝜇_𝑛→0, 𝜉_𝑛→0, as 𝑛→∞. For any given 𝑥,𝑦∈𝐶,  there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦), such that 𝑇^𝑛(𝑡)𝑥−𝑇^𝑛(𝑡)≤𝑦,𝑗𝑥−𝑦𝑥−𝑦²−𝜂(𝑡)𝑥−𝑦−𝑇^𝑛(𝑡)𝑥−𝑇^𝑛(𝑡)𝑦²+𝜇_𝑛𝜙𝑥−𝑦+𝜉_𝑛, (9) for any 𝑡≥0,  for all  𝑛≥1, where 𝜙∶[0,∞)→[0,∞) is continuous and strictly increasing function with 𝜙(0)=0.

A (𝜂,{𝜇_𝑛},{𝜉_𝑛},𝜙)-total asymptotically strict pseudocontractive semigroup is said to be uniformly  Lipschitzian, if there exists a bounded measurable function 𝐿∶[0,∞)→(0,∞), such that 𝑇^𝑛(𝑡)𝑥−𝑇^𝑛(𝑡)𝑦≤𝐿(𝑡)𝑥−𝑦,∀𝑥,𝑦∈𝐶,𝑡≥0,𝑛∈ℕ. (10)

Remark 4. According to the definitions, it is obvious that a pseudocontractive semigroup is a strict pseudocontractive semigroup with 𝜂(𝑡)=0, and a strict pseudocontractive semigroup is an asymptotically strict pseudocontractive semigroup with 𝑘_𝑛=1. An asymptotically strict pseudocontractive semigroup is a (𝜂,{𝜇_𝑛},{𝜉_𝑛},𝜙)-total asymptotically strict pseudocontractive semigroup with 𝜙(𝑡)=𝑡²,  𝜇_𝑛=𝑘_𝑛−1,  and  𝜉_𝑛=0.

Definition 5 (see [11]). The normalized duality mapping 𝐽 of a Banach space 𝐸 is said to be weakly sequential continuous; if for all {𝑥_𝑛}⊂𝐸, 𝑥_𝑛⇀𝑥, then there exist 𝑗(𝑥_𝑛)∈𝐽(𝑥_𝑛),  𝑗(𝑥)∈𝐽(𝑥) such that 𝑗(𝑥_𝑛)⇀̇𝑗(𝑥), where weak convergence and weak star convergence are denoted by ⇀ and ⇀̇, respectively.

In order to prove the main results of this paper, the following lemmas should be used.

Lemma 6 (see [4]). For any 𝑥,𝑦∈𝐸, one has 𝑥+𝑦²≤‖𝑥‖²,.+2𝑦,𝑗𝑥−𝑦∀𝑗𝑥−𝑦∈𝐽𝑥−𝑦 (11)

Lemma 7 (see [12]). Let {𝑎_𝑛},  {𝑏_𝑛}, and {𝛿_𝑛} be the sequences of ℝ⁺, which satisfy 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛,∀𝑛≥1. (12) If ∑^∞_𝑛=1𝛿_𝑛<∞,  ∑^∞_𝑛=1𝑏_𝑛<∞, then the limit lim_𝑛→∞𝑎_𝑛 exists.

3. Main Results

Theorem 8. Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸, and let 𝒯∶={𝑇(𝑡)∶𝐶→𝐶,𝑡≥0} be a uniformly Lipschitzian and (𝜂,{𝜇_𝑛},{𝜉_𝑛},𝜙)-total asymptotically strict pseudocontractive semigroup defined in Definition 3. Suppose that 𝐹(𝒯)∶=⋂_𝑡≥0𝐹(𝑇(𝑡))≠∅ and there exists a compact subset 𝐾 of 𝐸 such that ⋂_𝑡≥0𝑇(𝑡)(𝐶)⊆𝐾. We assume that there exist positive constants 𝑀 and 𝑀^∗, such that 𝜙(𝑥)≤𝑀^∗𝑥² for all 𝑥≥𝑀. Let {𝑥_𝑛} be the sequence defined by the modified Ishikawa type iteration algorithm: 𝑥₁𝑦∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛(𝑡)𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛(𝑡)𝑦_𝑛. (13) Then {𝑥_𝑛} converges strongly to a common fixed point 𝑥^∗∈𝐹(𝒯) in 𝐶, if the following conditions are satisfied:(i)∑^∞_𝑛=1𝛼²_𝑛<∞,  ∑^∞_𝑛=1𝛼_𝑛=∞,  ∑^∞_𝑛=1𝛼_𝑛𝜇_𝑛<∞,  and ∑^∞_𝑛=1𝛼_𝑛𝜉_𝑛<∞;(ii)𝛽_𝑛→0  as  𝑛→∞,  ∑^∞_𝑛=1𝛼_𝑛𝛽_𝑛<∞;(iii)𝜂=inf_𝑡≥0𝜂(𝑡)>0,  𝐿=sup_𝑡≥0𝐿(𝑡)<+∞.

Proof. We divide the proof into four steps.
Step  1.  Firstly, we prove that lim_𝑛→∞‖𝑥−𝑝‖ exists for any 𝑝∈𝐹(𝒯).
By the definitions of 𝑇(𝑡) and {𝑥_𝑛}, we have 𝑇^𝑛(𝑡)𝑥_𝑛−𝑝≤𝐿𝑥_𝑛,𝑦−𝑝_𝑛−𝑝=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛(𝑡)𝑥_𝑛≤−𝑝1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑇^𝑛(𝑡)𝑥_𝑛≤−𝑝1−𝛽_𝑛+𝛽_𝑛𝐿𝑥_𝑛≤(𝑥−𝑝1+𝐿)_𝑛,𝑇−𝑝_𝑛(𝑡)𝑦_𝑛−𝑝≤𝐿𝑦_𝑛−𝑝≤𝐿(1+𝐿)𝑥_𝑛+𝑝. (14) This follows from that 𝑥_𝑛+1−𝑝=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛(𝑡)𝑦_𝑛≤−𝑝1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑇^𝑛(𝑡)𝑦_𝑛≤−𝑝1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝐿(1+𝐿)𝑥_𝑛≤−𝑝1+𝐿+𝐿²𝑥_𝑛,𝑦−𝑝_𝑛−𝑥_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛(𝑡)𝑥_𝑛−𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑇^𝑛(𝑡)𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝+𝑇^𝑛(𝑡)𝑥_𝑛−𝑝≤𝛽_𝑛(1+𝐿)𝑥_𝑛,𝑇−𝑝_𝑛(𝑡)𝑦_𝑛−𝑥_𝑛≤𝑇^𝑛(𝑡)𝑦_𝑛−𝑝+𝑥_𝑛≤−𝑝1+𝐿+𝐿²𝑥_𝑛,𝑥−𝑝_𝑛+1−𝑥_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛(𝑡)𝑦_𝑛−𝑥_𝑛≤𝛼_𝑛𝑇^𝑛(𝑡)𝑦_𝑛−𝑥_𝑛≤𝛼_𝑛1+𝐿+𝐿²𝑥_𝑛−𝑝. (15)
Since 𝒯∶={𝑇(𝑡)∶𝐶→𝐶,𝑡≥0} is total asymptotically strict pseudocontractive semigroup, for any point 𝑥_𝑛+1∈𝐶 and 𝑝∈𝐹(𝒯), by (9), we have 𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1,𝑗𝑥_𝑛+1(𝑡𝑇−𝑝≤−𝜂)^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1+𝜇_𝑛𝜙𝑥_𝑛+1−𝑝+𝜉_𝑛. (16)
Since 𝜙 is an increasing function, it results in that 𝜙(𝑥)≤𝜙(𝑀), if 𝑥≤𝑀; 𝜙(𝑥)≤𝑀^∗𝑥², if 𝑥≥𝑀. In either case, we can obtain that 𝜙(𝑥)≤𝜙(𝑀)+𝑀^∗𝑥². (17)
Hence, by Lemma 6, we have 𝑥_𝑛+1−𝑝²=𝑥_𝑛−𝑝+𝛼_𝑛𝑇^𝑛(𝑡)𝑦_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝²+2𝛼_𝑛𝑇^𝑛(𝑡)𝑦_𝑛−𝑇^𝑛(𝑡)𝑥_𝑛,𝑗𝑥_𝑛+1−𝑝+2𝛼_𝑛𝑇^𝑛(𝑡)𝑥_𝑛−𝑇^𝑛(𝑡)𝑥_𝑛+1,𝑗𝑥_𝑛+1−𝑝+2𝛼_𝑛𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1,𝑗𝑥_𝑛+1−𝑝+2𝛼_𝑛𝑥_𝑛+1−𝑥_𝑛,𝑗𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑝²+2𝛼_𝑛𝐿𝑦_𝑛−𝑥_𝑛⋅𝑥_𝑛+1−𝑝+2𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1⋅𝑥_𝑛+1−𝑝−2𝛼_𝑛𝜂(𝑡)𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1²+2𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1⋅𝑥_𝑛+1−𝑝+2𝛼_𝑛𝜇_𝑛𝜙𝑥_𝑛+1−𝑝+2𝛼_𝑛𝜉_𝑛≤𝑥_𝑛−𝑝²+2𝛼_𝑛𝛽_𝑛𝐿(1+𝐿)1+𝐿+𝐿²𝑥_𝑛−𝑝²+2𝛼²_𝑛1+𝐿+𝐿²²𝑥_𝑛−𝑝²−2𝛼_𝑛𝜂(𝑡)𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1²+2𝛼²_𝑛1+𝐿+𝐿²²𝑥_𝑛−𝑝²+2𝛼_𝑛𝜇_𝑛𝜙(𝑀)+𝑀^∗1+𝐿+𝐿²𝑥_𝑛−𝑝²+2𝛼_𝑛𝜉_𝑛=1+𝛿_𝑛𝑥_𝑛−𝑝²+𝑏_𝑛, (18) where 𝛿_𝑛=[2𝛼_𝑛𝛽_𝑛𝐿(1+𝐿)+4𝛼²_𝑛(1+𝐿+𝐿²)+2𝑀^∗𝛼_𝑛𝜇_𝑛](1+𝐿+𝐿²),  𝑏_𝑛=2𝛼_𝑛𝜇_𝑛𝜙(𝑀)+2𝛼_𝑛𝜉_𝑛.
By the conditions (i) and (ii), we have ∑^∞_𝑛=1𝛿_𝑛<∞, ∑^∞_𝑛=1𝑏_𝑛<∞. Thus, by Lemma 7, we can obtain that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists.
Step  2.  Now we prove that liminf_𝑛→∞‖𝑥_𝑛−𝑇^𝑛(𝑡)𝑥_𝑛‖=0.
From (18), we know that 2𝛼_𝑛𝜂(𝑡)𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²+𝑏_𝑛. (19) As 𝜂=inf_𝑡≥0𝜂(𝑡)>0,  𝐴=sup_𝑛‖𝑥_𝑛−𝑝‖<∞, we can have ^𝑚_𝑛=12𝛼_𝑛𝜂𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1²≤^𝑚_𝑛=1𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛿_𝑛𝐴²+𝑏_𝑛≤𝑥₁−𝑝²+𝐴^2𝑚_𝑛=1𝛿_𝑛+^𝑚_𝑛=1𝑏_𝑛. (20) Then, ^∞_𝑛=12𝛼_𝑛𝜂𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1²≤^∞_𝑛=1𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛿_𝑛𝐴²+𝑏_𝑛≤𝑥₁−𝑝²+𝐴^2∞_𝑛=1𝛿_𝑛+^∞_𝑛=1𝑏_𝑛<∞. (21)
Since ∑^∞_𝑛=1𝛼_𝑛=∞, then (21) implies that liminf_𝑛→∞𝑥_𝑛+1−𝑇^𝑛(𝑡)𝑥_𝑛+1=0. (22)
Otherwise, if liminf_𝑛→∞‖𝑥_𝑛+1−𝑇^𝑛(𝑡)𝑥_𝑛+1‖=𝑐>0, then there exists an 𝑁, such that ‖𝑥_𝑛−𝑇^𝑛(𝑡)𝑥_𝑛‖≥𝑐/2, when 𝑛≥𝑁. So, we have ^∞_𝑛=12𝛼_𝑛𝜂𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1²=^𝑁_𝑛=12𝛼_𝑛𝜂𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1²+^∞_𝑛=𝑁2𝛼_𝑛𝜂𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1²≥^𝑁_𝑛=12𝛼_𝑛𝜂𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1²+𝑐²2𝜂^∞_𝑛=𝑁𝛼_𝑛=∞. (23) This is in contradiction with (21).
Because 𝑥_𝑛−𝑇^𝑛(𝑡)𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇^𝑛(𝑡)𝑥_𝑛+1+𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑇^𝑛(𝑡)𝑥_𝑛≤𝑥_𝑛+1−𝑇^𝑛(𝑡)𝑥_𝑛+1+(1+𝐿)𝑥_𝑛−𝑥_𝑛+1≤𝑥_𝑛+1−𝑇^𝑛(𝑡)𝑥_𝑛+1+𝛼_𝑛(1+𝐿)1+𝐿+𝐿²𝐴 (24) and lim_𝑛→∞𝛼_𝑛=0, we have liminf_𝑛→∞𝑥_𝑛−𝑇^𝑛(𝑡)𝑥_𝑛=0. (25)
Step  3.  Now we prove that liminf_𝑛→∞‖𝑥_𝑛−𝑇(𝑡)𝑥_𝑛‖=0.
Consider 𝑥_𝑛+1−𝑇(𝑡)𝑥_𝑛+1≤𝑥_𝑛+1−𝑇^𝑛+1(𝑡)𝑥_𝑛+1+𝑇^𝑛+1(𝑡)𝑥_𝑛+1−𝑇(𝑡)𝑥_𝑛+1≤𝑥_𝑛+1−𝑇^𝑛+1(𝑡)𝑥_𝑛+1+𝐿𝑇^𝑛(𝑡)𝑥_𝑛+1−𝑥_𝑛+1≤𝑥_𝑛+1−𝑇^𝑛+1(𝑡)𝑥_𝑛+1+𝐿𝑇^𝑛(𝑡)𝑥_𝑛−𝑇^𝑛(𝑡)𝑥_𝑛+𝑇^𝑛(𝑡)𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1≤𝑥_𝑛+1−𝑇^𝑛+1(𝑡)𝑥_𝑛+1+𝐿𝑇^𝑛(𝑡)𝑥_𝑛−𝑥_𝑛+𝐿(1+𝐿)𝑥_𝑛−𝑥_𝑛+1≤𝑥_𝑛+1−𝑇^𝑛+1(𝑡)𝑥_𝑛+1+𝐿𝑇^𝑛(𝑡)𝑥_𝑛−𝑥_𝑛+𝛼_𝑛𝐿(1+𝐿)1+𝐿+𝐿²𝐴. (26) Since lim_𝑛→∞𝛼_𝑛=0 and (25), we have liminf_𝑛→∞𝑥_𝑛−𝑇(𝑡)𝑥_𝑛=0. (27) Thus, there exists a subsequence {𝑥_𝑛𝑘}⊆{𝑥_𝑛} such that lim_𝑛→∞𝑥_𝑛𝑘−𝑇(𝑡)𝑥_𝑛𝑘=0. (28)
Step  4.  Finally, we prove the sequence {𝑥_𝑛} converges strongly to a common fixed point of the semigroup 𝒯∶={𝑇(𝑡)∶𝐶→𝐶,𝑡≥0}.
Since 𝐾 is a compact subset of 𝐸 and ⋂_𝑡≥0𝑇(𝑡)(𝐶)⊆𝐾, just as the proof in [9, 10], there exists a subsequence {𝑥_𝑛𝑘𝑖}⊆{𝑥_𝑛𝑘}⊆{𝑥_𝑛}⊆𝐶, such that 𝑇(𝑡)𝑥_𝑛𝑘𝑖→𝑥^∗∈𝐾. From (28), we have lim_{𝑛𝑘𝑖→∞}‖𝑇(𝑡)𝑥_𝑛𝑘𝑖−𝑥_𝑛𝑘𝑖‖=0, and 𝑥_𝑛𝑘𝑖−𝑥^∗≤𝑥_𝑛𝑘𝑖−𝑇(𝑡)𝑥_𝑛𝑘𝑖+𝑇(𝑡)𝑥_𝑛𝑘𝑖−𝑥^∗⟶0. (29) Hence we have that 𝑇(𝑡)𝑥^∗−𝑥^∗=lim_{𝑛𝑘𝑖→∞}𝑥_𝑛𝑘𝑖−𝑇(𝑡)𝑥_𝑛𝑘𝑖=0. (30) That is, 𝑥^∗∈𝐹(𝒯).
Since, for any 𝑝∈𝐹(𝒯), lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, lim_{𝑛𝑘𝑖→∞}‖𝑥_𝑛𝑘𝑖−𝑥^∗‖=0, and {𝑥_𝑛𝑘𝑖}⊆{𝑥_𝑛}, so we have lim_𝑛→∞‖𝑥_𝑛−𝑝‖=0; that is, 𝑥_𝑛 converges strongly to an element 𝑥^∗=𝑝 of 𝐹(𝒯).

Remark 9. (a) If we take 𝛽_𝑛=0 in the modified Ishikawa type iteration algorithm (13), then (13) is called the modified Mann type iteration algorithm in many articles, such as in [9, 10]. (b) In Theorem 8, because there is no limit to 𝑡 of 𝑇(𝑡), so our result is stronger and the conditions here are less than in [9, 10]. For example, the conditions “for any bounded subset 𝐷⊂𝐶, lim_𝑛→∞sup_{𝑥∈𝐷,𝑠∈ℝ+}𝑇^𝑛𝑠+𝑡_𝑛𝑥−𝑇^𝑛𝑡_𝑛𝑥=0” (31) in [9, 10] can be removed in Theorem 8. (c) The condition “there exists a compact subset 𝐾 of 𝐸 such that ⋂_𝑡≥0𝑇(𝑡)(𝐶)⊆𝐾” does not look natural. But it easy to see that this condition is established naturally when we assume 𝐶 is a compact subset of 𝐸. So, the result in Theorem 8 is still true if this condition is replaced by the condition “let 𝐶 be a compact subset of 𝐸.” If there is no compactness assumption, we can get the following weak convergence theorem.

Theorem 10. Let 𝐸 be a reflexive Banach space satisfying the opial condition and 𝐶 be a nonempty bounded closed convex subset of 𝐸. Let 𝒯∶={𝑇(𝑡)∶𝐶→𝐶,𝑡≥0} be a uniformly Lipschitzian and (𝜂,{𝜇_𝑛},{𝜉_𝑛},𝜙)-total asymptotically strict pseudocontractive semigroup defined by Definition 3. Suppose that there exist positive constants 𝑀 and 𝑀^∗, such that 𝜙(𝑥)≤𝑀^∗𝑥², for all 𝑥≥𝑀, and 𝐹(𝒯)∶=⋂_𝑡≥0𝐹(𝑇(𝑡))≠∅. Let {𝑥_𝑛} be the sequence defined by (13). Then {𝑥_𝑛}  converges weakly to a common fixed point 𝑥^∗∈𝐹(𝒯) in 𝐶, if the following conditions are satisfied. (i)∑^∞_𝑛=1𝛼²_𝑛<∞,  ∑^∞_𝑛=1𝛼_𝑛=∞,  ∑^∞_𝑛=1𝛼_𝑛𝜇_𝑛<∞,  ∑^∞_𝑛=1𝛼_𝑛𝜉_𝑛<∞.(ii)𝛽_𝑛→0 as 𝑛→∞,  and ∑^∞_𝑛=1𝛼_𝑛𝛽_𝑛<∞.(iii)𝜆=inf_𝑡≥0𝜆(𝑡)>0,  𝐿=sup_𝑡≥0𝐿(𝑡)<+∞.

Proof. It can be proved just like the proof in Theorem 8 that, for each 𝑝∈𝐹(𝒯), lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, and, for all 𝑡>0, 𝑇(𝑡)𝑥_𝑛  is bounded, liminf‖𝑇(𝑡)𝑥_𝑛−𝑥_𝑛‖=0. Thus, there exists a subsequence {𝑥_𝑛𝑘}⊆{𝑥_𝑛} such that lim_𝑛→∞‖𝑥_𝑛𝑘−𝑇(𝑡)𝑥_𝑛𝑘‖=0.
Now we prove that 𝐼−𝑇(𝑡) is demiclosed at zero (see [11]).
Since 𝐶 is a closed and convex subset of a reflexive Banach space 𝐸, there exists a subsequence {𝑥_𝑛𝑘𝑖}⊆{𝑥_𝑛𝑘}⊆{𝑥_𝑛}, such that 𝑥_𝑛𝑘𝑖⇀𝑥^∗∈𝐶. Without loss of generality, we can assume that {𝑥_𝑛} replaces {𝑥_𝑛𝑘𝑖} now.
In the following, we prove that 𝑥^∗=𝑇(𝑡)𝑥^∗.
Firstly, we choose 𝛼∈(0,1/(1+𝐿)) and 𝑦_𝑚=(1−𝛼)𝑥+𝑇^𝑚(𝑡)𝑥 for 𝑚≥1. Since 𝑇(𝑡) is uniformly Lipschitzian, we have 𝑥_𝑛−𝑇^𝑚(𝑡)𝑥_𝑛≤^𝑚−1_𝑘=0𝑇^𝑘(𝑡)𝑥_𝑛−𝑇^𝑘+1(𝑡)𝑥_𝑛≤^𝑚−1_𝑘=0𝐿𝑥_𝑛−𝑇(𝑡)𝑥_𝑛=𝑚𝐿𝑥_𝑛−𝑇(𝑡)𝑥_𝑛⟶0,𝑛⟶∞. (32)
Because 𝑇(𝑡) is totally asymptotically strictly pseudocontractive, we have 𝐼−𝑇^𝑚(𝑡)𝑦_𝑚,𝐽𝑥−𝑦_𝑚=𝐼−𝑇^𝑚(𝑡)𝑦_𝑚,𝐽𝑥−𝑦_𝑚−𝐽𝑥_𝑛−𝑦_𝑚+𝐼−𝑇^𝑚(𝑡)𝑦_𝑚,𝐽𝑥_𝑛−𝑦_𝑚=𝐼−𝑇^𝑚(𝑡)𝑦_𝑚,𝐽𝑥−𝑦_𝑚−𝐽𝑥_𝑛−𝑦_𝑚+𝐼−𝑇^𝑚(𝑡)𝑥_𝑛,𝐽𝑥_𝑛−𝑦_𝑚+𝐼−𝑇^𝑚(𝑡)𝑦_𝑚−𝐼−𝑇^𝑚(𝑡)𝑥_𝑛,𝐽𝑥_𝑛−𝑦_𝑚=𝐼−𝑇^𝑚(𝑡)𝑦_𝑚,𝐽𝑥−𝑦_𝑚−𝐽𝑥_𝑛−𝑦_𝑚+𝐼−𝑇^𝑚(𝑡)𝑥_𝑛,𝐽𝑥_𝑛−𝑦_𝑚−𝜂(𝑡)𝐼−𝑇^𝑚(𝑡)𝑥_𝑛−𝐼−𝑇^𝑚(𝑡)𝑦_𝑚²+𝜇_𝑛𝜙𝑥_𝑛−𝑦_𝑛+𝜉_𝑛≤𝐼−𝑇^𝑚(𝑡)𝑦_𝑚,𝐽𝑥−𝑦_𝑚−𝐽𝑥_𝑛−𝑦_𝑚+𝐼−𝑇^𝑚(𝑡)𝑥_𝑛,𝐽𝑥_𝑛−𝑦_𝑛+𝜇_𝑛𝑀+𝑀^∗𝑥_𝑛−𝑦_𝑚²+𝜉_𝑛. (33) Since 𝑥_𝑛⇀𝑥^∗, lim_𝑛→∞‖𝑥_𝑛−𝑇(𝑡)𝑥_𝑛‖=0 (note the 𝑥_𝑛 of here instead of 𝑥_𝑛𝑘𝑖), and 𝐽 is weakly sequential continuous duality mapping, we have 𝐼−𝑇^𝑚(𝑡)𝑥^∗−𝐼−𝑇^𝑚(𝑡)𝑦_𝑚,𝐽𝑥^∗−𝑦_𝑚≤(1+𝐿)𝑥^∗−𝑦_𝑚²≤(1+𝐿)𝛼²𝑥^∗−𝑇^𝑚(𝑡)𝑥^∗2. (34) Hence 𝑥^∗−𝑇^𝑚(𝑡)𝑥^∗2=𝑥^∗−𝑇^𝑚(𝑡)𝑥^∗,𝐽𝑥^∗−𝑇^𝑚(𝑡)𝑥^∗=1𝛼𝑥^∗−𝑇^𝑚(𝑡)𝑥^∗,𝐽𝑥^∗−𝑦_𝑚=1𝛼𝑥^∗−𝑇^𝑚(𝑡)𝑥^∗−𝑦_𝑚−𝑇^𝑚(𝑡)𝑦_𝑚,𝐽𝑥^∗−𝑦_𝑚+1𝛼𝑦_𝑚−𝑇^𝑚(𝑡)𝑦_𝑚,𝐽𝑥^∗−𝑦_𝑚≤𝛼(1+𝐿)𝑥^∗−𝑇^𝑚(𝑡)𝑥^∗2+1𝛼𝜇_𝑚𝑀+𝜇_𝑚𝑀^∗(diam𝐶)+𝜉_𝑛. (35) This implies that 𝛼(1−𝛼(1+𝐿))𝑥^∗−𝑇^𝑚(𝑡)𝑥^∗2≤𝜇_𝑚𝑀+𝜇_𝑚𝑀^∗(diam𝐶)+𝜉_𝑛,∀𝑚∈ℕ. (36) Let 𝑚→∞; then we have ‖𝑥^∗−𝑇^𝑚(𝑡)𝑥^∗‖→0, as 𝑚→∞, for 𝜇_𝑛→0,  𝜉_𝑛→0. Hence, 𝑇^𝑚(𝑡)𝑥^∗→𝑥^∗, as 𝑚→∞, and 𝑇^𝑚+1(𝑡)𝑥^∗→𝑇(𝑡)𝑥^∗. By the continuity of 𝑇(𝑡), we have 𝑇(𝑡)𝑥^∗=𝑥^∗.
Now, for the sequence {𝑥_𝑛} generated by (13), we prove that 𝑥_𝑛⇀𝑥^∗.
Suppose the contrary; if there exists another subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛}, such that 𝑥_𝑛𝑗⇀𝑦^∗ with 𝑦^∗≠𝑥^∗, then we have that lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ and lim_𝑛→∞‖𝑥_𝑛−𝑦^∗‖ exist. Since 𝐸 satisfies the Opial condition, we have liminf_{𝑛𝑘𝑖→∞}𝑥_𝑛𝑘𝑖−𝑥^∗<lim_{𝑛𝑘𝑖→∞}𝑥_𝑛𝑘𝑖−𝑦^∗=lim_𝑛→∞𝑥_𝑛−𝑦^∗=lim_{𝑛𝑗→∞}𝑥_𝑛𝑗−𝑦^∗<liminf_{𝑛𝑗→∞}𝑥_𝑛𝑗−𝑥^∗=lim_𝑛→∞𝑥_𝑛−𝑥^∗=liminf_{𝑛𝑘𝑖→∞}𝑥_𝑛𝑘𝑖−𝑥^∗. (37) This is a contraction, which shows 𝑥^∗=𝑦^∗. Therefore, 𝑥_𝑛⇀𝑥^∗∈𝐹(𝒯). This completes the proof.

Remark 11. (a) Our results extend many other results that have been proved for this important class of general pseudocontractive mappings. For example, we extend the total asymptotically strict pseudocontractive mapping in [1] and Lipschitzian pseudo-contraction semigroup in [8] to the total asymptotically strict pseudocontractive semigroup. (b) In addition, we study the weak convergence of the total asymptotically strict pseudocontractive semigroup by using the demiclosedness of 𝐼−𝑇(𝑡) which, in some way, extends the result in [11] in Banach spaces. (c) And the method by using the subsequence of a subsequence of the sequence {𝑥_𝑛} in this paper is different from the previous references.

Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and the National Natural Science Foundation of China (11271330).