Strong Convergence Theorems for a Countable Family of Total Quasi-𝝓-Asymptotically Nonexpansive Nonself Mappings

Abstract

The purpose of this paper is to introduce a class of total quasi-ϕ-asymptotically nonexpansive-nonself mappings and to study the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper extend and improve the corresponding results announced by some authors recently.

1. Introduction

Throughout this paper, we assume that 𝐸 is a real Banach space, 𝐶 is a nonempty closed and convex subset of 𝐸, 𝐸^∗ is the dual space of 𝐸, and 𝐽∶𝐸→2^𝐸∗ is the normalized duality mapping defined by 𝐽(𝑥)=𝑓^∗∈𝐸^∗,𝑥,𝑓^∗=‖𝑥‖²=𝑓²,𝑥∈𝐸. (1.1)

Recall that a Banach space 𝐸 is said to be strictly convex if ‖𝑥+𝑦‖/2<1 for all 𝑥,𝑦∈𝑈={𝑧∈𝐸∶||𝑧||=1} with 𝑥≠𝑦. 𝐸 is said to be uniformly convex, if for each 𝜖∈(0,2], there exists 𝛿>0 such that ‖𝑥+𝑦‖/2<1−𝛿 for all 𝑥,𝑦∈𝑈 with ‖𝑥−𝑦‖≥𝜖. 𝐸 is said to be smooth, if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (∗∗) exists for all 𝑥,𝑦∈𝑈. And 𝐸 is said to be uniformly smooth, if the above limit is exists uniformly for 𝑥,𝑦∈𝑈.

In the sequel, we shall denote the fixed point set of a mapping 𝑇 by 𝐹(𝑇). When {𝑥_𝑛} is a sequence in 𝐸, then 𝑥_𝑛→𝑥 (𝑥_𝑛⇀𝑥) will denote strong (weak) convergence of the sequence {𝑥_𝑛} to 𝑥.

A mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive, if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.2)

A mapping 𝑇∶𝐶→𝐶 is said to be asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝑘_𝑛𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.3)

Recall that a subset 𝐶 of 𝐸 is said to be retract of 𝐸, if there exists a continuous mapping 𝑃∶𝐸→𝐶 such that 𝑃𝑥=𝑥, for all 𝑥∈𝐶.

It is well known that every nonempty closed and convex subset of a uniformly convex Banach space is a retract of 𝐸. A mapping 𝑃∶𝐸→𝐶 is said to be a retraction, if 𝑃²=𝑃. It follows that if a mapping 𝑃 is a retraction, then 𝑃𝑦=𝑦 for all 𝑦 in the range of 𝑃. A mapping 𝑃∶𝐸→𝐶 is said to be a nonexpansive retraction, if it is nonexpansive and it is a retraction from 𝐸 to 𝐶.

In the sequel, we assume that 𝐸 is a smooth, strictly convex, and reflexive Banach space and 𝐶 is a nonempty closed convex subset of 𝐸. Throughout this paper we assume that 𝜙∶𝐸×𝐸→ℛ⁺ is the Lyapunov function which is defined by 𝜙𝑥,𝑦=‖𝑥‖−2⟨𝑥,𝐽𝑦⟩+𝑦²,∀𝑥,𝑦∈𝐸. (1.4)

It is obvious from the definition of 𝜙 that ‖𝑥‖−𝑦²≤𝜙𝑥,𝑦≤‖𝑥‖+𝑦²,∀𝑥,𝑦∈𝐸,𝜙𝑥,𝐽⁻¹𝜆𝐽𝑦+(1−𝜆)𝐽𝑧≤𝜆𝜙𝑥,𝑦+(1−𝜆)𝜙(𝑥,𝑧),∀𝑥,𝑦∈𝐸. (1.5)(1.6)

Following Alber [1], the generalized projection Π_𝐶∶𝐸→𝐶 is defined by Π_𝐶(𝑥)=arginf_𝑦∈𝐶𝜙𝑦,𝑥,∀𝑥∈𝐸. (1.7)

Lemma 1.1 (see [1]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and 𝐶 be a nonempty closed convex subset of 𝐸. Then the following conclusions hold: (1)𝜙(𝑥,Π_𝐶𝑦)+𝜙(Π_𝑐𝑦,𝑦)≤𝜙(𝑥,𝑦) for all 𝑥∈𝐶 and 𝑦∈𝐸; (2)If 𝑥∈𝐸 and 𝑧∈𝐶, then 𝑧=Π_𝐶𝑥⇔⟨𝑧−𝑦,𝐽𝑥−𝐽𝑧⟩≥0, for all 𝑦∈𝐶; (3)For 𝑥,𝑦∈𝐸,𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦.

Remark 1.2. If 𝐸 is a real Hilbert space 𝐻, then 𝜙(𝑥,𝑦)=||𝑥−𝑦||² and Π_𝐶=𝑃_𝐶 (the metric projection of 𝐻 onto 𝐶).

A mapping 𝑇∶𝐶→𝐶 is said to be closed, if for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, then 𝑇𝑥=𝑦.

Definition 1.3. Let 𝑃∶𝐸→𝐶 be the nonexpansive retraction. (1)𝑇∶𝐶→𝐸 is said to be quasi-𝜙-nonexpansive nonself mapping, if 𝐹(𝑇)≠∅ and 𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥),∀𝑥∈𝐶,𝑢∈𝐹(𝑇). (1.8)(2)𝑇∶𝐶→𝐸 is said to be quasi-𝜙-asymptotically nonexpansive nonself mapping, if 𝐹(𝑇)≠∅ and there exists a real sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that 𝜙𝑢,𝑇(𝑃𝑇)^𝑛−1𝑥≤𝑘_𝑛𝜙(𝑢,𝑥),∀𝑥∈𝐶,𝑢∈𝐹(𝑇),𝑛≥1. (1.9)(3)𝑇∶𝐶→𝐸 is said to be total quasi-𝜙-asymptotically nonexpansive nonself mapping, if 𝐹(𝑇)≠∅ and there exists nonnegative real sequence {𝜈_𝑛},{𝜇_𝑛} with 𝜈_𝑛→0,𝜇_𝑛→0(as 𝑛→∞) and a strictly increasing continuous function 𝜌∶ℛ⁺→ℛ⁺ with 𝜌(0)=0 such that for all 𝑥∈𝐶,𝑢∈𝐹(𝑇)𝜙𝑢,𝑇(𝑃𝑇)^𝑛−1𝑥≤𝜙(𝑢,𝑥)+𝜈_𝑛𝜌𝜙(𝑢,𝑥)+𝜇_𝑛,∀𝑛≥1. (1.10)(4) A countable family of nonself mappings {𝑇_𝑖}∶𝐶→𝐸 is said to be uniformly total quasi-𝜙-asymptotically nonexpansive, if ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅ and there exists nonnegative real sequence {𝜈_𝑛},{𝜇_𝑛} with 𝜈_𝑛→0,𝜇_𝑛→0 (as 𝑛→∞) and a strictly increasing continuous function 𝜌∶ℛ⁺→ℛ⁺ with 𝜌(0)=0 such that for each 𝑖≥1 and all 𝑥∈𝐶,𝑢∈⋂^∞_𝑖=1𝐹(𝑇_𝑖)𝜙𝑢,𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥≤𝜙(𝑢,𝑥)+𝜈_𝑛𝜌𝜙(𝑢,𝑥)+𝜇_𝑛,∀𝑛≥1. (1.11)

Remark 1.4. From the definitions, it is easy to know that(1) If 𝑇 is a quasi-𝜙-nonexpansive nonself mapping, then it must be a quasi-𝜙-asymptotically nonexpansive nonself mapping with {𝑘_𝑛=1}.(2) Taking 𝜌(𝑡)=𝑡,𝑡>0, 𝜈_𝑛=(𝑘_𝑛−1) and 𝜇_𝑛=0, then (1.9) can be rewritten as 𝜙𝑢,𝑇(𝑃𝑇)^𝑛−1𝑥≤𝜙(𝑢,𝑥)+𝜈_𝑛𝜌𝜙(𝑢,𝑥)+𝜇_𝑛,∀𝑛≥1,𝑥∈𝐶,𝑢∈𝐹(𝑇). (1.12)This implies that each quasi-𝜙-asymptotically nonexpansive nonself mapping must be a total quasi-𝜙-asymptotically nonexpansive nonself mapping, but the converse is not true.

A nonself mapping 𝑇∶𝐶→𝐸 is said to be uniformly L-Lipschitz continuous, if there exists a constant 𝐿>0 such that 𝑇(𝑃𝑇)^𝑛−1𝑥−𝑇(𝑃𝑇)^𝑛−1𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶,𝑛≥1. (1.13)

Lemma 1.5 (see [2]). Let 𝐸 be a smooth and uniformly convex Banach space and let {𝑥_𝑛}, {𝑦_𝑛} be two sequences of 𝐸. If 𝜙(𝑥_𝑛,𝑦_𝑛)→0(as 𝑛→∞) and either {𝑥_𝑛} or {𝑦_𝑛} is bounded, then ||𝑥_𝑛−𝑦_𝑛||→0(as 𝑛→∞).

Lemma 1.6. Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and 𝐶 be a nonempty closed and convex subset 𝐸. Let 𝑇∶𝐶→𝐸 be a closed and total quasi-𝜙-asymptotically nonexpansive nonself mapping with nonnegative real sequence {𝜈_𝑛},{𝜇_𝑛} and a strictly increasing continuous function 𝜌∶ℛ⁺→ℛ⁺ such that 𝜈_𝑛→0,𝜇_𝑛→0 and 𝜌(0)=0. Then the fixed point set 𝐹(𝑇) is a closed and convex subset of 𝐶.

Proof. Let {𝑥_𝑛} be a sequence in 𝐹(𝑇) such that 𝑥_𝑛→𝑢(as 𝑛→∞). Since 𝑇𝑥_𝑛=𝑥_𝑛→𝑢, by the closeness of 𝑇, we have 𝑢=𝑇𝑢, that is, 𝑢∈𝐹(𝑇). This shows that 𝐹(𝑇) is a closed set in 𝐶.
Next, we prove that 𝐹(𝑇) is convex. For any 𝑥,𝑦∈𝐹(𝑇),𝑡∈(0,1), putting 𝑞=𝑡𝑥+(1−𝑡)𝑦, we prove that 𝑞∈𝐹(𝑇). Indeed, let {𝑢_𝑛} be a sequence generated by 𝑢₁=𝑇𝑞,𝑢₂=𝑇𝑃𝑇𝑞=𝑇𝑃𝑢₁,𝑢₃=𝑇(𝑃𝑇)²𝑞=𝑇𝑃𝑢₂𝑢,…,_𝑛=𝑇(𝑃𝑇)^𝑛−1𝑞=𝑇𝑃𝑢_𝑛−1,…, (1.14) we have𝜙𝑞,𝑢_𝑛=𝑞²−2𝑞,𝐽𝑢_𝑛+𝑢_𝑛²=𝑞²−2𝑡𝑥,𝐽𝑢_𝑛−2(1−𝑡)𝑦,𝐽𝑢_𝑛+𝑢_𝑛²=𝑞²+𝑡𝜙𝑥,𝑢_𝑛+(1−𝑡)𝜙𝑦,𝑢_𝑛−𝑡‖𝑥‖²−(1−𝑡)𝑦². (1.15) Since 𝑡𝜙𝑥,𝑢_𝑛+(1−𝑡)𝜙𝑦,𝑢_𝑛≤𝑡𝜙𝑥,𝑞+𝜈_𝑛𝜌𝜙𝑥,𝑞+𝜇_𝑛+(1−𝑡)𝜙𝑦,𝑞+𝜈_𝑛𝜌𝜙𝑦,𝑞+𝜇_𝑛=𝑡‖𝑥‖²−2𝑥,𝐽𝑞+𝑞²+𝜈_𝑛𝜌𝜙𝑥,𝑞+𝜇_𝑛+(1−𝑡)𝑦²−2𝑦,𝐽𝑞+𝑞²+𝜈_𝑛𝜌𝜙𝑦,𝑞+𝜇_𝑛=𝑡‖𝑥‖²+(1−𝑡)𝑦²−𝑞²+𝑡𝜈_𝑛𝜌𝜙𝑥,𝑞+(1−𝑡)𝜈_𝑛𝜌𝜙𝑦,𝑞+𝜇_𝑛. (1.16) Substituting (1.16) into (1.15), and simplifying we have 𝜙𝑞,𝑢_𝑛≤𝑡𝜈_𝑛𝜌𝜙𝑥,𝑞+(1−𝑡)𝜈_𝑛𝜌𝜙𝑦,𝑞+𝜇_𝑛⟶0(𝑛⟶∞). (1.17) By Lemma 1.5, we have 𝑢_𝑛→𝑞(𝑛→∞). This implies that 𝑢_𝑛+1→𝑞(𝑛→∞).
Since 𝑢_𝑛+1=𝑇(𝑃𝑇)^𝑛𝑞=𝑇𝑃𝑇(𝑃𝑇)^𝑛−1𝑞=𝑇𝑃𝑢_𝑛 and 𝑇 is closed, we have 𝑞=𝑇𝑃𝑞. Since 𝑞∈𝐶, 𝑃𝑞=𝑞, thus 𝑞=𝑇𝑞. this implies that 𝐹(𝑇) is a convex set in 𝐶.

Concerning the strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasi-𝜙-nonexpansive and quasi-𝜙-asymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see e.g., [2–19]).

The purpose of this paper is to modify the Halpern and Mann-type iteration algorithm for a family of of total quasi-𝜙-asymptotically nonexpansive nonself mappings and to have the strong convergence under removing 𝐹(𝑇) is a convex set of condition and a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper extend and improve the corresponding results of Chang et al. [4–7], W. P. Guo and W. Guo [8], Hao et al. [9], Kamimura and Takahashi [10], Kiziltunc and Temir [11], Nilsrakoo and Saejung [2], Pathak et al. [12], Qin et al. [13], Su et al. [14], Thianwan [15], Wang et al. [16], Yıldırım and Özdemir [17], Yang and Xie [18], Zegeye et al. [19], Kanjanasamranwong et al. [20], Saewan and Kumam [21–24] and Wattanawitoon and Kumam [25].

2. Main Results

Theorem 2.1. Let 𝐸 be a real uniformly convex and uniformly smooth Banach space, and 𝐶 be a nonempty closed convex subset 𝐸. Let 𝑇_𝑖∶𝐶→𝐸,𝑖=1,2,… be a family of closed and uniformly total quasi-𝜙-asymptotically nonexpansive nonself mappings with nonnegative real sequence {𝜈_𝑛},{𝜇_𝑛} and a strictly increasing continuous function 𝜌∶ℛ⁺→ℛ⁺ such that 𝜈_𝑛→0,𝜇_𝑛→0 and 𝜌(0)=0, and for each 𝑖≥1, 𝑇_𝑖 be uniformly 𝐿_𝑖-Lipschitz continuous. Let {𝛼_𝑛} be a sequence in [0,1] and {𝛽_𝑛} be a sequence in (0,1) satisfying the following conditions: (a) lim_𝑛→∞𝛼_𝑛=0; (b)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Let {𝑥_𝑛} be a sequence generated by 𝑥₁∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦;𝐶₁𝑦=𝐶,_𝑛,𝑖=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛𝐶,𝑖≥1,_𝑛+1=𝑧∈𝐶_𝑛∶sup_𝑖≥1𝜙𝑧,𝑦_𝑛,𝑖≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝜃_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1. (2.1) where 𝜃_𝑛=𝜈_𝑛sup_𝑢∈ℱ𝜌(𝜙(𝑢,𝑥_𝑛))+𝜇_𝑛, for all 𝑛≥1, ℱ∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖). If ℱ is a nonempty-bounded subset in 𝐶, then {𝑥_𝑛} converges strongly to Π_ℱ𝑥₁.

Proof. We divide the proof of Theorem 2.1 into five steps.
(I)ℱ  and  𝐶_𝑛,𝑛≥1  are closed and convex subset in 𝐶.
In fact, it follows from Lemma 1.6 that 𝐹(𝑇_𝑖), 𝑖≥1 is closed and convex subset of 𝐶. Therefore ℱ is a closed and convex subset in 𝐶.
Again by the assumption that 𝐶₁=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for some 𝑛≥2. In view of the definition of 𝜙 we have that 𝐶_𝑛+1=𝑧∈𝐶_𝑛∶sup_𝑖≥1𝜙𝑧,𝑦_𝑛,𝑖≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝜃_𝑛=_𝑖≥1𝑧∈𝐶_𝑛∶𝜙𝑧,𝑦_𝑛,𝑖≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝜃_𝑛∩𝐶_𝑛=_𝑖≥1𝑧∈𝐶_𝑛∶2𝛼_𝑛𝑧,𝐽𝑥₁+21−𝛼_𝑛𝑧,𝐽𝑥_𝑛−2𝑧,𝐽𝑦_𝑛,𝑖≤𝛼_𝑛𝑥₁²+1−𝛼_𝑛𝑥_𝑛²−𝑦_𝑛,𝑖²+𝜃_𝑛∩𝐶_𝑛. (2.2) This implies that 𝐶_𝑛+1 is closed and convex. The conclusion is proved.
(II)   Now we prove that  ℱ⊂𝐶_𝑛,𝑛≥1.
In fact, it is obvious that ℱ⊂𝐶₁=𝐶. Suppose that ℱ⊂𝐶_𝑛 for some 𝑛≥2. Letting 𝑤_𝑛,𝑖=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛, (2.3) it follows from (1.6) that for any 𝑢∈ℱ⊂𝐶_𝑛 we have 𝜙𝑢,𝑦_𝑛,𝑖=𝜙𝑢,𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑤_𝑛,𝑖≤𝛼_𝑛𝜙𝑢,𝑥₁+1−𝛼_𝑛𝜙𝑢,𝑤_𝑛,𝑖,𝜙𝑢,𝑤_𝑛,𝑖=𝜙𝑢,𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛≤𝛽_𝑛𝜙𝑢,𝑥_𝑛+1−𝛽_𝑛𝜙𝑢,𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛≤𝛽_𝑛𝜙𝑢,𝑥_𝑛+1−𝛽_𝑛𝜙𝑢,𝑥_𝑛+𝜈_𝑛𝜌𝜙𝑢,𝑥_𝑛+𝜇_𝑛=𝜙𝑢,𝑥_𝑛+1−𝛽_𝑛𝜈_𝑛𝜌𝜙𝑢,𝑥_𝑛+𝜇_𝑛≤𝜙𝑢,𝑥_𝑛+𝜈_𝑛𝜌𝜙𝑢,𝑥_𝑛+𝜇_𝑛, (2.4)(2.5) therefore we have sup_𝑖≥1𝜙𝑢,𝑦_𝑛,𝑖≤𝛼_𝑛𝜙𝑢,𝑥₁+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛+𝜈_𝑛𝜌𝜙𝑢,𝑥_𝑛+𝜇_𝑛≤𝛼_𝑛𝜙𝑢,𝑥₁+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛+𝜈_𝑛sup_𝑢∈ℱ𝜌𝜙𝑢,𝑥_𝑛+𝜇_𝑛≤𝛼_𝑛𝜙𝑢,𝑥₁+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛+𝜃_𝑛, (2.6) where 𝜃_𝑛=𝜈_𝑛sup_𝑢∈ℱ𝜌(𝜙(𝑢,𝑥_𝑛))+𝜇_𝑛. This shows that 𝑢∈𝐶_𝑛+1, and so ℱ⊂𝐶_𝑛+1. The conclusion is proved.
(III)   Next we prove that  {𝑥_𝑛}  is a Cauchy sequence in 𝐶.
In fact, since 𝑥_𝑛=Π_𝐶𝑛𝑥₁, from Lemma 1.1(2) we have 𝑥_𝑛−𝑦,𝐽𝑥₁−𝐽𝑥_𝑛≥0,∀𝑦∈𝐶_𝑛. (2.7) Again since ℱ⊂𝐶_𝑛, for all 𝑛≥1, we have 𝑥_𝑛−𝑢,𝐽𝑥₁−𝐽𝑥_𝑛≥0,∀𝑢∈ℱ. (2.8) It follows from Lemma 1.1(1) that for each 𝑢∈ℱ and for each 𝑛≥1𝜙𝑥_𝑛,𝑥₁=𝜙Π_𝐶𝑛𝑥₁,𝑥₁≤𝜙𝑢,𝑥₁−𝜙𝑢,𝑥_𝑛≤𝜙𝑢,𝑥₁. (2.9) Therefore {𝜙(𝑥_𝑛,𝑥₁)} is bounded. By virtue of (1.5), {𝑥_𝑛} is also bounded.
Since 𝑥_𝑛=Π_𝐶𝑛𝑥₁ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝜙(𝑥_𝑛,𝑥₁)≤𝜙(𝑥_𝑛+1,𝑥₁), for all 𝑛≥1. This implies that {𝜙(𝑥_𝑛,𝑥₁)} is nondecreasing. Hence the limit lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₁) exists. By the construction of 𝐶_𝑛, for any positive integer 𝑚≥𝑛, we have 𝐶_𝑚⊂𝐶_𝑛 and 𝑥_𝑚=Π_𝐶𝑚𝑥₁∈𝐶_𝑛. This shows that 𝜙𝑥_𝑚,𝑥_𝑛=𝜙𝑥_𝑚,Π_𝐶𝑛𝑥₁≤𝜙𝑥_𝑚,𝑥₁−𝜙𝑥_𝑛,𝑥₁⟶0,as𝑛,𝑚⟶∞. (2.10) It follows from Lemma 1.5 that lim_{𝑚,𝑛→∞}||𝑥_𝑚−𝑥_𝑛||=0. Hence {𝑥_𝑛} is a Cauchy sequence in 𝐶. Since 𝐶 is a nonempty closed subset of Banach space 𝐸, it is complete, without loss of generality, we can assume that 𝑥_𝑛→𝑥^∗(𝑛→∞).
By the assumption, it is easy to see that lim_𝑛→∞𝜃_𝑛=lim_𝑛→∞𝜈_𝑛sup_𝑢∈ℱ𝜌𝜙𝑢,𝑥_𝑛+𝜇_𝑛=0. (2.11)
(IV)   Now we prove that  𝑥^∗∈ℱ.
In fact, since 𝑥_𝑛+1∈𝐶_𝑛+1 and 𝛼_𝑛→0, it follows from (2.1) and (2.11) that sup_𝑖≥1𝜙𝑥_𝑛+1,𝑦_𝑛,𝑖≤𝛼_𝑛𝜙𝑥_𝑛+1,𝑥₁+1−𝛼_𝑛𝜙𝑥_𝑛+1,𝑥_𝑛+𝜃_𝑛⟶0(as𝑛⟶∞). (2.12) Since 𝑥_𝑛→𝑥^∗, by virtue of Lemma 1.5 for each 𝑖≥1,we have lim_𝑛→∞𝑦_𝑛,𝑖=𝑥^∗. (2.13) Since {𝑥_𝑛} is bounded, {𝑇_𝑖}^∞_𝑖=1 is uniformly total quasi-𝜙-asymptotically nonexpansive nonself mappings with nonnegative real sequence {𝜈_𝑛},{𝜇_𝑛} and a strictly increasing continuous function 𝜌∶ℛ⁺→ℛ⁺ such that 𝜈_𝑛→0,𝜇_𝑛→0, and 𝜌(0)=0, for any given 𝑢∈ℱ, we have 𝜙𝑢,𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛≤𝜙𝑢,𝑥_𝑛+𝜈_𝑛𝜌𝜙𝑢,𝑥_𝑛+𝜇_𝑛. (2.14) This implies that {𝑇_𝑖(𝑃𝑇_𝑖)^𝑛−1𝑥_𝑛} is uniformly bounded. Since 𝑤_𝑛,𝑖=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛≤𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛≤𝑥_𝑛+𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛. (2.15) This implies that {𝑤_𝑛,𝑖} is also uniformly bounded.
Since 𝛼_𝑛→0, from (2.1), for each 𝑖≥1 we have lim_𝑛→∞𝐽𝑦_𝑛,𝑖−𝐽𝑤_𝑛,𝑖=lim_𝑛→∞𝛼_𝑛𝐽𝑥₁−𝐽𝑤_𝑛,𝑖=0. (2.16) Since 𝐽⁻¹ is uniformly continuous on each bounded subset of 𝐸^∗, it follows from (2.13) and (2.16) that lim_𝑛→∞𝑤_𝑛,𝑖=𝑥^∗foreach𝑖≥1. (2.17) Since 𝐽 is uniformly continuous on each bounded subset of 𝐸, we have 0=lim_𝑛→∞𝐽𝑤_𝑛,𝑖−𝐽𝑥^∗=lim_𝑛→∞𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛−𝐽𝑥^∗=lim_𝑛→∞𝛽_𝑛𝐽𝑥_𝑛−𝐽𝑥^∗+1−𝛽_𝑛𝐽𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛−𝐽𝑥^∗=lim_𝑛→∞1−𝛽_𝑛𝐽𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛−𝐽𝑥^∗. (2.18) By condition (b), we have that lim_𝑛→∞𝐽𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛−𝐽𝑥^∗=0. (2.19) Since 𝐽 is uniformly continuous, this shows that lim_𝑛→∞𝑇_𝑖(𝑃𝑇_𝑖)^𝑛−1𝑥_𝑛=𝑥^∗ uniformly in 𝑖≥1.
Again by the assumptions that for each 𝑖≥1, 𝑇_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous, thus we have 𝑇_𝑖𝑃𝑇_𝑖^𝑛𝑥_𝑛−𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛≤𝑇_𝑖𝑃𝑇_𝑖^𝑛𝑥_𝑛−𝑇_𝑖𝑃𝑇_𝑖^𝑛𝑥_𝑛+1+𝑇_𝑖𝑃𝑇_𝑖^𝑛𝑥_𝑛+1−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛≤𝐿_𝑖+1𝑥_𝑛−𝑥_𝑛+1+𝑇_𝑖𝑃𝑇_𝑖^𝑛𝑥_𝑛+1−𝑥_𝑛+1+𝑥_𝑛−𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛. (2.20) Since lim_𝑛→∞𝑇_𝑖(𝑃𝑇_𝑖)^𝑛−1𝑥_𝑛=𝑥^∗ and 𝑥_𝑛→𝑥^∗, these together with (2.20) imply that lim_𝑛→∞||𝑇_𝑖(𝑃𝑇_𝑖)^𝑛𝑥_𝑛−𝑇_𝑖(𝑃𝑇_𝑖)^𝑛−1𝑥_𝑛||=0 and lim_𝑛→∞𝑇_𝑖(𝑃𝑇_𝑖)^𝑛𝑥_𝑛=𝑥^∗, that is, lim_𝑛→∞𝑇_𝑖𝑃𝑃𝑇_𝑖^𝑛−1𝑥_𝑛=𝑥^∗. (2.21) In view continuity of 𝑇_𝑖𝑃, it yields that 𝑇_𝑖𝑃𝑥^∗=𝑥^∗. Since 𝑥^∗∈𝐶,𝑃𝑥^∗=𝑥^∗. This shows that 𝑇𝑥^∗=𝑥^∗. By the arbitrariness of 𝑖≥1, we have 𝑥^∗∈ℱ.
(V)  Finally we prove that  𝑥_𝑛→𝑥^∗=Π_ℱ𝑥₁.
Let 𝑤=Π_ℱ𝑥₁. Since 𝑤∈ℱ⊂𝐶_𝑛 and 𝑥_𝑛=Π_𝐶𝑛𝑥₁, we have 𝜙(𝑥_𝑛,𝑥₁)≤𝜙(𝑤,𝑥₁), for all 𝑛≥1. This implies that 𝜙𝑥^∗,𝑥₁=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₁≤𝜙𝑤,𝑥₁. (2.22) In view of the definition of Π_ℱ𝑥₁, from (2.22) we have 𝑥^∗=𝑤. Therefore 𝑥_𝑛→𝑥^∗=Π_ℱ𝑥₁. This completes the proof of Theorem 2.1.

Theorem 2.2. Let 𝐸, 𝐶, {𝛼_𝑛}, {𝛽_𝑛} be the same as in Theorem 2.1. Let 𝑇_𝑖∶𝐶→𝐸,𝑖=1,2,… be a family of closed and uniformly quasi-𝜙-asymptotically nonexpansive nonself mappings with sequence {𝑘_𝑛}⊂[1,∞),𝑘_𝑛→1, and for each 𝑖≥1, 𝑇_𝑖 be uniformly 𝐿_𝑖-Lipschitz continuous. Let {𝑥_𝑛} be a sequence generated by 𝑥₁∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦;𝐶₁𝑦=𝐶,_𝑛,𝑖=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_𝑖𝑃𝑇_𝑖^𝑛−1𝑥_𝑛𝐶,𝑖≥1,_𝑛+1=𝑧∈𝐶_𝑛∶sup_𝑖≥1𝜙𝑧,𝑦_𝑛,𝑖≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝜃_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1, (2.23) where 𝜃_𝑛=(𝑘_𝑛−1)sup_𝑢∈ℱ𝜙(𝑢,𝑥_𝑛), ℱ∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖). If ℱ is a nonempty bounded subset in 𝐶, then {𝑥_𝑛} converges strongly to Π_ℱ𝑥₁.

Proof. By Remark 1.4𝑇_𝑖∶𝐶→𝐸,𝑖=1,2,… be a family of closed and uniformly quasi-𝜙-asymptotically nonexpansive nonself mappings that it is a family of closed and uniformly total quasi-𝜙-asymptotically nonexpansive nonself mappings with taking 𝜌(𝑡)=𝑡,𝑡>0, 𝜈_𝑛=(𝑘_𝑛−1) and 𝜇_𝑛=0. Therefore all conditions in Theorem 2.1 are satisfied. By the similar methods as given in the proof of Theorem 2.1, we can prove that the sequence {𝑥_𝑛} defined by (2.23) converges strongly to Π_ℱ𝑥₁.

Theorem 2.3. Let 𝐸, 𝐶, {𝛼_𝑛}, {𝛽_𝑛} be the same as in Theorem 2.2. Let 𝑇_𝑖∶𝐶→𝐸,𝑖=1,2,… be a family of quasi-𝜙-nonexpansive nonself mappings such that ℱ∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅ and for each 𝑖≥1, 𝑇_𝑖 be uniformly 𝐿_𝑖-Lipschitz continuous. Let {𝑥_𝑛} be a sequence generated by 𝑥₁∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦;𝐶₁𝑦=𝐶,_𝑛,𝑖=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_𝑖𝑥_𝑛𝐶,𝑖≥1,_𝑛+1=𝑧∈𝐶_𝑛∶sup_𝑖≥1𝜙𝑧,𝑦_𝑛,𝑖≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1. (2.24) Then {𝑥_𝑛} converges strongly to Π_ℱ𝑥₁.

Proof. By Remark 1.4𝑇_𝑖∶𝐶→𝐸,𝑖=1,2,… be a family of quasi-𝜙-nonexpansive nonself mappings that it is a family of uniformly quasi-𝜙-asymptotically nonexpansive nonself mappings with sequence {𝑘_𝑛}={1}. Hence 𝜃_𝑛=(𝑘_𝑛−1)sup_𝑢∈ℱ𝜙(𝑢,𝑥_𝑛)=0 Therefore all conditions in Theorem 2.2 are satisfied. By the similar methods, we can prove that the sequence {𝑥_𝑛} defined by (2.24) converges strongly to Π_ℱ𝑥₁.

3. Application and Example

In this section we utilize the results presented in Section 2 to prove a strong convergence theorem concerning maximal monotone operators in Hilbert spaces.

Let 𝐸 be a real Hilbert space and let 𝐴 be a maximal monotone operator from 𝐸 to 𝐸. For each 𝑟>0, we can define a single valued mapping 𝐽^𝐴_𝑟∶𝐸→𝐸 by 𝐽^𝐴_𝑟=(𝐼+𝑟𝐴)⁻¹ and such a mapping 𝐽^𝐴_𝑟 is called the resolvent of 𝐴. It is easy to prove that 𝐽^𝐴_𝑟 is a nonexpansive mapping and 𝐴⁻¹(0)=𝐹(𝐽^𝐴_𝑟) for all 𝑟>0. Therefore it is a uniformly 1-Lipschitz continuous and quasi-𝜙-nonexpansive mapping. Hence for each 𝑝∈𝐹(𝐽^𝐴_𝑟) and 𝑤∈𝐸, we have 𝜙𝑝,𝐽^𝐴_𝑟𝑤≤𝜙𝑝,𝑤, (3.1) and 𝐹(𝐽^𝐴_𝑟)=𝐴⁻¹(0). These show that all conditions in Theorem 2.3 are satisfied. Hence from Theorem 2.3 we have the following.

Theorem 3.1. Let 𝐸 be a real Hilbert space. Let 𝐴₁,𝐴₂ be two maximal monotone operators from 𝐸 to 𝐸 such that ℱ=𝐴₁⁻¹(0)∩𝐴₂⁻¹(0)≠∅. Let 𝐽^𝐴1_𝑟 and 𝐽^𝐴2_𝑟 be the resolvent of 𝐴₁ and 𝐴₂, respectively, where 𝑟>0. Let {𝛼_𝑛}, {𝛽_𝑛} be the same as in Theorem 2.3 and {𝑥_𝑛} be the sequence defined by 𝑥₁∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦;𝐶₁𝑦=𝐸,_𝑛,𝑖=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝐽^𝐴𝑖_𝑟𝑥_𝑛𝐶,𝑖=1,2,_𝑛+1=𝑧∈𝐶_𝑛∶max_𝑖=1,2𝜙𝑧,𝑦_𝑛,𝑖≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₁,∀𝑛≥1, (3.2) where 𝑃_𝐶 is the metric projection from 𝐻 onto the subset 𝐶⊂𝐻. Then the sequence {𝑥_𝑛} defined by (3.2) converges strongly to 𝑃_ℱ𝑥₁.

Acknowledgments

The authors would like to express their thanks to the referees and the Editor for their helpful comments and suggestions. This Project was supported by the Scientific Research Fund of SiChuan Provincial Education Department (no. 11ZA222).