Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and Applications

Abstract

The purpose of this paper is to present the notion of weak relatively nonexpansive multi-valued mapping and to prove the strong convergence theorems of fixed point for weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper, the notion of multivalued resolvent of maximal monotone operator has been also presented which is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of maximal monotone operator.

1. Introduction and Preliminaries

Iterative methods for approximating fixed points of multivalued mappings in Banach spaces have been studied by some authors, see for instance [1–4]. Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑇∶𝐶→𝐶 a multivalued mapping such that 𝑇𝑥 is nonempty for all 𝑥∈𝐶. In [4], Homaeipour and Razani have defined the relatively nonexpansive multivalued mapping and have proved some convergence theorems.

Theorem SA 1 (see [4]). Let 𝐸 be a uniformly convex and uniformly smooth Banach space, and 𝐶 a nonempty closed convex subset of 𝐸. Suppose 𝑇∶𝐶→𝐶 is a relatively nonexpansive multivalued mapping. Let {𝛼_𝑛} be a sequence of real numbers such that 0≤𝛼_𝑛≤1 and liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0. For a given 𝑥₁∈𝐶, let {𝑥_𝑛} be the iterative sequence defined by 𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛∈𝑇𝑥_𝑛. (1.1) If 𝐽 is weakly sequentially continuous, then {𝑥_𝑛} converges weakly to a fixed point of 𝑇.

Theorem SA 2 (see [4]). Let 𝐸 be a uniformly convex and uniformly smooth Banach space, and 𝐶 a nonempty closed convex subset of 𝐸. Suppose 𝑇∶𝐶→𝐶 is a relatively nonexpansive multivalued mapping. Let {𝛼_𝑛} be a sequence of real numbers such that 0≤𝛼_𝑛≤1 and liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0. For a given 𝑥₁∈𝐶, let {𝑥_𝑛} be the iterative sequence defined by 𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛∈𝑇𝑥_𝑛. (1.2) If the interior of 𝐹(𝑇) is nonempty, then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Let 𝐸 be a Banach space with dual 𝐸^∗. We denote by 𝐽 the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽𝑥=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖²=𝑓², (1.3) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. It is well known that if 𝐸^∗ is uniformly convex, then 𝐽 is uniformly continuous on bounded subsets of 𝐸.

As we all know that if 𝐶 is a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑃_𝐶∶𝐻→𝐶 is the metric projection of 𝐻 onto 𝐶, then 𝑃_𝐶 is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently it is not available in more general Banach spaces. In this connection, Alber [5] recently introduced a generalized projection operator Π_𝐶 in a Banach space 𝐸 which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that 𝐸 is a smooth Banach space. Consider the functional defined as [5, 6] by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,for𝑥,𝑦∈𝐸. (1.4) Observe that, in a Hilbert space 𝐻, (1.4) reduces to 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖²,𝑥,𝑦∈𝐻.

The generalized projection Π_𝐶∶𝐸→𝐶 is a map that assigns to an arbitrary point 𝑥∈𝐸 the minimum point of the functional 𝜙(𝑥,𝑦), that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the following minimization problem: 𝜙(𝑥,𝑥)=min_𝑦∈𝐶𝜙𝑦,𝑥, (1.5) existence and uniqueness of the operator Π_𝐶 follow from the properties of the functional 𝜙(𝑥,𝑦) and strict monotonicity of the mapping 𝐽 (see, eg., [5–7]). In Hilbert space, Π_𝐶=𝑃_𝐶. It is obvious from the definition of function 𝜙 that 𝑦−‖𝑥‖²≤𝜙𝑦,𝑥≤𝑦+‖𝑥‖²,∀𝑥,𝑦∈𝐸. (1.6)

Remark 1.1. If 𝐸 is a reflexive strictly convex and smooth Banach space, then for 𝑥,𝑦∈𝐸, 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦. It is sufficient to show that if 𝜙(𝑥,𝑦)=0 then 𝑥=𝑦. From (1.6), we have ‖𝑥‖=‖𝑦‖. This implies ⟨𝑥,𝐽𝑦⟩=‖𝑥‖²=‖𝐽𝑦‖². From the definitions of 𝑗, we have 𝐽𝑥=𝐽𝑦, that is, 𝑥=𝑦, see [8, 9] for more details.

Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑇∶𝐶→𝐶 a multivalued mapping such that 𝑇𝑥 is nonempty for all 𝑥∈𝐶. A point 𝑝 is called a fixed point of 𝑇, if 𝑝∈𝑇𝑝. The set of fixed points of 𝑇 is represented by 𝐹(𝑇). A point 𝑝∈𝐶 is called an asymptotic fixed point of 𝑇, if there exists a sequence {𝑥_𝑛} in 𝐶 which converges weakly to 𝑝 and lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0, where 𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=inf_{𝑢∈𝑇𝑥𝑛}‖𝑥_𝑛−𝑢‖. The set of asymptotic fixed points of 𝑇 is represented by 𝐹(𝑇). Moreover, a multivalued mapping 𝑇∶𝐶→𝐶 is called relatively nonexpansive multivalued mapping, if the following conditions are satisfied:(1)𝐹(𝑇)≠∅,(2)𝜙(𝑝,𝑧)≤𝜙(𝑝,𝑥),  for  all   𝑝∈𝐹(𝑇),  for  all  𝑥∈𝐶,  for  all 𝑧∈𝑇𝑥,(3)𝐹(𝑇)=𝐹(𝑇).

In [4], authors also give an example which is a relatively nonexpansive multivalued mapping but not a nonexpansive multivalued mapping.

The purpose of this paper is to present the notion of weak relatively nonexpansive multivalued mapping and to prove the strong convergence theorems for the weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. We first give the definition of weak relatively nonexpansive multivalued mapping as follows.

Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑇∶𝐶→𝐶 a multivalued mapping such that 𝑇𝑥 is nonempty for all 𝑥∈𝐶. A point 𝑝∈𝐶 is called an strong asymptotic fixed point of 𝑇, if there exists a sequence {𝑥_𝑛} in 𝐶 which converges strongly to 𝑝 and lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0, where 𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=inf_{𝑢∈𝑇𝑥𝑛}‖𝑥_𝑛−𝑢‖. The set of the strong asymptotic fixed points of 𝑇 is represented by 𝐹(𝑇). Moreover, a multivalued mapping 𝑇∶𝐶→𝐶 is called weak relatively nonexpansive multivalued mapping, if the following conditions are satisfied: (I)𝐹(𝑇)≠∅, (II)𝜙(𝑝,𝑧)≤𝜙(𝑝,𝑥),  for  all  𝑝∈𝐹(𝑇),  for  all  𝑥∈𝐶,  for  all  𝑧∈𝑇𝑥, (III)𝐹(𝑇)=𝐹(𝑇).

We need the following Lemmas for the proof of our main results.

Lemma 1.2 (see Kamimura and Takahashi [7]). Let 𝐸 be a uniformly convex and smooth Banach space and let {𝑥_𝑛},{𝑦_𝑛} be two sequences of 𝐸. If 𝜙(𝑥_𝑛,𝑦_𝑛)→0 and either {𝑥_𝑛} or {𝑦_𝑛} is bounded, then 𝑥_𝑛−𝑦_𝑛→0.

Lemma 1.3 (see Alber [5]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑥∈𝐸. Then, 𝑥₀=Π_𝐶𝑥 if and only if 𝑥₀−𝑦,𝐽𝑥−𝐽𝑥₀≥0,for𝑦∈𝐶. (1.7)

Lemma 1.4 (see Alber [5]). Let 𝐸 be a reflexive, strictly convex and smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑥∈𝐸. Then 𝜙𝑦,Π_𝑐𝑥+𝜙Π_𝑐𝑥,𝑥≤𝜙𝑦,𝑥,∀𝑦∈𝐶. (1.8)

Lemma 1.5. Let 𝐸 be a uniformly convex and smooth Banach space, let 𝐶 be a closed convex subset of 𝐸, and let 𝑇∶𝐶→𝐶 be a weak relatively nonexpansive multivalued mapping. Then 𝐹(𝑇) is closed and convex.

Proof. First, we show 𝐹(𝑇) is closed. Let {𝑥_𝑛} be a sequence in 𝐹(𝑇) such that 𝑥_𝑛→𝑞. Since 𝑇 is weak relatively nonexpansive multivalued mapping, we have 𝜙𝑥_𝑛,𝑧≤𝜙𝑥_𝑛,𝑞,∀𝑧∈𝑇𝑞,𝑛=1,2,3,…. (1.9) Therefore 𝜙𝑞,𝑧=lim_𝑛→∞𝜙𝑥_𝑛,𝑧≤lim_𝑛→∞𝜙𝑥_𝑛,𝑞=𝜙𝑞,𝑞=0. (1.10) By Lemma 1.2, we have 𝑞=𝑧, hence, 𝑇𝑞={𝑞}, so 𝑞∈𝐹(𝑇). Therefore 𝐹(𝑇) is closed. Next, we show 𝐹(𝑇) is convex. Let 𝑥,𝑦∈𝐹(𝑇), put 𝑧=𝑡𝑥+(1−𝑡)𝑦 for any 𝑡∈(0,1). For 𝑤∈𝑇𝑧, we have 𝜙(𝑧,𝑤)=‖𝑧‖²−2⟨𝑧,𝐽𝑤⟩+‖𝑤‖²=‖𝑧‖²−2𝑡𝑥+(1−𝑡)𝑦,𝐽𝑤+‖𝑤‖²=‖𝑧‖²−2𝑡⟨𝑥,𝐽𝑤⟩−2(1−𝑡)𝑦,𝐽𝑤+‖𝑤‖²=‖𝑧‖²+𝑡𝜙(𝑥,𝑤)+(1−𝑡)𝜙𝑦,𝑤−𝑡‖𝑥‖²−(1−𝑡)𝑦²≤‖z‖²+𝑡𝜙(𝑥,𝑧)+(1−𝑡)𝜙𝑦,𝑧−𝑡‖𝑥‖²−(1−𝑡)𝑦²=‖𝑧‖²−2𝑡𝑥+(1−𝑡)𝑦,𝐽𝑧+‖𝑧‖²=‖𝑧‖²−2⟨𝑧,𝐽𝑧⟩+‖𝑧‖²=𝜙(𝑧,𝑧)=0. (1.11) By Lemma 1.2, we have 𝑧=𝑤, so 𝑧∈𝑇𝑧, that is 𝑧∈𝐹(𝑇). Therefore, 𝐹(𝑇) is convex. This completes the proof.

Remark 1.6. Let 𝐸 be a strictly convex and smooth Banach space, and 𝐶 a nonempty closed convex subset of 𝐸. Suppose 𝑇∶𝐶→𝐶 is a weak relatively nonexpansive multivalued mapping. If 𝑝∈𝐹(𝑇), then 𝑇(𝑝)={𝑝}.

2. An Example of Weak Relatively Nonexpansive Multivalued Mapping

Next, we give an example which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping.

Example 2.1. Let 𝐸=𝑙² and 𝑥₀=()𝑥1,0,0,0,…₁=()𝑥1,1,0,0,…₂=()𝑥1,0,1,0,0,…₃=()⋯𝑥1,0,0,1,0,0,…_𝑛=()⋯1,0,1,0,0,…,0,1,0,0,0,… (2.1) It is obvious that {𝑥_𝑛}^∞_𝑛=1 converges weakly to 𝑥₀. On the other hand, we have ‖𝑥_𝑛−𝑥_𝑚‖=√2 for any 𝑛≠𝑚. Define a mapping 𝑇∶𝐸→2^𝐸 as follows: 𝑇(𝑥)=𝑦=𝑡𝑥_𝑛∶𝑛𝑛𝑛+2≤𝑡≤𝑛+1,if𝑥=𝑥_𝑛(∃𝑛≥1),−𝑥,if𝑥≠𝑥_𝑛(∀𝑛≥1). (2.2) It is also obvious that 𝐹(𝑇)={0} and 0−𝑦=𝑦≤‖𝑥‖=‖0−𝑥‖,∀𝑥∈𝐸,𝑦∈𝑇𝑥. (2.3) Since 𝐸=𝐿² is a Hilbert space, we have 𝜙0,𝑦=0−𝑦²=𝑦²≤‖𝑥‖²=‖0−𝑥‖²=𝜙(0,𝑥),∀𝑥∈𝐸,𝑦∈𝑇𝑥. (2.4) Next, we prove 𝐹(𝑇)=𝐹(𝑇). In fact, that for any strong convergent sequence {𝑧_𝑛}⊂𝐸 such that 𝑧_𝑛→𝑧₀ and 𝑑(𝑧_𝑛,𝑇𝑧_𝑛)→0 as 𝑛→∞, then there exist sufficiently large nature number 𝑁 such that 𝑧_𝑛≠𝑥_𝑚, for any 𝑛,𝑚>𝑁. Then 𝑇𝑧_𝑛=−𝑧_𝑛 for 𝑛>𝑁, it follows from 𝑧_𝑛−𝑇𝑧_𝑛=𝑑𝑧_𝑛,𝑇𝑧_𝑛→0 (2.5) that 2𝑧_𝑛→0 and hence 𝑧_𝑛→𝑧₀=0, this implies 𝑧₀∈𝐹(𝑇), so that 𝐹(𝑇)=𝐹(𝑇). Then 𝑇 is a weak relatively multivalued nonexpansive mapping.
We second claim that 𝑇 is not relatively multivalued nonexpansive mapping. In fact, that 𝑥_𝑛⇀𝑥₀ and 𝑑𝑥_𝑛,𝑇𝑥_𝑛≤𝑛−𝑛𝑛+1𝑛+2→0 (2.6) as 𝑛→∞ hold, but 𝑥₀∈𝐹(𝑇).

3. Strong Convergence of Monotone Hybrid Algorithm

Theorem 3.1. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑇∶𝐶→𝐶 be a weak relatively multivalued nonexpansive mapping such that 𝐹(𝑇)≠∅. Assume that {𝛼_𝑛}^∞_𝑛=0,{𝛽_𝑛}^∞_𝑛=0,{𝛾_𝑛}^∞_𝑛=0 are three sequences in [0,1] such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, lim_𝑛→∞𝛼_𝑛=0 and 0<𝛾≤𝛾_𝑛≤1 for some constant 𝛾∈(0,1). Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀𝑦∈𝐶arbitrarily,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛,𝑧_𝑛∈𝑇𝑥_𝑛𝐶_𝑛=𝑧∈𝐶_𝑛−1∶𝜙𝑧,𝑦_𝑛≤1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝜙𝑧,𝑥₀𝐶,𝑛≥1,₀𝑥=𝐶,_𝑛+1=Π_𝐶𝑛𝑥₀, (3.1) then {𝑥_𝑛} converges to 𝑞=Π_𝐹(𝑇)𝑥₀.

Proof. We first show that 𝐶_𝑛 is closed and convex for all 𝑛≥0. From the definitions of 𝐶_𝑛, it is obvious that 𝐶_𝑛 is closed for all 𝑛≥0. Next, we prove that 𝐶_𝑛 is convex for all 𝑛≥0. Since 𝜙𝑧,𝑦_𝑛≤1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝜙𝑧,𝑥₀ (3.2) is equivalent to 2𝑧,1−𝛼_𝑛𝐽𝑥_𝑛+𝛼_𝑛𝐽𝑥₀−𝐽𝑦_𝑛≤1−𝛼_𝑛𝑥_𝑛²+𝛼_𝑛𝑥₀². (3.3) It is easy to get that 𝐶_𝑛 is convex for all 𝑛≥0.
Next, we show that 𝐹(𝑇)⊂𝐶_𝑛 for all 𝑛≥0. Indeed, for each 𝑝∈𝐹(𝑇), we have 𝜙𝑝,𝑦_𝑛=𝜙𝑝,𝐽⁻¹𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛=𝑝²−2𝑝,𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛²≤𝑝²−2𝛼_𝑛𝑝,𝐽𝑥₀−2𝛽_𝑛𝑝,𝐽𝑥_𝑛−2𝛾_𝑛𝑝,𝐽𝑧_𝑛+𝛼_𝑛𝑥₀²+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑧_𝑛²≤𝛼_𝑛𝜙𝑝,𝑥₀+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑧_𝑛≤𝛼_𝑛𝜙𝑝,𝑥₀+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑥_𝑛≤𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛. (3.4) So, 𝑝∈𝐶_𝑛, which implies that 𝐹(𝑇)⊂𝐶_𝑛 for all 𝑛≥0.
Since 𝑥_𝑛+1=Π_𝐶𝑛𝑥₀ and 𝐶_𝑛⊂𝐶_𝑛−1, then we get 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀,∀𝑛≥0. (3.5) Therefore, {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing. On the other hand, by Lemma 1.4 we have 𝜙𝑥_𝑛,𝑥₀=𝜙Π_𝐶𝑛−1𝑥₀,𝑥₀≤𝜙𝑝,𝑥₀−𝜙𝑝,𝑥_𝑛≤𝜙𝑝,𝑥₀, (3.6) for all 𝑝∈𝐹(𝑇)⊂𝐶_𝑛−1 and for all 𝑛≥1. Therefore, 𝜙(𝑥_𝑛,𝑥₀) is also bounded. This together with (3.5) implies that the limit of {𝜙(𝑥_𝑛,𝑥₀)} exists. Put lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀=𝑑. (3.7) From Lemma 1.4, we have, for any positive integer 𝑚, that 𝜙𝑥_𝑛+𝑚,𝑥_𝑛+1=𝜙𝑥_𝑛+𝑚,Π_𝐶𝑛𝑥₀≤𝜙𝑥_𝑛+𝑚,𝑥₀−𝜙Π_𝐶𝑛𝑥₀,𝑥₀=𝜙𝑥_𝑛+𝑚,𝑥₀−𝜙𝑥_𝑛+1,𝑥₀, (3.8) for all 𝑛≥0. This together with (3.7) implies that lim_𝑛→∞𝜙𝑥_𝑛+𝑚,𝑥_𝑛+1=0 (3.9) uniformly for all 𝑚, holds. By using Lemma 1.2, we get that lim_𝑛→∞𝑥_𝑛+𝑚−𝑥_𝑛+1=0 (3.10) uniformly for all 𝑚, holds. Then {𝑥_𝑛} is a Cauchy sequence, therefore there exists a point 𝑝∈𝐶 such that 𝑥_𝑛→𝑝.
Since 𝑥_𝑛+1=Π_𝐶𝑛𝑥₀∈𝐶_𝑛, from the definition of 𝐶_𝑛, we have 𝜙𝑥_𝑛+1,𝑦_𝑛≤1−𝛼_𝑛𝜙𝑥_𝑛+1,𝑥_𝑛+𝛼_𝑛𝜙𝑥_𝑛+1,𝑥₀. (3.11) This together with (3.9) and lim_𝑛→∞𝛼_𝑛=0 implies that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑦_𝑛=0. (3.12) Therefore, by using Lemma 1.2, we obtain lim_𝑛→∞𝑥_𝑛+1−𝑦_𝑛=0. (3.13) Since 𝑗 is uniformly norm-to-norm continuous on bounded sets, then we have lim_𝑛→∞𝐽𝑥_𝑛+1−𝐽𝑦_𝑛=lim_𝑛→∞𝐽𝑥_𝑛+1−𝐽𝑥_𝑛=0. (3.14) Noticing that 𝐽𝑥_𝑛+1−𝐽𝑦_𝑛=𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛=𝛼_𝑛𝐽𝑥_𝑛+1−𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+1−𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑥_𝑛+1−𝐽𝑧_𝑛≥𝛾_𝑛𝐽𝑥_𝑛+1−𝐽𝑧_𝑛−𝛼_𝑛𝐽𝑥_𝑛+1−𝐽𝑥₀−𝛽_𝑛𝐽𝑥_𝑛+1−𝐽𝑥_𝑛, (3.15) which leads to 𝐽𝑥_𝑛+1−𝐽𝑧_𝑛≤1𝛾_𝑛𝐽𝑥_𝑛+1−𝐽𝑦_𝑛+𝛼_𝑛𝐽𝑥₀−𝐽𝑥_𝑛+1+𝛽_𝑛𝐽𝑥_𝑛+1−𝐽𝑥_𝑛. (3.16) From (3.14) and lim_𝑛→∞𝛼_𝑛=0,0<𝛾≤𝛾_𝑛≤1, we obtain lim_𝑛→∞𝐽𝑥_𝑛+1−𝐽𝑧_𝑛=0. (3.17) Since 𝐽⁻¹ is also uniformly norm-to-norm continuous on bounded sets, then we obtain lim_𝑛→∞𝑥_𝑛+1−𝑧_𝑛=0. (3.18) Observe that 𝑥_𝑛−𝑧_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛+1−𝑧_𝑛. (3.19) It follows from (3.10) and (3.18) that 𝑧_𝑛−𝑥_𝑛𝑑𝑥→0,as𝑛→∞,_𝑛,𝑇𝑥_𝑛→0,as𝑛→∞. (3.20)(3.21) Since we have proved that 𝑥_𝑛→𝑝, which together with (3.21) and that, 𝑇 is weak relatively multivalued nonexpansive mapping, implies that 𝑝∈𝐹(𝑇).
Finally, we prove that 𝑝=Π_𝐹(𝑇)𝑥₀. From Lemma 1.4, we have 𝜙𝑝,Π_𝐹(𝑇)𝑥₀+𝜙Π_𝐹(𝑇)𝑥₀,𝑥₀≤𝜙𝑝,𝑥₀. (3.22) On the other hand, since 𝑥_𝑛+1=Π_𝐶𝑛𝑥₀ and 𝐹(𝑇)⊂𝐶_𝑛, for all 𝑛. Also from Lemma 1.4, we have 𝜙Π_𝐹(𝑇)𝑥₀,𝑥_𝑛+1+𝜙𝑥_𝑛+1,𝑥₀≤𝜙Π_𝐹(𝑇)𝑥₀,𝑥₀. (3.23) By the definition of 𝜙(𝑥,𝑦), we know that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥₀=𝜙𝑝,𝑥₀. (3.24) Combining (3.22), (3.23), and (3.24), we know that 𝜙(𝑝,𝑥₀)=𝜙(Π_𝐹(𝑇)𝑥₀,𝑥₀). Therefore, it follows from the uniqueness of Π_𝐹(𝑇)𝑥₀ that 𝑝=Π_𝐹(𝑇)𝑥₀. This completes the proof.

When 𝛼_𝑛≡0 in Theorem 3.1, we obtain the following result.

Theorem 3.2. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑇∶𝐶→𝐶 be a weak relatively multivalued nonexpansive mapping such that 𝐹(𝑇)≠∅. Assume that {𝛼_𝑛}^∞_𝑛=0 is a sequences in [0,1] such that 0≤𝛼_𝑛≤𝛼<1 for some constant 𝛼∈(0,1). Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀𝑦∈𝐶arbitrarily,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛∈𝑇𝑥_𝑛𝐶_𝑛=𝑧∈𝐶_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛𝐶,𝑛≥1,₀𝑥=𝐶,_𝑛+1=Π_𝐶𝑛𝑥₀, (3.25) then {𝑥_𝑛} converges to 𝑞=Π_𝐹(𝑇)𝑥₀.

4. Applications for Maximal Monotone Operators

In this section, we apply the above results to prove some strong convergence theorem concerning maximal monotone operators in a Banach space 𝐸.

Let 𝐴 be a multivalued operator from 𝐸 to 𝐸^∗ with domain 𝐷(𝐴)={𝑧∈𝐸∶𝐴𝑧≠∅} and range 𝑅(𝐴)={𝑧∈𝐸∶𝑧∈𝐷(𝐴)}. An operator 𝐴 is said to be monotone if 𝑥₁−𝑥₂,𝑦₁−𝑦₂≥0 (4.1) for each 𝑥₁,𝑥₂∈𝐷(𝐴) and 𝑦₁∈𝐴𝑥₁,𝑦₂∈𝐴𝑥₂. A monotone operator 𝐴 is said to be maximal if its graph 𝐺(𝐴)={(𝑥,𝑦)∶𝑦∈𝐴𝑥} is not properly contained in the graph of any other monotone operator. We know that if 𝐴 is a maximal monotone operator, then 𝐴⁻¹0 is closed and convex. The following result is also well known.

Theorem 4.1 (see Rockafellar [10]). Let 𝐸 be a reflexive, strictly convex and smooth Banach space and let 𝐴 be a monotone operator from 𝐸 to 𝐸^∗. Then 𝐴 is maximal if and only if 𝑅(𝐽+𝑟𝐴)=𝐸^∗ for all 𝑟>0.

Let 𝐸 be a reflexive, strictly convex and smooth Banach space, and let 𝐴 be a maximal monotone operator from 𝐸 to 𝐸^∗. Using Theorem 4.1 and strict convexity of 𝐸, we obtain that for every 𝑟>0 and 𝑥∈𝐸, there exists a unique 𝑥_𝑟 such that 𝐽𝑥∈𝐽𝑥_𝑟+𝑟𝐴𝑥_𝑟. (4.2) Then we can define a single valued mapping 𝐽_𝑟∶𝐸→𝐷(𝐴) by 𝐽_𝑟=(𝐽+𝑟𝐴)⁻¹𝐽 and such a 𝐽_𝑟 is called the resolvent of 𝐴. We know that 𝐴⁻¹0=𝐹(𝐽_𝑟) for all 𝑟>0, see [9, 11] for more details. Using Theorem 3.1, we can consider the problem of strong convergence concerning maximal monotone operators in a Banach space. Such a problem has been also studied in [1, 7, 10–20].

Theorem 4.2. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐴 be a maximal monotone operator from 𝐸 to 𝐸^∗ with 𝐴⁻¹0≠∅, and let 𝐽_𝑀∶𝐸→2^𝐸 be a multivalued mapping defined as follows: 𝐽_𝑀∶𝑥↦𝑦∈𝐸∶𝑦=𝐽_𝑟𝑥,𝑟∈𝑀,𝑥∈𝐸, (4.3) where 𝑀 is a set of real numbers such that inf𝑀>0. 𝐽_𝑀 is called the multivalued resolvent of 𝐴. Then 𝐽_𝑀 is a weak relatively nonexpansive multivalued mapping.

Proof. Since 𝐹𝐽_𝑀=𝑝∈𝐸∶𝐽_𝑟=𝑝=𝑝,𝑟∈𝑀𝑝∈𝐸∶𝑝∈𝐴⁻¹0,𝑟∈𝑀=𝐴⁻¹0, (4.4) so that 𝐹(𝐽_𝑀)≠∅. Next, we show 𝜙𝑝,𝑧≤𝜙𝑝,𝑥,∀𝑝∈𝐹𝐽_𝑀,∀𝑥∈𝐸,∀𝑧∈𝐽_𝑀𝑥. (4.5) From the monotonicity of 𝐴, we have 𝜙𝑝,𝑧=𝑝²−2𝑝,𝐽𝑧+‖𝑧‖²=𝑝²+2𝑝,𝐽𝑥−𝐽𝑧−𝐽𝑥+𝐽_𝑟𝑤²=𝑝²+2𝑝,𝐽𝑥−𝐽𝑧−2𝑝,𝐽𝑥+‖𝑧‖²=𝑝²−2𝑧−𝑝−𝑧,𝐽𝑥−𝐽𝑧−𝐽𝑥−2𝑝,𝐽𝑥+‖𝑧‖²=𝑝²⟨⟩+‖𝑧‖−2𝑧−𝑝,𝐽𝑥−𝐽𝑧−𝐽𝑥+2𝑧,𝐽𝑥−𝐽𝑧−2𝑝,𝐽𝑥²≤𝑝²+2⟨𝑧,𝐽𝑥−𝐽𝑧⟩−2𝑝,𝐽𝑥+‖𝑧‖²=𝑝²−2𝑝,𝐽𝑥+‖𝑥‖²−‖𝑧‖²+2⟨𝑧,𝐽𝑥⟩−‖𝑤‖²().=𝜙𝑝,𝑥−𝜙𝑧,𝑥≤𝜙𝑝,𝑥 (4.6) Finally, we show 𝐹(𝐽_𝑀)=𝐹(𝐽_𝑀). Observe that 𝐹(𝐽_𝑀)⊇𝐹(𝐽_𝑀) is obvious. Next, we show 𝐹(𝐽_𝑀)⊆𝐹(𝐽_𝑀). Let {𝑧_𝑛}⊂𝐸 be a sequence such that 𝑧_𝑛→𝑝 and lim_𝑛→∞𝑑(𝑧_𝑛,𝐽_𝑀𝑧_𝑛)=0. There exist sequences {𝑦_𝑛} in 𝐸 and {𝑟_𝑛}⊆𝑀 such that lim_𝑛→∞𝑧_𝑛−𝑦_𝑛=0,𝑦_𝑛=𝐽_𝑟𝑛𝑧_𝑛. (4.7) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we obtain 1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝑦_𝑛→0. (4.8) It follows from 1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝑦_𝑛∈𝐴𝐽_𝑟𝑛𝑧_𝑛 (4.9) and the monotonicity of 𝐴 that 𝑤−𝐽_𝑟𝑛𝑧_𝑛,𝑤^∗−1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛≥0 (4.10) for all 𝑤∈𝐷(𝐴) and 𝑤^∗∈𝐴𝑤. Letting 𝑛→∞, we have ⟨𝑤−𝑝,𝑤^∗⟩≥0 for all 𝑤∈𝐷(𝐴) and 𝑤^∗∈𝐴𝑤. Therefore, from the maximality of 𝐴, we obtain 𝑝∈𝐴⁻¹0=𝐹(𝐽_𝑀). Hence 𝐽_𝑀 is a weak relatively nonexpansive multivalued mapping. This completes the proof.

By using Theorems 3.1 and 4.2, we directly obtain the following result.

Theorem 4.3. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐴 be a maximal monotone operator from 𝐸 to 𝐸^∗ with 𝐴⁻¹0≠∅, let 𝐽_𝑀 be a multivalued resolvent of 𝐴, where inf𝑀>0, and let {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} be three sequences of real numbers such that 𝛼_𝑛→0 and 0<𝛾≤𝛾_𝑛≤1 for some constant 𝛾∈(0,1). Define a sequence {𝑥_𝑛} of 𝐶 as follows: 𝑥₀𝑦∈𝐶arbitrarily,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛,𝑧_𝑛∈𝐽_𝑀𝑥_𝑛𝐶_𝑛=𝑧∈𝐶_𝑛−1∶𝜙𝑧,𝑦_𝑛≤1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝜙𝑧,𝑥₀𝐶,𝑛≥1,₀𝑥=𝐶,_𝑛+1=Π_𝐶𝑛𝑥₀, (4.11) where 𝐽 is the duality mapping on 𝐸. Then {𝑥_𝑛} converges strongly to Π_𝐴⁻¹₀𝑥₀, where Π_𝐴⁻¹₀ is the generalized projection from 𝐸 onto 𝐴⁻¹0.

Acknowledgment

This project is supported by the National Natural Science Foundation of China under Grant (11071279).