An Implicit Algorithm for Maximal Monotone Operators and Pseudocontractive Mappings

Abstract

The purpose of this paper is to construct an implicit algorithm for finding the common solution of maximal monotone operators and strictly pseudocontractive mappings in Hilbert spaces. Some applications are also included.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻.

Recall that 𝑆 is said to be a strictly pseudo contractive mapping if there exists a constant 0≤𝜌<1 such that𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+𝜌(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (1.1) For such case, we also say that 𝑆 is a 𝜌-strictly pseudo-contractive mapping. When 𝜌=0, 𝑇 is said to be nonexpansive. It is clear that (1.1) is equivalent to𝑆𝑥−𝑆𝑦,𝑥−𝑦≤𝑥−𝑦²−1−𝜌2(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (1.2) We denote by 𝐹(𝑆) the set of fixed points of 𝑆.

A mapping 𝐴∶𝐶→𝐻 is said to be 𝛼-inverse strongly monotone if𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦², (1.3) for some 𝛼>0 and for all 𝑥, 𝑦∈𝐶. It is known that if 𝐴 is an 𝛼-inverse strongly monotone, then ‖𝐴𝑥−𝐴𝑦‖≤1/𝛼‖𝑥−𝑦‖for all 𝑥, 𝑦∈𝐶.

Let 𝐵 be a mapping of 𝐻 into 2^𝐻. The effective domain of 𝐵 is denoted by dom(𝐵), that is, dom(𝐵)={𝑥∈𝐻∶𝐵𝑥≠∅}. A multi valued mapping 𝐵 is said to be a monotone operator on 𝐻 iff𝑥−𝑦,𝑢−𝑣≥0, (1.4) for all 𝑥, 𝑦∈dom(𝐵), 𝑢∈𝐵𝑥, and 𝑣∈𝐵𝑦. A monotone operator 𝐵 on 𝐻 is said to be maximal if its graph is not strictly contained in the graph of any other monotone operator on 𝐻. Let 𝐵 be a maximal monotone operator on 𝐻, and let 𝐵⁻¹0={𝑥∈𝐻∶0∈𝐵𝑥}.

For a maximal monotone operator 𝐵 on 𝐻 and 𝜆>0, we may define a single-valued operator 𝐽^𝐵_𝜆=(𝐼+𝜆𝐵)⁻¹∶𝐻→dom(𝐵), which is called the resolvent of 𝐵 for 𝜆. It is known that the resolvent 𝐽^𝐵_𝜆 is firmly nonexpansive, that is,𝐽^𝐵_𝜆𝑥−𝐽^𝐵_𝜆𝑦²≤𝐽^𝐵_𝜆𝑥−𝐽^𝐵_𝜆𝑦,𝑥−𝑦, (1.5) for all 𝑥, 𝑦∈𝐶 and 𝐵⁻¹0=𝐹(𝐽^𝐵_𝜆) for all 𝜆>0.

Algorithms for finding the fixed points of nonlinear mappings or for finding the zero points of maximal monotone operators have been studied by many authors. The reader can refer to [1–24]. Especially, Takahashi et al. [6] recently gave the following convergence result.

Theorem 1.1. Let 𝐶 be a closed and convex subset of a real Hilbert space 𝐻. Let 𝐴 be an 𝛼-inverse strongly monotone mapping of 𝐶 into 𝐻 and let 𝐵 be a maximal monotone operator on 𝐻, such that the domain of 𝐵 is included in 𝐶. Let 𝐽^𝐵_𝜆=(𝐼+𝜆𝐵)⁻¹ be the resolvent of 𝐵 for 𝜆>0, and let 𝑆 be a nonexpansive mapping of 𝐶 into itself, such that 𝐹(𝑆)∩(𝐴+𝐵)⁻¹0≠∅. Let 𝑥₁=𝑥∈𝐶 and let {𝑥_𝑛}⊂𝐶, be a sequence generated by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝛼_𝑛𝑥+1−𝛼_𝑛𝐽^𝐵_𝜆𝑛𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛, (1.6) for all 𝑛≥0, where {𝜆_𝑛}⊂(0,2𝛼), {𝛼_𝑛}⊂(0,1) and {𝛽_𝑛}⊂(0,1) satisfy 0<𝑎≤𝜆_𝑛≤𝑏<2𝛼,<𝑐≤𝛽_𝑛≤𝑑<1,lim_𝑛→∞𝜆_𝑛+1−𝜆_𝑛=0,lim_𝑛→∞𝛼_𝑛=0,_𝑛𝛼_𝑛=∞, (1.7) then {𝑥_𝑛} generated by (1.6) converges strongly to a point of 𝐹(𝑆)∩(𝐴+𝐵)⁻¹0.

Motivated and inspired by the works in this field, the purpose of this paper is to construct an implicit algorithm for finding the common solution of maximal monotone operators and strictly-pseudocontractive mappings in Hilbert spaces. Some applications are also included.

2. Preliminaries

The following resolvent identity is well known: for 𝜆>0 and 𝜇>0, there holds the identity𝐽^𝐵_𝜆𝑥=𝐽^𝐵_𝜇𝜇𝜆𝜇𝑥+1−𝜆𝐽^𝐵_𝜆𝑥,𝑥∈𝐻. (2.1) We use the following notation:(i)𝑥_𝑛⇀𝑥 stands for the weak convergence of {𝑥_𝑛} to 𝑥;(ii)𝑥_𝑛→𝑥 stands for the strong convergence of {𝑥_𝑛} to 𝑥.

We need the following lemmas for the next section.

Lemma 2.1 (see[14]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑆∶𝐶→𝐻 be a 𝜌-strict pseudo contraction. Define 𝑇∶𝐶→𝐻 by 𝑇𝑥=𝛼𝑥+(1−𝛼)𝑇𝑥 for each 𝑥∈𝐶, then, as 𝛼∈[𝜌,1), 𝑇 is nonexpansive such that 𝐹(𝑆)=𝐹(𝑇).

Lemma 2.2 (see[15]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let the mapping 𝐴∶𝐶→𝐻 be 𝛼-inverse strongly monotone and 𝜆>0 a constant, then one has (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦²≤𝑥−𝑦²+𝜆(𝜆−2𝛼)𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (2.2) In particular, if 0≤𝜆≤2𝛼, then 𝐼−𝜆𝐴 is nonexpansive.

Lemma 2.3 (see[14]). Let 𝐶 be a nonempty, closed and convex of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a 𝜆-strictly pseudo-contractive mapping, then 𝐼−𝑇 is demi closed at 0, that is, if 𝑥_𝑛⇀𝑥∈𝐶 and 𝑥_𝑛−𝑇𝑥_𝑛→0, then 𝑥=𝑇𝑥.

Lemma 2.4 (see[16]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋, and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all 𝑛≥0 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0, then lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.5 (see[17]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛𝛾_𝑛, (2.3) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that(1)∑^∞_𝑛=1𝛾_𝑛=∞,(2)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=1|𝛿_𝑛𝛾_𝑛|<∞, then lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

In this section, we will prove our main results.

Theorem 3.1. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐴 be an 𝛼-inverse strongly monotone mapping of 𝐶 into 𝐻, and let 𝐵 be a maximal monotone operator on 𝐻, such that the domain of 𝐵 is included in 𝐶. Let 𝐽^𝐵_𝜆=(𝐼+𝜆𝐵)⁻¹ be the resolvent of 𝐵 for 𝜆 which satisfies 𝑎≤𝜆≤𝑏 where [𝑎,𝑏]⊂(0,2𝛼). Let 𝜅∈(0,1) be a constant and 𝑆∶𝐶→𝐶 a 𝜌-strict pseudocontraction with 𝜌∈[0,1) such that 𝐹(𝑆)∩(𝐴+𝐵)⁻¹0≠∅. For 𝑡∈(0,1−𝜆/2𝛼), let {𝑥_𝑡}⊂𝐶 be a net defined by 𝑥_𝑡=𝜅1−𝜌1−𝜅𝜌𝑆𝑥_𝑡+1−𝜅𝐽1−𝜅𝜌^𝐵𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡, (3.1) then the net {𝑥_𝑡} converges strongly, as 𝑡→0+, to a point 𝑥=𝑃_{𝐹(𝑆)∩(𝐴+𝐵)}⁻¹₀(0), where 𝑃 is the metric projection.

Remark 3.2. Now, we show that the net {𝑥_𝑡} defined by (3.1) is well defined. For any 𝑡∈(0,1−𝜆/2𝛼), we define a mapping 𝑊∶=𝜅(𝜌𝐼+(1−𝜌)𝑆)+(1−𝜅)𝐽^𝐵_𝜆((1−𝑡)𝐼−𝜆𝐴). Note that 𝜌𝐼+(1−𝜌)𝑆 (by Lemma 2.1), 𝐽^𝐵_𝜆, and 𝐼−𝜆/(1−𝑡)𝐴 (by Lemma 2.2) are nonexpansive. For any 𝑥, 𝑦∈𝐶, we have 𝑊𝑥−𝑊𝑦=𝜅𝜌𝑥+1−𝜌𝑆𝑥+(1−𝜅)𝐽^𝐵_𝜆))−()𝐽((1−𝑡𝑥−𝜆𝐴𝑥−𝜅𝜌𝑦+1−𝜌𝑆𝑦1−𝜅^𝐵_𝜆()𝜌++()𝜆1−𝑡𝑦−𝜆𝐴𝑦≤𝜅𝑥−𝑦1−𝜌𝑆𝑥−𝑆𝑦1−𝜅)(1−𝑡𝑥−−()𝜆1−𝑡𝐴𝑥1−𝑡𝑦−≤[()𝑡,1−𝑡𝐴𝑦1−1−𝜅]𝑥−𝑦 (3.2) which implies the mapping 𝑇 is a contraction on 𝐶. We use 𝑥_𝑡 to denote the unique fixed point of 𝑊 in 𝐶. Therefore, {𝑥_𝑡} is well defined. We can rewrite (3.1) as 𝑥_𝑡=𝜅𝜌𝑥_𝑡+1−𝜌𝑆𝑥_𝑡+(1−𝜅)𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡. (3.3)

In order to prove Theorem 3.1, we need the following propositions.

Proposition 3.3. Under the assumptions of Theorem 3.1, the net {𝑥_𝑡} defined by (3.1) and hence (3.3) is bounded.

Proof. Let 𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹0. It follows that 𝑧=𝑆𝑧=𝜌𝑧+1−𝜌𝑆𝑧=𝐽^𝐵_𝜆(𝑧−𝜆𝐴𝑧), (3.4) for all 𝜆>0. We can write 𝐽^𝐵_𝜆(𝑧−𝜆𝐴𝑧) as 𝐽^𝐵_𝜆(𝑡𝑧+(1−𝑡)(𝑧−𝜆𝐴𝑧/(1−𝑡))), for all 𝑡∈(0,1). Since 𝐽^𝐵_𝜆 is nonexpansive for all 𝜆>0, we have 𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²=𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡/(1−𝑡)−𝐽^𝐵_𝜆(𝑡𝑧+(1−𝑡)(𝑧−𝜆𝐴𝑧/(1−𝑡)))²≤(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡/(1−𝑡)−(𝑡𝑧+(1−𝑡)(𝑧−𝜆𝐴𝑧/(1−𝑡)))²=(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡/(1−𝑡)−(𝑧−𝜆𝐴𝑧/(1−𝑡))+𝑡(−𝑧)². (3.5) By using the convexity of ‖⋅‖ and the 𝛼-inverse strong monotonicity of 𝐴, we derive (1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡/(1−𝑡)−(𝑧−𝜆𝐴𝑧/(1−𝑡))+𝑡(−𝑧)²≤(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡/(1−𝑡)−(𝑧−𝜆𝐴𝑧/(1−𝑡))²+𝑡‖𝑧‖²=(1−𝑡)𝑥_𝑡−𝑧−𝜆𝐴𝑥_𝑡−𝐴𝑧/(1−𝑡)²+𝑡‖𝑧‖²=(1−𝑡)𝑥_𝑡−𝑧²−2𝜆1−𝑡𝐴𝑥_𝑡−𝐴𝑧,𝑥_𝑡+𝜆−𝑧²(1−𝑡)²𝐴𝑥_𝑡−𝐴𝑧²+𝑡‖𝑧‖²≤(1−𝑡)𝑥_𝑡−𝑧²−2𝛼𝜆1−𝑡𝐴𝑥_𝑡−𝐴𝑧²+𝜆²(1−𝑡)²𝐴𝑥_𝑡−𝐴𝑧²+𝑡‖𝑧‖²=(1−𝑡)𝑥_𝑡−𝑧²+𝜆(1−𝑡)²(𝜆−2(1−𝑡)𝛼)𝐴𝑥_𝑡−𝐴𝑧²+𝑡‖𝑧‖². (3.6) By the assumption, we have 𝜆−2(1−𝑡)𝛼≤0, for all 𝑡∈(0,1−𝜆/2𝛼). Then, from (3.5) and (3.6), we obtain 𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²≤(1−𝑡)𝑥_𝑡−𝑧²+𝜆(1−𝑡)²(𝜆−2(1−𝑡)𝛼)𝐴𝑥_𝑡−𝐴𝑧²+𝑡‖𝑧‖²≤(1−𝑡)𝑥_𝑡−𝑧²+𝑡‖𝑧‖². (3.7) It follows from (3.3) and (3.7) that 𝑥_𝑡−𝑧²≤𝜅(𝜌𝐼+(1−𝜌)𝑆)𝑥_𝑡−𝑧²+(1−𝜅)𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²≤𝜅𝑥_𝑡−𝑧²+(1−𝜅)𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²≤𝜅𝑥_𝑡−𝑧²+(1−𝜅)(1−𝑡)𝑥_𝑡−𝑧²+𝑡‖𝑧‖². (3.8) It follows that 𝑥_𝑡−𝑧≤‖𝑧‖. (3.9) Therefore, {𝑥_𝑡} is bounded.

Remark 3.4. Since 𝐴 is 𝛼-inverse strongly monotone, it is 1/𝛼-Lipschitz continuous. At the same time, 𝑆 is nonexpansive. So, from the boundedness, we deduce immediately that {𝐴𝑥_𝑡}, 𝐽^𝐵_𝜆((1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡), and {𝑆𝑥_𝑡} are also bounded.

Proposition 3.5. Assume that all conditions in Theorem 3.1 hold. Let {𝑥_𝑡} be the net defined by (3.1), then one has lim_𝑡→0+‖𝑥_𝑡−𝑆𝑥_𝑡‖=0 and lim_𝑡→0+‖𝑥_𝑡−𝐽^𝐵_𝜆((1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡)‖=0.

Proof. By (3.7) and (3.8), we obtain 𝑥_𝑡−𝑧²≤[1−(1−𝜅)𝑡]𝑥_𝑡−𝑧²+𝜆(1−𝜅)()(()𝛼1−𝑡𝜆−21−𝑡)𝐴𝑥_𝑡−𝐴𝑧²+(1−𝜅)𝑡‖𝑧‖². (3.10) So, 𝜆()(2()1−𝑡1−𝑡𝛼−𝜆)𝐴𝑥_𝑡−𝐴𝑧²≤𝑡‖𝑧‖²−𝑡𝑥_𝑡−𝑧²⟶0. (3.11) Since liminf_𝑡→0+(𝜆/(1−𝑡))(2(1−𝑡)𝛼−𝜆)>0, we obtain lim_𝑡→0+𝐴𝑥_𝑡−𝐴𝑧=0. (3.12) Next, we show ‖𝑥_𝑡−𝑆𝑥_𝑡‖→0. By using the firm nonexpansivity of 𝐽^𝐵_𝜆, we have 𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²=𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝐽^𝐵_𝜆(𝑧−𝜆𝐴𝑧)²≤(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−(𝑧−𝜆𝐴𝑧),𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡=1−𝑧2(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−(𝑧−𝜆𝐴𝑧)²+𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²−(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝜆𝐴𝑧−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡². (3.13) By the nonexpansivity of 𝐼−𝜆𝐴/(1−𝑡), we have (1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−(𝑧−𝜆𝐴𝑧)²=(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡/(1−𝑡)−(𝑧−𝜆𝐴𝑧/(1−𝑡))+𝑡(−𝑧)²≤(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡/(1−𝑡)−(𝑧−𝜆𝐴𝑧/(1−𝑡))²+𝑡‖𝑧‖²≤(1−𝑡)𝑥_𝑡−𝑧²+𝑡‖𝑧‖². (3.14) It follows that 𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²≤12(1−𝑡)𝑥_𝑡−𝑧²+𝑡‖𝑧‖²+𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²−(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝐴𝑧². (3.15) Thus, 𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²≤(1−𝑡)𝑥_𝑡−𝑧²+𝑡‖𝑧‖²−(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝐴𝑧²=(1−𝑡)𝑥_𝑡−𝑧²+𝑡‖𝑧‖²−(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡²+2𝜆(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡,𝐴𝑥_𝑡−𝐴𝑧−𝜆²𝐴𝑥_𝑡−𝐴𝑧²≤(1−𝑡)𝑥_𝑡−𝑧²+𝑡‖𝑧‖²−(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡²+2𝜆(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡𝐴𝑥_𝑡−𝐴𝑧. (3.16) This together with (3.8) implies that 𝑥_𝑡−𝑧²≤𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡−𝑧²≤(1−𝑡)𝑥_𝑡−𝑧²+𝑡‖𝑧‖²−(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡²+2𝜆(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡𝐴𝑥_𝑡−𝐴𝑧. (3.17) Hence, (1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡²≤𝑡‖𝑧‖²−𝑥_𝑡−𝑧²+2𝜆(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡𝐴𝑥_𝑡−𝐴𝑧. (3.18) Since ‖𝐴𝑥_𝑡−𝐴𝑧‖→0 (by (3.12)), we deduce lim_𝑡→0+(1−𝑡)𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡=0. (3.19) Therefore, lim_𝑡→0+𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡=0. (3.20) Hence, lim_𝑡→0+𝑥_𝑡−𝑆𝑥_𝑡=lim_𝑡→0+𝑥_𝑡−𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡=0. (3.21)

Finally, we prove Theorem 3.1.

Proof. From (3.5) and (3.8), we have 𝑥_𝑡−𝑧²≤(1−𝑡)𝑥_𝑡−𝜆1−𝑡𝐴𝑥_𝑡−𝜆𝑧−1−𝑡𝐴𝑧−𝑡𝑧²=(1−𝑡)²𝑥_𝑡−𝜆1−𝑡𝐴𝑥_𝑡−𝜆𝑧−1−𝑡𝐴𝑧²−2𝑡(1−𝑡)𝑧,𝑥_𝑡−𝜆1−𝑡𝐴𝑥_𝑡−𝜆𝑧−1−𝑡𝐴𝑧+𝑡²‖𝑧‖²≤(1−𝑡)²𝑥_𝑡−𝑧²−2𝑡(1−𝑡)𝑧,𝑥_𝑡−𝜆1−𝑡𝐴𝑥_𝑡−𝐴𝑧−𝑧+𝑡²‖𝑧‖²=(1−2𝑡)𝑥_𝑡−𝑧²+2𝑡−(1−𝑡)𝑧,𝑥_𝑡−𝜆1−𝑡𝐴𝑥_𝑡+𝑡−𝐴𝑧−𝑧2‖𝑧‖²+𝑥_𝑡−𝑧². (3.22) It follows that 𝑥_𝑡−𝑧²≤−𝑧,𝑥_𝑡−𝜆1−𝑡𝐴𝑥_𝑡+𝑡−𝐴𝑧−𝑧2‖𝑧‖²+𝑥_𝑡−𝑧²+𝑡‖𝑧‖𝑥_𝑡−𝜆1−𝑡𝐴𝑥_𝑡−𝐴𝑧−𝑧≤−𝑧,𝑥_𝑡−𝜆1−𝑡𝐴𝑥_𝑡−𝐴𝑧−𝑧+𝑡𝑀, (3.23) where 𝑀 is some constant such that sup_{()𝑡∈0,1−𝜆/2𝛼}‖𝑧‖²+𝑥_𝑡−𝑧²+‖𝑧‖𝑥_𝑡−𝜆1−𝑡𝐴𝑥_𝑡−𝐴𝑧−𝑧≤𝑀. (3.24) Next we show that {𝑥_𝑡} is relatively norm compact as 𝑡→0+. Assume that {𝑡_𝑛}⊂(0,1−𝜆/2𝛼) is such that 𝑡_𝑛→0+ as 𝑛→∞. Put 𝑥_𝑛∶=𝑥_𝑡𝑛. From (3.23), we have 𝑥_𝑛−𝑧²≤−𝑧,𝑥_𝑛−𝜆1−𝑡_𝑛𝐴𝑥_𝑛−𝐴𝑧−𝑧+𝑡_𝑛𝑀,𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹0. (3.25) Since {𝑥_𝑛} is bounded, without loss of generality, we may assume that 𝑥_𝑛⇀𝑥∈𝐶. From (3.21), we have lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (3.26) We can apply Lemma 2.3 to (3.26) to deduce 𝑥∈𝐹(𝑆). Further, we show that 𝑥 is also in (𝐴+𝐵)⁻¹0. Let 𝑣∈𝐵𝑢. Set 𝑧_𝑛=𝐽^𝐵_𝜆((1−𝑡_𝑛)𝑥_𝑛−𝜆𝐴𝑥_𝑛), for all 𝑛, then we have 1−𝑡_𝑛𝑥_𝑛−𝜆𝐴𝑥_𝑛∈(𝐼+𝜆𝐵)𝑧_𝑛⟹1−𝑡_𝑛𝜆𝑥_𝑛−𝐴𝑥_𝑛−𝑧_𝑛𝜆∈𝐵𝑧_𝑛. (3.27) Since 𝐵 is monotone, we have, for (𝑢,𝑣)∈𝐵, 1−𝑡_𝑛𝜆𝑥_𝑛−𝐴𝑥_𝑛−𝑧_𝑛𝜆−𝑣,𝑧_𝑛⟹−𝑢≥01−𝑡_𝑛𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑧_𝑛−𝜆𝑣,𝑧_𝑛⟹−𝑢≥0𝐴𝑥_𝑛+𝑣,𝑧_𝑛≤1−𝑢𝜆𝑥_𝑛−𝑧_𝑛,𝑧_𝑛−𝑡−𝑢^𝑛𝜆𝑥_𝑛,𝑧_𝑛⟹−𝑢𝐴𝑥+𝑣,𝑧_𝑛≤1−𝑢𝜆𝑥_𝑛−𝑧_𝑛,𝑧_𝑛−𝑡−𝑢^𝑛𝜆𝑥_𝑛,𝑧_𝑛−𝑢+𝐴𝑥−𝐴𝑥_𝑛,𝑧_𝑛⟹−𝑢𝐴𝑥+𝑣,𝑧_𝑛≤1−𝑢𝜆𝑥_𝑛−𝑧_𝑛𝑧_𝑛+𝑡−𝑢^𝑛𝜆𝑥_𝑛𝑧_𝑛−𝑢+𝐴𝑥−𝐴𝑥_𝑛𝑧_𝑛−𝑢. (3.28) It follows that 1⟨𝐴𝑥+𝑣,𝑥−𝑢⟩≤𝜆𝑥_𝑛−𝑧_𝑛𝑧_𝑛+𝑡−𝑢^𝑛𝜆𝑥_𝑛𝑧_𝑛+−𝑢𝐴𝑥−𝐴𝑥_𝑛𝑧_𝑛−𝑢+𝐴𝑥+𝑣,𝑥−𝑧_𝑛. (3.29) Since 𝑥_𝑛−𝑥,𝐴𝑥_𝑛−𝐴𝑥≥𝛼𝐴𝑥_𝑛−𝐴𝑥², (3.30)𝐴𝑥_𝑛→𝐴𝑧, and 𝑥_𝑛⇀𝑥, we have 𝐴𝑥_𝑛→𝐴𝑥. We also observe that 𝑡_𝑛→0, ‖𝑥_𝑛−𝑧_𝑛‖→0 and 𝑧_𝑛⇀𝑥. Then, from (3.29), we derive ⟨−𝐴𝑥−𝑣,𝑥−𝑢⟩≥0. (3.31) Since 𝐵 is maximal monotone, we have −𝐴𝑥∈𝐵𝑥. This shows that 0∈(𝐴+𝐵)𝑥. So, we have 𝑥∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹0. Hence, 𝑥_𝑛−(𝜆/1−𝑡_𝑛)(𝐴𝑥_𝑛−𝐴𝑧)⇀𝑥 because of ‖𝐴𝑥_𝑛−𝐴𝑧‖→0. Therefore, we can substitute 𝑥 for 𝑧 in (3.25) to get 𝑥_𝑛−𝑥²≤−𝑥,𝑥_𝑛−𝜆1−𝑡_𝑛𝐴𝑥_𝑛−𝐴𝑥−𝑥+𝑡_𝑛𝑀. (3.32) Consequently, the weak convergence of {𝑥_𝑛} to 𝑥 actually implies that 𝑥_𝑛→𝑥. This has proved the relative norm compactness of the net {𝑥_𝑡} as 𝑡→0+.
Now we return to (3.25) and take the limit as 𝑛→∞ to get ‖𝑥−𝑧‖²≤−⟨𝑧,𝑥−𝑧⟩,𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹0. (3.33) Equivalently, ‖𝑥‖²≤⟨𝑥,𝑧⟩,𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹0. (3.34) This clearly implies that ‖𝑥‖≤‖𝑧‖,𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹0. (3.35) Therefore, 𝑥 is the minimum norm element in 𝐹(𝑆)∩(𝐴+𝐵)⁻¹0. This completes the proof.

Corollary 3.6. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐴 be an 𝛼-inverse strongly monotone mapping of 𝐶 into H, and let 𝐵 be a maximal monotone operator on 𝐻, such that the domain of 𝐵 is included in 𝐶. Let 𝐽^𝐵_𝜆=(𝐼+𝜆𝐵)⁻¹ be the resolvent of 𝐵 for 𝜆 which satisfies 𝑎≤𝜆≤𝑏 where [𝑎,𝑏]⊂(0,2𝛼). Let 𝜅∈(0,1) be a constant and 𝑆∶𝐶→𝐶 a nonexpansive mapping such that 𝐹(𝑆)∩(𝐴+𝐵)⁻¹0≠∅. For 𝑡∈(0,1−𝜆/2𝛼), let {𝑥_𝑡}⊂𝐶 be a net defined by 𝑥_𝑡=𝜅1−𝜌1−𝜅𝜌𝑆𝑥_𝑡+1−𝜅𝐽1−𝜅𝜌^𝐵𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡, (3.36) then the net {𝑥_𝑡} converges strongly, as 𝑡→0+, to a point 𝑥=𝑃_{𝐹(𝑆)∩(𝐴+𝐵)}⁻¹₀(0).

Corollary 3.7. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐴 be an 𝛼-inverse strongly monotone mapping of 𝐶 into H, and let 𝐵 be a maximal monotone operator on 𝐻, such that the domain of 𝐵 is included in 𝐶. Let 𝐽^𝐵_𝜆=(𝐼+𝜆𝐵)⁻¹ be the resolvent of 𝐵 for 𝜆>0 such that (𝐴+𝐵)⁻¹0≠∅. Let 𝜆 be a constant satisfying 𝑎≤𝜆≤𝑏 where [𝑎,𝑏]⊂(0,2𝛼). For 𝑡∈(0,1−𝜆/2𝛼), let {𝑥_𝑡}⊂𝐶 be a net generated by 𝑥_𝑡=𝐽^𝐵_𝜆(1−𝑡)𝑥_𝑡−𝜆𝐴𝑥_𝑡, (3.37) then the net {𝑥_𝑡} converges strongly, as 𝑡→0+, to a point 𝑥=𝑃_(𝐴+𝐵)⁻¹₀(0).

4. Applications

Next, we consider the problem for finding the minimum norm solution of a mathematical model related to equilibrium problems. Let 𝐶 be a nonempty, closed, and convex subset of a Hilbert space, and let 𝐺∶𝐶×𝐶→𝑅 be a bifunction satisfying the following conditions:(E1)𝐺(𝑥,𝑥)=0, for all 𝑥∈𝐶,(E2)𝐺 is monotone, that is, 𝐺(𝑥,𝑦)+𝐺(𝑦,𝑥)≤0, for all 𝑥, 𝑦∈𝐶,(E3) for all 𝑥, 𝑦, 𝑧∈𝐶, limsup_𝑡↓0𝐺(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐺(𝑥,𝑦),(E4) for all 𝑥∈𝐶, 𝐺(𝑥,⋅) is convex and lower semicontinuous.

Then, the mathematical model related to equilibrium problems (with respect to 𝐶) is to find 𝑥∈𝐶 such that𝐺𝑥,𝑦≥0, (4.1) for all 𝑦∈𝐶. The set of such solutions 𝑥 is denoted by 𝐸𝑃(𝐺). The following lemma appears implicitly in Blum and Oettli [19].

Lemma 4.1. Let 𝐶 be a nonempty, closed, and convex subset of 𝐻, and let 𝐺 be a bifunction of 𝐶×𝐶 into 𝑅 satisfying (E1)–(E4). Let 𝑟>0 and 𝑥∈𝐻, then there exists 𝑧∈𝐶 such that 𝐺+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (4.2)

The following lemma was given in Combettes and Hirstoaga [20].

Lemma 4.2. Assume that 𝐺∶𝐶×𝐶→𝑅 satisfies (E1)–(E4). For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝐺𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶, (4.3) for all 𝑥∈𝐻. Then, the following hold:(1)𝑇_𝑟 is single valued,(2)𝑇_𝑟 is a firmly nonexpansive mapping, that is, for all 𝑥,𝑦∈𝐻, 𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦, (4.4)(3)𝐹(𝑇_𝑟)=𝐸𝑃(𝐺), (4)𝐸𝑃(𝐺) is closed and convex.
We call such 𝑇_𝑟 the resolvent of 𝐺 for 𝑟>0. Using Lemmas 4.1 and 4.2, we have the following lemma. See [18] for a more general result.

Lemma 4.3. Let 𝐻 be a Hilbert space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐻. Let 𝐺∶𝐶×𝐶→𝑅 satisfy (E1)–(E4). Let 𝐴_𝐺 be a multivalued mapping of 𝐻 into itself defined by 𝐴_𝐺≥𝑥=𝑧∈𝐻∶𝐺𝑥,𝑦𝑦−𝑥,𝑧,∀𝑦∈𝐶,𝑥∈𝐶,∅,𝑥∉𝐶, (4.5) then, 𝐸𝑃(𝐺)=𝐴_𝐺⁻¹(0), and 𝐴_𝐺 is a maximal monotone operator with 𝑑𝑜𝑚(𝐴_𝐺)⊂𝐶. Further, for any 𝑥∈𝐻 and 𝑟>0, the resolvent 𝑇_𝑟 of 𝐺 coincides with the resolvent of 𝐴_𝐺, that is, 𝑇_𝑟𝑥=𝐼+𝑟𝐴_𝐺⁻¹𝑥. (4.6)
From Lemma 4.3, Theorem 3.1, and Lemma 4.2, one has the following results.

Corollary 4.4. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝐺 be a bifunction from 𝐶×𝐶→𝑅 satisfying (E1)–(E4), and let 𝑇_𝑟 be the resolvent of 𝐺 for 𝑟>0. Let 𝜅∈(0,1) be a constant and 𝑆∶𝐶→𝐶 a 𝜌-strict pseudocontraction with 𝜌∈[0,1) such that 𝐹(𝑆)∩𝐸𝑃(𝐺)≠∅. For 𝑡∈(0,1), let {𝑥_𝑡}⊂𝐶 be a net defined by 𝑥_𝑡=𝜅1−𝜌1−𝜅𝜌𝑆𝑥_𝑡+1−𝜅𝑇1−𝜅𝜌^𝑟(1−𝑡)𝑥_𝑡, (4.7) then the net {𝑥_𝑡} converges strongly, as 𝑡→0+, to a point 𝑥=𝑃_{𝐹(𝑆)∩𝐸𝑃(𝐺)}(0).

Corollary 4.5. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝐺 be a bifunction from 𝐶×𝐶→𝑅 satisfying (E1)–(E4), and let 𝑇_𝑟 be the resolvent of 𝐺 for 𝑟>0. Let 𝜅∈(0,1) be a constant and 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹(𝑆)∩𝐸𝑃(𝐺)≠∅. For 𝑡∈(0,1), let {𝑥_𝑡}⊂𝐶 be a net defined by 𝑥_𝑡=𝜅1−𝜌1−𝜅𝜌𝑆𝑥_𝑡+1−𝜅𝑇1−𝜅𝜌^𝑟(1−𝑡)𝑥_𝑡, (4.8) then the net {𝑥_𝑡} converges strongly, as 𝑡→0+, to a point 𝑥=𝑃_{𝐹(𝑆)∩𝐸𝑃(𝐺)}(0).

Corollary 4.6. Let 𝐶 be a nonempty, closed and convex subset of a real Hilbert space 𝐻. Let 𝐺 be a bifunction from 𝐶×𝐶→𝑅 satisfying (E1)–(E4), and let 𝑇_𝑟 be the resolvent of 𝐺 for 𝑟>0. Suppose 𝐸𝑃(𝐺)≠∅. For 𝑡∈(0,1), let {𝑥_𝑡}⊂𝐶 be a net generated by 𝑥_𝑡=𝑇_𝑟(1−𝑡)𝑥_𝑡,𝑡∈(0,1), (4.9) then the net {𝑥_𝑡} converges strongly, as 𝑡→0+, to a point 𝑥=𝑃_{𝐸𝑃(𝐺)}(0).

Acknowledgments

H.-J. Li was supported in part by Colleges and Universities Science and Technology Development Foundation (20110322) of Tianjin. Y.-C. Liou was supported in part by NSC 100-2221-E-230-012. C.-L. Li was supported in part by NSFC 71161001-G0105. Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105.