Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article
Special Issue

Applications of Fixed Point and Approximate Algorithms

View this Special Issue

Research Article | Open Access

Volume 2012 |Article ID 506976 | https://doi.org/10.1155/2012/506976

Kamonrat Nammanee, Suthep Suantai, Prasit Cholamjiak, "A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions", Journal of Applied Mathematics, vol. 2012, Article ID 506976, 14 pages, 2012. https://doi.org/10.1155/2012/506976

A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions

Academic Editor: Giuseppe Marino
Received23 Nov 2011
Accepted27 Jan 2012
Published05 Apr 2012

Abstract

We study strong convergence of the sequence generated by implicit and explicit general iterative methods for a one-parameter nonexpansive semigroup in a reflexive Banach space which admits the duality mapping š½šœ‘, where šœ‘ is a gauge function on [0,āˆž). Our results improve and extend those announced by G. Marino and H.-K. Xu (2006) and many authors.

1. Introduction

Let šø be a real Banach space and šøāˆ— the dual space of šø. Let š¾ be a nonempty, closed, and convex subset of šø. A (one-parameter) nonexpansive semigroup is a family š”‰={š‘‡(š‘”)āˆ¶š‘”ā‰„0} of self-mappings of š¾ such that(i)š‘‡(0)š‘„=š‘„ for all š‘„āˆˆš¾,(ii)š‘‡(š‘”+š‘ )š‘„=š‘‡(š‘”)š‘‡(š‘ )š‘„ for all š‘”,š‘ ā‰„0 and š‘„āˆˆš¾,(iii)for each š‘„āˆˆš¾, the mapping š‘‡(ā‹…)š‘„ is continuous,(iv)for each š‘”ā‰„0, š‘‡(š‘”) is nonexpansive, that is,ā€–š‘‡(š‘”)š‘„āˆ’š‘‡(š‘”)š‘¦ā€–ā‰¤ā€–š‘„āˆ’š‘¦ā€–,āˆ€š‘„,š‘¦āˆˆš¾.(1.1) We denote š¹ by the common fixed points set of š”‰, that is, ā‹‚š¹āˆ¶=š‘”ā‰„0š¹(š‘‡(š‘”)).

In 1967, Halpern [1] introduced the following classical iteration for a nonexpansive mapping š‘‡āˆ¶š¾ā†’š¾ in a real Hilbert space:š‘„š‘›+1=š›¼š‘›ī€·š‘¢+1āˆ’š›¼š‘›ī€øš‘‡š‘„š‘›,š‘›ā‰„0,(1.2) where {š›¼š‘›}āŠ‚(0,1) and š‘¢āˆˆš¾.

In 1977, Lions [2] obtained a strong convergence provide the real sequence {š›¼š‘›} satisfies the following conditions:

C1: limš‘›ā†’āˆžš›¼š‘›=0; C2: āˆ‘āˆžš‘›=0š›¼š‘›=āˆž; C3: limš‘›ā†’āˆž(š›¼š‘›āˆ’š›¼š‘›āˆ’1)/š›¼2š‘›=0.

Reich [3] also extended the result of Halpern from Hilbert spaces to uniformly smooth Banach spaces. However, both Halpernā€™s and Lionā€™s conditions imposed on the real sequence {š›¼š‘›} excluded the canonical choice š›¼š‘›=1/(š‘›+1).

In 1992, Wittmann [4] proved that the sequence {š‘„š‘›} converges strongly to a fixed point of š‘‡ if {š›¼š‘›} satisfies the following conditions:

C1: limš‘›ā†’āˆžš›¼š‘›=0; C2: āˆ‘āˆžš‘›=0š›¼š‘›=āˆž; C3: āˆ‘āˆžš‘›=0|š›¼š‘›+1āˆ’š›¼š‘›|<āˆž.

Shioji and Takahashi [5] extended Wittmannā€™s result to real Banach spaces with uniformly GĆ¢teaux differentiable norms and in which each nonempty closed convex and bounded subset has the fixed point property for nonexpansive mappings. The concept of the Halpern iterative scheme has been widely used to approximate the fixed points for nonexpansive mappings (see, e.g., [6ā€“12] and the reference cited therein).

Let š‘“āˆ¶š¾ā†’š¾ be a contraction. In 2000, Moudafi [13] introduced the explicit viscosity approximation method for a nonexpansive mapping š‘‡ as follows:š‘„š‘›+1=š›¼š‘›š‘“ī€·š‘„š‘›ī€ø+ī€·1āˆ’š›¼š‘›ī€øš‘‡š‘„š‘›,š‘›ā‰„0,(1.3) where š›¼š‘›āˆˆ(0,1). Xu [14] also studied the iteration process (1.3) in uniformly smooth Banach spaces.

Let š“ be a strongly positive bounded linear operator on a real Hilbert space š», that is, there is a constant š›¾>0 such thatāŸØš“š‘„,š‘„āŸ©ā‰„š›¾ā€–š‘„ā€–2,āˆ€š‘„āˆˆš».(1.4)

A typical problem is to minimize a quadratic function over the fixed points set of a nonexpansive mapping on a Hilbert space š»:minš‘„āˆˆš¶12āŸØš“š‘„,š‘„āŸ©āˆ’āŸØš‘„,š‘āŸ©,(1.5) where š¶ is the fixed points set of a nonexpansive mapping š‘‡ on š» and š‘ is a given point in š».

In 2006, Marino and Xu [15] introduced the following general iterative method for a nonexpansive mapping š‘‡ in a Hilbert space š»:š‘„š‘›+1=š›¼š‘›ī€·š‘„š›¾š‘“š‘›ī€ø+ī€·š¼āˆ’š›¼š‘›š“ī€øš‘‡š‘„š‘›,š‘›ā‰„1,(1.6) where {š›¼š‘›}āŠ‚(0,1), š‘“ is a contraction on š», and š“ is a strongly positive bounded linear operator on š». They proved that the sequence {š‘„š‘›} generated by (1.6) converges strongly to a fixed point š‘„āˆ—āˆˆš¹(š‘‡) which also solves the variational inequalityāŸØ(š“āˆ’š›¾š‘“)š‘„āˆ—,š‘„āˆ’š‘„āˆ—āŸ©ā‰„0,āˆ€š‘„āˆˆš¹(š‘‡),(1.7) which is the optimality condition for the minimization problem: minš‘„āˆˆš¶(1/2)āŸØš“š‘„,š‘„āŸ©āˆ’ā„Ž(š‘„), where ā„Ž is a potential function for š›¾š‘“ (i.e., ā„Žā€²(š‘„)=š›¾š‘“(š‘„) for š‘„āˆˆš»).

Suzuki [16] first introduced the following implicit viscosity method for a nonexpansive semigroup {š‘‡(š‘”)āˆ¶š‘”ā‰„0} in a Hilbert space:š‘„š‘›=š›¼š‘›ī€·š‘¢+1āˆ’š›¼š‘›ī€øš‘‡ī€·š‘”š‘›ī€øš‘„š‘›,š‘›ā‰„1,(1.8) where {š›¼š‘›}āŠ‚(0,1) and š‘¢āˆˆš¾. He proved strong convergence of iteration (1.8) under suitable conditions. Subsequently, Xu [17] extended Suzukiā€™s [16] result from a Hilbert space to a uniformly convex Banach space which admits a weakly sequentially continuous normalized duality mapping.

Motivated by Chen and Song [18], in 2007, Chen and He [19] investigated the implicit and explicit viscosity methods for a nonexpansive semigroup without integral in a reflexive Banach space which admits a weakly sequentially continuous normalized duality mapping:š‘„š‘›=š›¼š‘›š‘“ī€·š‘„š‘›ī€ø+ī€·1āˆ’š›¼š‘›ī€øš‘‡ī€·š‘”š‘›ī€øš‘„š‘›š‘„,š‘›ā‰„1,(1.9)š‘›+1=š›¼š‘›š‘“ī€·š‘„š‘›ī€ø+ī€·1āˆ’š›¼š‘›ī€øš‘‡ī€·š‘”š‘›ī€øš‘„š‘›,š‘›ā‰„1,(1.10) where {š›¼š‘›}āŠ‚(0,1).

In 2008, Song and Xu [20] also studied the iterations (1.9) and (1.10) in a reflexive and strictly convex Banach space with a GĆ¢teaux differentiable norm. Subsequently, Cholamjiak and Suantai [21] extended Song and Xuā€™s results to a Banach space which admits duality mapping with a gauge function. Wangkeeree and Kamraksa [22] and Wangkeeree et al. [23] obtained the convergence results concerning the duality mapping with a gauge function in Banach spaces. The convergence of iterations for a nonexpansive semigroup and nonlinear mappings has been studied by many authors (see, e.g., [24ā€“38]).

Let šø be a real reflexive Banach space which admits the duality mapping š½šœ‘ with a gauge šœ‘. Let {š‘‡(š‘”)āˆ¶š‘”ā‰„0} be a nonexpansive semigroup on šø. Recall that an operator š“ is said to be strongly positive if there exists a constant š›¾>0 such thatī«š“š‘„,š½šœ‘ī¬ā‰„(š‘„)(š›¾ā€–š‘„ā€–šœ‘ā€–š‘„ā€–),ā€–š›¼š¼āˆ’š›½š“ā€–=supā€–š‘„ā€–ā‰¤1||ī«(š›¼š¼āˆ’š›½š“)š‘„,š½šœ‘ī¬||,(š‘„)(1.11) where š›¼āˆˆ[0,1] and š›½āˆˆ[āˆ’1,1].

Motivated by Chen and Song [18], Chen and He [19], Marino and Xu [15], Colao et al. [39], and Wangkeeree et al. [23], we study strong convergence of the following general iterative methods:š‘„š‘›=š›¼š‘›ī€·š‘„š›¾š‘“š‘›ī€ø+ī€·š¼āˆ’š›¼š‘›š“ī€øš‘‡ī€·š‘”š‘›ī€øš‘„š‘›š‘„,š‘›ā‰„1,(1.12)š‘›+1=š›¼š‘›ī€·š‘„š›¾š‘“š‘›ī€ø+ī€·š¼āˆ’š›¼š‘›š“ī€øš‘‡ī€·š‘”š‘›ī€øš‘„š‘›,š‘›ā‰„1,(1.13) where {š›¼š‘›}āŠ‚(0,1), š‘“ is a contraction on šø and š“ is a positive bounded linear operator on šø.

2. Preliminaries

A Banach space šø is called strictly convex if ā€–š‘„+š‘¦ā€–/2<1 for all š‘„,š‘¦āˆˆšø with ā€–š‘„ā€–=ā€–š‘¦ā€–=1 and š‘„ā‰ š‘¦. A Banach space šø is called uniformly convex if for each šœ–>0 there is a š›æ>0 such that for š‘„,š‘¦āˆˆšø with ā€–š‘„ā€–,ā€–š‘¦ā€–ā‰¤1 and ā€–š‘„āˆ’š‘¦ā€–ā‰„šœ–,ā€–š‘„+š‘¦ā€–ā‰¤2(1āˆ’š›æ) holds. The modulus of convexity of šø is defined byš›æšøī‚†ā€–ā€–ā€–1(šœ–)=inf1āˆ’2ā€–ā€–ā€–āˆ¶ī‚‡,(š‘„+š‘¦)ā€–š‘„ā€–,ā€–š‘¦ā€–ā‰¤1,ā€–š‘„āˆ’š‘¦ā€–ā‰„šœ–(2.1) for all šœ–āˆˆ[0,2]. šø is uniformly convex if š›æšø(0)=0, and š›æšø(šœ–)>0 for all 0<šœ–ā‰¤2. It is known that every uniformly convex Banach space is strictly convex and reflexive. Let š‘†(šø)={š‘„āˆˆšøāˆ¶ā€–š‘„ā€–=1}. Then the norm of šø is said to be GĆ¢teaux differentiable iflimš‘”ā†’0ā€–š‘„+š‘”š‘¦ā€–āˆ’ā€–š‘„ā€–š‘”(2.2) exists for each š‘„,š‘¦āˆˆš‘†(šø). In this case šø is called smooth. The norm of šø is said to be FrĆ©chet differentiable if for each š‘„āˆˆš‘†(šø), the limit is attained uniformly for š‘¦āˆˆš‘†(šø). The norm of šø is called uniformly FrĆ©chet differentiable, if the limit is attained uniformly for š‘„,š‘¦āˆˆš‘†(šø). It is well known that (uniformly) FrĆ©chet differentiability of the norm of šø implies (uniformly) GĆ¢teaux differentiability of the norm of šø.

Let šœŒšøāˆ¶[0,āˆž)ā†’[0,āˆž) be the modulus of smoothness of šø defined byšœŒšøī‚†1(š‘”)=sup2(ī‚‡.ā€–š‘„+š‘¦ā€–+ā€–š‘„āˆ’š‘¦ā€–)āˆ’1āˆ¶š‘„āˆˆš‘†(šø),ā€–š‘¦ā€–ā‰¤š‘”(2.3)

A Banach space šø is called uniformly smooth if šœŒšø(š‘”)/š‘”ā†’0 as š‘”ā†’0. See [40ā€“42] for more details.

We need the following definitions and results which can be found in [40, 41, 43].

Definition 2.1. A continuous strictly increasing function šœ‘āˆ¶[0,āˆž)ā†’[0,āˆž) is said to be gauge function if šœ‘(0)=0 and limš‘”ā†’āˆžšœ‘(š‘”)=āˆž.

Definition 2.2. Let šø be a normed space and šœ‘ a gauge function. Then the mapping š½šœ‘āˆ¶šøā†’2šøāˆ— defined by š½šœ‘ī€½š‘“(š‘„)=āˆ—āˆˆšøāˆ—āˆ¶āŸØš‘„,š‘“āˆ—(āŸ©=ā€–š‘„ā€–šœ‘ā€–š‘„ā€–),ā€–š‘“āˆ—()ī€¾ā€–=šœ‘ā€–š‘„ā€–,š‘„āˆˆšø,(2.4) is called the duality mapping with gauge function šœ‘.

In the particular case šœ‘(š‘”)=š‘”, the duality mapping š½šœ‘=š½ is called the normalized duality mapping.

In the case šœ‘(š‘”)=š‘”š‘žāˆ’1,š‘ž>1, the duality mapping š½šœ‘=š½š‘ž is called the generalized duality mapping. It follows from the definition that š½šœ‘(š‘„)=šœ‘(ā€–š‘„ā€–)/ā€–š‘„ā€–š½(š‘„) and š½š‘ž(š‘„)=ā€–š‘„ā€–š‘žāˆ’2š½(š‘„),š‘ž>1.

Remark 2.3. For the gauge function šœ‘, the function Ī¦āˆ¶[0,āˆž)ā†’[0,āˆž) defined by ī€œĪ¦(š‘”)=š‘”0šœ‘(š‘ )š‘‘š‘ (2.5) is a continuous convex and strictly increasing function on [0,āˆž). Therefore, Ī¦ has a continuous inverse function Ī¦āˆ’1.

It is noted that if 0ā‰¤š‘˜ā‰¤1, then šœ‘(š‘˜š‘„)ā‰¤šœ‘(š‘„). Furtherī€œĪ¦(š‘˜š‘”)=0š‘˜š‘”ī€œšœ‘(š‘ )š‘‘š‘ =š‘˜š‘”0ī€œšœ‘(š‘˜š‘„)š‘‘š‘„ā‰¤š‘˜š‘”0šœ‘(š‘„)š‘‘š‘„=š‘˜Ī¦(š‘”).(2.6)

Remark 2.4. For each š‘„ in a Banach space šø, š½šœ‘(š‘„)=šœ•Ī¦(ā€–š‘„ā€–), where šœ• denotes the sub-differential.

We also know the following facts:(i)š½šœ‘ is a nonempty, closed, and convex set in šøāˆ— for each š‘„āˆˆšø,(ii)š½šœ‘ is a function when šøāˆ— is strictly convex,(iii)If š½šœ‘ is single-valued, thenš½šœ‘(šœ†š‘„)=š‘ š‘–š‘”š‘›(šœ†)šœ‘(ā€–šœ†š‘„ā€–)š½šœ‘(ā€–š‘„ā€–)šœ‘ī«(š‘„),āˆ€š‘„āˆˆšø,šœ†āˆˆā„,š‘„āˆ’š‘¦,š½šœ‘(š‘„)āˆ’š½šœ‘ī¬(š‘¦)ā‰„(šœ‘(ā€–š‘„ā€–)āˆ’šœ‘(ā€–š‘¦ā€–))(ā€–š‘„ā€–āˆ’ā€–š‘¦ā€–),āˆ€š‘„,š‘¦āˆˆšø.(2.7) Following Browder [43], we say that a Banach space šø has a weakly continuous duality mapping if there exists a gauge šœ‘ for which the duality mapping š½šœ‘ is single-valued and continuous from the weak topology to the weak* topology, that is, for any {š‘„š‘›} with š‘„š‘›ā‡€š‘„, the sequence {š½šœ‘(š‘„š‘›)} converges weakly* to š½šœ‘(š‘„). It is known that the space ā„“š‘ has a weakly continuous duality mapping with a gauge function šœ‘(š‘”)=š‘”š‘āˆ’1 for all 1<š‘<āˆž. Moreover, šœ‘ is invariant on [0,1].

Lemma 2.5 (See [44]). Assume that a Banach space šø has a weakly continuous duality mapping š½šœ‘ with gauge šœ‘.(i)For all š‘„,š‘¦āˆˆšø, the following inequality holds: Ī¦(ī«ā€–š‘„+š‘¦ā€–)ā‰¤Ī¦(ā€–š‘„ā€–)+š‘¦,š½šœ‘ī¬.(š‘„+š‘¦)(2.8) In particular, for all š‘„,š‘¦āˆˆšø, ā€–š‘„+š‘¦ā€–2ā‰¤ā€–š‘„ā€–2+2āŸØš‘¦,š½(š‘„+š‘¦)āŸ©.(2.9)(ii)Assume that a sequence {š‘„š‘›} in šø converges weakly to a point š‘„āˆˆšø. Then the following holds: limsupš‘›ā†’āˆžĪ¦ī€·ā€–ā€–š‘„š‘›ā€–ā€–ī€øāˆ’š‘¦=limsupš‘›ā†’āˆžĪ¦ī€·ā€–ā€–š‘„š‘›ā€–ā€–ī€ø()āˆ’š‘„+Ī¦ā€–š‘„āˆ’š‘¦ā€–(2.10) for all š‘„,š‘¦āˆˆšø.

Lemma 2.6 (See [23]). Assume that a Banach space šø has a weakly continuous duality mapping š½šœ‘ with gauge šœ‘. Let š“ be a strongly positive bounded linear operator on šø with coefficient š›¾>0 and 0<šœŒā‰¤šœ‘(1)ā€–š“ā€–āˆ’1. Then ā€–š¼āˆ’šœŒš“ā€–ā‰¤šœ‘(1)(1āˆ’šœŒš›¾).

Lemma 2.7 (See [12]). Assume that {š‘Žš‘›} is a sequence of nonnegative real numbers such that š‘Žš‘›+1ā‰¤ī€·1āˆ’š›¾š‘›ī€øš‘Žš‘›+š›¾š‘›š›æš‘›,š‘›ā‰„1,(2.11) where {š›¾š‘›} is a sequence in (0,1) and {š›æš‘›} is a sequence in ā„ such that
(a) āˆ‘āˆžš‘›=1š›¾š‘›=āˆž; (b) limsupš‘›ā†’āˆžš›æš‘›ā‰¤0 or āˆ‘āˆžš‘›=1|š›¾š‘›š›æš‘›|<āˆž.
Then limš‘›ā†’āˆžš‘Žš‘›=0.

3. Implicit Iteration Scheme

In this section, we prove a strong convergence theorem of an implicit iterative method (1.12).

Theorem 3.1. Let šø be a reflexive which admits a weakly continuous duality mapping š½šœ‘ with gauge šœ‘ such that šœ‘ is invariant on [0,1]. Let š”‰={š‘‡(š‘”)āˆ¶š‘”ā‰„0} be a nonexpansive semigroup on šø such that š¹ā‰ āˆ…. Let š‘“ be a contraction on šø with the coefficient š›¼āˆˆ(0,1) and š“ a strongly positive bounded linear operator with coefficient š›¾>0 and 0<š›¾<š›¾šœ‘(1)/š›¼. Let {š›¼š‘›} and {š‘”š‘›} be real sequences satisfying 0<š›¼š‘›<1, š‘”š‘›>0 and limš‘›ā†’āˆžš‘”š‘›=limš‘›ā†’āˆžš›¼š‘›/š‘”š‘›=0. Then {š‘„š‘›} defined by (1.12) converges strongly to š‘žāˆˆš¹ which solves the following variational inequality: ī«(š“āˆ’š›¾š‘“)(š‘ž),š½šœ‘ī¬(š‘žāˆ’š‘¤)ā‰¤0,āˆ€š‘¤āˆˆš¹.(3.1)

Proof. First, we prove the uniqueness of the solution to the variational inequality (3.1) in š¹. Suppose that š‘,š‘žāˆˆš¹ satisfy (3.1), so we have ī«(š“āˆ’š›¾š‘“)(š‘),š½šœ‘ī¬ī«(š‘āˆ’š‘ž)ā‰¤0,(š“āˆ’š›¾š‘“)(š‘ž),š½šœ‘ī¬(š‘žāˆ’š‘)ā‰¤0.(3.2) Adding the above inequalities, we get ī«š“(š‘)āˆ’š“(š‘ž)āˆ’š›¾(š‘“(š‘)āˆ’š‘“(š‘ž)),š½šœ‘ī¬(š‘āˆ’š‘ž)ā‰¤0.(3.3) This shows that ī«š“(š‘āˆ’š‘ž),š½šœ‘ī¬ī«š‘“(š‘āˆ’š‘ž)ā‰¤š›¾(š‘)āˆ’š‘“(š‘ž),š½šœ‘ī¬,(š‘āˆ’š‘ž)(3.4) which implies by the strong positivity of š“(ī«š“š›¾ā€–š‘āˆ’š‘žā€–šœ‘ā€–š‘āˆ’š‘žā€–)ā‰¤(š‘āˆ’š‘ž),š½šœ‘ī¬((š‘āˆ’š‘ž)ā‰¤š›¾š›¼ā€–š‘āˆ’š‘žā€–šœ‘ā€–š‘āˆ’š‘žā€–).(3.5) Since šœ‘ is invariant on [0,1], šœ‘(1)š›¾ā€–š‘āˆ’š‘žā€–šœ‘(ā€–š‘āˆ’š‘žā€–)ā‰¤š›¾š›¼ā€–š‘āˆ’š‘žā€–šœ‘(ā€–š‘āˆ’š‘žā€–).(3.6) It follows that ī€·šœ‘(1)ī€ø(š›¾āˆ’š›¾š›¼ā€–š‘āˆ’š‘žā€–šœ‘ā€–š‘āˆ’š‘žā€–)ā‰¤0.(3.7) Therefore š‘=š‘ž since 0<š›¾<(š›¾šœ‘(1))/š›¼.
We next prove that {š‘„š‘›} is bounded. For each š‘¤āˆˆš¹, by Lemma 2.6, we haveā€–ā€–š‘„š‘›ā€–ā€–=ā€–ā€–š›¼āˆ’š‘¤š‘›ī€·š‘„š›¾š‘“š‘›ī€ø+ī€·š¼āˆ’š›¼š‘›š“ī€øš‘‡ī€·š‘”š‘›ī€øš‘„š‘›ā€–ā€–=ā€–ā€–ī€·āˆ’š‘¤š¼āˆ’š›¼š‘›š“ī€øš‘‡ī€·š‘”š‘›ī€øxš‘›āˆ’ī€·š¼āˆ’š›¼š‘›š“ī€øš‘¤+š›¼š‘›ī€·ī€·š‘„š›¾š‘“š‘›ī€øī€øā€–ā€–ī€·āˆ’š“(š‘¤)ā‰¤šœ‘(1)1āˆ’š›¼š‘›š›¾ī€øā€–ā€–š‘„š‘›ā€–ā€–āˆ’š‘¤+š›¼š‘›ī€·ā€–ā€–š‘„š›¾š›¼š‘›ā€–ā€–ā€–ī€øā‰¤ā€–ā€–š‘„āˆ’š‘¤+ā€–š›¾š‘“(š‘¤)āˆ’š“(š‘¤)š‘›ā€–ā€–āˆ’š‘¤āˆ’š›¼š‘›šœ‘(1)š›¾ā€–ā€–š‘„š‘›ā€–ā€–āˆ’š‘¤+š›¼š‘›ā€–ā€–š‘„š›¾š›¼š‘›ā€–ā€–āˆ’š‘¤+š›¼š‘›ā€–š›¾š‘“(š‘¤)āˆ’š“(š‘¤)ā€–,(3.8) which yields ā€–ā€–š‘„š‘›ā€–ā€–ā‰¤1āˆ’š‘¤šœ‘(1)š›¾āˆ’š›¾š›¼ā€–š›¾š‘“(š‘¤)āˆ’š“(š‘¤)ā€–.(3.9) Hence {š‘„š‘›} is bounded. So are {š‘“(š‘„š‘›)} and {š“š‘‡(š‘”š‘›)š‘„š‘›}.
We next prove that {š‘„š‘›} is relatively sequentially compact. By the reflexivity of šø and the boundedness of {š‘„š‘›}, there exists a subsequence {š‘„š‘›š‘—} of {š‘„š‘›} and a point š‘ in šø such that š‘„š‘›š‘—ā‡€š‘ as š‘—ā†’āˆž. Now we show that š‘āˆˆš¹. Put š‘„š‘—=š‘„š‘›š‘—, š›½š‘—=š›¼š‘›š‘— and š‘ š‘—=š‘”š‘›š‘— for š‘—āˆˆā„•, fix š‘”>0. We see thatā€–ā€–š‘„š‘—ā€–ā€–ā‰¤āˆ’š‘‡(š‘”)š‘[š‘”/š‘ š‘—]āˆ’1ī“š‘˜=0ā€–ā€–š‘‡ī€·(š‘˜+1)š‘ š‘—ī€øš‘„š‘—ī€·āˆ’š‘‡š‘˜š‘ š‘—ī€øš‘„š‘—+1ā€–ā€–+ā€–ā€–ā€–š‘‡š‘”ī‚µī‚øš‘ š‘—ī‚¹š‘ š‘—ī‚¶š‘„š‘—š‘”āˆ’š‘‡ī‚µī‚øš‘ š‘—ī‚¹š‘ š‘—ī‚¶š‘ā€–ā€–ā€–+ā€–ā€–ā€–š‘‡š‘”ī‚µī‚øš‘ š‘—ī‚¹š‘ š‘—ī‚¶ā€–ā€–ā€–ā‰¤ī‚øš‘”š‘āˆ’š‘‡(š‘”)š‘š‘ š‘—ī‚¹ā€–ā€–š‘‡ī€·š‘ š‘—ī€øš‘„š‘—āˆ’š‘„š‘—ā€–ā€–+ā€–ā€–š‘„š‘—ā€–ā€–+ā€–ā€–ā€–š‘‡ī‚µī‚øš‘”āˆ’š‘š‘”āˆ’š‘ š‘—ī‚¹š‘ š‘—ī‚¶ā€–ā€–ā€–=ī‚øš‘”š‘āˆ’š‘š‘ š‘—ī‚¹š›½š‘—ā€–ā€–ī€·š‘ š“š‘‡š‘—ī€øš‘„š‘—ī€·š‘„āˆ’š›¾š‘“š‘—ī€øā€–ā€–+ā€–ā€–š‘„š‘—ā€–ā€–+ā€–ā€–ā€–š‘‡ī‚µī‚øš‘”āˆ’š‘š‘”āˆ’š‘ š‘—ī‚¹š‘ š‘—ī‚¶ā€–ā€–ā€–ā‰¤š‘āˆ’š‘š‘”š›½š‘—š‘ š‘—ā€–ā€–ī€·š‘ š“š‘‡š‘—ī€øš‘„š‘—ī€·š‘„āˆ’š›¾š‘“š‘—ī€øā€–ā€–+ā€–ā€–š‘„š‘—ā€–ā€–ī€½āˆ’š‘+maxā€–š‘‡(š‘ )š‘āˆ’š‘ā€–āˆ¶0ā‰¤š‘ ā‰¤š‘ š‘—ī€¾.(3.10) So we have limsupš‘—ā†’āˆžĪ¦ī€·ā€–ā€–š‘„š‘—ā€–ā€–ī€øāˆ’š‘‡(š‘”)š‘ā‰¤limsupš‘—ā†’āˆžĪ¦ī€·ā€–ā€–š‘„š‘—ā€–ā€–ī€ø.āˆ’š‘(3.11) On the other hand, by Lemma 2.5 (ii), we have limsupš‘—ā†’āˆžĪ¦ī€·ā€–ā€–š‘„š‘—ā€–ā€–ī€øāˆ’š‘‡(š‘”)š‘=limsupš‘—ā†’āˆžĪ¦ī€·ā€–ā€–š‘„š‘—ā€–ā€–ī€ø(ā€–āˆ’š‘+Ī¦ā€–š‘‡(š‘”)š‘āˆ’š‘).(3.12) Combining (3.11) and (3.12), we have Ī¦(ā€–š‘‡(š‘”)š‘āˆ’š‘ā€–)ā‰¤0.(3.13) This implies that š‘āˆˆš¹. Further, we see that ā€–š‘„š‘—ī€·āˆ’š‘ā€–šœ‘ā€–š‘„š‘—ī€ø=ī«š‘„āˆ’š‘ā€–š‘—āˆ’š‘,š½šœ‘ī€·š‘„š‘—=āˆ’š‘ī€øī¬ī«ī€·š¼āˆ’š›½š‘—š“ī€øš‘‡ī€·š‘ š‘—ī€øš‘„š‘—āˆ’ī€·š¼āˆ’š›½š‘—š“ī€øš‘,š½šœ‘ī€·š‘„š‘—āˆ’š‘ī€øī¬+š›½š‘—ī«ī€·š‘„š›¾š‘“š‘—ī€øāˆ’š›¾š‘“(š‘),š½šœ‘ī€·š‘„š‘—āˆ’š‘ī€øī¬+š›½š‘—ī«š›¾š‘“(š‘)āˆ’š“(š‘),š½šœ‘ī€·š‘„š‘—ī€·āˆ’š‘ī€øī¬ā‰¤šœ‘(1)1āˆ’š›½š‘—š›¾ī€øā€–ā€–š‘„š‘—ā€–ā€–šœ‘ī€·ā€–ā€–š‘„āˆ’š‘š‘—ā€–ā€–ī€øāˆ’š‘+š›½š‘—ā€–ā€–š‘„š›¾š›¼š‘—ā€–ā€–šœ‘ī€·ā€–ā€–š‘„āˆ’š‘š‘—ā€–ā€–ī€øāˆ’š‘+š›½š‘—ī«š›¾š‘“(š‘)āˆ’š“(š‘),š½šœ‘ī€·š‘„š‘—.āˆ’š‘ī€øī¬(3.14) So we have ā€–ā€–š‘„š‘—ā€–ā€–šœ‘ī€·ā€–ā€–š‘„āˆ’š‘š‘—ā€–ā€–ī€øā‰¤1āˆ’š‘šœ‘(1)ī«š›¾āˆ’š›¾š›¼š›¾š‘“(š‘)āˆ’š“(š‘),š½šœ‘ī€·š‘„š‘—.āˆ’š‘ī€øī¬(3.15) By the definition of Ī¦, it is easily seen that Ī¦ī€·ā€–ā€–š‘„š‘—ā€–ā€–ī€øā‰¤ā€–ā€–š‘„āˆ’š‘š‘—ā€–ā€–šœ‘ī€·ā€–ā€–š‘„āˆ’š‘š‘—ā€–ā€–ī€ø.āˆ’š‘(3.16) Hence Ī¦ī€·ā€–ā€–š‘„š‘—ā€–ā€–ī€øā‰¤1āˆ’š‘šœ‘(1)ī«š›¾āˆ’š›¾š›¼š›¾š‘“(š‘)āˆ’š“(š‘),š½šœ‘ī€·š‘„š‘—.āˆ’š‘ī€øī¬(3.17) Therefore Ī¦(ā€–š‘„š‘—āˆ’š‘ā€–)ā†’0 as š‘—ā†’āˆž since š½šœ‘ is weakly continuous; consequently, š‘„š‘—ā†’š‘ as š‘—ā†’āˆž by the continuity of Ī¦. Hence {š‘„š‘›} is relatively sequentially compact.
Finally, we prove that š‘ is a solution in š¹ to the variational inequality (3.1). For any š‘¤āˆˆš¹, we see thatī€·š‘”ī«ī€·š¼āˆ’š‘‡š‘›š‘„ī€øī€øš‘›āˆ’ī€·ī€·š‘”š¼āˆ’š‘‡š‘›ī€øī€øš‘¤,š½šœ‘ī€·š‘„š‘›=ī«š‘„āˆ’š‘¤ī€øī¬š‘›āˆ’š‘¤,š½šœ‘ī€·š‘„š‘›āˆ’ī«š‘‡ī€·š‘”āˆ’š‘¤ī€øī¬š‘›ī€øš‘„š‘›ī€·š‘”āˆ’š‘‡š‘›ī€øš‘¤,š½šœ‘ī€·š‘„š‘›ā‰„ā€–ā€–š‘„āˆ’š‘¤ī€øī¬š‘›ā€–ā€–šœ‘ā€–ā€–š‘„āˆ’š‘¤š‘›ā€–ā€–āˆ’ā€–ā€–š‘‡ī€·š‘”āˆ’š‘¤š‘›ī€øš‘„š‘›ī€·š‘”āˆ’š‘‡š‘›ī€øš‘¤ā€–ā€–ā€–ā€–š½šœ‘ī€·š‘„š‘›ī€øā€–ā€–ā‰„ā€–ā€–š‘„āˆ’š‘¤š‘›ā€–ā€–šœ‘ā€–ā€–š‘„āˆ’š‘¤š‘›ā€–ā€–āˆ’ā€–ā€–š‘„āˆ’š‘¤š‘›ā€–ā€–ā€–ā€–š½āˆ’š‘¤šœ‘ī€·š‘„š‘›ī€øā€–ā€–āˆ’š‘¤=0.(3.18) On the other hand, we have ī€·š‘„(š“āˆ’š›¾š‘“)š‘›ī€ø1=āˆ’š›¼š‘›ī€·š¼āˆ’š›¼š‘›š“ī€·š‘”ī€øī€·š¼āˆ’š‘‡š‘›š‘„ī€øī€øš‘›,(3.19) which implies ī«ī€·š‘„(š“āˆ’š›¾š‘“)š‘›ī€ø,š½šœ‘ī€·š‘„š‘›1āˆ’š‘¤ī€øī¬=āˆ’š›¼š‘›ī€·š‘”ī«ī€·š¼āˆ’š‘‡š‘›š‘„ī€øī€øš‘›āˆ’ī€·ī€·š‘”š¼āˆ’š‘‡š‘›ī€øī€øš‘¤,š½šœ‘ī€·š‘„š‘›+ī«š“ī€·ī€·š‘”āˆ’š‘¤ī€øī¬š¼āˆ’š‘‡š‘›š‘„ī€øī€øš‘›,š½šœ‘ī€·š‘„š‘›ā‰¤ī«š“ī€·ī€·š‘”āˆ’š‘¤ī€øī¬š¼āˆ’š‘‡š‘›š‘„ī€øī€øš‘›,š½šœ‘ī€·š‘„š‘›.āˆ’š‘¤ī€øī¬(3.20) Observe ā€–ā€–š‘„š‘—ī€·š‘ āˆ’š‘‡š‘—ī€øš‘„š‘—ā€–ā€–=š›½š‘—ā€–ā€–ī€·š‘„š›¾š‘“š‘—ī€øī€·š‘ āˆ’š“š‘‡š‘—ī€øš‘„š‘—ā€–ā€–ā†’0,(3.21) as š‘—ā†’āˆž. Replacing š‘› by š‘›š‘— and letting š‘—ā†’āˆž in (3.20), we obtain ī«(š“āˆ’š›¾š‘“)(š‘),š½šœ‘ī¬(š‘āˆ’š‘¤)ā‰¤0,āˆ€š‘¤āˆˆš¹.(3.22) So š‘āˆˆš¹ is a solution of variational inequality (3.1); and hence š‘=š‘ž by the uniqueness. In a summary, we have proved that {š‘„š‘›} is relatively sequentially compact and each cluster point of {š‘„š‘›} (as š‘›ā†’āˆž) equals š‘ž. Therefore š‘„š‘›ā†’š‘ž as š‘›ā†’āˆž. This completes the proof.

4. Explicit Iteration Scheme

In this section, utilizing the implicit version in Theorem 3.1, we consider the explicit one in a reflexive Banach space which admits the duality mapping š½šœ‘.

Theorem 4.1. Let šø be a reflexive Banach space which admits a weakly continuous duality mapping š½šœ‘ with gauge šœ‘ such that šœ‘ is invariant on [0,1]. Let {š‘‡(š‘”)āˆ¶š‘”ā‰„0} be a nonexpansive semigroup on šø such that š¹ā‰ āˆ…. Let š‘“ be a contraction on šø with the coefficient š›¼āˆˆ(0,1) and š“ a strongly positive bounded linear operator with coefficient š›¾>0 and 0<š›¾<š›¾šœ‘(1)/š›¼. Let {š›¼š‘›} and {š‘”š‘›} be real sequences satisfying 0<š›¼š‘›<1, āˆ‘āˆžš‘›=1š›¼š‘›=āˆž, š‘”š‘›>0 and limš‘›ā†’āˆžš‘”š‘›=limš‘›ā†’āˆžš›¼š‘›/š‘”š‘›=0. Then {š‘„š‘›} defined by (1.13) converges strongly to š‘žāˆˆš¹ which also solves the variational inequality (3.1).

Proof. Since š›¼š‘›ā†’0, we may assume that š›¼š‘›<šœ‘(1)ā€–š“ā€–āˆ’1 and 1āˆ’š›¼š‘›(šœ‘(1)š›¾āˆ’š›¾š›¼)>0 for all š‘›. First we prove that {š‘„š‘›} is bounded. For each š‘¤āˆˆš¹, by Lemma 2.6, we have ā€–ā€–š‘„š‘›+1ā€–ā€–=ā€–ā€–š›¼āˆ’š‘¤š‘›ī€·š‘„š›¾š‘“š‘›ī€ø+ī€·š¼āˆ’š›¼š‘›š“ī€øš‘‡ī€·š‘”š‘›ī€øš‘„š‘›ā€–ā€–=ā€–ā€–ī€·āˆ’š‘¤š¼āˆ’š›¼š‘›š“ī€øš‘‡ī€·š‘”š‘›ī€øš‘„š‘›āˆ’ī€·š¼āˆ’š›¼š‘›š“ī€øš‘¤+š›¼š‘›ī€·ī€·š‘„š›¾š‘“š‘›ī€øī€øā€–ā€–ī€·āˆ’š“(š‘¤)ā‰¤šœ‘(1)1āˆ’š›¼š‘›š›¾ī€øā€–ā€–š‘„š‘›ā€–ā€–āˆ’š‘¤+š›¼š‘›ā€–ā€–š‘„š›¾š›¼š‘›ā€–ā€–āˆ’š‘¤+š›¼š‘›ā€–=ī€·ā€–š›¾š‘“(š‘¤)āˆ’š“(š‘¤)šœ‘(1)āˆ’š›¼š‘›ī€·šœ‘(1)ā€–ā€–š‘„š›¾āˆ’š›¾š›¼ī€øī€øš‘›ā€–ā€–āˆ’š‘¤+š›¼š‘›(ā‰¤ī€·ā€–š›¾š‘“š‘¤)āˆ’š“(š‘¤)ā€–1āˆ’š›¼š‘›ī€·šœ‘(1)ā€–ā€–š‘„š›¾āˆ’š›¾š›¼ī€øī€øš‘›ā€–ā€–āˆ’š‘¤+š›¼š‘›ī€·šœ‘(1)š›¾āˆ’š›¾š›¼ī€øī€øā€–š›¾š‘“(š‘¤)āˆ’š“(š‘¤)ā€–šœ‘(1).š›¾āˆ’š›¾š›¼(4.1) It follows from induction that ā€–ā€–š‘„š‘›+1ā€–ā€–ī‚»ā€–ā€–š‘„āˆ’š‘¤ā‰¤max1ā€–ā€–,(āˆ’š‘¤ā€–š›¾š‘“š‘¤)āˆ’š“(š‘¤)ā€–šœ‘(1)ī‚¼š›¾āˆ’š›¾š›¼,š‘›ā‰„1.(4.2) Thus {š‘„š‘›} is bounded, and hence so are {š‘“(š‘„š‘›)} and {š“š‘‡(š‘”š‘›)š‘„š‘›}. From Theorem 3.1, there is a unique solution š‘žāˆˆš¹ to the following variational inequality: ī«(š“āˆ’š›¾š‘“)š‘ž,š½šœ‘ī¬(š‘žāˆ’š‘¤)ā‰¤0,āˆ€š‘¤āˆˆš¹.(4.3) Next we prove that limsupš‘›ā†’āˆžī«(š“āˆ’š›¾š‘“)š‘ž,š½šœ‘ī€·š‘žāˆ’š‘„š‘›+1ī€øī¬ā‰¤0.(4.4) Indeed, we can choose a subsequence {š‘„š‘›š‘—} of {š‘„š‘›} such that limsupš‘›ā†’āˆžī«(š“āˆ’š›¾š‘“)š‘ž,š½šœ‘ī€·š‘žāˆ’š‘„š‘›ī€øī¬=limsupš‘—ā†’āˆžī‚¬(š“āˆ’š›¾š‘“)š‘ž,š½šœ‘ī‚€š‘žāˆ’š‘„š‘›š‘—.ī‚ī‚­(4.5) Further, we can assume that š‘„š‘›š‘—ā‡€š‘āˆˆšø by the reflexivity of šø and the boundedness of {š‘„š‘›}. Now we show that š‘āˆˆš¹. Put š‘„š‘—=š‘„š‘›š‘—,š›½š‘—=š›¼š‘›š‘— and š‘ š‘—=š‘”š‘›š‘— for š‘—āˆˆā„•, fix š‘”>0. We obtain ā€–ā€–š‘„š‘—+1ā€–ā€–ā‰¤āˆ’š‘‡(š‘”)š‘[š‘”/š‘ š‘—]āˆ’1ī“š‘˜=0ā€–ā€–š‘‡ī€·(š‘˜+1)š‘ š‘—ī€øš‘„š‘—ī€·āˆ’š‘‡š‘˜š‘ š‘—ī€øš‘„š‘—+1ā€–ā€–+ā€–ā€–ā€–š‘‡š‘”ī‚µī‚øš‘ š‘—ī‚¹š‘ š‘—ī‚¶š‘„š‘—š‘”āˆ’š‘‡ī‚µī‚øš‘ š‘—ī‚¹š‘ š‘—ī‚¶š‘ā€–ā€–ā€–+ā€–ā€–ā€–š‘‡š‘”ī‚µī‚øš‘ š‘—ī‚¹š‘ š‘—ī‚¶ā€–ā€–ā€–ā‰¤ī‚øš‘”š‘āˆ’š‘‡(š‘”)š‘sš‘—ī‚¹ā€–ā€–š‘‡ī€·š‘ š‘—ī€øš‘„š‘—āˆ’š‘„š‘—+1ā€–ā€–+ā€–ā€–š‘„š‘—ā€–ā€–+ā€–ā€–ā€–š‘‡ī‚µī‚øš‘”āˆ’š‘š‘”āˆ’š‘ š‘—ī‚¹š‘ š‘—ī‚¶ā€–ā€–ā€–=ī‚øš‘”š‘āˆ’š‘š‘ š‘—ī‚¹š›½š‘—ā€–ā€–ī€·š‘ š“š‘‡š‘—ī€øš‘„š‘—ī€·š‘„āˆ’š›¾š‘“š‘—ī€øā€–ā€–+ā€–ā€–š‘„š‘—ā€–ā€–+ā€–ā€–ā€–š‘‡ī‚µī‚øš‘”āˆ’š‘š‘”āˆ’š‘ š‘—ī‚¹š‘ š‘—ī‚¶ā€–ā€–ā€–ā‰¤š‘āˆ’š‘š‘”š›½š‘—š‘ š‘—ā€–ā€–ī€·š‘ š“š‘‡š‘—ī€øš‘„š‘—ī€·š‘„āˆ’š›¾š‘“š‘—ī€øā€–ā€–+ā€–ā€–š‘„š‘—ā€–ā€–ī€½āˆ’š‘+maxā€–š‘‡(š‘ )š‘āˆ’š‘ā€–āˆ¶0ā‰¤š‘ ā‰¤š‘ š‘—ī€¾.(4.6) It follows that limsupš‘›ā†’āˆžĪ¦(ā€–š‘„š‘—āˆ’š‘‡(š‘”)š‘ā€–)ā‰¤limsupš‘›ā†’āˆžĪ¦(ā€–š‘„š‘—āˆ’š‘ā€–). From Lemma 2.5 (ii) we have limsupš‘›ā†’āˆžĪ¦ī€·ā€–ā€–š‘„š‘—ā€–ā€–ī€øāˆ’š‘‡(š‘”)š‘=limsupš‘›ā†’āˆžĪ¦ī€·ā€–ā€–š‘„š‘—ā€–ā€–ī€ø(ā€–āˆ’š‘+Ī¦ā€–š‘‡(š‘”)š‘āˆ’š‘).(4.7) So we have Ī¦(ā€–š‘‡(š‘”)š‘āˆ’š‘ā€–)ā‰¤0 and hence š‘āˆˆš¹. Since the duality mapping š½šœ‘ is weakly sequentially continuous, limsupš‘›ā†’āˆžī«(š“āˆ’š›¾š‘“)š‘ž,š½šœ‘ī€·š‘žāˆ’š‘„š‘›+1ī€øī¬=limsupš‘—ā†’āˆžī‚¬(š“āˆ’š›¾š‘“)š‘ž,š½šœ‘ī‚€š‘žāˆ’š‘„š‘›š‘—+1=ī«ī‚ī‚­(š“āˆ’š›¾š‘“)š‘ž,š½šœ‘ī¬(š‘žāˆ’š‘)ā‰¤0.(4.8) Finally, we show that š‘„š‘›ā†’š‘ž. From Lemma 2.5 (i), we have Ī¦ī€·ā€–ā€–š‘„š‘›+1ā€–ā€–ī€øī€·ā€–ā€–ī€·āˆ’š‘ž=Ī¦š¼āˆ’š›¼š‘›š“ī€øš‘‡ī€·š‘”š‘›ī€øš‘„š‘›āˆ’ī€·š¼āˆ’š›¼š‘›š“ī€øš‘ž+š›¼š‘›ī€·ī€·š‘„š›¾š‘“š‘›ī€øī€øāˆ’š›¾š‘“(š‘ž)+š›¼š‘›(ā€–ā€–ī€øī€·ā€–ā€–ī€·š›¾š‘“(š‘ž)āˆ’š“(š‘ž))ā‰¤Ī¦š¼āˆ’š›¼š‘›š“š‘‡ī€·š‘”ī€øī€·š‘›ī€øš‘„š‘›ī€øāˆ’š‘ž+š›¼š‘›ī€·ī€·š‘„š›¾š‘“š‘›ī€øī€øā€–ā€–ī€øāˆ’š›¾š‘“(š‘ž)+š›¼š‘›ī«š›¾š‘“(š‘ž)āˆ’š“(š‘ž),š½šœ‘ī€·š‘„š‘›+1ī€·ī€·āˆ’š‘žī€øī¬ā‰¤Ī¦šœ‘(1)1āˆ’š›¼š‘›š›¾ī€øā€–ā€–š‘„š‘›ā€–ā€–āˆ’š‘ž+š›¼š‘›ā€–ā€–š‘„š›¾š›¼š‘›ā€–ā€–ī€øāˆ’š‘ž+š›¼š‘›ī«š›¾š‘“(š‘ž)āˆ’š“(š‘ž),š½šœ‘ī€·š‘„š‘›+1āˆ’š‘žī€øī¬=Ī¦ī€·ī€·šœ‘(1)āˆ’š›¼š‘›ī€·šœ‘(1)ā€–ā€–š‘„š›¾āˆ’š›¾š›¼ī€øī€øš‘›ā€–ā€–ī€øāˆ’š‘ž+š›¼š‘›ī«š›¾š‘“(š‘ž)āˆ’š“(š‘ž),š½šœ‘ī€·š‘„š‘›+1ā‰¤ī€·āˆ’š‘žī€øī¬1āˆ’š›¼š‘›ī€·šœ‘(1)Ī¦ī€·ā€–ā€–š‘„š›¾āˆ’š›¾š›¼ī€øī€øš‘›ā€–ā€–ī€øāˆ’š‘ž+š›¼š‘›ī«š›¾š‘“(š‘ž)āˆ’š“(š‘ž),š½šœ‘ī€·š‘„š‘›+1.āˆ’š‘žī€øī¬(4.9) Note that āˆ‘āˆžš‘›=1š›¼š‘›=āˆž and limsupš‘›ā†’āˆžāŸØš›¾š‘“(š‘ž)āˆ’š“(š‘ž),š½šœ‘(š‘„š‘›+1āˆ’š‘ž)āŸ©ā‰¤0. Using Lemma 2.7, we have š‘„š‘›ā†’š‘ž as š‘›ā†’āˆž by the continuity of Ī¦. This completes the proof.

Remark 4.2. Theorems 3.1 and 4.1 improve and extend the main results proved in [15] in the following senses:(i)from a nonexpansive mapping to a nonexpansive semigroup,(ii)from a real Hilbert space to a reflexive Banach space which admits a weakly continuous duality mapping with gauge functions.

Acknowledgments

The authors wish to thank the editor and the referee for valuable suggestions. K. Nammanee was supported by the Thailand Research Fund, the Commission on Higher Education, and the University of Phayao under Grant MRG5380202. S. Suantai and P. Cholamjiak wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, Thailand.

References

  1. B. Halpern, ā€œFixed points of nonexpanding maps,ā€ Bulletin of the American Mathematical Society, vol. 73, pp. 957ā€“961, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. P.-L. Lions, ā€œApproximation de points fixes de contractions,ā€ Comptes Rendus de l'AcadĆ©mie des Sciences, vol. 284, no. 21, pp. A1357ā€“A1359, 1977. View at: Google Scholar | Zentralblatt MATH
  3. S. Reich, ā€œApproximating fixed points of nonexpansive mappings,ā€ Panamerican Mathematical Journal, vol. 4, no. 2, pp. 23ā€“28, 1994. View at: Google Scholar | Zentralblatt MATH
  4. R. Wittmann, ā€œApproximation of fixed points of nonexpansive mappings,ā€ Archiv der Mathematik, vol. 58, no. 5, pp. 486ā€“491, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. N. Shioji and W. Takahashi, ā€œStrong convergence of approximated sequences for nonexpansive mappings in Banach spaces,ā€ Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641ā€“3645, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, ā€œApproximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 8, pp. 2350ā€“2360, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. C. E. Chidume and C. O. Chidume, ā€œIterative approximation of fixed points of nonexpansive mappings,ā€ Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 288ā€“295, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. Y. J. Cho, S. M. Kang, and H. Zhou, ā€œSome control conditions on iterative methods,ā€ Communications on Applied Nonlinear Analysis, vol. 12, no. 2, pp. 27ā€“34, 2005. View at: Google Scholar | Zentralblatt MATH
  9. T.-H. Kim and H.-K. Xu, ā€œStrong convergence of modified Mann iterations,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51ā€“60, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. S. Reich, ā€œStrong convergence theorems for resolvents of accretive operators in Banach spaces,ā€ Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287ā€“292, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. H.-K. Xu, ā€œAnother control condition in an iterative method for nonexpansive mappings,ā€ Bulletin of the Australian Mathematical Society, vol. 65, no. 1, pp. 109ā€“113, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. H.-K. Xu, ā€œIterative algorithms for nonlinear operators,ā€ Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240ā€“256, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. A. Moudafi, ā€œViscosity approximation methods for fixed-points problems,ā€ Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46ā€“55, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. H.-K. Xu, ā€œViscosity approximation methods for nonexpansive mappings,ā€ Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279ā€“291, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. G. Marino and H.-K. Xu, ā€œA general iterative method for nonexpansive mappings in Hilbert spaces,ā€ Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43ā€“52, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. T. Suzuki, ā€œOn strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,ā€ Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133ā€“2136, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  17. H.-K. Xu, ā€œA strong convergence theorem for contraction semigroups in Banach spaces,ā€ Bulletin of the Australian Mathematical Society, vol. 72, no. 3, pp. 371ā€“379, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  18. R. Chen and Y. Song, ā€œConvergence to common fixed point of nonexpansive semigroups,ā€ Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 566ā€“575, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  19. R. Chen and H. He, ā€œViscosity approximation of common fixed points of nonexpansive semigroups in Banach space,ā€ Applied Mathematics Letters, vol. 20, no. 7, pp. 751ā€“757, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. Y. Song and S. Xu, ā€œStrong convergence theorems for nonexpansive semigroup in Banach spaces,ā€ Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 152ā€“161, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  21. P. Cholamjiak and S. Suantai, ā€œViscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions,ā€ Journal of Global Optimization. In press. View at: Publisher Site | Google Scholar
  22. R. Wangkeeree and U. Kamraksa, ā€œStrong convergence theorems of viscosity iterative methods for a countable family of strict pseudo-contractions in Banach spaces,ā€ Fixed Point Theory and Applications, vol. 2010, Article ID 579725, 21 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. R. Wangkeeree, N. Petrot, and R. Wangkeeree, ā€œThe general iterative methods for nonexpansive mappings in Banach spaces,ā€ Journal of Global Optimization, vol. 51, no. 1, pp. 27ā€“46, 2011. View at: Publisher Site | Google Scholar
  24. I. K. Argyros, Y. J. Cho, and X. Qin, ā€œOn the implicit iterative process for strictly pseudo-contractive mappings in Banach spaces,ā€ Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 208ā€“216, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  25. S.-S. Chang, Y. J. Cho, H. W. J. Lee, and C. K. Chan, ā€œStrong convergence theorems for Lipschitzian demicontraction semigroups in Banach spaces,ā€ Fixed Point Theory and Applications, vol. 11, Article ID 583423, 10 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  26. Y. J. Cho, L. Ćirić, and S.-H. Wang, ā€œConvergence theorems for nonexpansive semigroups in CAT(0) spaces,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6050ā€“6059, 2011. View at: Publisher Site | Google Scholar
  27. Y. J. Cho, S. M. Kang, and X. Qin, ā€œSome results on k-strictly pseudo-contractive mappings in Hilbert spaces,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 5, pp. 1956ā€“1964, 2009. View at: Publisher Site | Google Scholar
  28. Y. J. Cho, S. M. Kang, and X. Qin, ā€œStrong convergence of an implicit iterative process for an infinite family of strict pseudocontractions,ā€ Bulletin of the Korean Mathematical Society, vol. 47, no. 6, pp. 1259ā€“1268, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  29. Y. J. Cho, X. Qin, and S. M. Kang, ā€œStrong convergence of the modified Halpern-type iterative algorithms in Banach spaces,ā€ Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 17, no. 1, pp. 51ā€“67, 2009. View at: Google Scholar
  30. W. Guo and Y. J. Cho, ā€œOn the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings,ā€ Applied Mathematics Letters, vol. 21, no. 10, pp. 1046ā€“1052, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  31. H. He, S. Liu, and Y. J. Cho, ā€œAn explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,ā€ Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4128ā€“4139, 2011. View at: Publisher Site | Google Scholar
  32. J. Kang, Y. Su, and X. Zhang, ā€œGeneral iterative algorithm for nonexpansive semigroups and variational inequalities in Hilbert spaces,ā€ journal of Inequalities and Applications, vol. 2010, Article ID 264052, 10 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  33. X. N. Li and J. S. Gu, ā€œStrong convergence of modified Ishikawa iteration for a nonexpansive semigroup in Banach spaces,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 4, pp. 1085ā€“1092, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  34. S. Li, L. Li, and Y. Su, ā€œGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 9, pp. 3065ā€“3071, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  35. Q. Lin, ā€œViscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mapping,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5451ā€“5457, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  36. S. Plubtieng and R. Punpaeng, ā€œFixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces,ā€ Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 279ā€“286, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  37. X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, ā€œConvergence of a modified Halpern-type iteration algorithm for quasi-Ļ•-nonexpansive mappings,ā€ Applied Mathematics Letters, vol. 22, no. 7, pp. 1051ā€“1055, 2009. View at: Publisher Site | Google Scholar
  38. T. Suzuki, ā€œStrong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,ā€ Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227ā€“239, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  39. V. Colao, G. Marino, and H.-K. Xu, ā€œAn iterative method for finding common solutions of equilibrium and fixed point problems,ā€ Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340ā€“352, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  40. R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, vol. 6, Springer, New York, NY, USA, 2009. View at: Zentralblatt MATH
  41. C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009.
  42. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
  43. F. E. Browder, ā€œConvergence theorems for sequences of nonlinear operators in Banach spaces,ā€ Mathematische Zeitschrift, vol. 100, pp. 201ā€“225, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  44. T.-C. Lim and H. K. Xu, ā€œFixed point theorems for asymptotically nonexpansive mappings,ā€ Nonlinear Analysis. Theory, Methods & Applications, vol. 22, no. 11, pp. 1345ā€“1355, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright Ā© 2012 Kamonrat Nammanee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views656
Downloads446
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.