Volume 2011, Article ID 754702,28pages doi:10.1155/2011/754702
Research Article
A Generalized Hybrid Steepest-Descent Method for Variational Inequalities in Banach Spaces
D. R. Sahu,
1N. C. Wong,
2and J. C. Yao
31Department of Mathematics, Banaras Hindu University, Varanasi 221005, India
2Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
3Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan
Correspondence should be addressed to N. C. Wong,wong@math.nsysu.edu.tw Received 13 September 2010; Accepted 9 December 2010
Academic Editor: S. Al-Homidan
Copyrightq2011 D. R. Sahu 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.
The hybrid steepest-descent method introduced by Yamada2001is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Lehdili and Moudafi1996introduced the new prox-Tikhonov regularization method for proximal point algorithm to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in Hilbert spaces. In this paper, motivated by Yamada’s hybrid steepest-descent and Lehdili and Moudafi’s algorithms, a generalized hybrid steepest-descent algorithm for computing the solutions of the variational inequality problem over the common fixed point set of sequence of nonexpansive-type mappings in the framework of Banach space is proposed. The strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions. Our strong convergence theorems extend and improve certain corresponding results in the recent literature.
1. Introduction
LetHbe a real Hilbert space with inner product·,·and norm · , respectively. LetCbe a nonempty closed convex subset ofHandDa nonempty closed convex subset ofC.
It is well known that the standard smooth convex optimization problem1, given a convex, Fr´echet-differentiable functionf:H → Rand a closed convex subsetCofH, find a pointx∗∈Csuch that
fx∗ min
x∈C:fx
1.1
can be formulated equivalently as the variational inequality problem VIP∇f,HoverCsee 2,3:
∇fx∗, v−x∗
≥0 ∀v∈C, 1.2
where∇f :H → His the gradient off.
In general, for a nonlinear mappingF : H → HoverC, the variational inequality problem VIPF, CoverDis to find a pointx∗∈Dsuch that
Fx∗, v−x∗ ≥0 ∀v∈D. 1.3 It is important to note that the theory of variational inequalities has been playing an important role in the study of many diverse disciplines, for instance, partial differential equations, optimal control, optimization, mathematical programming, mechanics, finance, and so forth, see, for example,1,2,4–6and references therein.
It is also known that ifFis Lipschitzian and strongly monotone, then for smallμ >0, the mappingPCI−μFis a contraction, wherePC is the metric projection fromHontoC seeSection 2.3. In this case, the Banach contraction principle guarantees that VIPF, Chas a unique solutionx∗and the sequence of Picard iteration process, given by,
xn1PC I−μF
xn ∀n∈N 1.4
converges strongly to x∗. This simplest iterative method for approximating the unique solution of VIPF, CoverCis called the projected gradient method1. It has been used widely in many practical problems, due, partially, to its fast convergence.
The projected gradient method was first proposed by Goldstein7and Levitin and Polyak8for solving convexly constrained minimization problems. This method is regarded as an extension of the steepest-descent or Cauchy algorithm for solving unconstrained optimization problems. It now has many variants in different settings, and supplies a prototype for various more advanced projection methods. In9, the first author introduced the normalS-iteration process and studied an iterative method for approximating the unique solution of VIPF, CoverCas follows:
xn1 PC
I−μF
1−αnxnαnPC
I−μF
xn ∀n∈N. 1.5
Note that the rate of convergence of iterative method1.5is faster than projected gradient method1.4, see9.
The projected gradient method requires repetitive use of PC, although the closed form expression of PC is not always known in many situations. In order to reduce the complexity probably caused by the projection mappingPC, Yamadasee6introduced a hybrid steepest-descent method for solving the problem VIPF,H. Here is the idea. Suppose T e.g.,T PCis a mapping from a Hilbert spaceHinto itself with a nonempty fixed point setFT, andFis a Lipschitzian and strongly monotone overH. Starting with an arbitrary initial guessx1inH, one generates a sequence{xn}by the following algorithm:
xn1:Txn−λnFxn ∀n∈N, 1.6
where{λn}is a slowly diminishing sequence. Yamada6, Theorem 3.3, page 486proved that the sequence{xn}defined by1.6converges strongly to a unique solution of VIPF,Hover FT.
Let X be a real Banach space with dual space X∗. We denote byJ the normalized duality mapping fromXinto 2X∗defined by
Jx:
f∗∈X∗:x, f∗x2f∗2
, x∈X, 1.7
where ·,· denotes the generalized duality pairing. It is well known that the normalized duality mapping is single-valued ifXsmooth, see10. LetCbe a nonempty subset of a real Banach spaceX. A mappingT :X → Xis said to be
1pseudocontractive overCif for eachx, y∈C, there existsjx−y∈Jx−ysatisfying Tx−Ty, j
x−y
≤ x−y2, 1.8
2δ-strongly accretive over C if for each x, y ∈ C, there exist a constant δ > 0 and jx−y∈Jx−ysatisfying
Tx−Ty, j x−y
≥δx−y2. 1.9
We consider the following general variational inequality problem over the fixed point set of nonlinear mapping in the framework of Banach space.
Problem 1.1. general variational inequality problem over the fixed point set of nonlinear mapping.
LetCbe a nonempty closed convex subset of a real smooth Banach spaceX. LetT :C → C be apossibly nonlinearmapping of which fixed point setFTis a nonempty closed convex set. Then for a given strongly accretive operatorF :X → X overC, the general variational inequality problem VIPF, CoverFTis
find a pointx∗∈FT such thatFx∗, Jv−x∗ ≥0 ∀v∈FT. 1.10 Recently, the method1.6has been applied successfully to signal processing, inverse problems, and so on11–13. This situation induces a natural question.
Question 1.2. Does sequence{xn}, defined by1.6, converges strongly a solution to a general variational inequality problem in the Banach space setting, that is, Problem1.1in a case where T :C → Cis given as such a nonexpansive mapping?
We now consider the following variational inclusion problem:
findz∈C such that 0∈Az, P
in the framework of Banach spaceX, whereA : X → 2X is a multivalued operator acting onC ⊆ X. In the sequel, we assume thatS A−10, the set of solutions of ProblemPis nonempty.
The Problem P can be regarded as a unified formulation of several important problems. For an appropriate choice of the operatorA, ProblemPcovers a wide range of mathematical applications; for example, variational inequalities, complementarity problems, and nonsmooth convex optimization. ProblemPhas applications in physics, economics, and in several areas of engineering. In particular, if ψ : H → R∪ {∞}is a proper, lower semicontinuous convex function, its subdifferential∂ψ Ais a maximal monotone operator, and a pointz∈ Hminimizesψif and only if 0∈∂ψz.
One of the most interesting and important problems in the theory of maximal monotone operators is to find an efficient iterative algorithm to compute approximately zeroes of maximal monotone operators. One method for solving zeros of maximal monotone operators is proximal point algorithm. Let A be a maximal monotone operator in a Hilbert spaceH. The proximal point algorithm generates, for startingx1 ∈ H, a sequence{xn}inH by
xn1Jcnxn ∀n∈N, 1.11
whereJcn : IcnA−1 is the resolvent operator associated with the operatorA, and{cn} is a regularization sequence in0,∞. This iterative procedure is based on the fact that the proximal mapJcn is single-valued and nonexpansive. This algorithm was first introduced by Martinet14. Ifψ :H → R∪ {∞}is a proper lower semicontinuous convex function, then the algorithm reduces to
xn1argmin
y∈H
ψ
y 1
2cnxn−y2
∀n∈N. 1.12
Rockafellar15studied the proximal point algorithm in the framework of Hilbert space and he proved the following.
Theorem 1.3. LetHbe a Hilbert space andA⊂ H × Ha maximal monotone operator. Let{xn}be a sequence inHdefined by1.11, where{cn}is a sequence in0,∞such that lim infn→ ∞cn >0.
If S/∅, then the sequence{xn}converges weakly to an element ofS.
Such weak convergence is global; that is, the just announced result holds in fact for anyx1∈ H.
Further, Rockafellar15posed an open question of whether the sequence generated by1.11converges strongly or not. This question was solved by G ¨uler16, who constructed an example for which the sequence generated by1.11converges weakly but not strongly.
This brings us to a natural question of how to modify the proximal point algorithm so that strongly convergent sequence is guaranteed. The Tikhonov method which generates a sequence {xn}by the rule
xnJμAnu ∀n∈N, 1.13
whereu∈ Handμn>0 such thatμn → ∞is studied by several authorssee, e.g., Takahashi 17and Wong et al.18to answer the above question.
In19, Lehdili and Moudafi combined the technique of the proximal map and the Tikhonov regularization to introduce the prox-Tikhonov method which generates the sequence {xn}by the algorithm
xn1JλAn
n xn ∀n∈N, 1.14
whereAn μnI A,μn > 0 is viewed as a Tikhonov regularization ofA. Note that An is strongly monotone, that is,x−x, y−y ≥μnx−x2for allx, y,x, y∈GAn, where GAnis graph ofAn.
Using the technique of variational distance, Lehdili and Moudafi19 were able to prove strong convergence of the algorithm1.14for solving ProblemPwhenAis maximal monotone operator on Hunder certain conditions imposed upon the sequences{λn}and {μn}.
It should be also noted thatAnis now a maximal monotone operator, hence{JλAn
n}is a sequence of nonexpansive mappings.
The main objective of this article is to solve the proposed Problem1.1. To achieve this goal, we present an existence theorem for Problem1.1. Motivated by Yamada’s hybrid steepest-descent and Lehdili and Moudafi’s algorithms1.6and1.14, we also present an iterative algorithm and investigate the convergence theory of the proposed algorithm for solving Problem1.1. The outline of this paper is as follows. InSection 2, we present some theoretical tools which are needed in the sequel. In Section 3, we present Theorem 3.3 the existence and uniqueness of solution of Problem 1.1 in a case when T : C → C is not necessarily nonexpansive mapping. In Section 4, we propose an iterative algorithm Algorithm 4.1, as a generalization of Yamada’s hybrid steepest-descent and Lehdili and Moudafi’s algorithms1.6and1.14, for computing to a unique solution of the variational inequality VIPF, C over
n∈NFTn in the framework of Banach space. In Section 5, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings and the solution of ProblemP. Our strong convergence theorems extend and improve corresponding results of Ceng et al.20; Ceng et al.21; Lehdili and Moudafi19; Sahu9; and Yamada6.
2. Preliminaries and Notations
2.1. Derivatives of FunctionalsLet X be a real Banach space. In the sequel, we always use SX to denote the unit sphere SX {x∈X:x1}. ThenXis said to be
istrictly convex ifx, y∈SXwithx /y⇒ 1−λxλy<1 for allλ∈0,1;
iismooth if the limit limt→0xty − x/texists for eachxandyinSX. In this case, the norm ofXis said to be Gˆateaux differentiable.
The norm ofX is said to be uniformly Gˆateaux differentiable if for each y ∈ SX, this limit is attained uniformly forx∈SX.
It is well known that every uniformly smooth spacee.g.,Lpspace, 1 < p < ∞has a uniformly Gˆateaux-differentiable normsee, e.g.,10.
LetUbe an open subset of a real Hilbert spaceH. Then, a functionΘ:H → R∪ {∞}
is called Gˆateaux differentiable22, page 135onUif for eachu∈U, there existsau∈ H such that
limt→0
Θuth−Θu
t au, h ∀h∈ H. 2.1
Then,Θ:U → H:u → auis called the Gˆateaux derivative ofΘonU.
Example 2.1 see6. Suppose that h ∈ H,β ∈ RandQ : H → His a bounded linear, self-adjoint, that is,Qx, y x, Qyfor allx, y ∈ H, and strongly positive mapping, that is,Qx, x ≥ αx2 for allx ∈ Hand for someα > 0. Define the quadratic function Θ:H → Rby
Θx: 1
2Qx, x − h, xβ ∀x∈ H. 2.2 Then, the Gˆateaux derivativeΘx Qx−βisQ-Lipschitzian andα-strongly monotone onH.
2.2. Lipschitzian Type Mappings
LetCbe a nonempty subset of a real Banach spaceXand letS1, S2 :C → Xbe two mappings.
We denoteBC, the collection of all bounded subsets ofC. The deviation betweenS1andS2
onB∈ BC, denoted byDBS1, S2, is defined by
DBS1, S2 sup{S1x−S2x:x∈B}. 2.3 A mappingT :C → Xis said to be
1L-Lipschitzian if there exists a constantL∈0,∞such thatTx−Ty ≤Lx−yfor allx, y∈C;
2nonexpansive ifTx−Ty ≤ x−yfor allx, y∈C;
3strongly pseudocontractive if for eachx, y ∈ C, there exist a constantk ∈0,1and jx−y∈Jx−ysatisfying
Tx−Ty, j x−y
≤kx−y2, 2.4
4λ-strictly pseudocontractivesee23if for eachx, y ∈ C, there exist a constant λ >0 andjx−y∈Jx−ysuch that
Tx−Ty, j x−y
≤ x−y2−λx−y−
Tx−Ty
2. 2.5
The inequality2.5can be restated as x−y−
Tx−Ty , j
x−y
≥λx−y−
Tx−Ty
2. 2.6
In Hilbert spaces,2.5 and so2.6is equivalent to the following inequality Tx−Ty2≤ x−y2kx−y−
Tx−Ty
2, 2.7
where k 1−2λ. From 2.6, one can prove that ifT isλ-strict pseudocontractive, then T is Lipschitz continuous with the Lipschitz constant L 1λ/λsee, Proposition 3.1.
Throughout the paper, we assume thatLλ,δ :
1−δ/λ.
Fact 2.2see10, Corollary 5.7.15. LetCbe a nonempty closed convex subset of a Banach spaceX and T : C → Ca continuous strongly pseudocontractive mapping. Then T has a unique fixed point inC.
Fix a sequence{an}in0,∞withan → 0 and let{Tn}be a sequence of mappings fromCintoX. Then{Tn}is called a sequence of asymptotically nonexpansive mappings if there exists a sequence{kn}in1,∞with limn→ ∞kn1 such that
Tnx−Tny ≤knx−y ∀x, y∈C, n∈N. 2.8
Motivated by the notion of nearly nonexpansive mappingssee10,24, we say{Tn}is a sequence of nearly nonexpansive mappings if
Tnx−Tny ≤ x−yan ∀x, y∈C, n∈N. 2.9
Remark 2.3. If{Tn} is a sequence of asymptotically nonexpansive mappings with bounded domain, then {Tn} is a sequence of nearly nonexpansive mappings. To see this, let {Tn} be a sequence of asymptotically nonexpansive mappings with sequence {kn} defined on a bounded setCwith diameter diamC. Fixan : kn−1diamC. Then,
Tnx−Tny ≤ x−y kn−1x−y ≤ x−yan 2.10
for allx, y∈Candn∈N.
We prove the following proposition.
Proposition 2.4. LetCbe a closed bounded set of a Banach spaceX and{Tn}a sequence of nearly nonexpansive self-mappings ofCwith sequence{an}such that∞
n1DCTn, Tn1 <
∞. Then, for each x ∈ C,{Tnx} converges strongly to some point ofC. Moreover, if T is a mapping ofCinto itself defined byTzlimn→ ∞Tnzfor allz∈C, thenT is nonexpansive and limn→ ∞DCTn, T 0.
Proof. The assumption∞
n1DCTn, Tn1 < ∞implies that ∞
n1Tnx−Tn1x < ∞for all z∈C. Hence{Tnz}is a Cauchy sequence for eachz∈C. Hence, forx∈C,{Tnx}converges strongly to some point inC. LetT be a mapping ofCinto itself defined byTzlimn→ ∞Tnz
for allz∈C. It is easy to see thatT is nonexpansive. Forz∈Candm, n∈Nwithm > n, we have
Tnx−Tmx ≤m−1
kn
Tkx−Tk1x
≤m−1
kn
DCTk, Tk1
≤∞
kn
DCTk, Tk1.
2.11
Then
Tnx−Tx lim
m→ ∞Tnx−Tmx ≤∞
kn
DCTk, Tk1 ∀x∈C, n∈N, 2.12
which implies that
DCTn, T≤∞
kn
DCTk, Tk1 ∀n∈N. 2.13
Therefore, limn→ ∞DCTn, T 0.
2.3. Nonexpansive Mappings and Fixed Points
A closed convex subsetCof a Banach space X is said to have the fixed-point property for nonexpansive self-mappings if every nonexpansive mapping of a nonempty closed convex bounded subsetMofCinto itself has a fixed point inM.
A closed convex subsetCof a Banach spaceXis said to have normal structure if for each closed convex bounded subset ofDofCwhich contains at least two points, there exists an elementx∈Dwhich is not a diametral point ofD. It is well known that a closed convex subset of a uniformly smooth Banach space has normal structure, see10for more details.
The following result was proved by Kirk25.
Fact 2.5Kirk25. LetXbe a reflexive Banach space and letCbe a nonempty closed convex bounded subset ofXwhich has normal structure. LetTbe a nonexpansive mapping ofCinto itself. ThenFTis nonempty.
A subset Cof a Banach spaceX is called a retract of X if there exists a continuous mappingP fromX ontoCsuch thatP x xfor allxinC. We call suchP a retraction ofX ontoC. It follows that if a mappingP is a retraction, thenP y yfor allyin the range of P. A retractionP is said to be sunny ifPP xtx−P x P x for eachxinX andt ≥ 0.
If a sunny retractionPis also nonexpansive, thenCis said to be a sunny nonexpansive retract of X.
LetCbe a nonempty subset of a Banach spaceXand letx∈X. An elementy0∈Cis said to be a best approximation toxifx−y0dx, C, wheredx, C infy∈Cx−y. The set of all best approximations fromxtoCis denoted by
PCx
y∈C:x−ydx, C
. 2.14
This defines a mapping PC from X into 2C and is called the nearest point projection mappingmetric projection mappingontoC. It is well known that ifCis a nonempty closed convex subset of a real Hilbert spaceH, then the nearest point projectionPC fromHontoC is the unique sunny nonexpansive retraction ofHontoC. It is also known thatPCx∈Cand
x−PCx, PCx−y
≥0 ∀x∈ H, y∈C. 2.15 LetF be a monotone mapping ofHintoHoverC ⊆ H. In the context of the variational inequality problem, the characterization of projection2.15implies
x∗∈VIPF, C⇐⇒x∗PC
x∗−μAx∗
∀μ >0. 2.16
We know the following fact concerning nonexpansive retraction.
Fact 2.6Goebel and Reich 26, Lemma 13.1. LetC be a convex subset of a real smooth Banach space X,D a nonempty subset of C, and P a retraction from Conto D. Then the following are equivalent:
aPis a sunny and nonexpansive.
bx−P x, Jz−P x ≤0 for allx∈C,z∈D.
cx−y, JP x−P y ≥ P x−P y2for allx, y∈C.
Fact 2.7Wong et al. 18, Proposition 6.1. LetC be a nonempty closed convex subset of a strictly convex Banach spaceX and letλi > 0 i 1,2, . . . , Nsuch thatN
i1λi 1. Let T1, T2, . . . , TN : C → Cbe nonexpansive mappings withN
i1FTi/∅and letT N
i1λiTi. ThenT is nonexpansive fromCinto itself andFT N
i1FTi.
Fact 2.8Bruck27. LetCbe a nonempty closed convex subset of a strictly convex Banach spaceX. Let{Sk}be a sequence nonexpansive mappings ofCinto itself with∞
k1FSk/∅ and{βk} sequence of positive real numbers such that∞
k1βk 1. Then the mappingT ∞
k1βkSkis well defined onCandFT ∞
k1FSk.
2.4. Accretive Operators and Zero
LetXbe a real Banach spaceX. For an operatorA:X → 2X, we define its domain, range, and graph as follows:
DA {x∈X:Ax /∅}, RA ∪{Az:z∈DA},
GT
x, y
∈X×X:x∈DA, y∈Ax
, 2.17
respectively. Thus, we writeA : X → 2X as follows:A ⊂ X×X. The inverseA−1 ofAis defined by
x∈A−1y⇐⇒y∈Ax. 2.18
The operatorAis said to be accretive if, for eachxi ∈DAandyi ∈ Axi i 1,2, there is j∈Jx1−x2such thaty1−y2, j ≥0. An accretive operatorAis said to be maximal accretive if there is no proper accretive extension ofAand m-accretive ifRIA Xit follows that RIrA Xfor allr >0. IfAism-accretive, then it is maximal accretivesee Fact2.10, but the converse is not true in general. IfAis accretive, then we can define, for eachλ >0, a nonexpansive single-valued mappingJλ:R1λA → DAbyJλ IλA−1. It is called the resolvent ofA. An accretive operatorAdefined onXis said to satisfy the range condition if DA⊂R1λAfor allλ >0, whereDAdenotes the closure of the domain ofA. It is well known that for an accretive operatorAwhich satisfies the range condition,A−10 FJλAfor allλ >0. We also define the Yosida approximationAr byAr I−JrA/r. We know thatArx∈ AJrAxfor allx∈RIrAandArx ≤ |Ax|inf{y:y∈Ax}for allx∈DA∩RIrA.
We also know the following28: for eachλ, μ >0 andx∈RIλA∩RIμA, it holds that
Jλx−Jμx ≤ λ−μ
λ x−Jλx. 2.19
Letfbe a continuous linear functional on∞. We usefnxnmto denote
fxm1, xm2, xm3, . . . , xmn, . . ., 2.20 form0,1,2, . . .. A continuous linear functionaljonl∞is called a Banach limit ifj∗j1 1 andjnxn jnxn1for eachx x1, x2, . . .inl∞.
Fix any Banach limit and denote it by LIM. Note thatLIM∗1, lim inf
n→ ∞ tn≤LIMntn≤lim sup
n→ ∞ tn, LIMntnLIMntn1, ∀tn∈l∞.
2.21
The following facts will be needed in the sequel for the proof of our main results.
Fact 2.9Ha and Jung29, Lemma 1. LetX be a Banach space with a uniformly Gˆateaux- differentiable norm,Ca nonempty closed convex subset ofX, and{xn}a bounded sequence inX. Let LIM be a Banach limit andy∈Csuch that LIMnyn−y2 infx∈CLIMnyn−x2. Then LIMnx−y, Jxn−y ≤0 for allx∈C.
Fact 2.10Cioranescu30. LetXbe a Banach space and letA:X → 2Xbe anm-accretive operator. ThenAis maximal accretive. IfHis a Hilbert space, thenA:H → 2His maximal accretive if and only if it ism-accretive.
3. Existence and Uniqueness of Solutions of VIPF, C
In this section, we deal with the existence and uniqueness of the solution of Problem1.1in a case whereT :C → Cis given as such a pseudocontractive mapping.
The following propositions will be used frequently throughout the paper.
Proposition 3.1. LetCbe a nonempty subset of a real smooth Banach spaceXandF:X → X an operator overC. Then
aifFisλ-strictly pseudocontractive, thenFis Lipschitzian with constant 11/λ;
bifFis bothδ-strongly accretive andλ-strictly pseudocontractive overCwithλδ >
1, thenI− Fis a contraction with Lipschitz constantLλ,δ;
cif τ ∈ 0,1 is a fixed number and F is both δ-strongly accretive and λ-strictly pseudocontractive overCwithλδ >1 andRI−τF⊆C, thenI−τF:C → Cis a contraction mapping with Lipschitz constant 1−1−Lλ,δτ.
Proof. aLetx, y∈C. From2.6, we have λx−y−
Fx− Fy
2≤ x−y−
Fx− Fy , J
x−y
≤ x−y−
Fx− Fy
x−y, 3.1
which gives us
x−y−
Fx− Fy ≤ 1
λx−y. 3.2
Thus,
Fx− Fy ≤ x−yx−y−
Fx− Fy ≤
1 1
λ
x−y. 3.3
Hence,Fis Lipschitzian with constant 11/λ.
bLetx, y∈C. Further, from2.6, we have λx−y−
Fx− Fy
2≤ x−y2− Fx− Fy, J x−y
≤1−δx−y2. 3.4
Observe that
λδ >1⇐⇒Lλ,δ∈0,1. 3.5
Hence
x−y−
Fx− Fy ≤
1−δ
λ x−yLλ,δx−y. 3.6
cLetx, y∈Cand fixed a numberτ ∈0,1. Assume thatλδ >1 andRI−τF⊆C.
SinceI− Fis a contraction with Lipschitz constantLλ,δ, we have I−τFx−I−τFy ≤ x−y−τ
Fx− Fy 1−τ
x−y τ
I− Fx−I− Fy
≤1−τx−yτI− Fx−I− Fy
≤1−1−Lλ,δτx−y.
3.7
Therefore,I−τF:C → Cis a contraction mapping with Lipschitz constant 1−1−Lλ,δτ.
Proposition 3.2. LetCbe a nonempty closed convex subset of a real smooth Banach spaceX.
LetT :C → Cbe a continuous pseudocontractive mapping and letF:X → X be bothδ- strongly accretive andλ-strictly pseudocontractive overCwithλδ >1 andRI−τF⊆Cfor eachτ ∈0,1. Assume thatChas the fixed-point property for nonexpansive self-mappings.
Then one has the following.
aFor eacht∈0,1, one chooses a numberμt∈0,1arbitrarily, there exists a unique pointvtofCdefined by
vt 1−tTvtt
I−μtF
vt. 3.8
bIfFT/∅andvtis a unique point ofCdefined by3.8, then i{vt}is bounded,
iiFvt, Jvt−v ≤0 for allv∈FT.
Proof. aFor eacht∈0,1, we choose a numberμt∈0,1arbitrarily and then the mapping Gt:C → Cdefined by
Gtv 1−tTvt
I−μtF
v ∀v∈C 3.9
is continuous and strongly pseudocontractive with constant 1−1−Lλ,δtμt. Indeed, for all x, y∈C, byProposition 3.1we have
Gtx−Gty, J x−y
1−tTx−Ty, J x−y
t
I−μtF x−
I−μtF y, J
x−y
≤1−tx−y2t
I−μtF x−
I−μtF
yx−y
≤
1−1−Lλ,δtμt x−y2.
3.10
By Fact2.2, there exists a unique fixed pointvtofGtinCdefined by vt 1−tTvtt
I−μtF
vt. 3.11
bAssume thatFT/∅. Take anyp∈FT. Using3.8, we have vt−
1−tpt
I−μtF vt , J
vt−p 1−tTvtt
I−μtF vt−
1−tpt
I−μtF vt , J
vt−p 1−tTvt−p, J
vt−p
≤1−tvt−p2.
3.12
Observe that vt−
1−tpt
I−μtF vt , J
vt−p
1−t vt−p
t vt−
I−μtF vt , J
vt−p 1−tvt−p2tμtFvt, J
vt−p .
3.13
Thus,
1−tvt−p2tμtFvt, J vt−p
vt−
1−tpt
I−μtF vt , J
vt−p
≤1−tvt−p2,
3.14
which implies that
Fvt, J vt−p
≤0. 3.15
SinceFisδ-strongly accretive, we have δvt−p2≤ Fvt− F
p , J
vt−p Fvt, J
vt−p − F
p , J
vt−p
≤ −F p
, J vt−p
≤ F p
vt−p,
3.16
which implies that
δvt−p ≤ F p
. 3.17
It shows that{vt}is bounded.
Now, we are ready to present the main result of this section.
Theorem 3.3. Let Cbe a nonempty closed convex subset of a real reflexive Banach space X with a uniformly Gˆateaux-differentiable norm. LetT :C → Cbe a continuous pseudocontractive mapping
withFT/∅and letF:X → Xbe bothδ-strongly accretive andλ-strictly pseudocontractive over Cwithλδ >1 andRI−τF⊆Cfor eachτ ∈0,1. Assume thatChas the fixed-point property for nonexpansive self-mappings. Then{vt}converges strongly ast → 0 to a unique solutionx∗of VIPF, CoverFT.
Proof. ByProposition 3.2,{vt : t ∈ 0,1}is bounded. SinceFis a Lipschitzian mapping, it follows that{Fvt:t∈0,1}is bounded. From3.8, we have
Tvtvt tμt
1−tFvt ∀t∈0,1. 3.18
and hence
Tvt ≤ vt tμt
1−tFvt ≤ vt t
1−tFvt ∀t∈0,1. 3.19 Noticing that limt→0t/1−t 0, there existst0 ∈0,1that{Tvt:t∈0, t0}is bounded.
This implies from3.18thatvt−Tvt → 0 as t → 0. The key is to show that{vt : t ∈ 0, t0}is relatively compact ast → 0. We may choose a sequence{tn}in0, t0such that limn→ ∞tn 0. Set vn : vtn. We will show that {vn} contains a subsequence converging strongly to an element ofC. Define the functionϕ:C → Rbyϕx:LIMnvn−x2,x∈C and let
M:
y∈C:ϕ y
inf
x∈Cϕx
. 3.20
SinceX is reflexive,ϕx → ∞ asx → ∞, and ϕis a continuous convex function. By Barbu and Precupanu31, Theorem 1.2, page 79, we have that the setMis nonempty. By Takahashi28, we see thatMis also closed, convex, and bounded.
From32, Theorem 6, we know that the mapping 2I−T has a nonexpansive inverse, denoted byg, which mapsCinto itself withFT Fg. Note that limn→ ∞vn−Tvn 0 implies that limn→ ∞vn −gvn 0. Moreover,Mis invariant underg, that is,Rg ⊆ M.
In fact, for eachy∈M, we have ϕ
gy
LIMnvn−gy2≤LIMngvn−gy2≤LIMnvn−y2ϕ y
, 3.21
and hencegy∈M. By assumption, we haveM∩Fg/∅. Lety∗∈M∩Fg. By Fact2.9, we have
LIMn
z−y∗, J
vn−y∗
≤0 ∀z∈C. 3.22
In particular, by takingzy∗− Fy∗, we have LIMn−F
y∗ , J
vn−y∗
≤0. 3.23
Using3.16and3.23, we have
δLIMnvn−y∗2≤LIMn−F y∗
, J
vn−y∗
≤0. 3.24
Thus, there exists a subsequence{vni}of{vn}such thatvni → y∗.
Assume that{vnj}is another subsequence of{vn}such thatvnj → z∗/y∗. It is easy to see thatz∗∈FT. Sincevni → y∗andJis norm to weak∗uniform continuous, we obtain fromProposition 3.2bthat
F y∗
, J
y∗−z∗
≤0. 3.25
Similarly, we have
Fz∗, J
z∗−y∗
≤0. 3.26
Adding the above two inequalities yields F
y∗
− Fz∗, J
y∗−z∗
≤0, 3.27
which implies that
δy∗−z∗2≤ F y∗
− Fz∗, J
y∗−z∗
≤0, 3.28
a contradiction. Hence,{vtn}converges strongly toy∗.
To see that the entire net{vt}actually converges strongly ast → 0, we assume that there is another sequence{sn}withsn ∈ 0, t0andsn → 0 asn → ∞such thatvsn → z asn → ∞, then,z∈FT. FromProposition 3.2b, we conclude thatzy∗. Therefore,{vt} converges strongly ast → 0toy∗∈FT. Noticing thaty∗∈FTis a solution of VIPF, C overFT. Indeed, fromProposition 3.2b, we have
F y∗
, J
y∗−v
≤0 ∀v∈FT. 3.29
One can easily see thaty∗is the unique solution of VIPF, CoverFT.
As the domain of operators considered in Theorem 3.3is not necessarily the entire spaceX,Theorem 3.3is more general in nature. It improves Ceng et al.20, Proposition 4.3 significantly and provides solutions of Problem1.1.
We now replace the fixed-point property assumption, mentioned inTheorem 3.3by imposing strict convexity on the underlying space.
Theorem 3.4. LetCbe a nonempty closed convex subset of a real strictly convex reflexive Banach space X with a uniformly Gˆateaux-differentiable norm. Let T : C → C be a continuous pseudocontractive mapping with FT/∅ and let F : X → X be both δ-strongly accretive and λ-strictly pseudocontractive overCwithλδ >1 andRI−τF⊆Cfor eachτ ∈0,1. Then{vt} converges strongly ast → 0to a unique solutionx∗of VIPF, CoverFT.
Proof. To be able to use the argument of the proof ofTheorem 3.3, we just need to show that the setM defined by3.20has a fixed point ofg. SinceFg/∅, letv ∈ Fg. SinceX is strictly convex, it follows from10, Proposition 2.1.10that the setM0defined byM0 {u∈ M:u−vinfx∈Mx−v}is a singleton. LetM0{u0}for someu0∈M. Observe that
gu0−vgu0−gv ≤ u0−v inf
x∈Mx−v. 3.30
Therefore,gu0u0.
4. Generalized Hybrid Steepest-Descent Algorithm
Motivated by Yamada’s hybrid steepest-descent and Lehdili and Moudafi’s algorithms,1.6 and 1.14, we introduce the following generalized hybrid steepest-descent algorithm for computing a unique solutionx∗of VIPF, Cover
n∈NFTn.
Algorithm 4.1. LetCbe a nonempty closed convex subset of a real smooth Banach spaceX and letF : X → X be an accretive operator overCsuch thatRI−τF ⊆ Cfor eachτ ∈ 0,1. Assume that {Tn}is a sequence of nearly nonexpansive mappings from Cinto itself with sequence{an}such that
n∈NFTn/∅. Starting with an arbitrary initial guessx1 ∈C, a sequence{xn}inCis generated via the following iterative scheme:
xn1Tnxn−αnFxn ∀n∈N, 4.1
where{αn}is a sequence in0,1.
We will study ourAlgorithm 4.1under the conditions:
C1limn→ ∞αn0,∞
n1αn∞, and either∞
n1|αn−αn1|<∞or limn→ ∞|1−αn/αn1| 0;
C2either∞
n1DDTn, Tn1<∞or limn→ ∞DDTn, Tn1/αn1 0 for eachD∈ BC;
C3limn→ ∞an/αn 0.
Now, we are ready to prove the main theorem for computing solution of VIPF, C over
n∈NFTnin the framework of Banach space.
Theorem 4.2. Let C be a nonempty closed convex subset of a reflexive Banach space X with a uniformly Gˆateaux-differentiable norm and{Tn}a sequence of nearly nonexpansive mappings from Cinto itself with sequence{an}such that
n∈NFTn/∅. LetTbe a mapping ofCinto itself defined byTz limn→ ∞Tnzfor allz ∈ Cand letF : X → Xbe bothδ-strongly accretive andλ-strictly pseudocontractive overCwithλδ >1 andRI−τF⊆Cfor eachτ ∈0,1. Assume thatChas the fixed-point property for nonexpansive self-mappings. For a givenx1∈C, let{xn}be a sequence in Cgenerated by4.1, where{αn}is a sequence in0,1satisfying conditions (C1)∼(C3). Then,{xn} converges strongly to a unique solutionx∗of VIPF, Cover
n∈NFTn.
Proof. LetTbe a mapping ofCinto itself defined byTzlimn→ ∞Tnzfor allz∈C. It is clear thatT is a nonexpansive mapping and
n∈NFTn ⊆ FT. So, we haveFT/∅. For each t∈0,1, we choose a numberμt∈0,1arbitrarily, letxtbe a unique point ofCsuch that
xt 1−tTxtt
I−μtF
xt. 4.2 It follows fromTheorem 3.3that{xt}converges strongly ast → 0to a unique solutionx∗of VIPF, Cover
n∈NFTn. Setyn:xn−αnFxn. We now proceed with the following steps.
Step 1. {xn}and{yn}are bounded.
Observe that
yn−x∗ ≤ xn−x∗αnFxn
≤ xn−x∗Fxn− Fx∗Fx∗
≤
21 λ
xn−x∗Fx∗ ∀n∈N.
4.3
Invoking4.3, we have
xn1−x∗Tnxn−αnFxn−x∗
≤ xn−αnFxn−x∗an
≤ I−αnFxn−I−αnFx∗αnFx∗an
≤1−1−Lλ,δαnxn−x∗αnFx∗an.
4.4
Note that limn→ ∞an/αn 0, so there exists a constantK >0 such that αnFx∗an
αn ≤K ∀n∈N. 4.5
By4.4, we have
xn1−x∗ ≤1−1−Lλ,δαnxn−x∗αnK
≤max
xn−x∗, K 1−Lλ,δ
∀n∈N. 4.6
Hence,{xn}is bounded and hence, from4.3,{yn}is bounded.
Step 2. yn−Tyn → 0 asn → ∞.
Note that the condition limn→ ∞αn 0 implies that yn−xn αnFxn → 0 as n → ∞. Observe that
yn−yn−1I−αnFxn−I−αnFxn−1 I−αnFxn−1−I−αn−1Fxn−1
≤1−1−Lλ,δαnxn−xn−1|αn−αn−1|Fxn−1
≤1−1−Lλ,δαnxn−xn−1|αn−αn−1|K1
4.7
for some constantK1>0. SetB:{yn}. ThenB∈ BC. It follows from4.1that
xn1−xnTnyn−Tn−1yn−1
≤ Tnyn−Tnyn−1Tnyn−1−Tn−1yn−1
≤ yn−yn−1DBTn, Tn−1 an
≤1−1−Lλ,δαnxn−xn−1DBTn, Tn−1 |αn−αn−1|K1an.
4.8
By conditionsC1∼C3and Xu33, Lemma 2.5, we obtain thatxn1−xn → 0 asn → ∞.
Hence,
xn1−TnxnTnyn−Tnxn ≤ yn−xnan−→0 asn−→ ∞,
xn−Tnxn ≤ xn−xn1xn1−Tnxn −→0 as n−→ ∞. 4.9
Moreover,
yn−Tnyn ≤ yn−xnxn−TnxnTnxn−Tnyn
≤2yn−xnxn−Tnxnan−→0 asn−→ ∞. 4.10
The definition ofTimplies that
Tyn−yn ≤ Tyn−Tnynxn1−xnxn−yn
≤ DBT, Tn xn1−xnxn−yn −→0 asn−→ ∞. 4.11
Step 3. lim supn→ ∞Fx∗, Jx∗−yn ≤0.
Sincext−yn 1−tTxt−yn tI−μtFxt−yn, we have xt−yn2 1−tTxt−yn, J
xt−yn t
I−μtF
xt−yn, J
xt−yn
≤1−tTxt−TynTyn−yn, J xt−yn
t
I−μtF
xt−xt, J xt−yn
xt−yn2
≤ xt−yn2 1−tTyn−yn, J
xt−yn
−tμtFxt, J
xt−yn
≤ xt−yn2 1−tTyn−ynxt−yn −tμtFxt, J xt−yn
,
4.12
which implies that
Fxt, J
xt−yn
≤ 1−t
tμt Tyn−ynxt−yn. 4.13 Since{xt}and{yn}are bounded andyn−Tyn → 0 asn → ∞, taking the superior limit in 4.13, we obtain that
lim sup
n→ ∞ Fxt, J
xt−yn
≤0. 4.14
Further, sincext → x∗ast → 0, the set{xt−yn}is bounded, and the duality mappingJis norm-to-weak∗uniformly continuous on bounded subsets ofX, it follows that
Fx∗, J
yn−x∗
− Fxt, J yn−xt
Fx∗, J
yn−x∗
−J
yn−xt
Fx∗− Fxt, J
yn−xt
≤Fx∗, J
yn−x∗
−J yn−xt
Fx∗− Fxtyn−xt −→0 as t−→0.
4.15
Letε >0. Then there existsδ1>0 such that Fx∗, J
x∗−yn
<
Fxt, J xt−yn
ε ∀n∈N, t∈0, δ1. 4.16
Using4.14, we get lim sup
n→ ∞ Fx∗, J
x∗−yn
≤lim sup
n→ ∞ Fxt, J
x∗−yn ε
≤ε.
4.17
Sinceεis arbitrary, we obtain that lim supn→ ∞Fx∗, Jx∗−yn ≤0.
Step 4. {xn}converges strongly tox∗.
