R E S E A R C H
Open Access
Convergence theorems for common
solutions of various problems with nonlinear
mapping
Kyung Soo Kim
1*, Jong Kyu Kim
2and Won Hee Lim
2*Correspondence:
[email protected] 1Graduate School of Education,
Mathematics Education, Kyungnam University, Changwon, Kyungnam 631-701, Republic of Korea Full list of author information is available at the end of the article
Abstract
In this paper, motivated and inspired by Zegeye and Shahzad (Nonlinear Anal. 70:2707-2716, 2009), Qinet al.(J. Comput. Appl. Math. 225(1):20-30, 2009) and Kimura and Takahashi (J. Math. Anal. Appl. 357:356-363, 2009), we introduce a new hybrid projection iterative scheme that converges strongly to a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of common fixed points for a family of hemi-relatively nonexpansive mappings in a Banach space.
MSC: 47H10; 47J20; 49J40
Keywords: common fixed point; generalized mixed equilibrium problem;
generalized projection; hemi-relatively nonexpansive; variational inequality problem
1 Introduction
A real Banach spaceEis said to bestrictly convexifx+y< for allx,y∈Ewithx=y= andx=y. It is said to beuniformly convexiflimn→∞xn–yn= for any two sequences {xn} and{yn}inE such thatxn=yn= andlimn→∞xn+yn= . It is known that a uniformly convex Banach space is reflexive and strictly convex. LetU={x∈E:x= }
be the unit sphere ofE. Then the Banach spaceEis said to besmoothif
lim t→
x+ty–x t
exists for eachx,y∈U. It is said to beuniformly smoothif the limit is attained uniformly forx,y∈E.
LetEbe a real Banach space with the norm · , and letE∗denote the dual space ofE. We denote byJthe normalized duality mapping fromEto E∗defined by
Jx=f∗∈E∗:x,f∗=x=f∗,
where·,· denotes the generalized duality pairing. It is well known that ifE∗ is strictly convex, thenJis single-valued, and ifEis uniformly smooth, thenJis uniformly norm-to-norm continuous on a bounded subset ofE. Moreover, ifEis a reflexive and strictly convex Banach space with a strictly convex dual, thenJ–is single-valued, one-to-one, surjective
and it is the duality mapping fromE∗intoEand thusJJ–=I
E∗andJ–J=IE(see []). We note thatJis the identity mapping in a Hilbert space.
A mappingA:D(A)⊂E→E∗is said to bemonotoneif for eachx,y∈D(A),
x–y,Ax–Ay ≥.
A mappingAis said to beγ-inverse strongly monotoneif there exists a positive real number
γ > such that
x–y,Ax–Ay ≥γAx–Ay, ∀x,y∈D(A).
If a mappingAisγ-inverse strongly monotone, then it is Lipschitz continuous with con-stant
γ,i.e.,
Ax–Ay ≤
γx–y, x,y∈D(A).
A mappingAis said to bestrongly monotone, if for eachx,y∈D(A), there existsk∈(, ) such that
x–y,Ax–Ay ≥kx–y.
A monotone mappingAis said to bemaximalif its graphG(A) ={(x,y) :y∈Ax}is not properly contained in the graph of any other monotone mapping. It is known that the monotone mappingAis maximal if and only if for (x,x∗)∈E×E∗,
x–y,x∗–y∗≥
for every (y,y∗)∈G(A) implies thatx∗∈Ax.
LetCbe a nonempty, closed convex subset of a Banach spaceE. For a bifunctionθ :
C×C→R, we assume thatθ satisfies the following conditions: (E) θ(x,x) = for allx∈C;
(E) θis monotone,i.e.,θ(x,y) +θ(y,x)≤for allx,y∈C; (E) for eachx,y,z∈C,
lim t↓θ
tz+ ( –t)x,y≤θ(x,y);
(E) for eachx∈C, the functiony→θ(x,y)is convex and lower semi-continuous. The generalized mixed equilibrium problem is to findx∈Csuch that
θ(x,y) +ψ(y) –ψ(x) +Ax,y–x ≥, ∀y∈C, (.)
IfA= , then problem (.) reduces to the following mixed equilibrium problem of find-ingx∈Csuch that
θ(x,y) +ψ(y) –ψ(x)≥, ∀y∈C,
which was considered by Ceng and Yao [].
Ifψ= , then problem (.) reduces to the following generalized equilibrium problem of findingx∈Csuch that
θ(x,y) +Ax,y–x ≥, ∀y∈C,
which was studied in [].
Ifψ= andA= , then problem (.) reduces to the following equilibrium problem of findingx∈Csuch that
θ(x,y)≥, ∀y∈C. (.)
The set of solutions of problem (.) is denoted byEP.
Ifθ= andψ= , then problem (.) reduces to the following classical variational in-equality problem of findingx∈Csuch that
Ax,y–x ≥, ∀y∈C. (.)
The set of solutions of problem (.) is denoted byVI(C,A).
Equilibrium problems, which were introduced in [] in , have had great impact and influence on the development of several branches of pure and applied sciences. They include numerous problems in economics, finance, physics, network, elasticity, optimiza-tion, variational inequalities, minimax problems, and semigroups; see, for instance, [–, –] and the references therein.
As well known, ifCis a nonempty, closed and convex subset of a Hilbert spaceHand
PC:H→Cis the metric projection ofHontoC, thenPCis nonexpansive. This fact actu-ally characterizes Hilbert spaces and, consequently, it is not true to more general Banach spaces. In this connection, Alber [] introduced a generalized projection operatorCin a Banach spaceEwhich is an analogue of the metric projection in Hilbert spaces. Consider the functional defined by
φ(x,y) =x– x,Jy +y forx,y∈E. (.)
Observe that, in a Hilbert spaceH, (.) reduces to
φ(x,y) =x–y forx,y∈H.
The generalized projectionC:E→Cis a mapping that assigns an arbitrary pointx∈E to the minimum point of the functionalφ(x,y), that is,Cx=x¯, wherex¯is the solution to the minimization problem
The existence and uniqueness of the mappingCfollows from the properties of the func-tionalφ(x,y) and strict monotonicity of the mappingJ(see, for example, [] and []). In a Hilbert space,C=PC. It is obvious from the definition of the functionφthat:
() (x–y)≤φ(x,y)≤(x+y)for allx,y∈E. () φ(x,y) =φ(x,z) +φ(z,y) + x–z,Jz–Jy for allx,y,z∈E.
() φ(x,y) =x,Jx–Jy +y–x,Jy ≤ xJx–Jy+y–xyfor allx,y∈E. () IfEis a reflexive, strictly convex and smooth Banach space, then, for allx,y∈E,
φ(x,y) = if and only if x=y.
Remark . In (), it is sufficient to show that ifφ(x,y) = thenx=y. In fact, from () we havex=y. This implies thatx,Jy =x=Jy. From the definition ofJ, we have Jx=Jy. Therefore, we havex=y. For more details, see [].
LetCbe a nonempty closed and convex subset ofE, and letT be a mapping fromC
into itself. We denote byF(T) be the set of fixed points ofT. A pointpinCis said to be aweak asymptotic fixed pointofT [] ifCcontains a sequence{xn} which converges weakly topsuch thatlimn→∞(Txn–xn) = . The set of asymptotic fixed points ofTwill be denoted byFˆ(T). A mappingTfromCinto itself is calledrelatively nonexpansive[–] ifFˆ(T) =F(T) andφ(p,Tx)≤φ(p,x) for allx∈Candp∈F(T). The asymptotic behavior of relatively nonexpansive mappings was studied in [, ] and [].
A point pinC is said to be astrong asymptotic fixed point ofT if Ccontains a se-quence{xn} which converges strongly topsuch thatlimn→∞(Txn–xn) = . The set of strong asymptotic fixed points of T is denoted byF˜(T). A mapping T fromC into it-self is calledrelatively weak nonexpansive if F˜(T) =F(T) and φ(p,Tx)≤φ(p,x) for all
x∈Candp∈F(T). A mappingTis calledhemi-relatively nonexpansiveifF(T)=∅and
φ(p,Tx)≤φ(p,x) for allx∈Candp∈F(T).
Remark . () It is obvious that a relatively nonexpansive mapping is a relatively weak nonexpansive mapping (see []). In fact, for any mappingT :C→C, we haveF(T)⊂
˜
F(T)⊂ ˆF(T). Therefore, ifT is a relatively nonexpansive mapping, thenF(T) =F˜(T) = ˆ
F(T).
() The class of hemi-relatively nonexpansive mappings is more general than the class of relatively weak nonexpansive mappings.
The converse of Remark . is not true. In order to explain this better, we give the fol-lowing example.
Example .([]) LetEbe any smooth Banach space, and letx= be any element ofE. We define a mappingT:E→Eas follows:
Tx=
⎧ ⎨ ⎩
(+n+)x ifx= (+n)x,
–x ifx= (+n)x
Remark . There are other examples of hemi-relatively nonexpansive mappings such as the generalized projections (or projections) from a smooth, strictly convex and reflexive Banach space, and others; see [].
A mappingT:C→Cis said to beclosed, if for any sequence{xn} ⊂Cwithxn→xand
Txn→y, thenTx=y.
In , Kimura and Takahashi [] proposed the following hybrid iteration method with a generalized projection for a family of relatively nonexpansive mappings{Tλ}in a Banach spaceE:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x=x∈C, chosen arbitrarily,
C=C,
yn(λ) =J–(αnJxn+ ( –αn)JTλxn) for allλ∈ ,
Cn+={z∈Cn:supλ∈ φ(z,yn(λ))≤φ(z,xn)}, xn+=Cn+x.
They proved that{xn}converges strongly toFx∈C, whereF=λ∈ F(Tλ) is the set of common fixed points ofTλ, andK is the generalized projection ofEonto a nonempty closed convex subsetKofE.
Recently, Zegeye and Shahzad [] introduced the following iterative scheme for finding a common element of the solution set of a variational inequality problem and a fixed point of a relatively weak nonexpansive mapping withγ-inverse strongly monotone mapping satisfyingAx ≤ Ax–Apfor allx∈Candp∈VI(C,A) (see,e.g., []):
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C, chosen arbitrarily, yn=C(J–(Jxn–αnAxn)),
zn=Tyn,
H={v∈C:φ(v,z)≤φ(v,y)≤φ(v,x)}, W=C,
Hn={v∈Hn–∩Wn–:φ(v,zn)≤φ(v,yn)≤φ(v,xn)}, Wn={v∈Hn–∩Wn–:xn–v,Jx–Jxn ≥}, xn+=Hn∩Wnx, n≥.
On the other hand, Qinet al.[] proposed the following hybrid iterative scheme:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E, chosen arbitrarily,
C=C, x=Cx,
yn=J–(αnJxn+ ( –αn)JTxn),
un∈Csuch thatf(un,y) +rny–un,Jun–Jyn ≥, ∀y∈C,
Cn+={z∈Cn:φ(z,un)≤φ(z,xn)},
whereT:C→Cis a closed hemi-relatively nonexpansive mapping. Under suitable condi-tions, they proved that the sequence{xn}converges strongly toF(T)∩EP(f)x, whereEP(f) is the solution of an equilibrium problem for a bifunctionf :C×C→R.
In this paper, we introduce a new hybrid projection iterative scheme that converges strongly to a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of common fixed points for a family of hemi-relatively nonexpansive mappings in a Banach space.
2 Preliminaries
LetE be a normed linear space withdimE≥. Themodulus of smoothnessofE is the functionρE: [,∞)→[,∞) defined by
ρE(τ) :=sup
x+y+x–y
– :x= ,y=τ
.
The spaceEis said to besmoothifρE(τ) > ,∀τ> andEis calleduniformly smoothif and only iflimt→ρEt(t) = . The modulus of convexity ofEis the functionδE: (, ]→[, ] defined by
δE(ε) :=inf
–x+y
:x=y= ,ε=x–y
.
Eis calleduniformly convexif and only ifδE(ε) > for everyε∈(, ]. Letp> . ThenEis said to bep-uniformly convexif there exists a constantc> such thatδ(ε)≥c·εpfor all
ε∈[, ]. Observe that everyp-uniformly convex space is uniformly convex. It is well known (see, for example, []) that
Lp(lp) or Wmp is
⎧ ⎨ ⎩
p-uniformly convex ifp≥; -uniformly convex if <p≤.
In the following, we shall need the following results.
Lemma .([]) Let E be a-uniformly convex and smooth Banach space.Then,for all x,y∈E,we have
x–y ≤
cJx–Jy,
where J is the normalized duality mapping of E,andc ( <c≤)is the-uniformly convex constant of E.
Lemma .([, ]) Let E be a real smooth,strictly convex and reflexive Banach space,
and let C be a nonempty closed convex subset of E.Then the following conclusions hold: (i) φ(y,Cx) +φ(Cx,x)≤φ(y,x),∀x∈E,y∈C.
(ii) Supposex∈Eandz∈C.Then
Lemma .([]) Let E be a strictly convex and smooth Banach space,C be a nonempty closed and convex subset of E and T:C→C be a hemi-relatively nonexpansive mapping.
Then F(T)is a closed convex subset of C.
Lemma .([]) Let E be a real smooth and uniformly convex Banach space,and let
{xn},{yn}be two sequences of E.Iflimn→∞φ(xn,yn) = and either{xn}or{yn}is bounded,
thenlimn→∞xn–yn= .
Lemma .([]) Let E be a real smooth Banach space and A:E→E∗ be a maximal monotone mapping.Then A–()is a closed and convex subset of E.
We denote the normal cone forCat a pointv∈CbyNC(v), that is,
NC(v) :=x∗∈E∗:v–y,x∗≥,∀y∈C.
Lemma .([]) Let C be a nonempty closed convex subset of a Banach space E,and let A be a monotone and hemicontinuous mapping of C into E∗with C=D(A).Let T⊂E×E∗ be a mapping defined as follows:
Tv:=
⎧ ⎨ ⎩
Av+NC(v), v∈C, ∅, v∈/C.
Then T is maximal monotone and T–() =VI(C,A).
Remark . It is well known that the monotone and hemicontinuous mappingAwith
D(A) =Eis maximal (see,e.g., []). Note that Lemma . is for the monotone and hemi-continuous mapping.
Remark . LetCbe a nonempty closed convex subset of a Banach spaceE, and letAbe a monotone and hemicontinuous mapping fromCintoE∗withC=D(A). Then
VI(C,A) =u∈C:v–u,Av ≥ for allv∈C.
It is obvious that the setVI(C,A) is a closed convex subset ofCand the setA– =VI(E,A) is a closed convex subset ofE(see []).
We make use of the functionV:E×E∗→Rdefined by
Vx,x∗=x– x,x∗+x∗
for allx∈Eandx∗∈E∗, which was studied by Alber []. That is,
Vx,x∗=φx,J–x∗
Lemma .([]) Let E be a reflexive,strictly convex and smooth Banach space with E∗ as its dual.Then
Vx,x∗+ J–x∗–x,y∗≤Vx,x∗+y∗
for all x∈E and x∗,y∗∈E∗.
Lemma .([]) Let C be a closed subset of a smooth,strictly convex and reflexive Banach space E.Let B:C→E∗ be a continuous and monotone mapping,ψ :C→Rbe a lower semicontinuous and convex function,andθ be a bifunction from C×C toRsatisfying
(E)-(E).Then,for r> and x∈E,there exists u∈C such that
θ(u,y) +Bu,y–u +ψ(y) –ψ(u) +
ry–u,Ju–Jx ≥, ∀y∈C.
Define a mappingTr:E→Cby
Tr(x) =
u∈C:θ(u,y) +Bu,y–u +ψ(y) –ψ(u)
+
ry–u,Ju–Jx ≥,∀y∈C
(.)
for allx∈E. Then the following conclusions hold: (i) Tris single-valued;
(ii) Tris a firmly nonexpansive type mapping [],i.e., for allx,y∈E,
Trx–Try,JTrx–JTry ≤ Trx–Try,Jx–Jy;
(iii) F(Tr) =GMEP=Fˆ(Tr);
(iv) GMEPis a closed and convex subset ofC; (v) φ(p,Trz) +φ(Trz,z)≤φ(p,z),∀p∈F(Tr),x∈E.
Remark .([]) The mappingTr:E→Cdefined by (.) is a relatively nonexpansive mapping. Thus, it is a hemi-relatively nonexpansive mapping.
3 An iterative scheme for a family of hemi-relatively nonexpansive mappings
In this section, we introduce a new hybrid iterative scheme for a common element of the solution set of problem (.), the solution set of problem (.) for an inverse strongly mono-tone mapping and the set of common fixed points of a family of hemi-relatively nonexpan-sive mappings.
Theorem . Let E be a real uniformly smooth and-uniformly convex Banach space and C be a nonempty,closed and convex subset of E.Let A:C→E∗be aγ-inverse strongly monotone mapping and B:C→E∗be a continuous and monotone mapping.Letψ:C→
Rbe a lower semicontinuous and convex function andθ be a bifunction from C×C to
Rsatisfying(E)-(E).Let{Tλ:λ∈ }be a family of closed hemi-relatively nonexpansive mappings of C into itself having
whereF=λ∈ F(Tλ)is the set of common fixed points of{Tλ}.Assume thatAx ≤ Ax– Apfor all x∈C and p∈VI(C,A).Suppose that <a<μn<b=c
γ
,where c is the constant in Lemma..Let{rn} ⊂[c∗, +∞)for some c∗> .Let{xn}be the sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C, chosen arbitrarily,
yn=CJ–(Jxn–μnAxn),
zn,λ=J–(αnJxn+ ( –αn)JTλyn),
un,λ=Trn,λzn,λ,
H,λ={v∈C:supλ∈ φ(v,u,λ)
≤αφ(v,x) + ( –α)φ(v,y)≤φ(v,x)}, Hn,λ={v∈Hn–,λ∩Wn–,λ:supλ∈ φ(v,un,λ)
≤αnφ(v,xn) + ( –αn)φ(v,yn)≤φ(v,xn)},
W,λ=C,
Wn,λ={v∈Hn–,λ∩Wn–,λ:xn–v,Jx–Jxn ≥},
xn+=Hn,λ∩Wn,λx, n≥,λ∈ ,
(.)
where J is the normalized duality mapping, and{αn} is a sequence in [, ] satisfying
lim infn→∞( –αn) > .Then{xn}converges tox,whereis the generalized projection of E onto.
Proof Step . We prove thatHn,λandWn,λboth are closed and convex subsets ofCand ⊂Hn,λ∩Wn,λwithn≥,λ∈ . In fact, it is obvious thatWn,λis closed and convex,
andHn,λis closed for eachn≥,λ∈ . Since
φ(v,un,λ)≤αnφ(v,xn) + ( –αn)φ(v,yn) ⇔ v,αnJxn+ ( –αn)Jyn–Jun,λ
≤( –αn)yn–un,λ+αnxn
and
αnφ(v,xn) + ( –αn)φ(v,yn)≤φ(v,xn) ⇔ φ(v,yn)≤φ(v,xn),
Hn,λis convex for eachn≥,λ∈ . HenceHn,λ∩Wn,λis closed and convex for alln≥, λ∈ .
Step . For any givenp∈, from (v) of Lemma . and thatTλis hemi-relatively non-expansive, we have
φ(p,u,λ) =φ(p,Tr,λz,λ)
≤φ(p,z,λ) =φp,J–αJx+ ( –α)JTλy
=p– p,αJx+ ( –α)JTλy
+αJx+ ( –α)JTλy
+αx+ ( –α)Tλy
=αφ(p,x) + ( –α)φ(p,Tλy)
≤αφ(p,x) + ( –α)φ(p,y) (.)
for eachλ∈ . From Lemma ., Lemma ., and the assumption ofA, we obtain
φ(p,y) =φp,CJ–(Jx–μAx)
≤φp,J–(Jx–μAx)
=V(p,Jx–μAx)
≤V(p,Jx–μAx+μAx)
– J–(Jx–μAx) –p,μAx
=V(p,Jx) – μ
J–(Jx–μAx) –J–(Jx),Ax
– μx–p,Ax
≤V(p,Jx) – μ
J–(Jx–μAx) –J–(Jx),Ax
– μx–p,Ax–Ap – μx–p,Ap
≤φ(p,x) +
μ c Ax
– μ
γAx–Ap
≤φ(p,x) + μ
μ c –γ
Ax–Ap
≤φ(p,x). (.)
From (.) and (.),
φ(p,u,λ)≤αφ(p,x) + ( –α)φ(p,y)≤φ(p,x) (.)
for eachλ∈ . Thus
sup
λ∈ φ(p,u,λ)≤αφ(p,x) + ( –α)φ(p,y)≤φ(p,x).
Therefore,p∈H,λandp∈H,λ∩W,λ. Suppose that⊂Hn–,λ∩Wn–,λ. Then, the
meth-ods in (.) and (.) imply that
φ(p,un,λ)≤αnφ(p,xn) + ( –αn)φ(p,yn)≤φ(p,xn), ∀λ∈ ,
which implies thatp∈Hn,λ. Sincexn=Hn–,λ∩Wn–,λx, it follows from Lemma . that xn–z,Jx–Jxn ≥, ∀z∈Hn–,λ∩Wn–,λ.
It implies thatxn–p,Jx–Jxn ≥. Hencep∈Wn,λ. Therefore,⊂Hn,λ∩Wn,λ. Then,
by induction onn,⊂Hn,λ∩Wn,λfor alln≥,λ∈ . From Lemma . and Lemma .,
Step . We prove that{xn}is a Cauchy sequence. Letp∈. From the definition ofHn,λ, Wn,λand Lemma ., we havexn=Hn–,λ∩Wn–,λxand
φ(p,xn) +φ(xn,x)≤φ(p,x).
Thus{xn}is bounded. Moreover, since
xn=Hn–,λ∩Wn–,λx, xn+=Hn,λ∩Wn,λx∈Hn,λ∩Wn,λ, we have
φ(xn+,xn) +φ(xn,x)≤φ(xn+,x),
which implies that{φ(xn,x)}is nondecreasing. It follows that the limit of{φ(xn,x)}exists. By the constructionHn,λ∩Wn,λ, one has that
Hm,λ∩Wm,λ⊂Hm–,λ∩Wm–,λ,
xm=Hm,λ∩Wm,λx∈Hn,λ∩Wn,λ
for any positive integerm≥n. From Lemma ., we have
φ(xm,xn) =φ(xm,Hn–,λ∩Wn–,λx) ≤φ(xm,x) –φ(Hn,λ∩Wn,λx,x)
=φ(xm,x) –φ(xn,x). (.)
Lettingm,n→ ∞in (.), we have
φ(xm,xn)→.
Thus, Lemma . implies that
lim
m,n→∞xm–xn= .
This implies that{xn}is a Cauchy sequence.
Step . Now, we prove that limn→∞yn–Tλyn = for each λ∈ and q ∈F =
λ∈ F(Tλ). Sincexn+∈Hn,λ, we obtain
φ(xn+,un,λ)≤φ(xn+,xn)
for allλ∈ and
φ(xn+,yn)≤φ(xn+,xn).
By (.) and Lemma ., we have
lim
Hence,
lim
n→∞xn–un,λ=nlim→∞xn–yn=nlim→∞un,λ–yn= (.)
for allλ∈ . By the methods in (.) and (.), we have
φ(p,un,λ)≤φ(p,zn,λ)≤φ(p,xn) (.)
for alln≥,λ∈ . SinceJis uniformly continuous on the bounded sets, it follows from Lemma .(v), (.) and (.) that for any givenp∈,
φ(un,λ,zn,λ) = φ(Trn,λzn,λ,zn,λ)
≤ φ(p,zn,λ) –φ(p,Trn,λzn,λ)
≤ φ(p,xn) –φ(p,un,λ)
= xn–un,λ– p,Jxn–Jun,λ
≤ xn–un,λ
xn+un,λ
+ pJxn–Jun,λ
→, asn→ ∞
for allλ∈ . From Lemma .,
lim
n→∞un,λ–zn,λ= . (.)
Thuslimn→∞zn,λ–xn= for eachλ∈ . SinceJis uniformly continuous on bounded
sets, we obtain
( –αn)Jxn–JTλyn = αnJxn+ ( –αn)JTλyn–Jxn
= Jzn,λ–Jxn
→
asn→ ∞. Sincelim infn→∞( –αn) > andJ–is uniformly continuous on bounded sets, we obtain
lim
n→∞xn–Tλyn= (.)
for allλ∈ . It follows from (.) and (.) that
yn–Tλyn ≤ yn–xn+xn–Tλyn →, asn→ ∞ (.)
for all λ∈ . Since {xn} is a Cauchy sequence, there exists a point q∈ C such that limn→∞xn=q. It follows from (.) thatyn→q. SinceTλis closed, from (.) we get
q∈F=λ∈ F(Tλ).
Step . Now, we show thatq∈VI(C,A)∩GMEP. LetS⊂E×E∗be a mapping as follows:
Sv=
⎧ ⎨ ⎩
By Lemma .,Sis maximal monotone, andS–() =VI(C,A). Let (v,w)∈G(S) (graph of S). Sincew∈Sv=Av+NC(v), we havew–Av∈NC(v). Moreover,yn∈Cimplies that
v–yn,w–Av ≥. (.)
On the other hand, fromyn=CJ–(Jxn–μnAxn) and Lemma ., we obtain that
v–yn,Jyn– (Jxn–μnAxn)
≥.
Hence,
v–yn,Jxn–Jyn
μn
–Axn
≤. (.)
From (.) and (.), we obtain
v–yn,w ≥ v–yn,Av
≥ v–yn,Av +
v–yn,Jxn–Jyn
μn
–Axn
=
v–yn,Av–Axn+
Jxn–Jyn
μn
=v–yn,Av–Ayn +v–yn,Ayn–Axn +
v–yn,Jxn–Jyn
μn
≥–v–ynAyn–Axn–v–yn
Jxn–Jyn
μn
≥–
γv–ynyn–xn–
μn
v–ynJxn–Jyn.
SinceJis uniformly continuous on bounded sets, by (.) we have
lim
n→∞v–yn,w =v–q,w ≥.
Thus, sinceSis maximal monotone, we haveq∈S–() andq∈VI(C,A). Next, we show
thatq∈GMEP=F(Tr,λ). Let
Hλ(un,λ,y) =θλ(un,λ,y) +Bun,λ,y–un,λ +ψ(y) –ψ(un,λ), ∀y∈C.
From (.) and (.), we obtainlimn→∞un,λ=qandlimn→∞zn,λ=qfor allλ∈ . SinceJis
uniformly continuous, from (.) we havelimn→∞Jun,λ–Jzn,λ= . Therefore, it follows
fromrn∈[c∗,∞) for somec∗> thatlimn→∞Jun,λr–nJzn,λ= . Sinceun,λ=Trn,λzn,λ, we have
Hλ(un,λ,y) +
rn
y–un,λ,Jun,λ–Jzn,λ ≥, ∀y∈C,λ∈ .
Combining the above inequality and (E), we get
y–un,λ ·
Jun,λ–Jzn,λ
rn ≥
rn
y–un,λ,Jun,λ–Jzn,λ = –Hλ(un,λ,y)
Taking the limit asn→ ∞in the above inequality and by (E), we haveHλ(y,q)≤ for all
y∈C,λ∈ . For anyt∈(, ) andy∈C, define
yt=ty+ ( –t)q∈C.
ThenHλ(yt,q)≤. From (E) and (E), we have
= Hλ(yt,yt) =Hλ
yt,ty+ ( –t)q
≤tHλ(yt,y) + ( –t)Hλ(yt,q) ≤tHλ(yt,y),
i.e.,Hλ(yt,y)≥, for allλ∈ . Thus, from (E) and lett↓, we haveHλ(q,y)≥ for all
y∈C,λ∈ . This implies thatq∈GMEP. Thereforeq∈.
Step . Finally, we prove thatq=x. Sincexn+=Hn,λ∩Wn,λxand by Lemma ., we have
xn+–z,Jx–Jxn+ ≥, ∀z∈Hn,λ∩Wn,λ. (.)
Taking the limit in (.) and from⊂Hn,λ∩Wn,λfor alln≥,λ∈ , we obtain
q–z,Jx–Jq ≥, ∀z∈.
Therefore, from Lemma ., we haveq=x.
Remark . An iterative scheme for finding a solution of the variational inequality prob-lem for a mappingAthat satisfies the following conditions in a -uniformly and uniformly smooth Banach spaceE:
() Ais inverse strongly monotone, () VI(C,A)=∅,
() Ax ≤ Ax–Aufor allx∈Candu∈VI(C,A).
If condition () holds, then we can prove a convergence theorem for variational inequal-ity problems. To consider the general variational inequalinequal-ity problem for inverse strongly monotone mappings, we have to assume condition () (see []).
For a practical case, we may apply this theorem to a finite number of mappings {T,T, . . . ,Tm}as follows.
Corollary . Let E be a real uniformly smooth and -uniformly convex Banach space and C be a nonempty, closed and convex subset of E. Let A:C→E∗ be a γ-inverse strongly monotone mapping and B:C→E∗be a continuous and monotone mapping.Let ψ:C→Rbe a lower semicontinuous and convex function andθbe a bifunction from C×C toRsatisfying(E)-(E).Let{T,T, . . . ,Tm}be a finite family of closed hemi-relatively non-expansive mappings of C into itself having
=F∩VI(C,A)∩GMEP=∅,
whereF=mk=F(Tk)is the set of common fixed points.Assume thatAx ≤ Ax–Ap for all x∈C and p∈VI(C,A).Suppose that <a<μn<b=c
γ
Lemma..Let{rn} ⊂[c∗, +∞)for some c∗> .Let{xn}be the sequence generated by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C, chosen arbitrarily, yn=CJ–(Jxn–μnAxn),
zn,k=J–(αnJxn+ ( –αn)JTkyn),
un,k=Trn,kzn,k,
H,k={v∈C:maxk=,,...,mφ(v,u,k)
≤αφ(v,x) + ( –α)φ(v,y)≤φ(v,x)}, Hn,k={v∈Hn–,k∩Wn–,k:maxk=,,...,mφ(v,un,k)
≤αnφ(v,xn) + ( –αn)φ(v,yn)≤φ(v,xn)},
W,k=C,
Wn,k={v∈Wn–,k∩Hn–,k:xn–v,Jx–Jxn ≥},
xn+=Hn,k∩Wn,kx, n≥,k= , , . . . ,m,
where J is the normalized duality mapping, and{αn} is a sequence in [, ] satisfying
lim infn→∞( –αn) > .Then{xn}converges tox,whereis the generalized projection of E onto.
IfE=His the Hilbert space, thenJ=J–=Iis the identity mapping onH. Then
Theo-rem . reduces to the following corollary.
Corollary . Let H be a real Hilbert space and C be a nonempty,closed and convex subset of H.Let A:C→H be a continuous and monotone mapping.Letψ:C→Rbe a lower semicontinuous and convex function andθbe a bifunction from C×C toRsatisfying(E)
-(E).Let{Tλ:C→C:λ∈ }be a family of closed hemi-relatively nonexpansive mappings with
=F∩VI(C,A)∩GMEP=∅,
whereF=λ∈ F(Tλ)is the set of common fixed points of{Tλ}.Assume thatAx ≤ Ax–
Apfor all x∈C and p∈VI(C,A).Suppose that <a<μn<b=c
γ
,where c is the constant in Lemma..Let{rn} ⊂[c∗, +∞)for some c∗> .Let{xn}be the sequence generated by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C, chosen arbitrarily, yn=PC(xn–μnAxn),
zn,λ=αnxn+ ( –αn)Tλyn,
un,λ=Trn,λzn,λ,
H,λ={v∈C:supλ∈ v–u,λ
≤αv–x+ ( –α)v–y≤ v–x}, Hn,λ={v∈Hn–,λ∩Wn–,λ:supλ∈ v–un,λ
≤αnv–xn+ ( –αn)v–yn≤ v–xn}, W,λ=C,
Wn,λ={v∈Wn–,λ∩Hn–,λ:xn–v,Jx–Jxn ≥},
where{αn}is a sequence in[, ]satisfyinglim infn→∞( –αn) > .Then{xn}converges to Px,where Pis the metric projection of H onto.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper was proposed by KSK. KSK prepared the manuscript initially, JKK and WHL confirmed all the steps of proofs in this research. All authors read and approved the final manuscript.
Author details
1Graduate School of Education, Mathematics Education, Kyungnam University, Changwon, Kyungnam 631-701, Republic
of Korea.2Department of Mathematics Education, Kyungnam University, Changwon, Kyungnam 631-701, Republic of Korea.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A4A01010526).
Received: 12 July 2013 Accepted: 1 December 2013 Published:02 Jan 2014
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10.1186/1029-242X-2014-2
