R E S E A R C H
Open Access
A new iteration process for equilibrium,
variational inequality, fixed point problems,
and zeros of maximal monotone operators in
a Banach space
Siwaporn Saewan
1*and Poom Kumam
2**Correspondence:
[email protected]; [email protected]
1Department of Mathematics and
Statistics, Faculty of Science, Thaksin University (TSU), Pa Phayom, Phatthalung, 93110, Thailand
2Department of Mathematics,
Faculty of Science, King Mongkut’s University of Technology Thonburi, Bang Mod, Bangkok, 10140, Thailand
Abstract
In this article, a new iterative process is introduced to approximate a common element of a fixed point set, the solutions of equilibrium problems, the solution set of variational inequality problems, and the set of zeros of maximal monotone operators in a uniformly smooth and strictly convex Banach space by using a hybrid projection method. Also, we prove new strong convergence theorems for this proposed iterative precess in a Banach space.
MSC: 47H05; 47H09; 47H10
Keywords: maximal monotone mappings; strong convergence; total quasi-φ-asymptotically nonexpansive mappings; hybrid scheme; equilibrium problem; variational inequality problems
1 Introduction
LetEbe a real Banach space,E*be the dual space ofE. A set-valued mappingA:D(A)⊂ E→E*with graphG(A) ={(x,x*) :x*∈Ax}, domainD(A) ={x∈E:Ax=∅}, and range R(A) =∪{Ax:x∈D(A)}.Ais said to bemonotoneifx–y,x*–y* ≥ wheneverx*∈Ax,
y*∈Ay. A monotone operatorAis said to bemaximal monotoneif its graph is not properly
contained in the graph of any other monotone operator. LetA⊂E×E*be a maximal
monotone operator. We consider the problem for findingx∈E
∈Ax, (.)
a pointx∈Eis called azero pointofA. Denote byA– the set of all pointsx∈Esuch
that ∈Ax. We know that ifAis maximal monotone, then the solution setA– ={x∈ D(A) : ∈Tx}is closed and convex. One popular algorithm for approximating a solution of this problem is called the proximal point algorithm which was first proposed by Mar-tinet [] and studied further by Rockafellar [] in Hilbert spaces. Since the proximal point algorithm weakly converges in general which is the proximal point algorithm is defined by
x∈Eand
xn+=Jrnxn, forn= , , , , . . . , (.)
where{rn} ⊂(,∞) andJrnare the resolvent ofA. Solovov and Svaitor [] proposed a
mod-ified proximal point algorithm which converges strongly to a solution of the equationA–
by using the projection method. Many problems in nonlinear analysis and optimization can be formulated by the proximal point algorithm (see [–]).
LetEbe a real Banach space with dualE*and letCbe a nonempty closed and convex
subset ofE. Letf :C×C→Rbe a bifunction. Theequilibrium problemis to findx∈C
such that
f(x,y)≥, ∀y∈C. (.)
The equilibrium problem is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, min-max problems, saddle point problem, fixed point problem, Nash EP. In , Takahashi and Zembayashi [, ] intro-duced iterative sequences for finding a common solution of an equilibrium problem and a fixed point problem.
A mappingA:D(A)⊂E→E*is said to beα-inverse-strongly monotoneif there exists a
constantα> such that
x–y,Ax–Ay ≥αAx–Ay, ∀x,y∈C.
IfAisα-inverse strongly monotone, then it is α-Lipschitz continuous,i.e.,
Ax–Ay ≤
αx–y, ∀x,y∈C.
LetC be a nonempty closed and convex subset of a real Banach spaceE. LetAbe a monotone operator fromCintoE.The variational inequality problemfor an operator A is to findzˆ∈Csuch that
y–ˆz,Azˆ ≥, ∀y∈C. (.)
The set of solutions of (.) is denoted byVI(A,C).
LetCbe a nonempty closed and convex subset ofE. A mappingT fromCinto itself is said to benonexpansiveif
Tx–Ty ≤ x–y, ∀x,y∈C.
T is said to betotal asymptotically nonexpansiveif there exist nonnegative real se-quencesνn,μnwithνn→,μn→ asn→ ∞and a strictly increasing continuous func-tionϕ:R+→R+withϕ() = such that
Tnx–Tny≤ x–y+μnψx–y+νn, ∀x,y∈C,∀n≥.
T [] ifCcontains a sequence{xn}which converges weakly topsuch thatlimn→∞xn–
Txn= . The asymptotic fixed point set ofTis denoted byF(T).
The value ofx*∈E*atx∈Ewill be denoted byx,x* orx*(x). For eachp> , the gener-alized duality mapping Jp:E→E
*
is defined by
Jp(x) =x*∈E*:x,x*=xp,x*=xp–
for all x∈E. In particular, J =J is called the normalized duality mapping. If E is a
Hilbert space, thenJ=I, whereIis the identity mapping. Consider the functional defined by
φ(x,y) =x– x,Jy +y forx,y∈E. (.)
If E is a Hilbert space, then φ(y,x) =y–x. It is obvious from the definition of φ
that
y–x≤φ(y,x)≤y+x, ∀x,y∈E. (.)
Tis said to beφ-nonexpansive[, ] if
φ(Tx,Ty)≤φ(x,y), ∀x,y∈C.
Tis said to bequasi-φ-nonexpansive[, ] ifF(T)=∅and φ(p,Tx)≤φ(p,x), ∀x∈Candp∈F(T).
Tis said to beasymptoticallyφ-nonexpansive[] if there exists a sequence{kn} ⊂[,∞) withkn→ asn→ ∞such that
φTnx,Tny≤knφ(x,y), ∀x,y∈C.
T is said to bequasi-φ-asymptotically nonexpansive[] if F(T)=∅and there exists a sequence{kn} ⊂[,∞) withkn→ asn→ ∞such that
φp,Tnx≤knφ(p,x), ∀x∈C,p∈F(T),∀n≥.
Tis said to betotal quasi-φ-asymptotically nonexpansiveifF(T)=∅and there exist non-negative real sequencesνn,μnwithνn→,μn→ asn→ ∞and a strictly increasing continuous functionϕ:R+→R+withϕ() = such that
φp,Tnx≤φ(p,x) +νnϕ
φ(p,x)+μn, ∀n≥,∀x∈C,p∈F(T).
A mappingTis said to beuniformly L-Lipschitz continuous, if there exists a constantL> such that
T is said to be closed if for any sequence {xn} ⊂ C such that limn→∞xn = x and
limn→∞Txn=y,Tx=y.
Remark . Every quasi-φ-nonexpansive mapping implies a quasi-φ-asymptotically
non-expansive mapping and a quasi-φ-asymptotically nonnon-expansive mapping implies a total quasi-φ-asymptotically nonexpansive mapping, but the converse is not true.
On the other hand, Alber [] introduced that thegeneralized projectionC:E→Cis a map that assigns to an arbitrary pointx∈Ethe minimum point of the functionalφ(x,y), that is,Cx=x¯, wherex¯is the solution of the minimization problem
φ(x¯,x) =inf
y∈Cφ(y,x). (.)
The existence and uniqueness of the operatorCfollow from the properties of the func-tionalφ(x,y) and strict monotonicity of the mappingJ. LetCbe the generalized projec-tion from a smooth strictly convex and reflexive Banach spaceEonto a nonempty closed convex subsetCofE. ThenCis a closed relatively quasi-nonexpansive mapping fromE ontoCwithF(C) =C.
Matsushita and Takahashi [] proposed the following hybrid iteration method with a generalized projection for a relatively nonexpansive mappingT in a Banach spaceE:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C chosen arbitrarily, yn=J–(αnJxn+ ( –αn)JTxn),
Cn={z∈C:φ(z,yn)≤φ(z,xn)},
Qn={z∈C:xn–z,Jx–Jxn ≥},
xn+=Cn∩Qnx.
(.)
They proved that{xn}converges strongly toF(T)x. Many authors studied the methods for approximating fixed points of a countable family of (relatively quasi-) nonexpansive mappings (see [–]).
Recently, Qinet al.[] considered a pair of asymptotically quasi-φ-nonexpansive map-pings. To be more precise, they proved the following results.
Theorem QCK Let E be a uniformly smooth and uniformly convex Banach space and C
be a nonempty closed and convex subset of E.Let T:C→C be a closed and asymptotically quasi-φ-nonexpansive mapping with the sequence{k(nt)} ⊂[,∞)such that kn(t)→as n→
∞and S:C→C be a closed and asymptotically quasi-φ-nonexpansive mapping with the sequence{k(st)} ⊂[,∞)such that k(ns)→as n→ ∞.Let{αn},{βn},{γn},and{δn}be real
following manner: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E chosen arbitrarily,
C=C,
x=Cx,
zn=J–(βnJxn+γnJ(Tnxn) +δnJ(Snxn)),
yn=J–(αnJxn+ ( –αn)Jzn),
Cn+={w∈Cn:φ(w,yn)≤φ(w,xn) + (kn– )Mn},
xn+=Cn+x,
(.)
where kn=max{kn(t),kn(s)}for each n≥,J is the duality mapping on E,Mn=sup{φ(z,xn) :
z∈}for each n≥.Assume that the control sequences{αn},{βn},{γn},and{δn}satisfy
the following restrictions: (a) βn+γn+δn= ,∀n≥;
(b) lim infn→∞γnδn,limn→∞βn= ; (c) ≤αn< andlim supn→∞αn< .
In , Alberet al.[] proved the strong convergence theorems to approximate a fixed point of a total asymptotically nonexpansive mapping in a Hilbert space. In , Changet al.[, ] proved the strong convergence theorems for finding the set of fixed points of a total quasi-φ-asymptotically nonexpansive mapping in the framework of Banach spaces.
Motivated and inspired by the work mentioned above, in this paper, we introduce a new hybrid projection algorithm for a pair of total quasi-φ-asymptotically nonexpansive mappings for finding a set of solutions of the equilibrium problem, a zero point of maxi-mal monotone operators, and a set of solutions of the variation inequality in a uniformly smooth and strictly convex Banach space.
2 Preliminaries
In this article, we denote the strong convergence and weak convergence of a sequence{xn} byxn→xandxnx, respectively.
A Banach spaceEwith the norm · is calledstrictly convexifx+y< for allx,y∈E
withx=y= andx=y. LetU={x∈E:x= }be the unit sphere ofE. A Banach spaceEis calledsmoothif the limitlimt→x+tyt–xexists for eachx,y∈U. It is also called uniformly smoothif the limit exists uniformly for allx,y∈U. Themodulus of convexityof
Eis the functionδ: [, ]→[, ] defined by
δ(ε) =inf
–x+y
:x,y∈E,x=y= ,x–y ≥ε
.
A Banach spaceEisuniformly convexif and only ifδ(ε) > for allε∈(, ]. Letpbe a fixed real number withp≥. A Banach spaceEis said to bep-uniformly convexif there exists a constantc> such thatδ(ε)≥cεpfor allε∈[, ]. Observe that everyp-uniformly convex is uniformly convex. One should note that no Banach space isp-uniformly convex for <p< .
• IfEis an arbitrary Banach space, thenJis monotone and bounded; • IfEis strictly convex, thenJis strictly monotone;
• IfEis smooth, thenJis single-valued and semi-continuous;
• IfEis uniformly smooth, thenJis uniformly norm-to-norm continuous on each bounded subset ofE;
• IfEis reflexive smooth and strictly convex, then the normalized duality mappingJis single-valued, one-to-one, and onto;
• IfEis a reflexive strictly convex and smooth Banach space andJis the duality mapping fromEintoE*, thenJ–is also single-valued, bijective and is also the duality
mapping fromE*intoE, and thusJJ–=IE* andJ–J=IE;
• IfEis uniformly smooth, thenEis smooth and reflexive; • IfEis a reflexive and strictly convex Banach space, thenJ–is
norm-weak*-continuous.
Remark . IfEis a reflexive strictly convex and smooth Banach space, thenφ(x,y) = if and only ifx=y. It is sufficient to show that ifφ(x,y) = , thenx=y. From (.), we have
x=y. This implies thatx,Jy =x=Jy. From the definition ofJ, one hasJx=Jy.
Therefore, we havex=y(see [–] for more details).
Recall that a Banach spaceEhas the Kadec-Klee property [, , ] if for any sequence
{xn} ⊂Eandx∈Ewithxnxandxn → x, thenxn–x → asn→ ∞. It is well known that ifEis a uniformly convex Banach space, thenEhas the Kadec-Klee property. The generalized projection [] fromEintoCis defined byC(x) =argminy∈Cφ(y,x). The existence and uniqueness of the operatorCfollow from the properties of the func-tionalφ(y,x) and the strict monotonicity of the mappingJ(see, for example, [, , , , ]). IfEis a Hilbert space, thenφ(x,y) =x–yand
Cbecomes the metric projection
PC:H→C. IfCis a nonempty closed and convex subset of a Hilbert spaceH, thenPC is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. We also need the following lemmas for the proof of our main results.
Lemma .(Alber []) Let C be a nonempty closed convex subset of a smooth Banach
space E and let x∈E.Then x=Cx if and only if
x–y,Jx–Jx ≥, ∀y∈C.
Lemma .(Alber []) Let E be a reflexive strictly convex and smooth Banach space,C
be a nonempty closed convex subset of E and let x∈E.Then
φ(y,Cx) +φ(Cx,x)≤φ(y,x), ∀y∈C.
Lemma . (Changeet al. []) Let C be a nonempty closed and convex subset of a
uniformly smooth and strictly convex Banach space E with the Kadec-Klee property.Let S:C→C be a closed and total quasi-φ-asymptotically nonexpansive mapping with non-negative real sequencesνnandμnwithνn→,μn→as n→ ∞and a strictly increasing
continuous functionζ :R+→R+withζ() = .Ifμn= ,then the fixed point set F(S)is a
For solving the equilibrium problem for a bifunctionf :C×C→R, let us assume that
f satisfies the following conditions: (A) f(x,x) = for allx∈C;
(A) f is monotone,i.e.,f(x,y) +f(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,
lim
t↓f
tz+ ( –t)x,y≤f(x,y);
(A) for eachx∈C,y→f(x,y)is convex and lower semi-continuous. The following result is in Blum and Oettli [].
Lemma .(Blum and Oettli []) Let C be a closed convex subset of a smooth,strictly
convex,and reflexive Banach space E, let f be a bifunction from C×C toRsatisfying
(A)-(A),and let r> and x∈E.Then there exists z∈C such that
f(z,y) +
ry–z,Jz–Jx ≥, ∀y∈C.
Lemma .(Takahashi and Zembayashi []) Let C be a closed convex subset of a
uni-formly smooth strictly convex and reflexive Banach space E and let f be a bifunction from C×C toRsatisfying conditions (A)-(A). For all r> and x∈E, define a mapping Kr:E→C as follows:
Krx=
z∈C:f(z,y) +
ry–z,Jz–Jx ≥,∀y∈C
.
Then the following hold: () Kris single-valued;
() Kris a firmly nonexpansive-type mapping[],that is,for allx,y∈E,
Krx–Kry,JKrx–JKry ≤ Krx–Kry,Jx–Jy;
() F(Kr) =EP(f);
() EP(f)is closed and convex.
Lemma .(Takahashi and Zembayashi []) Let C be a closed convex subset of a smooth
strictly convex and reflexive Banach space E,let f be a bifunction from C×C toRsatisfying
(A)-(A)and let r> .Then,for x∈E and q∈F(Kr), φ(q,Krx) +φ(Krx,x)≤φ(q,x).
Lemma .[] Let E be a uniformly convex Banach space and Br() ={x∈E:x ≤r} be a closed ball of E. Then there exists a continuous strictly increasing convex function g: [,∞)→[,∞)with g() = such that
λx+μy+γz≤ λx+|μy+γz–λμgx–y
LetEbe a smooth strictly convex and reflexive Banach space,Cbe a nonempty closed convex subset of E and A⊂E ×E* be a monotone operator satisfying D(A)⊂C⊂ J–(
λ>R(J+λA)). Then theresolvent Jλ:C→D(A) ofAis defined by
Jλx=
z∈D(A) :Jx∈Jz+λAz,∀x∈C .
Jλis a single-valued mapping fromE toD(A). For anyλ> , theYosida approximation
Aλ:C→E*ofAis defined byAλx=Jx–λJJλxfor allx∈C. We know thatAλx∈A(Jλx) for all λ> andx∈E.
Lemma .(Kohsaka and Takahashi []) Let E be a smooth strictly convex and reflexive
Banach space,C be a nonempty closed convex subset of E and A⊂E×E*be a monotone operator satisfying D(A)⊂C⊂J–(
λ>R(J+λA)).For anyλ> ,let Jλ and Aλ be the
resolvent and the Yosida approximation of A,respectively.Then the following hold: (a) φ(p,Jλx) +φ(Jλx,x)≤φ(p,x)for allx∈Candp∈A–;
(b) (Jλx,Aλx)∈Afor allx∈C; (c) F(Jλ) =A–.
Lemma .(Rockafellar []) Let E be a reflexive strictly convex and smooth Banach
space.Then an operator A⊂E×E*is maximal monotone if and only if R(J+λA) =E*for allλ> .
3 Main result
Theorem . Let C be a nonempty closed and convex subset of a uniformly smooth and
strictly uniformly convex Banach space E with the Kadec-Klee property. Let f be a bi-function from C×C toRsatisfying the conditions(A)-(A)and let A⊂E×E* be a maximal monotone operator satisfying D(A)⊂C and Jrn = (J+rnA)–J for all rn > . Let S:C→C be a closed and total quasi-φ-asymptotically nonexpansive mapping with nonnegative real sequences νnS, μSn with νnS→,μSn→ as n→ ∞and a strictly in-creasing continuous functionψS:R+→R+ withψS() = .Let T :C→C be a closed
and total quasi-φ-asymptotically nonexpansive mapping with nonnegative real sequences
νnT,μTn withνnT →,μTn →as n→ ∞and a strictly increasing continuous function
ψT:R+→R+withψT() = .Assume that S and T are uniformly L-Lipschitz continu-ous and F=F(S)∩F(T)∩EP(f)∩A–=∅.For an initial point x∈E,C=C,define the sequence{xn}by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
zn=Jrnxn, un=Krnxn,
yn=J–(αnJxn+βnJSnzn+γnJTnun),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn) +ζn},
xn+=Cn+x, n∈N,
(.)
where{αn},{βn},and{γn}are sequences in(, )such thatαn+βn+γn= ,{rn} ⊂[d,∞)
for some d> μn=sup{μSn,μTn}, νn=sup{νSn,νnT}, ψ =sup{ψS,ψT} for all n≥,ζ = νnsupq∈Fψ(φ(q,xn)) +μn.Iflimn→∞αnβn= andlim infn→∞αnγn< ,then{xn}converges
Proof First, we show thatCnis closed and convex for alln∈NsinceC=Cis convex. Suppose thatCnis convex for alln∈N. For anyv∈Cn, we know thatφ(v,yn)≤φ(v,xn) +ζn is equivalent to
v,Jxn–Jyn ≤ xn–yn+ζn.
That is,Cn+is convex for alln∈N. By the definition ofCn, it is obvious thatCnis closed for alln∈N.
We show that{xn}is well defined. It is obvious thatF⊂C=C. SupposeF⊂Cnfor
n∈N, from Lemma . and Lemma .,S,Tare total quasi-φ-asymptotically nonexpan-sive mappings. For eachq∈F⊂Cn, it follows that
φ(q,yn) =φq,J–αnJxn+βnJSnzn+γnJTnun
=q– q,αnJxn+βnJSnzn+γnJTnun
+αnJxn+βnJSnzn+γnJTnun
≤αnφ(q,xn) +βnφ
q,Snzn
+γnφ
q,Tnun
≤αnφ(q,xn) +βn
φ(q,zn) +νnSψSφ(q,zn)+μSn
+γn
φ(q,un) +νnTψTφ(q,un)+μTn
=αnφ(q,xn) +βnφ(q,zn) +βnνnSψS
φ(q,zn)
+βnμSn
+γnφ(q,un) +γnνnTψT
φ(q,un)+γnμTn
≤αnφ(q,xn) +βnφ(q,zn) +βnνnSψS
φ(q,xn)+βnμSn
+γnφ(q,un) +γnνnTψT
φ(q,xn)
+γnμTn
≤αnφ(q,xn) +βnφ(q,zn) +βnνnψ
φ(q,xn)+βnμn
+γnφ(q,un) +γnνnψ
φ(q,xn)+γnμn
≤αnφ(q,xn) +βnφ(q,zn) +γnφ(q,un) + ( –αn)νnψ
φ(q,xn)
+ ( –αn)μn
≤αnφ(q,xn) +βnφ(q,zn) +γnφ(q,un) +νnsup
q∈F
ψφ(q,xn)+μn
≤αnφ(q,xn) +βnφ(q,zn) +γnφ(q,un) +ζn
≤αnφ(q,xn) +βnφ(q,xn) +γnφ(q,un) +ζn
≤αnφ(q,xn) +βnφ(q,xn) +γnφ(q,Krnxn) +ζn
≤αnφ(q,xn) +βnφ(q,xn) +γnφ(q,xn) +ζn
≤φ(q,xn) +ζn, (.)
whereζn=νnsupq∈Fψ(φ(q,xn)) +μn. This shows thatq∈Cn+, thusF⊂Cn+. Hence,F⊂ Cnfor alln≥. This implies that the sequence{xn}is well defined.
We show thatlimn→∞xn=p. From the definition ofCn+ withxn=Cnx andxn+=
Cn+x∈Cn+⊂Cn, it follows that
By Lemma ., we get
φ(xn,x) =φ(Cnx,x)
≤φ(q,x) –φ(q,xn)
≤φ(q,x), ∀q∈F. (.)
From (.) and (.), we have thatlimn→∞φ(xn,x) exists. In particular, it follows from
(.) that the sequence{xn}is bounded and so are{zn},{un}, and{yn}. Sincexn∈Cn⊂E andEis reflexive, the sequence{xn}converges weakly to an element ofE, we assume that
xnp. Note thatCnis closed and convex andxn∈Cn. We have thatp∈Cn, that is,
xnp∈Cn asn→ ∞. (.)
Forp∈Cn, we have
lim inf
n→∞ φ(xn,x) =lim infn→∞
xn– xn,Jx +x
≥ p– p,Jx +x
=φ(p,x).
On the other hand,xn=Cnx, we have
φ(xn,x)≤φ(p,x), ∀p∈Cn.
It follows that
φ(p,x)≤lim inf
n→∞ φ(xn,x)≤lim supn→∞ φ(xn,x)≤φ(p,x).
This implies thatlimn→∞φ(xn,x) =φ(p,x). Hence, we get
xn → p asn→ ∞. (.)
From (.), (.), and the Kadec-Klee property ofE, we have
lim
n→∞xn=p. (.)
Therefore,
lim
n→∞ζn=nlim→∞νnsupq∈Fψ
φ(q,xn)+μn= . (.)
From (.), it follows that
lim
and hence
lim
n→∞Jxn–Jxn+= . (.)
We show thatp∈F(S)∩F(T)∩A–∩EP(f).
Now, we show thatp∈EP(f). Forxn+∈Cn+⊂Cnandxn=Cnx, it follows that
φ(xn+,xn) =φ(xn+,Cnx)
≤φ(xn+,x) –φ(Cnx,x)
=φ(xn+,x) –φ(xn,x).
Sincelimn→∞φ(xn,x) exists, we have
lim
n→∞φ(xn+,xn) = . (.)
Sincexn+⊂Cnand the definition ofCn+, we have φ(xn+,yn)≤φ(xn+,xn) +ζn. From
(.), we also have
lim
n→∞φ(xn+,yn) = . (.)
From (.) and (.), it follows that
yn → p asn→ ∞, (.)
and hence
Jyn → Jp asn→ ∞. (.)
This implies that{Jyn}is bounded. Note thatEis reflexive andE*is also reflexive, we can assume thatJynx*∈E*. SinceEis reflexive, we see thatJ(E) =E*. Hence, there exists
x∈Esuch thatJx=x*and we have
φ(xn+,yn) =xn+– xn+,Jyn +yn =xn+– xn+,Jyn +Jyn.
Takinglim infn→∞on the both sides of the equality above, in view of the weak lower
semi-continuity of the norm · , it follows that
≥ p– p,x*+x*
=p– p,Jx +Jx
=p– p,Jx +x
From Remark ., we havep=x, which implies thatJynJpasn→ ∞. From the Kadec-Klee property ofE*, we obtain that
Jyn→Jp asn→ ∞. (.)
Note thatJ–:E*→Eis demicontinuous, that is,y
npasn→ ∞. From the Kadec-Klee property ofE, it follows that
lim
n→∞yn=p. (.)
From (.), (.), and (.), it follows thatlimn→∞φ(q,un) =φ(q,p). Sinceun=Krnxnand
from Lemma ., we have
φ(un,xn) =φ(Krnxn,xn)≤φ(q,xn) –φ(q,Krnxn) =φ(q,xn) –φ(q,zn)→ asn→ ∞.
From (.), it follows that
un → p asn→ ∞. (.)
Since{un}is bounded andEis also reflexive, we can assume thatunu∈Eand we have φ(un,xn) =un– un,Jxn +xn.
Takinglim infn→∞on the both sides of the equality above, in view of the weak lower semi-continuity of the norm · , it follows that
≥ u– u,Jp +p
=φ(u,p).
From Remark ., we haveu=p, that is,unpasn→ ∞. From the Kadec-Klee property ofE, we obtain that
lim
n→∞un=p. (.)
Sincelimn→∞un=pandlimn→∞xn=p, we have that
lim
n→∞un–xn= . (.)
SinceJis uniformly norm-to-norm continuous, we obtain
lim
n→∞Jun–Jxn= .
Fromrn> , we haveJun–rnJxn→ asn→ ∞and
f(un,y) +
rn
By (A),
y–un
Jun–Jxn
rn ≥
rn
y–un,Jun–Jxn
≥–f(un,y)
≥f(y,un), ∀y∈C,
andun→p, we getf(y,p)≤ for ally∈C. For <t< , defineyt=ty+ ( –t)p. Then
yt∈C, which implies thatf(yt,p)≤. From (A), we obtain that
=f(yt,yt)≤tf(yt,y) + ( –t)f(yt,p)≤tf(yt,y).
Thusf(yt,y)≥. From (A), we havef(p,y)≥ for ally∈C. Hence,p∈EP(f).
Next, we show that p∈ A–. From (.), (.), (.), and (.), it follows that limn→∞φ(q,zn) =φ(q,p). Sincezn=Jrnxnand from Lemma ., we have
φ(zn,xn) =φ(Jrnxn,xn)≤φ(q,xn) –φ(q,Jrnxn) =φ(q,xn) –φ(q,zn)→ asn→ ∞.
From (.), it follows that
zn → p asn→ ∞. (.)
Since{un}is bounded andEis also reflexive, we can assume thatznz∈Eand we have
φ(zn,xn) =zn– zn,Jxn +xn.
Takinglim infn→∞on the both sides of the equality above, in view of the weak lower semi-continuity of the norm · , it follows that
≥ z– z,Jp +p
=φ(z,p).
From Remark ., we havez=p, that is,unpasn→ ∞. From the Kadec-Klee property ofE, we obtain that
lim
n→∞zn=p. (.)
Sincelimn→∞zn=pandlimn→∞xn=p, we have that
lim
n→∞zn–xn= , (.)
and hence
lim
From the condition{rn} ⊂[d,∞) for somed> , we have
lim
n→∞
rn
Jxn–Jzn= .
Thus, sincezn=Jrnxn, we have
lim
n→∞Arnxn=nlim→∞
rn
Jxn–Jzn= .
For any (w,w*)∈G(A), it follows from the monotonicity ofAthatw–zn,w*–A
rnxn ≥
for all n≥. Letting n→ ∞, we get w–p,w* ≥. Therefore, sinceAis maximal
monotone, we obtainp∈A–. On the other hand, we have
φ(q,xn) –φ(q,yn) =xn–yn– q,Jxn–Jun
≤ xn–yn
xn+yn
+ qJxn–Jyn.
In view ofxn–yn → andJxn–Jyn → asn→ ∞, we obtain that
φ(q,xn) –φ(q,yn)→ asn→ ∞. (.)
From Lemma ., we have
φ(q,yn) =φq,J–αnJxn+βnJSnzn+γnJTnun
≤ q– q,αnJxn+βnJSnzn+γnJTnun
+αnJxn+βnJSnzn+γnJTnun
–αnβng
Jxn–JSnzn
=αnφ(q,xn) +βnφq,Snzn
+γnφq,Tnun
–αnβngJxn–JSnzn
≤φ(q,xn) +ζn–αnβng
Jxn–JSnzn
. (.)
It follows fromlim infn→∞αnβn> , (.), (.), and the property ofgthat
lim
n→∞Jxn–JS nz
n= .
Sincexn→pasn→ ∞andJis uniformly continuous, it yields thatJxn→Jp, we have
JSnzn→Jp. (.)
SinceJ–is demicontinuous, we also have
Snznp. (.)
On the other hand, we observe that
Snzn–p=J
Snzn–Jp≤J
Snzn
we obtain thatSnz
n → p. SinceEhas the Kadee-Klee property, we get
lim
n→∞S nz
n=p. (.)
By the assumption thatSis uniformlyL-Lipschitz continuous, we have
Sn+zn–Snzn≤Sn+zn–Sn+zn++Sn+zn+–zn++zn+–zn+zn–Snzn
≤(L+ )zn+–zn+Sn+zn+–zn++zn–Snzn. (.)
Sincelimn→∞zn=pandlimn→∞Snzn=p, it yields thatSn+zn–Snzn →,n→ ∞. From
Snz
n→p, we getSn+zn→p, that is,SSnzn→p. In view of the closeness ofS, we have
Sp=p. This implies thatp∈F(S). By the same way, we have thatp∈F(T).
We show thatp=Fx. Fromxn=Cnx, we haveJx–Jxn,xn–v ≥,∀v∈Cn. Since
F⊂Cn, we also have
Jx–Jxn,xn–y ≥, ∀y∈F.
By taking limitn→ ∞, we obtain that
Jx–Jp,p–y ≥, ∀y∈F.
By Lemma ., we can conclude thatp=Fxandxn→pasn→ ∞. The proof is
com-pleted.
LetAbe a continuous and monotone operator ofCintoE*. Then we can find a solution
ofVI(A,C) in a uniformly smooth and strictly convex Banach spaceEwith the Kadec-Klee property by using the following lemma.
Lemma .(Zegeye and Shahzad []) Let C be a nonempty closed convex subset of a
uniformly smooth strictly convex real Banach space E.Let A:C→E* be a continuous monotone mapping.For any r> ,define a mapping Wr:E→C as follows:
Wrx=
z∈C:y–z,Az +
ry–z,Jz–Jx ≥,∀y∈C
for all x∈C.Then the following hold: () Wris single-valued;
() F(Wr) =VI(A,C);
() VI(A,C)is a closed and convex subset ofC; () φ(q,Wrx) +φ(Wrx,x)≤φ(q,x)for allq∈F(Wr).
Corrollary . Let C be a nonempty closed and convex subset of a uniformly smooth and
T:C→C be a closed and total quasi-φ-asymptotically nonexpansive mapping with non-negative real sequencesνnT,μTn withνnT→,μTn→as n→ ∞and a strictly increasing continuous functionψT:R+→R+withψT() = .Assume that S and T are uniformly
L-Lipschitz continuous and F =F(S)∩F(T)∩EP(f)∩VI(A,C)=∅.For an initial point x∈E,C=C,define the sequence{xn}by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
zn=Wrnxn,
un=Krnxn,
yn=J–(αnJxn+βnJSnzn+γnJTnun),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn) +ζn},
xn+=Cn+x, n∈N,
(.)
where{αn},{βn}and{γn}are sequences in(, )such thatαn+βn+γn= ,{rn} ⊂[d,∞)
for some d> μn=sup{μSn,μTn}, νn=sup{νSn,νnT}, ψ =sup{ψS,ψT} for all n≥,ζ = νnsupq∈Fψ(φ(q,xn)) +μn.Iflimn→∞αnβn= andlim infn→∞αnγn< ,then{xn}converges
strongly toFx.
Proof From the proof of Theorem ., we known thatlimn→∞zn=pandlimn→∞xn=p. We obtain that
lim
n→∞zn–xn= , (.)
and hence
lim
n→∞Jxn–Jzn= .
Since{rn} ⊂[d,∞) for somed> , we have
lim
n→∞
rn
Jxn–Jzn= . (.)
From the definition ofW, it follows that
y–zn,Anzn +
rn
y–zn,Jzn–Jxn ≥, ∀y∈C.
For <t< , defineyt=ty+ ( –t)p, thenyt∈C. We have
yt–zn,Anzn +
rn
yt–zn,Jzn–Jxn ≥, ∀yt∈C. (.)
It follows that
yt–zn,Anyt ≥ yt–zn,Anyt –yt–zn,Anzn +
rn
yt–zn,Jzn–Jxn
=yt–zn,Anyt–Anzn +
yt–zn,
Jzn–Jxn
rn
SinceAnare continuous and monotone mappings, we haveyt–zn,Anyt–Anzn ≥. From (.) and (.), it follows thatyt–zn,Anyt ≥. Take the limit asn→ ∞andzn→p. We getyt–p,Anyt ≥ for allyt∈C. Therefore,y–p,Anyt ≥ for ally∈C. Ift→, we have thaty–p,Anp ≥ for ally∈C. Hence,p∈VI(A,C). The proof is completed.
Corrollary . Let C be a nonempty closed and convex subset of a uniformly smooth and
strictly uniformly convex Banach space E with the Kadec-Klee property.Let A be a continu-ous and monotone operator of C into E*and let B⊂E×E*be a maximal monotone operator satisfying D(B)⊂C and Jrn= (J+rnB)–J for all rn> .Let S:C→C be a closed and
to-tal quasi-φ-asymptotically nonexpansive mapping with nonnegative real sequencesνS n,μSn
withνS
n→,μSn→as n→ ∞and a strictly increasing continuous functionψS:R+→
R+withψS() = .Let T:C→C be a closed and total quasi-φ-asymptotically
nonexpan-sive mapping with nonnegative real sequencesνnT,μTn withνnT→,μTn →as n→ ∞ and a strictly increasing continuous functionψT:R+→R+withψT() = .Assume that
S and T are uniformly L-Lipschitz continuous and F=F(S)∩F(T)∩VI(A,C)∩B–=∅. For an initial point x∈E,C=C,define the sequence{xn}by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
zn=Jrnxn, un=Wrnxn,
yn=J–(αnJxn+βnJSnzn+γnJTnun),
Cn+={v∈Cn:φ(v,yn)≤φ(v,xn) +ζn},
xn+=Cn+x, n∈N,
(.)
where{αn},{βn},and{γn}are sequences in(, )such thatαn+βn+γn= ,{rn} ⊂[d,∞)
for some d> μn=sup{μSn,μTn}, νn=sup{νSn,νnT}, ψ =sup{ψS,ψT} for all n≥,ζ = νnsupq∈Fψ(φ(q,xn)) +μn.Iflimn→∞αnβn= andlim infn→∞αnγn< ,then{xn}converges
strongly toFx.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to express their thanks to the reviewer for helpful suggestions and comments for the improvement of this paper. This work was supported by Thaksin University.
Received: 10 September 2012 Accepted: 27 December 2012 Published: 16 January 2013
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doi:10.1186/1029-242X-2013-23
Cite this article as:Saewan and Kumam:A new iteration process for equilibrium, variational inequality, fixed point problems, and zeros of maximal monotone operators in a Banach space.Journal of Inequalities and Applications2013
