R E S E A R C H
Open Access
Convergence results for a common solution
of a finite family of variational inequality
problems for monotone mappings with
Bregman distance function
Naseer Shahzad
1*, Habtu Zegeye
2and Abdullah Alotaibi
1*Correspondence:
1Department of Mathematics, King
Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article
Abstract
In this paper, we introduce an iterative process which converges strongly to a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function. Our convergence theorem is applied to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings.
MSC: 47H05; 47J05; 47J25
Keywords: Bregman projection; monotone mappings; strong convergence; variational inequality problems
1 Introduction
Throughout this paper,Eis a real reflexive Banach space withE∗ as its dual andf :E→
(–∞, +∞] is a proper, lower semicontinuous and convex function. We denote bydomf
the domain off, defined bydomf :={x∈E:f(x) < +∞}. For anyx∈int(domf) andy∈E, theright-hand derivativeoff atxin the direction ofyis defined by
f(x,y) := lim
t→+
f(x+ty) –f(x)
t . (.)
The functionf is said to beGâteaux differentiableatxiflimt→+(f(x+ty) –f(x))/texists for anyy∈E. In this case,f(x,y) coincides with∇f(x), the value of the gradient∇f off atx. The functionf is said to beGâteaux differentiableif it is Gâteaux differentiable for any x∈int(domf). The functionf is said to beFréchet differentiableatxif this limit is attained uniformly iny= . We say thatf isuniformly Fréchet differentiableon a subset
CofEif the limit is attained uniformly forx∈Candy= .
Letf :E→(–∞, +∞] be a Gâteaux differentiable function. The functionDf :domf ×
int(domf)→[, +∞) defined by
Df(x,y) :=f(x) –f(y) –
∇f(y),x–y
is called theBregman distance with respect to f [].
ABregman projectionwith respect tof [] ofx∈int(domf) onto the nonempty, closed and convex setC⊂int(domf) is the unique vectorPCf(x)∈Csatisfying
Df
PCf(x),x=infDf(y,x) :y∈C
.
Remark . IfEis a smooth and strictly convex Banach space andf(x) =xfor allx∈E, then we have that∇f(x) = J(x) for allx∈E, whereJthe normalized duality mapping from
EintoE∗, and henceDf(x,y) becomesDf(x,y) =x– x,Jy+yfor allx,y∈E, which
is the Lyapunov function introduced by Alber [] and studied by many authors (see,e.g., [–] and the references therein). In addition, under the same condition, the Bregman projectionPfC(x) reduces to the generalized projectionC(x) (see,e.g., []) which is defined
by
φC(x),x
=min
y∈Cφ(y,x). (.)
If E=H, a Hilbert space,Jis the identity mapping, and hence the Bregman projection
PfC(x) reduces to the metric projection ofHon toC,PC(x).
A mappingA:D(A)⊂E→E∗is said to beγ-inverse strongly monotoneif there exists a positive real numberγ such that
Ax–Ay,x–y ≥γAx–Ay for allx,y∈D(A). (.)
Ais said to bemonotoneif, for eachx,y∈D(A), the following inequality holds:
Ax–Ay,x–y ≥. (.)
Clearly, the class of monotone mappings includes the class ofγ-inverse strongly monotone mappings.
LetCbe a nonempty, closed and convex subset ofE andA:C→E∗ be a monotone mapping. The problem of finding
a pointu∈Csuch that Au,v–u ≥ for allv∈C (.)
is calledthe variational inequality problem. The set of solutions of the variational inequal-ity is denoted byVI(C,A).
Variational inequality problems are related with the convex minimization problem, the zero of monotone mappings and the complementarity problem. Consequently, many re-searchers (see,e.g., [, , –]) have made efforts to obtain iterative methods for approx-imating solutions of variational inequality problems.
IfE=H, a Hilbert space, Iidukaet al.[] introduced the following projection algorithm:
x=x∈C, xn+=PC(xn–αnAxn) for anyn≥, (.)
wherePCis the metric projection fromHontoCand{αn}is a sequence of positive real
numbers. They proved that the sequence{xn}generated by (.)converges weaklyto some
IfE is a -uniformly convex and uniformly smooth Banach space, andAisγ-inverse strongly monotone, Iiduka and Takahashi [] introduced the following iteration scheme for finding a solution of the variational inequality problem:
xn+=CJ–(Jxn–αnAxn) for anyn≥, (.)
whereCis the generalized projection fromEontoC,Jis the normalized duality mapping
from E intoE∗ and{αn} is a sequence of positive real numbers. They proved that the
sequence{xn}generated by (.)converges weaklyto some element ofVI(C,A).
It is worth mentioning that the convergence obtained above isweak convergence. Our concern now is to look for an iteration scheme which converges strongly to a solution of the variational inequality problem for a monotone mappingA.
In this regard, whenEis a -uniformly convex and uniformly smooth Banach space and
Ais aγ-inverse strongly monotone mapping satisfyingAu ≤ Ay–Aufor ally∈C
andu∈VI(C,A) forVI(C,A)=∅, Iiduka and Takahashi [] studied the following iterative scheme for a solution of the variational inequality problem:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈K chosen arbitrarily,
yn=CJ–(Jxn–αnAxn), Cn={z∈E:φ(z,yn)≤φ(z,xn)}, Qn={z∈E: xn–z,Jx–Jxn ≥}, xn+=Cn∩Qn(x), n≥,
(.)
where{αn}is a positive real sequence satisfying certain mild conditions andCn∩Qnis the
generalized projection fromEontoCn∩Qn,Jis the duality mapping fromEintoE∗. Then
they proved that the sequence{xn}converges strongly to an element ofVI(C,A).
Recently, Zegeye and Shahzad [] studied the following iterative scheme for a common point of a solution of two variational inequality problems for continuous monotone map-pings in a uniformly smooth and strictly convex real Banach spaceEwhich also enjoys the Kadec-Klee property:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C=C chosen arbitrarily,
un=T,γnxn; vn=T,γnxn, wn=J–(βJun+ ( –β)Jvn), Cn+={z∈Cn:φ(z,wn)≤φ(z,xn)}, xn+=Cn+(x), n≥,
(.)
where Ti,γx:={z∈C: Aiz,y–z+ γ y–z,Jz–Jx ≥,∀y∈C} for allx∈E, i= , ,
and β,γn∈(, ) satisfy certain mild conditions. Then they proved that the sequence {xn}converges strongly toF(x), whereF is the generalized projection fromE onto F:=i=VI(C,Ai)=∅.
In , Bregman [] discovered an elegant and effective technique for using the so-calledBregman distance function Df(·,·) in the process of designing and analyzing
feasibility and optimization problems but also for algorithms of solving nonlinear, equi-librium, variational inequality, fixed point problems and others (see,e.g., [–] and the references therein).
In , Reich and Sabach [] proposed an algorithm for finding a common zero point of a finite family of maximal monotone mappingsAi:E→E
∗
(i= , , . . . ,N) in a general reflexive Banach spaceEas follows:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈E chosen arbitrarily,
yi
n=ResλinAi(xn+e i n),
Cni ={z∈E:Df(z,yin)≤Df(z,xn+ein)}, Cn=
N
i=Cni, Qi
n={z∈E: ∇f(x) –∇f(xn),z–xn ≤}, xn+=PCfn∩Qn(x), ∀n≥,
(.)
where{λi
n}Ni=⊂(,∞),{ein}Ni=are error sequences inEwithein→ andP f
Cis the Bregman
projection with respect tof fromEonto a closed and convex subsetCofE. Those authors showed that the sequence{xn}defined by (.) converges strongly to a common element
inNi=A–(∗) =N
i=VI(E,Ai) under some mild conditions. Similar results are also
avail-able in [, ].
Remark . But it is worth mentioning that the iteration processes (.)-(.) seem diffi-cult in the sense that at each stage of iteration, the set(s)Cnand (or)Qnis (are) computed
and the next iterate is taken as the Bregman projection ofxonto the intersection ofCn
andQn(orQn). This seems difficult to do in applications.
It is our purpose in this paper to introduce an iterative scheme{xn}which converges
strongly to a common solution of a finite family of variational inequality problems for monotone mappings in real reflexive Banach spaces. Our scheme does not involve com-putations ofCnorQnfor eachn≥. Furthermore, we apply our convergence theorem to
a convex minimization problem. Our theorems extend and unify most of the results that have been proved for this important class of nonlinear operators.
2 Preliminaries
Letx∈int(domf). The subdifferential off atxis the convex set defined by∂f(x) ={x∗∈
E∗:f(x) + x∗,y–x ≤f(y),∀y∈E}, where theFenchel conjugateoff is the functionf∗:
E∗→(–∞, +∞] defined byf∗(x∗) =sup{ x∗,x–f(x) :x∈E}. The functionf is said to be:
(i) Essentially smoothif∂f is both locally bounded and single-valued on its domain. (ii) Essentially strictly convexif(∂f)–is locally bounded on its domain andf is strictly
convex on every convex subset ofdomf.
(iii) Legendreif it is both essentially smooth and essentially strictly convex.
We remark that we have the following:
(i) f is essentially smooth if and only iff∗is essentially strictly convex (see [], Theorem .).
(ii) (∂f)–=∂f∗(see []).
(iv) Iff is Legendre, then∇f is a bijection satisfying∇f = (∇f∗)–,
ran∇f =dom∇f∗=int(domf∗)andran∇f∗=dom∇f=int(domf)(see [], Theorem .).
When the subdifferential off is single-valued, then∂f =∇f (see []).
A function f on E iscoercive [] if the sublevel set of f is bounded; equivalently,
limx→∞f(x) =∞.
Letf:E→(–∞, +∞] be a convex and Gâteaux differentiable function. Themodulus of total convexityoff atx∈domf is the functionνf(x,·) : [, +∞)→[, +∞] defined by
νf(x,t) :=inf
Df(y,x) :y∈domf,y–x=t
.
The functionf is calledtotally convexatxifνf(x,t) > , whenevert> . The functionf
is called totally convex if it is totally convex at any pointx∈int(domf) and it is said to be totally convex on bounded sets ifνf(B,t) > for any nonempty bounded subsetBofEand t> , where the modulus of total convexity of the functionf on the setBis the function
νf :int(domf)×[, +∞)→[, +∞] defined by νf(B,t) :=inf
Vf(x,t) :x∈B∩domf
.
We know thatf is totally convex on bounded sets if and only iff is uniformly convex on bounded sets (see [], Theorem .). The following lemmas will be useful in the proof of our main result.
Lemma .[] The function f :E→(–∞, +∞]is totally convex on bounded subsets of E if and only if for any two sequences{xn}and{yn}inint(domf)anddomf,respectively,such that the first one is bounded,
lim
n→∞Df(yn,xn) = ⇒ nlim→∞yn–xn= .
Lemma .[] Let C be a nonempty,closed and convex subset of E.Let f:E→(–∞, +∞]
be a Gâteaux differentiable and totally convex function,and let x∈E.Then:
(i) z=PfC(x)if and only if ∇f(x) –∇f(z),y–z ≤,∀y∈C. (ii) Df(y,PCf(x)) +Df(PCf(x),x)≤Df(y,x),∀y∈C.
Lemma .[] Let f :E→(–∞, +∞]be a proper,lower semi-continuous and convex function,then f∗:E∗→(–∞, +∞]is a proper,weak∗lower semicontinuous and convex function.Thus,for all z∈E,we have
Df
z,∇f∗
N
i=
ti∇f(xi)
≤
N
i=
tiDf(z,xi). (.)
Lemma .[] Let f :E→Rbe Gâteaux differentiable onint(domf)such that∇f∗is bounded on bounded subsets ofdomf∗.Let x∗∈X and{xn} ⊂E.If{Df(x,xn)}is bounded, so is the sequence{xn}.
Letf :E→Rbe a Legendre and Gâteaux differentiable function. Following [] and [], we make use of the functionVf:E×E∗→[, +∞) associated withf, which is defined by
Vf
ThenVf is nonnegative and
Vf
x,x∗=Df
x,∇f∗x∗ for allx∈Eandx∗∈E∗. (.)
Moreover, by the subdifferential inequality,
Vf
x,x∗+y∗,∇f∗x∗–x≤Vf
x,x∗+y∗, (.)
∀x∈Eandx∗,y∗∈E∗(see []).
Lemma .[] Let{an}be a sequence of nonnegative real numbers satisfying the following
relation:
an+≤( –αn)an+αnδn, n≥n,
where{αn} ⊂(, )and{δn} ⊂Rsatisfy the following conditions:limn→∞αn= ,
∞
n=αn= ∞,andlim supn→∞δn≤.Thenlimn→∞an= .
Lemma .[] Let{an}be a sequence of real numbers such that there exists a subsequence {ni}of{n}such that ani<ani+for all i∈N.Then there exists an increasing sequence{mk} ⊂ Nsuch that mk→ ∞and the following properties are satisfied by all(sufficiently large) numbers k∈N:
amk ≤amk+ and ak≤amk+.
In fact,mkis the largest number n in the set{, , . . . ,k}such that the condition an≤an+
holds.
Following the agreement in [], we have the following lemma.
Lemma . Let f :E→(–∞, +∞]be a coercive Legendre function and C be a nonempty,
closed and convex subset of E.Let A:C→E∗be a continuous monotone mapping.For r>
and x∈E,define the mapping Fr:E→C as follows:
Frx:=
z∈C: Az,y–z+
r
∇f(z) –∇f(x),y–z≥,∀y∈C
for all x∈E.Then the following hold:
() Fris single-valued; () F(Fr) =VI(C,A);
() φ(p,Frx) +φ(Frx,x)≤φ(p,x)forp∈F(Fr); () VI(C,A)is closed and convex.
3 Main result
LetCbe a nonempty, closed and convex subset ofE. LetAi:C→E∗, fori= , , . . . ,N, be
∇f(x),y–z ≥,∀y∈C}for allx∈Eandi∈ {, , . . . ,N}, wheref is a Legendre and con-vex function fromEinto (–∞, +∞). Then, in what follows, we shall study the following iteration process: ⎧ ⎪ ⎨ ⎪ ⎩
x=u∈C chosen arbitrarily,
wn=TN,rn◦TN–,rn◦ · · · ◦T,rnxn,
xn+=PCf∇f∗(αn∇f(u) + ( –αn)∇f(wn)), ∀n≥,
(.)
where {αn} ⊂(, ) satisfieslimn→∞αn= and
∞
n= =∞, and{rn} ⊂[c,∞) for some c> .
Theorem . Let f :E→Rbe a strongly coercive Legendre function which is bounded,
uniformly Fréchet differentiable and totally convex on bounded subsets of E.Let C be a nonempty,closed and convex subset ofint(domf)and Ai:C→E∗,for i= , , . . . ,N,be a finite family of continuous monotone mappings withF:=Ni=VI(C,Ai)=∅.Let{xn}n≥be
a sequence defined by(.).Then{xn}converges strongly to x∗=PFf (u).
Proof By Lemma . we have that eachVI(C,Ai) for eachi∈ {, , . . . ,N}and hence F
are closed and convex. Thus, we can take x∗ :=PfFu. Let un,=T,rnxn,un,=T,rnun,,
. . . ,un,N–=TN–,rnun,N– andun,N =TN,rnun,N–=wn. Then, from (.), Lemmas ., .
and the property ofφ, we get that
Df
x∗,xn+
=Df
x∗,PCf∇f∗αn∇f(u) + ( –αn)∇f(wn)
≤Df
x∗,∇f∗αn∇f(u) + ( –αn)∇f(wn)
≤αnDf
x∗,u+ ( –αn)Df
x∗,wn
=αnDf
x∗,u+ ( –αn)Df
x∗,TN,rn◦TN–,rn◦ · · · ◦T,rnxn
≤αnDf
x∗,u+ ( –αn)Df
x∗,xn
. (.)
Thus, by induction,
Df
x∗,xn+
≤maxDf
x∗,xn
,Df
x∗,u, ∀n≥,
which implies by Lemma . that {xn} and hence {wn} are bounded. Now let zn = ∇f∗(αn∇f(u) + ( –αn)∇f(wn)). Then we have from (.) that xn+ =PfCzn. Using
Lem-mas ., ., .(), (.) and (.), we obtain that
Df
x∗,xn+
=Df
x∗,PCfzn
≤Df
x∗,zn
=Vx∗,∇f(zn)
≤Vx∗,∇f(zn) –αn
∇f(u) –∇fx∗+αn
∇f(u) –∇fx∗,zn–x∗
=Df
x∗,∇f∗αn∇f
x∗+ ( –αn)∇f(wn)
+αn
∇f(u) –∇fx∗,zn–x∗
≤αnφ
x∗,x∗+ ( –αn)Df
x∗,wn
+αn
∇f(u) –∇fx∗,zn–x∗
≤( –αn)Df
x∗,TN,rnun,N–
+αn
∇f(u) –∇fx∗,zn–x∗
which implies that
Df
x∗,xn+
≤( –αn)
Df
x∗,un,N–
–Df(wn,un,N–)
+αn
∇f(u) –∇fx∗,zn–x∗
≤( –αn)
Df
x∗,unN–
–Df(unN–,unN–)
– ( –αn)Df(wn,un,N–) +αn
∇f(u) –∇fx∗,zn–x∗
· · · ≤( –αn)Df
x∗,xn
– ( –αn)
Df(un,,xn) +Df(un,,un,)
+· · ·+Df(un,N–,un,N–) +Df(wn,un,N–)
+αn
∇f(u) –∇fx∗,zn–x∗
≤( –αn)Df
x∗,xn
+αn
∇f(u) –∇fx∗,zn–x∗
. (.)
Now, we consider two possible cases.
Case . Suppose that there existsn∈Nsuch that{Df(x∗,xn)}is decreasing. Then we
ob-tain that{Df(x∗,xn)}is convergent. Thus, from (.) we have thatDf(un,,xn),Df(un,,un,),
. . . ,Df(wn,un,N–)→ asn→ ∞, and hence by Lemma . we get that
un,–xn→, un,–un,→, . . . , wn–un,N–→ asn→ ∞. (.)
Furthermore, from the property ofDf(·,·) and the fact thatαn→ asn→ ∞, we have
that
Df(wn,zn) =Df
wn,∇f∗
αn∇f(u) + ( –αn)∇f(wn)
≤αnDf(wn,u) + ( –αn)Df(wn,wn)
≤αnDf(wn,u) + ( –αn)Df(wn,wn)→ asn→ ∞,
and hence from Lemma . we have thatwn–zn→ and this with (.) implies that
zn–un,N–→, zn–un,N–→, . . . , zn–un,→ asn→ ∞. (.)
Since{zn}is bounded andEis reflexive, we choose a subsequence{znk}of{zn}such that znkzandlim supn→∞ ∇f(u) –∇f(x∗),zn–x∗=limk→∞ ∇f(u) –∇f(x∗),znk–x
∗. Then,
from (.) and (.), we get thatunk,izfor eachi∈ {, , . . . ,N}.
Now, we show thatz∈VI(C,Ai) for eachi∈ {, , . . . ,N}. But from the definition ofun,i,
we have that
Aiun,i,y–un,i+
∇f(un,i) –∇f(xn) rn
,y–un,i
≥, ∀y∈C,
and hence
Aiunk,i,y–unk,i+ ∇
f(unk,i) –∇f(xnk) rnk
,y–unk,i
for eachi∈ {, , . . . ,N}. Setvt=ty+ ( –t)zfor allt∈(, ] andy∈C. Consequently, we
get thatvt∈C. Now, from (.) it follows that
Aivt,vt–unk,i ≥ Aivt,vt–unk,i– Aiunk,i,vt–unk,i
–
∇
f(unk,i) –∇f(xnk) rnk
,vt–unk,i
= Aivt–Aiunk,i,vt–unk,i
–
∇f
(unk,i) –∇f(xnk) rnk
,vt–unk,i
.
In addition, sincef is uniformly Fréchet differentiable and bounded, we have that∇f is uniformly continuous (see []). Thus, from (.) and the uniform continuity of∇f, we obtain that
∇f(unk,i) –∇f(xnk) rnk
→ ask→ ∞,
and sinceAis monotone, we also have that Aivt–Aiunk,i,vt–unk,i ≥. Thus, it follows
that
≤ lim
k→∞ Aivt,vt–unk,i= Aivt,vt–z,
and hence
Aivt,y–z ≥, ∀y∈C, for alli∈ {, , . . . ,N}.
Ift→, the continuity ofAiimplies that
Aiz,y–z ≥, ∀y∈C.
This implies thatz∈VI(C,Ai) for alli∈ {, , . . . ,N}.
Therefore, we obtain thatz∈Ni=VI(C,Ai). Thus, by Lemma ., we immediately
ob-tain thatlim supn→∞ ∇f(u) –∇f(x∗),zn–x∗= ∇f(u) –∇f(x∗),z–x∗ ≤. It follows from
Lemma . and (.) thatDf(x∗,xn)→ asn→ ∞. Consequently,xn→x∗.
Case . Suppose that there exists a subsequence{nj}of{n}such that
Df
x∗,xnj
<Df
x∗,xnj+
for allj∈N. Then, by Lemma ., there exists a nondecreasing sequence{mk} ⊂Nsuch
thatmk→ ∞,Df(x∗,xmk)≤Df(x∗,xmk+) andDf(x∗,xk)≤Df(x∗,xmk+) for allk∈N. From
(.) andαn→, we have
( –αmk)
Df(umk,,xmk) +· · ·+Df(wmk,umk,N–)
≤Df
x∗,xmk
–Df
x∗,xmk+
+αmkDf
x∗,xmk
+αmk
∇f(u) –∇fx∗,zmk–x
which implies that Df(umk,,xmk), . . . ,Df(wmk,umk,N–)→, and hence umk,–xmk →
, . . . ,wmk–umk,N–→ ask→ ∞. Thus, as in Case , we obtain that
lim sup
k→∞
∇f(u) –∇fx∗,zmk–x
∗≤. (.)
Furthermore, from (.) we have that
Df
x∗,xmk+
≤( –αmk)Df
x∗,xmk
+αmk
∇f(u) –∇fx∗,zmk–x
∗. (.)
Thus, sinceDf(x∗,xmk)≤Df(x∗,xmk+), we get that
αmkDf
x∗,xmk
≤Df
x∗,xmk
–Df
x∗,xmk+
+αmk
∇f(u) –∇fx∗,zmk–x
∗
≤αmk
∇f(u) –∇fx∗,zmk–x
∗.
Moreover, sinceαmk> , we obtain that
Df
x∗,xmk
≤∇f(u) –∇fx∗,zmk–x
∗.
It follows from (.) thatDf(x∗,xmk)→ ask→ ∞. This together with (.) implies that Df(x∗,xmk+)→. Therefore, sinceDf(x∗,xk)≤Df(x∗,xmk+) for all k∈N, we conclude
thatxk→x∗ask→ ∞. Hence, both cases imply that{xn}converges strongly tox∗=PfFu
and the proof is complete.
If in Theorem .N= , then we get the following corollary.
Corollary . Let f :E→Rbe a strongly coercive Legendre function which is bounded,
uniformly Fréchet differentiable and totally convex on bounded subsets of E.Let C be a nonempty,closed and convex subset ofint(domf),and let A:C→E∗be a continuous mono-tone mapping with VI(C,A)=∅.Let{xn}n≥be a sequence defined by(.),
⎧ ⎪ ⎨ ⎪ ⎩
x=u∈C chosen arbitrarily,
wn=Trnxn,
xn+=PCf∇f∗(αn∇f(u) + ( –αn)∇f(wn)),
(.)
where Tγx:={z∈C: Az,y–z+γ ∇f(z) –∇f(x),y–z ≥,∀y∈C}for all x∈E;αn∈
(, )satisfieslimn→∞αn= and
∞
n=αn=∞and{rn} ⊂[c,∞)for some c> .Then the
sequence{xn}n≥converges strongly to a point x∗=PVI(C,A)(u).
IfC=E, thenVI(C,A) =A–() and hence the following corollary holds.
Corollary . Let f :E→Rbe a strongly coercive Legendre function which is bounded,
uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let Ai : E→E∗,for i= , , . . . ,N,be a finite family of continuous monotone mappings.LetF:=
N
i=VI(C,Ai) =
N
If in Theorem . we assumeu= , then the scheme converges strongly to the common minimum-norm zero of a finite family of continuous monotone mappings. In fact, we have the following corollary.
Corollary . Let f :E→Rbe a strongly coercive Legendre function which is bounded,
uniformly Fréchet differentiable and totally convex on bounded subsets of E.Let C be a nonempty,closed and convex subset ofint(domf),and let Ai:C→E∗,for i= , , . . . ,N,be a finite family of continuous monotone mappings withF:=Ni=VI(C,Ai)=∅.Let{xn}n≥ be a sequence defined by(.)with u= .Then{xn}converges strongly to x∗=PfF(),which is the common minimum-norm(with respect to the Bregman distance)solution of the vari-ational inequalities.
4 Application
In this section, we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional in Banach spaces.
Letgi, fori= , , . . . ,N, be continuously Fréchet differentiable convex functionals such
that the gradients ofgi, (∇gi)|Care continuous and monotone. Forr> , letKi,rx:={z∈C: ∇gi(z),y–z+r ∇f(z) –∇f(x),y–z ≥,∀y∈C}for allx∈Eand for eachi∈ {, , . . . ,N}.
Then the following theorem holds.
Theorem . Let f :E→Rbe a strongly coercive Legendre function which is bounded,
uniformly Fréchet differentiable and totally convex on bounded subsets of E.Let gi, i=
, , . . . ,N, be continuously Fréchet differentiable convex functionals such that the gra-dients of gi, (∇gi)|C are continuous,monotone andF:=iN=arg miny∈Cgi(y)=∅,where
arg miny∈Cgi(y) :={z∈C:gi(z) =miny∈Cgi(y)}.Let{xn}n≥be a sequence defined by
⎧ ⎪ ⎨ ⎪ ⎩
x=u∈C chosen arbitrarily,
wn=KN,rn◦KN–,rn◦ · · · ◦K,rnxn,
xn+=PCf∇f∗(αn∇f(u) + ( –αn)∇f(wn)), ∀n≥,
(.)
whereαn∈(, )satisfieslimn→∞αn= and
∞
n=αn=∞and{rn} ⊂[c,∞)for some c> .Then the sequence{xn}converges strongly to p=PfF(u).
Proof We note that from the convexity and Fréchet differentiability of f, we have
VI(C, (∇gi)|C) =arg miny∈Cgi(y) for eachi∈ {, , . . . ,N}. Thus, by Theorem .,{xn}
con-verges strongly top=PfF(u).
Remark . Our results are new even if the convex functionfis chosen to bef(x) = pxp ( <p<∞) in uniformly smooth and uniformly convex spaces.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved final manuscript.
Author details
1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.2Department of
Mathematics, University of Botswana, Pvt. Bag 00704, Gaborone, Botswana.
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first and third authors acknowledge with thanks DSR for financial support. The second author undertook this work when he was visiting the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy, as a regular associate.
Received: 27 August 2013 Accepted: 14 November 2013 Published:13 Dec 2013
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