R E S E A R C H
Open Access
An algorithm for finding common solutions
of various problems in nonlinear operator
theory
Eric U Ofoedu
1, Jonathan N Odumegwu
1, Habtu Zegeye
2and Naseer Shahzad
3**Correspondence:
3Department 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, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone operators; the set of common fixed points of a finite family of quasi-nonexpansive mappings; and the set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty. Moreover, the common element is shown to be the metric projection of the initial guess on the intersection of these sets.
MSC: 47H06; 47H09; 47J05; 47J25
Keywords: classical equilibrium problem; generalized mixed equilibrium problem;
η
-inverse strongly monotone mapping; maximal monotone operator; nonexpansive mappings; real Hilbert space; pseudocontractive mappings; variational inequality problem1 Introduction
LetHbe a real Hilbert space. A mappingTwith domainD(T) and rangeR(T) inHis called anL-Lipschitzianmapping (or simply aLipschitzmapping) if and only if there existsL≥ such that for allx,y∈D(T),
Tx–Ty ≤Lx–y.
IfL∈[, ), thenTis calledstrict contractionor simplya contraction; and ifL= , thenT is called nonexpansive. A pointx∈D(T) is called afixed pointof an operatorTif and only ifTx=x. The set of fixed points of an operatorTis denoted byFix(T), that is,Fix(T) := {x∈D(T) :Tx=x}.
A mappingT with domainD(T) and rangeR(T) inHis called aquasi-nonexpansive mapping if and only ifFix(T)=∅and for anyx∈D(T), for anyu∈Fix(T),
Tx–u ≤ x–u.
Every nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive. The following examples show that the converse is not true.
Example .(see []) LetE= [–π,π] be a subspace of the set of real numbersR, endowed with the usual topology. DefineT:E→EbyTx=xcosxfor allx∈E. Clearly,F(T) ={}. Observe that
|Tx– |=|x| × |cosx| ≤ |x|=|x– |.
Thus,Tis quasi-nonexpansive. The mappingTis, however, not a nonexpansive mapping since forx=π andy=π,
|Tx–Ty|=π cos
π
–πcosπ=π.
But
|x–y|=π –π
=π
.
Example .(see [, ]) LetE=Rbe endowed with usual topology. DefineT:R→Rby
Tx=
x cos(
x), x= ,
, x= . (.)
It is easy to see thatF(T) ={}since forx= ,Tx=ximplies thatx cos
x=x. Thus, for any
x= ,cosx= , which is not possible. So,F(T) ={}. Next, observe that for anyx∈R,
|Tx– |=x
×cos
x
≤|x|
<|x|=|x– |.
So, the mappingTis quasi-nonexpansive. Finally, we show thatTis not nonexpansive. To see this, letx=π andy=π, then
|Tx–Ty|= πcos
π
– πcosπ
=
π.
But,
|x–y|= π–
π
=
π.
So,
|Tx–Ty|= π >
π =|x–y|.
Another important generalization of the class of nonexpansive mappings is the class of pseudocontractive mappings. These mappings are intimately connected with the impor-tant class of nonlinear accretive operators. This connection will be made precise in what follows.
A mappingT with domainD(T) and rangeR(T) inHis calledpseudocontractiveif and only if for allx,y∈D(T), the following inequality holds:
x–y ≤( +r)(x–y) –r(Tx–Ty) (.)
for allr> . As a consequence of a result of Kato [], the pseudocontractive mappings can also be defined in terms of the normalized duality mappings as follows: the mappingTis calledpseudocontractiveif and only if for allx,y∈D(T), we have that
Tx–Ty,x–y ≤ x–y. (.)
It now follows trivially from (.) that every nonexpansive mapping is pseudocontractive. We note immediately that the class of pseudocontractive mappings is larger than that of nonexpansive mappings. For examples of pseudocontractive mappings which are not non-expansive, the reader may see [].
To see the connection between the pseudocontractive mappings and the monotone mappings, we introduce the following definition: a mapping Awith domain D(A) and rangeR(A) inEis calledmonotoneif and only if for allx,y∈D(A), the following inequality is satisfied:
Ax–Ay,x–y ≥. (.)
The operatorAis calledη-inverse strongly monotoneif and only if there existsη∈(, ) such that for allx,y∈D(A), we have that
Ax–Ay,x–y ≥ηAx–Ay. (.)
It is easy to see from inequalities (.) and (.) that an operatorAis monotone if and only if the mappingT := (I–A) is pseudocontractive. Consequently, the fixed point theory for pseudocontractive mappings is intimately connected with the zero of monotone map-pings. For the importance of monotone mappings and their connections with evolution equations, the reader may consult any of the references [, ].
Due to the above connection, fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors.
LetCbe a closed convex nonempty subset of a real Hilbert spaceHwith inner product ·,·and norm · . Letf:C×C→Rbe a bifunction. The classicalequilibrium problem (EP) for a bifunctionf is to findu∗∈Csuch that
fu∗,y≥, ∀y∈C. (.)
The set of solutions forEP(.) is denoted by
The classical equilibrium problem (EP) includes as special cases the monotone inclusion problems, saddle point problems, variational inequality problems, minimization prob-lems, optimization probprob-lems, vector equilibrium probprob-lems, Nash equilibria in noncoop-erative games. Furthermore, there are several other problems, for example, the comple-mentarity problems and fixed point problems, which can also be written in the form of the classical equilibrium problem. In other words, the classical equilibrium problem is a unifying model for several problems arising from engineering, physics, statistics, com-puter science, optimization theory, operations research, economics and countless other fields. For the past years or so, many existence results have been published for various equilibrium problems (see,e.g., [–]). Approximation methods for such problems thus become a necessity.
Iterative approximation of fixed points and zeros of nonlinear mappings has been stud-ied extensively by many authors to solve nonlinear mapping equations as well as varia-tional inequality problems and their generalizations (see,e.g., [–]). Most published results on nonexpansive mappings (for example) focus on the iterative approximation of their fixed points or approximation of common fixed points of a given family of this class of mappings.
Some attempts to modify the Mann iteration method so that strong convergence is anteed have recently been made (we should recall that Mann iteration method only guar-antees weak convergence (see, for example, Bauschkeet al.[])). Nakajo and Takahashi [] formulated the following modification of the Mann iteration method for a nonexpan-sive mappingTdefined on a nonempty bounded closed and convex subsetCof a Hilbert spaceH:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C,
yn=αnxn+ ( –αn)Txn,
Cn={v∈C:yn–v≤ xn–v},
Qn={v∈C: xn–v,x–xn ≥},
xn+=PCn∩Qn(x), ∀n∈N,
(.)
wherePCdenotes the metric projection fromHonto a closed convex subsetCofH. They
proved that if the sequence{αn}n≥is bounded away from , then{xn}n≥defined by (.)
converges strongly toPF(T)(x).
Formulations similar to (.) for different classes of nonlinear problems had been pre-sented by Kim and Xu [], Nilsrakoo and Saejung [], Ofoeduet al.[], Yang and Su [], Zegeye and Shahzad [–].
2 Preliminary
In what follows, we shall make use of the following lemmas.
Lemma .(see,e.g., Kopecka and Reich []) Let C be a nonempty closed and convex subset of a real Hilbert space.Let x∈H and PC:H→C be the metric projection of H onto
C,then for any y∈C,
y–PCx+PCx–x≤ x–y.
Lemma . Let C be a closed convex nonempty subset of a real Hilbert space H;and let PC:H→C be the metric projection of H onto C.Let x∈H,then x=PCx if and only if
z–x,x–x ≤for all z∈C.
Lemma . Let H be a real Hilbert space,then for any x,y∈H,α∈[, ],
αx+ ( –α)y=αx+ ( –α)y–α( –α)x–y.
Lemma .(see Zegeye []) Let C be a nonempty closed convex subset of a real Hilbert space H.Let T:C→H be a continuous pseudocontractive mapping,then for all r> and x∈H,there exists z∈C such that
y–z,Tz– r
y–z, ( +r)z–x≤, ∀y∈C.
Lemma .(see Zegeye []) Let C be a nonempty closed convex subset of a real Hilbert space H.Let T:C→C be a continuous pseudocontractive mapping,then for all r> and x∈H,define a mapping Fr:H→C by
Frx=
z∈C: y–z,Tz– r
y–z, ( +r)z–x≤,∀y∈C
,
then the following hold:
() Fris single-valued;
() Fris firmly nonexpansive type mapping,i.e.,for allx,y∈H,
Frx–Fry≤ Frx–Fry,x–y;
() Fix(Fr)is closed and convex;andFix(Fr) =Fix(T)for allr> .
In the sequel, we shall require that the bifunctionf :C×C→Rsatisfies the following conditions:
(A) f(x,x) = ,∀x∈C;
(A) f is monotone in the sense thatf(x,y) +f(y,x)≤for allx,y∈C; (A) lim supt→+f(tz+ ( –t)x,y)≤f(x,y)for allx,y,z∈C;
(A) the functiony→f(x,y)is convex and lower semicontinuous for allx∈C.
r> and x∈H,there exists u∈C such that
f(u,y) +
r y–u,u–x ≥, ∀y∈C. (.)
Moreover,if for all x∈H we define a mapping Gr:H→Cby
Gr(x) =
u∈C:f(u,y) +
r y–u,u–x ≥,∀y∈C
, (.)
then the following hold:
() Gris single-valued for allr> ;
() Gris firmly nonexpansive,that is,for allx,z∈H,
Grx–Grz≤ Grx–Grz,x–z;
() Fix(Gr) =EP(f)for allr> ; () EP(f)is closed and convex.
Lemma .(see Ofoedu []) Let C be a nonempty closed convex subset of a real Hilbert space H.Let T:C→C be a continuous pseudocontractive mapping.For r> ,let Fr:H→
C be the mapping in Lemma.,then for any x∈H and for any p,q> ,
Fpx–Fqx ≤|p
–q| p
Fpx+x
.
Lemma .(Compare with Lemma of Ofoedu []) Let C be a closed convex nonempty subset of a real Hilbert space H.Let f :C×C→Rbe a bifunction satisfying conditions (A)-(A).Let r> and let Grbe the mapping in Lemma.,then for all p,q> and for
all x∈H,we have that
Gpx–Gqx ≤|
p–q| p
Gpx+x
.
3 Main results
LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetT,T, . . . ,Tm:
C→Cbemcontinuous pseudocontractive mappings; letS,S, . . . ,Sl:C→Cbel
con-tinuous quasi-nonexpansive mappings; letA,A, . . . ,Ad:C→Hbedγj-inverse strongly
monotone mappings with constantsγj∈(, ),j= , , . . . ,d; letf,f, . . . ,ft:C×C→Rbe
tbifunctions satisfying conditions (A)-(A). For allx∈E,i= , , . . . ,m, let
Fi,rx:=
z∈C: y–z,Tiz–
r
y–z, ( +r)z–x≤,∀y∈C
and for allx∈E,h= , , . . . ,t, let
Gh,r(x) =
u∈C:fh(u,y) +
r y–u,u–x ≥,∀y∈C
then in what follows we shall study the following iteration process: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C=C chosen arbitrarily,
zn=PC(xn–λnAn+xn),
yn=αnxn+ ( –αn)Sn+zn,
wn=η
m
i=βiFi,rnyn+ ( –η)
t
h=ξhGh,rnyn,
Cn+={z∈C:wn–z ≤ xn–z},
xn+= Cn+(x), n≥,
(.)
where An=An(modd),Sn=Sn(modl);{rn} ⊂(,∞) such thatlimn→∞rn=r> ;{αn}n≥ a
sequence in (, ) such that lim infn→∞αn( –αn) > ;{βi}mi=,{ξh}th= ⊂(, ) such that
m
i=βi= =
t
h=ξh; η∈(, ) and {λn} is a sequence in [a,b] for somea,b∈Rsuch
that <a<b< γ,γ =min≤j≤d{γj}.
Lemma . Let C be a nonempty closed convex subset of a real Hilbert space H. Let T,T, . . . ,Tm :C→C be m continuous pseudocontractive mappings; let S,S, . . . ,Sl :
C→C be l continuous quasi-nonexpansive mappings;let A,A, . . . ,Ad:C→H be dγj
-inverse strongly monotone mappings with constantsγj∈(, ),j= , , . . . ,d;let f,f, . . . ,ft:
C × C→ R be t bifunctions satisfying conditions (A)-(A). Let F :=mi=Fix(Ti)∩
d
j=A–j ()∩ l
k=Fix(Sk)∩ t
h=EP(fh)=∅.Let{xn}be a sequence defined by(.),then
the sequence{xn}is well defined for each n≥.
Proof We first show thatCnis closed and convex for eachn∈N∪{}. From the definitions
ofCnit is obvious thatCnis closed. Moreover, sincewn–z ≤ xn–zis equivalent to
z,xn–wn–xn+wn≤, it follows thatCnis convex for eachn∈N∪ {}. Next,
we prove thatF⊂Cnfor eachn∈N∪ {}. From the assumption, we see thatF⊂C=C.
Suppose thatF⊂Ckfor somek≥, then forp∈F, we obtain that
wk–p= η m i=
βiFi,rkyk+ ( –η)
m
h=
ξhGh,rkyk–p
≤ yk–p=αkxk+ ( –αk)Sk+zk–p
≤αkxk–p+ ( –αk)Sk+zk–p
≤αkxk–p+ ( –αk)zk–p. (.)
Furthermore,
zk–p =PC(xk–λkAk+xk) –p
≤ xk–λkAk+xk–p
=xk–p–λk(Ak+xk–Ak+p)
=xk–p– λk xk–p,Ak+xk–Ak+p+λkAk+xk–Ak+p
≤ xk–p+λk(λk– γ)Ak+xk–Ak+p
Thus,
zk–p ≤ xk–p. (.)
Using (.) in (.) gives
wk–p ≤ xk–p.
So,p∈Ck+. This implies, by induction, thatF⊂Cnso that the sequence generated by
(.) is well defined for alln≥.
Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let T,T, . . . ,Tm :C→C be m continuous pseudocontractive mappings; let S,S, . . . ,Sl :
C→C be l continuous quasi-nonexpansive mappings;let A,A, . . . ,Ad:C→H be dγj
-inverse strongly monotone mappings with constantsγj∈(, ),j= , , . . . ,d;let f,f, . . . ,ft:
C × C→ R be t bifunctions satisfying conditions (A)-(A). Let F :=mi=Fix(Ti)∩
d
j=A–j ()∩ l
k=Fix(Sk)∩ t
h=EP(fh)=∅.Let{xn}be a sequence defined by(.).Then
the sequence{xn}n≥converges strongly to the element of F nearest to x.
Proof From Lemma ., we obtain thatF⊂Cn,∀n≥ andxnis well defined for each
n≥. Fromxn=PCn(x) andxn+=PCn+(x)∈Cn+⊂Cn, we obtain that
xn+–xn,xn–x ≥ and xn–x ≤ xn+–x.
Besides, by Lemma .,
xn–p=PCnx–x ≤ x–p
–x
–xn≤ x–p.
Thus, the sequence{xn–x}n≥is a bounded nondecreasing sequence of real numbers.
So,limn→∞xn–xexists. Similarly, by Lemma ., we have that for any positive integer μ,
xn+μ–xn =xn+μ–PCnx
≤ xn+μ–x–PCnx–x
=xn+μ–x–xn–x for alln≥.
Sincelimn→∞xn–xexists, we have thatlimn→∞xn+μ–xn= and hence,{xn}n≥is
a Cauchy sequence inC. Therefore, there existsx∗∈Csuch thatlimn→∞xn=x∗. Since
xn+∈Cn+, we have that
wn–xn+ ≤ xn–xn+.
Thus,
lim
and hencexn–wn ≤ xn–xn++xn+–wn → asn→ ∞, which implies thatwn→x∗
asn→ ∞.
Next, we observe that forp∈Fand using Lemma .,
yn–p =αnxn+ ( –αn)Sn+zn–p
=αn(xn–p) + ( –αn)(Sn+zn–p)
=αnxn–p+ ( –αn)Sn+zn–p–αn( –αn)xn–Sn+zn (.)
≤αnxn–p+ ( –αn)zn–p–αn( –αn)xn–Sn+zn. (.)
But
zn–p≤ xn–p+λn(λn– γ)An+xn–An+p
=xn–p+λn(λn– γ)An+xn. (.)
So, using (.) in (.), we obtain that
yn–p≤αnxn–p+ ( –αn)
xn–p+λn(λn– γ)An+xn
–αn( –αn)xn–Sn+zn
=xn–p+ ( –αn)λn(λn– γ)An+xn
–αn( –αn)xn–Sn+zn. (.)
Moreover, we obtain that
wn–p = η
m
i=
βiFi,rnyn+ ( –η)
m
h=
ξhGh,rnyn–p
≤ yn–p. (.)
Using (.) in (.) we get that
wn–p≤ xn–p+ ( –αn)λn(λn– γ)An+xn
–αn( –αn)xn–Sn+zn. (.)
Now, using the fact thatλn< γ, inequality (.) gives (for some constantM> ) that
αn( –αn)xn–Sn+zn ≤ xn–p–wn–p≤Mxn–wn. (.)
Hence, we obtain from inequality (.) that
xn–Sn+zn → asn→ ∞. (.)
Moreover, from (.) we obtain that
which yields that
lim
n→∞An+xn= . (.)
Now,
xn–zn=xn–PC(xn–λnAn+xn)=PCxn–PC(xn–λnAn+xn)
≤ xn–xn+λnAn+xn=λnAn+xn
≤bAn+xn. (.)
It follows from (.) and (.) that
lim
n→∞xn–zn= ; (.)
and hencezn→x∗asn→ ∞.
We now show thatx∗∈lk=Fix(Sk). Observe that from (.) and (.) we obtain that
Sn+zn–zn ≤ Sn+zn–xn+zn–xn → asn→ ∞, (.)
so that
lim
n→∞Sn+zn=x
∗. (.)
Let{nσ}σ≥⊂Nbe such thatSnσ+=Sfor allσ∈N, then sinceznσ→x∗asσ→ ∞, we
obtain from (.), using the continuity ofS, that
x∗= lim
σ→∞Snσ+znσ =σlim→∞Sznσ=Sx ∗.
Similarly, if{nj}j≥⊂Nis such thatSnj+=Sfor allj∈N, then we have again that
x∗= lim
j→∞Snj+znj=jlim→∞Sznj=Sx
∗.
Continuing, we obtain thatSkx∗=x∗,k= , . . . ,l. Hence,x∗∈ l
k=F(Sk).
Next, we show that x∗∈dj=A–
j (). SinceAj isγ-inverse strongly monotone forj=
, , . . . ,d, we have thatAjisγ-Lipschitz continuous. Thus,
An+xn–An+x∗≤
γxn–x
∗→ asn→ ∞. (.)
Hence, from (.) and (.), we obtain that
An+x∗≤An+xn–An+x∗+An+xn → asn→ ∞.
As a result, we get that
lim
n→∞An+x
Let{ns}s≥⊂Nbe such thatAns+=Afor alls∈N. Then
Ax∗= lim
s→∞Ans+x∗= .
Similarly, we have thatAjx∗= forj= , . . . ,d. Thus,x∗∈
d
j=A–i ().
Furthermore, we show thatx∗∈mi=Fix(Ti) =
m
i=Fix(Fi,r),∀r> . Using the fact that
xn→x∗,zn→x∗asn→ ∞, we obtain that
F,rnyn–x∗≤yn–x∗
≤αnxn–x∗+ ( –αn)zn–x∗
≤xn–x∗+zn–x∗→ asn→ ∞. (.)
Thus, we obtain from (.) that
lim
n→∞F,rnyn=x
∗= lim
n→∞yn.
This implies thatlimn→∞F,rnyn–yn= . But by Lemma .,
F,rnyn–F,ryn ≤
|rn–r|
rn
F,rnyn+yn
→ asn→ ∞.
Thus,
lim
n→∞F,ryn=nlim→∞F,rnyn=x
∗.
So, the continuity ofF,rand the fact thatyn→x∗asn→ ∞give
x∗= lim
n→∞F,ryn=F,rx
∗.
A similar argument gives
x∗= lim
n→∞Fi,ryn=Fi,rx∗, i= , , . . . ,m.
Hence,
x∗∈
m
i=
Fix(Fi,r) =
m
i=
Fix(Ti).
Moreover, we show thatx∗∈th=EP(fh) =th=Fix(Gh,r). Observe that
G,rnyn–x
∗≤y
n–x∗
≤αnxn–x∗+ ( –αn)zn–x∗
Thus, we obtain from (.) that
lim
n→∞G,rnyn=x∗= lim n→∞yn.
This implies thatlimn→∞G,rnyn–yn= . But by Lemma .,
G,rnyn–G,ryn ≤
|rn–r|
rn
G,rnyn+yn
→ asn→ ∞.
Thus,
lim
n→∞G,ryn=nlim→∞G,rnyn=x
∗.
So, the continuity ofG,rand the fact thatyn→x∗asn→ ∞give
x∗= lim
n→∞G,ryn=G,rx∗.
A similar argument gives
x∗= lim
n→∞Gh,ryn=Gi,rx∗, h= , , . . . ,t.
Hence,
x∗∈
t
h=
Fix(Gh,r) =
t
h=
EP(fh).
Finally, we prove thatx∗=PF(x). Fromxn=PCn(x)n≥, we obtain that
x–xn,xn–z ≥, ∀z∈Cn.
SinceF⊂Cn, we also have that
x–xn,xn–p ≥, ∀p∈F. (.)
So,
≤ x–xn,xn–p=
x–x∗+x∗–xn,xn–x∗+x∗–p
=x–x∗,xn–x∗
+x–x∗,x∗–p
+x∗–xn,xn–x∗
+x∗–xn,x∗–p
≤x–x∗y,x∗–p
+x–x∗xn–x∗
+xn–x∗x∗–p–xn–x∗
. (.)
Inequality (.) implies that
≤x–x∗,x∗–p
By taking limit asn→ ∞in (.), we obtain that
x–x∗,x∗–p
≥, ∀p∈F.
Now, by Lemma . we have thatx∗=PF(x). This completes the proof.
Remark . We note thatx∗=PF(x) makes sense since it could be easily shown thatFis
closed and convex. In fact, it is enough to show that the set of zeros ofγ-inverse monotone mappings and a fixed point set of continuous quasi-nonexpansive mappings are convex sets. Closure of the two sets simply follows from the continuity of the mappings involved.
Remark . Several authors (see,e.g., [, ] and references therein) have studied the following problem: LetCbe a closed convex nonempty subset of a real Hilbert spaceH with inner product ·,·and norm·. Letf:C×C→Rbe a bifunction and:C→R∪ {+∞}be a proper extended real-valued function, whereRdenotes the set of real numbers. Let:C→Hbe a nonlinear monotone mapping. The generalized mixed equilibrium problem(abbreviatedGMEP) forf,andis to findu∗∈Csuch that
fu∗,y+(y) –u∗+u∗,y–u∗≥, ∀y∈C. (.)
The set of solutions forGMEP(.) is denoted by
GMEP(f,,) = u∈C:f(u,y) +(y) –(u) + u,y–u ≥,∀y∈C.
These authors always claim that if≡≡in (.), then (.) reduces to the classical equilibrium problem(abbreviatedEP), that is, the problem of findingu∗∈Csuch that
fu∗,y≥, ∀y∈C (.)
andGMEP(f, , ) is denoted byEP(f), where
EP(f) = u∈C:f(u,y)≥,∀y∈C.
If f ≡≡in (.), thenGMEP(.) reduces to the classicalvariational inequality problemandGMEP(, ,) is denoted byVI(,C), where
VI(,C) ={u∈C: u,y–u ≥,∀y∈C}.
Iff ≡≡, thenGMEP(.) reduces to the followingminimization problem:
findu∗∈Csuch that(y)≥u∗, ∀y∈C;
andGMEP(,, ) is denoted byArgmin(), where
If ≡, then (.) becomes themixed equilibrium problem(abbreviatedMEP) and GMEP(f,, ) is denoted byMEP(f,), where
MEP(f,) = u∈C:f(u,y) +(y) –(u)≥,∀y∈C.
If≡, then (.) reduces to thegeneralized equilibrium problem(abbreviatedGEP) and GMEP(f, ,) is denoted byGEP(f,), where
GEP(f,) = u∈C:f(u,y) + u,y–u ≥,∀y∈C.
Iff ≡, thenGMEP(.) reduces to the generalized variational inequality problem (ab-breviatedGVIP) andGMEP(,,) is denoted byGVIP(,,C), where
GVIP(,,C) = u∈K:(y) –(u) + u,y–u ≥,∀y∈C.
It is worthy to note that if we define:C×C→Rby
(x,y) =f(x,y) +(y) –(x) + x,y–x,
then it could be easily checked thatis a bifunction and satisfies properties (A)-(A). Thus, the so-called generalized mixed equilibrium problem reduces to the classical equi-librium problem for the bifunction . Thus, consideration of the so-called generalized mixed equilibrium problem in place of the classical equilibrium problem studied in this paper leads to no further generalization.
4 Application (convex differentiable optimization)
In Section , we defined a Lipschitz continuous mapping and an inverse strongly mono-tone mapping. Inverse strongly monomono-tone mappings arise in various areas of optimization and nonlinear analysis (see, for example, [–]). It follows from the Cauchy-Schwarz inequality that if a mappingA:D(A)⊆H→R(A)⊆His
L-inverse strongly monotone,
thenAisL-Lipschitz continuous. The converse of this statement, however, fails to be true. To see this, take for instanceA= –I, whereIis the identity mapping onH, thenAis L-Lipschitz continuous (withL= ) but not L-inverse strongly monotone (that is, not firmly nonexpansive in this case).
Baillon and Haddad [] showed in that ifD(A) =HandAis the gradient of a con-vex functional onH, thenAisL-inverse strongly monotone if and only ifAisL-Lipschitz continuous. This remarkable result, which has important applications in optimization the-ory (see, for example, [–]), has become known as the Baillon-Haddad theorem. In fact, we have the following theorem.
Theorem . (Baillon-Haddad) (see Corollary of []) Letφ :H →Rbe a convex Fréchet-differentiable functional on H such that ∇φ is L-Lipschitz continuous for some L∈(, +∞),then∇φ is a L-inverse strongly monotone mapping(where∇φ denotes the gradient of the functionalφ).
LetKbe a closed convex subset of a real Hilbert spaceH, consider the minimization problem given by
min
x∈Kφ(x), (.)
whereφis a Fréchet-differentiable convex functional. Let⊆K, the solution set of (.), be nonempty. It is known that a pointz∈if and only if the following optimality condition holds:
z∈K, ∇φ(z),x–z≥, x∈K. (.)
It is easy to see that ifK=H, then optimality condition (.) is equivalent toz∈if and only ifz∈(∇φ)–().
Thus, we obtain the following as a corollary of Theorem ..
Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H.Let T,T, . . . ,Tm:C→C be m continuous pseudocontractive mappings;let S,S, . . . ,Sl:C→
C be l continuous quasi-nonexpansive mappings;letφ,φ, . . . ,φd:H→H be d convex and
Fréchet-differentiable functionals on H such that(∇φ)jis Lj-Lipschitz continuous for some
Lj∈(, +∞),j= , , . . . ,d; let f,f, . . . ,ft:C×C→Rbe t bifunctions satisfying
condi-tions(A)-(A).Let F:=mi=Fix(Ti)∩
d
j=(∇φj)–()∩ l
k=Fix(Sk)∩ t
h=EP(fh)=∅.
Let{xn}n≥be a sequence defined by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C=C chosen arbitrarily,
zn=PC(xn–λn(∇φ)n+xn),
yn=αnxn+ ( –αn)Sn+zn,
wn=η
m
i=βiFi,rnyn+ ( –η)
t
h=ξhGh,rnyn,
Cn+={z∈Cn:wn–z ≤ xn–z},
xn+= Cn+(x), n≥,
where(∇φ)n= (∇φ)n(modd),Sn=Sn(modl);{rn} ⊂(,∞)such thatlimn→∞rn=r> ;{αn}n≥
a sequence in(, ) such thatlim infn→∞αn( –αn) > ; {βi}mi=,{ξh}th=⊂(, )such that
m
i=βi= =
t
h=ξh;η∈(, )and{λn}is a sequence in[a,b]for some a,b∈Rsuch that
<a<b<L,L=max≤j≤d{Lj}.Then the sequence{xn}n≥converges strongly to the element
of F nearest to x.
Proof Since, by our hypothesis, (∇φ)jisLj-Lipschitz continuous for someLj∈(, +∞),
j= , , . . . ,d, we obtain from Theorem . that (∇φ)jisLj-inverse strongly monotone,j=
, , . . . ,d; and sinceL=max≤j≤d{Lj}, it is then easy to see that (∇φ)jisL-inverse strongly
monotone,j= , , . . . ,d. The rest, therefore, follows as in the proof of Theorem . with
γ =L. This completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeria.2Bahir Dar
University, P.O. Box 859, Bahir Dar, Ethiopia. 3Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support.
Received: 27 August 2013 Accepted: 21 November 2013 Published:09 Jan 2014
References
1. Kim, GE: Weak and strong convergence for quasi-nonexpansive mappings in Banach spaces. Bull. Korean Math. Soc.
49(4), 799-813 (2012)
2. Dotson, WG: On the Mann iterative process. Trans. Am. Math. Soc.149, 63-73 (1970)
3. Diaz, JB, Metcalf, FT: On the set of subsequential limit points of successive approximations. Trans. Am. Math. Soc.135, 459-485 (1969)
4. Kato, T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn.19, 508-520 (1967)
5. Chidume, CE: Geometric Properties of Banach Spaces and Nonlinear Iterations. Lecture Notes in Mathematics, vol. 1965 (2009). ISBN:978-1-84882-189-7
6. Ofoedu, EU, Zegeye, H: Further investigation on iteration processes for pseudocontractive mappings with application. Nonlinear Anal. TMA75, 153-162 (2012)
7. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.63(1-4), 123-145 (1994)
8. Katchang, P, Jitpeera, T, Kumam, P: Strong convergence theorems for solving generalized mixed equilibrium problems and general system of variational inequalities by the hybrid method. Nonlinear Anal. Hybrid Syst. (2010). doi:10.1016/j.nahs.2010.07.001
9. Zegeye, H, Shahzad, N: Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems. Optimization, 1-17, iFirst (2012)
10. Zegeye, H, Shahzad, N: Convergence theorems for a common point of solutions of equilibrium and fixed point of relatively nonexpansive multi-valued mapping problems. Abstr. Appl. Anal.2012, Article ID 859598 (2012)
11. Censor, Y, Gibali, A, Reich, S, Sabach, S: Common solutions to variational inequalities. Set-Valued Var. Anal.20, 229-247 (2012)
12. Censor, Y, Gibali, A, Reich, S: A von Neumann alternating method for finding common solutions to variational inequalities. Nonlinear Anal.75, 4596-4603 (2012)
13. Ishikawa, S: Fixed point by a new iteration method. Proc. Am. Math. Soc.44, 147-150 (1974) 14. Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc.4, 506-510 (1953)
15. Martines-Yanes, C, Xu, HK: Strong convergence of CQ method for fixed point iteration. Nonlinear Anal.64, 2400-2411 (2006)
16. Nakajo, K, Takahashi, W: Strong convergence theorems for nonexpansive mappings. J. Math. Anal. Appl.279, 372-379 (2003)
17. Reich, S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl.67, 274-276 (1979)
18. Zegeye, H, Shahzad, N: Strong convergence theorems for a common zero of countably infinite family ofα-inverse strongly accretive mappings. Nonlinear Anal.71, 531-538 (2009)
19. Zegeye, H: Strong convergence theorems for maximal monotone mappings in Banach spaces. J. Math. Anal. Appl.
343, 663-671 (2008)
20. Bauschke, HH, Matouskova, E, Reich, S: Projections and proximal point methods: convergence results and counterexamples. Nonlinear Anal.56, 715-738 (2004)
21. Kim, TH, Xu, HK: Strong convergence of modified Mann iteration for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal.64, 1140-1152 (2006)
22. Nilsrakoo, W, Saejug, S: Weak and strong convergence theorems for countable Lipschitzian mappings and its applications. Nonlinear Anal. TMA69(8), 2695-2708 (2008)
23. Ofoedu, EU, Shehu, Y, Ezeora, JN: Solution by iteration of nonlinear variational inequalities involving finite family of asymptotically nonexpansive mappings and monotone mappings. Panam. Math. J.18(4), 61-75 (2008)
24. Yang, L, Su, Y: Strong convergence theorems for countable Lipschitzian mappings and its applications in equilibrium and optimization problems. Fixed Point Theory Appl.2009, Article ID 462489 (2009). doi:10.1155/2009/462489 25. Zegeye, H, Shahzad, N: Strong convergence theorems for a finite family of asymptotically nonexpansive mappings
and semigroups. Nonlinear Anal. (2007). doi:10.1016/j.na.2007.11.005
26. Zegeye, H, Shahzad, N: A hybrid approximation method for equilibrium, variational inequality and fixed point problems. Nonlinear Anal. Hybrid Syst.4, 619-630 (2010)
27. Zegeye, H, Shahzad, N: Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces. Optim. Lett.5(4), 691-704 (2011)
28. Kopecka, E, Reich, S: A note on alternating projections in Hilbert space. J. Fixed Point Theory Appl.12, 41-47 (2012) 29. Zegeye, H: An iterative approximation method for a common fixed point of two pseudo-contractive mappings. ISRN
Math. Anal.2011, Article ID 621901 (2011). doi:10.5402/2011/621901
30. Kassay, G, Reich, S, Sabach, S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim.21, 1319-1344 (2011)
31. Ofoedu, EU: A general approximation scheme for solutions of various problems in fixed point theory. Int. J. Anal.
2013, Article ID 762831 (2013). doi:10.1155/2013/762831 32. Byrne, CL: Applied Iterative Methods. AK Peters, Wellesley (2008)
33. Combettes, PL: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization
34. Dunn, JC: Convexity, monotonicity and gradient processes in Hilbert space. J. Math. Anal. Appl.53, 145-158 (1976) 35. Lions, PL, Mercier, B: Splitting algorithms for the sum two nonlinear operators. SIAM J. Numer. Anal.16, 964-979
(1979)
36. Liu, F, Nashed, MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal.6, 313-344 (1998)
37. Tseng, P: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim.29, 119-138 (1991)
38. Zhu, DL, Marcotte, P: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim.6, 714-726 (1996)
39. Baillon, JB, Haddad, G: Quelques propriétés des opérateurs angle-bornés etn-cycliquement monotones. Isr. J. Math.
26, 137-150 (1977)
40. Combettes, PL, Wajs, VR: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul.4, 1168-1200 (2005)
41. Yamada, I, Ogura, N: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim.25, 619-655 (2004)
42. Masad, E, Reich, S: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal.8, 367-371 (2007)
10.1186/1687-1812-2014-9
