R E S E A R C H
Open Access
Shrinking projection methods for solving
split equilibrium problems and fixed point
problems for asymptotically nonexpansive
mappings in Hilbert spaces
Uamporn Witthayarat
1, Afrah A N Abdou
2*and Yeol Je Cho
2,3**Correspondence:
[email protected]; [email protected]
2Department of Mathematics, King
Abdulaziz University, Jeddah, 21589, Saudi Arabia
3Department of Mathematics
Education and RINS, Gyeongsang National University, Jinju, 660-701, Korea
Full list of author information is available at the end of the article
Abstract
In this paper, we propose a new iterative sequence for solving common problems which consist of split equilibrium problems and fixed point problems for
asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems of the generated sequence{xn}by the shrinking
projection method. Our results improve and extend the previous results given in the literature.
MSC: 54E70; 47H25
Keywords: split equilibrium problem; asymptotically nonexpansive mapping; fixed point problem; Hilbert space
1 Introduction
Throughout this paper, letRandNdenote the set of all real numbers and the set of all positive integers, respectively. LetHbe a real Hilbert space andCbe a nonempty closed convex subset ofH.
A mappingT:C×C→Ris said to beasymptotically nonexpansiveif there exists a sequence{kn} ⊂[,∞) withlimn→∞kn= such that
Tnx–Tny≤knx–y
for allx,y∈C. It is easy to see that, ifkn≡, thenT is said to benonexpansive. We
de-note the set of fixed point ofT byF(T), that is,F(T) ={x∈C:Tx=x}. There are many iterative methods for solving a fixed point problem corresponding to an asymptotically nonexpansive mapping (see also [–]).
Recall that a Hilbert spaceHsatisfiesOpial’s condition[], that is, for any subsequence {xn} ⊂Hwithxnx, the following inequality
lim inf
n→∞ xn–x<lim infn→∞ xn–y
holds for ally∈Hwithy=x. Furthermore, a Hilbert spaceHhas a Kadec-Klee property, i.e.,xnxandxn → ximplyxn→x. In fact, from
xn–x=xn– xn,x+x,
we can conclude that a Hilbert space has a Kadec-Klee property.
In , Blum and Oettli [] introduced theequilibrium problemwhich is to findx∈C such that
F(x,y)≥ for ally∈C. (.)
They denoted the solution set of problem (.) asEP(F). Since the well-known problems were variational problems, complementary problems, fixed point problems, saddle point problems and other problems proposed from the equilibrium problem, it has become the most attractive topic for many mathematicians [–]. They have widely spread its applica-tions to other applied disciplines including physics, chemistry, economics and engineering (see, for example, [–]).
In , Combettes and Hirstoaga [] proposed an iterative method for solving problem (.) by the assumption thatEP(F)=∅. Moreover, there are many new iteratively generated sequences for solving this problem together with fixed point problems (see [–]).
Later, the so-calledsplit equilibrium problemwas introduced (shortly,SEP). LetH,H be two real Hilbert spaces. LetC,Qbe closed convex subsets ofHandH, respectively, and letA:H→Hbe a bounded linear operator. Further, letF:C×C→RandF: Q×Q→Rbe two bifunctions. TheSEPis to find the elementx∗∈Csuch that
F
x∗,y≥ for ally∈C (.)
and such that
Ax∗∈QsolvesF
Ax∗,v≥ for allv∈Q. (.)
The solution sets of problems (.) and (.) are symbolized byEP(F) andEP(F), re-spectively. Therefore, we denote={v∈C:v∈EP(F) such thatAv∈EP(F)}as the so-lution set ofSEP.
Clearly, theSEP contains two equilibrium problems, that is, we find out the solution of one equilibrium problem,i.e., its image under a given bounded linear operator, must be the solution of another equilibrium problem. In order to find a common solution of equilibrium problems, it has been mostly considered in the same spaces. However, we normally found that, in the real-life problems, it may be considered in different spaces. That is how theSEPworks very well for this case (see, for example, []). Moreover, the split variational inequality problem (shortly,SVIP) is its special case, which is to findx∗∈C such that
fx∗,x–x∗≥ for allx∈C, (.)
and corresponding to
wheref:H→Handg:H→Hare nonlinear mappings andA:H→His a bounded linear operator (see []).
In , He [] proposed the new algorithm for solving a split equilibrium problem and investigated the convergence behavior in several ways including both weak and strong convergence. Moreover, they gave some examples and mentioned that there exist many SEPs, and the new methods for solving it further need to be explored in the future. Later, in , Kazmi and Rizvi [] considered the iterative method to compute the common approximate solution of a split equilibrium problem, a variational inequality problem and a fixed point problem for a nonexpansive mapping in the framework of real Hilbert spaces. They generated the sequence iteratively as follows:
⎧ ⎪ ⎨ ⎪ ⎩
un=JrFn(I+γA∗(J
F
rn –I)A)xn,
yn=PC(un–λnDun),
xn+=αnv+βnxn+γnSyn
(.)
for eachn≥, whereA:H→His a bounded linear operator,D:C→His aτ-inverse strongly monotone mapping,F:C×C→R,F:Q×Q→Rare two bifunctions. They found that, under the sufficient conditions ofrn,λn,γ,βnandγn, the generated sequence
{xn}converges strongly to a common solution of all mentioned problems.
Recently, in , Bnouhachem [] introduced a new iterative method for solving split equilibrium problem and hierarchical fixed point problems by defining the sequence{xn}
as follows:
⎧ ⎪ ⎨ ⎪ ⎩
un=TrFn(I+γA∗(T
F
rn –I)A)xn,
yn=βnSxn+ ( –βn)un,
xn+=PC[αnρU(xn) + (I–αnμF)(T(yn))]
(.)
for eachn≥, whereS,Tare nonexpansive mappings,F:C→Cis ak-Lipschitz map-ping andη-strongly monotone,U:C→Cis aτ-Lipschitz mapping. Also, they proved some strong convergence theorems for the proposed iteration under some appropriate conditions.
In this paper, motivated and inspired by the results [, , ] and the recent works in this field, we introduce the shrinking projection method for solving split equilibrium prob-lems and fixed point probprob-lems for asymptotically nonexpansive mappings in the frame-work of Hilbert spaces and prove some strong convergence theorems for the proposed new iterative method. In fact, our results improve and extend the results given by some authors.
2 Preliminaries
In this section, we recall some concepts including the assumption which will be needed for the proof of our main result.
LetHbe a Hilbert space andCbe a nonempty closed convex subset ofH. For eachx∈H, there exists a unique nearest point ofC, denoted byPCx, such that
for ally∈C.PCis called themetric projectionfromHontoC. It is well known thatPCis
a firmly nonexpansive mapping fromHontoC, that is,
PCx–PCy≤ PCx–PCy,x–y
for allx,y∈H. Furthermore, for anyx∈Handz∈C,z=PCxif and only if
x–z,z–y ≥
for ally∈C. A mappingA:C→His calledα-inverse strongly monotoneif there exists α> such that
x–y,Ax–Ay ≥αAx–Ay
for allx,y∈H. Moreover, we can investigate that, for eachλ∈(, α],I–λAis a nonex-pansive mapping ofCintoH(see []).
Lemma . In a Hilbert space H,the following identity holds:
λx+ ( –λ)y=λx+ ( –λ)y–λ( –λ)x–y
for all x,y∈H andλ∈[, ].
Lemma .[] Let T be an asymptotically nonexpansive mapping defined on a bounded
closed convex subset C of a Hilbert space H.Assume that{xn}is a sequence in C with the
following properties:
() xnz; () Txn–xn→.
Then z∈F(T).
Assumption .[] LetF:C×C→Rbe a bifunction satisfying the following condi-tions:
(A) F(x,x) = for allx∈C;
(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,limt↓F(tz+ ( –t)x,y)≤F(x,y);
(A) for eachx∈C,y→F(x,y)is convex and lower semi-continuous.
Lemma .[] Let C be a nonempty closed convex subset of a Hilbert space H and F : C×C→Rbe a bifunction which satisfies conditions(A)-(A).For any x∈H and r> , define a mapping TF
r :H→C by
TrF(x) =
z∈C:F(z,y) +
ry–z,z–x ≥,∀y∈C
.
Then TF
r is well defined and the following hold: () TF
() TF
r is firmly nonexpansive,i.e.,for anyx,y∈H,
TrFx–TrFy≤TrFx–TrFy,x–y;
() F(TrF) =EP(F);
() EP(F)is closed and convex.
3 Main results
In this section, we prove some strong convergence theorems of an iterative algorithm for solving a split equilibrium together with a fixed point problem revolving an asymptotically nonexpansive mapping in the framework of Hilbert spaces.
Theorem . Let H,Hbe two real Hilbert spaces and C,Q be nonempty closed convex
subsets of Hilbert spaces H and H,respectively.Let F:C×C→Rand F:Q×Q→
Rbe two bifunctions satisfying conditions(A)-(A)and Fbe upper semi-continuous in the first argument.Let T:C→C be an asymptotically nonexpansive mapping and A: H→Hbe a bounded linear operator.Suppose that F(T)∩=∅,where={v∈C:v∈ EP(F)such that Av∈EP(F)},and let x∈C.For C=C and x=PCx,define a sequence {xn}iteratively as follows:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
un=TrFn(I–γA∗(I–T
F
rn)A)xn,
yn=αnxn+ ( –αn)Tnun,
Cn+={z∈Cn:yn–z≤ xn–z+θn},
xn+=PCn+x
(.)
for each n≥,whereθn= ( –αn)(kn– )sup{xn–z:z∈}, ≤αn≤a< for all
n∈N, <b≤rn<∞,γ ∈(, /L),L is the spectral radius of the operator A∗A and A∗
is the adjoint of A.Then the sequence{xn}generated by(.)strongly converges to a point
z∈F(T)∩.
Proof First of all, we investigate that, for eachn∈N,A∗(I–TF
rn)Ais a
L-inverse strongly
monotone mapping. SinceTF
rn is firmly nonexpansive and (I–T
F
rn) is
-inverse strongly monotone, it follows that
A∗I–TF
rn
Ax–A∗I–TF
rn
Ay
=A∗I–TF
rn
(Ax–Ay),A∗I–TF
rn
(Ax–Ay)
=I–TF
rn
(Ax–Ay),AA∗I–TF
rn
(Ax–Ay)
≤LI–TF
rn
(Ax–Ay),I–TF
rn
(Ax–Ay)
=LI–TF
rn
(Ax–Ay)
≤Lx–y,A∗I–TF
rn
(Ax–Ay)
for allx,y∈H, from which it can be concluded thatA∗(I–TF
rn)Ais a
L-inverse strongly
monotone mapping. Moreover, we claim that sinceγ ∈(,
L),I–γA∗(I–T F
rn)Ais
Next, we show thatF(T)∩⊂Cn+ for alln∈N. Letp∈F(T)∩,i.e.,TrFnp=pand
(I–γA∗(I–TF
rn)A)p=p. By mathematical induction, we havep∈C=Cand henceF(T)∩
⊂C. LetF(T)∩⊂Ckfor somek∈N. It follows that
uk–p=TrFk
I–γA∗I–TF
rk
Axk–TrFk
I–γA∗I–TF
rk
Ap
≤I–γA∗I–TF
rk
Axk–
I–γA∗I–TF
rk
Ap
≤ xk–p (.)
and
yk–p
=αkxk+ ( –αk)Tnuk–p
≤αkxk–p+ ( –αk)Tnuk–Tn–p
–αk( –αk)xk–p–
Tnuk–Tnp
≤αkxk–p+ ( –αk)uk–p–αk( –αk)xk–Tnuk
≤αkxk–p+ ( –αk)kkxk–p
=xk–p+ ( –αk)
kk– xk–p
≤ xk–p+ ( –αk)
kk– Mk
=xk–p+θk, (.)
whereMk=sup{xk–z:z∈}andθk= ( –αk)(kk– )Mk. It can be concluded that
p∈Ck+andF(T)∩⊂Ck+and, further,F(T)∩⊂Cnfor alln∈N.
Next, we show thatCnis closed and convex for alln∈N. First, it is obvious thatC=C is closed and convex. By induction, we suppose thatCk is closed and convex for some
k∈N. Let zm ∈Ck+ ⊂Ck with zm →z. SinceCk is closed, it follows thatx∈Ck and
yk–zm≤ zm–xk+θk. Then we have
yk–z =yk–zm+zm–z
=yk–zm+zm–z+ yk–zm,zm–z
≤ zm–xk+θk+zm–z+ yk–zmzm–z.
Lettingm→ ∞, we have
yk–z≤ z–xk+θk,
which means thatz∈Ck+. Letx,y∈Ck+⊂Ck andz=αx+ ( –α)yfor anyα∈[, ].
SinceCkis convex,z∈Ck,yk–x≤ x–xk+θkandyk–y≤ x–xk+θkand so
yk–z =yk–
αx+ ( –α)y
=α(yk–x) + ( –α)(yk–y)
=αyk–x+ ( –α)yk–y–α( –α)yk–x– (yk–y)
≤αx–xk+θk
+ ( –α)y–xk+θk
–α( –α)y–x
≤αx–xk+θk+ ( –α)y–xk–α( –α)(xk–x) – (xk–y)
=αx–xk+ ( –α)y–xk–α( –α)(xk–x) – (xk–y)+θk
=α(xk–x) + ( –α)(xk–y)
+θk
=xk–z+θk.
Therefore,z∈Ck+and henceCk+is closed and convex. It is immediately concluded that Cnis closed and convex for alln∈N, which implies that{xn}is well defined.
Next, fromxn=PCnx, we have
x–xn,xn–y ≥
for ally∈Cn. Sincep∈F(T)∩, we have
x–xn,xn–p ≥
for allp∈F(T)∩, that is, we have
≤ x–xn,xn–p ≤–xn–x+x–xnx–p.
This implies that
xn–x ≤ x–p (.)
for alln∈N. Fromxn=PCnxandxn+=PCn+x∈Cn+⊂Cn, we also have
x–xn,xn–xn+ ≥ (.)
for alln∈N, and so we have
≤ x–xn,xn–xn+
=x–xn,xn–x+x–xn+
≤–xn–x+x–xnx–xn+.
Hence we have
xn–x≤ x–xnx–xn+, (.)
that is,xn–x ≤ x–xn+for alln∈N. From (.), it follows that{xn}is bounded and
limn→∞xn–xexists.
Next, we show thatxn–xn+ →. From (.), we have
xn–xn+
=xn–x+ xn–x,x–xn++x–xn+
=xn–x+ xn–x,x–xn+ xn–x,xn–xn++x–xn+
≤ xn–x– xn–x+x–xn+
=x–xn+–xn–x.
Since the limit of{xn–x}exists, we have
lim
n→∞xn–xn+= . (.)
Thus, by (.) and (.), we have
yn–xn ≤ yn–xn++xn+–xn → (.)
asn→ ∞. Furthermore, sinceTF
rn is firmly nonexpansive, we have
un–p=TrFn
xn–γA∗
I–TF
rn
Axn
–TF
rn
p–γA∗I–TF
rn
Ap
≤I–γA∗I–TF
rn
Axn–
I–γA∗I–TF
rn
Ap
–I–TF
rn
I–γA∗I–TF
rn
Axn
–I–TF
rn
I–γA∗I–TF
rn
Ap
=xn–p–γ
A∗I–TF
rn
Axn–A∗
I–TF
rn
Ap–zn–TrFnzn
=xn–p– γ
xn–p,A∗
I–TF
rn
Axn–A∗
I–TF
rn
Ap
+γA∗I–TF
rn
Axn–A∗
I–TF
rn
Ap–zn–TrFnzn
≤ xn–p+γ
γ –
L
A∗I–TF
rn
Axn
–zn–TrFnzn
,
wherezn= (I–γA∗(I–TrFn)A)xn. Moreover,
yn–p =αnxn+ ( –αn)Tnun–p
≤αnxn–p+ ( –αn)kn
xn–p
+γ
γ – L
A∗I–TF
rn
Axn–zn–TrFnzn
=αnxn–p+ ( –αn)knxn–p– ( –αn)knzn–TrFnzn
+ ( –αn)knγ
γ–
L
A∗I–TF
rn
Axn
,
which leads to
( –αn)kn
γ
L–γ
A∗I–TF
rn
Axn+zn–TrFnzn
≤αn+ ( –αn)kn
Lettingρn=kn– . Then it is clear thatρn→ asn→ ∞and, by (.), we exactly have
( –αn)kn
γ
L –γ
A∗I–TF
rn
Axn)+zn–TrFnzn
≤αnxn–p+ ( –αn)(ρn+ )xn–p–yn–p
≤ xn–p–yn–p+ ( –αn)
ρn+ ρn
xn–p
≤xn–p+yn–p
xn–yn+ ( –αn)
ρn+ ρn
xn–p.
By (.) andρn→ asn→ ∞, we have
A∗I–TF
rn
Axn→, zn–TrFnzn→ (.)
asn→ ∞. Furthermore, sinceAis linear bounded and so isA∗, we can conclude that
lim
n→∞I–T
F
rn
Axn= . (.)
Next, we show thatun–xn →. We investigate the following:
un–xn=TrFnzn–xn
≤TF
rnzn–zn+zn–xn
=TF
rnzn–zn+I–γA
∗I–TF
rn
Axn–xn
=TF
rnzn–zn+γA
∗I–TF
rn
Axn. (.)
Consequently, by (.), we can conclude that
un–xn →. (.)
Next, we show thatTnx
n–xn →. We first consider
yn–xn=αnxn+ ( –αn)Tnun= ( –αn)Tnun–xn,
and sincexn+∈Cn+⊂Cn, we have
yn–xn+≤ xn–xn++θn,
which means that
yn–xn+ ≤ xn–xn++
θn. (.)
Hence,
Tnun–xn=
–αn
yn–xn
≤
–a
yn–xn++xn+–xn
≤
–a
xn–xn++
θn
+
and soTnu
n–xn →. Consider
Tnxn–xn≤Tnxn–Tnun+Tnun–xn
≤knxn–un+Tnun–xn.
Therefore, we haveTnx
n–xn → asn→ ∞. Puttingk∞ =sup{kn:n≥}<∞, we
deduce that
Txn–xn≤Txn–Tn+xn+Tn+xn–Tn+xn+
+Tn+xn+–xn++xn+–xn
≤k∞xn–Tnxn+ ( +k∞)xn–xn++Tn+xn+–xn+.
Hence we haveTxn–xn → asn→ ∞. Without loss of generality, since{xn}is bounded,
we may assume thatxnx∗. It is easy to see thatx∗∈Cnfor alln≥. On the other hand,
we have
xn–x ≤x∗–x.
It follows that
x∗–x≤lim inf
n→∞ xn–x ≤lim supn→∞ xn–x ≤x
∗–x
and so
lim
n→∞xn–x=x
∗–x .
Hencexn → x∗. Since every Hilbert space has the Kadec-Klee property, we
immedi-ately havexn→x∗.
Finally, we prove thatx∗∈F(T)∩. Sincexn→x∗andxn–Txn→ asn→ ∞, consider
x∗–Tx∗≤x∗–xn+xn–Txn+Txn–Tx∗
≤( +k)x∗–xn+xn–Txn.
We can see thatx∗–Tx∗= and, further,x∗∈F(T). Therefore, we havex∗∈F(T). Next, we show thatx∗∈. By (.),un=TrFn(I–γA∗(I–T
F
rn)), that is,
F(un,y) +
rn
y–un,un–xn–
rn
y–un,γA∗
TF
rn –I
Axn
≥
for ally∈C. From (A), it follows that
– rn
y–un,γA∗
TF
rn –I
Axn
+
rn
y–un,un–xn ≥F(y,un)
for ally∈C. SinceA∗(TF
rn –I)Axn →,un–xn → andxn–x∗ → asn→ ∞,
we have
F
for ally∈C. Letyt=ty+ ( –t)x∗ for any <t≤ andy∈C. It means thatyt∈Cand
hence
=F(yt,yt)≤tF(yt,y) + ( –t)F
yt,x∗
≤tF(yt,y),
and thenF(yt,y)≥. Lettingt→, we immediately haveF(x∗,y)≥,i.e.,x∗∈EP(F). Next, we show thatAx∗∈EP(F). SinceAis a bounded linear operator and (.), we have
TF
rnAxn–Ax
∗≤TAxn
rn –Axn–Ax
∗→
asn→ ∞, which yields thatTF
rnAxn→Ax∗. By the definition ofT
F
rn, we have
F
TF
rnAxn,y
+
rn
y–TF
rnAxn,T
F
rnAxn–Axn
≥ (.)
for all y∈C. SinceFis upper semi-continuous in the first argument, takinglim supin (.), it follows that
F
Ax∗,y≥
for allx,y∈C, from which it can be concluded thatAx∗∈EP(F). Consequently,x∗∈.
This completes the proof.
In Theorem ., if the mappingTis a nonexpansive mapping, then we immediately have the following.
Corollary . Let H,Hbe two real Hilbert spaces and C,Q be nonempty closed convex subsets of Hilbert spaces Hand H,respectively.Let F:C×C→Rand F:Q×Q→Rbe two bifunctions satisfying conditions(A)-(A)and Fbe upper semi-continuous in the first argument.Let T:C→C be a nonexpansive mapping and A:H→Hbe a bounded linear operator.Suppose that F(T)∩=∅,where={v∈C:v∈EP(F)such that Av∈EP(F)}, and let x∈C.For C=C and x=PCx,define a sequence{xn}iteratively as follows:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
un=TrFn(I–γA∗(I–T
F
rn)A)xn,
yn=αnxn+ ( –αn)Tun,
Cn+={z∈Cn:yn–z≤ xn–z+θn},
xn+=PCn+x
(.)
for each n∈N,where Mn=sup{xn–z:x∈}andθn= ( –αn)(kn– )Mn, ≤αn≤a<
for all n∈N, <b≤rn<∞,γ ∈(, /L),L is the spectral radius of the operator A∗A and
A∗ is the adjoint of A.Then the sequence{xn}generated by(.)strongly converges to a
point z∈F(T)∩.
IfH=H,C=QandA=Iin Theorem ., then we have the following.
and Fbe upper semi-continuous in the first argument.Let T:C→C be an asymptotically nonexpansive mapping.Suppose that F(T)∩=∅,where={v∈C:v∈EP(F)∩EP(F)}, and let x∈C.For C=C and x=PCx,define a sequence{xn}iteratively as follows:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
un=TrFnT
F
rnxn,
yn=αnxn+ ( –αn)Tnun,
Cn+={z∈Cn:yn–z≤ xn–z+θn},
xn+=PCn+x
(.)
for each n∈N,where Mn=sup{xn–z:z∈}andθn= ( –αn)(kn– )(Mn), ≤αn≤
a< and <b≤rn<∞for all n∈N.Then the sequence{xn}generated by(.)strongly
converges to a point z∈F(T)∩.
4 Applications
4.1 Applications to split variational inequality problems
Firstly, we point out the so-called variational inequality problem (shortly,VIP), which is to find a pointx∗∈Cwhich satisfies the following inequality:
Ax∗,z–x∗≥
for allz∈C. Its solution set is symbolized byVI(A,C).
In , Censoret al.[] proposed thesplit variational inequality problem(shortly, SVIP) which is formulated as follows:
Find a pointx∗∈Csuch thatfx∗,x–x∗≥ for allx∈C
and such that
y∗=Ax∗∈Qsolvesgy∗,y–y∗≥ for ally∈Q,
whereA:C→Cis a bounded linear operator. The solution set of split variational inequal-ity problem is denoted by theSVIP.
SettingF(x,y) =f(x),y–xandF(x,y) =g(x),y–x, it is clear thatF,Fsatisfy con-ditions (A)-(A), wheref andg are η- and η-inverse strongly monotone mappings, respectively. Then, by Theorem ., we get the following.
Theorem . Let H,H be two real Hilbert spaces and C,Q be nonempty closed
define a sequence{xn}iteratively as follows:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
un=TrFn(I–γA∗(I–T
F
rn)A)xn,
yn=αnxn+ ( –αn)Tnun,
Cn+={z∈Cn:yn–z≤ xn–z+θn},
xn+=PCn+x
(.)
for each n∈N,where Mn=sup{xn–z:z∈}andθn= ( –αn)(kn– )Mn, ≤αn≤a<
for all n∈N, <b≤rn<∞,γ ∈(, /L),L is the spectral radius of the operator A∗A and
A∗ is the adjoint of A.Then the sequence{xn}generated by(.)strongly converges to a
point z∈F(T)∩.
Proof The desired result can be proved directly through Theorem ..
4.2 Applications to split optimization problems
In this section, we mention applications to thesplit optimization problem, which is to find x∗∈Csuch that
fx∗≥f(x) for allx∈CsatisfyingAx∗=y∗∈Qsolvesgy∗≥g(y) (.)
for ally∈Q. We symbolizefor the solution set of the split optimization problem. Letf :C→Randg:Q→Rbe two functions satisfying the following assumption:
() for eachx,y∈C,f(tx+ ( –t)y)≤f(y), and for eachu,v∈Q,g(tu+ ( –t)v)≤g(v); () f(x)is concave and upper semi-continuous for allx∈Candg(u)is concave and
upper semi-continuous for allu∈Q.
LetF(x,y) =f(x) –f(y) for allx,y∈CandF(u,v) =g(u) –g(v) for allu,v∈Q. Iff andg satisfy conditions () and (), then it is clear thatF:C×C→RandF:Q×Q→Rare two bifunctions satisfying conditions (A)-(A). Therefore, by Theorem ., we have the following.
Theorem . Let H,Hbe two real Hilbert spaces and C,Q be nonempty closed convex subsets of Hilbert spaces H and H, respectively. Let f :C→Rand g:Q→Rbe two functions satisfying conditions()and().Let F:C×C→Rand F:Q×Q→Rbe two bifunctions satisfying conditions(A)-(A)and F be upper semi-continuous in the first argument.Let A:H→Hbe a bounded linear operator.Suppose that F(T)∩=∅and let x∈C.For C=C and x=PCx,define a sequence{xn}iteratively as follows:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
un=TrFn(I–γA∗(I–T
F
rn)A)xn,
yn=αnxn+ ( –αn)un,
Cn+={z∈Cn:yn–z ≤ xn–z},
xn+=PCn+x
(.)
for each n∈N,where≤αn≤a< , <b≤rn<∞,andγ ∈(, /L),L is the spectral
radius of the operator A∗A and A∗is the adjoint of A.Then the sequence{xn}generated by
(.)strongly converges to a point z∈F(T)∩.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics, School of Science, University of Phayao, Phayao, 56000, Thailand.2Department of
Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.3Department of Mathematics Education and RINS,
Gyeongsang National University, Jinju, 660-701, Korea.
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (18-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2014R1A2A2A01002100).
Received: 10 August 2015 Accepted: 27 October 2015 References
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