A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces

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R E S E A R C H

Open Access

A regularization projection algorithm for

various problems with nonlinear mappings in

Hilbert spaces

Buthinah A Bin Dehaish

1

_{, Abdul Latif}

2

_{, Huda O Bakodah}

1

_{and Xiaolong Qin}

2* *_{Correspondence: [email protected]}

2_{Department 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, a regularization projection algorithm is investigated for solving common elements of an equilibrium problem, a variational inequality problem and a ﬁxed point problem of a strictly pseudocontractive mapping. Strong convergence theorems are established in the framework of real Hilbert spaces.

Keywords: Hilbert space; equilibrium problem; variational inequality; nonexpansive mapping; ﬁxed point

1 Introduction and preliminaries

LetHbe a real Hilbert space with the inner product·,·and the norm · . LetCbe a nonempty closed convex subset ofH.PCstands for the metric projection fromHontoC. LetFbe a bifunction ofC×CintoR, whereRdenotes the set of real numbers. Consider the following equilibrium problem in the terminology of Blum and Oettli []:

Findx∈Csuch thatF(x,y)≥, ∀y∈C. (.)

In this paper, the solution set of problem (.) is denoted byFP(F).

To study problem (.), we may assume thatFsatisﬁes the following conditions: (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,

lim sup

t↓

Ftz+ ( –t)x,y≤F(x,y);

(A) for eachx∈C,y→F(x,y)is convex and lower semi-continuous.

LetSbe a self-mapping ofC.F(S) stands for the ﬁxed point set ofS. Recall thatSis said to beβ-contractive if there exists a constantβ∈[, ) such that

Sx–Sy ≤βx–y, ∀x,y∈C.

Sis said to be nonexpansive if

Sx–Sy ≤ x–y, ∀x,y∈C.

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Sis said to beκ-strictly pseudocontractive if there exists a constantκ∈[, ) such that

Sx–Sy≤ x–y+κ(x–Sx) – (y–Sy), ∀x,y∈C.

The class of κ-strictly pseudocontractive mappings was introduced by Browder and

Petryshyn []. It is clear that every nonexpansive mapping is a -strictly pseudocontractive mapping.

LetA:C→Hbe a mapping. Recall thatAis said to be monotone if

Ax–Ay,x–y ≥, ∀x,y∈C.

Ais said to be strongly monotone if there exists a constantα>  such that

Ax–Ay,x–y ≥αx–y, ∀x,y∈C.

For such a case, we say thatAisα-strongly monotone.Ais said to be inverse-strongly monotone if there exists a constantα>  such that

Ax–Ay,x–y ≥αAx–Ay_, _∀x_,_y_∈_C_.

For such a case, we say thatAisα-inverse-strongly monotone. It is clear thatAisα -in-verse-strongly monotone if and only ifA–_is_α_{-strongly monotone.}

Recall that the classical variational inequality is to ﬁndx∈Csuch that

Ax,y–x ≥, ∀y∈C. (.)

It is known thatx∈Cis a solution of (.) if and only ifxis a ﬁxed point ofPC(I–rA), wherer>  is a constant andIis an identity mapping. From now on, the solution set of (.) is denoted byVI(C,A).

A set-valued mappingT:H→His said to be monotone if for allx,y∈H,f ∈Txandg∈ Tyimplyx–y,f–g ≥. A monotone mappingT:H→H_{is maximal if the graph}_G₍_T₎ ofT is not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if, for any (x,f)∈H×H,x–y,f–g ≥ for all (y,g)∈G(T) impliesf ∈Tx.

Recently, Iiduka and Takahshi [] investigated variational inequality (.) and ﬁxed points of a nonexpansive mapping based on a Halpern-like algorithm. To be more clear, they proved the following result.

Theorem IT Let C be a closed convex subset of a real Hilbert space H. Let A be an

α-inverse-strongly monotone mapping of C into H,and let S be a nonexpansive mapping of C into itself such that F(S)∩VI(C,A)=∅.Suppose x=x∈C and{xn}is given by

xn+=αnx+ ( –αn)SPC(xn–λnAxn)

for every n= , , . . . ,where{αn}is a sequence in[, )and{λn}is a sequence in[, α].If

{αn}and{λn}are chosen so thatλn∈[a,b]for some a,b with <a<b< α,limn→∞αn= ,

∞

n=αn=∞,

∞

n=|αn+–αn|<∞and

∞

n=|λn+–λn|<∞,then{xn}converges strongly

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Since equilibrium problem (.) provides a unified model of several problems such as variational inequalities, fixed point problems and inclusion problems. In [], Takahashi and Takahashi further studied fixed points of a nonexpansive mapping and equilibrium problem (.) based on the viscosity approximation method, which was introduced by Moudafi []. To be more clear, they proved the following result.

Theorem TT Let C be a nonempty closed convex subset of a real Hilbert space H.Let F be a bifunction from C×C to R satisfying(A)-(A),and let S be a nonexpansive mapping of C into H such that F(S)∩EP(F)=∅.Let f be a contraction of H into itself,and let{xn}

and{un}be sequences generated by x∈H and ⎧

⎨ ⎩

F(un,y) +_rny–un,un–xn ≥, ∀y∈C,

xn+=αnf(xn) + ( –αn)Sun

for every n= , , . . . ,where{αn} ⊂[, ]and{rn} ⊂(,∞)satisfylimn→∞αn= ,∞n=αn=

∞,∞_n_=|αn+–αn|<∞,lim infn→∞rn> and∞n=|rn+–rn|<∞.Then{xn}and{un}

converge strongly to z∈F(S)∩EP(F),where z=PF(S)∩EP(F)f(z).

Subsequently, many authors investigated common solution problems based on hybrid projection methods due to real world applications, for example, image restoration and digital signal processing; see [–] and the references therein. In view of the complexity of convex, hybrid projection methods, they are not easy to implement. In this paper, we study a regularization projection algorithm for solving equilibrium problem (.), varia-tional inequality (.) and a ﬁxed point problem of aκ-strictly pseudocontractive map-ping. Possible computation errors are taken into account. Strong convergence theorems are established in the framework of real Hilbert spaces.

In order to prove our main results, we also need the following lemmas.

Lemma .[] Let C be a nonempty closed convex subset of H.Let A be a monotone mapping of C into H,and let NCv be a normal cone to C at v∈C,i.e.,

NCv= w∈H:v–u,w ≥,∀u∈C

and deﬁne a mapping T on C by

Tv=

⎧ ⎨ ⎩

Av+NCv, v∈C,

∅, v∈/C.

Then T is maximal monotone and∈Tv if and only ifAv,u–v ≥for all u∈C.

Lemma .[] Let C be a nonempty closed convex subset of H,and let S:C→H be a

κ-strictly pseudocontractive mapping.Then I–S is demi-closed at zero.

Lemma .[] Let C be a nonempty closed convex subset of H,and let F:C×C→Rbe a bifunction satisfying(A)-(A).Then,for any r> and x∈H,there exists z∈C such that

F(z,y) +

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Further,deﬁne

Trx=

z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

for all r> and x∈H.Then the following hold: (a) Tris single-valued;

(b) Tris ﬁrmly nonexpansive,i.e.,for anyx,y∈H,

Trx–Try≤ Trx–Try,x–y;

(c) F(Tr) =EP(F);

(d) EP(F)is closed and convex.

Lemma .[] Let C be a nonempty closed convex subset of H,and let S:C→H be a

κ-strictly pseudocontractive mapping.Deﬁne a mapping T by T=δI+ ( –δ)S,whereδis a constant in(, ).Ifδ∈[κ, )then T is nonexpansive and F(T) =F(S).

Lemma .[] Let{xn}and{yn}be bounded sequences in H,and let{βn}be a sequence

in[, ]with <lim inf_n_→∞βn≤lim supn→∞βn< .Suppose xn+= ( –βn)yn+βnxnfor all

integers n≥and

lim sup

n→∞

yn+–yn–xn+–xn

≤.

Thenlimn→∞yn–xn= .

Lemma .[] Assume that{αn}is a sequence of nonnegative real numbers such that

αn+≤( –γn)αn+δn+μn,

where{γn}is a sequence in(, )and{μn},{δn}are real sequences such that (i) ∞_n_=γn=∞and

∞

n=μn<∞; (ii) lim sup_n_→∞δn/γn≤or

∞

n=|δn|<∞.

Thenlimn→∞αn= .

2 Main results

Theorem . Let C be a closed convex subset of a real Hilbert space H.Let A:C→H be anα-inverse-strongly monotone mapping,and let F be a bifunction from C×C toRwhich satisﬁes(A)-(A).Let S:C→H be aκ-strictly pseudocontractive mapping,and let f be aβ-contraction on H.Assume that=F(S)∩VI(C,A)∩EP(F)=∅.Let{rn}and{sn}be

positive real number sequences.Let{αn},{βn},{γn}and{δn}be real number sequences in (, )such thatαn+βn+γn= .Let{xn}be a sequence generated in the following process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈H,

F(zn,z) +_rnz–zn,zn–xn ≥, ∀z∈C,

yn=PC(zn–snAzn+en),

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where{en}is a sequence in H.Assume that the control sequences satisfy the following

con-ditions:

(a) limn→∞αn= and

_∞

n=αn=∞; (b)  <lim infn→∞βn≤lim supn→∞βn< ;

(c) ∞_n_=en<∞,limn→∞|δn+–δn|= andκ≤δn≤δ< ; (d) lim_n_→∞|rn+–rn|= andlim infn→∞rn> ;

(e) limn→∞|sn+–sn|= , <s≤sn≤s< α,

whereδ,s,sare real constants.Then{xn}converges strongly to q,which is also a unique

solution to the variational inequality

f(x) –x,x–y≥, ∀y∈C.

Proof For anyx,y∈C, we see that

(I–snA)x– (I–snA)y

=x–y– snx–y,Ax–Ay+snAx–Ay

≤ x–y–sn(α–sn)Ax–Ay.

By using condition (e), we see that(I–snA)x– (I–snA)y ≤ x–y. This proves that

I–snAis nonexpansive. PutSn=δnI+ ( –δn)S. It follows from Lemma . thatSnis nonexpansive andF(Sn) =F(S). Letp∈be ﬁxed arbitrarily. Hence, we have

xn+–p ≤αnf(xn) –p+βnxn–p+γnSnyn–p

≤αnβxn–p+αnf(p) –p+βnxn–p+γnyn–p

≤ –αn( –β)

xn–p+αnf(p) –p+en

≤max

xn–p,

f(p) –p

 –β

+en.

It follows that xn–p ≤max{x –p,f(_–p)–βp}+

∞

n=en. This shows that {xn} is bounded, so are{yn}and{zn}. Letλn=xn+_––ββnnxn. It follows that

λn+–λn=

αn+f(xn+) +γn+Sn+yn+

 –βn+

–αnf(xn) +γnSnyn  –βn

= αn+(f(xn+) –Sn+yn+) + ( –βn+)Sn+yn  –βn+

–αn(f(xn) –Snyn) + ( –βn)Snyn  –βn

= αn+(f(xn+) –Sn+yn+)  –βn+

–αn(f(xn) –Snyn)  –βn

+Sn+yn+–Snyn.

Hence, we have

λn+–λn ≤

αn+f(xn+) –Sn+yn+

 –βn+

+αnf(xn) –Snyn  –βn

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Since zn=Trnxn, we ﬁnd thatF(zn,z) + _rnz–zn,zn–xn ≥,∀z∈C andF(zn+,z) + 

rn+z–zn+,zn+–xn+ ≥,∀z∈C. It follows thatF(zn,zn+) +



rnzn+–zn,zn–xn ≥ andF(zn+,zn) +_rn+ zn–zn+,zn+–xn+ ≥,∀z∈C. By using condition (A), we ﬁnd

thatzn+–zn,zn_rn–xn–zn+_rn+–xn+ ≥. Hence, we have

zn+–zn,zn+–xn–

rn

rn+

(zn+–xn+)

≥ zn+–zn.

This implies thatzn+–zn,xn+–xn+ ( –_rn+rn )(zn+–xn+) ≥ zn+–zn. Hence, we have

zn+–zn ≤ xn+–xn+|rn+ –rn|

rn+

Trn+xn–xn+.

It follows that

yn+–yn ≤PC(zn+–sn+Azn++en+) –PC(zn–sn+Azn+en+)

+PC(zn–sn+Azn+en+) –PC(zn–snAzn+en)

≤(I–sn+A)zn+– (I–sn+A)zn

+(zn–sn+Azn+en+) – (zn–snAzn+en)

≤ zn+–zn+|sn+–sn|Azn+en++en

≤ xn+–xn+|

rn+–rn|

rn+ Trn+ xn–xn

+|sn+–sn|Azn+en++en.

This implies that

Sn+yn+–Snyn

≤ Sn+yn+–Sn+yn+Sn+yn–Snyn

≤ yn+–yn+Sn+yn–Snyn

≤ xn+–xn+|rn+ –rn|

+|sn+–sn|Azn+en++en+|δn+–δn|Syn–yn. (.)

Substituting (.) into (.), we ﬁnd that

λn+–λn–xn+–xn

≤αn+f(xn+) –Sn+yn+

 –βn+

+αnf(xn) –Snyn  –βn

+|rn+–rn|

+|sn+–sn|Azn+en++en+|δn+–δn|Syn–yn.

It follows from conditions (a)-(e) that

lim sup

n→∞

λn+–λn–xn+–xn

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By using Lemma ., we see thatlimn→∞λn–xn= , which in turn implies that

lim

n→∞xn+–xn= . (.)

SinceTrnis ﬁrmly nonexpansive, we ﬁnd that

zn–p=Trnxn–Trnp

≤ xn–p,zn–p

= 

xn–p+zn–p–xn–zn

.

That is,

zn–p≤ xn–p–xn–zn.

It follows that

xn+–p≤αnf(xn) –p



+βnxn–p+γnSnyn–p

≤αnf(xn) –p



+βnxn–p+γnyn–p



+βnxn–p+γnzn–p+fn

≤αnf(xn) –p



+xn–p–γnxn–zn+fn,

wherefn=en(en+ zn–p). This further implies that

γnxn–zn≤αnf(xn) –p



+xn–p–xn+–p+fn

≤αnf(xn) –p+

xn–p+xn+–p

xn–xn++fn.

By using conditions (a) and (b), we ﬁnd from (.) that

lim

n→∞zn–xn= . (.)

SinceAisα-inverse-strongly monotone, we ﬁnd that

yn–p≤(zn–snAzn) – (p–snAp) +en



≤(zn–p) –sn(Azn–Ap)



+fn

≤ xn–p–sn(αn–sn)Azn–Ap+fn.

It follows that



+βnxn–p+γnyn–p

≤αnf(xn) –p



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This yields that

sn(α–sn)γnAzn–Ap≤αnf(xn) –p+xn–p–xn+–p+fn.

By using (.), we ﬁnd from conditions (a), (c) and (d) that

lim

n→∞Azn–Ap= . (.)

SincePCis ﬁrmly nonexpansive, we ﬁnd that

yn–p≤

(I–snA)zn+en– (I–snA)p,yn–p

= 

 (I–snA)zn+en– (I–snA)p



+yn–p

–(I–snA)zn+en– (I–snA)p– (yn–p)



≤ 

 zn–p

₊_f

n+yn–p–zn–yn–

sn(Azn–Ap) –en



= 

 zn–p

₊_f

n+yn–p–zn–yn

+ zn–yn,sn(Azn–Ap) –en

–sn(Azn–Ap) –en



≤ 

 xn–p

₊_f

n+yn–p–zn–yn

+ zn–ynsn(Azn–Ap) –en–sn(Azn–Ap) –en



,

which yields that

yn–p≤ xn–p+fn–zn–yn+ snzn–ynAzn–Ap

+ zn–ynen.

It follows that



≤αnf(xn) –p+βnxn–p+γnyn–p

≤αnf(xn) –p+xn–p+fn–γnzn–yn

+ snγnzn–ynAzn–Ap+ zn–ynen.

This in turn leads to

γnzn–yn≤αnf(xn) –p



+xn–p–xn+–p+fn

+ snγnzn–ynAzn–Ap+ zn–ynen.

By use of (.) and (.), we ﬁnd from restrictions (a), (b), (c) and (e) that

lim

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On the other hand, we have

γnSnyn–xn ≤ xn+–xn+αnxn–f(xn).

By using conditions (a) and (b), we ﬁnd from (.) that

lim

n→∞xn–Snyn= . (.)

SinceSnis nonexpansive, we ﬁnd that

Snxn–xn ≤ Snxn–Snyn+Snyn–xn

≤ xn–zn+zn–yn+Snyn–xn.

It follows from (.), (.) and (.) that

lim

n→∞xn–Snxn= . (.)

Notice that

Sxn–xn ≤Sxn–

δnxn+ ( –δn)Sxn+Snxn–xn

≤δnSxn–xn+Snxn–xn.

By using condition (c), we ﬁnd that

lim

n→∞xn–Sxn= . (.)

Now, we are in a position to showlim sup_n_→∞f(q) –q,xn–q ≤, whereq=Pf(q). To

show it, we can choose a subsequence{xni}of{xn}such that

lim sup

n→∞

f(q) –q,xn–q

= lim

i→∞

f(q) –q,xni–q

.

Since{xni}is bounded, we can choose a subsequence{xn_ij}of{xni}which converges weakly to some pointx¯. We may assume, without loss of generality, that{xni}converges weakly tox¯.

Next, we provex¯∈. First, we showx∈EP(F). Notice that

F(yn,y) + 

rn

y–yn,yn–xn ≥, ∀y∈C.

By using condition (A), we see that _rny–yn,yn–xn ≥F(y,yn),∀y∈C. Replacingnby

ni, we arrive at

y–yni,

yni–xni

rni

≥F(y,yni), ∀y∈C.

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we haveF(zt,x¯)≤. It follows that

 =F(zt,zt)≤tF(zt,z) + ( –t)F(zt,x¯)≤tF(zt,z),

which yields thatF(zt,z)≥,∀z∈C. Lettingt↓, we obtain from condition (A) that

F(x¯,z)≥,∀z∈C. This implies thatx¯∈EP(F).

Next, we show thatx¯∈VI(C,A). LetTbe a maximal monotone mapping deﬁned by

Tx=

⎧ ⎨ ⎩

Ax+NCx, x∈C,

∅, x∈/C.

For any given (x,y)∈Graph(T), we havey–Ax∈NCx. Sinceyn∈C, we havex–yn,y–

Ax ≥. Sinceyn=PC(zn–snAzn+en), we see thatx–yn,yn– (I–snA)zn–en ≥ and hence

x–yn,

yn–zn–en

sn

+Azn

≥.

It follows that

x–yni,y ≥ x–yni,Ax

≥ x–yni,Ax–

x–yni,

yni–zni–eni

sni

+Azni

=x–yni,Ax–Ayni+x–yni,Ayni–Azni

–

x–yni,

sni

≥ x–yni,Ayni–Azni–

x–yni,

sni

.

SinceAis Lipschitz continuous, we see thatx–x¯,y ≥. Notice thatTis maximal mono-tone and hence ∈Tx¯. This shows thatx¯∈VI(C,A). By using Lemma ., we ﬁnd that

¯

x∈F(S). It follows thatlim sup_n_→∞f(q) –q,xn–q ≤,

xn+–q ≤αn

f(xn) –q,xn+–q

+βnxn–qxn+–q+γnSnyn–qxn+–q

≤αn

f(xn) –f(q),xn+–q

+αn

f(q) –q,xn+–q

+βnxn–x¯xn+–q

+γn

xn–q+en

xn+–q

≤αnβ+βn+γn 

xn–q+xn+–q

+αn

+enxn+–q.

It follows that

xn+–q≤

 –αn( –β)xn–q+ αn

+ Knen,

whereK=sup_n_≥_{xn–q}. By using Lemma ., we ﬁnd thatlimn→∞xn–q= . This

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For nonexpansive mappings, we have the following result.

Corollary . Let C be a closed convex subset of a real Hilbert space H.Let A:C→H be anα-inverse-strongly monotone mapping,and let F be a bifunction from C×C toRwhich satisﬁes(A)-(A).Let S:C→H be nonexpansive,and let f be aβ-contraction on H.

Assume that=F(S)∩VI(C,A)∩EP(F)=∅.Let{rn}and{sn}be positive real number

se-quences.Let{αn},{βn}and{γn}be real number sequences in(, )such thatαn+βn+γn= .

Let{xn}be a sequence generated in the following process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈H,

yn=PC(zn–snAzn+en),

xn+=αnf(xn) +βnxn+γnSyn,

con-ditions:

_∞

(c) ∞_n_=en<∞;

(d) limn→∞|rn+–rn|= andlim infn→∞rn> ; (e) limn→∞|sn+–sn|= , <s≤sn≤s< α,

where s, s are real constants. Then{xn} converges strongly to q, which is also a unique

f(x) –x,x–y≥, ∀y∈C.

Further, ifSis an identity, we have the following result.

Corollary . Let C be a closed convex subset of a real Hilbert space H.Let A:C→H be anα-inverse-strongly monotone mapping,and let F be a bifunction from C×C toRwhich satisﬁes(A)-(A).Let f be aβ-contraction on H.Assume that=VI(C,A)∩EP(F)=∅.

Let{rn}and{sn}be positive real number sequences.Let{αn},{βn}and{γn}be real number

sequences in(, )such thatαn+βn+γn= .Let{xn}be a sequence generated in the following

process:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈H,

xn+=αnf(xn) +βnxn+γnPC(zn–snAzn+en),

con-ditions:

∞

(c) ∞_n_=en<∞;

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where s, s are real constants. Then{xn} converges strongly to q, which is also a unique

f(x) –x,x–y≥, ∀y∈C.

PuttingF(x,y) =  andrn= , we ﬁnd the following result.

Corollary . Let C be a closed convex subset of a real Hilbert space H.Let A:C→H be anα-inverse-strongly monotone mapping.Let S:C→H be aκ-strictly pseudocontractive mapping,and let f be aβ-contraction on H.Assume that=F(S)∩VI(C,A)=∅.Let{sn}

be a positive real number sequence.Let{αn},{βn},{γn}and{δn}be real number sequences

in(, )such thatαn+βn+γn= .Let{xn}be a sequence generated in the following process:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈H,

zn=PCxn,

yn=PC(zn–snAzn+en),

xn+=αnf(xn) +βnxn+γn(δnyn+ ( –δn)Syn),

con-ditions:

(a) lim_n_→∞αn= and

∞

(c) ∞_n_=en<∞,limn→∞|δn+–δn|= andκ≤δn≤δ< ; (d) limn→∞|sn+–sn|= , <s≤sn≤s< α,

whereδ,s,sare real constants.Then{xn}converges strongly to q,which is also a unique

f(x) –x,x–y≥, ∀y∈C.

Corollary . Let C be a closed convex subset of a real Hilbert space H.Let F be a

bifunc-tion from C×C toRwhich satisﬁes(A)-(A).Let S:C→H be aκ-strict pseudocontrac-tion,and let f be aβ-contraction on H.Assume that=F(S)∩EP(F)=∅.Let{rn}be a

positive real number sequence.Let{αn},{βn},{γn}and{δn}be real number sequences in (, )such thatαn+βn+γn= .Let{xn}be a sequence generated in the following process:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈H,

xn+=αnf(xn) +βnxn+γn(δnzn+ ( –δn)Szn),

con-ditions:

∞

n=αn=∞; (b)  <lim inf_n_→∞βn≤lim supn→∞βn< ;

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whereδis a real constant.Then{xn}converges strongly to q,which is also a unique solution

to the variational inequality

f(x) –x,x–y≥, ∀y∈C.

Remark . Comparing Theorem ., Corollaries .-. with Theorems IT, TT and the corresponding results in [–], we have the following.

(a) We extend the nonlinear mapping from the class of nonexpansive mappings to the class ofκ-strictly pseudocontractive mappings.

(b) Possible computation errors are taken into account.

(c) The conditions imposed on control sequences{αn}and{sn}are relaxed. (d) The common element is also a unique solution of variational inequality (.).

Theorem . Let C be a closed convex subset of a real Hilbert space H.Let A:C→H be anα-inverse-strongly monotone mapping,and let f be aβ-contraction on H.Assume that

VI(C,A)=∅.Let{sn}be a positive real number sequence.Let{αn},{βn}and{γn}be real

number sequences in(, )such thatαn+βn+γn= .Let{xn}be a sequence generated in

x∈C,xn+=αnf(xn) +βnxn+γnyn,where yn=PC(xn–snAxn+en),where{en}is a sequence

in H.Assume that the control sequences satisfy the following conditions: (a) limn→∞αn= and∞n=αn=∞;

(b)  <lim inf_n_→∞βn≤lim supn→∞βn< ;

(c) ∞_n_=en<∞,limn→∞|sn+–sn|= , <s≤sn≤s< α,

where s and sare real constants.Then{xn}converges strongly to q=PVI(C,A)f(q).

Remark . To construct a mathematical model which is as close as possible to a real complex problem, we often have to use more than one constraint. Solving such problems, we have to obtain some solution which is simultaneously the solution of two or more subproblems. This is a common problem in diverse areas of mathematics and physical sciences. It consists of trying to ﬁnd a solution satisfying certain constraints. In this pa-per, we investigate the problem of solving common solutions of an equilibrium problem, a variational inequality problem and a ﬁxed point problem of a strictly pseudocontractive mapping based on a regularization projection algorithm. Possible computation errors are also taken into account. Strong convergence theorems are established without compact assumptions and additional metric projections in the framework of real Hilbert spaces.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have contributed equally to the work. All authors read and approved the ﬁnal version of the manuscript.

Author details

1_{Department of Mathematics, Faculty of Science for Girls, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi}

Arabia. 2_{Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.}

Acknowledgements

This project was funded by the Deanship of Scientiﬁc Research (DSR), King Abdulaziz University under grant

No. (58-130-35-HiCi). The authors, therefore, acknowledge technical and ﬁnancial support of KAU. The authors are grateful to the editor and the anonymous reviewers for useful suggestions which improved the contents of the article.

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