R E S E A R C H
Open Access
Strong convergence of viscosity
approximation methods for the fixed-point of
pseudo-contractive and monotone mappings
Yan Tang
**Correspondence: [email protected] College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China
Abstract
In this paper, we introduce a viscosity iterative process, which converges strongly to a common element of the set of fixed points of a pseudo-contractive mapping and the set of solutions of a monotone mapping. We also prove that the common element is the unique solution of certain variational inequality. The strong convergence theorems are obtained under some mild conditions. The results presented in this paper extend and unify most of the results that have been proposed for this class of nonlinear mappings.
MSC: 47H09; 47H10; 47L25
Keywords: pseudo-contractive mappings; monotone mappings; fixed point;
variational inequalities; viscosity approximation
1 Introduction
LetCbe a closed convex subset of a real Hilbert spaceH. A mappingA:C→His called monotone if and only if
x–y,Ax–Ay ≥, ∀x,y∈C. (.)
A mappingA:C→His calledα-inverse strongly monotone if there exists a positive real numberα> such that
x–y,Ax–Ay ≥αAx–Ay, ∀x,y∈C. (.)
Obviously, the class of monotone mappings includes the class of theα-inverse strongly monotone mappings.
A mappingT:C→His called pseudo-contractive if∀x,y∈C, we have
Tx–Ty,x–y ≤ x–y. (.)
A mappingT:C→His calledκ-strict pseudo-contractive, if there exists a constant ≤κ≤ such that
x–y,Tx–Ty ≤ x–y–κ(I–T)x– (I–T)y, ∀x,y∈C. (.)
A mappingT:C→Cis called non-expansive if
Tx–Ty ≤ x–y, ∀x,y∈C. (.)
Clearly, the class of contractive mappings includes the class of strict pseudo-contractive mappings and non-expansive mappings. We denote byF(T) the set of fixed points ofT, that is,F(T) ={x∈C:Tx=x}.
A mappingf :C→Cis called contractive with a contraction coefficient if there exists a constantρ∈(, ) such that
f(x) –f(y)≤ρx–y, ∀x,y∈C. (.)
For finding an element of the set of fixed points of the non-expansive mappings, Halpern [] was the first to study the convergence of the scheme in
xn+=αn+u+ ( –αn+)T(xn). (.)
In , Moudafi [] introduced the viscosity approximation methods and proved the strong convergence of the following iterative algorithm under some suitable conditions
xn+=αnf(xn) + ( –αn)T(xn). (.)
Viscosity approximation methods are very important, because they are applied to con-vex optimization, linear programming, monotone inclusions and elliptic differential equa-tions. In a Hilbert space, many authors have studied the fixed points problems of the fixed points for the non-expansive mappings and monotone mappings by the viscosity approx-imation methods, and obtained a series of good results, see [–].
Suppose thatAis a monotone mapping fromCintoH. The classical variational inequal-ity problem is formulated as finding a pointu∈Csuch thatv–u,Au ≥,∀v∈C. The set of solutions of variational inequality problems is denoted byVI(C,A).
Takahashi [, ] introduced the following scheme and studied the weak and strong convergence theorem of the elements ofF(T)∩VI(C,A), respectively, under different con-ditions
xn+=αnxn+ ( –αn+)TPC(xn–λnxn), (.)
whereTis a non-expansive mapping,Ais anα-inverse strong monotone operator. Recently, Zegeye and Shahzad [] introduced the algorithms and obtained the strong convergence theorems in a Hilbert space, respectively,
xn+=αnf(xn) + ( –αn)Tnxn (.)
and
where Tn are asymptotically non-expansive mappings, andTrn, Frn are non-expansive
mappings.
Our concern is now the following: Is it possible to construct a new sequence that con-verges strongly to a common element of the intersection of the set of fixed points of a pseudo-contractive mapping and the solution set of a variational inequality problem for a monotone mapping?
2 Preliminaries
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH, a mappingPC: H→Cis called the metric projection, if∀x∈H, there exists a unique point inC, denote byPCxsuch that
x–PCx ≤ x–y, ∀y∈C.
It is well known thatPCis a non-expansive mapping, andPCxhave the property as follows:
x–PCx,y–PCx ≤, ∀x∈H,y∈C, (.)
x–y≥ x–PCx+y–PCx, ∀x∈H,y∈C. (.)
Lemma .[] Let{an}be a sequence of nonnegative real numbers satisfying the following relation:
an+≤( –θn)an+σn, n≥,
where{θn}is a sequence in(,)and{σn}is a real sequence such that (i) ∞n=θn=∞;
(ii) lim supn→∞σn
θn ≤or
∞
n=σn<∞. Thenlimn→∞an= .
Lemma . [] Let C be a closed convex subset of a Hilbert space H. Let A:C→H be a continuous monotone mapping,let T:C→C be a continuous pseudo-contractive mapping,define mappings Trand Fras follows:x∈H,r∈(,∞)
Tr(x) =
z∈C:y–z,Tz–
r
y–z, ( +r)z–x≤,∀y∈C
,
Fr(x) =
z∈C:y–z,Az+
ry–z,z–x ≥,∀y∈C
.
Then the following hold:
(i) TrandFrare single-valued;
(ii) TrandFrare firmly non-expansive mappings,i.e.,Trx–Try≤ Trx–Try,x–y, Frx–Fry≤ Frx–Fry,x–y;
Lemma .[] Let{xn}and{zn}be bounded sequence in a Banach space,and let{βn} be a sequence in[, ],which satisfies the following condition:
<lim inf
n→∞ βn<lim supn→∞ βn< . Suppose that
xn+=βnxn+ ( –βn)zn, n≥ and
lim
n→∞
zn+–zn–xn+–xn ≤.
Thenlimn→∞zn–xn= .
LetCbe a closed convex subset of a Hilbert spaceH. LetA:C→H be a continuous monotone mapping, letT:C→Cbe a continuous pseudo-contractive mapping. Then we define the mappings as follows: forx∈H,τn∈(,∞)
Tτn(x) =
z∈C:y–z,Tz–
τn
y–z, ( +τn)z–x
≤,∀y∈C
, (.)
Fτn(x) =
z∈C:y–z,Az+
τn
y–z,z–x ≥,∀y∈C
. (.)
3 Main results
Theorem . Let C be a nonempty closed convex subset of a Hilbert space H.Let T:C→C be a continuous pseudo-contractive mapping,let A:C→H be a continuous monotone mapping such that F =F(T)∩VI(C,A)=∅,let f :C→C be a contraction with a con-traction coefficient ρ∈(, ).The mappings Tτn and Fτn are defined as(.)and(.), respectively.Let{xn}be a sequence generated by x∈C
⎧ ⎨ ⎩
yn=λnxn+ ( –λn)Fτnxn, xn+=αnf(xn) +βnxn+γnTτnyn,
(.)
whereλn∈[, ],let{αn},{βn},{γn}be sequences of nonnegative real numbers in[, ]and (i) αn+βn+γn= ,n≥;
(ii) limn→∞αn= ,
∞
n=αn=∞;
(iii) <lim infn→∞λn<lim supn→∞λn< ; (iv) lim infn→∞τn> ,
∞
n=|τn+–τn|<∞.
Then the sequence{xn}converges strongly tox¯=PFf(¯x),and alsox is the unique solution¯ of the variational inequality
f(¯x) –x¯,y–x¯≤, ∀y∈F. (.)
Proof First, we prove that{xn}is bounded. Takep∈F, then we have from Lemmas .
that
Forn≥, becauseTτnandFτnare non-expansive, andf is contractive, we have
xn+–p
=αn
f(xn) –p +βn(xn–p) +γn(Tτnyn–p)
≤αnf(xn) –f(p)+αnf(p) –p+βnxn–p+γnTτnyn–p
≤ραnxn–p+αnf(p) –p+ ( –αn)xn–p
≤ – ( –ρ)αn
xn–p+αnf(p) –p
≤maxx–p,
f(p) –p
–ρ
.
Therefore,{xn}is bounded. Consequently, we get that{Fτnxn},{Tτnyn}and{yn},{f(xn)}
are bounded.
Next, we show thatxn+–xn →.
yn+–yn ≤λn+xn+–xn+ ( –λn+)Fτn+xn+–Fτnxn
+|λn+–λn|xn–Fτnxn. (.)
Letvn=Fτnxn,vn+=Fτn+xn+, by the definition of the mappingFτn, we have that
y–vn,Avn+
τn
y–vn,vn–xn ≥, ∀y∈C, (.)
y–vn+,Avn++
τn+
y–vn+,vn+–xn+ ≥, ∀y∈C. (.)
Puttingy:=vn+in (.), and lettingy:=vnin (.), we have that
vn+–vn,Avn+
τn
vn+–vn,vn–xn ≥, (.)
vn–vn+,Avn++
τn+
vn–vn+,vn+–xn+ ≥. (.)
Adding (.) and (.), we have that
vn+–vn,Avn–Avn++
vn+–vn,
vn–xn
τn
–vn+–xn+
τn+
≥.
SinceAis a monotone mapping, which implies that
vn+–vn,
vn–xn
τn
–vn+–xn+
τn+
≥.
Therefore, we have that
vn+–vn,vn–xn–
τn(vn+–xn+)
τn+
+vn+–vn+
i.e.,
vn+–vn≤
vn+–vn,xn+–xn+
– τn
τn+
(vn+–xn+)
≤ vn+–vn
xn+–xn+
– τn
τn+
vn+–xn+
. (.)
Without loss of generality, letbbe a real number such thatτn>b> ,∀n∈N, then we
have that
vn+–vn ≤ xn+–xn+
– τn
τn+
vn+–xn+
≤ xn+–xn+
b|τn+–τn|K, (.)
whereK=supvn+–xn+. Then we have from (.) and (.) that
yn+–yn ≤ xn+–xn+
( –λn+)|τn+–τn|
b K
+|λn+–λn|xn–Fτnxn. (.)
On the other hand, letun=Tτnyn,un+=Tτn+yn+, we have that
y–un,Tun–
τn
y–un, ( +τn)un–yn
≤, ∀y∈C, (.)
y–un+,Tun+–
τn+
y–un+, ( +τn+)un+–yn+
≤, ∀y∈C. (.)
Lety:=un+in (.), and lety:=unin (.), we have that
un+–un,Tun–
τn
un+–un, ( +τn)un–yn
≤, (.)
un–un+,Tun+–
τn+
un–un+, ( +τn+)un+–yn+
≤. (.)
Adding (.) and (.), and becauseT is pseudo-contractive, we have that
un+–un,
un–yn
τn
–un+–yn+
τn+
≥.
Therefore, we have
un+–un,un–yn–
τn(un+–yn+)
τn+
+un+–un+
≥.
Hence we have that
un+–un ≤ yn+–yn+
b|τn+–τn|M, (.)
Letxn+=βnxn+ ( –βn)zn, hence we have that
zn+–zn=
αn+ –βn+
f(xn+) –f(xn) +
αn+ –βn+
– αn –βn
f(xn)
+ γn+ –βn+
(un+–un) +
γn+ –βn+
– γn –βn
un. (.)
Hence we have from (.), (.), (.) and condition (iii) that
zn+–zn–xn+–xn
≤(ρ– )αn+ –βn+
xn+–xn+ αn+
–βn+ – αn
–βn
f(xn)+un
+ γn+ –βn+
|τn+–τn| b
( –λn+)K+M +
γn+ –βn+|
λn+–λn|xn–Fτnxn. (.)
Notice conditions (ii) and (iv), we have that
lim sup
n→∞
zn+–zn–xn+–xn = . (.)
Hence we have from Lemma . that
lim sup
n→∞ zn–xn= . (.)
Therefore, we have that
xn+–xn=| –βn|zn–xn →. (.)
Hence we have from (.) and (.) that
yn+–yn →, un+–un →, vn+–vn →. (.)
Sincexn=αn–f(xn–) +βn–xn–+γn–vn–, so we have that xn–vn ≤ xn–vn–+vn––vn
≤ –αn–( –ρ)
xn––vn–+vn––vn.
According to Lemma ., we have that
xn–vn →.
In the same way, we have that
xn–yn →, xn–un →.
Consequently, we have that
Now, we show thatlim supn→∞f(¯x) –x¯,xn+–x¯ ≤.
Since sequence{xn}is bounded, then there exists a sub-sequence{xnk}of{xn}andw∈C
such thatxnkw. Next, we show thatw∈F=F(T)∩VI(C,A). Sincevn=Fτnxn, we have that
y–vnk,Avnk+
τnk
y–vnk,vnk–xnk ≥, ∀y∈C.
Letvt=tv+ ( –t)w,t∈(, ),v∈C, then we have that
vt–vnk,Avt ≥ vt–vnk,Avt–Avnk–
vt–vnk,vnk–xnk
τnk
.
Sincexnk–vnk→, and alsoAis monotone, we have that
≤ lim
k→∞vt–vnk,Avt=vt–w,Avt.
Consequently,
vt–w,Avt ≥.
Ift→, by the continuity ofA, we havev–w,Aw ≥,∀v∈C. Thus,w∈VI(C,A). In addition, sinceun=Fτnyn, we have that
y–unk,Tunk–
τnk
y–unk, ( +τnk)unk–ynk
≤, ∀y∈C.
Letvt=tv+ ( –t)w,t∈(, ),v∈C, then we have that
unk–vt,Tvt ≥ vt–unk,Tunk–Tvt–
vt–unk, +τnk
τnk
unk–ynk
≥–vt–unk,vt–
τnk
vt–unk,unk–ynk.
Sinceynk–unk→, ift→, by the continuity ofT, we have that
–v–w,Tw ≥–v–w,w, ∀v∈C.
Let v=Tw, we havew=Tw, thus,w∈F(T). Consequently, we conclude thatw∈F=
F(T)∩VI(C,A).
Becausex¯=PFf(x¯), we have from (.) that
lim sup
n→∞
f(x¯) –x¯,xn–x¯
=f(x¯) –PFf(x¯),w–PFf(x¯)
≤. (.)
Next, we show thatxn→ ¯x. From formula (.), we have that
xn+–x¯=αn
f(xn) –x¯ +βn(xn–x¯) +γn(un–x¯)
≤αnf(xn) –x¯+βn(xn–x¯) +γn(un–x¯)
+ αn
f(xn) –x¯,βn(xn–x¯) +γn(un–x¯)
≤αnf(xn) –x¯+ ( –αn)xn–x¯+ αnβn
f(xn) –x¯,xn–x¯
+ αnγnf(xn) –x¯un–x¯
≤αnf(xn) –x¯+ ( –αn)xn–x¯+ αnβn
f(¯x) –x¯,xn–x¯
+ αnβn
f(xn) –f(¯x),xn–x¯+ αnγnf(xn) –x¯un–x¯
=( –αn)+ ραnβn
xn–x¯+αnf(xn) –x¯+ αnβn
f(¯x) –x¯,xn–x¯
+ αnγnf(xn) –x¯un–x¯
= – αn( –ρβn) xn–x¯+αnf(xn) –x¯
+xn–x¯
+ αnγnf(xn) –x¯un–x¯+ αnβn
f(¯x) –x¯,xn–x¯.
Let θn= αn( –ρβn), σn=αn[f(xn) –x¯+xn–x¯] + αnγnf(¯x) –x¯un–x¯+
αnβnf(¯x) – x¯,xn – x¯. According to Lemma . and formula (.), we have that
limn→∞xn–x¯= ,i.e., the sequence{xn}converges strongly tox¯∈F.
According to formula (.), we conclude thatx¯is the solution of the variational inequal-ity (.). Now, we show thatx¯is the unique solution of the variational inequality (.).
Suppose thaty¯∈Fis another solution of the variational inequality (.). Becausex¯is the solution of the variational inequality (.),i.e.,f(¯x) –x¯,y–x¯ ≤,∀y∈F. Becausey¯∈F, then we have
f(¯x) –x¯,y¯–x¯≤. (.)
On the other hand, to the solutiony¯∈F, sincex¯∈F, so
f(y¯) –y¯,x¯–y¯≤. (.)
Adding (.) and (.), we have that
¯
x–y¯–f(¯x) –f(¯y),x¯–y¯≤,
i.e.,
¯
x–¯y–f(¯x) –f(¯y) ,x¯–y¯≤f(¯x) –f(¯y),x¯–y¯.
Hence
¯x–y¯≤f(¯x) –f(¯y),x¯–¯y≤ρ¯x–y¯.
Becauseρ∈(, ), hence we conclude thatx¯=¯y, the uniqueness of the solution is
ob-tained.
mapping such that F =F(T)∩VI(C,A)=∅,let f :C→C be a contraction with a con-traction coefficient ρ∈(, ).The mappings Tτn and Fτn are defined as(.)and(.), respectively.Let{xn}be a sequence generated by x∈C
xn+=αnf(xn) +βnxn+γnTτnFτnxn, (.) where{αn},{βn},{γn}are sequences of nonnegative real numbers in[, ]and
(i) αn+βn+γn= ,n≥; (ii) limn→∞αn= ,
∞
n=αn=∞; (iii) lim infn→∞τn> ,
∞
n=|τn+–τn|<∞.
Then the sequence{xn}converges strongly tox¯=PFf(¯x),and alsox is the unique solution¯ of the variational inequality
f(¯x) –x¯,y–x¯≤, ∀y∈F. (.)
Proof Puttingλn= in Theorem ., we can obtain the result.
If in Theorem . and Theorem ., letf :≡u∈Cbe a constant mapping, we have the following theorems.
Theorem . Let C be a nonempty closed convex subset of a Hilbert space H.Let T:C→ C be a continuous pseudo-contractive mapping,let A:C→H be a continuous accretive mapping such that F=F(T)∩VI(C,A)=∅.The mappings Tτnand Fτnare defined as(.) and(.),respectively.Let{xn}be a sequence generated by
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x=x∈C,
yn=λnxn+ ( –λn)Fτnxn, xn+=αnu+βnxn+γnTτnyn,
(.)
whereλn∈[, ],and let{αn},{βn},{γn}be sequences of nonnegative real numbers in[, ] and
(i) αn+βn+γn= ,n≥; (ii) limn→∞αn= ,
∞
n=αn=∞;
(iii) <lim infn→∞λn<lim supn→∞λn< ; (iv) lim infn→∞τn> ,
∞
n=|τn+–τn|<∞.
Then the sequence{xn}converges strongly tox¯=PFu,and alsox is the unique solution of¯ the variational inequality
u–x¯,y–x¯ ≤, ∀y∈F. (.)
Theorem . Let C be a nonempty closed convex subset of a Hilbert space H.Let T:C→ C be a continuous pseudo-contractive mapping,let A:C→H be a continuous monotone mapping such that F=F(T)∩VI(C,A)=∅.The mappings Tτnand Fτnare defined as(.) and(.),respectively.Let{xn}be a sequence generated by x∈C
where{αn},{βn},{γn}are sequences of nonnegative real numbers in[, ]and (i) αn+βn+γn= ,n≥;
(ii) limn→∞αn= ,
∞
n=αn=∞; (iii) lim infn→∞τn> ,
∞
n=|τn+–τn|<∞.
Then the sequence {xn} converges tox¯=PFu, and alsox is the unique solution of the¯ variational inequality
u–x¯,y–x¯ ≤, ∀y∈F. (.)
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This work was supported by the Natural Science Foundation Project of Chongqing (CSTC, 2012jjA00039) and the Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130712, KJ130731).
Received: 15 October 2012 Accepted: 25 September 2013 Published:08 Nov 2013 References
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Cite this article as:Tang:Strong convergence of viscosity approximation methods for the fixed-point of
