# Viscosity Approximation Methods and Strong Convergence Theorems for the Fixed Point of Pseudocontractive and Monotone Mappings in Banach Spaces

Authors: Yan Tang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Suppose thatC is a nonempty closed convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone mappings. And the common element is the unique solution of certain variational inequality. The results presented in this paper extend most of the results that have been proposed for this class of nonlinear mappings.


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## Body

## 1. Introduction

LetE be a real Banach space with dual E*. A normalized duality mapping J:E→2E* is defined by
(1)Jx={f*∈E*:〈x,f*〉=∥x∥2=∥f*∥2},
where 〈·,·〉 denotes the generalized duality pairing between E and E*. It is well known that E is smooth if and only if J is single valued, and if E is uniformly smooth, then J is uniformly continuous on bounded subsets of E. Moreover, if E is reflexive and strictly convex Banach space with a strictly convex dual then J-1 is single valued, one-to-one surjective, and it is the duality mapping from E* into E, and then JJ-1=IE* and JJ-1=IE.A mappingA:D(A)⊂E→E* is said to be monotone if, for each x,y∈D(A), the following inequality holds:
(2)〈x-y,Ax-Ay〉≥0.A mappingA:D(A)⊂E→E* is said to be strongly monotone, if there exists a positive real number α>0 such that for all x,y∈D(A)(3)〈x-y,Ax-Ay〉≥α∥Ax-Ay∥2.Obviously, the class of monotone mappings includes the class of theα-inverse strongly monotone mappings.LetC be closed convex subset of Banach space E. A mapping T is said to be pseudocontractive if, for any x,y∈D(T), there exists j(x-y)∈J(x-y) such that
(4)〈Tx-Ty,j(x-y)〉≤∥x-y∥2.A mappingT is said to be κ-strictly pseudo-contractive, if, for any x,y∈D(T), there exist j(x-y)∈J(x-y) and a constant 0≤κ≤1 such that
(5)〈x-y-(Tx-Ty),j(x-y)〉≥κ∥(I-T)x-(I-T)y∥2.A mappingT:C→C is called nonexpansive if
(6)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.Clearly, the class of pseudo-contractive mappings includes the class of strict pseudo-contractive mappings and non-expansive mappings. We denote byF(T) the set of fixed points of T; that is, F(T)={x∈C:Tx=x}.A mappingf:C→C is called contractive with a contraction coefficient if there exists a constant ρ∈(0,1) such that
(7)∥f(x)-f(y)∥≤ρ∥x-y∥,∀x,y∈C.LetE be a real Banach space with dual E*. The norm on E is said to be uniformly Gateaux differentiable if for each y∈S1(0):={x∈E:∥x∥=1} the limit limt→0(∥x+ty∥-∥x∥)/t exists uniformly for x∈S1(0).For finding an element of the set of fixed points of the non-expansive mappings, Halpern [1] was the first to study the convergence of the scheme in 1967:
(8)xn+1=αn+1u+(1-αn+1)T(xn).Viscosity approximation methods are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations. In Hilbert spaces, many authors have studied the fixed points problems of the fixed points for the non-expansive mappings and monotone mappings by the viscosity approximation methods, and obtained a series of good results (see [2–17]).In 2000, Moudifi [18] introduced the viscosity approximation methods and proved the strong convergence of the following iterative algorithm in Hilbert spaces under some suitable conditions:
(9)xn+1=αnf(xn)+(1-αn)T(xn).Suppose thatA is monotone mapping from C into E. The classical variational inequality problem is formulated as finding a point u∈C such that 〈v-u,Au〉≥0, for all v∈C. The set of solutions of variational inequality problems is denoted by VI(C,A).Takahashi and Toyoda [19, 20] introduced the following scheme in Hilbert spaces and studied the weak and strong convergence theorem of the elements of F(T)∩VI(C,A), respectively, under different conditions:
(10)xn+1=αnxn+(1-αn+1)TPC(xn-λnxn),
where T is non-expansive mapping and A is α-inverse strong monotone operator.Recently, Zegeye and Shahzad [21] introduced the following algorithm and obtained the strong convergence theorem but still in Hilbert spaces:
(11)xn+1=αnf(xn)+(1-αn)TrnFrnxn,
where Trn, Frn are nonexpansive mappings.Our concern now is the following: is it possible to construct a new sequence in Banach spaces which converges strongly to a common element of fixed points of a finite family of pseudocontractive mappings and the solution set of a variational inequality problems for finite family of monotone mappings?


## 2. Preliminaries

In the sequel, we will use the following lemmas.Lemma 1 (see, e.g., [5]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(12)an+1≤(1-θn)an+σn,n≥0,
where {θn} is a sequence in (0,1) and {σn} is a real sequence such that (i)
∑n=0∞‍θn=∞;(ii)
limsupn→∞σn/θn≤0 or ∑n=0∞‍σn<∞.Then limn→∞an=0.Lemma 2 (see, e.g., [10]).
LetC be a nonempty, closed, and convex subset of uniformly smooth strictly convex real Banach space E with dual E*. Let A:C→E* be a continuous monotone mapping; define mapping Fr as follows: x∈E,r∈(0,∞)(13)Fr(x)={z∈C:〈y-z,Az〉+1r〈y-z,Jz-Jx〉≥0,∀y∈C1r}.
Then the following hold:(i)
Fr is well defined and single valued;(ii)
Fr is a firmly non-expansive mapping; that is,〈Frx-Fry,JFrx-JFry〉≤〈Frx-Fry,Jx-Jy〉;(iii)
F(Fr)=VI(C,A);(iv)
VI(C,A) is closed and convex.Lemma 3 (see, e.g., [22]).
LetC be a nonempty closed convex subset of uniformly smooth strictly convex real Banach space E. Let T:C→E, be a continuous strictly pseudocontractive mapping; define mapping Tr as follows: x∈E,r∈(0,∞)(14)Tr(x)={z∈C:〈y-z,Tz〉-1r〈y-z,(1+r)Jz-Jx〉≤0,∀y∈C1r}.
Then the following hold:(i)
Tr is single valued;(ii)
Tr is a firmly nonexpansive mapping; that is,〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;(iii)
F(Tr)=F(T);(iv)
F(T) is closed and convex.Lemma 4 (see, e.g., [23]).
Let{xn} and {zn} be bounded sequence in a Banach space, and let {βn} be a sequence in [0,1] which satisfies the following condition:
(15)0<liminfn→∞βn<limsupn→∞βn<1.
Suppose
(16)xn+1=βnxn+(1-βn)zn,n≥0,limn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
Then limn→∞∥zn-xn∥=0.Lemma 5 (see, e.g., [16]).
LetC be a nonempty closed and convex subset of a real smooth Banach space E. A mapping ΠC:E→C is a generalized projection. Let x∈E; then x0=ΠCx if and only if
(17)〈z-x0,Jx-Jx0〉≤0,∀z∈C.Lemma 6.
LetE be a real Banach space with dual E*. J:E→2E* is the generalized duality pairing; then, for all x,y∈E,
(18)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).


## 3. Main Results

LetC be a nonempty closed convex and bounded subset of a smooth, strictly convex and reflexive real Banach space E with dual E*. Let Ai:C→E*,i=1,2,…,m be a finite family of continuous monotone mappings, and let Ti:C→C,i=1,2,…,m, be a finite family of continuous strictly pseudo-contractive mappings. For the rest of this paper, Tirnx and Firnx are mappings defined as follows: for x∈E, rn∈(0,∞),
(19)Tirn(x):={z∈C:〈y-z,Tiz〉-1rn〈y-z,(1+rn)Jz-Jx〉≤0,∀y∈C1rn}.
Consider
(20)Firn(x):={z∈C:〈y-z,Aiz〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn}.
Denote F1=∩i=1mF(Tirn), F2=∩i=1mF(Firn).Lemma 7.
LetC be a nonempty closed convex and bounded subset of a smooth Banach space E, and let Γi:C→C,i=1,2,…,m, be a finite family of non-expansive mappings such that ∩i=1mF(Γi)≠∅. Suppose that α=inf{αi}>0; then there exists non-expansive mapping Γ:C→C such that F(Γ)=∩i=1mF(Γi).Proof.
Let{αi} be any sequence of positive real numbers satisfying ∑i=1m‍αi=1 and set Γ=∑i=1m‍αiΓi. Since each Γi is non-expansive for any i∈{1,2,…,m}, we have that Γ is well-defined nonexpansive mapping (see, e.g., [20]), and
(21)∥Γx-Γy∥=∥∑i=1m‍αiΓix-∑i=1m‍αiΓiy∥≤∑i=1m‍αi∥Γix-Γiy∥≤∥x-y∥.
Next, we claim that F(Γ)=∩i=1mF(Γi).
Clearly∩i=1mF(Γi)⊂F(Γ). Now, we prove that F(Γ)⊂∩i=1mF(Γi). Let x∈F(Γ) and p∈∩i=1mF(Γi). Then
(22)0=〈Γx-x,j(x-p)〉=〈∑i=1m‍αiΓix-x,j(x-p)〉=∑i=1m‍αi〈Γix-x,j(x-p)〉,
and non-expansivity of each Γi implies that 〈Γix-x,j(x-p)〉=0 for each i=1,2,…,m, which implies Γix=x for each i=1,2,…,m. Therefore, x∈∩i=1mF(Γi), and hence F(Γ)=∩i=1mF(Γi). The proof is complete.Theorem 8.
LetC be a nonempty closed convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Let Ti:C→C,i=1,2,…,m, be a finite family of continuous strictly pseudocontractive mappings, let Ai:C→E*,i=1,2,…,m, be a finite family of continuous monotone mappings such that F=F1∩F2≠∅, and let f:C→C be a contraction with a contraction coefficient ρ∈(0,1). Tirn and Firn are defined as (19) and (20), respectively. Let {xn} be a sequence generated by x0∈C(23)yn=λnxn+(1-λn)∑i=1m‍μiFirnxn,xn+1=αnf(xn)+βnxn+γn∑i=1m‍σiTirnyn,
where λn∈[0,1] and {αn}, {βn}, and {γn} are sequences of nonnegative real numbers in [0,1], and (i)
αn+βn+γn=1, n≥0, ∑i=1m‍μi=1, ∑i=1m‍σi=1, μi≥0, σi≥0;(ii)
limn→∞αn=0, ∑n=1∞‍αn=∞;(iii)
0<liminfn→∞βn<limsupn→∞βn<1;(iv)
limsupn→∞rn>0, ∑n=1∞‍|rn+1-rn|<∞.
Then the sequence{xn} converges strongly to an element x-=ΠFf(x-), and also x- is the unique solution of the variational inequality
(24)〈(f-I)(x-),j(y-x-)〉≤0,∀y∈F.Proof.
First we prove that{xn} is bounded. Take p∈F, because Firn is non-expansive; then we have that
(25)∥yn-p∥≤λn∥xn-p∥+(1-λn)×∑i=1m‍μi∥Firnxn-Firnp∥≤∥xn-p∥.
For n≥0, because Tirn and Firn are nonexpansive and f is contractive, we have from (25) that
(26)∥xn+1-p∥=∥αn(f(xn)-p)+βn(xn-p)+γn(∑i=1m‍σiTirnyn-p)∥≤αn∥f(xn)-f(p)∥+αn∥f(p)-p∥+βn∥xn-p∥+γn∥yn-p∥≤ραn∥xn-p∥+αn∥f(p)-p∥+(1-αn)∥xn-p∥≤[1-(1-ρ)αn]∥xn-p∥+αn∥f(p)-p∥≤max{∥x0-p∥,f(p)-p1-ρ}.
Therefore,{xn} is bounded. Consequently, we get that {Firnxn}, {Tirnyn} and {yn}, {f(xn)} are bounded.
Next, we show that∥xn+1-xn∥→0.
Consider(27)∥yn+1-yn∥≤λn+1∥xn+1-xn∥+(1-λn+1)∑i=1m‍μi∥Firn+1xn+1-Firnxn∥+|λn+1-λn|∥xn-∑i=1m‍μiFirnxn∥.
Letvin=Firnxn, vi,n+1=Firn+1xn+1; by the definition of mapping Firn, we have that
(28)〈y-vin,Aivin〉+1rn〈y-vin,Jvin-Jxn〉≥0,∀y∈C,(29)〈y-vi,n+1,Aivin+1〉+1rn+1〈y-vi,n+1,Jvi,n+1-Jxn+1〉≥0,∀y∈C.
Puttingy:=vi,n+1 in (28) and leting y:=vin in (29), we have that
(30)〈vi,n+1-vin,Aivin〉+1rn〈vi,n+1-vin,Jvin-Jxn〉≥0,〈vin-vi,n+1,Aivi,n+1〉+1rn+1〈vin-vi,n+1,Jvi,n+1-Jxn+1〉≥0.
Adding (30), we have that
(31)〈vi,n+1-vin,Aivin-Aivi,n+1〉+〈vi,n+1-vin,Jvin-Jxnrn-Jvi,n+1-Jxn+1rn+1〉≥0.
SinceAi, i∈{1,2,…,m} are monotone mappings, which implies that
(32)〈vi,n+1-vin,Jvin-Jxnrn-Jvi,n+1-Jxn+1rn+1〉≥0,
therefore we have that
(33)〈vi,n+1-vin,Jvin-Jxn-rn(Jvi,n+1-Jxn+1)rn+1+Jvi,n+1-Jvi,n+1rn(Jvi,n+1-Jxn+1)rn+1〉≥0.
That is,
(34)∥vi,n+1-vin∥2≤〈vi,n+1-vin,Jxn+1-Jxn+(1-rnrn+1)×(Jvi,n+1-Jxn+1)(1-rnrn+1)〉≤∥vi,n+1-vin∥{|1-rnrn+1|∥xn+1-xn∥+|1-rnrn+1|∥vi,n+1-xn+1∥}.
Without loss of generality, letb be a real number such that rn>b>0; then we have that
(35)∥vi,n+1-vin∥≤∥xn+1-xn∥+|1-rnrn+1|∥vi,n+1-xn+1∥≤∥xn+1-xn∥+1b|rn+1-rn|K,
where K=sup{∥vi,n+1-xn+1∥}.
Then we have from (35) and (27) that
(36)∥yn+1-yn∥≤∥xn+1-xn∥+(1-λn+1)|rn+1-rn|bK+|λn+1-λn|∥xn-∑i=1m‍μiFirnxn∥.
On the other hand, letuin=Tirnyn, ui,n+1=Tirn+1yn+1; we have that
(37)〈y-uin,Tiuin〉-1rn〈y-uin,(1+rn)Juin-Jyn〉≤0,∀y∈C,(38)〈y-ui,n+1,Tiui,n+1〉-1rn+1〈y-ui,n+1,(1+rn+1)×Jui,n+1-Jyn+1〉≤0,∀y∈C.
Lety:=ui,n+1 in (37), and let y:=uin in (38); we have that
(39)〈ui,n+1-uin,Tiuin〉-1rn〈ui,n+1-uin,(1+rn)Juin-Jyn〉≤0,〈uin-ui,n+1,Tiui,n+1〉-1rn+1〈uin-ui,n+1,(1+rn+1)Jui,n+1-Jyn+1〉≤0.
Adding (39) and because Ti,i∈{1,2,…,m} is pseudo-contractive, we have that
(40)〈ui,n+1-uin,Juin-Jynrn-Jui,n+1-Jyn+1rn+1〉≥0.
Therefore we have(41)〈ui,n+1-uin,Juin-Jyn-rn(Jui,n+1-Jyn+1)rn+1+Jui,n+1-Jui,n+1rn(Jui,n+1-Jyn+1)rn+1〉≥0.
Hence we have that(42)∥ui,n+1-uin∥≤∥yn+1-yn∥+1b|rn+1-rn|M,
where M=sup{∥uin-yn∥}.
Letxn+1=βnxn+(1-βn)zn. Hence we have that
(43)zn+1-zn=αn+11-βn+1(f(xn+1)-f(xn))+(αn+11-βn+1-αn1-βn)f(xn)+γn+11-βn+1(ui,n+1-uin)+(γn+11-βn+1-γn1-βn)uin.
Hence we have from (43), (42), and (36) that
(44)∥zn+1-zn∥-∥xn+1-xn∥≤(ρ-1)αn+11-βn+1∥xn+1-xn∥+|αn+11-βn+1-αn1-βn|×{∥f(xn)∥+∥uin∥}+γn+11-βn+1|rn+1-rn|b×((1-λn+1)K+M)+γn+11-βn+1|λn+1-λn|×∥xn-∑i=1m‍μiFirnxn∥.
Noticing the conditions (ii) and (iv), we have that(45)limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)=0.
Hence we have from Lemma4 that
(46)limsupn→∞∥zn-xn∥=0.
Therefore we have that(47)∥xn+1-xn∥=|1-βn|∥zn-xn∥⟶0.
Hence we have from (35), (36), and (42) that
(48)∥yn+1-yn∥⟶0,∥ui,n+1-uin∥⟶0,∥vi,n+1-vin∥⟶0.
In addition, sincexn+1=αnf(xn)+βnxn+γn∑i=1m‍σiuin, yn=λnxn+(1-λn)∑i=1m‍μivin, for all p∈F, we have from the monotonicity of Ai, the non-expansivity of Tirn, and the convexity of ∥·∥2 that
(49)∥xn+1-p∥2=∥αnf(xn)+βnxn+γn∑i=1m‍σiuin-p∥2≤∥αn(f(xn)-p)+βn(xn-p)∥2+γn∑i=1m‍σi∥uin-p∥2≤αn∥f(xn)-p∥2+βn∥xn-p∥2+γn∥yn-p∥2≤αn∥f(xn)-p∥2+βn∥xn-p∥2+γnλn∥xn-p∥2+(1-λn)γn∥vn-p∥2≤αn∥f(xn)-p∥2+(βn+γnλn)∥xn-p∥2+(1-λn)γn(∥xn-p∥2-∥xn-vin∥2)≤αn∥f(xn)-p∥2+∥xn-p∥2-(1-λn)γn∥xn-vin∥2.
So we have that(50)(1-λn)γn∥xn-vin∥2≤αn∥f(xn)-p∥2+∥xn-p∥2-∥xn+1-p∥2≤αn∥f(xn)-p∥2+∥xn-xn+1∥×(∥xn-p∥+∥xn+1-p∥).
Sinceαn→0, so we have from (47) that
(51)∥xn-vin∥⟶0.
In a similar way, we have that(52)∥xn-uin∥⟶0.
Consequently, we have that(53)∥yn-xn∥=|1-λn|∥xn-vin∥⟶0,∥yn-uin∥≤∥yn-xn∥+∥xn-uin∥⟶0.
SinceE is a reflexive real Banach space and the sequence {xn} is bounded, so there exists the subsequence {xnk} of {xn} and w∈C such that xnk⇀w. And because xn→vin, n→∞, therefore vink⇀w. Next we show that w∈F.
Becausevin=Firnxn, by the definition of mapping Firn, we have that
(54)〈y-vin,Aivin〉+1rn〈y-vin,Jvin-Jxn〉≥0,∀y∈C,〈y-vink,Aivink〉+〈y-vin,Jvink-Jxnkrn〉≥0,∀y∈C.
Letvt=tv+(1-t)w, t∈[0,1], for all v∈C; we have that
(55)〈vt-vink,Aivt〉≥〈vt-vink,Aivt〉-〈vt-vink,Aivink〉-〈vt-vink,Jvink-Jxnkrn〉=〈vt-vink,Aivt-Aivink〉-〈vt-vink,Jvink-Jxnkrn〉.
Becausexnk-vink→0, so (Jvink-Jxnk)/rn→0, and because Ai are monotone, we have that
(56)0≤limk→∞〈vt-vink,Aivt〉=〈vt-w,Aivt〉.
Consequently we have that(57)〈v-w,Aivt〉≥0.
Ift→0, by the continuity of Ai, we have that 〈v-w,Aiw〉≥0; that is, w∈VI(C,Ai), and then w∈F2.
Similarly, becauseuin=Tirnyn, by the definition of mapping Tirn, we have that
(58)〈y-uin,Tiuin〉-1rn〈y-uin,(1+rn)Juin-Jyn〉≤0,∀y∈C,〈y-uink,Tiuink〉-1rn〈y-uink,(1+rn)Juink-Jynk〉≤0,∀y∈C.
Letvt=tv+(1-t)w,t∈[0,1], for all v∈C. Because {Ti},i∈{1,2,…,m} are pseudocontractive mappings, we have that
(59)〈uink-vt,Tivt〉≥〈uink-vt,Tivt〉+〈vt-uink,Tiuink〉-1rn〈vt-uink,(1+rn)Juink-Jynk〉=〈vt-uink,Tiuink-Tivt〉-〈vt-uink,1+rnrnJuink-1rnJynk〉≥-∥vt-uink∥2-1rn〈vt-uink,Juink-Jynk〉-〈vt-uink,Juink〉=-〈vt-uink,Jvt〉-1rn〈vt-uink,Juink-Jynk〉.
Becauseynk-uink→0, so Juink-Jynk→0; we have that
(60)limk→∞〈uink-vt,Tivt〉≥limk→∞〈uink-vt,Jvt〉.
Consequently we have that(61)〈w-vt,Tivt〉≥〈w-vt,Jvt〉,〈v-w,Tivt〉≤〈v-w,Jvt〉.
Ift→0, by the continuity of Ti, we have that 〈v-w,Tiw-Jw〉≥0, for all v∈C; we conclude that w=Tiw; that is, w∈F(Ti), and then w∈F1. Consequently w∈F=F1∩F2.
Becausex-=ΠFf(x-), we have from Lemma 5 that
(62)limsupn→∞〈f(x-)-x-,j(xn+1-x-)〉=〈f(x-)-x-,j(w-x-)〉≤0.
Next we show thatxn→x-. Since uin=Tirnyn, from formula (23) and Lemma 6 we have that
(63)∥xn+1-x-∥2=∥αn(f(xn)-x-)+βn(xn-x-)+γn(∑i=1m‍σiTirnyn-x-)∥2≤∥βn(xn-x-)+γn(∑i=1m‍σiuin-x-)∥2+2αn〈f(xn)-x-,j(xn+1-x-)〉≤(1-αn)2∥xn-x-∥2+2αn〈f(xn)-f(x-),j(xn+1-x-)〉+2αn〈f(x-)-x-,j(xn+1-x-)〉≤(1-αn)2∥xn-x-∥2+2ραn∥xn-x-∥∥xn+1-x-∥+2αn〈f(x-)-x-,j(xn+1-x-)〉≤(1-2αn(1-ρ))∥xn-x-∥2+σn,
where M1=sup{∥xn-x-∥}, σn=αn2M12+2ραn∥xn+1-xn∥M1+2αn〈f(x-)-x-,j(xn+1-x-)〉. Let θn=2αn(1-ρ); according to Lemma 1 and formula (62), we have that limn→∞∥xn-x-∥=0; that is, the sequence {xn} converges strongly to x-∈F.
According to formula (62) we conclude that x- is the solution of the variational inequality (24). Now we show that x- is the unique solution of the variational inequality (24). Suppose that y-∈F is another solution of the variational inequality (24). Because x- is the solution of the variational inequality (24), that is, 〈(f-I)x-,j(y-x-)〉≤0, for all y∈F. And because that y-∈F, then we have
(64)〈(f-I)x-,j(y--x-)〉≤0.
On the other hand, to the solutiony-∈F, since x-∈F, so
(65)〈(f-I)y-,j(x--y-)〉≤0.
Adding (65) and (64), we have that
(66)〈x--y--(f(x-)-f(y-)),j(x--y-)〉≤0.
Hence
(67)(1-ρ)∥x--y-∥2≤0.
Becauseρ∈(0,1), hence we conclude that x-=y-, and the uniqueness of the solution is obtained.Theorem 9.
LetC be a nonempty closed convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Let Ti:C→C, i=1,2,…,m, be a finite family of continuous strictly pseudocontractive mappings, let Ai:C→E*, i=1,2,…,m, be a finite family of continuous monotone mappings such that F=F1∩F2≠∅, and let f:C→C be a contraction with a contraction coefficient ρ∈(0,1). Tirn and Firn are defined as (19) and (20), respectively. Let xn be a sequence generated by x0∈C(68)xn+1=αnf(xn)+βnxn+γn∑i=1m‍εiTirnFirnxn,
where {αn}, {βn}, and {γn} are sequences of nonnegative real numbers in [0,1], and (i)
αn+βn+γn=1, n≥0, ∑i=1m‍εi=1, εi≥0;(ii)
limn→∞αn=0, ∑n=1∞‍αn=∞;(iii)
0<liminfn→∞βn<limsupn→∞βn<1;(iv)
limsupn→∞rn>0, ∑n=1∞‍|rn+1-rn|<∞.
Then the sequencexn converges strongly to an element x-=ΠFf(x-), and also x- is the unique solution of the variational inequality
(69)〈(f-I)(x-),j(y-x-)〉≤0,∀y∈F.Proof.
Puttingλn=0 in Theorem 8, we can obtain the result.If, in Theorems8 and 9, we let f:≡u∈C be a constant mapping, we have the following corollaries.Corollary 10.
LetC be a nonempty closed convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Let Ti:C→C, i=1,2,…,m, be a finite family of continuous strictly pseudo-contractive mappings, and Ai:C→E*, i=1,2,…,m, be a finite family of continuous monotone mappings such that F=F1∩F2≠∅. Tirn and Firn are defined as (19) and (20), respectively. Let xn be a sequence generated by x0∈C(70)yn=λnxn+(1-λn)∑i=1m‍μiFirnxn,xn+1=αnu+βnxn+γn∑i=1m‍σiTirnyn,
where λn∈[0,1] and {αn},{βn}, and {γn} are sequences of nonnegative real numbers in [0,1], and (i)
αn+βn+γn=1, n≥0, ∑i=1m‍μi=1, ∑i=1m‍σi=1, μi≥0, σi≥0;(ii)
limn→∞αn=0, ∑n=1∞‍αn=∞;(iii)
0<liminfn→∞βn<limsupn→∞βn<1;(iv)
limsupn→∞rn>0, ∑n=1∞‍|rn+1-rn|<∞.
Then the sequencexn converges strongly to x-=ΠFu, and also x- is the unique solution of the variational inequality
(71)〈u-x-,j(y-x-)〉≤0,∀y∈F.Corollary 11.
LetC be a nonempty closed convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Let Ti:C→C,i=1,2,…,m, be a finite family of continuous strictly pseudocontractive mappings, and Ai:C→E*,i=1,2,…,m, be a finite family of continuous monotone mappings such that F=F1∩F2≠∅. Tirn and Firn are defined as (19) and (20), respectively. Let xn be a sequence generated by x0∈C(72)xn+1=αnu+βnxn+γn∑i=1m‍εiTirnFirnxn,
where {αn}, {βn}, and {γn} are sequences of nonnegative real numbers in [0,1], and (i)
αn+βn+γn=1,n≥0,∑i=1m‍εi=1,εi>0;(ii)
limn→∞αn=0,∑n=1∞‍αn=∞;(iii)
0<liminfn→∞βn<limsupn→∞βn<1;(iv)
limsupn→∞rn>0,∑n=1∞‍|rn+1-rn|<∞.
Then the sequencexn converges strongly to x-=ΠFu, and also x- is the unique solution of the variational inequality
(73)〈u-x-,j(y-x-)〉≤0,∀y∈F.



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*Source: 926078-2013-06-18.xml*