# A General Approximation Scheme for Solutions of Various Problems in Fixed Point Theory

Authors: Eric U. Ofoedu
Journal: International Journal of Analysis
 (2013)

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## Abstract

It is our aim to prove strong convergence of a new iterative sequence to a common element of the solution set of a generalized mixed equilibrium problem; the null space of an inverse strongly monotone operator; the set of common fixed points of a countable infinite family of nonexpansive mappings; and the set of fixed points of a continuous pseudocontractive mapping. Moreover, the common element is also a unique solution of a variational inequality problem and optimality condition for a certain minimization problem. Our theorems generalize, improve, and unify several recently announced results.


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## Body

## 1. Introduction

LetE be a real normed space with dual E*. The normalized duality mapping J from E to 2E* is defined by
(1)Jx:={x*∈E*:〈x,x*〉=∥x∥2,∥x∥=∥x*∥},
where 〈·,·〉 denotes the generalized duality pairing. It is well known (see, e.g., [1]) that if E* is strictly convex, then J is single-valued and if E=H is a Hilbert space, then J is the identity mapping. In the sequel, we shall denote the single-valued normalized duality mapping by j.A mappingT with domain D(T), and range R(T), in E is called a strict contraction or simply a contraction if and only if there exists γ0∈[0,1) such that for all x,y∈D(T),
(2)∥T(x)-T(y)∥≤γ0∥x-y∥,
and T is called nonexpansive if and only if for all x,y∈D(T),
(3)∥Tx-Ty∥≤∥x-y∥.
A point x∈D(T) is called a fixed point of an operator T if and only if Tx=x. The set of fixed points of an operator T is denoted by Fix(T), that is, Fix(T):={x∈D(T):Tx=x}.The most important generalization of the class of nonexpansive mappings is, perhaps, the class of pseudocontractive mappings. These mappings are intimately connected with the important class of nonlinear accretive operators. This connection will be made precise in what follows.A mappingT with domain D(T), and range R(T), in E is called pseudocontractive if and only if for all x,y∈D(T), the following inequality holds:
(4)∥x-y∥≤∥(1+r)(x-y)-r(Tx-Ty)∥,
for all r>0. As a consequence of a result of Kato [2], the pseudocontractive mappings can also be defined in terms of the normalized duality mappings as follows: the mapping T is called pseudocontractive if and only if for all x,y∈D(T), there exists j(x-y)∈J(x-y) such that
(5)〈Tx-Ty,j(x-y)〉≤∥x-y∥2.
It now follows trivially from (5) 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 nonexpansive, the reader may see [1].To see the connection between the pseudocontractive mappings and the accretive mappings, we introduce the following definition: a mappingA with domain, D(A), and range, R(A), in E is called accretive if and only if for all x,y∈D(A), the following inequality is satisfied:
(6)∥x-y∥≤∥x-y+r(Ax-Ay)∥,
for all r>0. Again, as a consequence of Kato [2], it follows that A is accretive if and only if for all x,y∈D(A), there exists j(x-y)∈J(x-y) such that
(7)〈Ax-Ay,j(x-y)〉≥0.
The operator A is called η-inverse strongly accretive  if and only if there exists η∈(0,1) such that for all x,y∈D(A), there exists j(x-y)∈J(x-y) such that
(8)〈Ax-Ay,j(x-y)〉≥η∥Ax-Ay∥2.
It is easy to see from inequalities (4) and (6) that an operator A is accretive if and only if the mapping T:=(I-A) is pseudocontractive. Consequently, the fixed point theory for pseudocontractive mappings is intimately connected with the mapping theory of accretive operators. For the importance of accretive operators and their connections with evolution equations, the reader may consult any of the references [1, 3].Due to the above connection, fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors. It is of interest to note that ifE=H is a Hilbert space, accretive (η-strongly accretive) operators coincide with the monotone (η-strongly monotone) operators, where an operator A with domain, D(A), and range, R(A), in H is called monotone if and only if for all x,y∈D(A), we have that
(9)〈Ax-Ay,x-y〉≥0,
and A is called η-inverse strongly monotone if and only if for all x,y∈D(A),
(10)〈Ax-Ay,x-y〉≥η∥Ax-Ay∥2.
A bounded linear operator A with domain, D(A), and range, R(A), in a Hilbert space H is called a strongly positive operator if and only if there exists a constant γ>0 such that for all x∈H,〈Ax,x〉≥γ∥x∥2.LetC be a closed convex nonempty subset of a real Hilbert space H with inner product 〈·,·〉 and norm ∥·∥. Let f:C×C→ℝ be a bifunction and Φ:C→ℝ∪{+∞} be a proper extended real valued function, where ℝ denotes the set of real numbers. Let Θ:C→H be a nonlinear monotone mapping. The generalized mixed equilibrium problem (GMEP) for f,Φ, and Θ is to find u*∈C such that
(11)f(u*,y)+Φ(y)-Φ(u*)+〈Θu*,y-u*〉≥0∀y∈C.
The set of solutions for GMEP (11) is denoted by
(12)GMEP(f,Φ,Θ)={u∈C:f(u,y)+Φ(y)-Φ(u)+〈Θu,y-u〉≥0∀y∈C}.
If Φ≡0≡Θ in (11), then (11) reduces to the classical equilibrium problem (EP), that is, the problem of finding u*∈C such that
(13)f(u*,y)≥0∀y∈C,
and GMEP(f,0,0) is denoted by EP(f), where
(14)EP(f)={u∈C:f(u,y)≥0∀y∈C}.
If f≡0≡Φ in (11), then GMEP (11) reduces to the classical variational inequality problem and GMEP(0,0,Θ) is denoted by VI(Θ,C), where
(15)VI(Θ,C)={u∈C:〈Θu,y-u〉≥0∀y∈C}.
If f≡0≡Θ, then GMEP (11) reduces to the following minimization problem:
(16)findu*∈CsuchthatΦ(y)≥Φ(u*)∀y∈C,
and GMEP(0,Φ,0) is denoted by Argmin(Φ), where
(17)Argmin(Φ)={u∈C:Φ(u)≤Φ(y)∀y∈C}.
If Θ≡0, then (11) becomes the mixed equilibrium problem (MEP) and GMEP(f,Φ,0) is denoted by MEP(f,Φ), where
(18)MEP(f,Φ)={u∈C:f(u,y)+Φ(y)-Φ(u)≥0∀y∈C}.
If Φ≡0, then (11) reduces to the generalized equilibrium problem (GEP) and GMEP(f,0,Θ) is denoted by GEP(f,Θ), where
(19)GEP(f,Θ)={u∈C:f(u,y)+〈Θu,y-u〉≥0∀y∈C}.
If f≡0, then GMEP (11) reduces to the generalized variational inequality problem (abbreviated GVIP) and GMEP(0,Φ,Θ) is denoted by GVI(Φ,Θ,C), where
(20)GVI(Φ,Θ,C)={u∈K:Φ(y)-Φ(u)+〈Θu,y-u〉≥0∀y∈C}.
The generalized mixed equilibrium problem (GMEP) includes special cases as the monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. Furthermore, there are several other problems, for example, the complementarity problems and fixed point problems, which can also be written in the form of the generalized mixed equilibrium problem. In other words, the generalized mixed equilibrium problem is a unifying model for several problems arising from engineering, physics, statistics, computer science, optimization theory, operations research, economics, and countless other fields. For the past 20 years or so, many existence results have been published for various equilibrium problems (see, e.g., [4–6]).Iterative approximation of fixed points and zeros of nonlinear operators has been studied extensively by many authors to solve nonlinear operator equations as well as variational inequality problems and their generalizations (see, e.g., [7–24]). Most published results on nonexpansive mappings (e.g.) focus on the iterative approximation of their fixed points or approximation of common fixed points of a given family of this class of mappings.LetC be a closed convex nonempty subset of H, let T:C→C be a nonexpansive mapping such that Fix(T)≠∅. Given that u∈C and a real sequence {αn}n≥0 in the interval (0,1), let a sequence {xn}n≥0 be defined by
(21)x0∈C,xn+1=αnu+(1-αn)Txn,n≥0.
Under appropriate conditions on the iterative parameter {αn}n≥0, it was shown by Bauschke [25], Halpern [26], Lions [27], and Wittmann [28] that {xn}n≥0 converges strongly to PF(T)u, the projection of u to the fixed point set, F(T), of T.Marino and Xu [11], proved that the iteration scheme given by
(22)x0∈H,xn+1=αnγf(xn)+(I-αnA)Txn,n≥0
converges strongly to a unique solution x′∈Fix(T) of the variational inequality
(23)〈(γf-A)x′,y-x′〉≤0∀y∈Fix(T),
which is the optimality condition for the minimization problem
(24)minx∈Fix(T)(12〈Ax,x〉-h(c)),
where h is a potential function for γf (i.e., h′(x)=γf(x) for all x∈H); provided f:H→H is a strict contraction, T:H→H is nonexpansive and the iterative parameter {αn}n≥0 that satisfies appropriate conditions.In [29], Yamada introduced the following hybrid iterative method:
(25)x0∈H,xn+1=Txn-μλnA(Txn),n≥0,
where T is nonexpansive, A is L-Lipschitzian and an η-strongly monotone operator. He proved that if μ and {λn}n≥1 satisfy appropriate conditions, then (25) converges strongly to a unique solution x′∈Fix(T) of the varational inequality 〈Ax′,y-x′〉≥0,forally∈F(T).Tian [30] introduced the following iterative method:
(26)x0∈H,xn+1=αnγf(xn)+(I-μαnA)Txn,n≥0,
and proved that if f:H→H is a contraction, A:H→H is an η-strongly monotone mapping, T:H→H a nonexpansive mapping and the parameter {αn}n≥1 satisfies appropriate conditions, then the sequence {xn}n≥1 converges strongly to a unique solution x′∈Fix(T) of the variational inequality
(27)〈(γf-μA)x′,y-x′〉≤0∀y∈Fix(T).
Let C be a convex nonempty subset of a real Banach space. Let {Tj}j≥1 be a countable infinite family of nonexpansive mappings of C into itself and let {λj}j≥1 be a sequence of real numbers such that λj∈(0,1) for all j∈ℕ. For all n∈ℕ, define a mapping Wn:C→C by
(28)Un,n+1∶=I,Un,n∶=λnTnUn,n+1+(1-λn)I,Un,n-1∶=λn-1Tn-1Un,n+(1-λn-1)I,⋮Un,k∶=λkTkUn,k+1+(1-λk)I,Un,k-1∶=λk-1Tk-1Un,k+(1-λk-1)I,⋮Un,2∶=λ2T2Un,3+(1-λ2)I,Wn∶=Un,1=λ1T1Un,2+(1-λ1)I.
The mapping Wn,n∈ℕ is called the W-mapping generated by the countable infinite family {Tj}j≥1 of nonexpansive mappings (see, e.g., [31]).In [32], Yao et al. studied the following problem. Let H be a real Hilbert space. Consider the iterative scheme
(29)x1∈H,xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)Wnxn,
where γ>0 is some constant, f:H→H is a given contractive mapping, and A is a strongly positive bounded linear operator on H. Assuming that F∶=⋂j=1∞Fix(Tj)≠∅, and under appropriate conditions on the iterative parameters, Yao et al. proved that the sequence {xn}n≥1 generated by (29) converges strongly to x*∈F, where x* is a unique solution of the variational inequality
(30)〈(A-γf)x*,y-x*〉≥0∀y∈F.
Colao and Marino [33] proved that if {xn}n≥1 and {un}n≥1 are sequences generated by x1∈H and
(31)G(un,y)+1rn〈y-un,un-xn〉≥0y∈H;xn+1=εnγf(xn)+βnxn+((1-βn)I-εnA)Wnun,n≥1,
(where εn, βn, rn, and γ satisfy appropriate conditions), then both {xn}n≥1 and {un}n≥1 converge strongly to an element x*∈⋂j=1∞Fix(Tj)∩EP(G), which is also a unique solution of the variational inequality
(32)〈(A-γf)x*,y-x*〉≥0∀y∈⋂‍∞j=1Fix(Tj)∩EP(G).
Let C be a nonempty closed and convex subset of real Hilbert space H. Let T:C→C be a continuous pseudocontractive mapping; A:C→H be a continuous monotone mapping; B:H→H an inverse strongly monoyone mapping. For each n∈ℕ, let Trn and Frn be defined for each x∈H by
(33)Trn∶={1rnz∈C:〈y-z,Tz〉-1rn〈y-z,(1+rn)z-x〉≤0,∀y∈C},Frn∶={1rnz∈C:〈y-z,Az〉+1rn〈y-z,z-x〉≤0,∀y∈C}.
Let {xn}n≥1 be a sequence generated iteratively by x1∈C,
(34)yn=Frnxn;xn+1=αnγf(xn)+δnxn+((1-δn)I-αnB)Trnyn,n≥1,
(where αn, δn, rn, and γ satisfy appropriate conditions), then it was recently proved by Chamnarnpan and Kumam [34] that the sequence {xn}n≥1 defined by (34) converges strongly to an element z∈Fix(T)∩VI(C,A), which is a unique solution of the variational inequality
(35)〈(B-γf)z,x-z〉≥0∀x∈Fix(T)∩VI(C,A).In this paper, motivated by the results of the authors mentioned above, it is our aim to prove strong convergence of a new iterative sequence to a common element of the solution set of a generalized mixed equilibrium problem; the null space of an inverse strongly monotone operator; the set of common fixed points of a countable infinite family of nonexpansive mappings; the set of fixed points of a continuous pseudocontractive mapping. Moreover, the element is also a unique solution of a variational inequality problem and optimality condition for a certain minimization problem. Our theorems generalize improve and unify several recently announced results.


## 2. Preliminary

In what follows, we shall make use of the following lemmas.Lemma 1 (see, e.g., [31]).
LetC be a closed convex nonempty subset of a strictly convex real Banach space. Let {Tj}j≥1 be a countable infinite family of nonexpansive mappings of C into itself and let {λj}j≥1 be a real sequence such that  0<λj≤b<1 for all j∈ℕ, for some constant b∈(0,1). Then, for all x∈C and k∈ℕ,limn→∞Un,kx exists (where Un,k is as defined in (28)).In particular, fork=1 in Lemma 1, we define a mapping W:C→C by
(36)Wx:=limn→∞Un,1x=limn→∞Wnx.Lemma 2 (see, e.g., [31]).
LetC be a closed convex nonempty subset of a strictly convex real Banach space. Let {Tj}j≥1 be a countable infinite family of nonexpansive mappings of C into itself such that ⋂j=1∞Fix(Tj)≠∅ and let {λj}j≥1 be a real sequence such that 0<λj≤b<1 for all j∈ℕ, for some constant b∈(0,1). Then, Fix(W)=⋂j=1∞Fix(Tj)=⋂n=1∞Fix(Wn).Lemma 3 (see [33]).
LetC be a closed convex nonempty subset of a strictly convex real Banach space. Let {Tj}j≥1 be a countable infinite family of nonexpansive mappings of C into itself such that ⋂j=1∞Fix(Tj)≠∅ and let {λj}j≥1 be a real sequence such that 0<λj≤b<1 for all j∈ℕ, for some constant b∈(0,1). Suppose that Wn:C→C is given by (28) for all n∈ℕ, then
(37)∥Wn+1x-Wnx∥≤2∏j=1n‍λj∥x-p∥,∀x∈C,p∈⋂‍∞j=1Fix(Tj).Lemma 4.
LetH be a real Hilbert space, then the following inequality holds:
(38)∥x+y∥2≤∥x∥2+2〈y,x+y〉∀x,y∈H.Lemma 5 (see, e.g., [35–37]).
Let{λn}n≥0 be a sequence of nonnegative real numbers satisfying the following condition:
(39)λn+1≤(1-αn)λn+σn,n≥0,
where {αn}n≥0 and {σn}n≥0 are sequences of real numbers such that {αn}n≥0⊂[0,1],∑n=0∞‍αn=+∞. Suppose that σn=o(αn),n≥0 (i.e., limn→∞(σn/αn)=0) or ∑n=0∞‍|σn|<+∞ or limsupn→∞(σn/αn)≤0, then λn→0 as n→∞.Lemma 6 (see, e.g., Suzuki [38]).
Let{tn}n≥0 and {yn}n≥0 be two bounded sequences in a real Banach space such that yn+1=βnyn+(1-βn)tn, for all n≥0, where {βn}n≥0 is a real sequence satisfying the condition 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that
(40)limsupn→∞(∥tn+1-tn∥-∥yn+1-yn∥)≤0,
then limn→∞∥tn-yn∥=0.Lemma 7 (see Zegeye [39]).
LetC 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>0 and x∈H, there exists z∈C such that
(41)〈y-z,T′z〉-1r〈y-z,(1+r)z-x〉≤0∀y∈C.Lemma 8 (see Zegeye [39]).
LetC 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>0 and x∈H, define a mapping Fr:H→C by
(42)Frx={1rz∈C:〈y-z,T′z〉-1r〈y-z,(1+r)z-x〉≤0∀y∈C},
then the following hold:(1)
Fr is single-valued;(2)
Fr is firmly nonexpansive type mapping, that is, for all x,y∈H,
(43)∥Frx-Fry∥2≤〈Frx-Fry,x-y〉;(3)
Fix(Fr) is closed and convex; and Fix(Fr)=Fix(T′) for all r>0.In the sequel, we shall require that the bifunctionf:C×C→ℝ satisfies the following conditions:(A1)
f(x,x)=0 for all x∈C;(A2)
f is monotone, in the sense that f(x,y)+f(y,x)≤0 for all x,y∈C;(A3)
limsupt→0+f(tz+(1-t)x,y)≤f(x,y) for all x,y,z∈C;(A4)
the functiony↦f(x,y) is convex and lower semicontinuous for all x∈C.Lemma 9 (Compare with Lemma 2.4 of [5]).
LetC be a closed convex nonempty subset of a real Hilbert space H. Let f:C×C→ℝ be a bifunction satisfying conditions (A1)–(A4);Θ:C→H a continuous monotone mapping and Φ:C→ℝ∪{+∞} a proper lower semicontinuous convex function. Then, for all r>0 and x∈H there exists u∈C such that
(44)f(u,y)+Φ(y)-Φ(u)+〈Θu,y-u〉+1r〈y-u,u-x〉≥0∀y∈K.
Moreover, if for all x∈H we define a mapping Gr:H→2C by
(45)Gr(x)={1ru∈C:f(u,y)+Φ(y)-Φ(u)+〈Θu,y-u〉+1r〈y-u,u-x〉≥0,∀y∈C1r},
then the following hold:  (1)
Gr is single-valued for all r>0;(2)
Gr is firmly nonexpansive, that is, for all x,z∈H,
(46)∥Grx-Grz∥2≤〈Grx-Grz,x-z〉;(3)
Fix(Gr)=GMEP(f,Φ,Θ) for all r>0;(4)
GMEP(f,Φ,Θ) is closed and convex.LetC be a closed convex nonempty subset of a uniformly convex real Banach space E, where E is called uniformly convex if and only if the modulus of convexity
(47)δE(ε)=inf{1-12∥x+y∥:∥x∥≤1,∥y∥≤1,∥x-y∥≥ε}
is positive for all ε∈(0,2]. Given a bounded sequence {xn}n≥1 in C, define
(48)rm(y):=sup{∥xk-y∥:k≥m},y∈E.
It is well known (see, e.g., [40]) that there exists unique cm∈C such that
(49)rm(cm)=inf{rm(y):y∈C}=rm∀m∈ℕ.
Clearly, rm+1≤rm and 0≤rm for all m∈ℕ, so that limm→∞rm=infm∈ℕrm exists. It is known (see [40]) that if limm→∞rm=infm∈ℕrm=0, then the sequence {xn}n≥1 converges.Definition 10 (see [40]).
If the sequence{cm}m≥1 converges, then c0=limm→∞cm is called the asymptotic center of {xn}n≥1 (with respect to C). Equivalently (see, e.g., [22]), c0∈C is called the asymptotic center of {xn}n≥1 (with respect to C) if and only if
(50)limsupn→∞∥xn-c0∥=infz∈Climsupn→∞∥xn-z∥.Lemma 11 (see Theorem 1 of Edelstein [40]).
LetC be a closed convex nonempty subset of a uniformly convex real Banach space E, Let {xn}n≥1 be a bounded sequence in C and {cm}m≥1 be as in (49), then c=limm→∞cm exists. In other words, the asymptotic center c of the sequence {xn}n≥1 (with respect to C) exists and is unique.


## 3. Main Results

Lemma 12.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T′:C→C be a continuous pseudocontractive mapping. For r>0, let Fr:H→C be the mapping in Lemma 8, then for any x∈H and for any p,q>0,
(51)∥Fpx-Fqx∥≤|p-q|p(∥Fpx∥+∥x∥).Proof.
Observe that for anyy∈C,
(52)〈y-Fpx,T′Fpx〉-1p〈y-Fpx,(1+p)Fpx-x〉≤0,(53)〈y-Fqx,T′Fqx〉-1q〈y-Fqx,(1+q)Fqx-x〉≤0.
So, we obtain (in particular) for y=Fqx in (52) and y=Fpx in (53) that
(54)〈Fqx-Fpx,T′Fpx〉-1p〈Fqx-Fpx,(1+p)Fpx-x〉≤0,(55)〈Fpx-Fqx,T′Fqx〉-1q〈Fpx-Fqx,(1+q)Fqx-x〉≤0,
respectively. Adding (54) and (55) gives
(56)-〈Fqx-Fpx,T′Fqx-T′Fpx〉-〈Fqx-Fpx,(1+p)Fpx-xp-(1+q)Fqx-xq〉≤0.
Using the fact that T′ is pseudocontractive, we obtain from (56) that
(57)-〈Fqx-Fpx,Fqx-Fpx〉-〈Fqx-Fpx,(1+p)Fpx-xp-(1+q)Fqx-xq〉≤0.
This implies that
(58)〈Fqx-Fpx,Fqx-Fpx〉+〈Fqx-Fpx,(1+p)Fpx-xp-(1+q)Fqx-xq〉≥0.
Equivalently, we obtain
(59)〈Fqx-Fpx,Fpx-xp-Fqx-xq〉≥0.
This implies that
(60)0≤〈Fpx-Fqx,Fqx-x-qp(Fpx-x)〉=〈Fpx-Fqx,Fqx-Fpx+Fpx-x-qp(Fpx-x)〉=〈Fpx-Fqx,Fqx-Fpx+(1-qp)(Fpx-x)〉≤-∥Fpx-Fqx∥2+|1-qp|∥Fpx-Fqx∥×(∥Fpx∥+∥x∥).
Thus,
(61)∥Fpx-Fqx∥≤|p-q|p(∥Fpx∥+∥x∥).Lemma 13.
LetC be a closed convex nonempty subset of a real Hilbert space H. Let f:C×C→ℝ be a bifunction satisfying conditions (A1)–(A4);Θ:C→H a monotone mapping and Φ:C→ℝ∪{+∞} a proper lower semicontinuous convex function. Let r>0 and let Gr be the mapping in Lemma 9, then for all p,q>0 and for all x∈H, one has that
(62)∥Gpx-Gqx∥≤|p-q|p(∥Gpx∥+∥x∥).Proof.
We observe that(63)f(Gpx,y)+Φ(y)-Φ(Gpx)+〈Θ(Gpx),y-Gpx〉+1p〈y-Gpx,Gpx-x〉≥0∀y∈H,(64)f(Gqx,y)+Φ(y)-Φ(Gqx)+〈Θ(Gqx),y-Gqx〉+1q〈y-Gqx,Gqx-x〉≥0∀y∈H.
In particular, we obtain from (63) and (64) (resp.) that
(65)f(Gpx,Gqx)+Φ(Gqx)-Φ(Gpx)+〈Θ(Gpx),Gqx-Gpx〉+1p〈Gqx-Gpx,Gpx-x〉≥0,f(Gqx,Gpx)+Φ(Gpx)-Φ(Gqx)+〈Θ(Gqx),Gpx-Gqx〉+1q〈Gpx-Gqx,Gqx-x〉≥0.
Adding (65) (and using the monotonicity of f and Θ), we get that
(66)0≤〈Gpx-Gqx,Gqx-xq-Gpx-xp〉.
This implies that
(67)0≤〈Gpx-Gqx,Gqx-x-qp(Gpx-x)〉=〈Gpx-Gqx,Gqx-Gpx+Gpx-x-qp(Gpx-x)〉=〈Gpx-Gqx,Gqx-Gpx+(1-qp)(Gpx-x)〉≤-∥Gpx-Gqx∥2+|1-qp|×∥Gpx-Gqx∥(∥Gpx∥+∥x∥).
Thus,
(68)∥Gpx-Gqx∥≤|p-q|p(∥Gpx∥+∥x∥).Lemma 14.
Let{an}n≥1 be a bounded sequence of nonnegative real numbers, then
(69)limsupn→∞an2=(limsupn→∞an)2.Proof.
It is well known that for any two bounded sequences{un}n≥1 and {vn}n≥1 of nonnegative real numbers, we have that
(70)limsupn→∞unvn≤limsupn→∞un×limsupn→∞vn.
It then follows that for any bounded sequence {an}n≥1 of nonnegative terms,
(71)limsupn→∞an2≤(limsupn→∞an)2.
To complete our proof, we show that
(72)limsupn→∞an2≥(limsupn→∞an)2.
Now, let {anj}j≥1 be a subsequence of {an}n≥1 such that limsupn→∞an=limj→∞anj, then by the continuity of the function f:ℝ→ℝ given by f(t)=t2, we obtain that
(73)(limsupn→∞an)2=(limj→∞anj)2=limj→∞anj2≤limsupn→∞an2.
The last inequality follows from the fact that limsupn→∞an2 equals supS, where S is the set of all subsequential limits of {an2}n≥1. This completes the proof.Remark 15.
We must note that Lemma14 holds trivially if the sequence {an}n≥1 was an unbounded sequence of nonnegative real numbers.Remark 16.
In the sequel, we shall make the following assumptions:H is a real Hilbert space; T′:H→H is a continuous pseudocontractive mapping; Tj:H→H,j=1,2,3,… is a countable infinite family of nonexpansive mappings; f:H×H→ℝ is a bifunction satisfying conditions (A1)–(A4);Φ:H→ℝ∪{+∞} a proper lower semicontinuous convex function; Θ:H→H a continuous monotone mapping; u∈H is a fixed vector; A:H→H is a strongly positive bounded linear operator with coefficient γ;B:H→His an η-inverse strongly monotone mapping. The sequences {rn}n≥1, {αn}n≥1,{βn}n≥1, and {λn}n≥1 are real sequences such that rn>0 for all n∈ℕ,limn→∞rn=r0>0;0<αn<1foralln∈ℕ,limn→∞αn=0,∑n=1∞‍αn=+∞;0<βn<1for alln∈ℕ,0<liminfn→∞βn≤limsupn→∞βn<1;0<λn≤b<1 for all n∈ℕ and for some constant b∈(0,1).ξ is a real constant such that 0<ξ<2η; Ω=Fix(T′)∩GMEP(f,Φ,Θ)∩B-1(0)∩⋂j=1∞Fix(Tj)≠∅. For r>0, Fr and Gr are as in Lemmas 8 and 9, respectively.We shall study the strong convergence of the sequence{xn}n≥1 generated iteratively from arbitrary x1∈H by
(74)xn+1=αnu+βnxn+((1-βn)I-αnA)Wn(I-ξB)FrnGrnxn,n≥1
to a unique solution x′∈Ω of the variational inequality
(75)〈u-Ax′,x-x′〉≤0∀x∈Ω.Lemma 17.
Suppose that the conditions of Remark16 are satisfied, then {xn}n≥1 defined by (74) is bounded.Proof.
Observe that for allx,y∈H,
(76)∥(x-ξBx)-(y-ξBy)∥2=∥x-y-ξ(Bx-By)∥2=∥x-y∥2-2ξ〈Bx-By,x-y〉+ξ2∥Bx-By∥2≤∥x-y∥2-2ξη∥Bx-By∥2+ξ2∥Bx-By∥2=∥x-y∥2-ξ(2η-ξ)∥Bx-By∥2.
Thus, since 0<ξ<2η, we get from (76) that
(77)∥(x-ξBx)-(y-ξBy)∥≤∥x-y∥∀x,y∈H.
It thus follows that for p∈Ω,
(78)∥(I-ξB)FrnGrnxn-p∥2=∥(I-ξB)FrnGrnxn-(I-ξB)p∥2≤∥FrnGrnxn-p∥2≤∥xn-p∥2.
Next, we may (without loss of generality) assume that αn<∥A∥-1(1-βn) for all n∈ℕ since limn→∞αn=0. So, for each x∈H such that ∥x∥=1, we have
(79)〈((1-βn)I-αnA)x,x〉=(1-βn)-αn〈Ax,x〉≥(1-βn)-αn∥A∥>(1-βn)-∥A∥-1(1-βn)∥A∥=0.
Thus, we have that for all n∈ℕ, the operator (1-βn)I-αnA is a positive bounded linear operator, so that
(80)∥(1-βn)I-αnA∥=sup∥x∥=1〈((1-βn)I-αnA)x,x〉=sup∥x∥=1(1-βn-αn〈Ax,x〉)≤1-βn-αnγ.
Therefore, for p∈Ω, we obtain from (74) using (78) and (80) that
(81)∥xn+1-p∥=∥αnu+βnxn+((1-βn)I-αnA)×Wn(I-ξB)FrnGrnxn-p∥=∥((1-βn)I-αnA)(Wn(I-ξB)FrnGrnxn-p)+αn(u-Ap)+βn(xn-p)Frn∥≤(1-βn-αnγ)∥Wn(I-ξB)FrnGrnxn-p∥+βn∥xn-p∥+αn∥u-Ap∥≤(1-αnγ)∥xn-p∥+αn∥u-Ap∥.
It is thus easy to show by mathematical induction (using (81)) that
(82)∥xn-p∥≤max{∥x1-p∥,1γ∥u-Ap∥}.
So, {xn}n≥1 is bounded. Consequently, {Wn(I-ξB)FrnGrnxn}n≥1,  {(I-ξB)FrnGrnxn}n≥1,  {FrnGrnxn}n≥1 and {Grnxn}n≥1 are all bounded.Lemma 18.
Let{xn}n≥1 be given by (74). Suppose that conditions of Remark 16 are satisfied, then
(83)limn→∞∥xn+1-xn∥=0.Proof.
Settn∶=(1/(1-βn))(xn+1-βnxn), then xn+1=βnxn+(1-βn)tn. Thus,
(84)tn+1-tn=11-βn+1(xn+2-βn+1xn+1)-11-βn(xn+1-βnxn)=11-βn+1(Frn+1αn+1u+[(1-βn+1)I-αn+1A]×Wn+1(I-ξB)Frn+1Grn+1xn+1)-11-βn(Frn+1αnu+[(1-βn)I-αnA]×Wn(I-ξB)FrnGrnxn)=αn+11-βn+1(u-AWn+1(I-ξB)Frn+1Grn+1xn+1)-αn1-βn(u-AWn(I-ξB)FrnGrnxn)+Wn+1(I-ξB)Frn+1Grn+1xn+1-Wn(I-ξB)FrnGrnxn+Wn+1(I-ξB)FrnGrnxn-Wn+1(I-ξB)FrnGrnxn.
Thus, forp∈Ω (using (84), Lemmas 3, 13, and 12), we obtain (for some constant M0>0) the following:
(85)∥tn+1-tn∥-∥xn+1-xn∥≤[αn+11-βn+1+αn1-βn]M0+∥Wn+1(I-ξB)FrnGrnxn-Wn(I-ξB)FrnGrnxn∥+∥Wn+1(I-ξB)Frn+1Grn+1xn+1-Wn+1(I-ξB)FrnGrnxn∥-∥xn+1-xn∥≤[αn+11-βn+1+αn1-βn]M0+2∏j=1n‍λj∥(I-ξB)FrnGrnxn-p∥+∥Frn+1Grn+1xn+1-Frn+1Grnxn∥+∥Frn+1Grnxn-FrnGrnxn∥-∥xn+1-xn∥≤[αn+11-βn+1+αn1-βn]M0+2∏j=1n‍λj∥FrnGrnxn-p∥+∥Grn+1xn+1-Grnxn∥+∥Frn+1Grnxn-FrnGrnxn∥-∥xn+1-xn∥≤[αn+11-βn+1+αn1-βn]M0+2∏j=1n‍λj∥FrnGrnxn-p∥+∥Grn+1xn+1-Grn+1xn∥+∥Grn+1xn-Grnxn∥+∥Frn+1Grnxn-FrnGrnxn∥-∥xn+1-xn∥≤[αn+11-βn+1+αn1-βn]M0+2∏j=1n‍λj∥FrnGrnxn-p∥+|rn+1-rn|rn×[(∥FrnGrnxn∥+2∥Grnxn∥+∥xn∥)].
Since 0<λn<b<1 for all n∈ℕ and for some constant b∈(0,1), it is thus easy to see that limn→∞∏j=1n‍λj=0. So, we obtain from (85) that
(86)limsupn→∞(∥tn+1-tn∥-∥xn+1-xn∥)≤0.
Lemma 6 therefore gives
(87)limn→∞∥tn-xn∥=0,
but
(88)xn+1-xn=(1-βn)(tn-xn).
Hence,
(89)limn→∞∥xn+1-xn∥=0.Lemma 19.
Let{xn}n≥1 be given by (74). Suppose conditions of Remark 16 are satisfied, then
(90)limn→∞∥Wn(I-ξB)FrnGrnxn-xn∥=limn→∞∥xn-FrnGrnxn∥=limn→∞∥BFrnGrnxn∥=limn→∞∥Wn(I-ξB)FrnGrnxn-FrnGrnxn∥=limn→∞∥xn-Grnxn∥=0.Proof.
Observe that for some constantM1>0,
(91)∥xn-Wn(I-ξB)FrnGrnxn∥≤∥xn-xn+1∥+∥xn+1-Wn(I-ξB)FrnGrnxn∥=∥xn-xn+1∥+∥Frnαnu+βnxn+[(1-βn)I-αnA]×Wn(I-ξB)FrnGrnxn-Wn(I-ξB)FrnGrnxn∥≤∥xn+1-xn∥+αn∥u-AWn(I-ξB)FrnGrnxn∥+βn∥xn-Wn(I-ξB)FrnGrnxn∥≤∥xn+1-xn∥+αnM1+βn∥xn-Wn(I-ξB)FrnGrnxn∥.
So, we obtain from (91) that
(92)∥Wn(I-ξB)FrnGrnxn-xn∥≤11-βn(∥xn-xn+1∥+αnM1).
Using (89) and the fact that limn→∞αn=0, we obtain from (92) that
(93)limn→∞∥Wn(I-ξB)FrnGrnxn-xn∥=0.
Next, observe that for fixed p∈Ω and using (2) of Lemma 9, we obtain the following:
(94)∥Grnxn-p∥2≤〈Grnxn-p,xn-p〉=12[∥Grnxn-p∥2+∥xn-p∥2-∥xn-Grnxn∥2],
so that
(95)∥Grnxn-p∥2≤∥xn-p∥2-∥xn-Grnxn∥2.
Furthermore, using (2) of Lemma 8, we obtain
(96)∥FrnGrnxn-p∥2≤〈FrnGrnxn-p,Grnxn-p〉=12[∥FrnGrnxn-p∥2+∥Grnxn-p∥2-∥Grnxn-FrnGrnxn∥2],
so that
(97)∥FrnGrnxn-p∥2≤∥Grnxn-p∥2-∥Grnxn-FrnGrnxn∥2.
Moreover, using the recursion formula (74), Lemma 4, and convexity of ∥·∥2, we have
(98)∥xn+1-p∥2=∥Frnαnu+βnxn+[(1-βn)I-αnA]×Wn(I-ξB)FrnGrnxn-p∥2≤βn∥xn-p∥2+(1-βn)×∥Wn(I-ξB)FrnGrnxn-p∥2+2αn〈Frnu-AWn(I-ξB)×FrnGrnxn,xn+1-p〉≤βn∥xn+1-p∥2+2βn∥xn+1-p∥∥xn-xn+1∥+βn∥xn+1-xn∥2×(1-βn)∥(I-ξB)FrnGrnxn-p∥2+2αn〈Frnu-AWn(I-ξB)×FrnGrnxn,xn+1-p〉.
Thus, using (76) and (97), we obtain from (98) the following:
(99)∥xn+1-p∥2≤∥(I-ξB)FrnGrnxn-p∥2+βn1-βn(2∥xn+1-p∥+∥xn+1-xn∥)×∥xn+1-xn∥+2αn1-βn×〈u-AWn(I-ξB)FrnGrnxn,xn+1-p〉≤∥FrnGrnxn-p∥2-ξ(2η-ξ)∥BFrnGrnxn∥2+βn1-βn(2∥xn+1-p∥+∥xn+1-xn∥)×∥xn+1-xn∥+2αn1-βn×〈u-AWn(I-ξB)FrnGrnxn,xn+1-p〉≤∥xn-p∥2-∥Grnxn-FrnGrnxn∥2-ξ(2η-ξ)∥BFrnGrnxn∥2+βn1-βn(2∥xn+1-p∥+∥xn+1-xn∥)×∥xn+1-xn∥+2αn1-βn×〈u-AWn(I-ξB)FrnGrnxn,xn+1-p〉.
So, from (99), we have (for some real constant M2>0)
(100)∥xn-FrnGrnxn∥2+ξ(2η-ξ)∥BFrnGrnxn∥2≤∥xn-p∥2-∥xn+1-p∥2+βn1-βn×(2∥xn+1-p∥+∥xn+1-xn∥)×∥xn+1-xn∥+2αn1-βn×〈u-AWn(I-ξB)FrnGrnxn,xn+1-p〉≤(αn+∥xn+1-xn∥)M2.
From (100), we have
(101)limn→∞(∥xn-FrnGrnxn∥2+ξ(2η-ξ)∥BFrnGrnxn∥2)=0,
and since ξ(2η-ξ)>0, then using (101) and applying sandwich theorem, we obtain
(102)limn→∞∥xn-FrnGrnxn∥=0,(103)limn→∞∥BFrnGrnxn∥=0.
Furthermore, observe that
(104)∥Wn(I-ξB)FrnGrnxn-FrnGrnxn∥≤∥Wn(I-ξB)FrnGrnxn-xn∥+∥xn-FrnGrnxn∥.
Hence, using (93), (102), (103), and (104), we have
(105)limn→∞∥Wn(I-ξB)FrnGrnxn-FrnGrnxn∥=0.
Besides, we obtain from the second line of inequality (99) (using (95) and the fact that Frn is firmly nonexpansive; thus nonexpansive) that for some constant M′>0,
(106)∥xn+1-p∥2≤∥FrnGrnxn-p∥2-ξ(2η-ξ)∥BFrnGrnxn∥2+βn1-βn(2∥xn+1-p∥+∥xn+1-xn∥)×∥xn+1-xn∥+2αn1-βn×〈u-AWn(I-ξB)FrnGrnxn,xn+1-p〉≤∥Grnxn-p∥2+[∥xn+1-xn∥+αn]M′≤∥xn-p∥2-∥xn-Grnxn∥2+[∥xn+1-xn∥+αn]M′.
So, we obtain from (106) that for some constant M′′>0,
(107)∥xn-Grnxn∥≤∥xn-p∥2-∥xn+1-p∥2+[∥xn+1-xn∥+αn]M′≤∥xn+1-xn∥M′′+[∥xn+1-xn∥+αn]M′.
Hence, limn→∞∥xn-Grnxn∥=0.Lemma 20.
Let{xn}n≥1 be given by (74). Suppose that conditions of Remark 16 hold, then
(108)limn→∞∥Wn(I-ξB)FrnGrnxn-(I-ξB)FrnGrnxn∥=0.Proof.
Letp∈Ω, then we obtain
(109)∥(I-ξB)FrnGrnxn-p∥2=∥FrnGrnxn-ξBFrnGrnxn-p∥2=∥(FrnGrnxn-ξBFrnGrnxn)-(p-ξBp)∥2=12[∥(FrnGrnxn-ξBFrnGrnxn)-(p-ξBp)∥2+∥(I-ξB)FrnGrnxn-p∥2-∥((FrnGrnxn-ξBFrnGrnxn)-(p-ξBp))-((I-ξB)FrnGrnxn-p)∥2]≤12[∥FrnGrnxn-p∥2+∥(I-ξB)FrnGrnxn-p∥2-∥(FrnGrnxn-p)-((I-ξB)FrnGrnxn-p)∥2+2ξ〈FrnGrnxn-(I-ξB)×FrnGrnxn,BFrnGrnxn〉-ξ2∥BFrnGrnxn∥2].
This inequality implies the following:
(110)∥(I-ξB)FrnGrnxn-p∥2≤∥xn-p∥2-∥FrnGrnxn-(I-ξB)FrnGrnxn∥2+2∥FrnGrnxn-(I-ξB)FrnGrnxn∥×∥BFrnGrnxn∥-ξ2∥BFrnGrnxn∥2≤∥xn-xn+1∥2+2∥xn+1-p∥×∥xn+1-xn∥+∥xn+1-p∥2-∥FrnGrnxn-(I-ξB)FrnGrnxn∥2+2∥FrnGrnxn-(I-ξB)FrnGrnxn∥×∥BFrnGrnxn∥.
But from the first line of inequality (99), we have
(111)∥xn+1-p∥2≤∥(I-ξB)FrnGrnxn-p∥2+βn1-βn(2∥xn+1-p∥+∥xn+1-xn∥)×∥xn+1-xn∥+2αn1-βn×〈u-AWn(I-ξB)FrnGrnxn,xn+1-p〉.
So, for some real constant M2>0, we obtain from (111) that
(112)∥xn+1-p∥2≤∥(I-ξB)FrnGrnxn-p∥2+(αn+∥xn+1-xn∥)M2.
Using (112) in (110) and rearranging terms, we obtain (for some real constant M3>0) that
(113)∥FrnGrnxn-(I-ξB)FrnGrnxn∥2≤(αn+∥xn+1-xn∥+∥BFrnGrnxn∥)M3.
So, (113) implies that
(114)limn→∞∥FrnGrnxn-(I-ξB)FrnGrnxn∥=0.
Hence,
(115)∥(I-ξB)FrnGrnxn-xn∥≤∥(I-ξB)FrnGrnxn-FrnGrnxn∥+∥FrnGrnxn-xn∥→asn→∞,∥Wn(I-ξB)FrnGrnxn-(I-ξB)FrnGrnxn∥≤∥Wn(I-ξB)FrnGrnxn-xn∥+∥xn-(I-ξB)FrnGrnxn∥.
The last inequality implies that
(116)limn→∞∥Wn(I-ξB)FrnGrnxn-(I-ξB)FrnGrnxn∥=0.Lemma 21.
Let{xn}n≥1 be given by (74) and let x′∈Ω be the unique solution of the variational inequality (75). Suppose that conditions of Remark 16 hold, then
(117)limsupn→∞〈u-Ax′,xn-x′〉≤0.Proof.
Let{xnj}j≥1 be a subsequence of {xn}n≥1 such that
(118)limsupn→∞〈u-Ax′,xn-x′〉=limj→∞〈u-Ax′,xnj-x′〉.
Since every real Hilbert space H is a uniformly convex real Banach space and {xn}n≥1 is a bounded sequence in H, then we obtain by Lemma 11 that there exists a unique x*∈H such that x* is the asymptotic center of the sequence {xnj}j≥1. We first show that x*∈Ω. Recall that r0=limn→∞rn and that by Lemma 1, Wx:=limn→∞Wnx exists for all x∈H and by Lemma 2 that Fix(W)=⋂n=1∞Fix(Wn)=⋂j=1∞Fix(Tj); thus,
(119)∥xnj-Wx*∥≤∥xnj+1-xnj∥+∥Grnjαnju+βnjxnj+[(1-βnj)I-αnjA]×Wnj(I-ξB)FrnjGrnjxnj-Wx*∥≤∥xnj+1-xnj∥+βnj∥xnj-Wx*∥+(1-βnj)×(∥Wnj(I-ξB)FrnjGrnjxnj-Wnjxnj∥+∥Wnjxnj-Wnjx*∥+∥Wnjx*-Wx*∥Grnj)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥≤∥xnj+1-xnj∥+βnj∥xnj-Wx*∥+(1-βnj)(∥(I-ξB)FrnjGrnjxnj-xnj∥+∥xnj-x*∥+∥Wnjx*-Wx*∥Grnj)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥.
Thus, we obtain from (119) that
(120)∥xnj-Wx*∥≤11-βnj∥xnj+1-xnj∥+∥(I-ξB)FrnjGrnjxnj-xnj∥+∥xnj-x*∥+∥Wnjx*-Wx*∥+αnj1-βnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥,
so that
(121)limsupj→∞∥xnj-Wx*∥≤limsupj→∞∥xnj-x*∥.
This implies by Definition 10 (with C=H) and uniqueness of x* that Wx*=x*; that is, x*∈⋂j=1∞Fix(Tj).
Next,(122)∥xnj-Fr0x*∥≤∥xnj+1-xnj∥+∥Frnjαnju+βnjxnj+[(1-βnj)I-αnjA]×Wnj(I-ξB)FrnjGrnjxnj-Fr0x*∥≤∥xnj+1-xnj∥+βnj∥xnj-Fr0x*∥+(1-βnj)×(∥Wnj(I-ξB)FrnjGrnjxnj-FrnjGrnjxnj∥+∥FrnjGrnjxnj-Frnjxnj∥+∥Frnjxnj-Frnjx*∥+∥Frnjx*-Fr0x*∥)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥≤∥xnj+1-xnj∥+βnj∥xnj-Fr0x*∥+(1-βnj)×(|rnj-r0|r0∥Wnj(I-ξB)FrnjGrnjxnj-FrnjGrnjxnj∥+∥Grnjxnj-xnj∥+∥xnj-x*∥+|rnj-r0|r0[∥Fr0x*∥+∥x*∥])+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥,
so that from (122), we obtain
(123)∥xnj-Fr0x*∥≤11-βnj∥xnj+1-xnj∥+∥Wnj(I-ξB)FrnjGrnjxnj-FrnjGrnjxnj∥+∥Grnjxnj-xnj∥+∥xnj-x*∥+|rnj-r0|r0[∥Fr0x*∥+∥x*∥]×αnj1-βnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥.
Thus,(124)limsupn→∞∥xnj-Fr0x*∥≤limsupn→∞∥xnj-x*∥.
So, by Definition 10, we obtain that Fr0x*=x*⇔T′x*=x*, that is, x*∈Fix(T′).
Moreover,(125)∥xnj-Gr0x*∥≤∥xnj+1-xnj∥+∥Frnjαnju+βnjxnj+[(1-βnj)I-αnjA]×Wnj(I-ξB)FrnjGrnjxnj-Gr0x*∥≤∥xnj+1-xnj∥+βnj∥xnj-Gr0x*∥+(1-βnj)×(∥Wnj(I-ξB)FrnjGrnjxnj-FrnjGrnjxnj∥+∥FrnjGrnjxnj-xnj∥+∥xnj-Grnjxnj∥+∥Grnjxnj-Grnjx*∥+∥Grnjx*-Gr0x*∥)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥≤∥xnj+1-xnj∥+βnj∥xnj-Gr0x*∥+(1-βnj)×(|rnj-r0|r0∥Wnj(I-ξB)FrnjGrnjxnj-FrnjGrnjxnj∥+∥FrnjGrnjxnj-xnj∥+∥xnj-Grnjxnj∥+∥xnj-x*∥+|rnj-r0|r0(∥Gr0x*∥+∥x*∥))+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥.
Thus, we obtain from (125) that
(126)∥xnj-Gr0x*∥≤11-βnj∥xnj+1-xnj∥+∥Wnj(I-ξB)FrnjGrnjxnj-FrnjGrnjxnj∥+∥FrnjGrnjxnj-xnj∥+∥xnj-Grnjxnj∥+∥xnj-x*∥+|rnj-r0|r0(∥Gr0x*∥+∥x*∥)+αnj1-βnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥.
Thus,
(127)limsupj→∞∥xnj-Gr0x*∥≤limsupj→∞∥xnj-x*∥.
Definition10 again implies that Gr0x*=x*. Hence x*∈GMEP(f,Φ,Θ).
Furthermore, since we have shown thatGr0x*=x*=Fr0x*, then making use of this fact and Lemma 13, we have
(128)∥xnj-(x*-ξBx*)∥≤∥xnj+1-xnj∥+∥Frnjαnju+βnjxnj+[(1-βnj)I-αnjA]×Wnj(I-ξB)FrnjGrnjxnj-(x*-ξBx*)∥≤∥xnj+1-xnj∥+βnj∥xnj-(x*-ξBx*)∥+(1-βnj)×(∥Wnj(I-ξB)FrnjGrnjxnj-(I-ξB)FrnjGrnjxnj∥+∥(I-ξB)FrnjGrnjxnj-(x*-ξBx*)∥Wnj(I-ξB)FrnjGrnjxnj∥)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥≤∥xnj+1-xnj∥+βnj∥xnj-(x*-ξBx*)∥+(1-βnj)×(∥Wnj(I-ξB)FrnjGrnjxnj-(I-ξB)FrnjGrnjxnj∥+∥FrnjGrnjxnj-x*∥)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥≤∥xnj+1-xnj∥+βnj∥xnj-(x*-ξBx*)∥+(1-βnj)×(∥Wnj(I-ξB)FrnjGrnjxnj-(I-ξB)FrnjGrnjxnj∥+∥FrnjGrnjxnj-FrnjGr0x*∥+∥FrnjGr0x*-Fr0Gr0x*∥)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥≤∥xnj+1-xnj∥+βnj∥xnj-(x*-ξBx*)∥+(1-βnj)×(∥Wnj(I-ξB)FrnjGrnjxnj-(I-ξB)FrnjGrnjxnj∥+∥Grnjxnj-Gr0x*∥+∥FrnjGr0x*-Fr0Gr0x*∥)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥.
This implies that
(129)∥xnj-(x*-ξBx*)∥≤∥xnj+1-xnj∥+βnj∥xnj-(x*-ξBx*)∥+(1-βnj)(∥Wnj(I-ξB)FrnjGrnjxnj-(I-ξB)FrnjGrnjxnj∥+∥Grnjxnj-Grnjx*∥+∥Grnjx*-Gr0x*∥+∥FrnjGr0x*-Fr0Gr0x*∥)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥≤∥xnj+1-xnj∥+βnj∥xnj-(x*-ξBx*)∥+(1-βnj)(|rnj-r0|r0∥Wnj(I-ξB)FrnjGrnjxnj-(I-ξB)FrnjGrnjxnj∥+∥xnj-x*∥+|rnj-r0|r0×(∥Fr0Gr0x*∥+2∥Gr0x*∥+∥x*∥)|rnj-r0|r0)+αnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥.
Inequality (129) thus gives
(130)∥xnj-(x*-ξBx*)∥≤11-βnj∥xnj+1-xnj∥+∥Wnj(I-ξB)FrnjGrnjxnj-(I-ξB)FrnjGrnjxnj∥+∥xnj-x*∥+4∥x*∥|rnj-r0|r0+αnj1-βnj∥u-AWnj(I-ξB)FrnjGrnjxnj∥.
Thus, we obtain from (130) that
(131)limsupj→∞∥xnj-(x*-ξBx*)∥≤limsupj→∞∥xnj-x*∥,
so that Definition 10 again gives x*-ξBx*=x* and this implies that x*∈B-1(0). Hence, x*∈Ω.
Now, letε∈(0,1) and x∈H be arbitrary, then, using Lemma 4 we have
(132)∥xnj-x*-ε(x-x*)∥2≤∥xnj-x*∥2-2ε〈x-x*,xnj-x*-ε(x-x*)〉=∥xnj-x*∥2+2ε〈x-x*,x*+ε(x-x*)-xnj〉.
Since x* is the asymptotic center of {xnj}j≥1, it is easy to see (using Definition 10 and Lemma 14) that
(133)limsupj→∞∥xnj-x*∥2≤limsupj→∞∥xnj-x*-ε(x-x*)∥2≤limsupj→∞∥xnj-x*∥2+2εlimsupj→∞〈x-x*,x*+ε(x-x*)-xnj〉.
Thus,
(134)0≤limsupj→∞〈x-x*,x*+ε(x-x*)-xnj〉,
But,
(135)〈x-x*,x*-xnj〉=〈x-x*,x*+ε(x-x*)-xnj〉-ε∥x-x*∥2.
So,
(136)limsupj→∞〈x-x*,x*-xnj〉=limsupj→∞(〈x-x*,x*+ε(x-x*)-xnj〉-ε∥x-x*∥2)=limsupj→∞〈x-x*,x*+ε(x-x*)-xnj〉-ε∥x-x*∥2≥-ε∥x-x*∥2.
Thus, since ε∈(0,1) is arbitrary, we obtain from (136) that
(137)limsupj→∞〈x-x*,x*-xnj〉≥0∀x∈H.
In particular, for x=x*+u-Ax′∈H and using the fact that x*∈Ω, we obtain from (137) that
(138)0≤limsupj→∞〈u-Ax′,x*-xnj〉≤〈u-Ax′,x*-x′〉+limsupj→∞〈u-Ax′,x′-xnj〉=〈u-Ax′,x*-x′〉+limj→∞〈u-Ax′,x′-xnj〉≤limj→∞〈u-Ax′,x′-xnj〉.
This implies that
(139)limj→∞〈u-Ax′,xnj-x′〉≤0.
Hence,
(140)limsupn→∞〈u-Ax′,xn-x′〉=limj→∞〈u-Ax′,xnj-x′〉≤0.Theorem 22.
Let{xn}n≥1 be given by (74). Suppose that the conditins of Remark 16 are satisfied, then {xn}n≥1 converges strongly to x′∈Ω which is a unique solution of the variational inequality (75).Proof.
From the recursion formula (74) and Lemma 4, we have
(141)∥xn+1-x′∥2=∥FrnGrnxn+βnxn-x′αnu+[(1-βn)I-αnA]×Wn(I-ξB)FrnGrnxn+βnxn-x′∥2=∥FrnGrnxn+βnxn-x′[(1-βn)I-αnA]×(Wn(I-ξB)FrnGrnxn-x′)+βn(xn-x′)+αn(u-Ax′)∥2≤∥FrnGrnxn+βnxn-x′[(1-βn)I-αnA]×(Wn(I-ξB)FrnGrnxn-x′)+βn(xn-x′)∥2+2αn〈u-Ax′,xn+1-x′〉=∥βn(xn-x′)[(1-βn)I-αnA]×(Wn(I-ξB)FrnGrnxn-x′)+βn(xn-x′)∥2+2αn〈u-Ax′,(1-βn)×(Wn(I-ξB)FrnGrnxn-x′)-αnA(Wn(I-ξB)FrnGrnxn-x′)+αn(u-Ax′)+βn(xn-x′)〉=∥βn(xn-x′)[(1-βn)I-αnA]×(Wn(I-ξB)FrnGrnxn-x′)+βn(xn-x′)∥2+2αn2∥u-Ax′∥2+2αn(1-βn)×〈u-Ax′,Wn(I-ξB)FrnGrnxn-x′〉-2αn2〈u-Ax′,A(Wn(I-ξB)FrnGrnxn-x′)〉+2αnβn〈u-Ax′,xn-x′〉≤(1-βn-αnγ+βn)2∥xn-x′∥2+2αn2∥u-Ax′∥2+2αn(1-βn)×〈u-Ax′,Wn(I-ξB)FrnGrnxn-xn〉+2αn(1-βn)〈u-Ax′,xn-x′〉-2αn2〈u-Ax′,A(Wn(I-ξB)FrnGrnxn-x′)〉+2βnαn〈u-Ax′,xn-x′〉=[1-2αnγ]∥xn-x′∥2+αn2γ2∥xn-x′∥2+2αn2∥u-Ax′∥2+2αn(1-βn)×〈u-Ax′,Wn(I-ξB)FrnGrnxn-xn〉-2αn2〈u-Ax′,AWn(I-ξB)FrnGrnxn-x′〉+2αn〈u-Ax′,xn-x′〉≤[1-2αnγ]∥xn-x′∥2+αn(αn+∥Wn(I-ξB)FrnGrnxn-xn∥)M4+2αn〈u-Ax′,xn-x′〉,
for some real constant M4>0. Thus,
(142)∥xn+1-x′∥2≤(1-δn)∥xn-x′∥2+σn,
where δn:=2αnγ and σn:=αn(αn+∥Wn(I-ξB)FrnGrnxn-xn∥)M4+2αn〈u-Ax′,xn-x′〉. It is easy to see that limsupn→∞(σn/δn)≤0 (using (93), Lemma 21 and the fact that limn→∞αn=0). Hence, we obtain by Lemma 5 that the sequence {xn}n≥1 converges strongly to x′∈Ω which is a unique solution of (75).Remark 23.
Letf:H→H be a strict contraction, then our method of proof easily carries over to the iteration method {xn}n≥1 given by x1∈H,
(143)xn+1=αnγf(xn)+βxn+((1-βn)I-αnA)Wn(I-ξB)FrnGrnxn,
making use of the fact that f is a contraction. Thus, we obatin the following theorem.Theorem 24.
Let{xn}n≥1 be given by (143). Suppose that the conditions of Remark 16 are satisfied, then {xn}n≥1 converges strongly to x′∈Ω which is a unique solution of the variational inequality
(144)〈(A-γf)x′,y-x′〉≥0,∀y∈Ω⟺x′=PΩ(I-A+γf)x′,
which is the optimality condition for the minimization problem minx∈Ω(1/2)〈Ax,x〉-h(x), where h is a potential function for γf (i.e., h′(x)=γf(x) for all x∈H), and one recalls that Ω=Fix(T′)∩GMEP(f,Φ,Θ)∩B-1(0)∩⋂j=1∞Fix(Tj).Remark 25.
Our theorems extend and unify most of the results that have been proved for the class of nonlinear operators studied in this paper. In particular, it is easy to see that Theorem24 extends, improves, generalizes, and unifies the corresponding results of Yao et al. [32], Colao and Marino [33], and Chamnarnpan and Kuman [34]. Furthermore, the setting of our theorems corrects the problem of well definedness which might arise in the results of Chamnarnpan and Kuman [34], and our method of proof is of independent interest.



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*Source: 762831-2013-01-15.xml*