# Convergence of a Hybrid Iterative Scheme for Fixed Points of Nonexpansive Maps, Solutions of Equilibrium, and Variational Inequalities Problems

Authors: Bashir Ali
Journal: Journal of Mathematics
 (2013)

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## Abstract

LetK be a closed, convex, and nonempty subset of a real q-uniformly smooth Banach space E, which is also uniformly convex. For some κ>0, let Ti:K→Ei∈ℕ and A:K→E be family of nonexpansive maps and κ-inverse strongly accretive map, respectively. Let G:K×K→ℝ be a bifunction satisfying some conditions. Let PK be a nonexpansive projection of E onto K. For some fixed real numbers δ∈(0,1), λ∈(0,(qκ/dq)1/(q-1)), and arbitrary but fixed vectors x1,u∈E, let {xn} and {yn} be sequences generated by G(yn,η)+(1/r)〈η-yn, jq(yn-xn)〉≥0, ∀η∈K, xn+1=αnu+(1-δ)(1-αn)xn+δ∑i≥1‍σinTiPK(yn-λAyn), n≥1, where r∈(0,1) is fixed, and {αn},{σi,n}⊂(0,1) are sequences satisfying appropriate conditions. If F:=[∩i=1∞F(Ti)]∩VI(K,A)∩EP(G)≠∅, under some mild conditions, we prove that the sequences {xn} and {yn} converge strongly to some element in F.


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## Body

## 1. Introduction

LetE be a real normed space and E* its dual space. For some real number q(1<q<∞), the generalized duality mappingJq:E→2E* is defined by
(1)Jq(x)={f*∈E*:〈x,f*〉=∥x∥q,∥f*∥=∥x∥q-1},
where 〈·,·〉 denotes the pairing between elements of E and elements of E*.Forq=2, J2 usually denoted by J is called the normalised duality mapping.LetE be a real Banach space; a map A:D(A)→E is said to be accretive if for all x,y∈D(A), there exists jq(x-y)∈Jq(x-y) such that
(2)〈Ax-Ay,jq(x-y)〉≥0.For some real numberκ>0, A is called κ-inverse strongly accretive if for all  x,y∈D(A), there exists jq(x-y)∈Jq(x-y) such that
(3)〈Ax-Ay,jq(x-y)〉≥κ∥Ax-Ay∥q.
Observe that a κ-inverse strongly accretive map is 1/κ-Lipschitzian.LetK be a nonempty, closed, and convex subset of E, and let A:K→E be an accretive mapping. A variational inequality problem is, searching for x*∈K such that for some jq(v-x*)∈Jq(v-x*)(4)〈Ax*,jq(v-x*)〉≥0,∀v∈K.
Let G:K×K→ℝ be a bifunction on a closed convex nonempty subset K of a real Banach space E; an equilibrium problem is searching for x*∈K such that
(5)G(x*,v)≥0,∀v∈K.A set of solutions of the problems (4) and (5) are denoted by VI(K,A) and EP(G), respectively.LetP be a mapping of E onto K. Then, P is said to be sunny if P(Px+t(x-Px))=Px for all x∈E and t≥0. A mapping P of E into E is said to be a retraction if P2=P. If a mapping P is a retraction, then Pz=z for every z∈R(P), range of P. A subset K is said to be sunny nonexpansive retract of E if there exists a sunny nonexpansive retraction of E onto K. A retraction P is said to be orthogonal if for each x, x-P(x) is normal to K in the sense of James [1].It is well known (see [2]) that if E is a Banach space; a projection mapping is a sunny nonexpansive retraction P of E onto K. If E is uniformly smooth and there exists a nonexpansive retraction of E onto K, then there exists a nonexpansive projection of E onto K. If E is a real smooth Banach space, then P is an orthogonal retraction of E onto K if and only if P(x)∈K and 〈P(x)-x,jq(P(x)-y)〉≤0 for all y∈K. It then follows that, for x,y∈K, we have 〈P(x)-x,jq(P(x)-P(y))〉≤0 and 〈P(y)-y,jq(P(y)-P(x))〉≤0 which implies
(6)∥P(x)-P(y)∥q≤〈x-y,jq(P(x)-P(y))〉.An accretive mappingA is said to be maximal if its graph GF(A) is not contained in the graph of any other accretive map. Equivalently, A is maximal accretive if for every (v0,w0)∈E×E such that 〈w-w0,jp(v-v0)〉≥0 holds for all w∈Av,v∈K; then, w0∈Av0. A mapping T with domain D(T) and range R(T) in E is said to be demiclosed at p if whenever {xn} is a sequence in D(T) such that xn⇀x*∈D(T) and Txn→p; then, Tx*=p. The following proposition is known to hold; see, for example, [3].Proposition 1.
LetA:K→E be a κ-inverse strongly accretive map. Let M be defined by
(7)Mv={Av+NKv,v∈K,∅,v∉K,
where NKv={w∈E:〈w,jq(v-u)〉≥0, for all u∈K}; then, M is maximal accretive and u∈M-1(0) if and only if u∈VI(K,A).Recently, Maingé [4] studied the Halpern-type scheme for approximation of a common fixed point of countable infinite family of nonexpansive mappings in a real Hilbert space.The present author [3] proved a strong convergence theorem for family of nonexpansive maps and solution of variational inequality problems. Kumam and Jaiboon [5] studied a hybrid iterative method for mixed equilibrium problem and variational inequality problem in the framework of a real Hilbert space.Various numerous authors studied the problem of approximating solutions of equilibrium and fixed point problems in the framework of a real Hilbert space; see, for example, [5–18] and the references contained therein. In [19], Ceng et al. studied this problem in the framework of a uniformly smooth and uniformly convex Banach space.Takahashi and Zembayashi [20] (see also [21–23]) studied the problem of approximating solutions of equilibrium problems and fixed points of some nonlinear maps in the framework of real Banach spaces.  It is our purpose in this paper to introduce a new hybrid iterative method for approximating a common element in the intersection of the set of fixed points of countable infinite family of nonexpansive mappings, the set of solutions of variational inequality problem, and the set of solutions of equilibrium problem in Banach spaces. Our theorems extend and improve some recent important results, and our method of proof in this paper is of independent interest.


## 2. Preliminaries

LetS:={x∈E:∥x∥=1} denote a unit sphere of the real Banach space E. E is said to have a Gâteaux differentiable norm if the limit
(8)limt→0∥x+ty∥-∥x∥t
exists for each x,y∈S; E is said to have a uniformly Gâteaux differentiable norm if for each y∈S, the limit is attained uniformly for x∈S. Let E be a normed space with dimE≥2. The modulus of smoothness of E is the function ρE:[0,∞)→[0,∞) defined by
(9)ρE(τ):=sup{∥x+y∥+∥x-y∥2-1:∥x∥=1;∥y∥=τ}.The spaceE is called uniformly smooth if and only if limt→0+(ρE(t)/t)=0. For some constant q>1, E is called q-uniformly smooth if there exists a constant c>0 such that ρE(t)≤ctq, t>0.The modulus of convexity ofE is the function δE:(0,2]→[0,1] defined by
(10)δE(ϵ)=inf{1-∥x+y2∥:∥x∥=∥y∥=1,ϵ=∥x-y∥}.E is called uniformly convex if and only if δE(ϵ)>0 for all ϵ∈(0,2].A Banach spaceE is said to be strictly convex if ∥x-y∥/2<1 for x,y∈E with ∥x∥=1=∥y∥ and x≠y.It is well known that ifE is smooth then the duality mapping is singled valued, and if E has uniformly Gâteaux differentiable norm then the duality mapping is norm-to-weak* uniformly continuous on bounded subset of E. Also, every q-uniformly smooth space is uniformly smooth and has a uniformly Gâteaux differentiable norm, and every uniformly convex space is strictly convex.In the sequel, we will make use of the following results.Lemma 2 (see Petryshyn [24]).
LetE be a real normed linear space. Then, the following inequality holds:
(11)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉∀x,y∈E,j(x+y)∈J(x+y).Theorem 3 (see Goebel and Kirk [25]).
LetE be a real uniformly convex Banach space, K a closed convex subset of E, and T:K→E a nonexpansive mapping. Then, (I-T) is demiclosed at zero, where I denotes the identity map.Lemma 4 (see Suzuki [26]).
Let{xn} and {yn} be bounded sequences in a Banach space E, and let {βn} be a sequence in [0,1] with 0<liminfβn≤limsupβn<1. Suppose that xn+1=βnyn+(1-βn)xn for all integers n≥0 and limsup(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, lim∥yn-xn∥=0.Lemma 5 (see Xu [27]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(12)an+1≤(1-αn)an+αnσn+γn,n≥0,
where (i) {αn}⊂[0,1], ∑‍αn=∞; (ii) limsupσn≤0; (iii) γn≥0; (n≥0), ∑‍γn<∞. Then, an→0 as n→∞.Lemma 6 (see Xu [28]).
LetE be a real q-uniformly smooth Banach space for some q>1; then, there exists some positive constant dq such that
(13)∥x+y∥q≤∥x∥q+q〈y,jq(x)〉+dq∥y∥q
for all x,y∈E and jq(x)∈Jq(x).Lemma 7 (see Kamimura and Takahashi [29]).
LetE be a real smooth and uniformly convex Banach space, and let R>0. Then, there exists a strictly increasing, continuous, and convex function g:[0,2R]→ℝ such that g(0)=0 and g(∥x-y∥)≤∥x∥2-2〈x,jy〉+∥y∥2 for all x,y∈BR.The following conditions are required on the bifunctionG:K×K→ℝ for solving equilibrium problems with respect to G:(A1)
G(x,x)=0 for all x∈K;(A2)
G is monotone; that is, G(x,y)+G(y,x)≤0 for all x,y∈K;(A3)
for allx,y,z∈K,limsupt→0+G(tz+(1-t)x,y)≤G(x,y);(A4)
for allx∈K,G(x,·) is convex and lower semicontinuous.Lemma 8 (see Blum and Oettli [30]).
LetE be a real smooth, strictly convex, and reflexive Banach space. Let G:K×K→ℝ be a bifunction satisfying (A1)–(A4), and let x∈E, r>0. Then, there exists z∈K such that
(14)G(z,y)+1r〈y-z,j(z-x)〉≥0,∀y∈K.Lemma 9.
LetK be a closed convex nonempty subset of a real uniformly smooth and strictly convex Banach space E. Let G:K×K→ℝ be a bifunction satisfying (A1)–(A4). For r>0 and x∈E, define a map Tr:E→K by
(15)Trx={1rz∈K:G(z,y)+1r〈y-z,j(z-x)〉≥0,∀y∈K}.
Then, the following hold:(i)
Tr is single-valued;(ii)
Fix(Tr)=EP(G);(iii)
ifTr is firmly nonexpansive-type, that is, for x,y∈E,
(16)〈Trx-Try,j(Trx-Try)〉≤〈x-y,j(Trx-Try)〉,then EP(G) is closed and convex.Proof.
(i) Letz1,z2∈Tr, then
(17)G(z1,z2)+1r〈z2-z1,j(z1-z2)〉≥0,G(z2,z1)+1r〈z1-z2,j(z2-z1)〉≥0.
Adding these inequalities and using (A2), we get
(18)〈z2-z1,j(z1-z2)〉≥0,
which implies z1=z2. Consider
(ii)(19)z∈F(Tr)⟺z=Trz⟺G(z,y)+1r〈y-z,j(z-z)〉≥0,∀y∈K,⟺G(z,y)≥0,∀y∈K,⟺z∈EP(G).
(iii)EP(G) is closed and convex follows from (ii) and the fact that every firmly nonexpansive map is nonexpansive and the fixed point set of nonexpansive map is closed and convex.LetE be a real q-uniformly smooth Banach space, and for some λ>0, let I:K→K and A:K→E be the identity and κ-inverse strongly accretive mappings, respectively. Then, for the map (I-λA):K→E, we have the following estimates:
(20)∥(I-λA)x-(I-λA)y∥q=∥x-y-λ(Ax-Ay)∥q≤∥x-y∥q-qλ〈Ax-Ay,j(x-y)〉+dqλq∥Ax-Ay∥q≤∥x-y∥q-λ(qκ-dqλq-1)∥Ax-Ay∥q.
If λ is chosen such that 0≤λ≤(qκ/dq)1/(q-1), we then have
(21)∥(I-λA)x-(I-λA)y∥≤∥x-y∥,
and so (I-λA) become a nonexpansive mapping of K into E.ForLp(1<p<∞) spaces, we have the following relation: if λ∈(0,2κ/(p-1)),
(22)∥(I-λA)x-(I-λA)y∥2=∥x-y-λ(Ax-Ay)∥2≤∥x-y∥2-2λ〈Ax-Ay,j(x-y)〉+(p-1)λ2∥Ax-Ay∥2≤∥x-y∥2-λ(2κ-(p-1)λ)∥Ax-Ay∥2≤∥x-y∥2.Also ifE is a Hilbert space and we choose λ∈(0,2κ), then (I-λA) is nonexpansive.


## 3. Path Convergence Theorems

In the sequel, we assume for eacht∈(0,1) that the sequence {σi,t}i satisfies ∑i≥1σi,t=1-t and the sequences {αn},{σi,n}i⊂(0,1), satisfy ∑i≥1σi,n=1-αn.For a countable family of nonexpansive mappings{Ti} of E, we denote a set 𝒩ℐ:={i∈ℕ:Ti≠I} (I being the identity mapping on E).LetK be a nonempty closed and convex subset of a real q-uniformly smooth Banach space E and PK a nonexpansive projection of E onto K. For some real number κ>0, let A:K→E be a κ-inverse strongly accretive mapping. For some real numbers δ∈(0,1), λ∈(0,(qκ/dq)1/(q-1)), and  r>0 arbitrarily chosen but fixed and for each t∈(0,1), define a map Tt:E→E by u∈E, arbitrary and fixed
(23)G(y,η)+1r〈η-y,j(y-x)〉≥0,∀η∈K,Ttx=tu+(1-δ)(1-t)x+δ∑i≥1σi,tTiPK(y-λAy),∀x∈E.Then,Tt is a strict contraction on E.Forx1,x2∈E, y1=Trx1, y2=Trx2, we have
(24)∥Ttx1-Ttx2∥≤(1-t)(1-δ)∥x1-x2∥+δ∑i≥1‍σi,t∥TiPK(y1-λAy1)+δ∑i≥1‍σi,t-TiPK(y2-λAy2)∥=(1-t)(1-δ)∥x1-x2∥+δ∑i≥1‍σi,t∥TiPK(I-λA)Trx1+δ∑i≥1‍σi,t-TiPK(I-λA)Trx2∥≤(1-t)∥x1-x2∥.Thus, for eacht∈(0,1), there exists a unique zt∈E such that
(25)G(y,η)+1r〈η-y,j(y-zt)〉≥0,∀η∈K,zt=tu+(1-δ)(1-t)zt+δ∑i≥1‍σi,tTiPK(y-λAy).Lemma 10.
LetE be a real q-uniformly smooth Banach space which is also uniformly convex. Let K be a closed, convex, and nonempty subset of E. For t∈(0,1), let {zt} be a net satisfying (25), and assume that ℱ:=⋂i=1∞F(TiPK(I-λA)Tr)≠∅. Then, {zt} is bounded and admits at most one accumulation point in ℱ as t→0.Proof.
Letx*∈ℱ. Then, using (25), we have
(26)∥zt-x*∥2=〈t(u-x*)+(1-t)(1-δ)(zt-x*)+δ∑i≥1‍σi,t(TiPK(Trzt-λATrzt)-x*),j(zt-x*)〉≤t〈u-x*,j(zt-x*)〉+(1-t)(1-δ)∥zt-x*∥2+δ∑i≥1‍σi,t∥zt-x*∥2=t〈u-x*,j(zt-x*)〉+(1-t)∥zt-x*∥2,
which implies
(27)∥zt-x*∥≤∥u-x*∥.
Thus, {zt} is bounded.
Now, assume for the sake of contradiction thatx′ and x* are two distinct accumulation points of {zt} in ℱ; then, there exists a subnet {zts} of {zt} such that zts→x′ as s→∞, and so we have the following estimates:
(28)∥zts-x*∥2=〈+δ∑i≥1‍σi,t(TiPK(zts-λAzts)-x*),j(zts-x*)ts(u-x*)+(1-ts)(1-δ)(zts-x*)+δ∑i≥1‍σi,t(TiPK(zts-λAzts)-x*),j(zts-x*)〉≤ts〈u-x*,j(zts-x*)〉+(1-ts)∥zts-x*∥2
so that
(29)∥zts-x*∥2≤〈u-x*,j(zts-x*)〉,
and since zts→x′ as s→∞, we get from (29)
(30)∥x′-x*∥2≤〈u-x*,j(x′-x*)〉.
Applying similar argument tox* as an accumulation point of {zt} in ℱ, we also get
(31)∥x*-x′∥2≤〈u-x′,j(x*-x′)〉.
Adding these last two inequalities, we get(32)2∥x*-x′∥2≤∥x*-x′∥2,
a contradiction, and thus x′=x*. This completes the proof.Lemma 11.
LetE be a real q-uniformly smooth Banach space which is also uniformly convex. Let K be a closed, convex, and nonempty subset of E. Let r∈(0,1) be fixed and {tn}⊂(0,1) such that limn→∞tn=0 and limn→∞(tn/σi,n)=0 for all i∈𝒩ℐ. Let {ztn} be a sequence satisfying (25), and let ℱ:=⋂i=1∞F(TiPK(I-λA)Tr)≠∅. Then, limn→∞∥ztn-(TiPK(I-λA))Trztn∥=0, for all i∈ℕ.Proof.
Fori∈ℕ and x*∈ℱ, we have the following estimates (using Lemma 7 establishing the existence of g):
(33)g(∥(TiPK(I-λA))Trztn-ztn∥)=g(∥[x*-(TiPK(I-λA))Trztn]-[x*-ztn]∥)≤∥x*-(TiPK(I-λA))Trztn∥2-2〈x*-(TiPK(I-λA))Trztn,j(x*-ztn)〉+∥x*-ztn∥2≤∥x*-ztn∥2-2×〈x*-ztn+ztn-(TiPK(I-λA))Trztn,j(x*-ztn)〉+∥x*-ztn∥2=2〈ztn-(TiPK(I-λA))Trztn,j(ztn-x*)〉.
Using (25), we have
(34)〈ztn-x*,j(ztn-x*)〉=tn〈u-x*,j(ztn-x*)〉+(1-tn)(1-δ)〈ztn-x*,j(ztn-x*)〉+δ∑i≥1‍σi,n×〈(TiPK(I-λA))Trztn-ztn+ztn-x*,j(ztn-x*)〉=tn〈u-x*,j(ztn-x*)〉+δ∑i≥1‍σi,n〈(TiPK(I-λA))Trztn&=∑i≥1‍σi,n-ztn,j(ztn-x*)〉+(1-tn)〈ztn-x*,j(ztn-x*)〉
which implies
(35)δ∑i≥1‍σi,n〈ztn-(TiPK(I-λA))Trztn,j(ztn-x*)〉=tn〈u-ztn,j(ztn-x*)〉.
Using this and (33), we get
(36)δ2∑i≥1‍σi,ng(∥(TiPK(I-λA))Trztn-ztn∥)≤tn〈u-ztn,j(ztn-x*)〉.
Thus,
(37)δ2g(∥(TiPK(I-λA))Trztn-ztn∥)≤tnσi,n〈u-ztn,j(ztn-x*)〉,∀i∈ℕ.
Since {ztn} is bounded and tn/σi,n→0 as n→∞, we have limn→∞g(∥(TiPK(I-λA))Trztn-ztn∥)=0 for all i∈ℕ. By property of g,  limn→∞∥(TiPK(I-λA))Trztn-ztn∥=0 for all i∈ℕ. This completes the proof.Theorem 12.
LetE be a real q-uniformly smooth Banach space which is also uniformly convex. Let K be a closed, convex, and nonempty subset of E. Let r∈(0,1) be fixed and {tn}⊂(0,1) such that limn→∞tn=0 and limn→∞(tn/σi,n)=0 for all i∈𝒩ℐ. Let {ztn} be a sequence satisfying (25), and let ℱ:=⋂i=1∞F(TiPK(I-λA)Tr)≠∅. If the duality mapping j of E is weakly sequentially continuous, then {ztn} converges strongly to an element in ℱ.Proof.
Since{ztn} is bounded, there exists a subsequence say {ztnk} of {ztn} that converges weakly to some point z∈K. Using demiclosedness property of [I-(TiPK(I-λA))Tr] at 0 for i∈ℕ, and the fact that limk→∞∥(TiPK(I-λA))Trztnk-ztnk∥=0, we get that z is a point in ℱ. We also observe from (33) that
(38)∥ztnk-z∥2=〈tnk(u-z)+(1-tnk)(1-δ)(ztnk-z)+δ∑i≥1‍σi,nk((TiPK(I-λA))Trztnk-z),j(ztnk-z)+δ∑i≥1‍σi,nk((TiPK(I-λA))Trztnk-z),j(ztnk-z)〉≤tnk〈u-z,j(ztnk-z)〉+(1-tnk)(1-δ)∥ztnk-z∥2+δ∑i≥1‍σi,nk∥ztnk-z∥2=tnk〈u-z,j(ztnk-z)〉+(1-tnk)∥ztnk-z∥2
which implies
(39)∥ztnk-z∥2≤〈u-z,j(ztnk-z)〉.
Sincej admits weak sequential continuity, the last inequality implies that the subsequence {ztnk} converges strongly to z, and since {ztn} admits unique accumulation point in ℱ, then it converges strongly to z. This completes the proof.The following corollary follows from Theorem12.Corollary 13.
LetE be a real Lp space, (1<p<∞). Let K, {tn}, ℱ, and {ztn} be as in Theorem 12. Then, {ztn} converges strongly to an element of ℱ.Theorem 14.
LetE be a real q-uniformly smooth Banach space which is also uniformly convex. Let K be a closed, convex, and nonempty subset of E. Let r∈(0,1) be fixed and {tn} a sequence in (0,1) such that limn→∞tn=0 and limn→∞(tn/σi,n)=0 for all i∈𝒩ℐ. Let {ztn} be a sequence satisfying (25), and let ℱ:=⋂i=1∞F(TiPK(I-λA)Tr)≠∅. If for at least one i in ℕ, TiPK(I-λA)Tr is demicompact, then {ztn} converges strongly to an element of ℱ.Proof.
For some fixedj0∈ℕ, let Tj0PK(I-λA)Tr be demicompact. Since limn→∞∥Tj0PK(I-λA)Trztn-ztn∥=0, there exists a subsequence say {ztnk} of {ztn} that converges strongly to some point z∈E. By continuity of TiPK(I-λA)Tr for all i∈ℕ, we have that z∈ℱ. But the sequence {ztn} admits unique accumulation point in ℱ; so, it converges strongly to z.The following corollaries follow from Theorem14.Corollary 15.
LetE be a real Lp space, (1<p<∞). Let K, {tn}, ℱ, and {ztn} be as in Theorem 14. If for at least one i∈ℕ, the map TiPK(I-λA)Tr is demicompact, then {ztn} converges strongly to an element of ℱ.Corollary 16.
LetK be a closed, convex, and nonempty subset of a real Hilbert space H. Let {tn}, ℱ, and {ztn} be as in Theorem 14. Then, {ztn} converges strongly to an element of ℱ.


## 4. Iterative Convergence Theorem

We now state and prove the following theorem.Theorem 17.
LetE be a real q-uniformly smooth Banach space which is also uniformly convex. Let K be a closed, convex, and nonempty subset of E. For some κ>0, let Ti:K→Ei∈ℕ and A:K→E be a family of nonexpansive maps and a κ-inverse strongly accretive map, respectively. Let G:K×K→ℝ be a bifunction satisfying (A1)–(A4). Let PK be a nonexpansive projection of E onto K. For some fixed real numbers r,δ∈(0,1) and λ∈(0,(qκ/dq)1/(q-1)), define a sequence {xn} iteratively by x1,u∈E and n∈ℕ as
(40)G(yn,η)+1r〈η-yn,j(yn-xn)〉≥0,∀η∈K,xn+1=αnu+(1-δ)(1-αn)xn+δ∑i≥1‍σi,nTiPK&xn+1×(yn-λAyn),
where {αn},{σi,n}⊂(0,1) are sequences satisfying the following conditions:(i)
limn→∞αn=0,(ii)
∑n=1∞αn=∞,(iii)
limn→∞∑i≥1‍|σi,n+1-σi,n|=0.
LetF:=[⋂i=1∞F(Ti)]⋂‍EP(G)⋂‍VI(K,A)≠∅. If either the duality map j of E is weakly sequentially continuous or for at least one i∈ℕ, TiPK(I-λA)Tr is demicompact, then {xn} converges strongly to some element in F.Proof.
Letx*∈ℱ then, we claim that ∥xn-x*∥≤max{∥u-x*∥,∥x1-x*∥} for all n≥1. It is clear that the claim is true for n=1. Assume that it is true for n=k for some k≥1,k∈ℕ. Then,
(41)∥xk+1-x*∥≤αk∥u-x*∥+(1-αk)(1-δ)∥xk-x*∥+δ∑i≥1‍σi,k∥TiPK(I-λA)Trxk-TiPK(I-λA)Trx*∥≤αk∥u-x*∥+(1-αk)∥xk-x*∥≤max{∥u-x*∥,∥x1-x*∥}.
Hence, the result, and so {xn} is bounded. Furthermore, {yn}, {TiPK(yn-λAyn)}, and {Ayn} are each bounded.
We now show thatlimn→∞∥xn+1-xn∥=0. Note that yn=Trxn, yn+1=Trxn+1, so that
(42)∥yn+1-yn∥=∥Trxn+1-Trxn∥≤∥xn+1-xn∥.
Define two sequences{βn} and {wn} by βn:=(1-δ)αn+δ and wn:=(xn+1-xn+βnxn)/βn. Then,
(43)wn=αnu+δ∑i≥1σi,nTiPK(yn-λAyn)βn.
Observe that {wn} is bounded and that
(44)∥wn+1-wn∥-∥xn+1-xn∥≤|αn+1βn+1-αnβn|∥u∥+|δ(1-αn+1)βn+1-1|∥xn+1-xn∥+δMβn+1∑i≥1‍|σi,n+1-σi,n|+δMβn+1βn|βn-βn+1|,
for some positive real number M. This implies
(45)limsupn→∞(∥wn+1-wn∥-∥xn+1-xn∥)≤0,
and by Lemma 4, limn→∞||wn-xn||=0. Hence,
(46)∥xn+1-xn∥=βn∥wn-xn∥→0asn→∞.
From (42) and (46), we have
(47)∥yn+1-yn∥→0asn→∞.
From (40), we have xn+1-xn=αn(u-xn)+δ∑i≥1‍σi,n[TiPK(yn-λAyn)-xn] which implies
(48)δ∥∑i≥1‍σi,n[TiPK(yn-λAyn)-xn]∥≤∥xn+1-xn∥+αn∥u-xn∥,
and thus limn→∞∥∑i≥1‍σi,n[TiPK(yn-λAyn)-xn]∥=0.  Let {tn} be a real sequence in (0,1) satisfying the following conditions:
(49)limn→∞tn=0,∑i≥1‍σi,n=(1-tn),limn→∞∥∑i≥1σi,n[TiPK(yn-λAyn)-xn]∥tn=0.
Letztn∈K be the unique fixed point satisfying (25) for each n∈ℕ, and let ztn→z∈ℱ as n→∞. Using (25) and Lemma 2, we have the following estimates:
(50)∥ztn-xn∥2≤∥(1-δ)(1-tn)(ztn-xn)∑i≥1‍+δ∑i≥1‍σi,n(TiPK(I-λA)Trztn-TiPK(I-λA)Trxn∑i≥1‍+TiPK(I-λA)Trxn-xn)∥2+2tn〈u-xn,j(ztn-xn)〉≤[(1-δ)(1-tn)∥ztn-xn∥+δ(1-tn)∥ztn-xn∥∑i≥1‍+∥δ∑i≥1‍σi,n[TiPK(I-λA)Trxn-xn]∥]2+2tn〈u-xn,j(ztn-xn)〉=[(1-tn)∥ztn-xn∥∑i≥1‍+∥δ∑i≥1‍σi,n[TiPK(I-λA)Trxn-xn]∥]2+2tn〈u-xn,j(ztn-xn)〉.
This implies(51)〈u-ztn,j(xn-ztn)〉≤tn2∥ztn-xn∥2+(1-tn)∥ztn-xn∥×(δ∥∑i≥1σi,n[TiPK(I-λA)Trxn-xn]∥tn)+δ2∥∑i≥1σi,n[TiPK(I-λA)Trxn-xn]∥22tn,
and, hence,
(52)limsupn→∞〈u-ztn,j(xn-ztn)〉≤0.
Moreover,
(53)〈u-ztn,j(xn-ztn)〉=〈u-z,j(xn-z)〉+〈u-z,j(xn-ztn)-j(xn-z)〉+〈z-ztn,j(xn-ztn)〉,
and since j is norm-to-weak* uniformly continuous on bounded sets, we have
(54)limsupn→∞〈u-z,j(xn-z)〉≤0.
From the recursion formula (40) and Lemma 2, we have the following:
(55)∥xn+1-z∥2=∥αn(u-z)+(1-αn)(1-δ)(xn-z)∑i≥1‍+δ∑i≥1‍σi,n[TiPK(I-λA)Trxn-TiPK(I-λA)Trz]∥2≤∥(1-αn)(1-δ)(xn-z)∑i≥1‍+δ∑i≥1‍σi,n[TiPK(I-λA)Trxn-TiPK(I-λA)Trz]∥2+2αn〈u-z,j(xn+1-z)〉≤(1-αn)∥xn-z∥2+2αn〈u-z,j(xn+1-z)〉,
and by Lemma 5, we get that {xn} converges strongly to z∈ℱ.
To complete the proof, we show thatz∈EP(G)∩VI(K,A).
We start by showing thatz∈EP(G).
Letx*∈F; then,
(56)∥yn-x*∥2=∥Trxn-Trx*∥2≤〈Trxn-Trx*,j(xn-x*)〉=〈yn-x*,j(xn-x*)〉≤12[∥xn-x*∥2+∥yn-x*∥2-g(∥xn-yn∥)];
thus,
(57)∥yn-x*∥2≤∥xn-x*∥2-g(∥xn-yn∥).
Using (40) and (57), we have
(58)∥xn+1-x*∥2=∥αn(u-x*)+(1-αn)(1-δ)(xn-x*)∑i≥1‍+δ∑i≥1‍σi,n[TiPK(I-λA)yn-TiPK(I-λA)x*]∥2≤αn∥u-x*∥2+(1-αn)(1-δ)∥xn-x*∥2+δ(1-αn)∥yn-x*∥2≤αn∥u-x*∥2+(1-αn)(1-δ)∥xn-x*∥2+δ(1-αn)[∥xn-x*∥2-g(∥xn-yn∥)].
This implies
(59)δ(1-αn)g(∥xn-yn∥)≤αn∥u-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2≤αn∥u-x*∥2+∥xn-xn+1∥[∥xn-x*∥+∥xn+1-x*∥],
and thus limn→∞δ(1-αn)g(∥xn-yn∥)=0. Using property of g, we get
(60)limn→∞∥xn-yn∥=0.
From (60), we have yn→z and j(xn-yn)→0 as n→∞. Since yn=Trxn, we have
(61)G(yn,η)+1r〈η-yn,j(yn-xn)〉≥0,∀η∈K,
It follows from (A2) that
(62)〈η-yn,j(yn-xn)r〉≥G(η,yn),
and so using (A4), we have G(η,z)≤0 for all η∈K. For real number t, 0<t≤1, and η∈K, let ηt=tη+(1-t)z. Clearly, ηt∈K. So, using (A1) and (A4), we have
(63)0=G(ηt,ηt)≤tG(ηt,η)+(1-t)G(ηt,z)≤tG(ηt,η).
This impliesG(ηt,η)≥0, and using this and (A3), we have that G(z,η)≥0 for all η∈K; hence, z∈EP(G).
Next, we show thatz∈VI(K,B).
Letx*∈F and bn:=PK(yn-λAyn); then,
(64)∥bn-x*∥q=∥PK(yn-λAyn)-PK(x*-λAx*)∥q≤∥yn-x*-λ(Ayn-Ax*)∥q≤∥yn-x*∥q+λ(dqλq-1-qκ)∥Ayn-Ax*∥q.
Using recursion formula (40), we have the following estimates:
(65)∥xn+1-x*∥q=∥αn(u-x*)+(1-αn)(1-δ)(xn-x*)∑i≥1‍+δ∑i≥1‍σi,n[Tibn-x*]∥q≤αn∥u-x*∥q+(1-αn)(1-δ)∥xn-x*∥q+(1-αn)δ∥bn-x*∥q≤αn∥u-x*∥q+(1-αn)(1-δ)∥xn-x*∥q+(1-αn)δ×[∥xn-x*∥q+λ(dqλq-1-qκ)∥Ayn-Ax*∥q]≤αn∥u-x*∥q+∥xn-x*∥q+(1-αn)δλ(dqλq-1-qκ)×∥Ayn-Ax*∥q,
which implies, by Mean Value Theorem, that
(66)-(1-αn)δλ(dqλq-1-qκ)∥Ayn-Ax*∥q≤αn∥u-x*∥q+∥xn-x*∥q-∥xn+1-x*∥q≤αn∥u-x*∥q+qσnq-1|∥xn-x*∥-∥xn+1-x*∥|≤αn∥u-x*∥q+qσnq-1∥xn-xn+1∥,
where σn is some nonnegative real number between ∥xn-x*∥ and ∥xn+1-x*∥ for each n. Since {xn} is bounded, αn→0, and ∥xn-xn+1∥→0 as n→∞, we have ∥Ayn-Ax*∥→0 as n→∞.
We also have the following:(67)∥bn-x*∥2=∥PK(yn-λAyn)-PK(x*-λAx*)∥2≤〈yn-λAyn-(x*-λAx*),j(bn-x*)〉=〈yn-x*,j(bn-x*)〉-λ〈Ayn-Ax*,j(bn-x*)〉≤12[∥bn-x*∥2+∥yn-x*∥2-g(∥bn-yn∥)]-λ〈Ayn-Ax*,j(bn-x*)〉
so that
(68)∥bn-x*∥2≤∥xn-x*∥2-g(∥bn-yn∥)-2λ〈Ayn-Ax*,j(bn-x*)〉.
We then have(69)∥xn+1-x*∥2=∥αn(u-x*)+(1-αn)(1-δ)(xn-x*)∑i≥1‍+δ∑i≥1‍σi,n[Tibn-x*]∥2≤αn∥u-x*∥2+(1-αn)(1-δ)∥xn-x*∥2+(1-αn)δ∥bn-x*∥2≤αn∥u-x*∥2+(1-αn)(1-δ)∥xn-x*∥2+(1-αn)δ[∥xn-x*∥2-g(∥bn-xn∥)∥xn-x*∥2-2λ〈Ayn-Ax*,j(bn-x*)〉]=αn∥u-x*∥2+(1-αn)∥xn-x*∥2-(1-αn)×δg(∥bn-yn∥)-2δλ(1-αn)〈Ayn-Ax*,j(bn-x*)〉,
and so
(70)(1-αn)δg(∥bn-yn∥)≤αn∥u-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2-2δλ(1-αn)〈Ayn-Ax*,j(bn-x*)〉.
Asαn→0,∥xn-xn+1∥→0 and ∥Axn-Ax*∥→0 as n→∞, we get
(71)limn→∞(1-αn)δg(∥bn-yn∥)=0,
which implies
(72)limn→∞∥bn-yn∥=0.
Let(73)Mv={Av+NKv,v∈K,∅,v∉K.
Then,M is maximal accretive. Let GF(M) denote the graph of M.
Let(v,w)∈GF(M). Since w-Av∈NKv and bn∈K, we have 〈w-Av,jq(v-bn)〉≥0 by definition of NKv. Also, as bn=PK(yn-λAyn) (using property of the projection PK), we have
(74)〈bn-(yn-λAyn),jq(v-bn)〉≥0
and, hence,
(75)〈bn-ynλ+Ayn,jq(v-bn)〉≥0.
Using this, we obtain the following estimates:(76)〈w,jq(v-bn)〉≥〈Av,jq(v-bn)〉≥〈Av,jq(v-bn)〉-〈bn-ynλ+Ayn,jq(v-bn)〉=〈Av-bn-ynλ-Ayn,jq(v-bn)〉=〈Av-Abn,jq(v-bn)〉+〈Abn-Ayn,jq(v-bn)〉-〈bn-ynλ,jq(v-bn)〉≥〈Abn-Ayn,jq(v-bn)〉-〈bn-ynλ,jq(v-bn)〉,
which implies 〈w,jq(v-z)〉≥0 (letting n→∞).
SinceM is maximal accretive, we obtained that z∈M-1(0), and, hence, z∈VI(K,A). This completes the proof.The following corollaries follow from Theorem17.Corollary 18.
LetE=Lp  space  (1<p<∞). Let K, PK, δ, A, r, and Ti, i=1,2,… be as in Theorem 17. Let λ∈(0,(2κ/(p-1))), and define sequences {xn} and {yn} by (40). Then, {xn} and {yn} converge strongly to some element in F.Corollary 19.
LetE=Lp  space  (1<p<∞). Let K, PK, δ, A, r, and Ti, i=1,2,… be as in Theorem 17. Let λ∈(0,(2κ/(p-1))), and define sequences {xn} and {yn} by (40). If for at least one i in ℕ,TiPK(I-λA)Tr is demicompact, then the sequences {xn} and {yn} both converge strongly to some element in F.Corollary 20.
LetE=H be a real Hilbert space. Let K, PK, δ, A, r, and Ti, i=1,2,… be as in Theorem 17. Let λ∈(0,2κ), and define sequences {xn} and {yn} by (40). Then, the sequences {xn} and {yn} both converge strongly to some element in F.Remark 21.
Prototypes of the sequences{αn} and {σi,n} in our theorems are the following:
(77)αn:=1n+1,σi,n:=n2i(n+1),∀i∈ℕ.



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*Source: 370143-2013-03-24.xml*