# On Variational Inclusion and Common Fixed Point Problems inq-Uniformly Smooth Banach Spaces

Authors: Yanlai Song; Huiying Hu; Luchuan Ceng
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce a general iterative algorithm for finding a common element of the common fixed-point set of an infinite family ofλi-strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in a q-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, development, and extension of the corresponding results in some references to a great extent.


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## Body

## 1. Introduction

Throughout this paper, we denote byE and E* a real Banach space and the dual space of E, respectively. Let C be a subset of E and T a mapping on C. We use F(T) to denote the set of fixed points of T. Let q>1 be a real number. The (generalized) duality mapping Jq:E→2E* is defined by
(1.1)Jq(x)={x*∈E*:〈x,x*〉=∥x∥q,∥x*∥=∥x∥q-1}
for all x∈E, where 〈·,·〉 denotes the generalized duality pairing between E and E*. In particular, J=J2 is called the normalized duality mapping and Jq(x)=∥x∥q-2J2(x) for x≠0. If E is a Hilbert space, then J=I, where I is the identity mapping. It is well known that if E is smooth, then Jq is single-valued, which is denoted by jq.The norm of a Banach spaceE is said to be Gâteaux differentiable if the limit
(1.2)limt→0∥x+ty∥-∥x∥t
exists for all x, y on the unit sphere S(E)={x∈E:∥x∥=1}. If, for each y∈S(E), the limit (1.2) is uniformly attained for x∈S(E), then the norm of E is said to be uniformly Gâteaux differentiable. The norm of E is said to be Fréchet differentiable if, for each x∈S(E), the limit (1.2) is attained uniformly for y∈S(E).LetρE:[0,1)→[0,1) be the modulus of smoothness of E defined by
(1.3)ρE(t)=sup{12(∥x+y∥+∥x-y∥)-1:x∈S(E),∥y∥≤t}.A Banach spaceE is said to be uniformly smooth if ρE(t)/t→0 as t→0. Let q>1. A Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c>0 such that ρE(t)≤ctq. It is well known that E is uniformly smooth if and only if the norm of E is uniformly Fréchet differentiable. If E is q-uniformly smooth, then q≤2 and E is uniformly smooth, and hence the norm of E is uniformly Fréchet differentiable; in particular, the norm of E is Fréchet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are Lp, where p>1. More precisely, Lp is min{p,2}-uniformly smooth for every p>1.A Banach spaceE is said to be uniformly convex if, for any ε∈(0,2], there exists δ>0 such that, for any x, y∈S(E), ∥x-y∥≥ε implies ∥(x+y)/2∥≤1-δ. It is known that a uniformly convex Banach space is reflexive and strictly convex.Recall that ifC and D are nonempty subsets of a Banach space E such that C is nonempty closed convex and D⊂C, then a mapping Q:C→D is sunny (see [1]) provided that
(1.4)Q(x+t(x-Q(x)))=Q(x)
for all x∈C and t≥0, whenever Qx+t(x-Q(x))∈C. A mapping Q:C→D is called a retraction if Qx=x for all x∈D. Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. The following proposition concerns the sunny nonexpansive retraction.Proposition 1.1 (see [1]).
LetC be a closed convex subset of a smooth Banach space E. Let D be a nonempty subset of C. Let Q:C→D be a retraction and let J be the normalized duality mapping on E. Then the following are equivalent:(a)
Q is sunny and nonexpansive,(b)
∥Qx-Qy∥2≤〈x-y,J(Qx-Qy)〉, for all x, y∈C,(c)
〈x-Qx,J(y-Qx)〉≤0, for all x∈C, y∈D.Among nonlinear mappings, the classes of nonexpansive mappings and strict pseudocontractions are two kinds of the most important nonlinear mappings. The studies on them have a very long history (see, e.g., [1–29] and the references therein). Recall that a mapping T:C→E is said to be nonexpansive, if
(1.5)∥Tx-Ty∥≤∥x-y∥forallx,y∈C.A mappingT:C→E is said to be λ-strict pseudocontractive in the terminology of Browder and Petryshyn (see [2–4]), if there exists a constant λ>0 such that
(1.6)〈Tx-Ty,jq(x-y)〉≤∥x-y∥q-λ∥(I-T)x-(I-T)y∥q,
for every x, y∈C and for some jq(x-y)∈Jq(x-y). It is clear that (1.6) is equivalent to the following:
(1.7)〈(I-T)x-(I-T)y,jq(x-y)〉≥λ∥(I-T)x-(I-T)y∥q.A mappingT:C→E is said to be L-Lipschitz if for all x,y∈C there exists a constant L>0 such that
(1.8)∥Tx-Ty∥≤L∥x-y∥forallx,y∈C.
In particular, if 0<L<1, then T is called contractive and if L=1, then T reduces to a nonexpansive mapping.A mappingT:C→E is said to be accretive if for all x, y∈C there exists jq(x-y)∈Jq(x-y) such that
(1.9)〈Tx-Ty,jq(x-y)〉≥0.For someη>0, T:C→E is said to be η-strongly accretive if for all x, y∈C, there exists jq(x-y)∈Jq(x-y) such that
(1.10)〈Tx-Ty,jq(x-y)〉≥η∥x-y∥q.For someμ>0,T:C→E is said to be μ-inverse strongly accretive if for all x, y∈C there exists jq(x-y)∈Jq(x-y) such that
(1.11)〈Tx-Ty,jq(x-y)〉≥μ∥Tx-Ty∥q.A set-valued mappingT:D(T)⊆E→2E is said to be accretive if for any x, y∈D(T), there exists j(x-y)∈J(x-y), such that for all u∈T(x) and v∈T(y)(1.12)〈u-v,j(x-y)〉≥0.A set-valued mappingT:D(T)⊆E→2E is said to be m-accretive if T is accretive and (I+ρT)(D(T))=E for every (equivalently, for some) ρ>0, where I is the identity mapping.LetM:D(M)→2E be m-accretive. The mapping JM,ρ:E→D(M) defined by
(1.13)JM,ρ(u)=(I+ρM)-1(u),∀u∈E
is called the resolvent operator associated with M, where ρ is any positive number and I is the identity mapping. It is well known that JM,ρ is single valued and nonexpansive (see [5]).In order to find the common element of the solutions set of a variational inclusion and the set of fixed points of a nonexpansive mappingS, Zhang et al. [6] introduced the following new iterative scheme in a Hilbert space H. Starting with an arbitrary point x1=x∈H, define sequences {xn} by
(1.14)yn=JM,λ(xn-λAxn),xn+1=αnx+(1-αn)Syn,
where A:H→H is an α-cocoercive mapping, M:H→2H is a maximal monotone mapping, S:H→H is a nonexpansive mapping, and {αn} is a sequence in [0,1]. Under mild conditions, they obtained a strong convergence theorem.LetC be a nonempty closed convex subset of a real reflexive, strictly convex, and q-uniformly smooth Banach space E. In this paper, we consider the general system of finding (x*,y*)∈C×C such that
(1.15)θ∈x*-y*+ρ1(Ay*+M1x*),θ∈y*-x*+ρ2(Bx*+M2y*),
where A,B:C→E, M1:D(M1)→2E and M2:D(M2)→2E are nonlinear mappings.In the case whereC=E, a uniformly convex and 2-uniformly smooth Banach space, Qin et al. [8] introduced the following scheme for finding a common element of the solution set of the variational inclusions and the fixed-point set of a λ-strict pseudocontraction. Starting with an arbitrary point x1=u∈E, define sequences {xn} by
(1.16)zn=JM2,ρ2(xn-ρ2A2xn),yn=JM1,ρ1(zn-ρ1A1zn),xn+1=αnu+βnxn+(1-αn-βn)[μSxn+(1-μ)yn],n≥1,
where A1,A2:E→E are two inverse strongly accretive operators, M1, M2:E→2E are two maximal monotone mappings, T:E→E is a λ-strict pseudocontraction, and S:E→E is defined as Sx=(1-λ/K2)x+(λ/K2)Tx, for all x∈E. Then they proved a strong convergence theorem under mild conditions.In this paper, motivated by Zhang et al. [6], Qin et al. [8], Yao et al. [9], Hao [10], Yao and Yao [11], and Takahashi and Toyoda [12], we consider a relaxed extragradient-type method for finding a common element of the solution set of a general system of variational inclusions for inverse strongly accretive mappings and the common fixed-point set of an infinite family of λi-strict pseudocontractions. Furthermore, we obtain strong convergence theorems under mild conditions. The results presented by us improve and extend the corresponding results announced by many others.


## 2. Preliminaries

In order to prove our main results, we need the following lemmas.Lemma 2.1 (see [16]).
LetC be a closed convex subset of a strictly convex Banach space E. Let T1 and T2 be two nonexpansive mappings from C into itself with F(T1)⋂‍F(T2)≠∅. Define a mapping S by
(2.1)Sx=λT1x+(1-λ)T2x,∀x∈C,
where λ is a constant in (0,1). Then S is nonexpansive and F(S)=F(T1)⋂‍F(T2).Lemma 2.2 (see [30]).
Let{αn} be a sequence of nonnegative numbers satisfying the property:
(2.2)αn+1≤(1-γn)αn+bn+γncn,n≥0,
where {γn}, {bn}, and {cn} satisfy the restrictions: (i)
limn→∞γn=0, ∑n=1∞‍γn=∞,(ii)
bn≥0,∑n=1∞‍bn<∞,(iii)
limsupn→∞cn≤0.
Then,limn→∞αn=0.Lemma 2.3 (see [31, page 63]).
Letq>1. Then the following inequality holds:
(2.3)ab≤1qaq+q-1qbq/(q-1)
for arbitrary positive real numbers a, b.Lemma 2.4 (see [17]).
LetE be a real q-uniformly smooth Banach space, then there exists a constant Cq>0 such that
(2.4)∥x+y∥q≤∥x∥q+q〈y,Jq(x)〉+Cq∥y∥q,∀x,y∈E.
In particular, if E is a real 2-uniformly smooth Banach space, then there exists a best smooth constant K>0 such that
(2.5)∥x+y∥2≤∥x∥2+2〈y,J(x)〉+2K∥y∥2,∀x,y∈E.Lemma 2.5 (see [20]).
LetC be a nonempty convex subset of a real q-uniformly smooth Banach space E and let T:C→C be a λ-strict pseudocontraction. For α∈(0,1), one defines Tαx=(1-α)x+αTx. Then, as α∈(0,μ], μ=min{1,{qλ/Cq}1/(q-1)}, Tα:C→C is nonexpansive such that F(Tα)=F(T).Lemma 2.6 (see [21]).
LetC be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space E which admits weakly sequentially continuous generalized duality mapping jq from E into E*. Let T:C→C be a nonexpansive mapping. Then, for all {xn}⊂C, if xn⇀x and xn-Txn→0, then x=Tx.Lemma 2.7 (see [21]).
LetC be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space E. Let V:C→E be a k-Lipschitzian and η-strongly accretive operator with constants k,η>0. Let 0<μ<(qη/Cqkq)1/(q-1) and τ=μ(η-Cqμq-1kq/q). Then for each t∈(0,min{1,1/τ}), the mapping S:C→E defined by S:=(I-tμV) is a contraction with a constant 1-tτ.Lemma 2.8 (see [21]).
LetC be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space E. Let QC be a sunny nonexpansive retraction from E onto C. Let V:C→E be a k-Lipschitzian and η-strongly accretive operator with constants k,η>0, f:C→E a L-Lipschitzian mapping with constant L≥0, and T:C→C a nonexpansive mapping such that F(T)≠∅. Let 0<μ<(qη/Cqkq)1/(q-1) and 0≤γL<τ, where τ=μ(η-Cqμq-1kq/q). Then {xt} defined by
(2.6)xt=QC[tγfxt+(I-tμV)Txt].
Has the following properties:(i)
{xt} is bounded for each t∈(0,min{1,1/τ}),(ii)
limt→0∥xt-Txt∥=0,(iii)
{xt} defines a continuous curve from (0,min{1,1/τ}) into C.Lemma 2.9.
LetC be a closed convex subset of a smooth Banach space E. Let D be a nonempty subset of C. Let Q:C→D be a retraction and let j, jq be the normalized duality mapping and generalized duality mapping on E, respectively. Then the following are equivalent:(a)
Q is sunny and nonexpansive,(b)
∥Qx-Qy∥2≤〈x-y,j(Qx-Qy)〉, for all x, y∈C,(c)
〈x-Qx,j(y-Qx)〉≤0, for all x∈C, y∈D,(d)
〈x-Qx,jq(y-Qx)〉≤0, for all x∈C, y∈D.Proof.
From Proposition1.1, we have a⇔b⇔c. We need only to prove c⇔d.
Indeed, ify-Qx≠0, then 〈x-Qx,j(y-Qx)〉≤0⇔〈x-Qx,jq(y-Qx)〉≤0, for all x∈C, y∈D (by the fact that jq(x)=∥x∥q-2j(x),∀x≠0).
Ify-Qx=0, then 〈x-Qx,j(y-Qx)〉=〈x-Qx,jq(y-Qx)〉=0, for all x∈C, y∈D. This completes the proof.Lemma 2.10.
LetC be a nonempty, closed, and convex subset of a q-uniformly smooth Banach space E which admits a weakly sequentially continuous generalized duality mapping jq from E into E*. Let QC be a sunny nonexpansive retraction from E onto C. Let V:C→E be a k-Lipschitzian and η-strongly accretive operator with constants k,η>0, f:C→E a L-Lipschitzian with constant L≥0, and T:C→C a nonexpansive mapping such that F(T)≠∅. Let 0<μ<(qη/Cqkq)1/(q-1) and 0≤γL<τ, where τ=μ(η-Cqμq-1kq/q). For each t∈(0,min{1,1/τ}), let {xt} be defined by (2.6), then {xt} converges strongly to x*∈F(T) as t→0, which is the unique solution of the following variational inequality:
(2.7)〈γfx*-μVx*,jq(p-x*)〉≤0,∀p∈F(T).Proof.
We first show the uniqueness of a solution of the variational inequality (2.7). Suppose both x^∈F(T) and x*∈F(T) are solutions of (2.7). It follows that
(2.8)〈γfx*-μVx*,jq(x^-x*)〉≤0,〈γfx^-μVx^,jq(x*-x^)〉≤0.
Adding up (2.8), we have
(2.9)〈(γf-μV)x^-(γf-μV)x*,jq(x*-x^)〉≤0.
On the other hand, we have that
(2.10)〈(γf-μV)x^-(γf-μV)x*,jq(x*-x^)〉=μ〈Vx*-Vx^,jq(x*-x^)〉-γ〈fx*-fx^,jq(x*-x^)〉≥μη∥x*-x^∥q-γL∥x*-x^∥q≥(μη-γL)∥x*-x^∥q≥(τ-γL)∥x*-x^∥q>0.
It is a contradiction. Therefore, x*=x^ and the uniqueness is proved. Below we use x* to denote the unique solution of (2.7).
Next, we prove thatxt→x* as t→0.
SinceE is reflexive and {xt} is bounded due to Lemma 2.8 (i), there exists a subsequence {xtn} of {xt} and some point x~∈C such that xtn⇀x~. By Lemma 2.8(ii), we have limt→0∥xtn-Txtn∥=0. Taken together with Lemma 2.6, we can get that x~∈F(T). Setting yt=tγfxt+(I-tμV)Txt, where t∈(0,min{1,1/τ}), then we can rewrite (2.6) as xt=QCyt.
We claim∥xtn-x~∥→0.
From Lemma2.9, we have
(2.11)〈yt-QCyt,jq(x~-QCyt)〉≤0.
It follows from (2.11) and Lemma 2.7 that
(2.12)∥xtn-x~∥q=〈QCytn-ytn,jq(xtn-x~)〉+〈ytn-x~,jq(xtn-x~)〉≤〈ytn-x~,jq(xtn-x~)〉=〈(I-tnμV)Txtn-(I-tnμV)x~,jq(xtn-x~)〉+tn〈γfxtn-μVx~,jq(xtn-x~)〉≤(1-tnτ)∥xtn-x~∥q+tn〈γfxtn-μVx~,jq(xtn-x~)〉.
It follows that
(2.13)∥xtn-x~∥q≤1τ〈γfxtn-μVx~,jq(xtn-x~)〉=1τ[〈γfxtn-γfx~,jq(xtn-x~)〉+〈γfx~-μVx~,jq(xtn-x~)〉]≤1τ[γL∥xtn-x~∥q+〈γfx~-μVx~,jq(xtn-x~)〉].
Therefore, we get
(2.14)∥xtn-x~∥q≤〈γfx~-μVx~,jq(xtn-x~)〉τ-γL.
Using that the duality map jq is weakly sequentially continuous from E to E* and noticing (2.14), we get that
(2.15)limn→∞∥xtn-x~∥=0.
We prove thatx~ solves the variational inequality (2.7). Since
(2.16)xt=QCyt=QCyt-yt+tγfxt+(I-tμV)Txt,
we derive that
(2.17)(μV-γf)xt=1t(QCyt-yt)-1t(I-T)xt+μ(Vxt-VTxt).
For all z∈F(T), note that
(2.18)〈(I-T)xt-(I-T)z,jq(xt-z)〉≥∥xt-z∥q-∥Txt-Tz∥∥xt-z∥q-1≥∥xt-z∥q-∥xt-z∥q=0.
It follows from Lemma 2.9 and (2.18) that
(2.19)〈(μV-γf)xt,jq(xt-z)〉=1t〈QCyt-yt,jq(xt-z)〉-1t〈(I-T)xt,jq(xt-z)〉+〈μ(Vxt-VTxt),jq(xt-z)〉=1t〈QCyt-yt,jq(xt-z)〉-1t〈(I-T)xt-(I-T)z,jq(xt-z)〉+〈μ(Vxt-VTxt),jq(xt-z)〉≤μ〈Vxt-VTxt,jq(xt-z)〉≤∥xt-Txt∥M,
where M=supn≥0{μk∥xt-z∥q-1}<∞.
Now replacingt in (2.19) with tn and letting n→∞, from (2.15) and Lemma 2.8 (ii), we obtain 〈(μV-γf)x~,jq(x~-z)〉≤0, that is, x~∈F(T) is a solution of (2.7). Hence x~=x* by uniqueness. Therefore, xtn→x* as n→∞. And consequently, xt→x* as t→0.Lemma 2.11.
LetC be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Let the mapping A:C→E be a α-inverse-strongly accretive operator. Then the following inequality holds:
(2.20)∥(I-λA)x-(I-λA)y∥q≤∥x-y∥q-λ(qα-Cqλq-1)∥Ax-Ay∥q.
In particular, if 0<λ≤(qα/Cq)1/(q-1), then I-λA is nonexpansive.Proof.
Indeed, for allx,y∈C, it follows from Lemma 2.4 that
(2.21)∥(I-λA)x-(I-λA)y∥q=∥(x-y)-λ(Ax-Ay)∥q≤∥x-y∥q-qλ〈Ax-Ay,jq(x-y)〉+Cqλq∥Ax-Ay∥q≤∥x-y∥q-qαλ∥Ax-Ay∥q+Cqλq∥Ax-Ay∥q≤∥x-y∥q-λ(qα-Cqλq-1)∥Ax-Ay∥q.
It is clear that if 0<λ≤(qα/Cq)1/(q-1), then I-λA is nonexpansive. This completes the proof.Lemma 2.12.
LetC be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Suppose M1,M2:C→2E are two m-accretive mappings and ρ1, ρ2 are two arbitrary positive constants. Let A,B:C→E be α-inverse strongly accretive and β-inverse strongly accretive, respectively. Let G:C→C be a mapping defined by
(2.22)G(x)=JM1,ρ1[JM2,ρ2(x-ρ2Bx)-ρ1AJM2,ρ2(x-ρ2Bx)],∀x∈C.
If 0<ρ1≤(qα/Cq)1/(q-1) and 0<ρ2≤(qβ/Cq)1/(q-1), then G:C→C is nonexpansive.Proof.
For allx, y∈C, by Lemma 2.11, we have
(2.23)∥Gx-Gy∥=∥JM1,ρ1[JM2,ρ2(x-ρ2Bx)-ρ1AJM2,ρ2(x-ρ2Bx)]-JM1,ρ1[JM2,ρ2(y-ρ2By)-ρ1AJM2,ρ2(y-ρ2By)]∥≤∥[JM2,ρ2(x-ρ2Bx)-ρ1AJM2,ρ2(x-ρ2Bx)]-[JM2,ρ2(y-ρ2By)-ρ1AJM2,ρ2(y-ρ2By)]∥≤∥(I-ρ1A)JM2,ρ2(x-ρ2Bx)-(I-ρ1A)JM2,ρ2(y-ρ2By)∥≤∥JM2,ρ2(x-ρ2Bx)-JM2,ρ2(y-ρ2By)∥≤∥(x-ρ2Bx)-(y-ρ2By)∥≤∥x-y∥,
which implies that G:C→C is nonexpansive. This completes the proof.Lemma 2.13.
LetC be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Suppose A,B:C→E are two inverse strongly accretive operators, M1, M2:C→2E are two m-accretive mappings, and ρ1, ρ2 are two arbitrary positive constants. Then (x*,y*)∈C×C is a solution of general system (1.15) if and only if x*=Gx*, where G is defined by Lemma 2.12.Proof.
Note that(2.24)θ∈x*-y*+ρ1(Ay*+M1x*)θ∈y*-x*+ρ2(Bx*+M2y*)⇕x*=JM1,ρ1(y*-ρ1Ay*)y*=JM2,ρ2(x*-ρ2Bx*)⇕G(x*)=JM1,ρ1[JM2,ρ2(x*-ρ2Bx*)-ρ1AJM2,ρ2(x*-ρ2Bx*)]=x*.
This completes the proof.Lemma 2.14 (see [18]).
LetE be a q-uniformly smooth Banach space and C a nonempty convex subset of E. Assume for each i≥0, Ti:C→E is a λi-strict pseudocontraction with λi∈(0,1). Assume inf{λi:i≥1}=λ>0 and F=⋂i=1∞‍F(Ti)≠∅. Let {ξi}i=1∞ be a positive sequence such that ∑i=1∞‍ξi=1, then ∑i=1∞‍ξiTi:C→E is a λ-strict pseudocontraction and F(∑i=1∞‍ξiTi)=F.Remark 2.15.
Under the assumptions of Lemma2.14, if for each i≥1 the mapping Ti:C→E is replaced by Ti:C→C, respectively, where C is a nonempty closed convex subset of E, then noticing the fact
(2.25)∑i=1∞‍ξiTix=limn→∞∑i=1n‍ξiTix=limn→∞1∑i=1nξi∑i=1n‍ξiTix∈C,
by Lemma 2.14, we deduce that ∑i=1∞‍ξiTi:C→C is a λ-strict pseudocontraction with λ=inf{λi:i≥1} and F(∑i=1∞‍ξiTi)=F.


## 3. Main Results

Theorem 3.1.
LetC be a nonempty closed convex subset of a strictly convex, and uniformly smooth Banach space E which admits a weakly sequentially continuous generalized duality mapping jq:E→E*. Let QC be a sunny nonexpansive retraction from E onto C. Assume the mappings A, B:C→E are α-inverse strongly accretive and β-inverse strongly accretive, respectively. Let M1, M2:C→2E two m-accretive mappings and ρ1, ρ2 be two arbitrary positive constants. Suppose V:C→E is k-Lipschitz and η-strongly accretive with constants k,η>0,f:C→E being L-Lipschitz with constant L≥0. Let {Si:C→C}i=0∞ be an infinite family of λi-strict pseudocontractions with {λi}⊂(0,1) and inf{λi:i≥0}=λ>0. Let 0<μ<(qη/Cqkq)1/(q-1), 0<ρ1<(qα/Cq)1/(q-1), 0<ρ2<(qβ/Cq)1/(q-1), 0≤γL<τ,  0<σ≤d, where τ=μ(η-Cqμq-1kq/q) and d=min{1,{qλ/Cq}1/(q-1)}. Assume {ξi}⊂(0,1) and ∑i=0∞‍ξi=1. Define a mapping Tx:=(1-σ)x+σ∑i=0∞‍ξiSix, for all x∈C. For arbitrarily given x0∈C and δ∈(0,1), let {xn} be the sequence generated iteratively by
(3.1)xn+1=QC[αnγfxn+γnxn+((1-γn)I-αnμV)(δTxn+(1-δ)yn)],yn=(1-βn)xn+βnkn,kn=JM1,ρ1(zn-ρ1Azn),zn=JM2,ρ2(xn-ρ2Bxn),n≥0.
Assume that {αn}, {βn}, and {γn} are three sequences in (0,1) satisfying the following conditions: (i)
∑i=0∞αn=∞,limn→∞αn=0,∑n=0∞|αn+1-αn|<∞,(ii)
0<liminfn→∞γn≤limsupn→∞γn<1,∑n=0∞|γn+1-γn|<∞,(iii)
∑n=0∞|βn+1-βn|<∞,limn→∞βn=β>0.
Suppose in addition thatF:=⋂i=0∞‍F(Si)⋂‍F(G)≠∅. Then {xn} converges strongly to some point x*∈F, which is the unique solution of the following variational inequality:
(3.2)〈γfx*-μVx*,jq(p-x*)〉≤0,∀p∈F.Proof.
We divide the proof into several steps.
Step  1. First, we show that sequences {xn} are bounded. From limn→∞αn=0 and 0<liminfn→∞γn≤limsupn→∞γn<1, there exist some a,b∈(0,1) such that {γn}⊂[a,b]. We may assume, without loss of generality, that {αn}⊂(0,(1-b)min{1,1/τ}). From Lemma 2.7, we deduce that
(3.3)∥(1-γn)I-αnμV∥≤(1-γn)-αnτ.
Taking x*∈F, it follows from Lemma 2.13 that
(3.4)x*=JM1,ρ1[JM2,ρ2(x*-ρ2Bx*)-ρ1AJM2,ρ2(x*-ρ2Bx*)].
Putting y*=JM2,ρ2(x*-ρ2Bx*), then we can deduce that x*=JM1,ρ1(y*-ρ1Ay*). By Lemma 2.11, we obtain
(3.5)∥kn-x*∥=∥JM1,ρ1(zn-ρ1Azn)-JM1,ρ1(y*-ρ1Ay*)∥≤∥(I-ρ1A)zn-(I-ρ1A)y*∥≤∥zn-y*∥=∥JM2,ρ2(xn-ρ2Bxn)-JM2,ρ2(x*-ρ2Bx*)∥≤∥(I-ρ2B)xn-(I-ρ2B)x*∥≤∥xn-x*∥.
It follows from (3.5) that
(3.6)∥yn-x*∥=∥[(1-βn)xn+βnkn]-x*∥≤(1-βn)∥xn-x*∥+βn∥kn-x*∥≤(1-βn)∥xn-x*∥+βn∥xn-x*∥≤∥xn-x*∥.
In view of Remark 2.15, let S:C→C be the mapping defined by Sx=∑i=0∞‍ξiSix for all x∈C, then we can deduce that S:C→C is a λ-strict pseudocontraction and F(S)=⋂i=0∞‍F(Si). By virtue of Lemma 2.5 and 0<σ≤d, where d=min{1,{qλ/Cq}1/(q-1)}, we can get that T:C→C is nonexpansive and F(T)=F(S)=⋂i=0∞‍F(Si). Putting ln=δTxn+(1-δ)yn, it follows that
(3.7)∥ln-x*∥=∥δTxn+(1-δ)yn-x*∥≤δ∥Txn-x*∥+(1-δ)∥yn-x*∥≤δ∥Txn-Tx*∥+(1-δ)∥yn-x*∥≤δ∥xn-x*∥+(1-δ)∥xn-x*∥=∥xn-x*∥.
It follows from (3.7) that
(3.8)∥xn+1-x*∥=∥QC[αnγfxn+γnxn+((1-γn)I-αnμV)ln]-x*∥≤∥αnγfxn+γnxn+[(1-γn)I-αnμV]ln-x*∥=∥[(1-γn)I-αnμV](ln-x*)+αn(γfxn-μVx*)+γn(xn-x*)∥≤(1-γn-αnτ)∥ln-x*∥+αn∥γfxn-μVx*∥+γn∥xn-x*∥≤(1-γn-αnτ)∥ln-x*∥+αnγ∥fxn-fx*∥+αn∥γfx*-μVx*∥+γn∥xn-x*∥≤(1-γn-αnτ)∥xn-x*∥+αnγL∥xn-x*∥+αn∥γfx*-μVx*∥+γn∥xn-x*∥≤[1-αn(τ-γL)]∥xn-x*∥+αn∥γfx*-μVx*∥≤max{∥x0-x*∥,∥γfx*-μVx*∥τ-γL}.
Hence, {xn} is bounded, so are {yn}, {kn}, {zn}, and {ln}.
Step  2. In this part, we will claim that ∥xn+1-xn∥→0, as n→∞.
We observe that(3.9)∥kn+1-kn∥=∥JM1,ρ1(zn+1-ρ1Azn+1)-JM1,ρ1(zn-ρ1Azn)∥≤∥(I-ρ1A)zn+1-(I-ρ1A)zn∥≤∥zn+1-zn∥=∥JM2,ρ2(xn+1-ρ2Bxn+1)-JM2,ρ2(xn-ρ2Bxn)∥≤∥(I-ρ2B)xn+1-(I-ρ2B)xn∥≤∥xn+1-xn∥.
It follows from (3.9) that
(3.10)∥yn+1-yn∥=∥[(1-βn+1)xn+1+βn+1kn+1]-[(1-βn)xn+βnkn]∥=∥(1-βn+1)(xn+1-xn)+βn+1(kn+1-kn)+(βn+1-βn)(kn-xn)∥≤(1-βn+1)∥xn+1-xn∥+βn+1∥kn+1-kn∥+|βn+1-βn|∥kn-xn∥≤(1-βn+1)∥xn+1-xn∥+βn+1∥xn+1-xn∥+|βn+1-βn|∥kn-xn∥≤∥xn+1-xn∥+|βn+1-βn|∥kn-xn∥.
Again from (3.1), we have
(3.11)∥xn+2-xn+1∥≤∥[αn+1γfxn+1+γn+1xn+1+((1-γn+1)I-αn+1μV)ln+1]-[αnγfxn+γnxn+((1-γn)I-αnμV)ln]∥≤αn+1γ∥fxn+1-fxn∥+γn+1∥xn+1-xn∥+∥((1-γn+1)I-αn+1μV)(ln+1-ln)∥+|αn+1-αn|γ∥fxn∥+|αn+1-αn|μ∥Vln∥+|γn+1-γn|∥ln-xn∥≤αn+1γL∥xn+1-xn∥+γn+1∥xn+1-xn∥+[(1-γn+1)-αn+1τ]∥ln+1-ln∥+|αn+1-αn|γ∥fxn∥+|αn+1-αn|μ∥Vln∥+|γn+1-γn|∥ln-xn∥.
It follows from (3.10) that
(3.12)∥ln+1-ln∥=∥δTxn+1+(1-δ)yn+1-δTxn-(1-δ)yn∥≤δ∥Txn+1-Txn∥+(1-δ)∥yn+1-yn∥≤δ∥xn+1-xn∥+(1-δ)∥yn+1-yn∥≤∥xn+1-xn∥+|βn+1-βn|∥kn-xn∥.
Substituting (3.12) into (3.11), we have
(3.13)∥xn+2-xn+1∥≤[(1-γn+1)-αn+1τ](∥xn+1-xn∥+|βn+1-βn|∥kn-xn∥)+|αn+1-αn|γ∥fxn∥+αn+1γL∥xn+1-xn∥+γn+1∥xn+1-xn∥+|αn+1-αn|μ∥Vln∥+|γn+1-γn|∥ln-xn∥≤[1-αn+1(τ-γL)]∥xn+1-xn∥+(|αn+1-αn|+|γn+1-γn|+|βn+1-βn|)M′,
where M'=supn≥0{μ∥Vln∥+γ∥fxn∥,∥ln-xn∥,∥kn-xn∥}<∞. From (i), (ii), (iii), (3.13), and Lemma 2.2, we deduce that
(3.14)limn→∞∥xn+1-xn∥=0.
We observe that
(3.15)∥ln-xn∥≤∥xn+1-xn∥+∥xn+1-ln∥=∥xn+1-xn∥+∥QC[αnγfxn+γnxn+((1-γn)I-αnμV)ln]-ln∥≤∥xn+1-xn∥+∥[αnγfxn+γnxn+((1-γn)I-αnμV)ln]-ln∥≤∥xn+1-xn∥+∥αn(γfxn-μVln)+γn(xn-ln)∥≤∥xn+1-xn∥+αn∥γfxn-μVln∥+γn∥xn-ln∥,
which implies that
(3.16)∥ln-xn∥≤11-γn(∥xn+1-xn∥+αn∥γfxn-μVln∥).
Noticing conditions (i) and (ii) and (3.14), we have
(3.17)limn→∞∥ln-xn∥=0.
Let
(3.18)Wx=δTx+(1-δ)[(1-β)x+βJM1,ρ1(I-ρ1A)JM2,ρ2(I-ρ2B)x],∀x∈C.
In view of Lemma 2.1, we see that W:C→C is nonexpansive such that
(3.19)F(W)=F(T)⋂‍F(JM1,ρ1(I-ρ1A)JM2,ρ2(I-ρ2B))=⋂i=0∞‍Si⋂‍F(G)=F.
Noticing that
(3.20)Wxn-ln=δTxn+(1-δ)[(1-β)xn+βJM1,ρ1(I-ρ1A)JM2,ρ2(I-ρ2B)xn]-δTxn-(1-δ)[(1-βn)xn+βnJM1,ρ1(I-ρ1A)JM2,ρ2(I-ρ2B)xn]=(1-δ)(β-βn)(JM1,ρ1(I-ρ1A)JM2,ρ2(I-ρ2B)xn-xn),
one has
(3.21)∥Wxn-xn∥≤∥Wxn-ln∥+∥ln-xn∥≤(1-δ)|β-βn|∥JM1,ρ1(I-ρ1A)JM2,ρ2(I-ρ2B)xn-xn∥+∥ln-xn∥.
In view of (3.17), (iii) and (3.21), we deduce that
(3.22)∥Wxn-xn∥→0asn→∞.
We define xt=QC[tγfxt+(I-tμV)Wxt], then it follows from Lemma 2.10 that {xt} converges strongly to some point x*∈F(W)=F, which is the unique solution of the variational inequality (3.2).
Step  3. We show that
(3.23)limsupn→∞〈γfx*-μVx*,jq(xn-x*)〉≤0,
where x* is the solution of the variational inequality of (3.2). To show this, we take a subsequence {xni} of {xn} such that
(3.24)limsupn→∞〈γfx*-μVx*,jq(xn-x*)〉=limi→∞〈γfx*-μVx*,jq(xni-x*)〉.
Without loss of generality, we may further assume that xni⇀z for some point z∈C due to reflexivity of the Banach space E and boundness of {xn}, it follows from (3.22) and Lemma 2.6 that z∈F(W)=F. Since the Banach space E has a weakly sequentially continuous generalized duality mapping jp:E→E*, we obtain that
(3.25)limsupn→∞〈γfx*-μVx*,jq(xn-x*)〉=limi→∞〈γfx*-μVx*,jq(xni-x*)〉=〈γfx*-μVx*,jq(z-x*)〉≤0.Step  4. We prove that limn→∞∥xn-x*∥. Setting hn=αnγfxn+γnxn+[(1-γn)I-αnμV]ln, for all n≥0. It follows from (3.1) that xn+1=QChn. In view of Lemmas 2.3, 2.7, and 2.9, we have
(3.26)∥xn+1-x*∥q=〈QChn-hn,jq(xn+1-x*)〉+〈hn-x*,jq(xn+1-x*)〉≤〈hn-x*,jq(xn+1-x*)〉=〈[(1-γn)I-αnμV](ln-x*),jq(xn+1-x*)〉+γn〈xn-x*,jq(xn+1-x*)〉+αn〈γfxn-μVx*,jq(xn+1-x*)〉≤[(1-γn)-αnτ]∥ln-x*∥∥xn+1-x*∥q-1+γn∥xn-x*∥∥xn+1-x*∥q-1+αn〈γfxn-γfx*,jq(xn+1-x*)〉+αn〈γfx*-μVx*,jq(xn+1-x*)〉≤[(1-γn)-αnτ]∥xn-x*∥∥xn+1-x*∥q-1+γn∥xn-x*∥∥xn+1-x*∥q-1+αnγL∥xn-x*∥∥xn+1-x*∥q-1+αn〈γfx*-μVx*,jq(xn+1-x*)〉≤[1-αn(τ-γL)]∥xn-x*∥∥xn+1-x*∥q-1+αn〈γfx*-μVx*,jq(xn+1-x*)〉≤[1-αn(τ-γL)][1q∥xn-x*∥q+q-1q∥xn+1-x*∥q]+αn〈γfx*-μVx*,jq(xn+1-x*)〉
which implies
(3.27)∥xn+1-x*∥q≤1-αn(τ-γL)1+(q-1)(τ-γL)αn∥xn-x*∥q+qαn1+(q-1)(τ-γL)αn×〈γfx*-μVx*,jq(xn+1-x*)〉≤[1-αn(τ-γL)]∥xn-x*∥q+qαn1+(q-1)(τ-γL)αn〈γfx*-μVx*,jq(xn+1-x*)〉.
Put an=αn(τ-γL) and cn=q〈γfx*-μVx*,jq(xn+1-x*)〉/[1+(q-1)(τ-γL)αn](τ-γL). Apply Lemma 2.2 to (3.27) to obtain xn→x*∈F as n→∞. This completes the proof.Remark 3.2.
Compared with the known results in the literature, our results are very different from those in the following aspects.(i)
The results in this paper improve and extend corresponding results in [6–13]. Especially, our result extends their results from 2-uniformly smooth Banach space or Hilbert space to more general q-uniformly smooth Banach space.(ii)
Our Theorem3.1 extends one nonexpansive mapping in [6, Theorem 2.1], one λ-strict pseudocontraction in [8, Theorem 3.1], and an infinite family of nonexpansive mappings in [10, Theorem 3.1] to an infinite family of λi-strict pseudocontractions. And our Theorem 3.1 gets a common element of the common fixed-point set of an infinite family of λi-strict pseudocontractions and the solution set of the general system of variational inclusions for two inverse strongly accretive mappings in a q-uniformly smooth Banach space.(iii)
We byf(xn) replace the u which is a fixed element in iterative scheme (1.16), where f is a L-Lipschitzian. And we also add a Lipschitz and strong accretive operator V in our scheme (3.1). In particular, whenever C=E, f=u, V=I, {Tn}n=0∞={T} and q=2, our scheme (3.1) reduces to (1.16).(iv)
It is worth noting that the Banach spaceE does not have to be uniformly convex in our Theorem 3.1. However, it is very necessary in Theorem 3.1 of Qin et al. [8] and many other literature.Corollary 3.3.
LetC be a nonempty closed convex subset of a strictly convex, and 2-reflexive E which admits a weakly sequentially continuous normalized duality mapping j:E→E*. Let QC be a sunny nonexpansive retraction from E onto C. Assume the mappings A, B:C→E are α-inverse strongly accretive and β-inverse strongly accretive, respectively. Let M1, M2:C→2E be two m-accretive operators and ρ1, ρ2 two arbitrary positive constants. Suppose V:C→E is a k-Lipschitzian and η-strongly accretive operator with constants k, η>0, f:C→E being a L-Lipschitzian with constant L≥0. Let 0<μ<η/K2k2, 0<ρ1<α/K2, 0<ρ2<β/K2 and 0≤γL<τ, where τ=μ(η-K2μk2). Let T:C→C be a nonexpansive with F=F(T)⋂‍F(G)≠∅. For arbitrarily given δ∈(0,1) and x0∈C, let {xn} be the sequence generated iteratively by
(3.28)xn+1=QC[αnγfxn+γnxn+((1-γn)I-αnμV)(δTxn+(1-δ)yn)],yn=(1-βn)xn+βnkn,kn=JM1,ρ1(zn-ρ1Azn),zn=JM2,ρ2(xn-ρ2Bxn),n≥0.
Assume that {αn}, {βn}, and {γn} are three sequences in (0,1) satisfying the following conditions: (i)
∑i=0∞αn=∞, limn→∞αn=0, ∑n=0∞|αn+1-αn|<∞,(ii)
0<liminfn→∞γn≤limsupn→∞γn<1, ∑n=0∞|γn+1-γn|<∞,(iii)
∑n=0∞|βn+1-βn|<∞, limn→∞βn=β>0.
Then{xn} converges strongly to x*∈F, which is the unique solution of the following variational inequality:
(3.29)〈γfx*-μVx*,j(p-x*)〉≤0,∀p∈F.



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