# Two General Algorithms for Computing Fixed Points of Nonexpansive Mappings in Banach Spaces

Authors: Shuang Wang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Recently, Yao et al. (2011) introduced two algorithms for solving a system of nonlinear variational inequalities. In this paper, we consider two general algorithms and obtain the extension results for computing fixed points of nonexpansive mappings in Banach spaces. Moreover, the fixed points solve the same system of nonlinear variational inequalities.


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## Body

## 1. Introduction

LetX be a real Banach space and let C be a nonempty closed convex subset of X. Recall that a mapping T:C→C is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥,  for all x,y∈C. We denote by Fix(T) the set of fixed points of T.Recently, Yao et al. [1] considered the following algorithms:(1.1)xt=ΠC(I-tF)ΠC(I-λA)ΠC(I-μB)xt,
and for an arbitrary point x0∈C,(1.2)xn+1=βnxn+(1-βn)ΠC(I-αnF)ΠC(I-λA)ΠC(I-μB)xn,n≥0,
where ΠC:X→C is a sunny nonexpansive retraction, F:C→X is a strongly positive bounded linear operator and A,B:C→X are α-inverse-strongly accretive and β-inverse-strongly accretive operators, respectively. They proved that the {xt} defined by (1.1) and {xn} defined by (1.2) converge strongly to a unique solution x¯ of the variational inequality 〈F(x¯),j(x¯-z)〉≤0. Furthermore, they proved that the above algorithms converge strongly to some solutions of a system of nonlinear inequalities, which involves finding (x*,y*)∈C×C such that(1.3)〈λAy*+x*-y*,j(x-x*)〉≥0,∀x∈C,〈μBx*+y*-x*,j(x-y*)〉≥0,∀x∈C.
For related works, please see [2–5] and the references therein.In this paper, we introduce two general algorithms (3.3) and (3.22) (defined below) and prove that the proposed algorithms strongly converge to x*∈Fix(T) which solves the variational inequality 〈F¯x*,j(x*-u)〉≤0,u∈Fix(T), where F¯:C→X is a β-Lipschitzian and η-strongly accretive operator. It is worth pointing out that our proofs contain some new techniques.


## 2. Preliminaries

LetX be a real Banach space with norm ∥·∥ and let X* be its dual space. The value of f∈X* and x∈X will be denoted by 〈x,f〉. For the sequence {xn} in X, we write xn⇀x to indicate that the sequence {xn} converges weakly to x. xn→x means that {xn} converges strongly to x.Letη>0, a mapping F¯ of C into X is said to be η-strongly accretive if there exists j(x-y)∈J(x-y) such that(2.1)〈F¯x-F¯y,j(x-y)〉≥η‖x-y‖2,
for all x,y∈C. A mapping F¯ from C into X is said to be β-Lipschitzian if, for β>0,(2.2)‖F¯x-F¯y‖≤β‖x-y‖,
for all x,y∈C. From the definition of F (see [1]), we note that a strongly positive bounded linear operator F is a ∥F∥-Lipschitzian and γ¯-strongly accretive operator.LetU={x∈X:∥x∥=1}. A Banach space X is said to be uniformly convex if for each ϵ∈(0,2], there exists δ>0 such that for any x,y∈U,(2.3)‖x-y‖≥ϵimplies‖x+y2‖≤1-δ.
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space X is said to be smooth if the limit(2.4)lim⁡t→0‖x+ty‖-‖x‖t,
exists for all x,y∈U. It is said to be uniformly smooth if the limit (2.4) is attained uniformly for x,y∈U. Also, we define a function ρ:[0,∞)→[0,∞) called the modulus of smoothness of X as follows:(2.5)ρ(τ)=sup⁡{12(‖x+y‖+‖x-y‖)-1:x,y∈X,‖x‖=1,‖y‖=τ}.
It is known that X is uniformly smooth if and only if limτ→0ρ(τ)/τ=0. Let q be a fixed real number with 1<q≤2. Then a Banach space X is said to be q-uniformly smooth if there exists a constant c>0 such that ρ(τ)≤cτq for all τ>0.In order to prove our main results, we need the following lemmas.Lemma 2.1 (see [6]).
Letq be a given real number with 1<q≤2 and let X be a q-uniformly smooth Banach space. Then
(2.6)‖x+y‖q≤‖x‖q+q〈y,Jq(x)〉+2‖Ky‖q,
for all x,y∈X, where K is the q-uniformly smooth constant of X and Jq is the generalized duality mapping from X into 2X* defined by
(2.7)Jq(x)={f∈X*:〈x,f〉=‖x‖q,‖f‖=‖x‖q-1},
for all x∈X.Lemma 2.2 (see [7]).
LetC be a closed convex subset of a smooth Banach space X, let D be a nonempty subset of C and Π be a retraction from C onto D. Then Π is sunny and nonexpansive if and only if
(2.8)〈u-Π(u),j(y-Π(u))〉≤0,
for all u∈C and y∈D.Lemma 2.3 (see [8]).
LetC be a nonempty bounded closed convex subset of a uniformly convex Banach space X and let T be a nonexpansive mapping of C into itself. If {xn} is a sequence of C such that xn⇀x and xn-Txn→0, then x is a fixed point of T.Lemma 2.4 (see [9, 10]).
Let{sn} be a sequence of nonnegative real numbers satisfying
(2.9)sn+1≤(1-λn)sn+λnδn+γn,n≥0,
where {λn}, {δn} and {γn} satisfy the following conditions:  (i)   {λn}⊂[0,1] and ∑n=0∞λn=∞,  (ii)  limsupn→∞δn≤0 or ∑n=0∞λnδn<∞, and  (iii)  γn≥0(n≥0), ∑n=0∞γn<∞. Then limn→∞sn=0.Lemma 2.5 (see [11]).
Let{xn} and {zn} be bounded sequences in Banach space E and {γn} be a sequence in [0,1] which satisfies the following condition:
(2.10)0<lim⁡inf⁡n→∞γn≤limsup⁡n→∞γn<1.
Suppose that xn+1=γnxn+(1-γn)zn,n≥0, and limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0. Then limn→∞∥zn-xn∥=0.In addition, we need the following extension of Lemma 2.5 in Wang and Hu [2] in a 2-uniformly smooth Banach space.Lemma 2.6.
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let F¯:C→X be a β-Lipschitzian and η-strongly accretive operator with 0<η≤2βK and 0<t<η/2β2K2. Then S=(I-tF¯):C→X is a contraction with contraction coefficient τt=1-2t(η-tβ2K2).Proof.
By Lemma2.1, we have
(2.11)‖Sx-Sy‖2=‖(x-y)-t(F¯x-F¯y)‖2=‖x-y‖2-2t〈F¯x-F¯y,j(x-y)〉+2t2K2‖F¯x-F¯y‖2≤‖x-y‖2-2tη‖x-y‖2+2t2β2K2‖x-y‖2=[1-2t(η-tβ2K2)]‖x-y‖2,
for all x,y∈C. From 0<η≤2βK and 0<t<η/2β2K2, we have
(2.12)‖Sx-Sy‖≤τt‖x-y‖,
where τt=1-2t(η-tβ2K2)∈(0,1). Hence S is a contraction with contraction coefficient τt.


## 3. Main Results

LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let T:C→C be a nonexpansive mapping with Fix(T)≠∅. Let F¯:C→X be a β-Lipschitzian and η-strongly accretive operator with 0<η≤2βK. Let t∈(0,η/2β2K2) and τt=1-2t(η-tβ2K2), consider a mapping St on C defined by(3.1)Stx=ΠC(I-tF¯)Tx,x∈C,
where ΠC is a sunny nonexpansive retraction from X onto C. It is easy to see that St is a contraction. Indeed, from Lemma 2.6, we have(3.2)‖Stx-Sty‖≤‖ΠC(I-tF¯)Tx-ΠC(I-tF¯)Ty‖≤‖(I-tF¯)Tx-(I-tF¯)Ty‖≤τt‖Tx-Ty‖≤τt‖x-y‖,
for all x,y∈C. Therefore, the following implicit method is well defined:(3.3)xt=ΠC(I-tF¯)Txt,xt∈C.Theorem 3.1.
The net{xt} generated by the implicit method (3.3) converges in norm, as t→0+ to the unique solution x*∈Fix(T) of the variational inequality:
(3.4)〈F¯x*,j(x*-u)〉≤0,u∈Fix⁡(T).Proof.
We first show that the solution set of (3.4) is singleton. As a matter of fact, we assume that x*∈Fix(T) and x̃∈Fix(T) both are solutions to (3.4), then
(3.5)〈F¯x*,j(x*-x̃)〉≤0,(3.6)〈F¯x̃,j(x̃-x*)〉≤0.
Adding (3.5) to (3.6), we get
(3.7)〈F¯x*-F¯x̃,j(x*-x̃)〉≤0.
The strong accretive of F¯ implies that x*=x̃, and the uniqueness is proved. Below we use x*∈Fix(T) to denote the unique solution of (3.4).
Next, we prove that{xt} is bounded. Taking u∈Fix(T), from (3.3) and using Lemma 2.6, we have(3.8)‖xt-u‖=‖ΠC(I-tF¯)Txt-ΠCu‖≤‖(I-tF¯)Txt-(I-tF¯)Tu-tF¯Tu‖≤‖(I-tF¯)Txt-(I-tF¯)Tu‖+t‖F¯u‖≤τt‖xt-u‖+t‖F¯u‖,
that is,
(3.9)‖xt-u‖≤t1-τt‖F¯u‖.
Observe that
(3.10)lim⁡t→0+t1-τt=1η.
From t→0+, we may assume, without loss of generality, that t≤η/2β2K2-ϵ, where ϵ is an arbitrarily small positive number. Thus, we have t/(1-τt) to be continuous, for all  t∈[0,η/2β2K2-ϵ]. Therefore, we obtain
(3.11)M1=sup⁡{t1-τt:t∈(0,η2β2K2-ϵ]}<+∞.
From (3.9) and (3.11), we have {xt} bounded and so is {F¯Txt}.
On the other hand, from (3.3), we obtain(3.12)‖xt-Txt‖=‖ΠC(I-tF¯)Txt-ΠCTxt‖≤‖(I-tF¯)Txt-Txt‖=t‖F¯Txt‖⟶0(t⟶0+).
Next, we show that{xt} is relatively norm-compact as t→0+. Assume that {tn}∈(0,η/2β2K2) such that tn→0+ as n→∞. Put xn:=xtn. It follows from (3.12) that(3.13)‖xn-Txn‖⟶0(n⟶∞).
For a givenu∈Fix(T), by (3.3) and using Lemma 2.2, we have(3.14)〈xt-(I-tF¯)Txt,j(xt-u)〉≤0.
By (3.14) and using Lemma 2.6, we have
(3.15)‖xt-u‖2=〈xt-u,j(xt-u)〉=〈xt-(I-tF¯)Txt,j(xt-u)〉+〈(I-tF¯)Txt-u,j(xt-u)〉≤〈(I-tF¯)Txt-u,j(xt-u)〉≤〈(I-tF¯)Txt-(I-tF¯)Tu,j(xt-u)〉+t〈F¯u,j(u-xt)〉≤τt‖xt-u‖2+t〈F¯u,j(u-xt)〉,
that is,
(3.16)‖xt-u‖2≤t1-τt〈F¯u,j(u-xt)〉≤M1〈F¯u,j(u-xt)〉.
In particular,
(3.17)‖xn-u‖2≤M1〈F¯u,j(u-xn)〉.
Since{xt} is bounded, without loss of generality, we may assume that {xn} converges weakly to a point x̃. Noticing (3.13) we can use Lemma 2.3 to get x̃∈Fix(T). Therefore we can substitute x̃ for u in (3.17) to get(3.18)‖xn-x̃‖≤M1〈F¯x̃,j(x̃-xn)〉.
Consequently, the weak convergence of {xn} to x̃ actually implies that xn→x̃. This has proved the relative norm compactness of the net {xt} as t→0+.
We next show thatx̃ solves the variational inequality (3.4). Observe that(3.19)xt=ΠC(I-tF¯)Txt-(I-tF¯)Txt-(I-tF¯)xt+(I-tF¯)Txt+xt-tF¯(xt)⟹F¯(xt)=1t[ΠC(I-tF¯)Txt-(I-tF¯)Txt-(I-tF¯)xt+(I-tF¯)Txt].
For any u∈Fix(T), we have
(3.20)〈F¯xt,j(xt-u)〉=1t〈ΠC(I-tF¯)Txt-(I-tF¯)Txt,j(xt-u)〉-1t〈(I-tF¯)xt-(I-tF¯)Txt,j(xt-u)〉≤-1t〈xt-Txt,j(xt-u)〉+〈F¯xt-F¯Txt,j(xt-u)〉≤-1t〈(I-T)xt-(I-T)u,j(xt-u)〉+β‖xt-Txt‖‖xt-u‖≤βM2‖xt-Txt‖,
where M2=sup{∥xt-u∥,t∈(0,η/2β2K2)}.
Now replacingt in (3.20) with tn and letting n→∞, we have(3.21)〈F¯x̃,j(x̃-u)〉≤0.
That is, x̃∈Fix(T) is a solution of (3.4), hence x̃=x* by uniqueness. In summary, we have shown that each cluster point of {xt} (at t→0) equals x*. Therefore, xt→x* as t→0.Theorem 3.2.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X with a weakly sequentially continuous duality mapping j. Let F¯:C→X be a β-Lipschitzian and η-strongly accretive operator with 0<η≤2βK. Suppose that T:C→C is a nonexpansive mapping with Fix(T)≠∅. Let ΠC be a sunny nonexpansive retraction from X onto C. Let {αn} and {βn} be two real sequences in (0,1) and satisfy the conditions:(A1)
limn→∞αn=0 and ∑n=0∞αn=∞,(A2)
0<liminfn→∞βn≤limsupn→∞βn<1.
For givenx1∈C arbitrarily, let the sequence {xn} be generated by
(3.22)yn=ΠC(I-αnF¯)Txn,xn+1=βnxn+(1-βn)yn,n≥0.
Then the sequence {xn} strongly converges to x*∈Fix(T) which solves the variational inequality (3.4).Proof.
We proceed with the following steps.
Step 1.
We claim that{xn} is bounded. From limn→∞αn=0, we may assume, without loss of generality, that 0<αn≤η/2β2K2-ϵ for all n. In fact, let u∈Fix(T), from (3.22) and using Lemma 2.6, we have
(3.23)‖yn-u‖=‖ΠC(I-αnF¯)Txn-ΠCu‖≤‖(I-αnF¯)Txn-(I-αnF¯)Tu-αnF¯u‖≤ταn‖xn-u‖+αn‖F¯u‖,
where ταn=1-2αn(η-αnβ2K2)∈(0,1). Then from (3.22) and (3.23), we obtain
(3.24)‖xn+1-u‖≤βn‖xn-u‖+(1-βn)‖yn-u‖≤βn‖xn-u‖+(1-βn)(ταn‖xn-u‖+αn‖F¯u‖)≤[1-(1-βn)(1-ταn)]‖xn-u‖+(1-βn)αn‖F¯u‖≤max⁡{‖xn-u‖,αn‖F¯u‖1-ταn}.
By induction, we have
(3.25)‖xn-u‖≤max⁡{‖x1-u‖,M3‖F¯u‖},
where M3=sup{αn/(1-ταn):0<αn≤η/2β2K2-ϵ}<+∞. Therefore, {xn} is bounded. We also obtain that {yn} and {F¯Txn} are bounded.Step 2.
We claim thatlimn→∞∥xn-yn∥=0. Observe that
(3.26)‖yn+1-yn‖=‖ΠC(I-αn+1F¯)Txn+1-ΠC(I-αnF¯)Txn‖≤‖(I-αn+1F¯)Txn+1-(I-αnF¯)Txn‖≤‖Txn+1-Txn‖+αn+1‖F¯Txn+1‖+αn‖F¯Txn‖≤‖xn+1-xn‖+αn+1‖F¯Txn+1‖+αn‖F¯Txn‖.
Therefore, we have
(3.27)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0.
From (3.22), (3.27), and using Lemma 2.5, we have limn→∞∥xn-yn∥=0.Step 3.
We claim thatlimn→∞∥yn-Tyn∥=0. Observe that
(3.28)‖yn-Tyn‖=‖ΠC(I-αnF¯)Txn-ΠCTyn‖≤‖Txn-Tyn‖+αn‖F¯Txn‖≤‖xn-yn‖+αn‖F¯Txn‖.
Hence, from Step 2 and limn→∞αn=0, we have
(3.29)lim⁡n→∞‖yn-Tyn‖=0.Step 4.
We claim thatlimsupn→∞〈F¯x*,j(x*-yn)〉≤0, where x*=limt→0xt and xt is defined by (3.3). Since yn is bounded, there exists a subsequence {ynk} of {yn} which converges weakly to ω. From Step 3, we obtain Tynk⇀ω. From Lemma 2.3, we have ω∈Fix(T). Hence, using Theorem 3.1, we have x*∈Fix(T) and
(3.30)limsup⁡n→∞〈F¯x*,j(x*-yn)〉=lim⁡k→∞〈F¯x*,j(x*-ynk)〉=〈F¯x*,j(x*-ω)〉≤0.Step 5.
We claim that{xn} converges strongly to x*∈Fix(T). From (3.22) and using Lemma 2.2, we have
(3.31)〈ΠC(I-αnF¯)Txn-(I-αnF¯)Txn,j(yn-x*)〉≤0.
Observe that
(3.32)‖yn-x*‖2=〈ΠC(I-αnF¯)Txn-x*,j(yn-x*)〉=〈ΠC(I-αnF¯)Txn-(I-αnF¯)Txn,j(yn-x*)〉+〈(I-αnF¯)Txn-x*,j(yn-x*)〉≤〈(I-αnF¯)Txn-x*,j(yn-x*)〉≤〈(I-αnF¯)Txn-(I-αnF¯)Tx*,j(yn-x*)〉+αn〈F¯x*,j(x*-yn)〉≤‖(I-αnF¯)Txn-(I-αnF¯)Tx*‖‖yn-x*‖+αn〈F¯x*,j(x*-yn)〉≤ταn‖xn-x*‖‖yn-x*‖+αn〈F¯x*,j(x*-yn)〉≤ταn22‖xn-x*‖2+12‖yn-x*‖2+αn〈F¯x*,j(x*-yn)〉,
that is,
(3.33)‖yn-x*‖2≤ταn‖xn-x*‖2+2αn〈F¯x*,j(x*-yn)〉.
By (3.22) and (3.33), we have
(3.34)‖xn+1-x*‖2≤βn‖xn-x*‖2+(1-βn)‖yn-x*‖2≤βn‖xn-x*‖2+(1-βn)[ταn‖xn-x*‖2+2αn〈F¯x*,j(x*-yn)〉]≤[1-(1-βn)(1-ταn)]‖xn-x*‖2+2M3(1-βn)(1-ταn)〈F¯x*,j(x*-yn)〉=(1-λn)‖xn-x*‖2+λnδn,
where λn=(1-βn)(1-ταn), δn=2M3〈F¯x*,j(x*-yn)〉. It is easy to see that λn→0, ∑n=1∞λn=∞ and limsupn→∞δn≤0. Hence, by Lemma 2.4, the sequence {xn} converges strongly to x*∈Fix(T). From x*=limt→0xt and Theorem 3.1, we have x* to be the unique solution of the variational inequality (3.4).TakingT=ΠC(I-λA)ΠC(I-μB) and F¯=F, where 0<λ≤α/K2 and 0<μ≤η/K2, we obtain the following theorems immediately.Corollary 3.3 (see [1, Theorem 3.5]).
The net{xt} generated by the implicit method (1.1) converges in norm, as t→0+, to the unique solution x̃ of variational inequality(3.35)x̃∈Ω,〈F(x̃),j(x̃-z)〉≤0,z∈Ω.Corollary 3.4 (see [1, Theorem 3.7]).
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X and let ΠC be a sunny nonexpansive retraction from X onto C. Let the mappings A,B:C→X be α-inverse-strongly accretive and β-inverse-strongly accretive operators, respectively. Let F:C→H be a strongly positive linear bounded operator with coefficient γ¯>0. For given x0∈C, let the sequence {xn} be generated iteratively by (1.2). Suppose that the sequences {αn} and {βn} satisfy the conditions (A1) and (A2), then {xn} converges strongly to x̃∈Ω which solves the variational inequality (3.35).



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