Gradient projection method with a new step size for the split feasibility problem

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R E S E A R C H

Open Access

Gradient projection method with a new

step size for the split feasibility problem

Pattanapong Tianchai

1*

*_{Correspondence:}

[email protected]

1_{Faculty of Science, Maejo}

University, Chiangmai, Thailand

Abstract

In this paper, we introduce an iterative scheme using the gradient projection method with a new step size, which is not depend on the related matrix inverses and the largest eigenvalue (or the spectral radius of the self-adjoint operator) of the related matrix, based on Moudaﬁ’s viscosity approximation method for solving thesplit feasibility problem(SFP), which is to ﬁnd a point in a given closed convex subset of a real Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another real Hilbert space. We suggest and analyze this iterative scheme under some appropriate conditions imposed on the parameters such that another strong convergence theorems for the SFP are obtained. The results presented in this paper improve and extend the main results of Tian and Zhang (J. Inequal. Appl. 2017:Article ID 13, 2017), and Tang et al. (Acta Math. Sci.

36B(2):602–613, 2016) (in a single-step regularized method) with a new step size, and many others. The examples of the proposed SFP are also shown through numerical results.

MSC: 47H05; 47H09; 47H20

Keywords: Split feasibility problem; Split common ﬁxed point problem; Gradient projection method

1 Introduction

Throughout this paper, we always assume thatCandQbe closed convex subsets of two real Hilbert spacesHandK, respectively with inner product and norm are denoted by·,· and · , respectively. LetA:H→K be a bounded linear operator. Thesplit feasibility problem(SFP) which was ﬁrst introduced by Censor and Elfving [3] is to ﬁnd

x∗∈C such that Ax∗∈Q. (1.1)

Suppose thatPCandPQare the orthogonal projections onto the setsCandQ, respectively. Assume that SFP (1.1) is consistent. We observe thatx∗∈Csolves the SFP (1.1) if and only if it solves the ﬁxed point equation

x∗=PC

I–γA∗(I–PQ)A

x∗,

whereγ > 0 is any positive constant,Iis the identity operator onHorK, andA∗denotes the adjoint ofA. To solve SFP (1.1) in the setting of a ﬁnite-dimensional real Hilbert space

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case, Byrne [4] proposed the so-calledCQalgorithm based on the Picard iteration method as follows:

xn+1=PC

I–γAt(I–PQ)A

xn, ∀n= 0, 1, 2, . . . , (1.2)

whereγ∈(0,2_L) such thatLbeing the largest eigenvalue of the matrixAt_A_as_At_{stands for} matrix transposition ofA. He proved that the sequence{xn}generated by (1.2) converges strongly to the SFP (1.1).

In [5], Yang presented a relaxedCQalgorithm for solving the SFP (1.1), where he used two halfspacesCnandQnin place ofCandQ, respectively, and at thenth iteration, the orthogonal projections ontoCnandQnare easily executed.

Both theCQalgorithm and the relaxedCQalgorithm used a ﬁxed step size related to the largest eigenvalue of the matrixA∗Aor the spectral radius of the self-adjoint operator

A∗A, which sometimes aﬀects convergence of the algorithms. In [6], Qu and Xiu presented a modiﬁcation of theCQalgorithm and the relaxedCQalgorithm by adopting the Armijo-like searches, which need not to compute the matrix inverses and the largest eigenvalue of the matrixA∗A. TheCQ-like algorithms are also proposed subsequently [3, 7–10].

In all theseCQ-like algorithms for the SFP (1.1), in order to get the step size, one has to compute the largest eigenvalue of the related matrix or use some line search scheme which usually requires many inner iterations to search for a suitable step size in every iteration. We note that a pointxsolves the SFP (1.1) means that there is an elementx∈Csuch that Ax–x∗= 0 for somex∗∈Q. This motivates us to consider the distance function

d(Ax,x∗) =Ax–x∗for allx∈C, and the constrained convex minimization problem:

min

x∈C,x∗∈Q 1 2Ax–x

∗2

such that minimizing with respect tox∗∈Q. Letg:C→Rbe a continuous diﬀerentiable function. First makes us consider the minimization:

min

x∈Cg(x) := 1

2Ax–PQAx

2 _(1.3)

is ill-posed. Therefore, Xu [11] considered the following Tikhonov regularization problem:

min

x∈Cg(x) :=g(x) + 2x

2₌1

2Ax–PQAx

2₊

2x

2_,

where> 0 is the regularization parameter. We know that the gradient∇gofgas

∇g(x) =∇g(x) +I=A∗(I–PQ)A+I, ∀x∈C,

such that ∇g(x) is ( +A2)-Lipschitzian continuous (that is, ∇g(x) –∇g(y) ≤ (+A2₎_x_–_y_{for all}_x_,_y_∈_C_{) and}_{-strongly monotone (that is,}_∇_g

(x) –∇g(y),x–y ≥

x–y2_{for all}_x_,_y_∈_C_).

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the setting of an inﬁnite-dimensional real Hilbert space as follows:

xn+1=PC(I–γn∇gn)xn =PC

I–γn

A∗(I–PQ)A+nI

xn, ∀n= 0, 1, 2, . . . . (1.4)

He proved that the sequence{xn}generated by (1.4) converges in norm to the minimum-norm solution (1.3) of the SFP (1.1), provided the parameters{n},{γn} ⊂(0, 1) satisfy the following conditions:

(i) limn→∞n= 0and0 <γn≤_A2n₊

n, (ii) ∞_n₌₀nγn=∞,

(iii) limn→∞|γn+1–₍γn|+γn|n+1–n| n+1γn+1)2 = 0.

In [1], Tian and Zhang suggested a single-step regularized method based on the Picard iteration method in the setting of an inﬁnite-dimensional real Hilbert space as follows:

xn+1=PC(I–λ∇gn)xn =PC

I–λA∗(I–PQ)A+nI

xn, ∀n= 0, 1, 2, . . . . (1.5)

They proved that the sequence{xn}generated by (1.5) converges in norm to the minimum-norm solution (1.3) of the SFP (1.1), provided the parameters{n} ⊂(0, 1) andλsatisfy the following conditions:

(i) 0 <λ< 2

A2₊₂,

(ii) limn→∞n= 0and∞_n₌₀n=∞, (iii) ∞_n₌₀|n+1–n|<∞.

We observe that in the proof of their proposed results,∞_n₌₀|n+1–n|<∞of the control

condition (iii) can be removed using the NST-condition (II) [12] (see also [13, 14]). The SFP (1.1) is an important and has been widely studied because it plays a prominent role in the signal processing and image reconstruction problem. Initiated by the SFP, sev-eral split type problems have been investigated and studied, for examples, the split vari-ational inequality problem (SVIP) (see [15]) and the split common null point problem (SCNP). We will consolidate these problems. LetS:H→HandT:K→Kbe two oper-ators with nonempty fixed point setsFix(S) :={x∈H:x=S(x)}andFix(T), respectively. IfSbe a nonexpansive mapping (that is,Sx–Sy ≤ x–yfor allx,y∈H), thenFix(S) is closed and convex (see [16]). Thesplit common fixed point problem(SCFP) is to find

x∗∈Fix(S) such that Ax∗∈Fix(T). (1.6)

IfS=PCandT=PQthenFix(S) =CandFix(T) =Q, and hence the SCFP (1.6) immedi-ately reduces to the SFP (1.1).

Assume that the SCFP (1.6) is consistent. In [17], Censor and Segal proposed and proved a strong convergence theorem for the SCFP (1.6) based on the Picard iteration method, in the case that the directed operatorsS(that is,x–Sx,Sx–y ≥0 for ally∈Fix(S) and

x∈H, for instanceS=PC) andT, still in a ﬁnite-dimensional real Hilbert space to extend the iteration method (1.2) of Byrne as follows:

xn+1=S

I–γAt(I–T)Axn, ∀n= 0, 1, 2, . . . ,

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In [18], Kraikaew and Saejung proposed and proved a strong convergence theorem for the SCFP (1.6) based on the Halpern iteration method, in the case that the quasi-nonexpansive mappingsS (that is,Sx–p ≤ x–pfor allx∈Handp∈Fix(S)) and

T such that bothI–SandI–T are demiclosed at zero on an inﬁnite-dimensional real Hilbert space as follows:

xn+1=αnx0+ (1 –αn)S

I–γA∗(I–T)Axn, ∀n= 0, 1, 2, . . . ,

whereγ ∈(0,1

L),{αn} ⊂(0, 1),limn→∞αn= 0 and

∞

n=0αn=∞.

In [2], Tang et al. proposed and proved a strong convergence theorem for the SCFP (1.6) based on the viscosity approximation method, in the case that the ﬁrmly nonexpansive mappingsS(that is,Sx–Sy2_≤_x_–_y2_–₍_I_–_S₎_x_{– (}_I_–_S₎_y2_{for all}_x_,_y_∈_H_{) and}_T_such

that bothI–SandI–Tare demiclosed at zero, and anα-contraction mappingh:H→H

withα∈(0, 1) (that is,h(x) –h(y) ≤αx–yfor allx,y∈H) on an inﬁnite-dimensional real Hilbert space in a single-step regularized method as follows:

xn+1=αnxn+βnh(xn) +γnS

I–ξnA∗(I–T)A

xn, ∀n= 0, 1, 2, . . . ,

whereαn+βn+γn= 1,ξn= ρn(I–T)Axn

2

2A∗(I–T)Axn2,{ρn} ⊂(0, 4),{αn},{βn},{γn} ⊂(0, 1) satisfy the

following conditions: (i) limn→∞βn= 0and

∞

n=0βn=∞, (ii) lim inf_n_→∞αnγn> 0.

We observe that in the proof of their proposed results, the sequence{ξn}may not con-verges to the zero, for instance ifA=Iandρn= 2 for alln= 0, 1, 2, . . . then the sequence {ξn}converges to the integer 1. The relaxedCQ-like algorithms are also proposed subse-quently [19–21].

In this paper, we modify in all these algorithms to solve the SCFP (1.6) and also solve the SFP (1.1) based on Moudafi’s viscosity approximation method [22], in the case that the firmly nonexpansive mappingsSandT such that bothI–SandI–Tare demiclosed at zero, and anα-contraction mappingh:H→Hwithα∈(0, 1) on an infinite-dimensional real Hilbert space in a single-step regularized method with the regularization parameter as follows:

xn+1=αnxn+βnh(xn) +γnS

I–λn

A∗(I–T)A+nI

xn, ∀n= 0, 1, 2, . . . . (1.7)

We suggest and analyze this iterative scheme (1.7) under some appropriate conditions imposed on the parameters with a new step size, which is not depend on the related matrix inverses and the largest eigenvalue (or the spectral radius of the self-adjoint operator) of the related matrix such that another strong convergence theorems for the SCFP (1.6) and the SFP (1.1) are obtained.

2 Preliminaries

LetH andK be two real Hilbert spaces,A:H→K be a bounded linear operator, A∗

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the notation:→to denote the strong convergency, to denote the weak convergency,

ωw(xn) =

x:∃{xnk} ⊂ {xn}such thatxnk x

to denote the weak limit set of{xn}andFix(T) ={x:x=Tx}to denote the ﬁxed point set of the mappingT.

LetC be a nonempty closed convex subset of a real Hilbert spaceH. Recall that the metric projection PC:H→C is deﬁned as follows: for eachx∈H, PCxis the unique point inCsatisfying

x–PCx=inf

x–y:y∈C.

Letf :H→Rbe a function. Recall that a functionf is called convex if

fλx+ (1 –λ)y≤λf(x) + (1 –λ)f(y), ∀λ∈[0, 1],∀x,y∈H.

A diﬀerentiable functionf is convex if and only if for eachx∈H, we have the inequality:

f(z)≥f(x) +∇f(x),z–x, ∀z∈H.

An elementg∈His said to be a subgradient off atx∈Hif we have the subdiﬀerentiable inequality

f(z)≥f(x) +g,z–x, ∀z∈H.

A functionf is said to be subdifferentiable atx∈H, if it has at least one subgradient atx. The set of subgradients off atx∈His called the subdifferentiable off atx, and it is de-noted by∂f(x). A functionf is called subdifferentiable, if it is subdifferentiable at allx∈H. If a functionf is differentiable and convex, then its gradient and subgradient coincide. A functionf is called lower semi-continuous (lsc) for allx∈Hif for eacha∈R, the set {x∈H:f(x)≤a}is a closed set, and a functionf is called weakly lower semi-continuous (w-lsc) atx∈Hiff is lsc atxfor a sequence{xn} ⊂Hsuch thatxn x.

We collect some known lemmas and deﬁnitions which are our main tools in proving the our results.

Lemma 2.1 Let H be a real Hilbert space.Then,for all x,y∈H,

(i) x+y2=x2+ 2x,y+y2, (ii) x+y2_≤_x2_{+ 2}_y_,_x₊_y_.

Lemma 2.2 ([23]) Let C be a nonempty closed convex subset of a real Hilbert space H.

Then:

(i) z=PCx⇔ x–z,z–y ≥0,∀x∈H,y∈C,

(ii) z=PCx⇔ x–z2≤ x–y2–y–z2,∀x∈H,y∈C, (iii) PCx–PCy2≤ x–y,PCx–PCy,∀x,y∈H.

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(i) monotone if

x–y,Tx–Ty ≥0, ∀x,y∈H,

(ii) L-Lipschitzian withL> 0if

Tx–Ty ≤Lx–y, ∀x,y∈H,

(iii) α-contraction if it isα-Lipschitzian withα∈(0, 1), (iv) nonexpansive if it is1-Lipschitzian,

(v) ﬁrmly nonexpansive if

Tx–Ty2≤ x–y2–(I–T)x– (I–T)y2, ∀x,y∈H.

Lemma 2.4([24]) Let H and K be two real Hilbert spaces and let T:K→K be a ﬁrmly nonexpansive mapping such that(I–T)xis a convex function from K toR= [–∞, +∞].

Let A:H→K be a bounded linear operator and f(x) =1₂(I–T)Ax2for all x∈H.Then: (i) ∇f(x) =A∗(I–T)Ax,∀x∈H,

(ii) ∇f isA2-Lipschitzian.

Lemma 2.5([24]) Let H be a real Hilbert space and T:H→H be an operator.The fol-lowing statements are equivalent:

(i) Tis ﬁrmly nonexpansive,

(ii) Tx–Ty2_≤_x_–_y_,_Tx_–_Ty_,_∀_x_,_y_∈_H_,

(iii) I–Tis ﬁrmly nonexpansive.

Lemma 2.6([25]) Let H be a real Hilbert space and let{xn}be a sequence in H.Then,for any given sequence{λn}∞_n₌₁⊂(0, 1)with∞_n₌₁λn= 1and for any positive integer i,j with

i<j,

∞

i=1

λnxn

2

≤

∞

i=1

λnxn2–λiλjxi–xj2.

Lemma 2.7([26]) Let{an}be a sequence of nonnegative real numbers such that

an+1≤(1 –γn)an+γnσn, ∀n= 0, 1, 2, . . . ,

where{γn}is a sequence in(0, 1)and{σn}is a sequence inRsuch that

(i) ∞_n₌₀γn=∞, (ii) lim sup_n_→∞σn≤0or

∞

n=0|γnσn|<∞.

Thenlimn→∞an= 0.

Lemma 2.8([27]) Let{tn}be a sequence of real numbers such that there exists a subse-quence{ni}of{n}such that tni<tni+1 for all i∈N.Then there exists a nondecreasing

se-quence{τ(n)} ⊂Nsuch thatτ(n)→ ∞,and the following properties are satisﬁed by all

(suﬃciently large)numbers n∈N:

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In fact,

τ(n) =max{k≤n:tk<tk+1}.

Lemma 2.9([28] (Demiclosedness principle)) Let C be a nonempty closed convex subset of a real Hilbert space H and let S:C→C be a nonexpansive mapping withFix(S)=∅.If the sequence{xn} ⊂C converges weakly to x and the sequence{(I–S)xn}converges strongly to y.Then(I–S)x=y;in particular,if y= 0then x∈Fix(S).

Lemma 2.10([16]) Let C be a nonempty closed convex subset of a Hilbert space H and let f be a proper convex lower semi-continuous function of C into(–∞,∞].If{xn} be a bounded sequence of C such that xn x0.Then f(x0)≤lim infn→∞f(xn).

3 Main result

Throughout this paper, we let:={x∈Fix(S) :Ax∈Fix(T)}. It is clear thatis closed and convex.

Theorem 3.1 Let H and K be two real Hilbert spaces and let S:H→H and T:K→K be two ﬁrmly nonexpansive mappings such that both I–S and I–T are demiclosed at zero.Let(I–T)xbe a convex function from K toRand A:H→K be a bounded linear operator,and let h:H→H be anα-contraction mapping.Assume that the SCFP(1.6)has a nonempty solution setand let{xn} ⊂H be a sequence generated by

⎧ ⎨ ⎩

x0∈H,

xn+1=αnxn+βnh(xn) +γnS(xn–λn∇gn(xn)), ∀n= 0, 1, 2, . . . ,

whereαn+βn+γn= 1and gn(xn) = ₂1(I–T)Axn2+₂nxn2such that

∇gn(xn) =A∗(I–T)Axn+nxn= 0, λn=

ρngn(xn)

∇gn(xn)2+∇gn(xn)+ρngn(xn)

for all n= 0, 1, 2, . . . .If the sequences{ρn} ⊂[a,b]for some a,b∈(0, 2),{αn} ⊂[c,d]for some c,d∈(0, 1)and{βn},{γn} ⊂(0, 1)satisfy the following conditions:

(i) 0≤n≤βn2msuch thatm> 1for alln= 0, 1, 2, . . .,

(ii) limn→∞βn= 0and

∞

n=0βn=∞,

then the sequence{xn}converges strongly to x∗∈where x∗=Ph(x∗).

Proof Fixn∈N∪ {0}. Note thatgn(x) =1₂(I–T)Ax2+

n

2x

2_{is Gáteaux diﬀerentiable}

by the convexity of(I–T)Axfor allx∈H. First, we show that{xn}is bounded. Pick

p∈. We havep∈Fix(S) andAp∈Fix(T). Observing thatI–Tis ﬁrmly nonexpansive, by Lemma 2.5 we have

xn–p,∇gn(xn) =

xn–p,A∗(I–T)Axn+nxn

=xn–p,A∗(I–T)Axn +nxn–p,xn

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+n 2

xn–p2+xn2–p2

≥(I–T)Axn– (I–T)Ap

2

+n 2

xn2_–_p2

≥1

2(I–T)Axn

2

+n 2

xn2–p2

≥gn(xn) – β2m

n 2 p

2_. _(3.1)

Therefore, by the nonexpansiveness ofSwe have

Sxn–λn∇gn(xn)

–p2

=Sxn–λn∇gn(xn)

–Sp2

≤xn–λn∇gn(xn) –p

2

=xn–p2+λn2∇gn(xn)

2

– 2λn

xn–p,∇gn(xn)

≤ xn–p2+λn2∇gn(xn)

2

– 2λngn(xn) +λnβn2mp2 ≤ xn–p2+

ρ_n2g2

n(xn)(∇gn(xn)2+∇gn(xn)+ρngn(xn)) (∇gn(xn)2+∇gn(xn)+ρngn(xn))2

– 2ρng

2

n(xn)

+β_n2mp2

=xn–p2–ρn(2 –ρn)

g2

n(xn)

+β_n2mp2

≤ xn–p2+βn2mp2 ≤xn–p+βnmp

2

. (3.2)

This implies that

–p≤ xn–p+βnmp. (3.3)

Hence,

xn+1–p=αnxn+βnh(xn) +γnS

xn–λn∇gn(xn)

–p

≤αnxn–p+βnh(xn) –p+γnS

xn–λn∇gn(xn)

–p

≤αnxn–p+βnh(xn) –h(p)+βnh(p) –p

+γn

xn–p+βnmp

≤αnxn–p+βnαxn–p+βnh(p) –p+γnxn–p+βnp

=1 – (1 –α)βn

xn–p+ (1 –α)βn

h(p) –p+p

1 –α

≤max

xn–p,

h(p) –p+p

1 –α

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By induction, we have

xn–p ≤max

x0–p,

h(p) –p+p

1 –α

.

This implies that{xn}is bounded, and so are{h(xn)}and{gn(xn)}. Using Lemma 2.6 and (3.2), we have

xn+1–p2=αnxn+βnh(xn) +γnS

xn–λn∇gn(xn)

–p2

=αn(xn–p) +βn

h(xn) –p

+γn

–p2

≤αnxn–p2+βnh(xn) –p

2

+γnS

xn–λn∇gn(xn)

–p2

–αnγnS

xn–λn∇gn(xn)

–xn

2

≤αnxn–p2+βnh(xn) –p

2

+γn

xn–p2

–ρn(2 –ρn) g

2

n(xn)

+β_n2mp2

–αnγnS

xn–λn∇gn(xn)

–xn2 ≤ xn–p2+βnh(xn) –p

2

–γnρn(2 –ρn) g

2

n(xn)

+βnp2

–αnγnS

xn–λn∇gn(xn)

–xn2. Therefore,

γnρn(2 –ρn)

g2

n(xn)

∇gn(xn)2+∇gn(xn)+ρngn(xn) +αnγnS

xn–λn∇gn(xn)

–xn

2

≤ xn–p2–xn+1–p2+βnh(xn) –p

2

+βnp2. (3.4)

SincePhis a contraction onH, by the Banach contraction principle there exists a unique elementx∗∈Hsuch that x∗=Ph(x∗). That isx∗∈. Now, we show thatxn→x∗ as

n→ ∞. We consider into two cases.

Case 1. Assume that{xn–p}is a monotone sequence. In other words, for n0large

enough,{xn–p}n≥n0is either nondecreasing or nonincreasing. As{xn–p}is bounded,

so{xn–p}is convergent. Therefore, by (3.4) we have

lim

n→∞γnρn(2 –ρn)

g2

n(xn)

= 0 (3.5)

and

lim

n→∞αnγnS

xn–λn∇gn(xn)

–xn

2

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By Lemma 2.4, we have

_∇_g_n₍_x_n₎_≤_∇_g_n₍_x_n_{) –}_∇_g_n₍_p₎₊_∇_g_n₍_p₎

≤A∗(I–T)Axn–A∗(I–T)Ap+nxn–p+∇gn(p) ≤A2+n

xn–p+∇gn(p) ≤A2+ 1xn–p+∇gn(p).

This implies that{∇gn(xn)}is bounded, and so is{∇gn(xn)2+∇gn(xn)+ρngn(xn)}. Hence,∇gn(xn)2+∇gn(xn)+ρngn(xn) <δfor someδ> 0. Since

(1 –d–βn)a(2 –b)

g2

n(xn) δ

≤γnρn(2 –ρn)

g2

n(xn)

∇gn(xn)2+∇gn(xn)+ρngn(xn) ,

by (3.5) we have

lim

n→∞gn(xn) = 0. (3.7)

Since

c(1 –d–βn)S

xn–λn∇gn(xn)

–xn

2

≤αnγnS

xn–λn∇gn(xn)

–xn

2

,

by (3.6) we have

lim

n→∞S

xn–λn∇gn(xn)

–xn= 0. (3.8)

Here we havelim_n_→∞n= 0 by the condition (ii). Therefore, in the same way, we have

lim

n→∞(I–T)Axn= 0.

Consider a subsequence{xnk}of{xn}. As{xn}is bounded, so{xnk}is bounded, there exists a subsequence{xn_kl}of{xnk}which converges weakly tow∈H. Without loss of generality, we can assume thatxnk wask→ ∞. Therefore,Axnk Awask→ ∞and

lim

k→∞(I–T)Axnk= 0.

By the demiclosedness at zero, we haveAw∈Fix(T). Next, we show thatw∈Fix(S). By Lemma 2.1(2), the ﬁrmly nonexpansiveness ofSand (3.3), we have

–p2

=Sxn–λn∇gn(xn)

–Sxn

+ (Sxn–Sp)

2

≤ Sxn–Sp2+ 2

–Sxn,S

xn–λn∇gn(xn)

–p

≤ xn–p2–(I–S)xn– (I–S)p

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+ 2Sxn–λn∇gn(xn)

–SxnS

xn–λn∇gn(xn)

–p

≤ xn–p2–(I–S)xn

2

+ 2λn∇gn(xn)xn–p+β m np

.

Therefore,

(I–S)xn

2

≤ xn–p2–S

xn–λn∇gn(xn)

–p2+ 2λn∇gn(xn)xn–p+βnmp

≤ xn–p2–S

xn–λn∇gn(xn)

–xn–xn–p 2

+ 2λn∇gn(xn)xn–p+βnp

≤2Sxn–λn∇gn(xn)

–xnxn–p+ 2λn∇gn(xn)xn–p+βnp

= 2Sxn–λn∇gn(xn)

–xnxn–p

+ 2ρngn(xn)∇gn(xn) ∇gn(xn)2+∇gn(xn)+ρngn(xn)

xn–p+βnp

≤2Sxn–λn∇gn(xn)

–xnxn–p+ 4gn(xn)

xn–p+βnp

.

It follows by (3.7) and (3.8) that

lim

n→∞(I–S)xn= 0.

Hence,limk→∞(I–S)xnk= 0. Therefore, by the demiclosedness at zero, we havew∈

Fix(S). That isw∈. Applying the characteristic ofPin Lemma 2.2(i) andx∗=Ph(x∗), we have

lim sup

n→∞

hx∗–x∗,xn+1–x∗ = max

w∈ωw(xn)

hx∗–x∗,w–x∗ ≤0.

Finally, we show thatxn→x∗asn→ ∞. By Lemma 2.1(ii) and (3.3), we have

xn+1–x∗ 2

=αnxn+βnh(xn) +γnS

xn–λn∇gn(xn)

–x∗2

=αn

xn–x∗

+βn

h(xn) –x∗

+γn

–x∗2

≤αn

xn–x∗

+γn

–x∗2

+ 2βn

h(xn) –x∗,xn+1–x∗

≤αnxn–x∗+γnS

xn–λn∇gn(xn)

–x∗2

+ 2βn

h(xn) –h

x∗,xn+1–x∗ + 2βn

hx∗–x∗,xn+1–x∗

≤αnxn–x∗+γnxn–x∗+βnmx∗

2

+ 2βnαxn–x∗xn+1–x∗+ 2βn

≤(1 –βn)xn–x∗+βnmx∗

2

+βnαxn–x∗

2

+xn+1–x∗ 2

+ 2βn

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This implies that

xn+1–x∗2≤

(1 –βn)2+βnα 1 –βnα

xn–x∗2+ 1 1 –βnα

2(1 –βn)βnmxn–x∗x∗

+β_n2mx∗2+ 2βn

≤

1 –2(1 –α)βn 1 –βnα

xn–x∗2+

(1 –α)βn 1 –βnα

βn

1 –αxn–x

∗2

+2β m–1

n

1 –α xn–x

∗_x∗₊βn2m–1 1 –αx

∗2

+ 2

1 –α

≤

1 –(1 –α)βn 1 –βnα

xn–x∗2+

(1 –α)βn 1 –βnα

βn 1 –αM

2

+2β m–1

n 1 –α Mx

∗₊βn2m–1 1 –αx

∗2

+ 2

1 –α

= (1 –ηn)xn–x∗2+ηnδn,

whereM=sup_n_≥₀xn–x∗<∞,ηn=(1–_1–αβn)βαn∈(0, 1) and

δn= βn 1 –αM

2₊2βnm–1 1 –α Mx

∗₊βn2m–1 1 –αx

∗2

+ 2

1 –α

hx∗–x∗,xn+1–x∗.

It is easy to see that∞_n₌₀ηn=∞andlim supn→∞δn≤0. Hence, by Lemma 2.7, the se-quence{xn}converges strongly tox∗=Ph(x∗).

Case 2. Assume that{xn–p}is not a monotone sequence. Then we can deﬁne an integer sequence{τ(n)}for alln≥n0(for somen0large enough) by

τ(n) =maxk∈N,k≤n:xk–p<xk+1–p

.

Clearly,{τ(n)}is a nondecreasing sequence such thatτ(n)→ ∞asn→ ∞, and for all

n≥n0we have

xτ(n)–x∗<xτ(n)+1–x∗.

From (3.4), we obtain

γτ(n)ρτ(n)(2 –ρτ(n))

g2

τ(n)(xτ(n))

∇g_τ(n)(xτ(n))2+∇gτ(n)(xτ(n))+ρτ(n)gτ(n)(xτ(n))

+ατ(n)γτ(n)S

xτ(n)–λτ(n)∇gτ(n)(xτ(n))

–xτ(n)

2

≤xτ(n)–x∗ 2

–xτ(n)+1–x∗ 2

+βτ(n)h(xτ(n)) –x∗ 2

+βτ(n)x∗ 2

≤βτ(n)h(xτ(n)) –x∗2+βτ(n)x∗2.

Aslimn→∞βτ(n)= 0 and{h(xτ(n))}is bounded, we get

lim

n→∞γτ(n)ρτ(n)(2 –ρτ(n))

g2τ(n)(xτ(n))

∇gτ(n)(xτ(n))2+∇gτ(n)(xτ(n))+ρτ(n)gτ(n)(xτ(n))

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and

lim

n→∞ατ(n)γτ(n)S

xτ(n)–λτ(n)∇gτ(n)(xτ(n))

–xτ(n)2= 0.

Following similar arguments to that in Case 1, we haveωw(xτ(n))⊂. Applying the

char-acteristic ofPin Lemma 2.2(i) andx∗=Ph(x∗), we have

lim sup

n→∞

hx∗–x∗,xτ(n)+1–x∗ = max

w∈ωw(xτ(n))

hx∗–x∗,w–x∗ ≤0,

and by similar arguments, we have

xτ(n)+1–x∗2≤(1 –ητ(n))xτ(n)–x∗2+ητ(n)δτ(n),

whereητ(n)∈(0, 1),

∞

τ(n)ητ(n)=∞andlim supn→∞δτ(n)≤0. Hence, by Lemma 2.7, we

havelimn→∞xτ(n)–x∗= 0, and thenlimn→∞xτ(n)+1–x∗= 0. Thus, by Lemma 2.8, we

have

0≤xn–x∗≤maxxτ(n)–x∗,xn–x∗≤xτ(n)+1–x∗.

Therefore, the sequence {xn} converges strongly to x∗ =Ph(x∗). This completes the

proof.

Corollary 3.2 Let H and K be two real Hilbert spaces and let S:H→H and T:K→K be two ﬁrmly nonexpansive mappings such that both I–S and I–T are demiclosed at zero.Let(I–T)xbe a convex function from K toRand A:H→K be a bounded linear operator,and let h:H→H be anα-contraction mapping.Assume that the SCFP(1.6)has a nonempty solution setand let{xn} ⊂H be a sequence generated by

⎧ ⎨ ⎩

x0∈H,

xn+1=αnxn+βnh(xn) +γnS(xn–λn∇g(xn)), ∀n= 0, 1, 2, . . . ,

whereαn+βn+γn= 1and g(xn) = 1₂(I–T)Axn2such that

∇g(xn) =A∗(I–T)Axn= 0, λn=

ρng(xn)

∇g(xn)2+∇g(xn)+ρng(xn)

for all n= 0, 1, 2, . . . .If the sequences{ρn} ⊂[a,b]for some a,b∈(0, 2),{αn} ⊂[c,d]for some c,d∈(0, 1)and{βn},{γn} ⊂(0, 1)satisfy the following conditions:limn→∞βn= 0and

_∞

n=0βn=∞,then the sequence{xn}converges strongly to x∗∈where x∗=Ph(x∗). Let:={x∈C:Ax∈Q}. It is clear thatis closed and convex. TakeS=PCandT=PQ into Theorem 3.1. We have the following consequences.

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α-contraction mapping.Assume that the SFP(1.1)has a nonempty solution setand let

{xn} ⊂H be a sequence generated by

⎧ ⎨ ⎩

x0∈H,

xn+1=αnxn+βnh(xn) +γnPC(xn–λn∇gn(xn)), ∀n= 0, 1, 2, . . . ,

whereαn+βn+γn= 1and gn(xn) =

1

2(I–PQ)Axn 2₊n

2xn

2_{such that}

∇gn(xn) =A

∗₍_I_–_P

Q)Axn+nxn= 0, λn=

ρngn(xn)

(ii) lim_n_→∞βn= 0and

∞

n=0βn=∞,

Corollary 3.4 Let C and Q be two nonempty closed convex subsets of real Hilbert spaces H and K,respectively.Let A:H→K be a bounded linear operator and h:H→H be an

α-contraction mapping.Assume that the SFP(1.1)has a nonempty solution setand let

{xn} ⊂H be a sequence generated by

⎧ ⎨ ⎩

x0∈H,

xn+1=αnxn+βnh(xn) +γnPC(xn–λn∇g(xn)), ∀n= 0, 1, 2, . . . ,

whereαn+βn+γn= 1and g(xn) = 1₂(I–PQ)Axn2such that

∇g(xn) =A∗(I–PQ)Axn= 0, λn=

ρng(xn)

for all n= 0, 1, 2, . . . .If the sequences{ρn} ⊂[a,b]for some a,b∈(0, 2),{αn} ⊂[c,d]for some c,d∈(0, 1)and{βn},{γn} ⊂(0, 1)satisfy the following conditions:limn→∞βn= 0and

∞

n=0βn=∞,then the sequence{xn}converges strongly to x∗∈where x∗=Ph(x∗). 4 Applications

In this section, assuming that the projectionsPC andPQare not easily calculated. We present a perturbation technique. Carefully speaking, the convex setsCandQsatisfy the following assumptions:

(H1) The setsCandQare given by

C=x∈H:c(x)≤0=∅,

Q=y∈K:q(y)≤0=∅,

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(H2) For anyx∈H, at least one subgradientξ∈∂c(x)can be calculated, and for any

y∈K, at least one subgradientη∈∂q(y)can be calculated, where∂c(x)and∂q(y) are a generalized gradient ofc(x)atxand a generalized gradient ofq(y)aty, respectively, which are deﬁned as follows:

∂c(x) =ξ∈H:c(z)≥c(x) +ξ,z–x,∀z∈H,

∂q(y) =η∈K:q(u)≥q(y) +η,u–y,∀u∈K.

We note that in (H1), the differentiability ofcandqare not assumed. The representations ofCandQin (H1) are therefore general enough, because any system of inequalitiesci(x)≤ 0,i∈Iand any system of inequalitiesqj(y)≤0,j∈J, whereci,qjare convex (not necessarily differentiable) functions, andI,Jare two arbitrary index sets, can be reformulated as the single inequalityc(x)≤0 and the single inequalityq(y)≤0 withc(x) =sup{ci(x) :i∈I} andq(y) =sup{qj(y) :j∈J}, respectively. Moreover, every convex functions defined on a finite-dimensional Hilbert space is subdifferentiable and its subdifferential operator is a bounded operator on any bounded subset in Hilbert space (see [29]).

Theorem 4.1 Let C and Q be two nonempty closed convex subsets of real Hilbert spaces

H and K,respectively,satisfy the conditions(H1)and(H2).Let A:H→K be a bounded linear operator and h:H→H be anα-contraction mapping.Assume that the SFP(1.1)

has a nonempty solution setand let{xn} ⊂H be a sequence generated by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x0∈H,

Cn={x∈H:c(xn) +ξn,x–xn ≤0},

Qn={y∈K:q(Axn) +ηn,y–Axn ≤0},

xn+1=αnxn+βnh(xn) +γnPCn(xn–λn∇gn(xn)), ∀n= 0, 1, 2, . . . ,

whereξn∈∂c(xn),ηn∈∂q(Axn),αn+βn+γn= 1and

gn(xn) = 1

2(I–PQn)Axn

2

+n 2xn

2_,

∇gn(xn) =A∗(I–PQn)Axn+nxn= 0, λn=

ρngn(xn)

(ii) lim_n_→∞βn= 0and

∞

n=0βn=∞,

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Case 1. Assume that{xn–p}is a monotone sequence. By similar arguments, there exists a subsequence{xnk}of{xn}which converges weakly tow∈H, and we have

lim

k→∞(I–PQnk)Axnk= 0, klim→∞(I–PCnk)xnk= 0.

By the deﬁnitions ofCnkandQnk we have

c(xnk)≤ ξnk,xnk–PC_nkxnk ≤ξ(I–PC_nk)xnk

and

q(Axnk)≤

ηnk,Axnk–PQ_nk(Axnk) ≤η(I–PQ_nk)Axnk,

whereξn ≤ξ<∞andηn ≤η<∞for alln= 0, 1, 2, . . . . Hence,

lim

k→∞c(xnk)≤0, klim→∞q(Axnk)≤0.

Therefore, by the w-lsc ofcandqatwandAw, respectively, applying Lemma 2.10, we have

c(w)≤lim inf

k→∞ c(xnk)≤0, q(Aw)≤lim infk→∞ q(Axnk)≤0.

It follows thatw∈CandAw∈Q. That is,w∈. By similar arguments, we have the se-quence{xn}converges strongly tox∗=Ph(x∗).

Case 2. Assume that{xn–p}is not a monotone sequence. Following similar arguments to those in Case 1 and Case 2 of the proof in Theorem 3.1, we haveωw(xτ(n))⊂. By similar

arguments, we have the sequence{xn}converging strongly tox∗=Ph(x∗). This completes

the proof.

Corollary 4.2 Let C and Q be two nonempty closed convex subsets of real Hilbert spaces H and K,respectively,satisfy the conditions(H1)and(H2).Let A:H→K be a bounded linear operator and h:H→H be anα-contraction mapping.Assume that the SFP(1.1)

has a nonempty solution setand let{xn} ⊂H be a sequence generated by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x0∈H,

Cn={x∈H:c(xn) +ξn,x–xn ≤0},

Qn={y∈K:q(Axn) +ηn,y–Axn ≤0},

xn+1=αnxn+βnh(xn) +γnPCn(xn–λn∇g(xn)), ∀n= 0, 1, 2, . . . ,

whereξn∈∂c(xn),ηn∈∂q(Axn),αn+βn+γn= 1and g(xn) =1₂(I–PQn)Axn2,

∇g(xn) =A∗(I–PQn)Axn= 0, λn=

ρng(xn)

for all n= 0, 1, 2, . . . .If the sequences{ρn} ⊂[a,b]for some a,b∈(0, 2),{αn} ⊂[c,d]for some c,d∈(0, 1)and{βn},{γn} ⊂(0, 1)satisfy the following conditions:lim_n_→∞βn= 0and

∞

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5 Numerical results

In this section, we give some insight into the behavior of the algorithms presented in Corollary 3.4 and Corollary 4.2. We implemented them in Mathematica to solve and run on a computer Pentium(R) mobile processor 1.50 GHz. We usexn+1–xn2< as the

stopping criterion.

Throughout the computational experiments, the parameters used in those algorithms were sets as= 10–6_,_ρ

n= 1,αn=1₂,βn=_n₊₃1 andγn= 1 –αn–βnfor alln= 0, 1, 2, . . . . In the results report below, all CPU times reported are in seconds. The approximate solution is referred to the last iteration.

In the computation, the projectionPCwhereCis a closed ball inRN, and in the projec-tionsPCnandPQnwhereCnandQnare the halfspaces inRN andRM, respectively, we use the formulation as follows.

Proposition 5.1 Forρ> 0and C={x∈RN_:_x

2≤ρ},we have

PCx= ⎧ ⎨ ⎩

ρx

x2, x∈/C; x, x∈C.

Proposition 5.2([30, 31]) Let H= (RN_,_·

2)and K= (RM, · 2).Assume that C and Q

satisfy the conditions(H1)and(H2),and deﬁne the halfspaces Cnand Qnas algorithm in

Corollary4.2.For any z∈RN _{and for each n}_{= 0, 1, 2, . . .}_{we have}

PCn(z) = ⎧ ⎨ ⎩

z–c(xn)+ξn,z–xn

ξn2 ξn, c(xn) +ξn,z–xn> 0;

z, c(xn) +ξn,z–xn ≤0,

and

PQn(Az) = ⎧ ⎨ ⎩

Az–q(Axn)+ηn,Az–Axn

ηn2 ηn, q(Axn) +ηn,Az–Axn> 0;

Az, q(Axn) +ηn,Az–Axn ≤0.

Example5.3 (A projection point problem) LetC={a∈R4_:_a

2≤3}andu∈R4chosen

arbitrarily. Find a unique solution x∗∈Cwhich is nearest the point uand satisﬁes the following system of linear equations:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x+ 2y+ 3z+w= 1,

x–y+z– 2w= 2,

x+y– 2z+w= 3,

wherex,y,z,w∈R. LetH= (R4_,_·

2) andK= (R3, · 2). Take

A= ⎛ ⎜ ⎝

1 2 3 1

1 –1 1 –2

1 1 –2 1

⎞ ⎟

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[image:18.595.118.479.97.253.2]

Table 1 Results for Example 5.3 using the algorithm in Corollary 3.4

Choicenessu Starting pointsx0

Number of iterations

CPU (s) Approximate solutionx∗(approximates ofAx∗andx∗2)

(0, 0, 0, 0)T _{(0, 0, 0, 0)}T ₉₅₀₀ _5.317 _{(1.974662, 0.512779, –0.498849, –0.508389)}T

(0, 0, 0, 0)T _{(1, 1, 1, 1)}T ₉₅₀₁ _5.187 _{(1.974695, 0.512718, –0.498837, –0.508336)}T

(0, 0, 0, 0)T _{(1, 2, 3, 4)}T ₉₅₀₅ _5.217 _{(1.974803, 0.512523, –0.498798, –0.508168)}T Ax∗= (0.99528, 1.97981, 2.97675)T_,_x∗₂_{= 2.16091}

(1, 1, 1, 1)T _{(0, 0, 0, 0)}T ₉₄₀₆ _5.488 _{(2.130245, 0.226710, –0.439453, –0.255758)}T

(1, 1, 1, 1)T _{(1, 1, 1, 1)}T ₉₄₀₆ _5.888 _{(2.130278, 0.226649, –0.439440, –0.255705)}T

(1, 1, 1, 1)T _{(1, 2, 3, 4)}T ₉₄₀₉ _5.568 _{(2.130386, 0.226451, –0.439402, –0.255534)}T Ax∗= (1.00955, 1.97560, 2.98010)T_,_x∗₂_{= 2.20179}

(1, 2, 3, 4)T (0, 0, 0, 0)T 12,180 4.977 (2.641308, –0.734424, –0.244857, 0.583179)T (1, 2, 3, 4)T (1, 1, 1, 1)T 12,179 5.127 (2.641332, –0.734471, –0.244847, 0.583221)T (1, 2, 3, 4)T _{(1, 2, 3, 4)}T _12,178 _5.387 _{(2.641411, –0.734620, –0.244817, 0.583350)}T

Ax∗= (1.02107, 1.96452, 2.97978)T_,_x∗

2= 2.81353

and

h(x) =u

for allx∈R4_{into Corollary 3.4, we have}

⎧ ⎨ ⎩

x0∈R4 chosen arbitrarily,

xn+1=αnxn+βnu+γnPC(xn–λn(A∗Axn–A∗b)), ∀n= 0, 1, 2, . . . ,

whereb= (1, 2, 3)T_,_λ

n= Axn–b

2

2A∗Axn–A∗b2₊₂_A∗_Axn_–_A∗_b₊_Axn_–_b2 ifAxn= bandλn= 0 ifAxn=b for alln= 0, 1, 2, . . . . Asn→ ∞, we havexn→x∗such thatx∗is the our solution, which de-pends on the pointuandx0. The numerical results are listed in Table 1 using the diﬀerent

pointsuand the diﬀerent starting pointsx0.

Example5.4 (A split feasibility problem) Let

A= ⎛ ⎜ ⎝

2 –1 3

4 2 5

2 0 2

⎞ ⎟

⎠, C=(x,y,z)∈R3:x+y2+ 2z≤0

and

Q=(x,y,z)∈R3:x2+y–z≤0. Find some pointx∗∈CwithAx∗∈Q.

LetH= (R3_,_·

2) andK= (R3, · 2). Takec(x,y,z) =x+y2+ 2z,q(x,y,z) =x2+y–z

andh(x,y,z) = 0 for allx,y,z∈Rinto Corollary 4.2, we have ⎧

⎨ ⎩

xn+1=αnxn+γnPCn(xn–λn(A∗Axn–A∗PQn(Axn))), ∀n= 0, 1, 2, . . . ,

whereλn= Axn–PQn(Axn)

2

2A∗Axn–A∗P_Qn(Axn)2₊₂_A∗_Axn_–_A∗_P

Qn(Axn)+Axn–PQn(Axn)2 ifAxn∈/ Qnandλn= 0 if

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[image:19.595.115.483.98.156.2]

Table 2 Results for Example 5.4 using Qu and Xiu method in [7]

Starting pointsx0 Number of iterations

CPU (s) Approximate solutionx∗ Approximate ofc(x∗)

Approximate ofq(Ax∗)

(1, 2, 3, 0, 0, 0)T ₁₈₉₀ _2.7740 _{(–0.1203, 0.0285, 0.0582)}T _{–0.00308775} _–0.0152279

(1, 1, 1, 1, 1, 1)T ₂₉₇₈ _4.2860 _{(0.8603, –0.1658, –0.5073)}T _–0.12681 _1.08162

[image:19.595.116.481.190.246.2]

(1, 2, 3, 4, 5, 6)T ₃₃₁₇ _4.8570 _{(3.6522, –0.1526, –2.3719)}T _–1.06831 _15.5579

Table 3 Results for Example 5.4 using Qu and Xiu method in [6]

(1, 2, 3)T ₆₄ _0.1570 _{(–0.4019, 0.0674, 0.1967)}T _{–0.00395724} _0.0322236

(1, 1, 1)T ₈₁ _0.0940 _{(0.3568, 0.0343, –0.2652)}T _–0.172424 _0.426806

rand(3, 1)∗10 105 0.0940 (0.8747, 0.0795, –0.6876)T _–0.49418 _1.5322

Table 4 Results for Example 5.4 using Li method in Algorithm 1 [8]

(1, 2, 3)T 4 0.1410 (–0.4024, 0.0658, 0.1958)T –0.00647036 0.0319258

(1, 1, 1)T 5 0.0940 (0.3532, 0.0392, –0.2707)T –0.186663 0.43465

rand(3, 1)∗10 8 0.0940 (0.8768, 0.0604, –0.6844)T _–0.488352 _1.51358

Table 5 Results for Example 5.4 using Algorithm in Corollary 4.2

(1, 2, 3)T ₁₂₂₀ _1.492 _{(–0.0009, 0.0007, 0.0002)}T _{–0.00049951} _0.00050081

(1, 1, 1)T ₁₀₆₂ _1.342 _{(0.0004, 0.0007, –0.0006)}T _{–0.00079951} _0.00130016

(4, 5, 6)T ₁₂₂₅ _1.642 _{(0.0006, 0.0007, –0.0007)}T _{–0.00079951} _0.00140036

rand(3, 1)∗10 1197 1.832 (0.0008, 0.0004, –0.0007)T _{–0.00059984} _0.00110064

(6, 5, 4)T ₂₅₆₉ _4.346 _{(0.0020, 0.0004, –0.0015)}T _{–0.00099984} _0.00190400

(2, 2, 2)T ₁₃₆₅ _2.083 _{(0.0007, 0.0006, –0.0008)}T _{–0.00089964} _0.00140049

(3, 2, 1)T 2093 3.525 (0.0015, 0.0005, –0.0013)T –0.00109975 0.00180225

which depends on the zero point andx0. The numerical results are listed in Table 5 using

the diﬀerent starting pointsx0, and we compare the results of Qu and Xiu [6, 7], and Li

[8], which are listed in Table 2, Table 3 and Table 4, respectively. We found that in the calculation approximate value ofq(Ax∗) using the our algorithm method was ﬁt to the solution than the algorithms method of Qu and Xiu, and Li.

Example5.5 (A convex feasibility problem) LetC={a∈R3: 2≤ a2≤3}. Find some

pointx∗∈Cwhich satisﬁes the following system of nonlinear inequalities: ⎧

⎨ ⎩

y2₊_z2_{– 4}_≤_0,

–x2₊_z_{– 1}_≤_0,

wherex,y,z∈R. LetH= (R3_,_·

2),K= (R3, · 2) andu∈C. TakeA=I,h(x,y,z) =uand

C=(x,y,z)∈R3:c(x,y,z) =supc1(x,y,z),c2(x,y,z)

≤0,

Q=(x,y,z)∈R3:q(x,y,z) =supq1(x,y,z),q2(x,y,z)

[image:19.595.115.480.281.338.2] [image:19.595.117.478.372.472.2]

(20)

[image:20.595.117.479.98.216.2]

ChoicenessuStarting pointsx0

CPU (s) Approximate solutionx∗ Approximate ofq1(x∗)

Approximate ofq2(x∗)

Approximate ofx∗2

(2, 0, 0)T _{(1, 1, 1)}T ₁₉₀₁ _3.295 _{(1.999999, 0.001343, 0.001343)}T _–4.00000 _–4.99865 _2.00000 (2, 0, 0)T _{(2, 2, 2)}T ₂₃₃₆ _4.486 _{(1.999998, 0.001651, 0.001651)}T _–3.99999 _–4.99834 _2.00000 (2, 0, 0)T _{(1, 2, 3)}T ₂₇₁₆ _4.576 _{(1.999998, 0.001506, 0.002259)}T _–3.99999 _–4.99773 _2.00000

(3, 0, 0)T _{(1, 1, 1)}T ₂₂₀₉ _3.935 _{(2.998244, 0.000948, 0.000948)}T _–4.00000 _–9.98852 _2.99824 (3, 0, 0)T _{(2, 2, 2)}T ₂₄₁₇ _3.505 _{(2.999133, 0.001595, 0.001595)}T _–3.99999 _–9.99320 _2.99913 (3, 0, 0)T _{(1, 2, 3)}T ₂₈₂₁ _3.955 _{(2.998563, 0.001346, 0.002019)}T _–3.99999 _–9.98936 _2.99856

(1, 2, 0)T _{(1, 1, 1)}T ₁₆₆₇ _3.375 _{(1.000131, 1.998964, 0.001299)}T _–0.00414 _–1.99896 _2.23520 (1, 2, 0)T _{(2, 2, 2)}T ₂₀₆₈ _3.745 _{(1.000919, 1.999901, 0.001849)}T _–0.00039 _–1.99999 _2.23639 (1, 2, 0)T _{(1, 2, 3)}T ₂₃₆₃ _4.566 _{(0.999977, 1.999833, 0.002357)}T _–0.00066 _–1.99760 _2.23591

such thatc1(x,y,z) =x2+y2+z2– 9,c2(x,y,z) = –x2–y2–z2+ 4,q1(x,y,z) =y2+z2– 4 and

q2(x,y,z) = –x2+z– 1 for allx,y,z∈Rinto Corollary 4.2, we have

⎧ ⎨ ⎩

xn+1=αnxn+βnu+γnPCn(xn–λn(xn–PQnxn)), ∀n= 0, 1, 2, . . . ,

whereλn=

xn–P_Qnxn

3xn–P_Qnxn+2 for alln= 0, 1, 2, . . . . Asn→ ∞, we havexn→x∗such thatx∗is

the our solution, which depends on the pointuandx0. The numerical results are listed in

Table 6 using the diﬀerent pointsu∈Cand the diﬀerent starting pointsx0.

Example5.6 (A convex minimization problem) Find minimize of f(x,y,z) = (x– 2)2+ (y– 2)2_{+ (}_z_{– 3)}2_{subject to the constraint}_g₍_x_,_y_,_z_{) =}_x2₊_y2₊_z2_{– 4 = 0 where}_x_,_y_,_z_∈_R_.

Deﬁne the Lagrange functionL(x,y,z,λ) as follows:

L(x,y,z,λ) =f(x,y,z) +λg(x,y,z)

= (x– 2)2+ (y– 2)2+ (z– 3)2+λx2+y2+z2– 4,

where x,y,z,λ∈R. Hence, the our solution set is equivalent to the solution set of the following system of nonlinear equations:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2(x– 2) + 2λx=Lx= 0, 2(y– 2) + 2λy=Ly= 0, 2(z– 3) + 2λz=Lz= 0,

x2₊_y2₊_z2_{– 4 =}_L

λ= 0.

LetH= (R4_,_·

2) andK= (R4, · 2). TakeA=I,h(x,y,z,λ) = 0 and

C=(x,y,z,λ)∈R4:c(x,y,z,λ) = sup

1≤i≤4

–qi(x,y,z,λ)≤0

,

Q=

(x,y,z,λ)∈R4:q(x,y,z,λ) = sup

1≤i≤4

qi(x,y,z,λ)≤0

,

such that

q1(x,y,z,λ) = 2(x– 2) + 2λx, q2(x,y,z,λ) = 2(y– 2) + 2λy,

(21)

[image:21.595.157.440.97.164.2]

Starting pointsx0

CPU (s) Approximate solution Lagrange

multiplier

(1, 2, 1, 0)T _15,562 _17.745 _{(0.969812, 0.969905, 1.455039)}T _1.061319

(2, 2, 2, 0)T _15,566 _14.511 _{(0.969905, 0.969812, 1.455039)}T _1.061319

(1, 2, 3, 0)T _15,567 _11.666 _{(0.969812, 0.969905, 1.455039)}T _1.061319

(4, 5, 6, 0)T _15,566 _17.495 _{(0.969812, 0.969905, 1.455039)}T _1.061319

for allx,y,z,λ∈Rinto Corollary 4.2, we have ⎧

⎨ ⎩

xn+1=αnxn+γnPCn(xn–λn(xn–PQnxn)), ∀n= 0, 1, 2, . . . ,

whereλn=

xn–P_Qnxn

3xn–P_Qnxn+2 for alln= 0, 1, 2, . . . . Asn→ ∞, we havexn→x∗ such that the

our solution is _√2

17(2, 2, 3)

T_{with some Lagrange multiplier}_λ_{, which depends on the zero} point andx0. The numerical results are listed in Table 7 using the diﬀerent starting points

x0, and we switch the stopping criterionto 10–4for the veriﬁcation to the our solution.

6 Conclusion

In this paper, we obtain an iterative scheme using the gradient projection method with a new step size, which is not depend on the related matrix inverses and the largest eigen-value (or the spectral radius of the self-adjoint operator) of the related matrix, based on Moudafi’s viscosity approximation method for solving thesplit common fixed point prob-lem (SCFP) for two firmly nonexpansive mappings and also solving thesplit feasibility problem(SFP) such that other strong convergence theorems for the SCFP and the SFP are obtained.

Acknowledgements

The author would like to thank the Faculty of Science, Maejo University for its ﬁnancial support.

Competing interests

The author declares that he has no competing interests.

Authors’ contributions

The author read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 12 December 2017 Accepted: 10 May 2018 References

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