Fixed Point Theory and Applications Volume 2010, Article ID 501293,12pages doi:10.1155/2010/501293
Research Article
An Ishikawa-Hybrid Proximal Point Algorithm for
Nonlinear Set-Valued Inclusions Problem Based on
A, η
-Accretive Framework
Hong Gang Li,
1An Jian Xu,
1and Mao Ming Jin
21Institute of Applied Mathematics Research, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China
2Institute of Nonlinear Analysis Research, Changjiang Normal University, Fuling Chongqing,
400803, China
Correspondence should be addressed to Hong Gang Li,[email protected]
Received 30 April 2010; Accepted 8 June 2010
Academic Editor: Massimo Furi
Copyrightq2010 Hong Gang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A general nonlinear framework for an Ishikawa-hybrid proximal point algorithm using the notion ofA, η-accretive is developed. Convergence analysis for the algorithm of solving a nonlinear set-valued inclusions problem and existence analysis of solution for the nonlinear set-valued inclusions problem are explored along with some results on the resolvent operator corresponding toA, η-accretive mapping due to Lan-Cho-Verma in Banach space. The result that sequence{xn} generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem with the convergence rateθis proved.
1. Introduction
resolvent operators associated with them, respectively. Moreover, by using the resolvent operator technique, many authors constructed some approximation algorithms for some nonlinear variational inclusions in Hilbert spaces or Banach spaces. Recently, Verma has developed a hybrid version of the Eckstein and Bertsekas 11 proximal point algorithm, introduced the algorithm based on theA, η-maximal monotonicity framework12, and studied convergence of the algorithm.
On the other hand, in 2008, Li13studied the existence of solutions and the stability of perturbed Ishikawa iterative algorithm for nonlinear mixed quasivariational inclusions involving A, η-accretive mappings in Banach spaces by using the resolvent operator technique in14.
Inspired and motivated by recent research work in this field, in this paper, a general nonlinear framework for a Ishikawa-hybrid proximal point algorithm using the notion of
A, η-accretive is developed. Convergence analysis for the algorithm of solving a nonlinear valued inclusions problem and existence analysis of solution for the nonlinear set-valued inclusions problem are explored along with some results on the resolvent operator corresponding toA, η-accretive mapping due to Lan et al. in Banach space. The result that sequence{xn}generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem as the convergence rateθis proved.
2. Preliminaries
LetX be a real Banach space with dual spaceX∗ and·,·and let the dual pair betweenX andX∗, 2X denote the family of all the nonempty subsets ofX and CBXthe family of all nonempty closed bounded subsets ofX. The generalized duality mappingJq : X → 2X∗ is defined by
Jqx
f∗∈X∗:x, f∗ xq, f∗xq−1, ∀x∈X, 2.1
whereq >1 is a constant.
The modulus of smoothness ofXis the functionρX:0,∞ → 0,∞defined by
ρXt sup
1
2xyx−y −1 :x ≤1, y≤t
. 2.2
A Banach spaceXis called uniformly smooth if
lim t→0
ρXt
t 0. 2.3
Xis calledq-uniformly smooth if there exists a constantc >0 such that
Remark 2.1. In particular,J2is the usual normalized duality mapping, andJqx xq−2J2x
for allx /0. IfX∗is strictly convex15, orX is uniformly smooth Banach space, thenJq is single valued. In what follows we always denote the single-valued generalized duality mapping byJqin real uniformly smooth Banach spaceXunless otherwise stated.
LetA, Q, g:X → X;η, N:X×X → Xbe single-valued mappings. LetM:X×X →
2Xbe a set-valuedA, η-accretive mapping. We considernonlinear set-valued mixed variational
inclusions problem withA, η-accretive mappings (NSVMVIP).
For anyu∈X, findingx∈X,yQxsuch that
u∈Ny, gx My . 2.5
Remark 2.2. A special case of problem2.5is the following.
iIf X X∗ is a Hilbert space, N 0 is the zero operator in X, Q I is the identity operator in X, and u 0, then problem 2.5 becomes the parametric usual variational inclusion 0 ∈ Mx with aA, η-maximal monotone mapping
M, which was studied by Verma12.
iiIfX is a real Banach space,Q I is the identity operator inX, andu 0, then problem2.5becomes the parametric usual variational inclusionu∈Nx, gx
Mxwith aA, η-accretive mapping, which was studied by Li13.
It is easy to see that a number of known special classes of variational inclusions and variational inequalities in the problem2.5are studiedsee2,7,12–14.
Let us recall the following results and concepts.
Definition 2.3. A single-valued mappingη :X×X → Xis said to beτ-Lipschitz continuous
if there exists a constantτ >0 such that
ηx, y ≤τx−y, ∀x, y∈X. 2.6
Definition 2.4. A single-valued mappingA:X → Xis said to be
iaccretive if
Ax1−Ax2, Jqx1−x2
≥0, ∀x1, x2∈X, 2.7
iistrictly accretive, ifAis accretive andAx1−Ax2, Jqx1−x20 if and only if x1x2 for allx1, x2∈X,
iiir-stronglyη-accretive if there exists a constantr >0 such that
Ax1−Ax2, Jqηx1, x2 ≥rx1−x2q, ∀x1, x2∈X, 2.8
ivα-Lipschitz continuous if there exists a constantα >0 such that
Definition 2.5. A single-valued mappingN:X×X → Xis said to be
i μ, ν-Lipschitz continuous if there exist constantsμ, ν >0 such that
Nx1, y1 −N
x2, y2 ≤μx1−x2νy1−y2 ∀xi, yi∈X, i1,2, 2.10
ii ψ, κ-Q-relaxed cocoercive with respect toAQ in the first argument if there exist constantsψ, κ >0, and for allxi∈X, yiQxii1,2such that
Ny1,· −N
y2,· , Jq
Ay1 −A
y2
≥ −ψNy1,·−Ny2,·qκx1−x2q,
2.11
whereA, Q:X → Xare single-valued mappings.
Definition 2.6. LetA : X → X, and letη : X ×X → X be single-valued mappings. A
set-valued mappingM:X → 2Xis said to be
iaccretive if
u1−u2, Jqx1−x2
≥0, ∀x1, x2∈X, u1∈Mx1, u2∈Mx2; 2.12
iiη-accretive if
u1−u2, Jqηx1, x2 ≥0, ∀x1, x2∈X, u1∈Mx1, u2∈Mx2; 2.13
iiir-strongly accretive if there exists a constantr >0 such that
y1−y2, Jqx1−x2
≥rx1−x2q, ∀xi ∈X, yi∈Mxi i1,2; 2.14
ivm-relaxedη-accretive if there exists a constantm >0 such that
u1−u2, Jqηx1, x2 ≥ −mx1−x2q, ∀x1, x2∈X, u1∈Mx1, u2∈Mx2, 2.15
vA-accretive, ifMis accretive andAρMX Xfor allρ >0,
Based on the literature8, we can define the resolvent operatorRA,ηρ,Mas follows.
Definition 2.7see8. Letη:X×X → Xbe a single-valued mapping,A:X → Xa strictly
η-accretive single-valued mapping andM : X×X → 2X a A, η-accretive mapping. The resolvent operatorRA,ηρ,M :X → Xis defined by
RA,ηρ,Mx AρM −1x, ∀x∈X,
2.16
whereρ >0 is a constant.
Remark 2.8. TheA, η-accretive mappings are more general thanH, η-monotone mappings
and m-accretive mappings in Banach space or Hilbert space, and the resolvent operators associated with A, η-accretive mappings include as special cases the corresponding resolvent operators associated with H, η-monotone operators, m-accretive mappings, A -monotone operators,η-subdifferential operators3–14,16,17.
Lemma 2.9see8. Letη : X×X → X beτ-Lipschtiz continuous mapping,A : X → X be
anr-stronglyη-accretive mapping, andM : X×X → 2XanA, η-accretive mapping. Then the
generalized resolvent operatorRA,ηρ,M:X → Xisτq−1/r−mρ-Lipschitz continuous, that is,
RA,ηρ,Mx−Rρ,MA,ηy ≤ rτ−q−mρ1 x−y, ∀x, y∈X, 2.17
whereρ∈0, r/m.
In the study of characteristic inequalities inq-uniformly smooth Banach spaces, Xu
18proved the following result.
Lemma 2.10 see18. LetX be a real uniformly smooth Banach space. Then X isq-uniformly smooth if and only if there exists a constantcq >0such that for allx, y∈X,
xyq
≤ xqqy, Jqxcqyq. 2.18
3. The Existence of Solutions
Now, we are studing the existence for solutions of problem2.5.
Lemma 3.1. LetX be a Banach space. Letη :X×X → X be aτ-Lipschtiz continuous mapping,
A:X → Xbe anr-stronglyη-accretive mapping, andM:X → 2XanA, η-accretive mapping.
Then the following statements are mutually equivalent.
iAn elementx∈Xis a solution of problem2.5.
iiFor ax∈Xand any1> λ >0, there existsyQxsuch that
x 1−λxλx−yRA,ηρ,MAy −ρNy, gx ρu , 3.1
whereρ >0is a constant.
Theorem 3.2. LetX be aq-uniformly smooth Banach space. LetA, Q, g : X → X;η, N : X ×
X → Xbe single-valued mappings, andηbe aτ-Lipschtiz continuous mapping,Aar-strongly
η-accretive andα-Lipschitz continuous mapping,Qbe aγ-strongly accretive andχ-Lipschitz continuous
mapping, and g aϕ-Lipschitz continuous mapping, respectively. LetN : X×X → X beμ, ν
-Lipschitz continuous, andψ, κ-Q-relaxed cocoercive with respect toAQin the first argument. Let M:X → 2Xbe a set-valuedA, η-accretive mapping. If the following condition holds:
τqαqc
qρqμqχqqρψμqχq−qρκ 1/qρνϕ
< τr−mρ 1−1cqχq−qγ 1/q
, 3.2
wherecq >0is the same as inLemma 2.10, andρ∈0, r/m, then the problem2.5has a solution
x∗∈X.
Proof. Define a mappingF:X → Xas follows:
Fx 1−λxλx−yRA,ηρ,MAy −ρNy, gx ρu , ∀x∈X. 3.3
For elementsx1, x2 ∈X, if we letyiQxiand
siAyi −ρNyi, gxi ρu i1,2, 3.4
then by3.1,3.3, andLemma 2.10, we have
Fx1−Fx21−λx1λx1−y1RA,ηρ,Ms1−1−λx2−λx2−y2RA,ηρ,Ms2
≤1−λx1−x2λx1−x2−y1−y2 λRA,ηρ,Ms2−RA,ηρ,Ms2
≤1−λx1−x2λ τq−1
r−mρ
ρNy2, gx1 −N
y2, gx2
Ay1 −A
y2 −ρ
Ny1, gx1 −N
y2, gx1 λx1−x2−
y1−y2 .
3.5
SinceN·,·isψ, κ-Q-relaxed cocoercive with respect toAQin the first argument andQis aχ-Lipschitz continuous mapping so we obtain
Ay1 −A
y2 −ρ
Ny1, gx1 −N
y2, gx1 q
≤Ay1 −A
y2 qcqρqNy1, gx1 −N
y2, gx1 q
−qρNy1,· −N
y2,· , Jq
Ay1 −A
y2
≤αqχqc
qρqμqχqqρψμqχq−qρκx1−x2q,
3.6
N
Byγ-strongly accretivity ofQ, we have
x1−x2−
y1−y2 q
≤ x1−x2qcqy1−y2q
−qy1−y2, Jqx1−x2
≤1cqχq−qγ x1−x2q.
3.8
Combining3.5,3.6,3.7, and3.8, we can get
Fx1−Fx2 ≤1−λ λθx1−x2, 3.9
where
θ1cqχq−qγ 1/q τ
q−1
r−mρ
αqc
qρqμqχqqρψμqχq−qρκ 1/qρνϕ
. 3.10
It follows from3.2and3.9thatFhas a fixed point inX, that is, there exists a pointx∗∈X such thatx∗∈Gx∗, and
x∗ 1−λx∗λx∗−y∗RA,η ρ,M
Ay∗ −ρNy∗, gx∗ ρu , 3.11
wherey∗Qx∗. This completes the proof.
4. Ishikawa-Hybrid Proximal Point Algorithm
Based onLemma 3.1, we develop an Ishikawa-hybrid proximal point algorithm for finding an iterative sequence solving problem2.5as follows.
Algorithm 4.1. Letx∗ be a solution of problem 2.5. Let {αn}∞n0,{βn}n∞0,{an}∞n0,{bn}∞n0 and
{ρn}∞n0, be five nonnegative sequences such that
lim
n→ ∞annlim→ ∞bn0, alim supn→ ∞ αn<1, blim supn→ ∞ βn<1, ρn↑ρ≤ ∞,
n0,1,2, . . ..
Step 1. For an arbitrarily initial pointx0∈X, we choose suitablez0∈X, letting
y0
1−β0 x0β0z0,
z0−RA,ηρ0,M
AQx0−ρ0N
Qx0, gx0 ρ0u ≤b0z0−x0,
x1 1−α0x0α0w0,
w0−RA,ηρ
0,M
AQy0 −ρ0NQy0 , gx0 ρ0u ≤a0w0−y0.
4.2
Step 2. The sequences{xn}and{yn}are generated by an iterative procedure
yn1−βn xnβnzn,
zn−RA,ηρn,M
AQxn−ρnNQxn, gxn ρnu ≤bnzn−xn,
xn1 1−αnxnαnwn,
wn−RA,ηρn,MAQyn −ρnNQyn , gxn ρnu ≤anwn−yn,
4.3
wheren1,2, . . . .
Remark 4.2. For a suitable choice of the mappingsA, η, Q, N, g, M, spaceX, and nonnegative
sequences {an}∞n0, {bn}∞n0, Algorithm 4.1 can be degenerated to a number of algorithms
involving many known algorithms which are due to classes of variational inequalities and variational inclusions12–14.
Theorem 4.3. LetX, A, N, Q, g, and Mbe the same as inTheorem 3.2, then condition3.2holds. Let {αn}∞n0, {βn}∞n0, {an}n∞0, {bn}∞n0 and {ρn}∞n0 be the same as in Algorithm 4.1. Then the sequence{xn}generated by hybrid proximal pointAlgorithm 4.1converges linearly to a solutionx∗
of problem2.5as
τqαqc
qρqμqχqqρψμqχq−qρκ 1/qρνϕ
< τr−mρ 1−1cqχq−qγ 1/q
, 4.4
wherecq>0is the same as inLemma 2.10,ρ∈0, r/m, and the convergence rate is
θ 1−a a
τq−1
r−mρ
q
αqqcqψμqρ−qκρμqρqρνϕ1cqχq−qγ 1/q
. 4.5
Proof. Suppose that the sequence{xn}is the the sequence generated by the Ishikawa-hybrid
proximal pointAlgorithm 4.1, and thatx∗is a solution of problem2.5. FromLemma 3.1and conditionαn ∈0,1, we can get
x∗ 1−αnx∗αnx∗−y∗RA,η ρn,M
Ay∗ −ρnNy∗, gx∗ ρnu , 4.6
For alln≥0, andynQxn, setting
un1 1−αnxnαn
xn−ynRA,ηρn,MAyn −ρnNyn, gxn ρnu , 4.7
we find the estimation
un1−x∗ ≤1−αnxn−x∗αn
×RA,ηρn,MAyn −ρnNyn, gxn ρnu
−RA,ηρn,MAy∗ −ρ
nNy∗, gx∗ ρnu
αnx∗−xn−y∗−yn ≤1−αnxn−x∗αn τ q−1 r−mρn
×Ayn −Ay∗ −ρnNyn, gxn −Ny∗, gxn
ρnNy∗, gxn −Ny∗, gx∗ αnx∗−xn−y∗−yn .
4.8
By the conditions andLemma 2.10, we have
A
yn −Ay∗ −ρnNyn, gxn −Ny∗, gxn q
≤Ayn−Ay∗qcqρqnNyn, gxn−Ny∗, gxn q
−qρnNyn, gxn −Ny∗, gxn , JqAyn −Ay∗
≤αqχqc
qρnqμqχqqρnψμqχq−qρnκ
xn−x∗q, Nx∗, gx
n −Nx∗, gx∗ ≤νϕxn−x∗,
4.9
x∗−x
n−y∗−yn ≤1cqχq−qγ x∗−xnq. 4.10
It follows from4.8–4.10that
un1−x∗ ≤θnxn−x∗, 4.11
where
θn 1−αn αnhn, 4.12
hn τ
q−1 r−mρn
q
αqχqcqρq
nμqχqqρnψμqχq−qρnκρnνϕ
1cqχq−qγ 1/q.
Sincexn1 1−αnxnαnwnand4.3,xn1−xnαnwn−xnand
xn1−un1 ≤1−αnxnαnwn
−1−αnxnαn
xn−ynRA,ηρn,MAyn −ρnNyn, gxn ρnu
≤αnwn−RA,ηρn,MAyn −ρnNyn, gxn ρnu αnxn−yn
≤αnanwn−ynαnxn−yn.
4.14
Next, we calculate
xn1−x∗ ≤ un1−x∗xn1−un1 ≤ un1−x∗αnanwn−ynαnxn−yn
≤ un1−x∗αnanwn−xnyn−xn αnxn−yn
≤ un1−x∗αnanwn−xnαnan1yn−xn
≤θnxn−x∗anx∗−xnanxn1−x∗αnan1yn−x∗x∗−xn
≤θnxn−x∗anxn1−x∗αnan1yn−x∗ anαnan1x∗−xn.
4.15
This implies that
xn1−x∗ ≤
θnan 1αnan
1−an xn−x
∗1αnan 1−an
yn−x∗, 4.16
letting
x∗1−β
n x∗βn
x∗−y∗RA,η ρn,M
Ay∗ −ρ
nNy∗, gx∗ ρnu . 4.17
For alln≥0, set
vn1−βn xnβn
xn−ynRA,ηρn,MAyn −ρnNyn, gxn ρnu . 4.18
For the same reason,
vn−x∗ ≤ϑnxn−x∗, 4.19
where
yn−x∗≤ϑnxn−x∗bnbnβnβn x∗−xnx∗−yn . 4.21
Furthermore,
yn−x∗≤
ϑnbnbnβnβn
1−bnbnβnβn xn−x
∗. 4.22
Combining4.16-4.22, then we have
xn1−x∗ ≤
θnan 1αnan
1−an
1αnan
1−an
ϑnbnbnβnβn 1−bnbnβnβn
xn−x∗.
4.23
By4.4and the condition limn→ ∞anlimn→ ∞bn0, we can see that
θϑlim sup n→ ∞ θn
1−a a
τq−1
r−mρ
q
αqqcqψμqρ−qκρμqρqρνϕ
1cqχq−qγ 1/q
<1,
4.24
and the convergence rate is θ.By 4.4, if h limn→ ∞hn, then it follows that 0 < h < 1 and 0 < θ < 1. Therefor, the sequence{xn}generated hybrid proximal pointAlgorithm 4.1 converges linearly to a solutionx∗of problem2.5with convergence rateθ. This completes the proof.
Remark 4.4. For a suitable choice of the mappingsA, η, Q, g, N,andM, we can obtain several
known results12–14,17as special cases ofTheorem 3.2andTheorem 4.3.
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