Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 840978, 12 pages
doi:10.1155/2011/840978
Research Article
The Over-Relaxed A-Proximal Point
Algorithm for General Nonlinear Mixed Set-Valued
Inclusion Framework
Xian Bing Pan,
1
HongGangLi,
2
and An Jian Xu
3
1
Yitong College, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2
Institute of Applied Mathematics Research, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China
3
College of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China
Correspondence should be addressed to Xian Bing Pan, [email protected]
Received 16 November 2010; Accepted 10 January 2011
Academic Editor: T. Benavides
Copyright q 2011 Xian Bing Pan 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.
The purpose of this paper is 1 a general nonlinear mixed set-valued inclusion framework for
the over-relaxed A-proximal point algorithm based on the A, η-accretive mapping is introduced,
and 2 it is applied to the approximation solvability of a general class of inclusions problems
using the generalized resolvent operator technique due to Lan-Cho-Verma, and the convergence of
and X
∗
,let2
X
denote the family of all the nonempty subsets of X,andletCBX denote
the family of all nonempty closed bounded subsets of X. The generalized duality mapping
J
q
: X → 2
X
∗
is single-valued if X
∗
is strictly convex 14,orX is uniformly smooth space. In
what follows we always denote the single-valued generalized duality mapping by J
q
in real
uniformly smooth Banach space X unless otherwise stated. We consider the following general
nonlinear mixed set-valued inclusion problem with A, η-accretive mappings (GNMSVIP).
Finding x ∈ X such that
0 ∈ F
A
x
M
x
x
1
,x
2
≥ r
x
1
− x
2
q
, ∀y
i
∈ M
x
i
,i 1, 2; 2.2
ii m-relaxed η-accretive, if there exists a constant m>0 such that
y
1
− y
2
,J
q
i 1, 2
; 2.3
iii c-cocoercive, if there exists a constant c such that
y
1
− y
2
,J
q
η
x
1
,x
2
≥ c
y
1
− y
2
q
ρ,M
: X → X is defined by
R
A,η
ρ,M
x
A ρM
−1
x
∀x ∈ X
,
2.5
where ρ>0 is a constant.
Remark 2.3. The A, η-accretive mappings are more general than H, 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 operators 1–7, 11–13.
Lemma 2.4 see 6. Let η : X × X → X be τ-Lipschtiz continuous mapping, A : X → X be an
r-strongly η-accretive mapping, and M : X → 2
X
be an A, η-accretive set-valued mapping. Then
x − y
∀x, y ∈ X
,
2.6
where ρ ∈ 0,r/m.
In the study of characteristic inequalities in q-uniformly smooth Banach spaces, Xu
14 proved the following result.
Lemma 2.5. Let X be a real uniformly smooth Banach space. Then X is q-uniformly smooth if and
only if there exists a constant c
q
> 0 such that for all x, y ∈ X,
x y
q
≤
x
q
q
y, J
q
X
be a set-valued mapping. The map M
−1
, the inverse of
M : X → 2
X
, is said to be general u, t-Lipschitz continuous at 0 if, and only if there exist
two constants u, t ≥ 0 for any w ∈ B
t
{w : w≤t, w ∈ X},asolutionx
∗
of the inclusion
0 ∈ Mxx
∗
∈ M
−1
0 exist and the x
∗
such that
x − x
∗
≤ u
w
∀x ∈ M
−1
2
,J
q
A
R
A,η
ρ,M
A
x
1
− ρF
A
x
1
− A
R
A,η
ρ,M
− ρF
A
x
1
− A
R
A,η
ρ,M
A
x
2
− ρF
A
x
2
− A
R
A,η
ρ,M
A
x
2
− ρF
A
x
2
q
I
k
A − ρFA and
s
i
Ax
i
− ρFAx
i
x
i
∈ X, i 1, 2, then by using Definition 2.2, Lemmas 2.4, 2.5,and
3.2, we can have
I
k
x
1
− I
k
x
2
q
Ax
1
− A
Fixed Point Theory and Applications 5
≤ c
q
Ax
1
− Ax
2
q
− q
A
x
1
− A
x
2
,J
q
A
R
A,η
− ρF
A
x
1
− A
R
A,η
ρ,M
A
x
2
− ρF
A
x
2
− ρF
A
x
1
− A
R
A,η
ρ,M
A
x
2
− ρF
A
x
2
R
A,η
ρ,M
A
x
2
− ρF
A
x
2
q
≤ c
q
Ax
1
− Ax
2
R
A,η
ρ,M
A
x
2
− ρF
A
x
2
q
.
3.4
Therefore, 3.3 holds.
Lemma 3.3. Let X be a q-uniformly smooth Banach space, η : X × X → X be a τ-Lipschtiz
continuous mapping, A : X → X be an r-strongly η-accretive and nonexpansive mapping,
F : X → X be an ξ-Lipschtiz continuous mapping, and I
k
∗
.
3.5
iii For a x
∗
∈ X, holds
I
k
x
∗
A
x
∗
− A
R
A,η
ρ,M
A
x
∗
q
1 c
q
ρ
q
ξ
q
− qρβ
<τ
r − mρ
1 c
q
ρ
q
ξ
q
>qρβ
,
3.7
6 Fixed Point Theory and Applications
where c
q
> 0 isthesameasinLemma 2.5, and ρ ∈ 0,r/m. Then the problem 2.1 has a solution
x
∗
∈ X.
x
i
− ρF
A
x
i
i 1, 2
, 3.9
then by 3.1 and 3.3, we have
N
x
1
− N
x
2
− A
x
2
− ρ
F
A
x
1
− F
A
x
2
.
3.10
By using r-strongly η-accretive of A, β-strongly η-accretive of F,andLemma 2.5,weobtain
Ax
q
c
q
ρ
q
FAx
1
− F
A
x
2
q
− qρ
F
A
x
1
− F
Ax
1
− Ax
2
q
.
3.11
Combining 3.10-3.11, by using nonexpansivity of A, we have
N
x
1
− N
x
2
≤ θ
∗
x
1
− x
2
∈ X
such that x
∗
Nx
∗
,and
x
∗
N
x
∗
R
A,η
ρ,M
A
x
∗
− ρF
A
x
∗
n
}
∞
n0
be three nonnegative sequences such that
∞
n1
b
n
< ∞,a lim sup
n →∞
a
n
≥ 1,ρ
n
↑ ρ ≤∞,
3.15
where ρ
n
, ρ ∈ 0,r/mn 0, 1, 2, ·, ·, · and each satisfies condition 3.7.
Step 1. For an arbitrarily chosen initial point x
0
∈ X,set
A
x
1
A
x
0
− ρ
0
F
A
x
0
≤ b
0
y
0
− A
x
0
y
n
− A
R
A,η
ρ
n
,M
A
x
n
− ρ
n
F
A
x
n
i η : X × X → X is τ-Lipschtiz continuous;
ii A : X → X be an r-strongly η-accretive mapping and nonexpansive;
iii F : X → X be an ξ-Lipschtiz continuous and β-strongly η-accretive mapping;
iv M : X → 2
X
be an A, η-accretive set-valued mapping;
8 Fixed Point Theory and Applications
v the FA M
−1
be u, t-Lipschitz continuous at 0u ≥ 0;
vi {a
n
}
∞
n0
a
n
≥ 1, {b
n
}
∞
n0
and {ρ
n
}
∞
n0
be three nonnegative sequences such that
∞
,J
q
A
R
A,η
ρ,M
A
x
n
− ρF
A
x
n
− A
R
A,η
ρ,M
A
x
n
− A
R
A,η
ρ,M
A
x
∗
− ρF
A
x
∗
q
c
q
a − 1
q
a
q
q
1 − a
aγ
d
q
,
a lim sup
n →∞
a
n
,d lim sup
n →∞
d
n
lim sup
be a solution of the Framework 2.1 for the conditions i–iv and
Lemma 3.4. Suppose that the sequence {x
n
} which generated by the hybrid proximal point
Algorithm 3.5 is bounded, from Lemma 3.4, we have
A
x
∗
1 − a
n
A
x
∗
a
n
A
R
A,η
ρ
n
,M
A,η
ρ
n
,M
A
x
n
− ρ
n
F
A
x
n
− x
∗
≤ d
n
A
,M
Ax
∗
− ρ
n
FAx
∗
x
∗
.
Fixed Point Theory and Applications 9
For I
k
A − AR
A,η
ρ,M
A − ρ
n
FA, and under the assumptions including the condition
vii3.21, then I
k
x
n
→ 0n →∞ since the FAM
−1
is u, t-Lipschitz continuous at 0.
Indeed, it follows that R
A,η
ρ
n
n
−ρ
n
FAx
n
. Next, by using the condition iv and 3.1, and setting
w ρ
−1
n
I
k
x
n
and z R
A,η
ρ
n
,M
Ax
n
− ρ
n
FAx
n
, we have
R
A,η
n
I
k
x
n
, ∀n>n
. 3.26
Now applying Lemma 3.3,weget
R
A,η
ρ
n
,M
A
x
n
− ρ
n
n
F
A
x
n
− R
A,η
ρ
n
,M
A
x
∗
− ρ
n
F
A
x
∗
u
ρ
n
q
I
k
x
n
− I
k
x
∗
q
≤
u
ρ
n
q
−
qγ − 1
n
,M
A
x
∗
− ρ
n
F
A
x
∗
q
c
q
Ax
n
− Ax
∗
− x
∗
≤ d
n
A
x
n
− A
x
∗
, 3.28
where d
n
q
c
q
u
q
/2γ − 1r
A
x
n
a
n
A
R
A,η
ρ
n
,M
A
x
n
− ρ
n
F
A
x
n
q
≤
1 − a
n
A
x
n
a
n
A
R
A,η
ρ
n
,M
s
n
1 − a
n
A
x
n
− A
x
∗
a
n
A
R
A,η
ρ
n
,M
s
n
− A
x
∗
q
a
n
A
R
A,η
ρ
n
,M
s
n
− A
R
A,η
,J
q
A
R
A,η
ρ
n
,M
s
n
− A
R
A,η
ρ
n
,M
s
∗
≤ c
q
,M
s
∗
q
q
1 − a
n
a
n
γ
R
A,η
ρ
n
,M
s
n
− R
A,η
ρ
− q
1 − a
n
a
n
γ
R
A,η
ρ
n
,M
A
x
n
− ρ
n
F
A
x
n
γ
d
q
n
Ax
n
− Ax
∗
q
.
3.30
Thus, we have
Az
n1
− Ax
∗
q
≤ θ
n
Ax
a
n
γ
d
q
n
< 1,
3.32
and a
q
n
q1 − a
n
a
n
γ>0, a
n
≥ 1,
∞
n1
b
n
< ∞,andd
n
q
− Ax
n
.It
follows that
A
x
n1
− A
z
n1
≤
1 − a
n
A
x
n
a
A
x
n
≤ a
n
y
n
− R
A,η
ρ
n
,M
A
x
n
− ρ
Next, we can obtain
A
x
n1
− A
x
∗
≤
A
z
n1
− A
x
∗
A
− A
x
n
≤
A
z
n1
− A
x
∗
b
n
A
x
n1
− A
b
n
A
x
∗
− A
x
n
.
3.34
This implies from 3.38 and 3.39 that
A
x
n1
− A
x
∗
q
c
q
a
n
− 1
q
a
q
n
q
1 − a
n
a
n
γ
d
q
n
< 1,
3.36
where a
n
1 − b
n
lim sup
n →∞
θ
n
q
c
q
a − 1
q
a
q
q
1 − a
aγ
d
q
.
3.37
}
∞
n0
and {ρ
n
}
∞
n0
be the same as in Algorithm 3.5.If
0 < 1 − a
2
1 − γd
2
− a
1 −
2γ − 1
d
2
< 1, 3.38
then the bounded sequence {x
n
} generated by the general over-relaxed A-proximal point algorithm
converges linearly to a solution x
u
2
/2γ − 1r
2
u
2
ρ
2
n
u
2
/2γ − 1r
2
u
2
ρ
2
, a
lim sup
n →∞
a
n
,
∞
n1
b
n
tions
, vol. 13, no. 2, pp. 277–289, 2008.
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inclusions involving A, η-accretive mappings,” Advances in Nonlinear Variational Inequalities, vol. 11,
no. 1, pp. 41–50, 2008.
13 H Y. Lan, “On multivalued nonlinear variational inclusion problems with A, η-accretive mappings
in Banach spaces,” Journal of Inequalities and Applications, vol. 2006, Article ID 59836, 12 pages, 2006.
14 H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 16, no. 12, pp. 1127–1138, 1991.
15 X. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,” Proceedings of the
American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991.
16 R. U. Verma, “A hybrid proximal point algorithm based on the A, η-maximal monotonicity
framework,” Applied Mathematics Letters, vol. 21, no. 2, pp. 142–147, 2008.
17 H. G. Li, A. J. Xu, and M. M. Jin, “An Ishikawa-hybrid proximal point algorithm for nonlinear set-
valued inclusions problem based on A, η-accretive framework,” Fixed Point Theory and Applications,
vol. 2010, Article ID 501293, 12 pages, 2010.
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