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Iterative algorithms with errors for zero points of $m$-accretive operators
Fixed Point Theory and Applications2013, 2013:148 doi:10.1186/1687-1812-2013-148
Sun Young Cho ([email protected])Xiaolong Qin ([email protected])Lin Wang ([email protected])
ISSN 1687-1812
Article type Research
Submission date 20 March 2013
Acceptance date 21 May 2013
Publication date 6 June 2013
Article URL http://www.fixedpointtheoryandapplications.com/content/2013/1/148
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ITERATIVE ALGORITHMS WITH ERRORS FOR ZERO POINTS OF
m-ACCRETIVE OPERATORS
XIAOLONG QIN1, SUN YOUNG CHO2, LIN WANG3
Abstract. In this paper, we study the convergence of paths for continuous pseudocontractions in a real
Banach space. As an application, we consider the problem of finding zeros ofm-accretive operators based
on an iterative algorithm with errors. Strong convergence theorems for zeros of m-accretive operators
are established in a real Banach space.
Keywords: accretive operator; fixed point; nonexpansive mapping; iterative algorithm; zero point.
2000 AMS Subject Classification: 47H05, 47H09, 47J25, 65J15.
1. Introduction and preliminaries
Let C be a nonempty closed convex subset of a real Banach space E and let E be the
dual space of E. Let , denote the pairing between E and E. The normalized duality
mapping J : E 2E
is defined by
J(x) = {f E : x, f = x2 = f2}
for all x E. In the sequel, we use j to denote the single-valued normalized duality
mapping. Let UE = {x E : x = 1}. E is said to be smooth or is said to have a
Gateaux differentiable norm if the limit
limt0
x + ty xt
exists for each x, y UE. E is said to have a uniformly Gateaux differentiable norm
if for each y UE, the limit is attained uniformly for all x UE. E is said to be
uniformly smooth or is said to have a uniformly Frechet differentiable norm if the limit is
attained uniformly for x, y UE. It is known that if the norm of E is uniformly Gateaux
differentiable, then the duality mapping J is single-valued and uniformly norm to weak
continuous on each bounded subset of E.1
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2 X. QIN, S.Y. CHO, L. WANG
Recall that a closed convex subset C of a Banach space E is said to have a normal
structure if for each bounded closed convex subset K of C which contains at least two
points, there exists an element x of K which is not a diametral point of K, i.e.,
sup{x y : y K} < d(K),
where d(K) is the diameter of K. It is well known that a closed convex subset of a
uniformly convex Banach space has the normal structure and a compact convex subset of
a Banach space has the normal structure; see [1] for more details.
Let T : C C be a nonlinear mapping. In this paper, we use F(T) to denote the
set of fixed points of T. Recall that T is said to be contractive if there exits a constant
(0, 1) such that
T x T y x y, x, y C.
For such a case, we also call T an -contraction. T is said to be a strong contraction if
there exits a constant (0, 1) and some j(x y) J(x y) such that
T x T y , j(x y) x y2, x, y C.
For such a case, we also call T an -strong pseudocontraction. T is said to be nonexpansive
if
T x T y x y, x, y C.
T is said to be pseudocontractive if there exists some j(x y) J(x y) such that
T x T y , j(x y) x y2, x, y C.
For the class of nonexpansive mappings, one classical way to study nonexpansive map-
pings is to use contractions to approximate a nonexpansive mapping ([2],[3]). More pre-
cisely, take t (0, 1) and define a contraction Tt : C C by
Ttx = tu + (1 t)T x, x C,
where u C is a fixed point. Banachs contraction mapping principle guarantees that Tt
has a unique fixed point xt in C. That is,
xt = tu + (1 t)T xt.
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 3
In the case that T enjoys a fixed point, Browder [2] proved that if E is a Hilbert space,
then xt converges strongly to a fixed point of T. Reich [3] extended Browders result to
the setting of Banach spaces and proved that if E is a uniformly smooth Banach space,then xt converges strongly to a fixed point of T.
Let D be a nonempty subset of C. Let Q : C D. Q is said to be contraction if
Q2 = Q; sunny if for each x C and t (0, 1), we have Q
tx + (1 t)Qx
= Qx; sunny
nonexpansive retraction if Q is sunny, nonexpansive, and contraction. K is said to be a
nonexpansive retract of C if there exists a nonexpansive retraction from C onto D.
The following result, which was established in [4,5], describes a characterization of
sunny nonexpansive retractions on a smooth Banach space.
Let E be a smooth Banach space and let C be a nonempty subset ofE. Let Q : E C
be a retraction and let j be the normalized duality mapping on E. Then the following
are equivalent:
(1) Q is sunny and nonexpansive;
(2) Qx Qy2 x y, j(Qx Qy), x, y E;
(3) x Qx,j(y Qx) 0, x E, y C.
Recently, Xu [6] improved Reichs results by considering the viscosity approximation
method which was first introduced by Moudafi [7]. To be more precise, he studied the
following continuous scheme:
xt = tf(xt) + (1 t)T xt, (X)
where f is an -contraction and T is a nonexpansive mapping. He showed that the
sequence {xt} converges strongly to a point in F(T) in a uniformly smooth Banach space.
If one defines Q : C F(T), where C denotes the set of -contractions, by
Q(f) := limt
xt,
then Q(f) solves the following variational inequality:
(I f)Q(f), j(Q(f) p) 0, p F(T).
The questions that naturally arise in connection with Xus results are as follows.
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4 X. QIN, S.Y. CHO, L. WANG
(1) Can one extend the framework of the space from uniformly smooth Banach spaces
to a more general Banach space ?
(2) Can one extend the viscosity approximation method by considering strong pseu-docontractions instead of contractions ?
(3) Do Xus results still hold for a larger class of nonlinear mappings ?
In Section 2, we give an affirmative answer to the above questions.
Let I denote the identity operator on E. An operator A E E with domain
D(A) = {z E : Az = } and range R(A) = {Az : z D(A)} is said to be accretive if
for each xi D(A) and yi Axi, i = 1, 2, there exists j(x1 x2) J(x1 x2) such that
y1 y2, j(x1 x2) 0.
An accretive operator A is said to be m-accretiveifR(I+ rA) = E for all r > 0. In a real
Hilbert space, an operator A is m-accretive if and only if A is maximal monotone. In this
paper, we use A1(0) to denote the set of zeros of A. Interest in accretive operators, which
is an important class of nonlinear operators, stems mainly from their firm connection
with equations of evolution. It is known that many physically significant problems can
be modeled by initial value problems of the form
x(t) + Ax(t) = 0, x(0) = x0, (1.1)
where A is an accretive operator in an appropriate Banach space. Typical examples where
such evolution equations occur can be found in the heat, wave or Schrodinger equations.
If x(t) is dependent on t, then (1.1) is reduced to
Au = 0, (1.2)
whose solutions correspond to the equilibrium points of the system (1.1). Consequently,
considerable research efforts have been devoted, especially within the past 40 years or so,
to methods for finding approximate solutions (when they exist) of equation (1.2). An
early fundamental result in the theory of accretive operators, due to Browder [8], states
that the initial value problem (1.1) is solvable ifA is locally Lipschitz and accretive on E.
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 5
For an accretive operator A, we can define a nonexpansive single-valued mapping Jr :
R(I+ rA) D(A) by Jr = (I+ rA)1 for each r > 0, which is called the resolvent of A.
One of classical methods for studying the problem 0 Ax, where A E E is anaccretive operator, is the following:
x0 E, xn+1 = Jrnxn, n 0,
where Jrn = (I + rnA)1 and {rn} is a sequence of positive real numbers.
As we know, many well-known problems arising in various branches of science can
be studied by using algorithms which are iterative in their nature. As an example, in
computer tomography with limited data, each piece of information implies the existence
of a convex set in which the required solution lies. The problem of finding a point in the
intersection of the sets is then of crucial interest, and it cannot be usually solved directly.
Therefore, an iterative algorithm must be used to approximate such a point.
Recall that the normal Mann iterative process was introduced by Mann [9] in 1953.
The normal Mann iterative process generates a sequence {xn} in the following manner:
x1 C, xn+1 = (1 n)xn + nT xn, n 1, (M)
where {n} is a sequence in the interval (0, 1). IfT is a nonexpansive mapping with a fixed
point and the control sequence {n} is chosen so that n=1
n(1 n) = , then the
sequence {xn} generated in the normal Mann iterative process converges weakly to a fixed
point of T. In an infinite-dimensional Hilbert space, the normal Mann iteration process
has only weak convergence, in general, even for nonexpansive mappings. Therefore, many
authors try to modify the normal Mann iterative process to have strong convergence for
nonexpansive mappings; see, e.g., [10-24] and the references therein.
Recently, Qin and Su [18] studied the problem of modifying the normal Mann iterative
process to have strong convergence for m-accretive operators. To be more precise, they
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6 X. QIN, S.Y. CHO, L. WANG
considered the following iterative process:
x1 C,
yn = nxn + (1 n)Jrnxn,
xn+1 = u + (1 n)yn, n 1,
(QS)
where u is a fixed element in C and Jrn = (I + rnA)1. They proved that the sequence
{xn} generated in the above iterative process converges strongly to a zero of A.
In this paper, we study the convergence of paths for continuous pseudocontractions in
a real Banach space by viscosity approximation methods. As applications, we consider
the problem of finding zeros of m-accretive operators based on an iterative process with
errors. Strong convergence theorems of zeros are established in a real Banach space.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1 [25] Let {an}, {bn}, and {cn} be three nonnegative real sequences satisfying
an+1 (1 tn)an + bn + cn, n 0,
where {tn} is a sequence in (0, 1). Assume that the following conditions are satisfied:(a)
n=0
tn = and bn = o(tn);
(b)
n=0cn < .
Then limn an = 0.
Lemma 1.2 [26] Let C be a nonempty, bounded, closed, and convex subset of a reflexive
Banach space E which also has the normal structure. Let T be a nonexpansive mapping
of C into itself. Then F(T) is nonempty.
Lemma 1.3 [27] Let C be a nonempty, closed, and convex subset of a Banach space E,
and let T : C C be a continuous and strong pseudocontraction. Then T has a unique
fixed point in C.
Lemma 1.4 [28] In a Banach space E, there holds the inequality
x + y2 x2 + 2y, j(x + y), x, y E,
where j(x + y) J(x + y).
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 7
Set A = (0, 1), let B(A) denote the Banach space of all bounded real-valued functions
on A with a supremum norm and let X be a subspace of B(A).
Lemma 1.5 [1] Let C be a nonempty, closed, and convex subset of a Banach space E.
Suppose that the norm of E is uniformly Gateaux differentiable. Let {xt} be a bounded
set in E, and z C. Let t be a mean on X. Then txt z2 = minyCxt y if and
only if ty z, xt z 0 for all y C.
Lemma 1.6 [29] Let E be a Banach space and let A be an m-accretive operator. For
> 0, > 0, and x E, we have
Jx = J
x +
1
Jx
,
where J = (I + A)1 and J = (I + A)
1.
Lemma 1.7 [30] Let {xn} and {yn} be bounded sequences in a Banach space E, and let
{n} be a sequence in (0, 1) with
0 < lim infn
n lim supn
n < 1.
Suppose that xn+1 = (1 n)yn + nxn, n 1 and
lim supn
(yn+1 yn xn+1 xn) 0.
Then limn yn xn = 0.
2. Main results
Now, we are in a position to prove the strong convergence of paths for continuous
pseudocontractions.
Theorem 2.1. LetE be a real reflexive Banach space with the uniformly Gateaux differ-
entiable norm and the normal structure, and let C be a nonempty closed convex subset of
E. LetT : C C be a continuous pseudocontraction with a fixed point, and let f : C C
be a fixed continuous bounded and strong pseudocontraction with the coefficient (0, 1).
Let {xt} be a sequence generated by the following:
xt = tf(xt) + (1 t)T xt, (2.1)
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8 X. QIN, S.Y. CHO, L. WANG
where t (0, 1). Then {xt} converges strongly as t 0 to a fixed point x of T, which is
the unique solution in F(T) to the following variational inequality:
f(x) x, j(x p) 0, p F(T).
Proof. For t (0, 1), define a mapping Tft : C C by
Tft x = tf(x) + (1 t)T x.
Then Tft : C C is a continuous, strong pseudocontraction for each t (0, 1). Indeed,
we have
T
f
t x T
f
t y, j(x y)
= tf(x) f(y), j(x y) + (1 t)T x T y , j(x y)
tx y2 + (1 t)x y2
=
1 t(1 )
x y2, x, y C.
In view of Lemma 1.3, we see that Tft has a unique fixed point xt in C for each t (0, 1).
Hence (2.1) is well defined. Next, we show that {xt} is bounded. For any p F(T) and
t (0, 1), we see that
xt p2 = xt p, j(xt p)
= tf(xt) p, j(xt p) + (1 t)T xt p, j(xt p)
= tf(xt) f(p), j(xt p) + tf(p) p, j(xt p)
+ (1 t)T xt p, j(xt p)
txt p2 + tf(p) p, j(xt p)
+ (1 t)xt p2
1 t(1 )
xt p2 + tf(p) p, j(xt p),
which implies that
xt p2
1
1 f(p) p, j(xt p). (2.2)
It follows that
xt p
1
1 f(p) p. (2.3)
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 9
This shows that {xt} is bounded, so is {T xt}. On the other hand, we see from (2.1) that
xt T xt = tf(xt) T xt.Since f and {xt} are bounded, we obtain that f(xt) is bounded. It follows that
limt0
xt T xt = 0. (2.4)
Define g(x) = txt x2, x C, where t is a Banach limit. Then g(x) is continuous,
convex and g(x) as x . We see that g attains its infinimum over C (see, e.g.,
[1]). Set
D =
x C : g(x) = infyC
g(y)
.
Then D is a nonempty bounded closed convex subset of C. Next, we show that there
exits a point x D such that T x = x.
Indeed, for any y C, define a mapping S : C C by
Sx =1
2y +
1
2T x, x C.
It is easy to see that S is a continuous strong pseudocontraction. From Lemma 1.3, wesee that S has a unique fixed point x in C, that is, x = 1
2y + 1
2T x. This implies that
y = (2I T)x (2I T)(C).
This shows that C (2I T)(C). Define another mapping h : C C by
h(x) = (2I T)1x, x C.
We see that h is a nonexpansive mapping. Indeed, we have
h(x) h(y) = (2I T)1x (2I T)1y
I + (I T)1
x
I + (I T)1
y
x y, x, y C.
We also see that F(h) = F(T). Indeed,
x = h(x) 2x T x = x x = T x.
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10 X. QIN, S.Y. CHO, L. WANG
On the other hand, we have
x h(x) = hh1(x) h(x)
h1(x) x
= (2I T)(x) x
= x T x.
It follows from (2.4) that
limt0
xt h(xt) = 0.
Fixing x D, we have
g(h(x)) = txt h(x)2
= th(xt) h(x)2
txt x2
= g(x).
This implies that h(x) D, that is, h(D) D. Since C is a nonempty closed convexsubset of E which has the normal structure, we see that h has a fixed point. We denote
it by x. It follows from F(h) = F(T) that T x = x. From Lemma 1.5, we see that
tp x, j(xt x
) 0, p C.
In particular, we have
tf(x
) x
, j(xt x
) 0. (2.5)
In view of (2.2), we arrive at
txt x2 0.
This implies that there exists a subnet {xt} of {xt} such that xt x.
On the other hand, we have
xt f(xt) = (1 t)
T xt f(xt)
.
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 11
For any p F(T), we see that
xt f(xt), j(xt p)
= (1 t)T xt f(xt), j(xt p)
= (1 t)T xt xt, j(xt p) + (1 t)xt f(xt), j(xt p)
= (1 t)T xt p, j(xt p) + (1 t)p xt, j(xt p)
+ (1 t)xt f(xt), j(xt p)
(1 t)xt f(xt), j(xt p),
which implies that
xt f(xt), j(xt p) 0, p F(T). (2.6)
In particular, we have
xt f(xt), j(xt p) 0, p F(T). (2.7)
It follows that
x
f
(x
), j
(x
p
) 0,
p
F
(T
).
(2.8)
If there exits another subnet {xt} of {xt} such that xt x F(T). From (2.8), we
arrive at
x f(x), j(x x) 0. (2.9)
It is also from (2.6) that
xt f(xt), j(xt x) 0. (2.10)
It follows that
x f(x), j(x x) 0. (2.11)
Adding up (2.9) and (2.11), we obtain that
x f(x) x + f(x), j(x x) 0.
This implies that
x
x
2
x
x
2
.
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12 X. QIN, S.Y. CHO, L. WANG
Note that (0, 1). We see that x = x. This shows that {xt} converges strongly to
x F(T), which is the unique solution to the variational inequality
(f I)x, p x 0, p F(T).
This completes the proof.
Remark 2.2. Note that uniformly smooth Banach spaces enjoy the uniformly Gateaux
differentiable norm and the uniform normal structure. Theorem 2.1, which gives an
affirmative answer to the questions presented in Section 1, improves Theorem 4.1 of Xu
[6] in the following aspects:
(1) improves the framework of spaces from uniformly smooth Banach spaces to the
Banach space with the uniformly Gateaux differentiable norm.
(2) improves the mapping f from the class of contractions to the class of strongly
pseudocontractions.
(3) improves the mapping T from the class of nonexpansive mappings to the class of
pseudocontractions.
Remark 2.3. Under the conditions of Theorem 2.1, we can define a mapping Q : C F(T), where C denotes the set of continuous bounded and strong pseudocontractions,
by
Q(f) := limt0
xt, f C.
From Proposition 1.1, we see that Q is a sunny nonexpansive retraction from C onto
F(T).
Remark 2.4. If T is a nonexpansive mapping and f(x) = u, a fixed element for every
x C in Theorem 2.1, then {xt} defined by
xt = tu + (1 t)T xt,
where t (0, 1), converges strongly as t 0 to a fixed point Q(u) of T, where Q is the
sunny nonexpansive retraction from C onto F(T). The point Q(u) is the unique solution
in F(T) to the following variational inequality:
u Q(u), j(Q(u) p) 0, p F(T).
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 13
Next, we prove strong convergence of iterative processes with errors for m-accretive
operators.
Theorem 2.5. Let E be a real reflexive Banach space with the uniformly Gateaux dif-
ferentiable norm and let A be an m-accretive operator in E. Assume that C := D(A)
is convex and has the normal structure. Let {n}, {n}, {n}, and {n} be real number
sequences in (0, 1). Let QC be a sunny nonexpansive retraction from E onto C and let
{xn} be a sequence generated in the following manner:
x0 C,
yn = nxn + nJrn(xn + en+1) + nQCfn,
xn+1 = nu + (1 n)yn, n 0,
()
where {en} is a sequence in E, {fn} is a bounded sequence in E, {rn} is a positive
real numbers sequence, u is a fixed element in C and Jrn = (I + rnA)1. Assume that
A1(0) = and the above control sequences satisfy the following restrictions:
(a) n + n + n = 1 for each n 0;
(b) limn n = 0 and
n=1n = ;
(c) 0 < lim infn n lim supn n < 1;
(d)
n=1 en < and
n=0n < ;
(e) rn for each n 1 and limn |rn rn+1| = 0.
Then the sequence {xn} generated by () converges strongly to a zero Q(u) of A, which
is the unique solution to the following variational inequality:
(f I)Q(u), p Q(u) 0, p A1(0).
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14 X. QIN, S.Y. CHO, L. WANG
Proof. First, we prove that {xn} is bounded. Fixing p A1(0), we see that
x1 p
= 0(u p) + (1 0)(y0 p)
0u p + (1 0)y0 p
= 0u p + (1 0)
0(x0 p) + 0
Jr0(x0 + e1) p
+ 0(QCf0 p)
0u p + (1 0)
0x0 p + 0Jr0(x0 + e1) p + 0f0 p
0u p + (1 0)
(1 0)x0 p + e1 + 0f0 p
K,
where K = u p + x0 p + e1 + f0 p < . Put
M1 = max{K, supn0
fn p}.
Next, we prove that
xn p M1 +
ni=1
ei, n 1. (2.12)
It is easy to see that (2.12) holds for n = 1. We assume that the result holds for some n.
It follows that
xn+1 p
nu p + (1 n)yn p
= nu p + (1 n)
n(xn p) + n
Jrn(xn + en+1) p
+ n(QCfn p)
nu p + (1 n)
nxn p + nJrn(xn + en+1) p + nfn p
nu p + (1 n)
(1 n)xn p + en+1 + nfn p
nM1 + (1 n)
(1 n)
M1 +
ni=1
ei
+ en+1 + nM1
= M1 +n+1
i=1
ei.
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 15
This shows that (2.12) holds for all n 1. If rn+1 rn, we see from Lemma 1.6 that
Jrn(xn + en+1) Jrn+1(xn+1 + en+2)
rn
rn+1(xn + en+1) +
1
rn
rn+1
Jrn+1(xn + en+1) (xn+1 + en+2)
= rn
rn+1
(xn + en+1) (xn+1 + en+2)
+
rn+1 rnrn+1
Jrn+1(xn + en+1) (xn+1 + en+2)
xn xn+1 + en+1 + en+2 +
M2
(rn+1 rn),
(2.13)
where M2 is an appropriate constant such that
M2 supn1
{Jrn+1(xn + en+1) (xn+1 + en+2)}.
Put gn =xn+1nxn
1n. This implies that
xn+1 = (1 n)gn + nxn, n 0. (2.14)
Now, we compute gn+1 gn. Note that
gn+1 gn =n+1u + (1 n+1)yn+1 n+1xn+1
1 n+1
nu + (1 n)yn nxn1 n
= n+1(u yn+1) + n+1Jrn+1(xn+1 + en+2) + n+1QCfn+11 n+1
n(u yn) + nJrn(xn + en+1) + nQCfn
1 n
=n+1(u yn+1) + n+1
QCfn+1 Jrn+1(xn+1 + en+2)
1 n+1
n(u yn) + n
QCfn Jrn(xn + en+1)
1 n
+ Jrn+1(xn+1 + en+2) Jrn(xn + en+1).
(2.15)
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16 X. QIN, S.Y. CHO, L. WANG
It follows that
gn+1 gn
n+11 n+1
u yn+1 + n1 n
yn u
+ n+1
1 n+1
QCfn+1 Jrn+1(xn+1 + en+2)
n
1 n
QCfn Jrn(xn + en+1)
+ Jrn+1(xn+1 + en+2) Jrn(xn + en+1)
n+1
1 n+1u yn+1 +
n
1 nyn u +
n+1
1 n+1QCfn+1 Jrn+1(xn+1 + en+2)
+n
1 nQCfn Jrn(xn + en+1) + Jrn+1(xn+1 + en+2) Jrn(xn + en+1).
(2.16)
Substituting (2.13) into (2.16), we arrive at
gn+1 gn xn xn+1
n+1
1 n+1u yn+1 +
n
1 nyn u + en+1 + en+2
+M2
|rn+1 rn| +
n
1 nyn u +
n+1
1 n+1QCfn+1 Jrn+1(xn+1 + en+2)
+ n1 n
QCfn Jrn(xn + en+1).
(2.17)
In a similar way, we can obtain (2.17) when rn rn+1. From the conditions (b)-(e), we
see that
lim supn
gn+1 gn xn xn+1
0.
It follows from Lemma 1.7 that
limn
gn xn = 0. (2.18)
In view of (2.14), we have
xn+1 xn = (1 n)(gn xn),
which combines with the condition (c) and (2.18) gives that
limnxn+1 xn = 0. (2.19)
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 17
On the other hand, we see from the algorithm () that
xn Jrn(xn + en+1)
xn xn+1 + xn+1 yn + yn Jrn(xn + en+1)
xn xn+1 + nu yn + nxn Jrn(xn + en+1) + nQCfn xn.
This implies that
(1 n)xn Jrn(xn + en+1) xn xn+1 + nu yn + nQCfn xn.
From the conditions (b)-(d) and (2.19), we conclude that
limn
xn Jrn(xn + en+1) = 0. (2.20)
Since
n=1 en < , we see that
limn
xn Jrnxn = 0. (2.21)
Take a fixed number r such that > r > 0. From Lemma 1.6, we obtain that
Jrnxn Jrxn = Jr r
rnxn + (1
r
rn)Jrnxn
Jrxn
(1 rrn
)(Jrnxn xn)
Jrnxn xn.
(2.22)
Note that
xn Jrxn = xn Jrnxn + Jrnxn Jrxn
xn Jrnxn + Jrnxn Jrxn
2xn Jrnxn.
From (2.21), we see that
limn
xn Jrxn = 0. (2.23)
Next, we claim that
lim supn
u Q(u), j(xn Q(u)) 0, (2.24)
where
Qu = limt0 zt, u C
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18 X. QIN, S.Y. CHO, L. WANG
and zt solves the fixed point equation
zt = tu + (1 t)Jrzt, t (0, 1),
from which it follows that
zt xn = (1 t)(Jrzt xn) + t(u xn).
For any t (0, 1), we see that
zt xn2 = (1 t)Jrzt xn, j(zt xn) + tu xn, j(zt xn)
= (1 t)
Jrzt Jrxn, j(zt xn) + Jrxn xn, j(zt xn)
+ tu zt, j(zt xn) + tzt xn, j(zt xn)
(1 t)
zt xn2 + Jrxn xnzt xn
+ tu zt, j(zt xn) + tzt xn2
zt xn2 + Jrxn xnzt xn + tu zt, j(zt xn).
It follows that
zt u, j(zt xn) 1
tJrxn xnzt xn t (0, 1).
In view of (2.23), we see that
lim supn
zt u, j(zt xn) 0. (2.25)
Since zt Q(u) as t 0 and the fact that j is strong to weak uniformly continuous on
bounded subsets of E, we see that
|u Q(u), j(xn Q(u)) zt u, j(zt xn)|
|u Q(u), j(xn Q(u)) u Q(u), j(xn zt)|
+ |u Q(u), j(xn zt) zt u, j(zt xn)|
|u Q(u), j(xn Q(u)) j(xn zt)| + |zt Q(u), J(xn zt)|
u Q(u)j(xn Q(u)) j(xn zt) + zt Q(u)xn zt 0 as t 0.
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 19
Hence, for any > 0, there exists > 0 such that t (0, ) the following inequality
holds:
u Q(u), j(xn Q(u)) zt u, j(zt xn) + .
This implies that
lim supn
u Q(u), j(xn Q(u)) lim supn
zt u, j(zt xn) + .
Since is arbitrary and (2.25), we see that lim supnu Q(u), j(xn Q(u)) 0. That
is,
lim supn
u Q(u), j(xn+1 Q(u)) 0. (2.26)
Finally, we show that xn Q(u) as n . Note that
yn Q(u) = nxn + nJrn(xn + en+1) + nQCfn Q(u)
nxn Q(u) + nJrn(xn + en+1) Q(u) + nQCfn Q(u)
(n + n)xn Q(u) + en+1 + nfn Q(u)
xn Q(u) + n,
where n = en+1 + nM1. From the condition (d), we see that
n=0n < . It follows
from Lemma 1.5 that
xn+1 Q(u)2
= (1 n)(yn Q(u)) + n(u Q(u))2
(1 n)2yn Q(u)
2 + 2nu Q(u), J(xn+1 Q(u))
(1 n)(xn Q(u) + n)2 + 2nu Q(u), J(xn+1 Q(u))
= (1 n)(xn Q(u)2 + 2nxn Q(u) +
2n)
+ 2nu Q(u), J(xn+1 Q(u))
(1 n)xn Q(u)2 + nM3 + 2nu Q(u), J(xn+1 Q(u)),
(2.27)
where M3 is an appropriate constant such that M3 supn0{2xn Q(u) + n}. Let
n+1 = max{uQ(u), J(xn+1Q(u)), 0}. Next, we show that limn n+1 = 0. Indeed,
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20 X. QIN, S.Y. CHO, L. WANG
from (2.26), for any given > 0, there exists a positive integer n1 such that
u Q(u), J(xn+1 Q(u)) < , n n1.
This implies that 0 n+1 < n n1. Since > 0 is arbitrary, we see that
limn n+1 = 0. It follows from (2.27) that
xn+1 Q(u)2 (1 n)xn Q(u)
2 + nM3 + 2nn+1.
Put an = xn Q(u)2, tn = n, bn = 2nn+1 and cn = nM3 for every n 0. In view
of Lemma 1.1, we can obtain the desired conclusion easily.
As an application of Theorem 2.5, we have the following results.
Corollary 2.6. Let E be a real reflexive Banach space with the uniformly Gateaux dif-
ferentiable norm and let A be an m-accretive operator in E. Assume that C := D(A)
is convex and has the normal structure. Let {n} and {n} be real number sequences in
(0, 1). Let {xn} be a sequence generated by (QS), where {rn} is a positive real numbers
sequence, u is a fixed element in C and Jrn = (I + rnA)1. Assume that A1(0) = and
the above control sequences satisfy the following restrictions:(a) limn n = 0 and
n=1
n = ;
(b) 0 < lim infn n lim supn n < 1;
(c) rn for each n 1 and limn |rn rn+1| = 0.
Then the sequence {xn} converges strongly to a zero of A.
Remark 2.7. Corollary 2.6 improves Theorem 2.1 of Qin and Su [18]. To be more
precise, we partially relax the restrictions on the parameters and extend the framework
of the space; see [18] for more details.
Remark 2.8. It is of interest to design an explicit iterative process to approximate zeros of
accretive operators by Moudafis viscosity approximation method with continuous strong
pseudocontractions.
Competing interests: The authors declare that they have no competing interests.
Authors contributions: All authors contributed equally and significantly in writing this
paper. All authors read and approved the final manuscript.
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ITERATIVE ALGORITHMS WITH ERRORS FOR m-ACCRETIVE OPERATORS 21
Author details: 1Department of Mathematics, Hangzhou Normal University, Hangzhou
310036, China; 2Department of Mathematics, Gyeongsang National University, Jinju 660-
701, Korea;3
College of Statistics and Mathematics, Yunnan University of Finance andEconomics, Kunming 650221, China
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