J. Appl. Math. & Computing Vol. 17(2005), No. 1 - 2, pp. 329 - 341
APPROXIMATING RANDOM COMMON FIXED POINT OF
RANDOM SET-VALUED STRONGLY
PSEUDO-CONTRACTIVE MAPPINGS
JUN LI AND NAN-JING HUANG∗
Abstract. In this paper, we introduce new random iterative sequences
with errors approximating a unique random common fixed point for three
random set-valued strongly pseudo-contractive mappings and show the convergence of the random iterative sequences with errors by using an approximation method in real uniformly smooth separable Banach spaces. As
applications, we study the existence of random solutions for some kind of
random nonlinear operator equations group in separable Hilbert spaces.
AMS Mathematics Subject Classification : 54H25, 47H07, 47H10, 46S50.
Key words and phrases : Random strongly pseudo-contractive mapping,
iteration with errors, random common fixed point, set-valued mapping.
1. Introduction
Concerning the stability and the convergence problems of Ishikawa, Mann, Liu
and Xu iteration process for single-valued and set-valued accretive and pseudocontractive mapping have been studied extensively by many authors for approximating the fixed points of some nonlinear mappings and for approximating
solutions of some nonlinear operator equations in Banach spaces(see, for example, [1, 3-5, 11, 12, 16, 25]). On the other hand, the study of random fixed
points was initiated by the Prague school of probabilistic in the fifties. Recently,
random fixed point theory has received much attention for the last decade (see,
for example, [2, 6, 7, 9, 10, 14, 15, 17, 18, 20-22, 24]).
Received November 19, 2003. ∗ Corresponding author.
The work was supported by the Scientific Research Foundation for the Returned Overseas Chinese
Scholars, State Education Ministry, and the China West Normal University Research Fund 2003-3.
Corresponding author: Nan-jing Huang (e-mail: [email protected])
c 2005 Korean Socity for Computational & Applied Mathematics and Korean SIGCAM.
329
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Jun Li and Nan-jing Huang
Recently Choudhury [6, 7] has suggested and analyzed random Mann iterative
sequence in separable Hilbert spaces for finding random solutions and random
fixed points for some kind of random equations and random operators.
Motivated and inspired by the above works, the purpose of this paper is
to introduce a new random iterative sequences with errors approximating the
unique random common fixed point for three random set-valued strongly pseudocontractive mappings and show the convergence of the random iterative sequences with errors by using an approximation method in real uniformly smooth
separable Banach spaces. As applications, we shall utilize the result presented
in Theorem 3.1 to study the existence of random solutions for some kind of random nonlinear operator equations group in separable Hilbert spaces. The results
presented in this paper extend, improve and unify the corresponding results in
[3, 5].
2. Preliminaries
Now, we recall the following iterative processes due to Ishikawa [13], Liu [16]
and Xu [25], respectively.
(I) Let K be a nonempty convex subset of a Banach space X and T : K → K
be a mapping. The sequence {xn }∞
n=0 in K defined by

 x0 ∈ K,
zn = (1 − βn )xn + βn T xn , n ≥ 0.

xn+1 = (1 − αn )xn + αn T zn ,
∞
is called the Ishikawa iterative process [13], where {αn }∞
n=0 , {βn }n=0 are sequences in [0, 1] satisfying some conditions.
(II) For a nonempty convex subset K of a Banach space X and a mapping
T : K → X, the sequence {xn }∞
n=0 in K defined by

 x0 ∈ K,
zn = (1 − βn )xn + βn T xn + vn , n ≥ 0.

xn+1 = (1 − αn )xn + αn T zn + un ,
∞
is called the Ishikawa iterative process with errors [16], where {un }∞
n=0 , {vn }n=0
∞
∞
X
X
are summable sequences in X (i.e.,
kun k < ∞ and
kvn k < ∞) and
n=0
n=0
∞
{αn }∞
n=0 , {βn }n=0 are sequences in [0, 1] satisfying some conditions.
Approximating random common fixed point
331
(III) For a nonempty convex subset K of a Banach space X and a mapping
T : K → K, the sequence {xn }∞
n=0 in K defined by

 x0 ∈ K,
zn = (1 − βn − δn )xn + βn T xn + δn vn , n ≥ 0,

xn+1 = (1 − αn − γn )xn + αn T zn + γn un ,
∞
is called the Ishikawa iterative process with errors [25], where {un }∞
n=0 , {vn }n=0
∞
∞
∞
are arbitrary bounded sequences in K and {αn }n=0 , {βn }n=0 , {γn }n=0 , and
{δn }∞
n=0 are sequences in [0, 1] satisfying some conditions.
Throughout this paper, let (Ω,A) be a measurable space and X an arbitrary
∗
real uniformly smooth separable Banach space. We denote by 2X , X ∗ , 2X , and
B(X) the family of all nonempty subset of X, the duality space of X, the family
of all nonempty subset of X ∗ , and the class of Borel σ-fields in X, respectively.
Definition 2.1. A mapping x : Ω → X is said to be measurable if for each
B ∈ B(X), {t ∈ Ω : x(t) ∈ B} ∈ A.
Definition 2.2. A mapping T : Ω × X → X is called a random operator if for
each x ∈ X, T (t, x) = x(t) is measurable.
Definition 2.3. A set-valued mapping T : Ω → 2X is said to be measurable if
for any B ∈ B(X), T −1 (B) = {t ∈ Ω : T (t) ∩ B 6= ∅} ∈ A.
Definition 2.4. A mapping T : Ω × X → 2X is called a random set-valued
mapping if for each x ∈ X, T (·, x) : Ω → 2X is measurable.
Definition 2.5. Let J denote the normalized duality mapping from X into 2X
by
J(x) = {j(x) : hx, j(x)i = kxk2 = kj(x)k2 }
for all x ∈ X, where h·, ·i denotes the duality pairing between X and X ∗ .
∗
Remark 2.1. X is a uniformly smooth Banach space (equivalently, X ∗ is a
uniformly convex Banach space)if and only if J is single-valued and uniformly
continuous on any bounded subset of X.
Definition 2.6. Let K be a nonempty subset of X and T : Ω × K → 2X a
random set-valued mapping. T is said to be
i) random accretive if for any x, y ∈ K, there exists j(x − y) ∈ J(x − y)
such that for each t ∈ Ω ,
hu − v, j(x − y)i ≥ 0,
∀u ∈ T (t, x), ∀v ∈ T (t, y).
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Jun Li and Nan-jing Huang
ii) random strongly accretive if there exists a function k : Ω → (0, 1) such
that, for any x, y ∈ K, there exists j(x − y) ∈ J(x − y) such that for
each t ∈ Ω,
hu − v, j(x − y)i ≥ k(t)kx − yk2 ,
∀u ∈ T (t, x), ∀v ∈ T (t, y),
where k(t) is called the random strongly accretive coefficient of T .
iii) random (strongly) pseudo-contractive if I − T : Ω × K → 2X is random
(strongly) accretive, where I : Ω × K → K satisfies I(t, x) = x for each
t ∈ Ω and x ∈ K. Obviously, if T is random strongly pseudo-contractive,
then there exists a function k : Ω → (0, 1) such that, for any x, y ∈ K,
there exists j(x − y) ∈ J(x − y) such that for each t ∈ Ω,
hu − v, j(x − y)i ≤ (1 − k(t))kx − yk2 ,
∀u ∈ T (t, x), ∀v ∈ T (t, y).
Remark 2.2. If T (t, x) = T x for all t ∈ Ω and x ∈ K, then random (strongly)
accretive mapping and random (strongly) pseudo-contractive mapping reduce
to (strongly) accretive mapping and (strongly) pseudo-contractive mapping respectively.
∗
Lemma 2.1([19]). Let X be a real Banach space and J : X → 2X be a
normalized duality mapping, then for any given x, y ∈ X, there exists j(x + y) ∈
J(x + y) such that
kx + yk2 ≤ kxk2 + 2hy, j(x + y)i.
∞
Lemma 2.2([23]). Let {an }∞
n=0 , {bn }n=0 be two nonnegative real sequences and
∞
X
{λn }∞
a
real
sequence
in
[0,
1]
such
that
λn = ∞ and bn → 0 as n → ∞.
n=0
n=0
If there exists a positive integer N such that
an+1 ≤ (1 − λn )an + λn bn
for all n ≥ N , then we have limn→∞ an = 0.
3. Convergence of random iterative sequences
Theorem 3.1. Let X be a real uniformly smooth separable Banach space and
Ti : Ω × X → 2X (i = 1, 2, 3) three random set-valued strongly pseudo-contractive
mappings with functions ki : Ω → (0, 1) respectively.
Assume T1 , T2 and T3 have random common fixed points in X, and for each
n ≥ 0, un , vn , wn : Ω → X are measurable mappings and αn , βn , γn : Ω → [0, 1]
Approximating random common fixed point
333
are measurable functions. For any given measurable mapping x0 : Ω → X, the
random iterative sequence {xn (t)}∞
n=0 is defined by:

 sn (t) = (1 − γn (t))xn (t) + γn (t)pn (t) + vn (t)
zn (t) = (1 − βn (t))xn (t) + βn (t)hn (t) + wn (t)
n = 0, 1, 2, · · ·, (3.1)

xn+1 (t) = (1 − αn (t))xn (t) + αn (t)gn (t) + αn (t)un (t)
for each t ∈ Ω, and some gn (t) ∈ T1 (t, zn (t)), hn (t) ∈ T2 (t, sn (t)), pn (t) ∈
T3 (t, xn (t)). If the sequences {un }, {vn }, {wn }, {αn }, {βn } and {γn } satisfy the
following conditions:
(i) limn→∞ un (t) = limn→∞ vn (t) = limn→∞ wn (t) = 0
∞
X
(ii)
αn (t) = ∞ for each t ∈ Ω,
for each t ∈ Ω,
n=0
(iii) limn→∞ αn (t) = limn→∞ βn (t) = limn→∞ γn (t) = 0
for each t ∈ Ω,
and the sequences {gn }, {hn } and {pn } are measurable and bounded, then the
random iterative sequence {xn (t)}∞
n=0 defined by (3.1) converges strongly to the
unique random common fixed point of T1 , T2 and T3 in X.
Proof. It is easy to see that {xn }, {zn } and {sn } are sequences of measurable
mappings from Ω into X. Since T1 , T2 and T3 have random common fixed
points in X, there exists a measurable mapping x∗ : Ω → X such that x∗ (t) ∈
Ti (t, x∗ (t)) for each t ∈ Ω and i = 1, 2, 3. By iii) of Definition 2.6, it is easy to
prove that x∗ is the unique random common fixed point of T1 , T2 and T3 .
Next, we have to prove that, for each t ∈ Ω, {xn (t)}∞
n=0 converges strongly
to x∗ (t). For each t ∈ Ω, it follows from Lemma 2.1 and (3.1) that
kxn+1 (t) − x∗ (t)k2
= k(1 − αn (t))xn (t) + αn (t)gn (t) + αn (t)un (t) − x∗ (t)k2
= k(1 − αn (t))(xn (t) − x∗ (t)) + αn (t)(gn (t) − x∗ (t)) + αn (t)un (t)k2
≤ (1 − αn (t))2 kxn (t) − x∗ (t)k2 + 2hαn (t)(gn (t) − x∗ (t))
+αn (t)un (t), j(xn+1 (t) − x∗ (t))i
= (1 − αn (t))2 kxn (t) − x∗ (t)k2 + 2αn (t)hgn (t) − x∗ (t), j(zn (t) − x∗ (t))i
+2αn (t)cn (t) + 2αn (t)hun (t), j(xn+1 (t) − x∗ (t))i,
(3.2)
where
cn (t) = hgn (t) − x∗ (t), j(xn+1 (t) − x∗ (t)) − j(zn (t) − x∗ (t))i.
It follows from iii) of Definition 2.6 that
hgn (t) − x∗ (t), j(zn (t) − x∗ (t))i ≤ (1 − k1 (t))kzn (t) − x∗ (t)k2 , ∀t ∈ Ω.
(3.3)
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Jun Li and Nan-jing Huang
From (3.2) and (3.3), we have
kxn+1 (t) − x∗ (t)k2
≤ (1 − αn (t))2 kxn (t) − x∗ (t)k2 + 2αn (t)(1 − k1 (t))kzn (t) − x∗ (t)k2
+2αn (t)cn (t) + 2αn (t)hun (t), j(xn+1 (t) − x∗ (t))i
(3.4)
Now, we prove that, for each t ∈ Ω, cn (t) → 0 and hun (t), j(xn+1 (t)−x∗ (t))i → 0
as n → ∞. In fact, since xn (t), gn (t), hn (t) and x∗ (t) are bounded in X, then
∞
∞
∗
∞
∗
∞
{xn (t)}∞
n=0 , {gn (t)}n=0 , {hn (t)}n=0 , {xn+1 (t)−x (t)}n=0 and {gn (t)−x (t)}n=0
are all bounded sequences in X. It follows from conditions (i), (ii) and (iii) that,
for each t ∈ Ω,
xn+1 (t) − x∗ (t) − (zn (t) − x∗ (t))
= (βn (t) − αn (t))xn (t) + αn (t)gn (t) − βn (t)hn (t) + αn (t)un (t) − wn (t)
→0
(as n → ∞).
In view of uniform continuity of J on any bounded subset of X, we have, for
each t ∈ Ω,
j(xn+1 (t) − x∗ (t)) − j(zn (t) − x∗ (t)) → 0
as n → ∞
and so
cn (t) → 0
and hun (t), j(xn+1 (t) − x∗ (t))i → 0
as n → ∞.
(3.5)
From (3.1) and Lemma 2.1, we have
kzn (t) − x∗ (t)k2
= k(1 − βn (t))xn (t) + βn (t)hn (t) + wn (t) − x∗ (t)k2
= k(1 − βn (t))(xn (t) − x∗ (t)) + βn (t)(hn (t) − x∗ (t)) + wn (t)k2
≤ (1 − αn (t))2 kxn (t) − x∗ (t)k2
+2hβn (t)(hn (t) − x∗ (t)) + wn (t), j(zn (t) − x∗ (t))i
= (1 − βn (t))2 kxn (t) − x∗ (t)k2 + 2βn (t)hhn (t) − x∗ (t), j(sn (t) − x∗ (t))i
+2βn (t)dn (t) + 2hwn (t), j(zn (t) − x∗ (t))i
≤ (1 − βn (t))2 kxn (t) − x∗ (t)k2 + 2βn (t)(1 − k2 (t))ksn (t) − x∗ (t)k2
+2βn (t)dn (t) + 2hwn (t), j(zn (t) − x∗ (t))i
(3.6)
for each t ∈ Ω, where
dn (t) = hhn (t) − x∗ (t), j(zn (t) − x∗ (t)) − j(sn (t) − x∗ (t))i,
∀t ∈ Ω.
As in the proof of (3.5), we have, for each t ∈ Ω,
dn (t) → 0,
hwn (t), j(zn (t) − x∗ (t))i → 0
as
n → ∞.
(3.7)
Approximating random common fixed point
335
Similarly,
ksn (t) − x∗ (t)k2
= k(1 − γn (t))xn (t) + γn (t)pn (t) + vn (t) − x∗ (t)k2
= k(1 − γn (t))(xn (t) − x∗ (t)) + γn (t)(pn (t) − x∗ (t)) + vn (t)k2
≤ (1 − γn (t))2 kxn (t) − x∗ (t)k2
+2hγn (t)(pn (t) − x∗ (t)) + vn (t), j(sn (t) − x∗ (t))i
= (1 − γn (t))2 kxn (t) − x∗ (t)k2 + 2γn (t)hpn (t) − x∗ (t), j(xn (t) − x∗ (t))i
+2γn (t)en (t) + 2hvn (t), j(sn (t) − x∗ (t))i
= (1 − γn (t))2 kxn (t) − x∗ (t)k2 + 2γn (t)(1 − k3 (t))kxn (t) − x∗ (t)k2
+2γn (t)en (t) + 2hvn (t), j(sn (t) − x∗ (t))i
(3.8)
for each t ∈ Ω, where
en (t) = hpn (t) − x∗ (t), j(sn (t) − x∗ (t)) − j(xn (t) − x∗ (t))i,
∀t ∈ Ω,
and for each t ∈ Ω,
en (t) → 0,
hvn (t), j(sn (t) − x∗ (t))i → 0
as n → ∞.
(3.9)
Let k(t) = min{k1 (t), k2 (t), k3 (t)} for each t ∈ Ω. It follows from (3.6), (3.8)
and (3.4) that
kxn+1 (t) − x∗ (t)k2
≤ {(1 − αn (t))2 + 2αn (t)(1 − k(t))[(1 − βn (t))2
+2βn (t)(1 − k(t))]}kxn (t) − x∗ (t)k2 + αn (t)fn (t),
(3.10)
where
fn (t) = 8βn (t)γn (t)(1 − k(t))3 M
+4βn (t)(1 − k(t))2 [2γn (t)en (t) + 2hvn (t), j(sn (t) − x∗ (t))i]
+4βn (t)(1 − k(t))dn (t) + 2cn (t)
+4(1 − k(t))hwn (t), j(zn (t) − x∗ (t))i
+2hun (t), j(xn+1 (t) − x∗ (t))i
(3.11)
for each t ∈ Ω and
M (t) = sup kxn (t) − x∗ (t)k2 < ∞
(3.12)
n≥0
for each t ∈ Ω. It follows from conditions (i), (ii), (3.5), (3.7), (3.9), (3.11) and
(3.12) that
fn (t) → 0 as n → ∞
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Jun Li and Nan-jing Huang
for each t ∈ Ω. Obviously,
0 ≤ (1 − βn (t))2 + 2βn (t)(1 − k(t))
≤ 1 + βn2 (t)
n = 0, 1, 2, · · ·,
(3.13)
and
0 ≤ (1 − αn (t))2 + 2αn (t)(1 − k(t))
≤ 1 − k(t)αn (t) + α2n (t)
n = 0, 1, 2, · · ·,
(3.14)
for each t ∈ Ω. From (3.13), (3.14) and (3.10), we have, for each t ∈ Ω,
kxn+1 (t) − x∗ (t)k2 ≤ (1 − k(t)αn (t))kxn (t) − x∗ (t)k2
+αn (t)[αn (t) + 2βn2 (t)(1 − k(t))M (t) + fn (t)].
Let
an (t) = kxn (t) − x∗ (t)k2 ,
λn (t) = k(t)αn (t),
αn (t) + 2βn2 (t)(1 − k(t))M (t) + fn (t)
bn (t) =
k(t)
for each t ∈ Ω, then
an+1 (t) ≤ (1 − λn (t))an (t) + λn (t)bn (t)
for each t ∈ Ω and n = 0, 1, 2, · · ·. It follows from conditions (i), (ii), (iii) and
Lemma 2.2 that, for each t ∈ Ω, an (t) → 0 as n → ∞, that is xn (t) → x∗ (t) as
n → ∞. This completes the proof.
Letting T1 = T2 = T3 in Theorem 3.1, we have the following result:
Theorem 3.2. Let X be a real uniformly smooth separable Banach space and
T : Ω × X → 2X a random set-valued strongly pseudo-contractive mapping with
function k : Ω → (0, 1).
Assume T has random fixed points in X, and for each n ≥ 0, un , vn , wn :
Ω → X are measurable mappings and αn , βn , γn : Ω → [0, 1] are measurable
functions. For any given measurable mapping x0 : Ω → X, the random iterative
sequence {xn (t)}∞
n=0 is defined by:

 sn (t) = (1 − γn (t))xn (t) + γn (t)pn (t) + vn (t)
zn (t) = (1 − βn (t))xn (t) + βn (t)hn (t) + wn (t)
n = 0, 1, 2, · · ·,

xn+1 (t) = (1 − αn (t))xn (t) + αn (t)gn (t) + αn (t)un (t)
for each t ∈ Ω, and some gn (t) ∈ T (t, zn (t)), hn (t) ∈ T (t, sn (t)), pn (t) ∈
T (t, xn (t)).
Approximating random common fixed point
337
If the sequences {un }, {vn }, {wn }, {αn }, {βn } and {γn } satisfy conditions (i),
(ii) and (iii) of Theorem 3.1, and the sequences {gn }, {hn } and {pn } are measurable and bounded, then the random iterative sequence {xn (t)}∞
n=0 defined above
converges strongly to the unique random common fixed point of T1 , T2 and T3 in
X.
Letting un (t) = vn (t) = wn (t) = 0 for each t ∈ Ω in Theorem 3.1, we obtain
the following:
Theorem 3.3. Let X be a real uniformly smooth separable Banach space and
Ti : Ω × X → 2X (i = 1, 2, 3) three random set-valued strongly pseudo-contractive
mappings with functions ki : Ω → (0, 1) respectively.
Assume T1 , T2 and T3 have random common fixed points in X, and for each
n ≥ 0, αn , βn , γn : Ω → [0, 1] are measurable functions. For any given measurable mapping x0 : Ω → X, the random iterative sequence {xn (t)}∞
n=0 is defined
by:

 sn (t) = (1 − γn (t))xn (t) + γn (t)pn (t)
zn (t) = (1 − βn (t))xn (t) + βn (t)hn (t)
n = 0, 1, 2, · · ·,
(3.15)

xn+1 (t) = (1 − αn (t))xn (t) + αn (t)gn (t)
for each t ∈ Ω, and some gn (t) ∈ T1 (t, zn (t)), hn (t) ∈ T2 (t, sn (t)), pn (t) ∈
T3 (t, xn (t)).
If the sequences {αn }, {βn } and {γn } satisfy conditions (ii) and (iii) of Theorem 3.1, and the sequences {gn }, {hn } and {pn } are measurable and bounded,
then the random iterative sequence {xn (t)}∞
n=0 defined by (3.15) converges strongly
to the unique random common fixed point of T1 , T2 and T3 in X.
Theorem 3.4. Let X be a real uniformly smooth separable Banach space, K a
nonempty bounded closed convex subset of X and Ti : Ω × K → 2K (i = 1, 2, 3)
three random set-valued strongly pseudo-contractive mappings with functions ki :
Ω → (0, 1), respectively. Assume T1 , T2 and T3 have random common fixed
points in K, and for each n ≥ 0, αn , βn , γn : Ω → [0, 1] are measurable functions
satisfying conditions (ii) and (iii) of Theorem 3.1. If the sequences {gn }, {hn }
and {pn } are measurable, then the random iterative sequence {xn (t)}∞
n=0 defined
by (3.15) converges strongly to the unique random common fixed point of T1 , T2
and T3 in K.
Proof. Since T1 , T2 and T3 have random common fixed points in K, there exists
a measurable mapping x∗ : Ω → K such that x∗ (t) ∈ Ti (t, x∗ (t)) for each t ∈ Ω
and i = 1, 2, 3. By iii) of Definition 2.6, it is easy to prove that x∗ is the
unique random common fixed point of T1 , T2 and T3 . Since xn (t), gn (t), hn (t)
and x∗ (t) ∈ K, then it follows from the boundedness of K that
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Jun Li and Nan-jing Huang
∞
∞
∞
{xn (t)}∞
n=0 , {gn (t)}n=0 , {hn (t)}n=0 , {pn (t)}n=0 ,
∗
∞
∗
∞
{xn+1 (t) − x (t)}n=0 , {gn (t) − x (t)}n=0 , and
∗
∞
{hn (t) − x∗ (t)}∞
n=0 , {pn (t) − x (t)}n=0
are all bounded sequences in K. The rest of proof follows as those of Theorem
3.1 only setting un (t) = vn (t) = wn (t) = 0 for each t ∈ Ω and therefore is
omitted. This completes the proof.
4. Applications
As applications, we shall utilize the results presented in section 3 to study
the existence of random solutions for some kinds of random nonlinear operator
equations system in separable Hilbert spaces.
Theorem 4.1. Let X be a real separable Hilbert space and S : Ω × X → X a
continuous random strongly accretive mapping with function k : Ω → (0, 1). Let
f : Ω → X be any given measurable mapping and T : Ω × X → X be denoted by
T = I − S + f.
For each n ≥ 0, assume un , vn , wn : Ω → X are measurable mappings and
αn , βn , γn : Ω → [0, 1] are measurable functions.
For any given measurable mapping x0 : Ω → X, the random iterative sequence
{xn (t)}∞
n=0 is defined by:

 sn (t) = (1 − γn (t))xn (t) + γn (t)T (t, xn (t)) + vn (t)
zn (t) = (1 − βn (t))xn (t) + βn (t)T (t, sn (t)) + wn (t)
n = 0, 1, 2, · · ·, (4.1)

xn+1 (t) = (1 − αn (t))xn (t) + αn (t)T (t, zn (t)) + αn (t)un (t)
for each t ∈ Ω.
If the range R(T ) of T is bounded and the sequences {un }, {vn }, {wn }, {αn },
{βn } and {γn } satisfy conditions (i), (ii) and (iii) of Theorem 3.1, then, for any
given measurable mapping f : Ω → X, the random operator equation
f (·) = S(·, x(·))
(4.2)
∗
has a unique random solution x : Ω → X, and the random iterative sequence
∗
{xn (t)}∞
n=0 defined by (4.1) converges strongly to x .
Proof. By the assumption, T : Ω × X → X is a continuous random strongly
pseudo-contractive mapping with a function h : Ω → (0, 1) defined by h(t) =
1 − k(t). It is easy to see that {sn }, {zn } and {xn } are sequences of measurable
mappings from Ω into X. Then, for any given t ∈ Ω, T (t, ·) : X → X is a
strongly pseudo-contractive mapping with a number h(t) ∈ (0, 1). By the wellknown result in [8], T (t, ·) : X → X has a unique fixed point, that is, there
exists a unique
Approximating random common fixed point
339
point x∗ (t) ∈ X such that
x∗ (t) = T (t, x∗ (t)),
that is,
f (t) = S(t, x∗ (t)),
which implies that x∗ is a unique random solution of the random operator equation (4.2). Since the range R(T ) of T is bounded, as in the proof of Theorem
3.1, we can prove that the random iterative sequence {xn (t)}∞
n=0 defined by
(4.1) converges strongly to the unique random fixed point x∗ : Ω → X. This
completes the proof.
From Theorem 3.4 and Theorem 4.1, we have the following:
Theorem 4.2. Let X be a real separable Hilbert space, K a nonempty bounded
closed convex subset of X and S : Ω × K → K a continuous random strongly
accretive mappings with function k : Ω → (0, 1). Let f : Ω → K be any given
measurable mapping and T : Ω × K → K be denoted by T = I − S + f .
For each n ≥ 0, assume αn , βn , γn : Ω → [0, 1] are measurable functions.
For any given measurable mapping x0 : Ω → K, the random iterative sequence
{xn (t)}∞
n=0 is defined by:

 sn (t) = (1 − γn (t))xn (t) + γn (t)T (t, xn (t))
zn (t) = (1 − βn (t))xn (t) + βn (t)T (t, sn (t))
n = 0, 1, 2, · · ·,
(4.3)

xn+1 (t) = (1 − αn (t))xn (t) + αn (t)T (t, zn (t))
for each t ∈ Ω.
If the sequences {αn }, {βn } and {γn } satisfy conditions (ii) and (iii) of Theorem 3.1, then, for any given measurable mapping f : Ω → K, the random
operator equation defined by (4.2) has a unique random solution x∗ : Ω → K,
and the random iterative sequence {xn (t)}∞
n=0 defined by (4.3) converges strongly
to x∗ .
Remark 4.1. The results presented in this paper extend, improve and unify
the corresponding results in [3, 5].
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Approximating random common fixed point
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Jun Li received his MS from Sichuan University. Since 1998 he has been at China West
Normal University. In 2003, he received a lecture from China West Normal University.
His research interests center on the optimization theory and nonlinear functional analysis
with applications.
Department of Mathematics, China West Normal University, Nanchong, Sichuan 637002,
P. R. China
e-mail: [email protected]
Nan-jing Huang received his BS and MS and Ph. D from Sichuan University in 1983
and 1990 and 1997, respectively. Since 1990 he has work at Sichuan University. In 1998,
he received a Full Professor from Sichuan University. His main areas of research are in
optimization theory, fixed point theory, nonlinear analysis with applications.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China
e-mail: [email protected]