Báo cáo hóa học: " CONVERGENCE THEOREMS FOR FIXED POINTS OF DEMICONTINUOUS PSEUDOCONTRACTIVE MAPPINGS" pdf

CONVERGENCE THEOREMS FOR FIXED POINTS OF
DEMICONTINUOUS PSEUDOCONTRACTIVE MAPPINGS
C. E. CHIDUME AND H. ZEGEYE
Received 26 August 2004
Let D be an open subset of a real uniformly smooth Banach space E.SupposeT :
¯
D →
E is a demicontinuous pseudocontractive mapping satisfying an appropriate condition,
where
¯
D denotes the closure of D. Then, it is proved that (i)
¯
D ⊆᏾ (I + r(I −T)) for every
r>0; (ii) for a given y
0
∈ D, there exists a unique path t → y
t
∈
¯
D, t ∈ [0,1], satisfying
y
t
:= tT y
t
+(1−t)y
0
.Moreover,ifF(T) =∅or there exists y
0
∈ D such that the set
K :={y ∈ D : Ty= λy+(1−λ)y
0

for λ>1} is bounded, then it is proved that, as t →1
−
,
the path
{y
t
} converges strongly to a fixed point of T. Furthermore, explicit iteration
procedures with bounded error terms are proved to converge strongly to a fixed point
of T.
1. Introduction
Let D be a nonempty subset of a real linear space E.AmappingT : D
→ E is called
a contraction mapping if there exists L ∈ [0,1)suchthatTx −Ty≤Lx − y for all
x, y ∈D.IfL = 1thenT is called nonexpansive. T is called pseudocontractive if there exists
j(x − y) ∈J(x − y)suchthat

Tx−Ty, j(x − y)

≤x − y
2
, ∀x, y ∈K, (1.1)
where J is the normalized duality mapping from E to 2
E
∗
defined by
Jx :=

f
∗
∈ E

∗
:

x, f
∗

=
x
2
=


f
∗


2

. (1.2)
T is called strongly pseudocontractive if there exists k
∈ (0,1) such that

Tx−Ty, j(x − y)

≤ kx − y
2
, ∀x, y ∈K. (1.3)
Clearlytheclassofnonexpansivemappingsisasubsetofclassofpseudocontractivemap-
pings. T is said to be demicontinuous if
{x

n
}⊆D and x
n
→ x ∈ D together imply that
Tx
n
 Tx,where→ and  denote the strong and weak convergences, respectively. We
denote by F(T) the set of fixed points of T.
Copyright © 2005 Hindawi Publishing Corporation
Fixed Point Theory and Applications 2005:1 (2005) 67–77
DOI: 10.1155/FPTA.2005.67
68 Fixed points of demicontinuous pseudocontractive maps
Closely related to the class of pseudocontractive mappings is the class of accretive map-
pings. A mapping A : D(A) ⊆E → E is called accretive if T := (I −A) is pseudocontractive.
If E is a Hilbert space, accretive operators are also called monotone.AnoperatorA is called
m-accretive if it is accretive and ᏾(I + rA), the range of (I + rA), is E for all r>0; and A
is said to satisfy the range condition if cl(D(A)) ⊆᏾( I + rA), for all r>0, where cl(D(A))
denotes the closure of the domain of A.
Let z ∈D,thenforeacht ∈(0,1), and for a nonexpansive map T, there exists a unique
point x
t
∈ D satisfying the condition,
x
t
= tTx
t
+(1−t)z (1.4)
since the mapping x → tTx +(1−t)z is a contraction. When E is a Hilbert space and T is
aself-map,Browder[1] showed that {x
t

} converges strongly to an element of F(T) which
is nearest to u as t →1
−
. This result was extended to various more general Banach spaces
by Reich [10], Takahashi and Ueda [11], and a host of other authors. Recently, Morales
and Jung [7] proved the existence and convergence of a continuous path to a fixed point
of a continuous pseudocontractive mapping in reflexive Banach spaces. More precisely,
they proved the following theorem.
Theorem 1.1 [7, Proposition 2(iv), Theorem 1]. Suppose D is a nonempty closed con-
vex subset of a reflexive Banach space E and T : D → E is a continuous pseudocontractive
mapping satisfying the weakly inward condition. Then for z ∈ D, there exists a unique path
t → y
t
∈ D, t ∈[0, 1), satisfying the following condition,
y
t
= tTy
t
+(1−t)z. (1.5)
Further more, suppose E is assumed to have a uniformly G
ˆ
ateaux differentiable norm and
is such that every closed convex and bounded subset of D has the fixed point propert y for
nonexpansive self-mappings. If F(T) =∅or there exists x
0
∈ D such that the set K :={x ∈
D : Tx =λx +(1−λ)x
0
for λ>1} is bounded, then as t →1
−

, the path converges strongly to
afixedpointofT.
From Theorem 1.1, one question arises quite naturally.
Question. Can the continuity of T be weakened to demicontinuity of T?
In connection with this, Lan and Wu [3] proved the following theorem in the Hilbert
space setting.
Theorem 1.2 [3, Theorems 2.3 and 2.5]. Le t E be a Hilbert space. Suppose D is a nonempty
closed convex subset of E and T : D
→ E is a demicontinuous pseudocontractive mapping
satisfying the weakly inward condit ion. Then for z ∈ D, there exists a unique path t → y
t
∈
D, t ∈(0,1), satisfying the following condition:
y
t
= tTy
t
+(1−t)z. (1.6)
Moreover, if (i) D is bounded then F(T) =∅and {y
t
} converges strongly to a fixed point
of T as t → 1
−
; (ii) D is unbounded and F(T) =∅then {y
t
} converges strongly to a fixed
point of T as t →1
−
.
C. E. Chidume and H. Zegeye 69

Let D be a nonempty open and convex subset of a real uniformly smooth Banach space
E.SupposeT :
¯
D →E is a demicontinuous pseudocontractive mapping which satisfies
for some z ∈D, Tx−z = λ(x −z)forx ∈∂D, λ>1, (1.7)
where
¯
D is the closure of D.
It is our purpose in this paper to give sufficient conditions to ensure that
¯
D
⊆ (I +
r(I −T))(
¯
D)foreveryr>0 and to prove the existence and convergence of a path to a
fixed point of a demicontinuous pseudocontractive mapping in spaces more general than
Hilbert spaces. More precisely, we prove that for a given y
0
∈ D, there exists a unique path
t → y
t
∈
¯
D, t ∈ (0,1), satisfying y
t
:= tTy
t
+(1−t)y
0
.Moreover,ifF(T) =∅or there ex-

ists y
0
∈ D such that the set K :={y ∈ D : Ty = λy +(1−λ)y
0
for λ>1} is bounded,
then the path {y
t
} converges strongly to a fixed point of T. Furthermore, the sequence
{x
n
} generated from x
1
∈ K by x
n+1
:= (1 −λ
n
)x
n
+ λ
n
Tx
n
−λ
n
θ
n
(x
n
−x
1

), for all integers
n
≥ 1, where {λ
n
} and {θ
n
} are real sequences satisfying appropriate conditions, con-
verges strongly to a fixed point of T.Ourtheoremsprovideanaffirmativeanswertothe
above question in uniformly smooth Banach spaces and extend Theorem 1.2 to uniformly
smooth spaces provided that the interior of D,int(D), is nonempty.
2. Preliminaries
Let E be a real normed linear space of dimension ≥ 2. The modulus of smoothness of E is
defined by
ρ
E
(τ):= sup

x + y+ x − y
2
−1:x=1, y=τ

, τ>0. (2.1)
If there exist a constant c>0 and a real number 1 <q<∞,suchthatρ
E
(τ) ≤ cτ
q
,then
E is said to be q-uniformly smooth. Typical examples of such spaces are L
p
and the Sobolev

spaces W
m
p
for 1<p<∞.ABanachspaceE is called uniformly smooth if lim
τ→0
(ρ
E
(τ)/τ) =
0. If E is a real uniformly smooth Banach space, then
x + y
2
≤x
2
+2

y, j(x)

+max

x,1

yb

y

(2.2)
holds for every x, y ∈ E where b :[0,∞) → [0,∞) is a continuous strictly increasing func-
tion satisfying the following conditions:
(i) b(ct)
≤ cb(t), ∀c ≥1,

(ii) lim
t→0
b(t) =0. (See, e.g., [8].)
Let D beanonemptysubsetofaBanachspaceE.Forx ∈D,theinward set of x, I
D
(x),
is defined by I
D
(x):={x + λ(u −x):u ∈D, λ ≥1}.AmappingT : D → E is called weakly
inward if Tx ∈cl[I
D
(x)] for all x ∈ D,wherecl[I
D
(x)] denotes the closure of the inward
set. Every self-map is trivially weakly inward.
Let D ⊆ E be closed convex and let Q be a mapping of E onto D.AmappingQ of E
into E is said to be a retraction if Q
2
= Q.IfamappingQ is a retraction, then Qz =z for
every z ∈ R(Q), range of Q.AsubsetD of E is said to be a nonexpansive ret ract of E if
there exists a nonexpansive retraction of E onto D. If E = H, the metric projection P
D
is a
nonexpansive retraction from H to any closed convex subset D of H.
70 Fixed points of demicontinuous pseudocontractive maps
In what follows, we will make use of the following lemma and theorems.
Lemma 2.1 [2]. Let {λ
n
}, {γ
n

},and{α
n
} be sequences of nonnegative numbers satisfying

∞
1
α
n
=∞and γ
n
/α
n
→ 0,asn →∞. Let the recursive inequality
λ
n+1
≤ λ
n
−2α
n
ψ

λ
n

+ γ
n
, n =1,2, , (2.3)
be given where ψ :[0,∞) → [0,∞) is a nondecreasing function such that it is positive on
(0,∞) and ψ(0) =0. Then λ
n

→ 0,asn →∞.
Theorem 2.2 [6]. Let E be a uniformly smooth Banach space and let D be an open subset
of E.SupposeT :
¯
D → E is a demicontinuous strongly pseudocontractive mapping which
satisfies
for some z ∈ D : Tx−z = λ(x −z) for x ∈∂D, λ>1. (2.4)
Then T has a unique fixed point in
¯
D.
Remark 2.3. We observe that, in Theorem 2.2, if, in addition, D is convex, then any weakly
inward map satisfies condition (2.4).
Theorem 2.4 (Reich [10]). Let E be uniformly smooth. Let A ⊂ E ×E be accretive with
cl(D(A)) convex. Suppose A satisfies the range condition. Le t J
t
:= (I + tA)
−1
, t>0 be the
resolvent of A and assume that A
−1
(0) is nonempty. Then, for each x ∈ ᏾(I + rA)(
¯
D),
lim
t→∞
J
t
x = Px ∈A
−1
(0),whereP is the sunny nonexpansive retraction of cl(D(A)) onto

A
−1
(0).
Remark 2.5. Fr om the proof of Theorem 2.4, we observe that we may replace the as-
sumption that A
−1
(0) =∅with the assumption that x
t
= J
t
x is bounded, for each x ∈
᏾(I + tA)andt>0.
3. Main results
We first prove the following results which will be used in the sequel.
Proposition 3.1. Let D be an open subset of a real uniformly smooth Banach space E
and let T :
¯
D
→ E be a demicontinuous pseudocontractive mapping which satisfies condition
(2.4). Let A
T
:
¯
D →E be defined by A
T
:= I + r(I −T) for any r>0. Then
¯
D ⊆A
T
[

¯
D].
Proof. Let z ∈
¯
D.Thenitsuffices to show that there exists x ∈
¯
D such that z = A
T
(x).
Define g :
¯
D → E by g(x):= (1/(1 + r))(rT(x)+z)forsomer>0. Then clearly g is demi-
continuous and for x, y ∈
¯
D we have that g(x) −g(y), j(x − y)≤(r/(1 + r))x − y
2
.
Thus, g is a strongly pseudocontractive mapping which satisfies condition (2.4). There-
fore, by Theorem 2.2, there exists x ∈
¯
D such that g(x) =x, that is, z = A
T
(x). The proof
is complete. 
Corollary 3.2. Le t E be a real uniformly smooth Banach space and let A : E →E be demi-
continuous accretive mapping. Then A is m-accretive.
C. E. Chidume and H. Zegeye 71
Proof. Set T := (I −A). Then, we obtain that T is a demicontinuous pseudocontractive
self-map of E. Clearly, condition (2.4) is satisfied. The conclusion follows from
Proposition 3.1. 

Corollary 3.2 was proved by Minty [5] in a Hilbert space setting for continuous accre-
tive mappings and this was extended to general Banach spaces by Martin [4].
We now prove the following theorems.
Theorem 3.3. Let D be an open and convex subset of a real uniformly smooth Banach space
E.LetT :
¯
D →E be a demicontinuous pseudocontractive mapping satisfying condition (2.4).
Then for a given y
0
∈ D, there exists a unique path t → y
t
∈
¯
D, t ∈(0,1), satisfy ing
y
t
= tTy
t
+(1−t)y
0
. (3.1)
Furthermore, if F(T) =∅or there exists z ∈ D such that the set K :={y ∈ D : Ty= λy +
(1 −λ)z for λ>1} is bounded, then the path {y
t
} described by (3.1) converges strongly to a
fixed point of T as t →1
−
.
Proof. For each t ∈ (0,1) the mapping T
t

defined by T
t
x := tT(t
n
)x +(1−t)y
0
is demi-
continuous and strongly pseudocontractive. By Theorem 2.2, it has a unique fixed point
y
t
in
¯
D, that is, for each t ∈(0,1) there exists y
t
∈
¯
D satisfying (3.1). Continuity of y
t
fol-
lows as in [7]. Now we show the convergence of {y
t
} to a fixed point of T.LetA :=I −T.
Then A is accretive and by Proposition 3.1,
¯
D ⊆(I + rA)(
¯
D)forallr>0 and hence A sat-
isfies the range condition. Moreover, from (3.1), y
t
+(t/(1 −t))Ay

t
= y
0
. But this implies
that y
t
= (I +(t/(1 −t))A)
−1
y
0
= J
(t/(1−t))
y
0
. Furthermore, since A
−1
(0) =∅or the fact
that K is bounded implies that {y
t
} is bounded (see, e.g., [7]), we have by Theorem 2.4
that y
t
→ y
∗
∈ A
−1
(0) and hence y
t
→ y
∗

∈ F(T)ast → 1
−
. This completes the proof of
the theorem. 
Remark 3.4. We note that, in Theorem 3.3, the requirement that T satisfies condition
(2.4) may be replaced with the weakly inward condition. Fur thermore, Theorem 3.3
extends [3, Theorems 2.3 and 2.5] to the more general Banach spaces which include
l
p
,L
p
,W
m
p
,1 <p<∞, spaces, provided that int(D)isnonempty.
For our next theorem and corollary, {λ
n
}, {θ
n
},and{c
n
} are real sequences in [0,1]
satisfying the following conditions:
(i) lim
n→∞
θ
n
= 0;
(ii)


∞
n=1
λ
n
θ
n
=∞,lim
n→∞
(b(λ
n
)/θ
n
) =0;
(iii) lim
n→∞
((θ
n−1
/θ
n
−1)/λ
n
θ
n
) =0, c
n
= o(λ
n
θ
n
).

Theorem 3.5. Let D be an open and convex subset of a real uniformly smooth Banach space
E.SupposeT :
¯
D → E is a bounded demicontinuous pseudocontractive mapping satisfying
condition (2.4). Suppose
¯
D is a nonexpansive re tract of E with Q as the nonexpansive retrac-
tion. Let a sequence {x
n
} be generated from x
0
∈ E by
x
n+1
= Q

1 −λ
n

x
n
+ λ
n
Tx
n
−λ
n
θ
n


x
n
−x
0

−c
n

x
n
−u
n

, (3.2)
72 Fixed points of demicontinuous pseudocontractive maps
for all positive integers n,where{u
n
}is a sequence of bounded error terms. If either F(T) =∅
or the set K :={x ∈ D : Tx = λx +(1−λ)x
0
for λ>1} is bounded, then there exists d>0
such that whenever λ
n
≤ d and c
n
/λ
n
θ
n
,b(λ

n
)/θ
n
≤ d
2
for all n ≥0, {x
n
} converges strongly
to a fixed point of T.
Proof. By Theorem 3.3, F(T) =∅.Letx
∗
∈ F(T). Let r>1besufficiently large such that
x
0
∈ B
r/2
(x
∗
).
Claim 3.6. {x
n
} is bounded.
It suffices to show by induction that {x
n
} belongs to B = B
r
(x
∗
) for all positive inte-
gers. Now, x

0
∈ B by assumption. Hence we may assume that x
n
∈ B and set M := 2r +
sup{(I −T)x
i
+ x
i
−u
i
,fori ≤n}.Weprovethatx
n+1
∈ B.Supposex
n+1
is not in B.
Then
x
n+1
−x
∗
 >rand thus from (3.2)wehavethatx
n+1
−x
∗
≤x
n
−x
∗
−λ
n

((I −
T)x
n
+ θ
n
(x
n
−x
0
))−c
n
(x
n
−u
n
)≤x
n
−x
∗
+ λ
n
(I −T)x
n
+ θ
n
(x
n
−x
0
)+(c

n
/λ
n
)(x
n
−
u
n
)≤r + M. Moreover, from (3.2) and inequality (2.2), and using the fact that θ
n
≤ 1,
we get that


x
n+1
−x
∗


2
=


Q

1 −λ
n

x

n
+ λ
n
Tx
n
−λ
n
θ
n

x
n
−x
0

−c
n

x
n
−u
n

−x
∗


≤



x
n
−x
∗
−λ
n

(I −T)x
n
+ θ
n

x
n
−x
0

−c
n
(x
n
−u
n
)


2
≤



x
n
−x
∗


2
−2λ
n

(I −T)x
n
, j

x
n
−x
∗

−2λ
n
θ
n

x
n
−x
0
, j


x
n
−x
∗

−2c
n

x
n
−u
n
, j

x
n
−x
∗

+max



x
n
−x
∗


,1


λ
n




(I −T)x
n
+ θ
n

x
n
−x
0

+
c
n
λ
n

x
n
−u
n






×b

λ
n




(I −T)x
n
+ θ
n

x
n
−x
0

+
c
n
λ
n

x
n
−u
n







≤


x
n
−x
∗


2
−2λ
n

(I −T)x
n
, j

x
n
−x
∗

−2λ
n

θ
n

x
n
−x
0
, j

x
n
−x
∗

−2c
n

x
n
−u
n
, j

x
n
−x
∗

+(r +1)λ
n

Mb

λ
n
M

.
(3.3)
Since T is pseudocontractive and x
∗
∈ F(T), we have (I −T)x
n
, j(x
n
−x
∗
)≥0. Hence
(3.3)gives


x
n+1
−x
∗


2
≤



x
n
−x
∗


2
−2λ
n
θ
n

x
n
−x
0
, j

x
n
−x
∗

+2c
n


x
n
−u

n


·


x
n
−x
∗


+(r +1)λ
n
M
2
b

λ
n

.
(3.4)
Choose L>0sufficiently small such that L
≤ r
2
/(2
√
D
∗

+2M)
2
,whereD
∗
= (r +1)M
2
.
Set d :=
√
L. Then since x
n+1
−x
∗
 > x
n
−x
∗
 by our assumption, from (3.4)weget
that 2λ
n
θ
n
x
n
−x
0
, j(x
n
−x
∗

)≤(r +1)M
2
λ
n
b(λ
n
)+2c
n
Mr which gives x
n
−x
0
, j(x
n
−
x
∗
)≤D
∗
L, since c
n
/λ
n
θ
n
,b(λ
n
)/θ
n
≤ L = d

2
,foralln ≥ 1 by our assumption.
C. E. Chidume and H. Zegeye 73
Now adding x
0
−x
∗
, j(x
n
−x
∗
) to both sides of this inequality, we get that


x
n
−x
∗


2
≤ LD
∗
+

x
0
−x
∗
, j


x
n
−x
∗

≤ LD
∗
+


x
0
−x
∗




x
n
−x
∗


≤ LD
∗
+
r
2



x
n
−x
∗


.
(3.5)
Solving this quadratic inequality for x
n
−x
∗
 and using the estimate
√
r
2
/16 + LD
∗
≤
r/4+
√
LD
∗
,weobtainthatx
n
−x
∗
≤r/2+

√
LD
∗
. But in any case, x
n+1
−x
∗
≤
x
n
−x
∗
+ λ
n
(I −T)x
n
+ θ
n
(x
n
−x
0
)+(c
n
/λ
n
)(x
n
−u
n

) so that x
n+1
−x
∗
≤r/2+
√
LD
∗
+ λ
n
M ≤ r, by the original choices of L and λ
n
, and this contradicts the assumption
that x
n+1
is not in B. Therefore, x
n
∈ B for all positive integers n.Thus{x
n
} is bounded.
Now we show that x
n
→ x
∗
.Let{y
n
} be a subsequence of {y
t
: t ∈[0,1)},suchthaty
n

:=
y
t
n
, t
n
= 1/(1 + θ
n
). Then from (3.2) and inequality (2.2) and using the fact that y
n
∈
¯
D
for all n ≥0, we get


x
n+1
− y
n


2
=


Q

1 −λ
n


x
n
+ λ
n
Tx
n
−λ
n
θ
n

x
n
−x
0

−c
n

x
n
−u
n

− y
n


2

≤


x
n
− y
n
−λ
n

(I −T)x
n
+ θ
n

x
n
−x
0

−c
n

x
n
−u
n




2
≤


x
n
− y
n


2
−2λ
n

(I −T)x
n
+ θ
n

x
n
−x
0

, j

x
n
− y
n


−2c
n

x
n
−u
n
, j

x
n
− y
n

+max



x
n
− y
n


,1

λ
n





(I −T)x
n
+ θ
n

x
n
−x
0

+
c
n
λ
n

x
n
−u
n





×b


λ
n




(I −T)x
n
+ θ
n

x
n
−x
0

+
c
n
λ
n

x
n
−u
n







≤

1 −2λ
n
θ
n



x
n
− y
n


2
−2λ
n

(I −T)x
n
+ θ
n

y
n
−x
0


, j

x
n
− y
n

+2c
n


x
n
−u
n


·


x
n
− y
n


+max




x
n
− y
n


,1

λ
n




(I −T)x
n
+ θ
n

x
n
−x
0

+
c
n
λ
n


x
n
−u
n





×
b

λ
n




(I −T)x
n
+ θ
n

x
n
−x
0

+

c
n
λ
n

x
n
−u
n






.
(3.6)
Since Ty
n
= y
n
+ θ
n
(y
n
−x
0
)andT is pseudocontractive, we get that (I −T)x
n
+ θ

n
(y
n
−
x
0
), j(x
n
− y
n
)≥0. Moreover, since {x
n
}, {y
n
}, and hence {Tx
n
}, are bounded, there ex-
ists M
0
> 0 such that max {x
n
− y
n
,1,x
n
− y
n
·x
n
−u

n
,(I −T)x
n
+ θ
n
(x
n
−x
0
)+
(c
n
/λ
n
)(x
n
−u
n
)} ≤M
0
. Therefore, (3.6)withpropertyofb gives


x
n+1
− y
n


2

≤

1 −2λ
n
θ
n



x
n
− y
n


2
+ M
0
λ
n
b

λ
n

+ c
n
M
0
. (3.7)

On the other hand, by the pseudocontractivity of T and the fact that θ
n
(y
n
−x
0
)+(y
n
−
Ty
n
) =0, we have that


y
n−1
− y
n


≤




y
n−1
− y
n
+

1
θ
n

(I −T)y
n−1
−(I −T)y
n





≤
θ
n−1
−θ
n
θ
n



y
n−1


+ z

=


θ
n−1
θ
n
−1




y
n−1


+ z

.
(3.8)
74 Fixed points of demicontinuous pseudocontractive maps
However,


x
n
− y
n


2
≤



x
n
− y
n−1


2
+


y
n−1
− y
n





y
n−1
− y
n


+2



y
n−1
−x
n



. (3.9)
Therefore, these estimates with (3.7)givethat


x
n+1
− y
n


2
≤

1 −2λ
n
θ
n



x
n
− y

n−1


2
+ M
1

θ
n−1
θ
n
−1

+ M
1
λ
n
b

λ
n

+ c
n
M
1
, (3.10)
for some M
1
> 0. Thus, by Lemma 2.1, x

n+1
− y
n
→ 0. Hence, since y
n
→ x
∗
by Theorem
3.3,wehavethatx
n
→ x
∗
, this completes the proof of the theorem. 
With the help of Remark 2.3 and Theorem 3.5 we obtain the following corollary.
Corollary 3.7. Let D be an ope n and convex subset of a real uniformly smooth Banach
space E.SupposeT :
¯
D → E is a bounded demicontinuous pseudocontractive mapping satis-
fy ing the weakly inward condition. Suppose
¯
D is a nonexpansive retract of E with Q as the
nonexpansive retraction. Let a sequence {x
n
} be generated from x
0
∈E by
x
n+1
= Q


1 −λ
n

x
n
+ λ
n
Tx
n
−λ
n
θ
n

x
n
−x
0

−c
n

x
n
−u
n

, (3.11)
for all positive integers n,where{u
n

} isasequenceoferrorterms.IfeitherF(T) =∅or the
set K :={x ∈D : Tx = λx +(1−λ)x
0
for λ>1} is bounded then, there exists d>0 such that
whenever λ
n
≤ d and c
n
/λ
n
θ
n
,b(λ
n
)/θ
n
≤ d
2
for all n ≥0, {x
n
} converges strongly to a fixed
point of T.
Remark 3.8. For the case where E is q-uniformly smooth, where q>1, and t ≤ M for
some M>0, the function b in (2.2)isestimatedbyb(t) ≤ct
q−1
for some c>0 (see [9]).
Thus, we have the following corollary.
Corollary 3.9. Let D be an open and convex subs et of a real q-uniformly smooth Banach
space E.SupposeT :
¯

D
→ E is a bounded demicontinuous pseudocontractive mapping satis-
fy ing condition (2.4). Suppose
¯
D is a nonexpansive retract of E with Q as the nonexpansive
retraction and let {λ
n
}, {θ
n
},and{c
n
} be real sequences in (0,1] satisfying the following
conditions:
(i) lim
n→∞
θ
n
= 0;
(ii)

∞
n=1
λ
n
θ
n
=∞, lim
n→∞
(λ
(q−1)

n
/θ
n
) =0;
(iii) lim
n→∞
((θ
n−1
/θ
n
−1)/λ
n
θ
n
) =0, c
n
= o(λ
n
θ
n
).
Let a sequence {x
n
} be generated from x
0
∈ E by
x
n+1
= Q


1 −λ
n

x
n
+ λ
n
Tx
n
−λ
n
θ
n

x
n
−x
0

−c
n

x
n
−u
n

, (3.12)
for all positive integers n,where{u
n

}is a bounded sequence of error terms. If either F(T) =∅
or the set K :={x ∈ D : Tx = λx +(1−λ)x
0
for λ>1} is bounded, then there exists d>0
such that whenever λ
n
≤ d and c
n
/λ
n
θ
n
,λ
(q−1)
n
/θ
n
≤ d
2
for all n ≥0, {x
n
} converges strongly
to a fixed point of T.
C. E. Chidume and H. Zegeye 75
Remark 3.10. Examples of sequences {λ
n
} and {θ
n
} satisfying conditions of Corollary 3.9
are as follows: λ

n
= 2(n +1)
−a
, θ
n
= 2(n +1)
−b
,andc
n
= 2(n +1)
−1
with 0 <b<aand
a + b<1if2≤q<∞, and with 0 <b<a(q −1) and a + b(q −1) < 1if1<q<2.
If in Theorem 3.5, T is a self-map of
¯
D, then the projection operator Q is replaced with
I, the identity map on E.Moreover,T satisfies condition (2.4). As a consequence, we have
the following corollaries.
Corollary 3.11. Let D be an open and convex subset of a real uniformly smooth Ba-
nach space E.SupposeT :
¯
D →
¯
D is a bounded demicontinuous pseudocontractive mapping.
Suppose {λ
n
}, {θ
n
},and{c
n

} are real seque nces in (0, 1] satisfying conditions (i)–(iii) of
Theorem 3.5 and λ
n
(1 + θ
n
)+c
n
≤ 1, ∀n ≥0.Letasequence{x
n
} be generated from x
0
∈E
by
x
n+1
=

1 −λ
n

x
n
+ λ
n
Tx
n
−λ
n
θ
n


x
n
−x
0

−c
n

x
n
−u
n

, (3.13)
for all positive integers n,where{u
n
}is a sequence of bounded error terms. If either F(T) =∅
or the set K :={x ∈ D : Tx = λx +(1−λ)x
0
for λ>1} is bounded, then there exists d>0
such that whenever λ
n
≤ d and c
n
/λ
n
θ
n
,b(λ

n
)/θ
n
≤ d
2
for all n ≥0, {x
n
} converges strongly
to a fixed point of T.
Proof. The conditions on λ
n
, θ
n
,andc
n
imply that the sequence {x
n
} is well defined.
Thus, the proof follows from Theorem 3.5. 
If in Theorem 3.5, D is assumed to be b ounded, then the conditions λ
n
≤ d and
c
n
/λ
n
θ
n
,b(λ
n

)/θ
n
≤ d
2
for some d>0 which guarantee the boundedness of the sequence
{x
n
} are not needed. In fact, we have the following corollary.
Corollary 3.12. Let D be an open convex and bounded subset of a real uniformly smooth
Banach space E.SupposeT :
¯
D → E is a bounded demicontinuous pseudocontractive map-
ping satisfying the weakly inward condition. Suppose
¯
D is a nonexpansive retract of E with
Q as the nonexpansive retraction and let {λ
n
}, {θ
n
},and{c
n
} be real sequences in (0,1)
satisfying conditions (i)–(iii) of Theorem 3.5.Letasequence{x
n
} be generated from x
0
∈ E
by
x
n+1

= Q

1 −λ
n

x
n
+ λ
n
Tx
n
−λ
n
θ
n

x
n
−x
0

−c
n

x
n
−u
n

, (3.14)

for all positive integers n,where{u
n
} isasequenceoferrorterms.Then{x
n
} converges
strongly to a fixed point of T.
Proof. Since D, and hence
¯
D, is bounded we have that {x
n
} is bounded. Thus the conclu-
sion follows from Theorem 3.5. 
Corollary 3.13. Let D be an open convex and bounded subset of a real uniformly smooth
Banach space E.SupposeT :
¯
D →
¯
D is a bounded demicontinuous pseudocontractive map-
ping. Let {λ
n
}, {θ
n
},and{c
n
} be real sequences in (0, 1] satisfying conditions (i)–(iii) of
Theorem 3.5 and λ
n
(1 + θ
n
)+c

n
≤ 1, ∀n ≥0.Letasequence{x
n
} be generated from x
0
∈E
76 Fixed points of demicontinuous pseudocontractive maps
by
x
n+1
=

1 −λ
n

x
n
+ λ
n
Tx
n
−λ
n
θ
n

x
n
−x
0


−c
n

x
n
−u
n

, (3.15)
for all positive integers n,where{u
n
} isasequenceoferrorterms.Then{x
n
} converges
strongly to a fixed point of T.
Remark 3.14. If in Theorem 3.5, D is bounded, T is a self-map, and c
n
≡ 1foralln ≥1,
that is, the error term is ignored, then the following corollary holds.
Corollary 3.15. Let D be an open convex and bounded subset of a real uniformly smooth
Banach space E.SupposeT :
¯
D →
¯
D is a bounded demicontinuous pseudocontractive map-
ping. Let {λ
n
} and {θ
n

} be real sequences in (0,1] satisfying conditions (i)–(iii) of Theorem
3.5 with c
n
≡ 0 for all n ≥1 and λ
n
(1 + θ
n
) ≤1,foralln ≥ 0.Letasequence{x
n
} be gener-
ated from x
0
∈ E by
x
n+1
=

1 −λ
n

x
n
+ λ
n
Tx
n
−λ
n
θ
n


x
n
−x
0

, (3.16)
for all positive integers n. Then {x
n
} converges strongly to a fixed point of T.
The following convergence theorem is for the approximation of solution of demicon-
tinuous accretive mappings.
Theorem 3.16. Let D be an open and convex subset of a real uniformly smooth Banach
space E.SupposeA :
¯
D
→ E is a bounded demicontinuous accretive mapping which satisfies,
for some x
0
∈ D, Ax =t(x −x
0
) for all x ∈ ∂D and t<0.Suppose
¯
D is a nonexpansive retract
of E with Q as the nonexpansive retraction and let {λ
n
}, {θ
n
},and{c
n

} be real sequences in
(0,1] satisfying conditions (i)–(iii) of Theorem 3.5.Letasequence{x
n
} be generated from
x
0
∈ E by
x
n+1
= Q

x
n
−λ
n

Ax
n
+ θ
n

x
n
−x
0

−c
n

x

n
−u
n

, (3.17)
for all positive integers n,where{u
n
} is a s equence of bounded error ter ms. Suppose ei-
ther N(A) =∅(N(A) is the null space of A)or the set K :={x ∈ D :(I − A)x = λx +
(1 −λ)x
0
for λ>1} is bounded. Then there exists d>0 such that whenever λ
n
≤ d and
c
n
/λ
n
θ
n
,b(λ
n
)/θ
n
≤ d
2
for all n ≥0, {x
n
} converges strongly to a zero of A.
Proof. Set T := (I −A). Then, we have that for some x

0
∈ D,(I − T)x = t(x −x
0
)for
x ∈∂D and t<0. This implies that Tx−x
0
= λ(x −x
0
)forallx ∈∂D and λ>1. Moreover,
F(T) =∅or the set K ={x ∈ D : Tx = λx +(1−λ)x
0
,forλ = (1 −t) > 1} is bounded.
Therefore, by Theorem 3.5, {x
n
} converges strongly to x
∗
∈ F(T). But F(T) = N(A).
Hence, {x
n
} converges strongly to x
∗
∈ N(A). The proof of the theorem is complete. 
The following corollary follows from Theorem 3.16.
Corollary 3.17. Let E be a real uniformly smooth Banach space and suppose A : E → E is
a bounded demicontinuous accretive mapping. Let {λ
n
}, {θ
n
},and{c
n

} be real sequences in
(0,1] satisfying conditions (i)–(iii) of Theorem 3.5.Letasequence{x
n
} be generated from
C. E. Chidume and H. Zegeye 77
x
0
∈ E by
x
n+1
= x
n
−λ
n

Ax
n
+ θ
n

x
n
−x
0

−c
n

x
n

−u
n

, (3.18)
for all positive integers n,where
{u
n
} isasequenceofboundederrorterms.IfeitherN(A) =
∅ or the set K :={x ∈ E :(I −A)x = λx +(1−λ)x
0
for λ>1} is bounded, then there exists
d>0 such that whenever λ
n
≤ d and c
n
/λ
n
θ
n
,b(λ
n
)/θ
n
≤ d
2
for all n ≥ 0, {x
n
} converges
strongly to a point of N(A).
Acknowledgment

The second author undertook this work when he was visiting The Abdus Salam Interna-
tional Centre for Theoretical Physics, Trieste, Italy, as a Junior Associate.
References
[1] F.E.Browder,Convergence of approximants to fixed points of nonexpansive non-linear mappings
in Banach spaces, Arch. Rational Mech. Anal. 24 (1967), 82–90.
[2] C. E. Chidume and H. Zegeye, Approximation of solutions of nonlinear equations of monotone
and Hammer s tein type,Appl.Anal.82 (2003), no. 8, 747–758.
[3] K.Q.LanandJ.H.Wu,Conver gence of approximants for demicontinuous pseudo-contractive
maps in Hilbert spaces , Nonlinear Anal. Ser. A: Theory Methods 49 (2002), no. 6, 737–746.
[4] R. H. Mar tin Jr., A global existence theorem for autonomous differential equations in a Banach
space,Proc.Amer.Math.Soc.26 (1970), 307–314.
[5] G.J.Minty,Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346.
[6] C.H.Morales,Zeros for accretive operators satisfying certain boundary conditions, J. Math. Anal.
Appl. 105 (1985), no. 1, 167–175.
[7] C.H.MoralesandJ.S.Jung,Convergence of paths for pseudocontractive mappings in Banach
spaces,Proc.Amer.Math.Soc.128 (2000), no. 11, 3411–3419.
[8] S. Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear
Anal. 2 (1978), no. 1, 85–92.
[9]
, Constructive techniques for accretive and monotone operators, Applied Nonlinear Anal-
ysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978), Academic Press, New
York, 1979, pp. 335–345.
[10]
, Strong conver gence theorems for resolvents of accretive operators in Banach spaces,J.
Math. Anal. Appl. 75 (1980), no. 1, 287–292.
[11] W. Takahashi and Y. Ueda, On Reich’s strong convergence theorems for resolvents of accretive
operators,J.Math.Anal.Appl.104 (1984), no. 2, 546–553.
C. E. Chidume: The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy
E-mail address:
H. Zegeye: The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy

E-mail address: