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Fixed point and weak convergence theorems for point-dependent lambda-hybrid
mappings in Banach spaces
Fixed Point Theory and Applications 2011, 2011:105 doi:10.1186/1687-1812-2011-105
Young-Ye Huang ()
Jyh-Chung Jeng ()
Tian-Yuan Kuo ()
Chung-Chien Hong ()
ISSN 1687-1812
Article type Research
Submission date 25 August 2011
Acceptance date 23 December 2011
Publication date 23 December 2011
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Fixed point and weak convergence theorems for
point-dependent λ-hybrid mappings in Banach
spaces
Young-Ye Huang
1
, Jyh-Chung Jeng
2
, Tian-Yuan Kuo

3
and Chung-Chien Hong
∗4
1
Center for General Education, Southern Taiwan University, 1 Nantai St., Yongkang
Dist., Tainan 71005, Taiwan
2
Nanjeon Institute of Technology, 178 Chaoqin Rd., Yenshui Dist., Tainan 73746,
Taiwan
3
Fooyin University, 151 Jinxue Rd., Daliao Dist., Kaohsiung 83102, Taiwan
4
Department of Industrial Management, National Pingtung University of Science
and Technology, 1 Shuefu Rd., Neopu, Pingtung 91201, Taiwan
∗
Corresponding author:
Email addresses:
YYH:
JCJ:
TYK:
1
Abstract
The purpose of this article is to study the fixed point and weak convergence
problem for the new defined class of point-dependent λ-hybrid mappings
relative to a Bregman distance D
f
in a Banach space. We at first extend
the Aoyama–Iemoto–Kohsaka–Takahashi fixed point theorem for λ-hybrid
mappings in Hilbert spaces in 2010 to this much wider class of nonlinear
mappings in Banach spaces. Secondly, we derive an Opial-like inequality

for the Bregman distance and apply it to establish a weak convergence
theorem for this new class of nonlinear mappings. Some concrete examples
in a Hilbert space showing that our extension is proper are also given.
Keywords: fixed point, Banach limit, Bregman distance, Gˆateaux differ-
entiable, subdifferential.
2010 MSC: 47H09; 47H10.
2
1 Introduction
Let C be a nonempty subset of a Hilbert space H. A mapping T : C → H is
said to be
(1.1) nonexpansive if T x −Ty ≤ x −y, ∀x, y ∈ C, cf. [1, 2];
(1.2) nonspreading if T x −Ty
2
≤ x −y
2
+ 2 x − T x, y −Ty, ∀x, y ∈ C,
cf. [3–5];
(1.3) hybrid if T x − T y 
2
≤ x − y
2
+ x − Tx, y − T y, ∀x, y ∈ C, cf.
[3, 5–7].
As shown in [3], (1.2) is equivalent to
2T x −T y
2
≤ T x −y
2
+ x −T y
2

for all x, y ∈ C.
In 1965, Browder [1] established the following
Browder fixed point Theorem. Let C be a nonempty closed convex subset of
a Hilbert space H, and let T : C → C be a nonexpansive mapping. Then, the
following are equivalent:
(a) There exists x ∈ C such that {T
n
x}
n∈N
is bounded;
(b) T has a fixed point.
The above result is still true for nonspreading mappings which was shown
in Kohsaka and Takahashi [4]. (We call it the Kohsaka–Takahashi fixed point
theorem.)
3
Recently, Aoyama et al. [8] introduced a new class of nonlinear mappings in
a Hilbert space containing the classes of nonexpansive mappings, nonspreading
mappings and hybrid mappings. For λ ∈ R, they call a mapping T : C → H
(1.4) λ-hybrid if T x −Ty
2
≤ x − y
2
+ λ x − Tx, y − T y, ∀x, y ∈ C.
And, among other things, they establish the following
Aoyama–Iemoto–Kohsaka–Takahashi fixed point Theorem. [8] Let C be
a nonempty closed convex subset of a Hilbert space H, and let T : C → C be a
λ-hybrid mapping. Then, the following are equivalent:
(a) There exists x ∈ C such that {T
n
x}

n∈N
is bounded;
(b) T has a fixed point.
Obviously, T is nonexpansive if and only if it is 0-hybrid; T is nonspreading
if and only if it is 2-hybrid; T is hybrid if and only if it is 1-hybrid.
Motivated by the above works, we extend the concept of λ-hybrid from Hilbert
spaces to Banach spaces in the following way:
Definition 1.1. For a nonempty subset C of a Banach space X, a Gˆateaux
differentiable convex function f : X → (−∞, ∞] and a function λ : C → R, a
mapping T : C → X is said to be point-dependent λ-hybrid relative to D
f
if
(1.5) D
f
(T x, T y) ≤ D
f
(x, y) + λ(y) x −Tx, f

(y) −f(T y), ∀x, y ∈ C,
where D
f
is the Bregman distance associated with f and f

(x) denotes the Gˆateaux
derivative of f at x.
In this article, we study the fixed point and weak convergence problem for
4
mappings satisfying (1.5). This article is organized in the following way: Sec-
tion 2 provides preliminaries. We investigate the fixed point problem for point-
dependent λ-hybrid mappings in Section 3, and we give some concrete examples

showing that even in the setting of a Hilbert space, our fixed point theorem gen-
eralizes the Aoyama–Iemoto–Kohsaka–Takahashi fixed point theorem properly in
Section 4. Section 5 is devoting to studying the weak convergence problem for
this new class of nonlinear mappings.
2 Preliminaries
In what follows, X will be a real Banach space with topological dual X
∗
and
f : X → (−∞, ∞] will be a convex function. D denotes the domain of f, that is,
D = {x ∈ X : f(x) < ∞},
and D
◦
denotes the algebraic interior of D, i.e., the subset of D consisting of
all those points x ∈ D such that, for any y ∈ X \ {x}, there is z in the open
segment (x, y) with [x, z] ⊆ D. The topological interior of D, denoted by Int(D),
is contained in D
◦
. f is said to be proper provided that D = ∅. f is called lower
semicontinuous (l.s.c.) at x ∈ X if f(x) ≤ lim inf
y→x
f(y). f is strictly convex if
f(αx + (1 − α)y) < αf(x) + (1 −α)f(y)
for all x, y ∈ X and α ∈ (0, 1).
The function f : X → (−∞, ∞] is said to be Gˆateaux differentiable at x ∈ X
5
if there is f

(x) ∈ X
∗
such that

lim
t→0
f(x + ty) −f(x)
t
= y, f

(x)
for all y ∈ X.
The Bregman distance D
f
associated with a proper convex function f is the
function D
f
: D × D → [0, ∞] defined by
D
f
(y, x) = f(y) −f(x) + f
◦
(x, x − y), (1)
where f
◦
(x, x − y) = lim
t→0
+
f(x + t(x − y)) − f(x)/t. D
f
(y, x) is finite valued
if and only if x ∈ D
◦
, cf. Proposition 1.1.2 (iv) of [9]. When f is Gˆateaux

differentiable on D, (1) becomes
D
f
(y, x) = f(y) −f(x) −y − x, f

(x), (2)
and then the modulus of total convexity is the function ν
f
: D
◦
×[0, ∞) → [0, ∞]
defined by
ν
f
(x, t) = inf{D
f
(y, x) : y ∈ D, y − x = t}.
It is known that
ν
f
(x, ct) ≥ cν
f
(x, t) (3)
for all t ≥ 0 and c ≥ 1, cf. Proposition 1.2.2 (ii) of [9]. By definition it follows
that
D
f
(y, x) ≥ ν
f
(x, y − x). (4)

6
The modulus of uniform convexity of f is the function δ
f
: [0, ∞) → [0, ∞]
defined by
δ
f
(t) = inf

f(x) + f(y) −2f

x + y
2

: x, y ∈ D, x − y ≥ t

.
The function f is called uniformly convex if δ
f
(t) > 0 for all t > 0. If f is
uniformly convex then for any ε > 0 there is δ > 0 such that
f

x + y
2

≤
f(x)
2
+

f(y)
2
− δ (5)
for all x, y ∈ D with x −y ≥ ε.
Note that for y ∈ D and x ∈ D
◦
, we have
f(x) + f(y) −2f

x + y
2

=f(y) − f(x) −
f

x +
y−x
2

− f(x)
1
2
≤f(y) − f(x) −f
◦
(x, y − x) ≤ D
f
(y, x),
where the first inequality follows from the fact that the function t → f(x + tz) −
f(x)/t is nondecreasing on (0, ∞). Therefore,
ν

f
(x, t) ≥ δ
f
(t) (6)
whenever x ∈ D
◦
and t ≥ 0. For other properties of the Bregman distance D
f
,
we refer readers to [9].
The normalized duality mapping J from X to 2
X
∗
is defined by
Jx = {x
∗
∈ X
∗
: x, x
∗
 = x
2
= x
∗

2
}
for all x ∈ X.
7
When f(x) = x

2
in a smooth Banach space X, it is known that f

(x) =
2J(x) for x ∈ X, cf. Corollaries 1.2.7 and 1.4.5 of [10]. Hence, we have
D
f
(y, x) = y
2
− x
2
− y − x, f

(x)
= y
2
− x
2
− 2 y − x, Jx
= y
2
+ x
2
− 2 y, Jx.
Moreover, as the normalized duality mapping J in a Hilbert space H is the
identity operator, we have
D
f
(y, x) = y
2

+ x
2
− 2 y, x = y − x
2
.
Thus, in case λ is a constant function and f(x) = x
2
in a Hilbert space, (1.5)
coincides with (1.4). However, in general, they are different.
A function g : X → (−∞, ∞] is said to be subdifferentiable at a point x ∈ X
if there exists a linear functional x
∗
∈ X
∗
such that
g(y) − g(x) ≥ y − x, x
∗
, ∀y ∈ X.
We call such x
∗
the subgradient of g at x. The set of all subgradients of g at x
is denoted by ∂g(x) and the mapping ∂g : X → 2
X
∗
is called the subdifferential
of g. For a l.s.c. convex function f, ∂f is bounded on bounded subsets of Int(D)
if and only if f is bounded on bounded subsets there, cf. Proposition 1.1.11 of [9].
A proper convex l.s.c. function f is Gˆateaux differentiable at x ∈ Int(D) if and
only if it has a unique subgradient at x; in such case ∂f(x) = f


(x), cf. Corollary
1.2.7 of [10].
8
The following lemma will be quoted in the sequel.
Lemma 2.1. (Proposition 1.1.9 of [9]) If a proper convex function f : X →
(−∞, ∞] is Gˆateaux differentiable on Int(D) in a Banach space X, then the
following statements are equivalent:
(a) The function f is strictly convex on Int(D).
(b) For any two distinct points x, y ∈ Int(D), one has D
f
(y, x) > 0.
(c) For any two distinct points x, y ∈ Int(D), one has
x −y, f

(x) −f

(y) > 0.
Throughout this article, F (T ) will denote the set of all fixed points of a
mapping T .
3 Fixed point theorems
In this section, we apply Lemma 2.1 to study the fixed point problem for
mappings satisfying (1.5).
Theorem 3.1. Let X be a reflexive Banach space and let f : X → (−∞, ∞] be
a l.s.c. strictly convex function so that it is Gˆateaux differentiable on Int(D) and
is bounded on bounded subsets of Int(D). Suppose C ⊆ Int(D) is a nonempty
closed convex subset of X and T : C → C is point-dependent λ-hybrid relative to
D
f
for some function λ : C → R. For x ∈ C and any n ∈ N define
S

n
x =
1
n
n−1

k=0
T
k
x,
9
where T
0
is the identity mapping on C. If {T
n
x}
n∈N
is bounded, then every weak
cluster point of {S
n
x}
n∈N
is a fixed point of T.
Proof. Since T is point-dependent λ -hybrid relative to D
f
, we have, for any y ∈ C
and k ∈ N ∪ {0},
0 ≤ D
f
(T

k
x, y) − D
f
(T
k+1
x, T y) + λ(y)

T
k
x −T
k+1
x, f

(y) −f

(T y)

= f(T
k
x) −f(y) −

T
k
x −y, f

(y)

− f(T
k+1
x) + f(T y) +


T
k+1
x −Ty, f

(T y)

+ λ(y)

T
k
x −T
k+1
x, f

(y) −f

(T y)

=

f(T
k
x) −f(T
k+1
x)

+ [f(T y) −f(y)] +

λ(y)(T

k
x −T
k+1
x) −T
k
x + y, f

(y)

+

T
k+1
x −Ty −λ(y)(T
k
x −T
k+1
x), f

(T y)

.
Summing up these inequalities with respect to k = 0, 1, . . . , n −1, we get
0 ≤ [f(x) −f(T
n
x)] + n [f(T y) −f(y)] + λ(y)(x − T
n
x) + ny − nS
n
x, f


(y)
+ (n + 1)S
n+1
x −x − nT y − λ(y)(x −T
n
x), f

(T y).
Dividing the above inequality by n, we have
0 ≤
f(x) − f(T
n
x)
n
+ [f(T y) −f(y)] +

λ(y)(x −T
n
x)
n
+ y − S
n
x, f

(y)

+

n + 1

n
S
n+1
x −
x
n
− Ty −
λ(y)(x −T
n
x)
n
, f

(T y)

. (7)
Since {T
n
x}
n∈N
is bounded, {S
n
x}
n∈N
is bounded, and so, in view of X being
reflexive, it has a subsequence {S
n
i
x}
i∈N

so that S
n
i
x converges weakly to some
v ∈ C as n
i
→ ∞. Replacing n by n
i
in (7), and letting n
i
→ ∞, we obtain from
the fact that {T
n
x}
n∈N
and {f (T
n
x)}
n∈N
are bounded that
0 ≤ f(T y) − f(y) + y − v, f

(y)+ v − T y, f

(T y). (8)
10
Putting y = v in (8), we get
0 ≤ f(T v) − f(v) + v − T v, f

(T v),

that is,
0 ≤ −D
f
(v, T v),
from which follows that D
f
(v, T v) = 0. Therefore T v = v by Lemma 2.1. 
The following theorem comes from Theorem 3.1 immediately.
Theorem 3.2. Let X be a reflexive Banach space and let f : X → (−∞, ∞] be
a l.s.c. strictly convex function so that it is Gˆateaux differentiable on Int(D) and
is bounded on bounded subsets of Int(D). Suppose C ⊆ Int(D) is a nonempty
closed convex subset of X and T : C → C is point-dependent λ-hybrid relative
to D
f
for some function λ : C → R. Then, the following two statements are
equivalent:
(a) There is a point x ∈ C such that {T
n
x}
n∈N
is bounded.
(b) F(T ) = ∅.
Taking λ(x) = λ, a constant real number, for all x ∈ C and noting the function
f(x) = x
2
in a Hilbert space H satisfies all the requirements of Theorem 3.2,
the corollary below follows immediately.
Corollary 3.3. [8] Let C be a nonempty closed convex subset of Hilbert space
H and suppose T : C → C is λ-hybrid. Then, the following two statements are
equivalent:

(a) There exists x ∈ C such that {T
n
(x)}
n∈N
is bounded.
11
(b) T has a fixed point.
We now show that the fixed point set F (T ) is closed and convex under the
assumptions of Theorem 3.2.
A mapping T : C → X is said to be quasi-nonexpansive with respect to D
f
if
F (T) = ∅ and D
f
(v, T x) ≤ D
f
(v, x) for all x ∈ C and all v ∈ F (T ).
Lemma 3.4. Let f : X → (−∞, ∞] be a proper strictly convex function on a
Banach space X so that it is Gˆateaux differentiable on Int(D), and let C ⊆ Int( D)
be a nonempty closed convex subset of X. If T : C → C is quasi-nonexpansive
with respect to D
f
, then F (T) is a closed convex subset.
Proof. Let x ∈ F (T ) and choose {x
n
}
n∈N
⊆ F (T) such that x
n
→ x as n → ∞.

By the continuity of D
f
(·, T x) and D
f
(x
n
, T x) ≤ D
f
(x
n
, x), we have
D
f
(x, T x) = lim
n→∞
D
f
(x
n
, T x) ≤ lim
n→∞
D
f
(x
n
, x) = D
f
(x, x) = 0.
Thus, due to the strict convexity of f , it follows from Lemma 2.2 that T x = x.
This shows F (T ) is closed. Next, let x, y ∈ F(T ) and α ∈ [0, 1]. Put z =

αx + (1 −α)y. We show that T z = z to conclude F (T) is convex. Indeed,
12
D
f
(z, T z)
=f(z) − f(T z) −z −T z, f

(T z)
=f(z) + [αf(x) + (1 −α)f(y)] − f(T z) −z − T z, f

(T z)−[αf (x) + (1 − α)f(y)]
=f(z) + α[f(x) − f(T z) −x − T z, f

(T z)]
+ (1 −α)[f(y) − f(T z) −y − T z, f

(T z)] −[αf (x) + (1 − α)f(y)]
=f(z) + αD
f
(x, T z) + (1 −α)D
f
(y, T z) −[αf(x) + (1 − α)f(y)]
≤f(z) + αD
f
(x, z) + (1 −α)D
f
(y, z) − [αf(x) + (1 −α )f (y)]
=f(z) + α[f(x) − f(z) − x −z, f

(z)] + (1 − α)[f(y) − f(z) −y − z, f


(z)]
− [αf(x) + (1 −α)f(y)]
=f(z) + αf(x) − αf(z) − αx −αz, f

(z) + (1 − α)f(y) − (1 − α)f(z)
− (1 −α)y − (1 − α)z, f

(z) −[αf(x) + (1 −α)f (y)]
= −αx + (1 −α)y − (αz + (1 −α)z), f

(z)
= −0, f

(z) = 0.
Therefore, T z = z by the strictly convex of f. This completes the proof. 
Proposition 3.5. Let f : X → (−∞, ∞] be a proper strictly convex function on
a reflexive Banach space X so that it is Gˆateaux differentiable on Int(D) and is
bounded on bounded subsets of Int(D), and let C ⊆ Int(D) be a nonempty closed
convex subset of X. Suppose T : C → C is point-dependent λ-hybrid relative to
D
f
for some function λ : C → R and has a point x
0
∈ C such that {T
n
(x
0
)}
n∈N

is
bounded. Then, T is quasi-nonexpansive with respect to D
f
, and therefore, F (T )
13
is a nonempty closed convex subset of C.
Proof. In view of Theorem 3.2, F (T) = ∅. Now, for any v ∈ F (T ) and any
y ∈ C, as T is point-dependent λ-hybrid relative to D
f
, we have
D
f
(v, T y) = D
f
(T v, T y)
≤ D
f
(v, y) + λ(y) v −Tv, f

(y) −f

(T y)
= D
f
(v, y)
for all y ∈ C, so T is quasi-nonexpansive with respect to D
f
, and hence, F (T ) is
a nonempty closed convex subset of C by Lemma 3.4. 
For the remainder of this section, we establish a common fixed point theorem

for a commutative family of point-dependent λ-hybrid mappings relative to D
f
.
Lemma 3.6. Let X be a reflexive Banach space and let f : X → (−∞, ∞] be a
l.s.c. strictly convex function so that it is Gˆateaux differentiable on Int(D) and
is bounded on bounded subsets of Int(D). Suppose C ⊆ Int(D) is a nonempty
bounded closed convex subset of X and {T
1
, T
2
, . . . , T
N
} is a commutative finite
family of point-dependent λ-hybrid mappings relative to D
f
for some function
λ : C → R from C into itself. Then {T
1
, T
2
, . . . , T
N
} has a common fixed point.
Proof. We prove this lemma by induction with respect to N. To begin with,
we deal with the case that N = 2. By Proposition 3.5, we see that F (T
1
) and
F (T
2
) are nonempty bounded closed convex subsets of X. Moreover, F (T

1
) is T
2
-
invariant. Indeed, for any v ∈ F (T
1
), it follows from T
1
T
2
= T
2
T
1
that T
1
T
2
v =
T
2
T
1
v = T
2
v, which shows that T
2
v ∈ F (T
1
). Consequently, the restriction of T

2
to F (T
1
) is point-dependent λ-hybrid relative to D
f
, and hence by Theorem 3.2,
14
T
2
has a fixed point u ∈ F (T
1
), that is, u ∈ F (T
1
) ∩F(T
2
).
By induction hypothesis, assume that for some n ≥ 2, E = ∩
n
k=1
F (T
k
) is
nonempty. Then, E is a nonempty closed convex subset of X and the restriction of
T
n+1
to E is a point-dependent λ-hybrid mapping relative to D
f
from E into itself.
By Theorem 3.2, T
n+1

has a fixed point in X. This shows that E ∩F(T
n+1
) = ∅,
that is, ∩
n+1
k=1
F (T
k
) = ∅, completing the proof. .
Theorem 3.7. Let X be a reflexive Banach space and let f : X → (−∞, ∞]
be a l.s.c. strictly convex function so that it is Gˆateaux differentiable on Int(D).
Suppose C ⊆ Int(D) is a nonempty bounded closed convex subset of X and {T
i
}
i∈I
is a commutative family of point-dependent λ-hybrid mappings relative to D
f
for
some function λ : C → R from C into itself. Then, {T
i
}
i∈I
has a common fixed
point.
Proof. Since C is a nonempty bounded closed convex subset of the reflexive
Banach space X, it is weakly compact. By Proposition 3.5, each F (T
i
) is a
nonempty weakly compact subset of C. Therefore, the conclusion follows once
we note that {F (T

i
)}
i∈I
has the finite intersection property by Lemma 3.6. .
4 Examples
In this section, we give some concrete examples for our fixed point theorem.
At first, we need a lemma.
Lemma 4.1. Let h and k be two real numbers in [0, 1]. Then, the following two
statements are true.
15
(a) (h
2
− k
2
)
2
− (h −k)
2
≥ 0, if
h+k
2
> 0.5.
(b) (h
2
− k
2
)
2
− (h −k)
2

≤ 0, if
h+k
2
≤ 0.5.
Proof. First, we represent h and k by
h = 0.5 + a, where − 0.5 ≤ a ≤ 0.5,
and
k = 0.5 + b, where − 0.5 ≤ b ≤ 0.5.
Then, we have
(h
2
− k
2
)
2
− (h −k)
2
= (a − b)
2
(a + b)(a + b + 2).
If
h+k
2
> 0.5, then a + b > 0, and so through the above equation, we obtain
that (h
2
− k
2
)
2

− (h −k)
2
≥ 0. On the other hand,
h+k
2
≤ 0.5 implies a + b ≤ 0,
and hence, (h
2
− k
2
)
2
− (h −k)
2
≤ 0. 
Example 4.2. Let C =

x ∈ l
2
(N) : x = (x
1
, x
2
, , x
n
, ), 0 ≤ x
i
≤ 1 −
1
i+1


and δ be a positive number so small that
√
δ < 0.5. Define a mapping T : C → C
by
T x = (T x
1
, T x
2
, . . . , T x
n
, . . . ) : Tx
i
=















x

2
i
, if
√
δ < x
i
≤ 1 −
1
i+1
;
δ, if δ < x
i
≤
√
δ;
x
i
, if 0 ≤ x
i
≤ δ.
Then for any λ ∈ R, T is not λ-hybrid. However, for each x ∈ C, if we let
n
x
= min

n :

∞
i=n+1
x

2
i
≤ δ
2

and define λ : C → R by
16
λ(x) =
1

1
n
x
+1
−
1
(n
x
+1)
2

2
,
then T is point-dependent λ-hybrid, that is,
T x −T y
2
≤ x − y
2
+ λ(y) x −Tx, y − Ty (9)
for all x, y ∈ C. Therefore, we can apply Theorem 3.2 to conclude that T has

a fixed point, while the Aoyama–Iemoto–Kohsaka–Takahashi fixed point theorem
fails to give us the desired conclusion.
Proof. Let x and y be two elements from C so that the m
th
coordinate of x is
1 −
1
m+1
, the m
th
coordinate of y is 0.5 and the rest coordinates of x and y are
zero. We have
T x −T y
2
− x −y
2
− m x − Tx, y − T y
=


1 −
1
m + 1

2
− (0.5)
2

2
−


1 −
1
m + 1

− 0.5

2
− m


1 −
1
m + 1

−

1 −
1
m + 1

2


0.5 −(0.5)
2

=
9
16

−
2
m + 1
+
9
2(m + 1)
2
−
4
(m + 1)
3
+
1
(m + 1)
4
−
m
2
4(m + 1)
2
→
5
16
as m → ∞.
Since the value of above equality is always positive as m is large enough, we
conclude that there is no constant λ to satisfy the inequality:
T x −T y
2
≤ x − y
2

+ λ x − Tx, y − T y
for all x, y ∈ C.
17
It remains to show that T satisfies the inequality (9). We can rewrite the
inequality as
∞

i=1
(T x
i
− Ty
i
)
2
≤
∞

i=1
(x
i
− y
i
)
2
+
∞

i=1
λ(y)(x
i

− Tx
i
)(y
i
− Ty
i
).
Thus, if we can show that for all i ∈ N,
(T x
i
− Ty
i
)
2
≤ (x
i
− y
i
)
2
+ λ(y)(x
i
− Tx
i
)(y
i
− Ty
i
), (10)
then the assertion follows. We prove inequality (10) holds for all i ∈ N by

considering the following two cases: (I) i > min{n
x
, n
y
} and (II) i ≤ min{n
x
, n
y
}.
• Case (I). i > min{n
x
, n
y
}.
In this case, at least one of x
i
and y
i
is less than or equal to δ. Suppose
that 0 ≤ x
i
≤ δ. There are three subcases to discuss.
(I-1): If
√
δ < y
i
≤ 1 −
1
i+1
, then we have

(T x
i
− Ty
i
)
2
= (x
i
− y
2
i
)
2
≤ (x
i
− y
i
)
2
≤ (x
i
− y
i
)
2
+ λ(y)(x
i
− Tx
i
)(y

i
− Ty
i
).
(I-2): δ < y
i
≤
√
δ, then we have
(T x
i
− Ty
i
)
2
= (x
i
− δ)
2
≤ (x
i
− y
i
)
2
≤ (x
i
− y
i
)

2
+ λ(y)(x
i
− Tx
i
)(y
i
− Ty
i
).
(I-3): If 0 ≤ y
i
≤ δ, then we have
(T x
i
− Ty
i
)
2
= (x
i
− y
i
)
2
≤ (x
i
− y
i
)

2
+ λ(y)(x
i
− Tx
i
)(y
i
− Ty
i
).
18
The case that 0 ≤ y
i
≤ δ can be proved in the same manner.
• Case (II). i ≤ min{n
x
, n
y
}.
In this case, there are 9 subcases to discuss.
(II-1):
√
δ < x
i
≤ 1 −
1
i+1
and
√
δ < y

i
≤ 1 −
1
i+1
.
If
x
i
+y
i
2
≤ 0.5, it follows from Lemma 4.1 that
(T x
i
− Ty
i
)
2
= (x
2
i
− y
2
i
)
2
≤ (x
i
− y
i

)
2
≤ (x
i
− y
i
)
2
+ λ(y)(x
i
− Tx
i
)(y
i
− Ty
i
).
If
x
i
+y
i
2
> 0.5, then both x
i
and y
i
are greater than
1
i+1

, and so by
considering the graph of the function g(z) = z − z
2
in R, which is
symmetric to the line L : x = 0.5, we have

x
i
− x
2
i

≥

1
i + 1

−

1
i + 1

2
≥

1
n
y
+ 1


−

1
n
y
+ 1

2
and

y
i
− y
2
i

≥

1
i + 1

−

1
i + 1

2
≥

1

n
y
+ 1

−

1
n
y
+ 1

2
.
Consequently, we obtain
(T x
i
− Ty
i
)
2
=

x
2
i
− y
2
i

2

≤ 1 ≤
1

1
n
y
+1
−
1
(n
y
+1)
2

2
(x
i
− x
2
i
)(y
i
− y
2
i
)
≤ (x
i
− y
i

)
2
+ λ(y)(x
i
− Tx
i
)(y
i
− Ty
i
).
(II-2): δ < x
i
≤
√
δ and
√
δ < y
i
≤ 1 −
1
i+1
.
19
If y
i
≤ 0.5, then
x
i
+y

i
2
< 0.5. Thus, from Lemma 4.1, we have
(T x
i
− Ty
i
)
2
= (δ − y
2
i
)
2
≤ (x
2
i
− y
2
i
)
2
≤ (x
i
− y
i
)
2
≤ (x
i

− y
i
)
2
+ λ(y)(x
i
− Tx
i
)(y
i
− Ty
i
).
If y
i
> 0.5, we have either
δ < x
i
≤ δ +

1
i + 1

−

1
i + 1

2
or

δ +

1
i + 1

−

1
i + 1

2
< x
i
≤
√
δ.
When δ < x
i
≤ δ + (
1
i+1
) − (
1
i+1
)
2
, by considering the graph of the
function g(z) = z − z
2
in R, we have

y
i
− y
2
i
≥

1
i + 1

−

1
i + 1

2
≥ x
i
− δ.
and thus, we obtain
y
i
− x
i
≥ y
2
i
− δ > 0.
Therefore,
(T x

i
− Ty
i
)
2
=(δ − y
2
i
)
2
≤(x
i
− y
i
)
2
≤ (x
i
− y
i
)
2
+ λ(y)(x
i
− Tx
i
)(y
i
− Ty
i

).
When δ+

1
i+1

−

1
i+1

2
< x
i
≤
√
δ, both of x
i
−δ and y
i
−y
2
i
are greater
than

1
i+1

−


1
i+1

2
and thus also greater than

1
n
y
+1

−

1
n
y
+1

2
.
20
Therefore,
(T x
i
− Ty
i
)
2
= (δ − y

2
i
)
2
≤ 1 ≤
1

1
n
y
+1
−
1
(n
y
+1)
2

2
(x
i
− δ)(y
i
− y
2
i
)
≤ (x
i
− y

i
)
2
+ λ(y) (x
i
− Tx
i
) (y
i
− Ty
i
) .
21
Likely, we can prove the case:
(II-3):
√
δ < x
i
≤ 1 −
1
i+1
and δ < y
i
≤
√
δ.
(II-4): 0 ≤ x
i
≤ δ and
√

δ < y
i
≤ 1 −
1
i+1
.
Then, we have
(T x
i
− Ty
i
)
2
= (x
i
− y
2
i
)
2
≤ (x
i
− y
i
)
2
≤ (x
i
− y
i

)
2
+ λ(y)(x
i
− Tx
i
)(y
i
− Ty
i
).
Similarly, we can prove the case:
(II-5):
√
δ < x
i
≤ 1 −
1
i+1
and 0 ≤ y
i
≤ δ.
(II-6): δ < x
i
≤
√
δ and δ < y
i
≤
√

δ.
In this case, we have
(T x
i
− Ty
i
)
2
= (δ − δ)
2
= 0 ≤ (x
i
− y
i
)
2
+ λ(y)(x
i
− Tx
i
)(y
i
− Ty
i
).
(II-7): 0 ≤ x
i
≤ δ and δ < y
i
≤

√
δ.
This case can be treated as (I-2).
(II-8): 0 ≤ x
i
≤ δ and 0 ≤ y
i
≤ δ.
This case can be treated as (I-3).
(II-9): δ < x
i
≤
√
δ and 0 ≤ y
i
≤ δ.
This case can be treated as (I-2). 
To end this section, we give another example which shows that the concept
of a nonspreading mapping in the sense of (1.2) is generally different from that
of a 2-hybrid mapping relative to some D
f
in Hilbert spaces.
22
Example 4.3. Define f : R → R by f(x) = x
10
for all x ∈ R, and define
T : [0, 0.85] → [0, 0.85] by T x = x
2
for all x ∈ [0, 0.85]. Then, T is neither
nonexpansive nor nonspreading, but it is λ-hybrid relative to D

f
for any λ ≥ 0.
Thus, we can apply Theorem 3.2 to conclude T has a fixed point, while both of the
Browder Fixed Point Theorem and the Kohsaka–Takahashi fixed point theorem
fail.
Proof. It is easy to check that T is not nonexpansive. As for not nonspreading,
taking x = 0.85 and y = 0 .5, we have
T x −T y
2
= (x
2
− y
2
)
2
=

(0.85)
2
− (0.5)
2

2
= 0.22325625
while
x −y
2
+ 2 x − Tx, y − T y
=(x −y)
2

+ 2(x −x
2
)(y − y
2
)
=(0.85 −0.5)
2
+ 2

0.85 −(0.85)
2

0.5 −(0.5)
2

= 0.18625.
Hence, T is not nonspreading in the sense of (1.2). It remains to show that for
any λ ≥ 0, T is λ-hybrid relative to D
f
. Note at first that, for all λ ≥ 0 and for
all x, y ∈ [0, 0.85],
λ x − T x, f

(y) −f

(T y)
=λ

x −x
2


10y
9
− 10y
18

≥ 0.
Hence, it suffices to prove that T is 0-hybrid relative to D
f
, that is, to show that
D
f
(T x, T y) −D
f
(x, y) ≤ 0, ∀x, y ∈ [0, 0.85].
23
Fixed any x ∈ [0, 0.85], let h(y) = D
f
(T x, T y) −D
f
(x, y). Then
h(y) = f (T x) − f(T y) − T x −Ty, f

(T y) −[f(x) −f(y) − x − y, f

(y)]
= x
20
+ 9y
20

− 10x
2
y
18
− x
10
− 9y
10
+ 10xy
9
.
We have
h

(y) = 180y
19
− 180x
2
y
17
− 90y
9
+ 90xy
8
= 90y
8

2y
11
− 2x

2
y
9
− y + x

= 90y
8

2y
9
(y
2
− x
2
) −(y − x)

= 90y
8

2y
9
(y + x)(y − x) −(y − x)

= 90y
8
(y − x)

2y
9
(y + x) −1


.
Since y and x are in [0, 0.85], one has
2y
9
(y + x) −1 < 2(0.85)
9
(0.85 + 0.85) −1 < 0,
and hence
h

(y)







≥ 0 , if y ≤ x;
≤ 0 , if y > x.
Moreover, we know h(y) = 0 if x = y. Therefore, h(y) is always less than or equal
to zero and we have proved that D
f
(T x, T y)−D
f
(x, y) ≤ 0 for all x, y ∈ [0, 0.85].

5 Weak convergence theorems
In this section, we discuss the demiclosedness and the weak convergence prob-

24