Open Access

Fixed point and weak convergence theorems for point-dependent λ-hybrid mappings in Banach spaces

  • Young-Ye Huang1,
  • Jyh-Chung Jeng2,
  • Tian-Yuan Kuo3 and
  • Chung-Chien Hong4Email author
Fixed Point Theory and Applications20112011:105

https://doi.org/10.1186/1687-1812-2011-105

Received: 25 August 2011

Accepted: 23 December 2011

Published: 23 December 2011

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.

2010 MSC: 47H09; 47H10.

Keywords

fixed pointBregman distanceGâteaux differentiablesubdifferential

1 Introduction

Let C be a nonempty subset of a Hilbert space H. A mapping T : CH is said to be

(1.1) nonexpansive if ||Tx - Ty|| ≤ ||x - y||, x, y C, cf. [1, 2];

(1.2) nonspreading if ||Tx - Ty||2 ≤ ||x - y||2 + 2 〈x - Tx, y - Ty〉, x, y C, cf. [35];

(1.3) hybrid if ||Tx - Ty||2 ≤ ||x - y||2 + 〈x - Tx, y - Ty〉, x, y C, cf. [3, 57].

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 : CC be a nonexpansive mapping. Then, the following are equivalent:
  1. (a)

    There exists x C such that {T n x} n is bounded;

     
  2. (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.)

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 , they call a mapping T : CH

(1.4) λ-hybrid if ||Tx - Ty||2 ≤ ||x - y||2 + λx - Tx, y - Ty〉, 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 : CC be a λ-hybrid mapping. Then, the following are equivalent:
  1. (a)

    There exists x C such that {T n x} n is bounded;

     
  2. (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 Gâteaux differentiable convex function f : X → (-∞,∞] and a function λ : C, a mapping T : CX is said to be point-dependent λ-hybrid relative to D f if

(1.5) D f (Tx, Ty) ≤ D f (x, y) + λ(y) 〈x - Tx, f'(y) - f(Ty)〉, x, y C,

where D f is the Bregman distance associated with f and f'(x) denotes the Gâteaux derivative of f at x.

In this article, we study the fixed point and weak convergence problem for mappings satisfying (1.5). This article is organized in the following way: Section 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 generalizes 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 ( α x + ( 1 - α ) y ) < α f ( x ) + ( 1 - α ) f ( y )

for all x, y X and α (0, 1).

The function f : X → (-∞, ∞] is said to be Gâteaux differentiable at x X if there is f'(x) X* such that
lim t 0 f ( x + t y ) - 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 o , cf. Proposition 1.1.2 (iv) of [9]. When f is Gâteaux 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 , c t ) c ν 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)
The modulus of uniform convexity of f is the function δ f : [0, ∞) → [0, ∞] defined by
δ f ( t ) = inf f ( x ) + f ( y ) - 2 f 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 ) - 2 f 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 tf(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
J x = { x * X * : x , x * = | | x | | 2 = | | x * | | 2 }

for all x X.

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 , J x = | | y | | 2 + | | x | | 2 - 2 y , J x .
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 Gâteaux 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].

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 Gâteaux differentiable on I n t ( D ) in a Banach space X, then the following statements are equivalent:
  1. (a)

    The function f is strictly convex on I n t ( D ) .

     
  2. (b)

    For any two distinct points x , y I n t ( D ) , one has D f (y, x) > 0.

     
  3. (c)
    For any two distinct points x , y I n t ( 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 Gâteaux differentiable on I n t ( D ) and is bounded on bounded subsets of I n t ( D ) . Suppose C I n t ( D ) is a nonempty closed convex subset of X and T: CC is point-dependent λ-hybrid relative to D f for some function λ : C. For x C and any n define
S n x = 1 n k = 0 n - 1 T k x ,

where T0is the identity mapping on C. If {T n x}nis bounded, then every weak cluster point of {S n x}nis 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 {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 - T y , 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 - T y - 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 ) + n y - n S n x , f ( y ) + ( n + 1 ) S n + 1 x - x - n T 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 - T y - λ ( y ) ( x - T n x ) n , f ( T y ) .
(7)
Since {T n x}nis bounded, {S n x}nis bounded, and so, in view of X being reflexive, it has a subsequence { S n i x } i 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}nand {f(T n x)}nare bounded that
0 f ( T y ) - f ( y ) + y - v , f ( y ) + v - T y , f ( T y ) .
(8)
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, Tv) = 0. Therefore Tv = 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 Gâteaux differentiable on I n t ( D ) and is bounded on bounded subsets of I n t ( D ) . Suppose C I n t ( D ) is a nonempty closed convex subset of X and T: CC is point-dependent λ-hybrid relative to D f for some function λ : C. Then, the following two statements are equivalent:
  1. (a)

    There is a point x C such that {T n x}n is bounded.

     
  2. (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 : CC is λ-hybrid. Then, the following two statements are equivalent:
  1. (a)

    There exists x C such that {T n (x)}n is bounded.

     
  2. (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 : CX is said to be quasi-nonexpansive with respect to D f if F(T) ≠ and D f (v, Tx) ≤ 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 Gâteaux differentiable on I n t ( D ) , and let C I n t ( D ) be a nonempty closed convex subset of X. If T: CC 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 F(T) such that x n x as n → ∞. By the continuity of D f (·, Tx) 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 Tx = 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 Tz = z to conclude F(T) is convex. Indeed,
D f ( z , T z ) = f ( z ) - f ( T z ) - z - T z , f ( T z ) = f ( z ) + [ α f ( x ) + ( 1 - α ) f ( y ) ] - f ( T z ) - z - T z , f ( T z ) - [ α f ( 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 ) ] - [ α f ( x ) + ( 1 - α ) f ( y ) ] = f ( z ) + α D f ( x , T z ) + ( 1 - α ) D f ( y , T z ) - [ α f ( x ) + ( 1 - α ) f ( y ) ] f ( z ) + α D f ( x , z ) + ( 1 - α ) D f ( y , z ) - [ α f ( 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 ) ] - [ α f ( x ) + ( 1 - α ) f ( y ) ] = f ( z ) + α f ( x ) - α f ( z ) - α x - α z , f ( z ) + ( 1 - α ) f ( y ) - ( 1 - α ) f ( z ) - ( 1 - α ) y - ( 1 - α ) z , f ( z ) - [ α f ( x ) + ( 1 - α ) f ( y ) ] = - α x + ( 1 - α ) y - ( α z + ( 1 - α ) z ) , f ( z ) = - 0 , f ( z ) = 0 .

Therefore, Tz = 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 Gâteaux differentiable on I n t ( D ) and is bounded on bounded subsets of Int(D), and let C I n t ( D ) be a nonempty closed convex subset of X. Suppose T: CC is point-dependent λ-hybrid relative to D f for some function λ : C and has a point x0 C such that {T n (x0)}nis bounded. Then, T is quasi-nonexpansive with respect to D f , and therefore, F(T) 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 - T v , 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 Gâteaux differentiable on I n t ( D ) and is bounded on bounded subsets of I n t ( D ) . Suppose C I n t ( D ) is a nonempty bounded closed convex subset of X and{T1, T2,..., T N } is a commutative finite family of point-dependent λ-hybrid mappings relative to D f for some function λ : C from C into itself. Then {T1, T2,..., 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(T1) and F(T2) are nonempty bounded closed convex subsets of X. Moreover, F(T1) is T2-invariant. Indeed, for any v F(T1), it follows from T1T2 = T2T1 that T1T2v = T2T1v = T2v, which shows that T2v F(T1). Consequently, the restriction of T2 to F(T1) is point-dependent λ-hybrid relative to D f , and hence by Theorem 3.2, T2 has a fixed point u F(T1), that is, u F(T1) ∩ F(T2).

By induction hypothesis, assume that for some n ≥ 2, E = k = 1 n 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 EF(T n+1 ) ≠ , that is, k = 1 n + 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 Gâteaux differentiable on I n t ( D ) . Suppose C I n t ( D ) is a nonempty bounded closed convex subset of X and{T i } iI is a commutative family of point-dependent λ-hybrid mappings relative to D f for some function λ : C from C into itself. Then, {T i } iI 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 )} iI 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.
  1. (a)

    (h 2 - k 2)2 - (h - k)2 ≥ 0, if h + k 2 > 0 . 5 .

     
  2. (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 (h2 - k2)2 - (h - k)2 ≥ 0. On the other hand, h + k 2 0 . 5 implies a + b ≤ 0, and hence, (h2 - k2)2 - (h - k)2 ≤ 0.

Example 4.2. Let C = { x l 2 ( ) : 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: CC by
T x = ( T x 1 , T x 2 , , T x n , ) : T x i = x i 2 , i f δ < x i 1 - 1 i + 1 ; δ , i f δ < x i δ ; x i , i f 0 x i δ .
Then for any λ , T is not λ-hybrid. However, for each x C, if we let n x = min { n : i = n + 1 x i 2 δ 2 } and define λ: C by
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 - T x , y - T y
(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 - T x , 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 1 6 - 2 m + 1 + 9 2 ( m + 1 ) 2 - 4 ( m + 1 ) 3 + 1 ( m + 1 ) 4 - m 2 4 ( m + 1 ) 2 5 1 6  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 - T x , y - T y

for all x, y C.

It remains to show that T satisfies the inequality (9). We can rewrite the inequality as
i = 1 ( T x i - T y i ) 2 i = 1 ( x i - y i ) 2 + i = 1 λ ( y ) ( x i - T x i ) ( y i - T y i ) .
Thus, if we can show that for all i ,
( T x i - T y i ) 2 ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y i ) ,
(10)

then the assertion follows. We prove inequality (10) holds for all i 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 - T y i ) 2 = ( x i - y i 2 ) 2 ( x i - y i ) 2 ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y i ) .
(I-2): δ < y i δ , then we have
( T x i - T y i ) 2 = ( x i - δ ) 2 ( x i - y i ) 2 ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y i ) .
(I-3): If 0 ≤ y i δ, then we have
( T x i - T y i ) 2 = ( x i - y i ) 2 ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y i ) .

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 - T y i ) 2 = ( x i 2 - y i 2 ) 2 ( x i - y i ) 2 ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y 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 - z2 in , which is symmetric to the line L : x = 0.5, we have
( x i - x i 2 ) 1 i + 1 - 1 i + 1 2 1 n y + 1 - 1 n y + 1 2
and
( y i - y i 2 ) 1 i + 1 - 1 i + 1 2 1 n y + 1 - 1 n y + 1 2 .
Consequently, we obtain
( T x i - T y i ) 2 = ( x i 2 - y i 2 ) 2 1 1 1 n y + 1 - 1 ( n y + 1 ) 2 2 ( x i - x i 2 ) ( y i - y i 2 ) ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y i ) .

(II-2): δ < x i δ and δ < y i 1 - 1 i + 1 .

If y i ≤ 0.5, then x i + y i 2 < 0 . 5 . Thus, from Lemma 4.1, we have
( T x i - T y i ) 2 = ( δ - y i 2 ) 2 ( x i 2 - y i 2 ) 2 ( x i - y i ) 2 ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y 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 - z2 in , we have
y i - y i 2 1 i + 1 - 1 i + 1 2 x i - δ .
and thus, we obtain
y i - x i y i 2 - δ > 0 .
Therefore,
( T x i - T y i ) 2 = ( δ - y i 2 ) 2 ( x i - y i ) 2 ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y i ) .

When δ + 1 i + 1 - 1 i + 1 2 < x i δ , both of x i -δ and y i - y i 2 are greater than 1 i + 1 - 1 i + 1 2 and thus also greater than 1 n y + 1 - 1 n y + 1 2 .

Therefore,
( T x i - T y i ) 2 = ( δ - y i 2 ) 2 1 1 1 n y + 1 - 1 ( n y + 1 ) 2 2 ( x i - δ ) ( y i - y i 2 ) ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y i ) .

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 - T y i ) 2 = ( x i - y i 2 ) 2 ( x i - y i ) 2 ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y 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 - T y i ) 2 = ( δ - δ ) 2 = 0 ( x i - y i ) 2 + λ ( y ) ( x i - T x i ) ( y i - T y 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.

Example 4.3. Define f : by f(x) = x10for all x , and define T : [0, 0.85] → [0, 0.85] by Tx = x2for 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 . 8 5 ) 2 - ( 0 . 5 ) 2 ] 2 = 0 . 2 2 3 2 5 6 2 5
while
| | x - y | | 2 + 2 x - T x , y - T y = ( x - y ) 2 + 2 ( x - x 2 ) ( y - y 2 ) = ( 0 . 8 5 - 0 . 5 ) 2 + 2 [ 0 . 8 5 - ( 0 . 8 5 ) 2 ] [ 0 . 5 - ( 0 . 5 ) 2 ] = 0 . 1 8 6 2 5 .
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 ) ( 1 0 y 9 - 1 0 y 1 8 ) 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 . 8 5 ] .
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 - T y , f ( T y ) - [ f ( x ) - f ( y ) - x - y , f ( y ) ] = x 2 0 + 9 y 2 0 - 1 0 x 2 y 1 8 - x 1 0 - 9 y 1 0 + 1 0 x y 9 .
We have
h ( y ) = 180 y 19 - 180 x 2 y 17 - 90 y 9 + 90 x y 8 = 90 y 8 ( 2 y 1 1 - 2 x 2 y 9 - y + x ) = 90 y 8 [ 2 y 9 ( y 2 - x 2 ) - ( y - x ) ] = 90 y 8 [ 2 y 9 ( y + x ) ( y - x ) - ( y - x ) ] = 90 y 8 ( y - x ) [ 2 y 9 ( y + x ) - 1 ] .
Since y and x are in [0, 0.85], one has
2 y 9 ( y + x ) - 1 < 2 ( 0 . 85 ) 9 ( 0 . 85 + 0 . 85 ) - 1 < 0 ,
and hence
h ( y ) 0 , i f y x ; 0 , i f 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 (Tx, Ty) - 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 problem of point-dependent λ-hybrid relative to D f . We denote the weak convergence and strong convergence of a sequence {x n } to v in a Banach space by x n v and x n v, respectively. For a nonempty closed convex subset C of a Banach space X, a mapping T : CX is demiclosed if for any sequence {x n } in C with x n v and x n - Tx n → 0, one has Tv = v.

We first derive an Opial-like inequality for the Bregman distance. For the Opial's inequality, we refer readers to Lemma 1 of [11].

Lemma 5.1. Suppose f : X → (-∞,∞] is a proper strictly convex function so that it is Gâteaux differentiable on I n t ( D ) in a Banach space X and{x n }nis a sequence in D such that x n v for some v I n t ( D ) . Then
liminf n D f ( x n , v ) < liminf n D f ( x n , y ) , y Int ( D ) w i t h y v .
Proof. Since
D f ( x n , v ) D f ( x n , y ) = f ( x n ) f ( v ) x n v , f ( v ) [ f ( x n ) f ( y ) x n y , f ( y ) ] = f ( x n ) f ( v ) x n v , f ( v ) f ( x n ) + f ( y ) + x n y , f ( y ) ] + x n v , f ( y ) x n v , f ( y ) = [ f ( v ) f ( y ) v y , f ( y ) ] + x n v , f ( y ) f ( v ) = D f ( v , y ) + x n v , f ( y ) f ( v )
and x n v, we have
lim n [ D f ( x n , v ) - D f ( x n , y ) ] = - D f ( v , y ) .
Consequently,
liminf n D f ( x n , v ) = liminf n [ ( D f ( x n , v ) - D f ( x n , y ) ) + D f ( x n , y ) ] = lim n ( D f ( x n , v ) - D f ( x n , y ) ) + liminf n D f ( x n , y ) = - D f ( v , y ) + liminf n D f ( x n , y ) ,
and hence in view of D f (v, y) > 0 for yv we obtain
liminf n D f ( x n , v ) < liminf n D f ( x n , y ) .

Proposition 5.2. Let f : X → (-∞,∞] be a strictly convex function so that it is Gâteaux differentiable on I n t ( D ) and is bounded on bounded subsets of I n t ( D ) . Suppose C is a closed convex subset of I n t ( D ) and T: CC is point-dependent λ-hybrid relative to D f for some λ : C. Then T is demiclosed.

Proof. Let {x n } be any sequence in C with x n v and x n - Tx n → 0. We have to show that Tv = v. Since f is bounded on bounded subsets, by Proposition 1.1.11 of [9] there exists a constant M > 0 such that
max { sup { | | f ( x n ) | | : n } , | | λ ( v ) | | , | | f ( T v ) | | , | | f ( v ) | | } M .
Rewrite D f (x n , Tv) as
D f ( x n , T v ) = f ( x n ) - f ( T v ) - x n - T v , f ( T v ) = f ( x n ) + f ( T x n ) - f ( T x n ) - f ( T v ) - x n - T v , f ( T v ) + T x n - T v , f ( T v ) - T x n - T v , f ( T v ) = [ f ( T x n ) - f ( T v ) - T x n - T v , f ( T v ) ] + f ( x n ) - f ( T x n ) + T x n - x n , f ( T v ) = D f ( T x n , T v ) + f ( x n ) - f ( T x n ) + T x n - x n , f ( T v ) .
(11)
Noting f(x n ) - f(Tx n ) ≤ 〈x n - Tx n , f'(x n )〉 and T is point-dependent λ-hybrid relative to D f , we have from (11) that
D f ( x n , T v ) D f ( T x n , T v ) + x n - T x n , f ( x n ) - x n - T x n , f ( T v ) D f ( x n , v ) + λ ( v ) x n - T x n , f ( v ) - f ( T v ) + x n - T x n , f ( x n ) - f ( T v ) D f ( x n , v ) + [ | λ ( v ) | ( | | f ( v ) | | + | | f ( T v ) | | ) + ( | | f ( x n ) | | + | | f ( T v ) | | ) ] | | x n - T x n | | D f ( x n , v ) + 2 M ( M + 1 ) | | x n - T x n | | .
(12)
If Tvv, then Lemma 5.1 and (12) imply that
liminf n D f ( x n , v ) < liminf n D f ( x n , T v ) liminf n [ D f ( x n , v ) + 2 M ( M + 1 ) | | x n - T x n | | ] = liminf n D f ( x n , v ) ,

a contradiction. This completes the proof. □

A mapping T : CC is said to be asymptotically regular if, for any x C, the sequence {T n+1 x - T n x} tends to zero as n → ∞.

Theorem 5.3. Suppose the following conditions hold:

(5.3.1) f : X → (-∞,∞] is l.s.c. uniformly convex function so that it is Gâteaux differentiable on I n t ( D ) and is bounded on bounded subsets of I n t ( D ) in a reflexive Banach space X.

(5.3.2) C I n t ( D ) is a closed convex subset of X.

(5.3.3) T : CC is point-dependent λ-hybrid relative to D f for some λ : C and is asymptotically regular with a bounded sequence {T n x0}nfor some x0 C.

(5.3.4) The mapping xf'(x) for x X is weak-to-weak* continuous.

Then for any x C, {T n x}nis weakly convergent to an element v F(T).

Proof. Let v F(T) and x C. If {T n x}nis not bounded, then there is a subsequence { T n i x } i such that | | v - T n i x | | 1 for all i and | | v - T n i x | | as i → ∞. From (5.3.3), for any n , we have
D f ( v , T n + 1 x ) = D f ( T v , T n + 1 x ) D f ( v , T n x ) + λ ( T n x ) { v - T v , f ( T n x ) - f ( T n + 1 x ) } = D f ( v , T n x ) D f ( v , x ) ,
which in conjunction with (3), (4), and (6) implies that
D f ( v , x ) D f ( v , T n i x ) ν f ( T n i x , | | v - T n i x | | ) | | v - T n i x | | ν f ( T n i x , 1 ) | | v - T n i x | | δ f ( 1 ) , as  i ,
a contradiction. Therefore, for any x X, {T n x}nis bounded, and so it has a subsequence { T n j x } j which is weakly convergent to w for some w C. As T n j x - T n j + 1 x 0 , it follows from the demiclosedness of T that w F(T). It remains to show that T n x w as n → ∞. Let { T n k x } n be any subsequence of {T n x}nso that T n k x u for some u C. Then u F(T). Since both of {D f (w, T n x)}nand {D f (u, T n x)}nare decreasing, we have
lim n [ D f ( w , T n x ) - D f ( u , T n x ) ] = lim n [ f ( w ) - f ( u ) - w - u , f ( T n x ) ] = a
for some a . Particularly, from (5.3.4) we obtain
a = lim n j [ f ( w ) - f ( u ) - w - u , f ( T n j x ) ] = f ( w ) - f ( u ) - w - u , f ( w )
and
a = lim n k [ f ( w ) - f ( u ) - w - u , f ( T n k x ) ] = f ( w ) - f ( u ) - w - u , f ( u ) .

Consequently, 〈w - u, f'(w) - f'(u)〉 = 0, and hence w = u by the strict convexity of f. This shows that T n x w for some w F(T).□

Adopting the technique of [8], we have the following ergodic theorem for point-dependent λ-hybrid mappings in Hilbert spaces.

Theorem 5.4. Suppose

(5.4.1) C is nonempty closed convex subset of a Hilbert space H.

(5.4.2) T : CC is a point-dependent λ-hybrid mapping for some function λ : C, that is,
| | T x - T y | | 2 | | x - y | | 2 + λ ( y ) x - T x , y - T y , x , y C .

(5.4.3) F(T) ≠ .

Then for any x C, the sequence {S n (x)}ndefined by
S n ( x ) = 1 n k = 0 n - 1 T k x

converges weakly to some point v F(T).

Declarations

Authors’ Affiliations

(1)
Center for General Education, Southern Taiwan University
(2)
Nanjeon Institute of Technology
(3)
Fooyin University
(4)
Department of Industrial Management, National Pingtung University of Science and Technology

References

  1. Browder FE: Fixed point theorems for noncompact mappings in a Hilbert space. Proc Nat Acad Sci USA 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleGoogle Scholar
  2. Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar
  3. Iemoto S, Takahashi W: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal 2009, 71: 2082–2089. 10.1016/j.na.2009.03.064MathSciNetView ArticleGoogle Scholar
  4. Kohsaka F, Takahashi W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch Math 2008, 91: 166–177. 10.1007/s00013-008-2545-8MathSciNetView ArticleGoogle Scholar
  5. Takahashi W, Yao JC: Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces. Taiwanese J Math 2011, 15: 457–472.MathSciNetGoogle Scholar
  6. Kocourek P, Takahashi W, Yao JC: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwanese J Math 2010, 14: 2497–2511.MathSciNetGoogle Scholar
  7. Takahashi W: Fixed point theorems for new nonlinear mappings in a Hilbert space. J Nonlinear Convex Anal 2010, 11: 79–88.MathSciNetGoogle Scholar
  8. Aoyama K, Iemoto S, Kohsaka F, Takahashi W: Fixed point and ergodic theorems for λ -hybrid mappings in Hilbert spaces. J Nonlinear Convex Anal 2010, 11: 335–343.MathSciNetGoogle Scholar
  9. Butnariu D, Iusem AN: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic Publishers, The Netherlands; 2000.View ArticleGoogle Scholar
  10. Ciorãnescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, The Netherlands; 1990.View ArticleGoogle Scholar
  11. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive. mappings Bull Amer Math Soc 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleGoogle Scholar

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© Huang et al; licensee Springer. 2011

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