Fixed Point and Weak Convergence Theorems for α , β-Hybrid Mappings in Banach Spaces

and Applied Analysis 3 where f◦ x, x − y limt→ 0 f x t x − y − f x /t. Df y, x is finite valued if and only if x ∈ D◦, compare Proposition 1.1.2 iv of 5 . When f is Gâteaux differentiable on D, 2.4 becomes Df ( y, x ) f ( y ) − f x − 〈y − x, f ′ x 〉, 2.5 and then the modulus of total convexity is the function νf : D◦ × 0,∞ → 0,∞ defined by νf x, t inf { Df ( y, x ) : y ∈ D,∥∥y − x∥∥ t}. 2.6 It is known that νf x, ct ≥ cνf x, t 2.7 for all t ≥ 0 and c ≥ 1, compare Proposition 1.2.2 ii of 5 . By definition it follows that Df ( y, x ) ≥ νf ( x, ∥ ∥y − x∥∥). 2.8 The modulus of uniform convexity of f is the function δf : 0,∞ → 0,∞ defined by δf t inf { f x f ( y ) − 2f ( x y 2 ) : x, y ∈ D,∥∥x − y∥∥ ≥ t } . 2.9 The function f is called uniformly convex if δf 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 )


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 Tx − Ty ≤ x − y , for all x, y ∈ C; 1.2 nonspreading if 2 Tx − Ty 2 ≤ Tx − y 2 Ty − x 2 , for all x, y ∈ C, cf. 1, 2 ; for all x, y ∈ C, where D f is the Bregman distance associated with f.Sections 3 and 4 are devoted to investigation of the fixed point and weak convergence problem for such type of mappings, respectively.Our fixed point theorem extends that of 4 .

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, and D • denotes the algebraic interior of D, that is, 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 for all y ∈ X.
The Bregman distance D f associated with a proper convex function f is the function and then the modulus of total convexity is the function It is known that for all t ≥ 0 and c ≥ 1, compare Proposition 1.2.2 ii of 5 .By definition it follows that The modulus of uniform convexity of f is the function δ f : 0, ∞ → 0, ∞ defined by 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 for all x, y ∈ D with x − y ≥ ε.Note that for y ∈ D and x ∈ D • , we have

2.11
where the first inequality follows from the fact that the function t → f x tz − f x /t is nondecreasing on 0, ∞ .Therefore 12 whenever x ∈ D • and t ≥ 0. For other properties of the Bregman distance D f , we refer readers to 5 .
The normalized duality mapping J from X to 2 X * is defined by x * 2 2.13 for all x ∈ X.When f x x 2 in a smooth Banach space, it is known that f x 2J x for x ∈ X, compare Corollaries 1.2.7 and 1.4.5 of 6 .Hence we have

2.14
Moreover, as the normalized duality mapping J in a Hilbert space H is the identity operator, we have

2.15
Thus, in case f x x 2 in a Hilbert space, 1.5 coincides with 1.4 .However, in general they are different as the following example shows.

2.17
If T were α, β -hybrid for some α, β ∈ R, then we would have

2.19
we see that T is not α, β -hybrid for any α, β ∈ R.But some simple computations show that T is 1, 0 -hybrid relative to D f .
A function g : X → −∞, ∞ is said to be subdifferentiable at a point x ∈ X if there exists a linear functional x * ∈ X * such that

2.20
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  c For any two distinct points x, y ∈ Int D , one has 2.21 Throughout this paper, F T will denote the set of all fixed points of a mapping T .

Fixed Point Theorems
In this section, we apply Lemma 2.2 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 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 α, β -hybrid relative to D f .Then the following two statements are equivalent.
a There is a point Proof.If F T / ∅, then {T n v} n∈N {v} is bounded for any v ∈ F T .Now assume that {T n x} n∈N is bounded for some x ∈ C, and for any n ∈ N, let S n x 1/n n−1 k 0 T k x.Then {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 → ∞.Since T is α, β -hybrid relative to D f , we have, for any y ∈ C and k ∈ N ∪ {0},

3.2
Similarly, we also have

3.3
Consequently, we obtain from 3.1 that

3.4
Abstract and Applied Analysis 7 Summing up these inequalities with respect to k 0, 1, . . ., n − 1, we get

3.5
Dividing the above inequality by n, we have

3.6
Replacing n by n i and letting n i → ∞, we obtain from the fact that {f T n x } n∈N is bounded that Putting y v in 3.7 , we get from which follows that D f v, Tv 0. Therefore Tv v by Lemma 2.2.
Since the function f x x 2 in a Hilbert space H satisfies all the requirements of Theorem 3.1, the corollary below follows immediately.We now show that the fixed point set F T is closed and convex under the assumptions of Theorem 3.1.
A mapping T : C → X is said to be quasi-nonexpansive with respect to The following lemma is shown in Proof.In view of Theorem 3.1, F T / ∅.Now, for any v ∈ F T and any y ∈ C, as T is α, βhybrid relative to D f , we have so T is quasi-nonexpansive with respect to D f , and hence F T is a nonempty closed convex subset of C by Lemma 3.3.
For the remainder of this section, we establish a common fixed point theorem for a commutative family of α, β -hybrid mappings relative to D f .Lemma 3.5.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 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 α, β -hybrid mappings relative to D f 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 3.4, we see that F T 1 and F T 2 are nonempty bounded closed convex subsets of X.Moreover, Consequently, the restriction of T 2 to F T 1 is α, β -hybrid relative to D f , and so by Theorem 3.1, 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 Here we would like to remark that in the above lemma, the assumption f is bounded on bounded subsets of Int D can be dropped by checking the proof of Theorem 3.1 and noting that we have assumed C is bounded.Theorem 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 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 α, β -hybrid mappings relative to D f 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 3.4, 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 3.5.

Weak Convergence Theorems
In this section, we discuss the demiclosedness and the weak convergence problem of α, βhybrid mappings relative to D f .We denote the weak convergence and strong convergence of a sequence {x n } to v in a Banach space X by x n v and x n → v, respectively.For a nonempty closed convex subset C of a Banach space X, a mapping T : C → X is demiclosed if for any sequence {x n } in C with x n v and x n − Tx n → 0, one has Tv v.The following Opial-like inequality for the Bregman distance is proved in 7 .For the Opial's inequality we refer readers to Lemma 1 of 8 .
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, by Proposition 1.1.11of 5 there exists a constant M > 0 such that Note that, for α ∈ R, Similarly, for β ∈ R, we have Thus we obtain from the α, β -hybrid of T that which implies that

4.6
Consequently, if Tv / v, then Lemma 4.1 implies that lim inf a contradiction.This completes the proof.
A mapping T : C → C 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 4.3.Suppose the following conditions hold.
which in conjunction with 2.7 , 2.8 , and 2.12 implies that 4.9 a contradiction.Therefore, for any x ∈ X, {T n x} n∈N is bounded, and so it has a subsequence {T n j x} j∈N 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} k∈N be any subsequence of {T n x} n∈N so that T n k x u for some u ∈ C. Then u ∈ F T .Since both of {D f w, T n x } n∈N and {D f u, T n x } n∈N are decreasing, we have for some a ∈ R. Particularly, from 4.3.4we obtain a lim

4.11
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 .

Conclusion
In this paper, we have introduced the Bregman distance D f and a new class of mappings, α, β -hybrid mappings relative to D f in Banach spaces.We also have given and proved a necessary and sufficient condition for the existence of fixed points of the introduced mappings and some properties of the mappings.In fact, our result properly extends the Kocourek-Takahashi-Yao fixed point theorems for α, β -hybrid mappings in Hilbert spaces in 2010 4 .Since the Kocourek-Takahashi-Yao fixed point theorems can be applied to study the nonexpansive mappings, the nonspreading mappings, and the hybrid mappings, our theorems in this paper are also good for these famous mappings in the field of fixed point theory.
According to 7 , the fixed point theorems in this paper can be expected to discuss in a wider class of mappings, called point-dependent α, β -hybrid mappings relative to D f in Banach spaces.In a point-dependent α, β -hybrid mapping, the α and the β are not constant again but two functions from a nonempty subset of a Banach space to real numbers.Therefore, inequality 1.4 for point-dependent α, β -hybrid mappings becomes α y Tx − Ty In addition, Noor 9-11 provides algorithms to search the fixed points of nonexpansive mappings and then combines the result with general variational inequalities to study applied mathematical problems.We are motivated by that and expect to develop algorithms from the theorems of this paper to approach the fixed points of the introduced mappings.Through the combination of the fixed point theorems and the corresponding algorithms, the introduced mappings of this paper would be able to be applied to more fields of applied mathematics.

1.3 hybrid if 3 2 Tx − y 2 Ty − x 2 ,
Tx − Ty 2 ≤ x − y for all x, y ∈ C, cf. 3 .Recently, Kocourek et al. 4 introduced a new class of nonlinear mappings in a Hilbert space containing the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings and established some fixed point and ergodic theorems for mappings in this new class.For α, β ∈ R, they call a mapping T : C → H 1.5 α, β -hybrid relative to D f if there are α, β ∈ R such that αD f Tx, Ty 1 − α D f x, T y ≤ βD f Tx, y 1 − β D f x, y 1.2

Example 2 . 1 .
Let f x |x| for x ∈ R. f is a continuous convex function with

Lemma 2 . 2
see Proposition 1.1.9 of 5 .If a proper convex function f : X → −∞, ∞ is Gâteaux 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.

Corollary 3 . 2
see 4 .Let C be a nonempty closed convex subset of Hilbert space H and Suppose T : C → C is α, β -hybrid.Then the following two statements are equivalent.a There exists x ∈ C such that {T n x } n∈N is bounded.b T has a fixed point.
Huang et al. 7. Let f : X → −∞, ∞ be a proper strictly convex function on a reflexive Banach space X so that it is Gâteaux 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 α, β -hybrid relative to D f 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 is a nonempty closed convex subset of C.
Lemma 3.3.Let f : X → −∞, ∞ be a proper strictly convex function on a Banach space X so that it is Gâteaux 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.Proposition 3.4.
. uniformly convex function so that it is Gâteaux differentiable on Int D and is bounded on bounded subsets of Int D in a reflexive Banach space X. T : C → C is α, β -hybrid relative to D f and is asymptotically regular with a bounded sequence {T n x 0 } n∈N for some x 0 ∈ C. The mapping x → f x for x ∈ X is weak-to-weak * continuous.Then for any x ∈ C, {T n x} n∈N is weakly convergent to an element v ∈ F T .Proof.Let v ∈ F T and x ∈ C. If {T n x} n∈N is not bounded, then there is a subsequence {T n i x} i∈N such that v − T n i x ≥ 1 for all i ∈ N and v − T n i x → ∞ as i → ∞.From 4.3.3, for any n ∈ N ∪ {0} we have