#### Abstract

We introduce the class of -hybrid mappings relative to a Bregman distance in a Banach space, and then we study the fixed point and weak convergence problem for such mappings.

#### 1. Introduction

Let be a nonempty subset of a Hilbert space . A mapping is said to be (1.1) nonexpansive if ;(1.2) nonspreading if , cf. [1, 2];(1.3) hybrid if , 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 , they call a mapping (1.4)-hybrid if

for all .

Obviously, is nonexpansive if and only if it is -hybrid; is nonspreading if and only if it is -hybrid; is hybrid if and only if it is -hybrid.

Motivated by the above works, we extend the concept of -hybrid from Hilbert spaces to Banach spaces. For a nonempty subset of a Banach space , a Gâteaux differentiable convex function and , a mapping is said to be(1.5)-hybrid relative to if there are such that

for all , where is the Bregman distance associated with . 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].

#### 2. Preliminaries

In what follows, will be a real Banach space with topological dual , and will be a convex function. denotes the domain of , that is, and denotes the algebraic interior of , that is, the subset of consisting of all those points such that, for any , there is in the open segment with . The topological interior of , denoted by , is contained in . is said to be proper provided that . is called lower semicontinuous (l.s.c.) at if . is strictly convex if for all and .

The function is said to be Gâteaux differentiable at if there is such that for all .

The Bregman distance associated with a proper convex function is the function defined by where . is finite valued if and only if , compare Proposition 1.1.2 (iv) of [5]. When is Gâteaux differentiable on , (2.4) becomes and then the modulus of total convexity is the function defined by It is known that for all and , compare Proposition 1.2.2 (ii) of [5]. By definition it follows that

The modulus of uniform convexity of is the function defined by

The function is called uniformly convex if for all . If is uniformly convex then for any there is such that for all with .

Note that for and , we have where the first inequality follows from the fact that the function is nondecreasing on . Therefore whenever and . For other properties of the Bregman distance , we refer readers to [5].

The normalized duality mapping from to is defined by for all .

When in a smooth Banach space, it is known that for , compare Corollaries 1.2.7 and 1.4.5 of [6]. Hence we have Moreover, as the normalized duality mapping in a Hilbert space is the identity operator, we have Thus, in case in a Hilbert space, (1.5) coincides with (1.4). However, in general they are different as the following example shows.

*Example 2.1. *Let for . is a continuous convex function with
Let and define by
If were -hybrid for some , then we would have
Since
we see that is not -hybrid for any . But some simple computations show that is -hybrid relative to .

A function is said to be subdifferentiable at a point if there exists a linear functional such that We call such the subgradient of at . The set of all subgradients of at is denoted by , and the mapping is called the subdifferential of . For a l.s.c. convex function is bounded on bounded subsets of if and only if is bounded on bounded subsets there, compare Proposition 1.1.11 of [5]. A proper convex l.s.c. function is Gâteaux differentiable at if and only if it has a unique subgradient at ; in such case , compare Corollary 1.2.7 of [6].

The following lemma will be quoted in the sequel.

Lemma 2.2 (see Proposition 1.1.9 of [5]). *If a proper convex function is Gâteaux differentiable on in a Banach space , then the following statements are equivalent.*(a)*The function is strictly convex on .*(b)*For any two distinct points , one has .*(c)*For any two distinct points , one has *

Throughout this paper, will denote the set of all fixed points of a mapping .

#### 3. 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 be a reflexive Banach space and let be a l.s.c. strictly convex function so that it is Gâteaux differentiable on and is bounded on bounded subsets of . Suppose is a nonempty closed convex subset of and is -hybrid relative to . Then the following two statements are equivalent.*(a)*There is a point such that is bounded.*(b)*. *

*Proof. * If , then is bounded for any . Now assume that is bounded for some , and for any , let . Then is bounded, and so, in view of being reflexive, it has a subsequence so that converges weakly to some as . Since is -hybrid relative to , we have, for any and ,
Rewrite as
Similarly, we also have
Consequently, we obtain from (3.1) that
Summing up these inequalities with respect to , we get
Dividing the above inequality by , we have
Replacing by and letting , we obtain from the fact that is bounded that
Putting in (3.7), we get
that is,
from which follows that . Therefore by Lemma 2.2.

Since the function in a Hilbert space satisfies all the requirements of Theorem 3.1, the corollary below follows immediately.

Corollary 3.2 (see [4]). *Let be a nonempty closed convex subset of Hilbert space and Suppose is -hybrid. Then the following two statements are equivalent.*(a)*There exists such that is bounded.*(b)* has a fixed point.*

We now show that the fixed point set is closed and convex under the assumptions of Theorem 3.1.

A mapping is said to be quasi-nonexpansive with respect to if and for all and all .

The following lemma is shown in Huang et al. [7].

Lemma 3.3. *Let be a proper strictly convex function on a Banach space so that it is Gâteaux differentiable on , and let be a nonempty closed convex subset of . If is quasi-nonexpansive with respect to , then is a closed convex subset.*

Proposition 3.4. *Let be a proper strictly convex function on a reflexive Banach space so that it is Gâteaux differentiable on and is bounded on bounded subsets of , and let be a nonempty closed convex subset of . Suppose is -hybrid relative to and has a point such that is bounded. Then is quasi-nonexpansive with respect to , and therefore is a nonempty closed convex subset of .*

*Proof. * In view of Theorem 3.1, . Now, for any and any , as is -hybrid relative to , we have
so is quasi-nonexpansive with respect to , and hence is a nonempty closed convex subset of by Lemma 3.3.

For the remainder of this section, we establish a common fixed point theorem for a commutative family of -hybrid mappings relative to .

Lemma 3.5. *Let be a reflexive Banach space and let be a l.s.c. strictly convex function so that it is Gâteaux differentiable on and is bounded on bounded subsets of . Suppose is a nonempty bounded closed convex subset of and is a commutative finite family of -hybrid mappings relative to from into itself. Then has a common fixed point.*

*Proof. *We prove this lemma by induction with respect to . To begin with, we deal with the case that . By Proposition 3.4, we see that and are nonempty bounded closed convex subsets of . Moreover, is -invariant. Indeed, for any , it follows from that , which shows that . Consequently, the restriction of to is -hybrid relative to , and so by Theorem 3.1, has a fixed point , that is, .

By induction hypothesis, assume that for some , is nonempty. Then is a nonempty closed convex subset of , and the restriction of to is a -hybrid mapping relative to from into itself. By Theorem 3.1, has a fixed point in . This shows that , that is, , completing the proof.

Here we would like to remark that in the above lemma, the assumption is bounded on bounded subsets of can be dropped by checking the proof of Theorem 3.1 and noting that we have assumed is bounded.

Theorem 3.6. *Let be a reflexive Banach space and let be a l.s.c. strictly convex function so that it is Gâteaux differentiable on . Suppose is a nonempty bounded closed convex subset of and is a commutative family of -hybrid mappings relative to from into itself. Then has a common fixed point.*

*Proof. *Since is a nonempty bounded closed convex subset of the reflexive Banach space , it is weakly compact. By Proposition 3.4, each is a nonempty weakly compact subset of . Therefore, the conclusion follows once we note that has the finite intersection property by Lemma 3.5.

#### 4. Weak Convergence Theorems

In this section, we discuss the demiclosedness and the weak convergence problem of -hybrid mappings relative to . We denote the weak convergence and strong convergence of a sequence to in a Banach space by and , respectively. For a nonempty closed convex subset of a Banach space , a mapping is demiclosed if for any sequence in with and , one has .

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].

Lemma 4.1. *Suppose is a proper strictly convex function so that it is Gâteaux differentiable on in a Banach space and is a sequence in such that for some . Then
*

Proposition 4.2. *Suppose is a strictly convex function so that it is Gâteaux differentiable on and is bounded on bounded subsets of in a Banach space , and suppose is a closed convex subset of . If is -hybrid relative to , then is demiclosed.*

*Proof. * Let be any sequence in with and . We have to show that . Since is bounded, by Proposition 1.1.11 of [5] there exists a constant such that
Note that, for ,
Similarly, for , we have
Thus we obtain from the -hybrid of that
which implies that
Consequently, if , then Lemma 4.1 implies that
a contradiction. This completes the proof.

A mapping is said to be asymptotically regular if, for any , the sequence tends to zero as .

Theorem 4.3. *Suppose the following conditions hold. *(4.3.1)* is l.s.c. uniformly convex function so that it is Gâteaux differentiable on and is bounded on bounded subsets of in a reflexive Banach space .*(4.3.2)* is a closed convex subset of .*(4.3.3)* is -hybrid relative to and is asymptotically regular with a bounded sequence for some .*(4.3.4)* The mapping for is weak-to-weak* continuous.**Then for any , is weakly convergent to an element .*

*Proof. *Let and . If is not bounded, then there is a subsequence such that for all and as . From (4.3.3), for any we have
which in conjunction with (2.7), (2.8), and (2.12) implies that
a contradiction. Therefore, for any , is bounded, and so it has a subsequence which is weakly convergent to for some . As , it follows from the demiclosedness of that . It remains to show that as . Let be any subsequence of so that for some . Then . Since both of and are decreasing, we have
for some . Particularly, from (4.3.4) we obtain
Consequently, , and hence by the strict convexity of . This shows that for some .

#### 5. Conclusion

In this paper, we have introduced the Bregman distance and a new class of mappings, -hybrid mappings relative to 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 -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 -hybrid mappings relative to in Banach spaces. In a point-dependent -hybrid mapping, the and the 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 -hybrid mappings becomes for all .

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.