#### Abstract

Using Bregman functions, we introduce the new concept of Bregman generalized -projection operator , where is a reflexive Banach space with dual space is a proper, convex, lower semicontinuous and bounded from below function; is a strictly convex and Gâteaux differentiable function; and is a nonempty, closed, and convex subset of . The existence of a solution for a class of variational inequalities in Banach spaces is presented.

#### 1. Introduction

Many nonlinear problems in functional analysis can be reduced to the search of fixed points of nonlinear operators. See, for example, [1–14] and the references therein. Let be a (real) Banach space with norm and dual space . For any in , we denote the value of in at by . When is a sequence in , we denote the strong convergence of to by and the weak convergence by . Let be a nonempty subset of and be a mapping. We denote by the set of* fixed points* of . Let be a nonempty, closed, and convex subset of a smooth Banach space ; let be a mapping from into itself. A point is said to be an* asymptotic fixed point* [15] of if there exists a sequence in which converges weakly to and . We denote the set of all asymptotic fixed points of by . A point is called a* strong asymptotic fixed point* of if there exists a sequence in which converges strongly to and . We denote the set of all strong asymptotic fixed points of by .

We recall the definition of Bregman distances. Let be a strictly convex and Gâteaux differentiable function on a Banach space . The* Bregman distance* [16] (see also [17, 18]) corresponding to is the function defined by
It follows from the strict convexity of that for all in . However, might not be symmetric and might not satisfy the triangular inequality.

When is a smooth Banach space, setting for all in , we have that for all in . Here is the normalized duality mapping from into . Hence, reduces to the usual map as If is a Hilbert space, then .

Let be strictly convex and Gâteaux differentiable and be nonempty. A mapping is said to be(i)Bregman nonexpansive if
(ii)*Bregman quasi-nonexpansive* if and
(iii)*Bregman relatively nonexpansive* if the following conditions are satisfied:(1) is nonempty;(2), ;(3);(iv)*Bregman weak relatively nonexpansive* if the following conditions are satisfied:(1) is nonempty;(2), ;(3).

It is clear that any Bregman relatively nonexpansive mapping is a Bregman quasi-nonexpansive mapping. It is also obvious that every Bregman relatively nonexpansive mapping is a Bregman weak relatively nonexpansive mapping, but the converse is not true in general; see, for example, [19]. Indeed, for any mapping we have . If is Bregman relatively nonexpansive, then .

Let be a reflexive Banach space, let be a proper, convex, lower semicontinuous function, let be strictly convex and Gâteaux differentiable, and let be nonempty. We define a functional by It could easily be seen that satisfies the following properties:(1) is convex and continuous with respect to when is fixed;(2) is convex and lower semicontinuous with respect to when is fixed.

*Definition 1. *Let be a Banach space with dual space , let be a proper, convex, lower semicontinuous function, let be strictly convex and Gâteaux differentiable, and let be a nonempty, closed subset of . We say that is a Bregman generalized -projection operator if

In this paper, using Bregman functions, we introduce the new concept of Bregman generalized -projection operator , where is a reflexive Banach space with dual space , is a proper, convex, lower semicontinuous, and bounded from below function, is a strictly convex and Gâteaux differentiable function, and is a nonempty, closed, and convex subset of . The existence of a solution for a class of variational inequalities in Banach spaces is presented. Our results improve and generalize some known results in the current literature; see, for example, [20, 21].

#### 2. Properties of Bregman Functions and Bregman Distances

Let be a (real) Banach space, and let . For any in , the* gradient * is defined to be the linear functional in such that
The function is said to be* Gâteaux differentiable* at if is well defined, and is* Gâteaux differentiable* if it is Gâteaux differentiable everywhere on . We call Fréchet differentiable at (see, for example, [22, page 13] or [23, page 508]) if, for all , there exists such that
The function is said to be* Fréchet differentiable* if it is Fréchet differentiable everywhere.

For any , let . A function is said to be(i)strongly coercive if
(ii)*locally bounded* if is bounded for all ;(iii)*locally uniformly smooth* on ([24, pages 207, 221]) if the function , defined by
satisfies
(iv)*locally uniformly convex* on (or* uniformly convex on bounded subsets* of ([24, pages 203, 221])) if the gauge of* uniform convexity* of , defined by
satisfies

For a locally uniformly convex map , we have for all in and for all in .

Let be a Banach space and a strictly convex and Gâteaux differentiable function. By (1), the Bregman distance satisfies [16] In particular,

We call a function * lower semicontinuous* if is closed for all in . For a lower semicontinuous convex function , the* subdifferential * of is defined by
for all in . It is well known that is maximal monotone [25, 26]. For any lower semicontinuous convex function , the* conjugate function * of is defined by
It is well known that
We also know that if is a proper lower semicontinuous convex function, then is a proper weak* lower semicontinuous convex function. Here, saying is* proper* we mean that .

The following definition is slightly different from that in Butnariu and Iusem [22].

*Definition 2 (see [23]). *Let be a Banach space. A function is said to be a* Bregman function* if the following conditions are satisfied: (1) is continuous, strictly convex, and Gâteaux differentiable;(2)the set is bounded for all in and .

The following lemma follows from Butnariu and Iusem [22] and Zlinescu [24].

Lemma 3. * Let be a reflexive Banach space and a strongly coercive Bregman function. Then *(1)* is one-to-one, onto, and norm-to-weak* continuous;*(2)* if and only if ;*(3)* is bounded for all in and ;*(4)* is Gâteaux differentiable and .*

The following two results follow from [24, Proposition 3.6.4].

Proposition 4. * Let be a reflexive Banach space and let be a convex function which is locally bounded. The following assertions are equivalent:*(1)* is strongly coercive and locally uniformly convex on **;*(2)* is locally bounded and locally uniformly smooth on **;*(3)* is Fréchet differentiable and ** is uniformly norm-to-norm continuous on bounded subsets of **.*

Proposition 5. * Let be a reflexive Banach space and a continuous convex function which is strongly coercive. The following assertions are equivalent: *(1)* is locally bounded and locally uniformly smooth on **;*(2)* is Fréchet differentiable and ** is uniformly norm-to-norm continuous on bounded subsets of **;*(3)* is strongly coercive and locally uniformly convex on **.*

Let be a Banach space and let be a nonempty convex subset of . Let be a strictly convex and Gâteaux differentiable function. Then, we know from [27] that for in and in , we have
Further, if is a nonempty, closed, and convex subset of a reflexive Banach space and is a strongly coercive Bregman function, then, for each in , there exists a unique in such that
The* Bregman projection * from onto defined by has the following property:
See [22] for details.

Lemma 6 (see [9]). *Let be a Banach space and a Gâteaux differentiable function which is locally uniformly convex on . Let and be bounded sequences in . Then the following assertions are equivalent:*(1)*;*(2)*.*

Lemma 7 (see [23, 28]). *Let be a reflexive Banach space, let be a strongly coercive Bregman function, and let be the function defined by
**
The following assertions hold: *(1)* for all in and in ;*(2)* for all in and in .*

It also follows from the definition that is convex in the second variable , and

Lemma 8 (see [29, Proposition 23.1]). *Let be a real Banach space and let be a lower semicontinuous convex function. Then there exist and such that
*

#### 3. Properties of Bregman -Projection Operator

Theorem 9. * Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be a proper, convex, lower semicontinuous function and let be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded, and locally uniformly convex on . Then for all .*

*Proof. *Let and . Then there exists a sequence such that . We consider the following two possible cases.*Case 1*. If is bounded, then there exists a subsequence of and such that as . Since is convex and lower semicontinuous with respect to , we deduce that is convex and weakly lower semicontinuous with respect to . This implies that
and hence . This shows that .*Case 2.* Assume that is unbounded. Since is proper, convex, and lower semicontinuous, we know that the function , defined by
is proper, convex, and lower semicontinuous. In view of Lemma 8, there exist and such that
This implies that for any and
Next, we show that is bounded. If not, then there exists a subsequence of such that as . Since is strongly coercive, we conclude that
This implies that
Since is proper in , we obtain that which contradicts (31). By a similar argument, as in Case 1, we can prove that which completes the proof.

Theorem 10. *Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded, and locally uniformly convex on . Then the following assertions hold:*(i)*for any given , is a nonempty, closed, and convex subset of ;*(ii)* is monotone; that is, for any , and ,
*(iii)*For any given , if and only if
*

*Proof. *(i) Let be fixed. In view of Theorem 9, we conclude that . According to (20) we have . Let us prove that is closed. Let and as . In view of (6), we deduce that
This implies that and hence is closed. Next, we show that is convex. Let and . By the property (2) of the functional , we obtain
Thus, we have and hence is convex.

(ii) Let , , and . Then we have
In view of (37), we conclude that is monotone.

(iii) It is a simple matter to see that implies that
To this end, let and be arbitrarily chosen. By the definition of we see that
Therefore,
and hence
On the other hand, by the definition of Bregman distance, we obtain that
This, together with (41), implies that
Since is demi-continuous, letting in (43), we conclude that
Conversely, assume that
This implies that

#### 4. Applications to Variational Inequalities

In this section, we investigate the existence of solution to the following variational inequality problem: find the point such that where is a nonempty, closed, and convex subset of the Banach space , and and are two mappings.

*Definition 11 (KKM mapping [30]). *Let be a nonempty subset of a linear space . A set-valued mapping is called a KKM mapping if, for any finite subset of , we have
where denotes the convex hull of .

Lemma 12 (Fan KKM Theorem [30]). *Let be a nonempty convex subset of a Hausdorff topological vector and let be a KKM mapping with closed values. If there exists a point such that is a compact subset of , then .*

Theorem 13. *Let be a nonempty, closed, and convex subset of a reflexive Banach space with dual space . Let be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded and locally uniformly convex on . Let be a continuous mapping and be a proper, convex, lower semicontinuous function. If there exists an element such that
**
is a compact subset of , then the variational inequality (47) has a solution.*

*Proof. *In view of Theorem 10, we need to prove that the following inclusion has a solution:
We define a set-valued mapping by
It is obvious that, for any , . Let us prove that is closed for any . Let and as . Then,
This implies that
Since and are continuous and is lower semicontinuous, we conclude that
Therefore,
which implies that . Now, we prove that is a KKM mapping. Indeed, suppose and with . Let . In view of the property (2) of , we obtain
and hence
Hence there exists at least one number , such that
that is, . Thus, is a KKM mapping.

If , then . By the definition of , we obtain
which is equivalent to
Therefore,
In view of (49), we deduce that is compact. It follows from Lemma 12 that . Hence there exists at least one ; that is,
In view of the definition of Bregman -projection operator , we conclude that
This completes the proof.

Theorem 14. *Let be a reflexive Banach space and a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of . Let be a proper, convex, lower semicontinuous function. Let be a nonempty, closed, and convex subset of and let be a Bregman weak relatively nonexpansive mapping. Let be a sequence in such that . Let be a sequence generated by
**
where is the gradient of . Then , , and converge strongly to .*

*Proof. *We divide the proof into several steps.*Step 1*. We prove that is closed and convex for each .

It is clear that is closed and convex. Let be closed and convex for some . For , we see that
is equivalent to
It could easily be seen that is closed and convex. Therefore, is closed and convex for each .*Step 2*. We claim that for all .

It is obvious that . Assume now that for some . Employing Lemma 7, for any , we obtain
This proves that and hence for all .*Step **3.* We prove that , and are bounded sequences in .

Since , we get that
for each . This implies that the sequence is bounded and hence there exists such that
We claim that the sequence is bounded. Assume on the contrary that as . In view of Lemma 8, there exist and such that
From the definition of Bregman distance, it follows that
Without loss of generality, we may assume that for each . This implies that
Since is strongly coercive, by letting in (72), we conclude that , which is a contradiction. Therefore, is bounded. Since is an infinite family of Bregman weak relatively nonexpansive mappings from into itself, we have for any that
This, together with Definition 2 and the boundedness of , implies that the sequence is bounded.*Step 4.* We show that for some , where .

From Step 3 we know that is bounded. By the construction of , we conclude that and for any positive integer . This, together with (23), implies that
In view of (21), we conclude that
It follows from (75) that the sequence is bounded and hence there exists such that
In view of (64), we conclude that
This proves that is an increasing sequence in and hence the limit exists. Letting in (74), we deduce that . In view of Lemma 6, we obtain that as . This means that is a Cauchy sequence. Since is a Banach space and is closed and convex, we conclude that there exists such that
Now, we show that . In view of Lemma 6 and (78), we obtain
Since , we conclude that
This, together with (79), implies that
It follows from Lemma 6, (79), and (81) that
In view of (78), we get
From (78) and (83), it follows that
Since is uniformly norm-to-norm continuous on any bounded subset of , we obtain
Applying Lemma 6 we derive that
It follows from the three-point identity (see (14)) that for any
as .

The function is bounded on bounded subsets of and, thus, is also bounded on bounded subsets of (see, e.g., [22, Proposition 1.1.11], for more details). This implies that the sequences , , and are bounded in .

In view of Proposition 4(3), we know that dom and is strongly coercive and uniformly convex on bounded subsets of . Let and be the gauge of uniform convexity of the conjugate function . We prove that for any

Let us show (88). For any given , in view of the definition of the Bregman distance (see (2)) and Lemma 6, we obtain
In view of (87), we get that
In view of (87) and (88), we conclude that
as . From the assumption , we get
Therefore, from the property of we deduce that
Since is uniformly norm-to-norm continuous on bounded subsets of , we arrive at
This implies that .

Finally, we show that . From , we conclude that
Since for each , we obtain
Letting in (96), we deduce that
In view of (21), we have , which completes the proof.

*Remark 15. *Theorem 14 improves Theorem 4.1 of [20] in the following aspects.(1)For the structure of Banach spaces, we extend the duality mapping to more general case, that is, a convex, continuous, and strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets.(2)For the mappings, we extend the mapping from a relatively nonexpansive mapping to a Bregman weak relatively nonexpansive mapping. We remove the assumption on the mapping and extend the result to a Bregman weak relatively nonexpansive mapping, where is the set of asymptotic fixed points of the mapping .(3)Theorems 9 and 10 extend and improve corresponding results of [20].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publishing of this paper.

#### Acknowledgment

This research was partially supported by a grant from NSC.