Using Bregman functions, we introduce the new concept of Bregman generalized f-projection operator , where E is a reflexive Banach space with dual space E^*; f : E → ℝ ∪ {+∞} is a proper, convex, lower semicontinuous and bounded from below function; g : E → ℝ is a strictly convex and Gâteaux differentiable function; and C is a nonempty, closed, and convex subset of E. 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 E be a (real) Banach space with norm ∥·∥ and dual space E^*. For any x in E, we denote the value of x^* in E^* at x by 〈x, x^*〉. When {x_n} _n∈ℕ is a sequence in E, we denote the strong convergence of {x_n} _n∈ℕ to x ∈ E by x_n → x and the weak convergence by x_n⇀x. Let C be a nonempty subset of E and T : C → E be a mapping. We denote by F(T) = {x ∈ C : Tx = x} the set of fixed points of T. Let C be a nonempty, closed, and convex subset of a smooth Banach space E; let T be a mapping from C into itself. A point p ∈ C is said to be an asymptotic fixed point [15] of T if there exists a sequence {x_n} _n∈ℕ in C which converges weakly to p and lim⁡_n→∞∥x_n − Tx_n∥ = 0. We denote the set of all asymptotic fixed points of T by . A point p ∈ C is called a strong asymptotic fixed point of T if there exists a sequence {x_n} _n∈ℕ in C which converges strongly to p and lim⁡_n→∞∥x_n − Tx_n∥ = 0. We denote the set of all strong asymptotic fixed points of T by .

We recall the definition of Bregman distances. Let g : E → ℝ be a strictly convex and Gâteaux differentiable function on a Banach space E. The Bregman distance [16] (see also [17, 18]) corresponding to g is the function D_g : E × E → ℝ defined by

(1)

It follows from the strict convexity of g that D_g(x, y) ≥ 0 for all x, y in E. However, D_g might not be symmetric and D_g might not satisfy the triangular inequality.

When E is a smooth Banach space, setting g(x) = ∥x∥² for all x in E, we have that ∇g(x) = 2Jx for all x in E. Here J is the normalized duality mapping from E into E^*. Hence, D_g(·, ·) reduces to the usual map ϕ(·, ·) as

(2)

If E is a Hilbert space, then D_g(x, y) = ∥x−y∥².

Let g : E → ℝ be strictly convex and Gâteaux differentiable and C⊆E be nonempty. A mapping T : C → E is said to be

(i)
Bregman nonexpansive if
(3)
(ii)
Bregman quasi-nonexpansive if F(T) ≠ ∅ and
(4)
(iii)
Bregman relatively nonexpansive if the following conditions are satisfied:
- (1)
  F(T) is nonempty;
- (2)
  D_g(p, Tv) ≤ D_g(p, v), ∀p ∈ F(T), v ∈ C;
- (3)
  ;
(iv)
Bregman weak relatively nonexpansive if the following conditions are satisfied:
- (1)
  F(T) is nonempty;
- (2)
  D_g(p, Tv) ≤ D_g(p, v), ∀p ∈ F(T), v ∈ C;
- (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 T : C → C we have . If T is Bregman relatively nonexpansive, then .

Let E be a reflexive Banach space, let f : E → ℝ ∪ {+∞} be a proper, convex, lower semicontinuous function, let g : E → ℝ be strictly convex and Gâteaux differentiable, and let C⊆E be nonempty. We define a functional H : E × E^* → ℝ ∪ {+∞} by

(5)

It could easily be seen that H satisfies the following properties:

(1)
H(x, x^*) is convex and continuous with respect to x^* when x is fixed;
(2)
H(x, x^*) is convex and lower semicontinuous with respect to x when x^* is fixed.

Definition 1. Let E be a Banach space with dual space E^*, let f : E → ℝ ∪ {+∞} be a proper, convex, lower semicontinuous function, let g : E → ℝ be strictly convex and Gâteaux differentiable, and let C be a nonempty, closed subset of E. We say that is a Bregman generalized f-projection operator if

(6)

In this paper, using Bregman functions, we introduce the new concept of Bregman generalized f-projection operator , where E is a reflexive Banach space with dual space E^*, f : E → ℝ ∪ {+∞} is a proper, convex, lower semicontinuous, and bounded from below function, g : E → ℝ is a strictly convex and Gâteaux differentiable function, and C is a nonempty, closed, and convex subset of E. 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 E be a (real) Banach space, and let g : E → ℝ. For any x in E, the gradient ∇g(x) is defined to be the linear functional in E^* such that

(7)

The function g is said to be Gâteaux differentiable at x if ∇g(x) is well defined, and g is Gâteaux differentiable if it is Gâteaux differentiable everywhere on E. We call g Fréchet differentiable at x (see, for example, [22, page 13] or [23, page 508]) if, for all ϵ > 0, there exists δ > 0 such that

(8)

The function g is said to be Fréchet differentiable if it is Fréchet differentiable everywhere.

For any r > 0, let B_r : = {z ∈ E : ∥z∥ ≤ r}. A function g : E → ℝ is said to be

(i)
strongly coercive if
(9)
(ii)
locally bounded if g(B_r) is bounded for all r > 0;
(iii)
locally uniformly smooth on E ([24, pages 207, 221]) if the function σ_r : [0, +∞)→[0, +∞], defined by
(10)
satisfies
(11)
(iv)
locally uniformly convex on E (or uniformly convex on bounded subsets of E ([24, pages 203, 221])) if the gauge ρ_r : [0, +∞)→[0, +∞] of uniform convexity of g, defined by
(12)
satisfies
(13)

For a locally uniformly convex map g : E → ℝ, we have

(14)

for all x, y in B_r and for all α in (0,1).

Let E be a Banach space and g : E → ℝ a strictly convex and Gâteaux differentiable function. By (1), the Bregman distance satisfies [16]

(15)

In particular,

(16)

We call a function g : E → (−∞, +∞] lower semicontinuous if {x ∈ E : g(x) ≤ r} is closed for all r in ℝ. For a lower semicontinuous convex function g : E → ℝ, the subdifferential ∂g of g is defined by

(17)

for all x in E. It is well known that ∂g ⊂ E × E^* is maximal monotone [25, 26]. For any lower semicontinuous convex function g : E → (−∞, +∞], the conjugate function g^* of g is defined by

(18)

It is well known that

(19)

(20)

We also know that if g : E → (−∞, +∞] is a proper lower semicontinuous convex function, then g^* : E^* → (−∞, +∞] is a proper weak* lower semicontinuous convex function. Here, saying g is proper we mean that dom⁡ g : = {x ∈ E : g(x)<+∞} ≠ ∅.

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

Definition 2 (see [23].)Let E be a Banach space. A function g : E → ℝ is said to be a Bregman function if the following conditions are satisfied:

(1)
g is continuous, strictly convex, and Gâteaux differentiable;
(2)
the set {y ∈ E : D_g(x, y) ≤ r} is bounded for all x in E and r > 0.

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

Lemma 3. Let E be a reflexive Banach space and g : E → ℝ a strongly coercive Bregman function. Then

(1)
∇g : E → E^* is one-to-one, onto, and norm-to-weak* continuous;
(2)
〈x − y, ∇g(x)−∇g(y)〉 = 0 if and only if x = y;
(3)
{x ∈ E : D_g(x, y) ≤ r} is bounded for all y in E and r > 0;
(4)
dom g^* = E^*, g^* is Gâteaux differentiable and ∇g^* = (∇g) ⁻¹.

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

Proposition 4. Let E be a reflexive Banach space and let g : E → ℝ be a convex function which is locally bounded. The following assertions are equivalent:

(1)
g is strongly coercive and locally uniformly convex on E;
(2)
dom g^* = E^*, g^* is locally bounded and locally uniformly smooth on E;
(3)
dom g^* = E^*, g^* is Fréchet differentiable and ∇g^* is uniformly norm-to-norm continuous on bounded subsets of E^*.

Proposition 5. Let E be a reflexive Banach space and g : E → ℝ a continuous convex function which is strongly coercive. The following assertions are equivalent:

(1)
g is locally bounded and locally uniformly smooth on E;
(2)
g^* is Fréchet differentiable and ∇g^* is uniformly norm-to-norm continuous on bounded subsets of E;
(3)
dom g^* = E^*, g^* is strongly coercive and locally uniformly convex on E.

Let E be a Banach space and let C be a nonempty convex subset of E. Let g : E → ℝ be a strictly convex and Gâteaux differentiable function. Then, we know from [27] that for x in E and x₀ in C, we have

(21)

Further, if C is a nonempty, closed, and convex subset of a reflexive Banach space E and g : E → ℝ is a strongly coercive Bregman function, then, for each x in E, there exists a unique x₀ in C such that

(22)

The Bregman projection

from E onto C defined by

has the following property:

(23)

See [22] for details.

Lemma 6 (see [9].)Let E be a Banach space and g : E → ℝ a Gâteaux differentiable function which is locally uniformly convex on E. Let {x_n} _n∈ℕ and {y_n} _n∈ℕ be bounded sequences in E. Then the following assertions are equivalent:

(1)
lim⁡_n→∞D_g(x_n, y_n) = 0;
(2)
lim⁡_n→∞∥x_n − y_n∥ = 0.

Lemma 7 (see [23], [28].)Let E be a reflexive Banach space, let g : E → ℝ be a strongly coercive Bregman function, and let V be the function defined by

(24)

The following assertions hold:

(1)
D_g(x, ∇g^*(x^*)) = V(x, x^*) for all x in E and x^* in E^*;
(2)
V(x, x^*)+〈∇g^*(x^*) − x, y^*〉≤V(x, x^* + y^*) for all x in E and x^*, y^* in E^*.

It also follows from the definition that V is convex in the second variable x^*, and

(25)

Lemma 8 (see [29], Proposition 23.1.)Let E be a real Banach space and let f : E → ℝ ∪ {+∞} be a lower semicontinuous convex function. Then there exist x^* ∈ E^* and a ∈ ℝ such that

(26)

3. Properties of Bregman f-Projection Operator

Theorem 9. Let C be a nonempty, closed, and convex subset of a reflexive Banach space E. Let f : E → ℝ ∪ {+∞} be a proper, convex, lower semicontinuous function and let g : E → ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded, and locally uniformly convex on E. Then for all x^* ∈ E^*.

Proof. Let x^* ∈ E^* and λ = inf⁡_y∈CH(y, x^*). Then there exists a sequence {x_n} _n∈ℕ ⊂ C such that λ = lim⁡_n→∞H(x_n, x^*). We consider the following two possible cases.

Case 1. If C is bounded, then there exists a subsequence of {x_n} _n∈ℕ and x ∈ C such that as j → ∞. Since H(z, x^*) is convex and lower semicontinuous with respect to z, we deduce that H(z, x^*) is convex and weakly lower semicontinuous with respect to z. This implies that

(27)

and hence

. This shows that

Case 2. Assume that C is unbounded. Since f : C → ℝ ∪ {+∞} is proper, convex, and lower semicontinuous, we know that the function f_C : E → ℝ ∪ {+∞}, defined by

(28)

is proper, convex, and lower semicontinuous. In view of Lemma 8, there exist x^* ∈ E^* and a ∈ ℝ such that

(29)

This implies that for any x^* ∈ E^* and x ∈ C

(30)

Next, we show that {x_n} _n∈ℕ is bounded. If not, then there exists a subsequence

of {x_n} _n∈ℕ such that

as k → ∞. Since g is strongly coercive, we conclude that

(31)

This implies that

(32)

Since f is proper in C, we obtain that λ = inf⁡_y∈CH(y, x^*) = lim⁡_n→∞H(x_n, x^*)<+∞ which contradicts (31). By a similar argument, as in Case 1, we can prove that

which completes the proof.

Theorem 10. Let C be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g : E → ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded, and locally uniformly convex on E. Then the following assertions hold:

(i)
for any given x^* ∈ E^*, is a nonempty, closed, and convex subset of C;
(ii)
is monotone; that is, for any x^*, y^* ∈ E^*, and ,
(33)
(iii)
For any given x^* ∈ E^*, if and only if
(34)

Proof. (i) Let x^* ∈ E^* be fixed. In view of Theorem 9, we conclude that . According to (20) we have g(x) + g^*(x^*)−〈x, x^*〉≥0, ∀ (x, x^*) ∈ E × E^*. Let us prove that is closed. Let and x_n → x as n → ∞. In view of (6), we deduce that

(35)

This implies that

and hence

is closed. Next, we show that

is convex. Let

and 0 ≤ t ≤ 1. By the property (2) of the functional H, we obtain

(36)

Thus, we have

and hence

is convex.

(ii) Let , , and . Then we have

(37)

In view of (37), we conclude that

is monotone.

(iii) It is a simple matter to see that implies that

(38)

To this end, let y ∈ C and t ∈ (0,1] be arbitrarily chosen. By the definition of

we see that

(39)

Therefore,

(40)

and hence

(41)

On the other hand, by the definition of Bregman distance, we obtain that

(42)

This, together with (41), implies that

(43)

Since ∇g is demi-continuous, letting t → 0 in (43), we conclude that

(44)

Conversely, assume that

(45)

This implies that

(46)

4. Applications to Variational Inequalities

In this section, we investigate the existence of solution to the following variational inequality problem: find the point x ∈ C such that

(47)

where C is a nonempty, closed, and convex subset of the Banach space E, and A : C → E^* and f : C → ℝ ∪ {+∞} are two mappings.

Definition 11 (KKM mapping [30]). Let C be a nonempty subset of a linear space X. A set-valued mapping G : C → 2^X is called a KKM mapping if, for any finite subset {y₁, y₂, …, y_n} of C, we have

(48)

where co⁡{y₁, y₂, …, y_n} denotes the convex hull of {y₁, y₂, …, y_n}.

Lemma 12 (Fan KKM Theorem [30]). Let C be a nonempty convex subset of a Hausdorff topological vector X and let G : C → 2^X be a KKM mapping with closed values. If there exists a point y₀ ∈ C such that G(y₀) is a compact subset of C, then ⋂_y∈CG(y) ≠ ∅.

Theorem 13. Let C be a nonempty, closed, and convex subset of a reflexive Banach space E with dual space E^*. Let g : E → ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded and locally uniformly convex on E. Let A : C → E^* be a continuous mapping and f : E → ℝ ∪ {+∞} be a proper, convex, lower semicontinuous function. If there exists an element y₀ ∈ C such that

(49)

is a compact subset of C, 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:

(50)

We define a set-valued mapping V : C → 2^C by

(51)

It is obvious that, for any y ∈ C, V(y) ≠ ∅. Let us prove that V(y) is closed for any y ∈ C. Let {x_n} _n∈ℕ ⊂ V(y) and x_n → x as n → ∞. Then,

(52)

This implies that

(53)

Since ∇g and A are continuous and f is lower semicontinuous, we conclude that

(54)

Therefore,

(55)

which implies that x ∈ V(y). Now, we prove that V : C → 2^C is a KKM mapping. Indeed, suppose y₁, y₂, …, y_n ∈ C and 0 < a₁, a₂, …, a_n ≤ 1 with

. Let

. In view of the property (2) of H, we obtain

(56)

and hence

(57)

Hence there exists at least one number j = 1,2, …, n, such that

(58)

that is, z ∈ V(y). Thus, V is a KKM mapping.

If x ∈ V(y₀), then H(z, ∇g(z) − Az) ≤ H(y₀, ∇g(z) − Az). By the definition of H, we obtain

(59)

which is equivalent to

(60)

Therefore,

(61)

In view of (49), we deduce that V(y₀) is compact. It follows from Lemma 12 that ⋂_y∈CV(y) ≠ ∅. Hence there exists at least one x₀ ∈ ⋂_y∈CV(y)); that is,

(62)

In view of the definition of Bregman f-projection operator

, we conclude that

(63)

This completes the proof.

Theorem 14. Let E be a reflexive Banach space and g : E → ℝ a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let f : E → ℝ ∪ {+∞} be a proper, convex, lower semicontinuous function. Let C be a nonempty, closed, and convex subset of E and let T : C → C be a Bregman weak relatively nonexpansive mapping. Let {α_n} _{n∈ℕ∪{0}} be a sequence in (0,1) such that liminf⁡_n→∞α_n(1 − α_n) > 0. Let {x_n} _{n∈ℕ∪{0}} be a sequence generated by

(64)

where ∇g is the gradient of g. Then {x_n} _n∈ℕ, {Tx_n} _n∈ℕ, and {y_n} _n∈ℕ converge strongly to

Proof. We divide the proof into several steps.

Step 1. We prove that C_n is closed and convex for each n ∈ ℕ ∪ {0}.

It is clear that C₀ = C is closed and convex. Let C_m be closed and convex for some m ∈ ℕ. For z ∈ C_m, we see that

(65)

is equivalent to

(66)

It could easily be seen that C_m+1 is closed and convex. Therefore, C_n is closed and convex for each n ∈ ℕ ∪ {0}.

Step 2. We claim that F ⊂ C_n for all n ∈ ℕ ∪ {0}.

It is obvious that F ⊂ C₀ = C. Assume now that F ⊂ C_m for some m ∈ ℕ. Employing Lemma 7, for any w ∈ F ⊂ C_m, we obtain

(67)

This proves that w ∈ C_m+1 and hence F ⊂ C_n for all n ∈ ℕ ∪ {0}.

Step 3. We prove that {x_n} _n∈ℕ, {y_n} _n∈ℕ, and {Tx_n} _n∈ℕ are bounded sequences in C.

Since , we get that

(68)

for each w ∈ F(T). This implies that the sequence {H(w, ∇g(x_n))} _n∈ℕ is bounded and hence there exists M₁ > 0 such that

(69)

We claim that the sequence {x_n} _n∈ℕ is bounded. Assume on the contrary that ∥x_n∥→∞ as n → ∞. In view of Lemma 8, there exist x^* ∈ E^* and a ∈ ℝ such that

(70)

From the definition of Bregman distance, it follows that

(71)

Without loss of generality, we may assume that ∥x_n∥ ≠ 0 for each n ∈ ℕ. This implies that

(72)

Since g is strongly coercive, by letting n → ∞ in (72), we conclude that 0 ≥ ∞, which is a contradiction. Therefore, {x_n} _n∈ℕ is bounded. Since {T_n} _n∈ℕ is an infinite family of Bregman weak relatively nonexpansive mappings from C into itself, we have for any q ∈ F that

(73)

This, together with Definition 2 and the boundedness of {x_n} _n∈ℕ, implies that the sequence {T_nx_n} _n∈ℕ is bounded.

Step 4. We show that x_n → v for some v ∈ F, where .

From Step 3 we know that {x_n} _n∈ℕ is bounded. By the construction of C_n, we conclude that C_m ⊂ C_n and for any positive integer m ≥ n. This, together with (23), implies that

(74)

In view of (21), we conclude that

(75)

It follows from (75) that the sequence {D_g(x_n, x)} _n∈ℕ is bounded and hence there exists M₂ > 0 such that

(76)

In view of (64), we conclude that

(77)

This proves that {D_g(x_n, x)} _n∈ℕ is an increasing sequence in ℝ and hence the limit lim⁡_n→∞D_g(x_n, x) exists. Letting m, n → ∞ in (74), we deduce that D_g(x_m, x_n) → 0. In view of Lemma 6, we obtain that ∥x_m − x_n∥ → 0 as m, n → ∞. This means that {x_n} _n∈ℕ is a Cauchy sequence. Since E is a Banach space and C is closed and convex, we conclude that there exists v ∈ C such that

(78)

Now, we show that v ∈ F. In view of Lemma 6 and (78), we obtain

(79)

Since x_n+1 ∈ C_n+1, we conclude that

(80)

This, together with (79), implies that

(81)

It follows from Lemma 6, (79), and (81) that

(82)

In view of (78), we get

(83)

From (78) and (83), it follows that

(84)

Since ∇g is uniformly norm-to-norm continuous on any bounded subset of E, we obtain

(85)

Applying Lemma 6 we derive that

(86)

It follows from the three-point identity (see (14)) that for any w ∈ F

(87)

as n → ∞.

The function g is bounded on bounded subsets of E and, thus, ∇g is also bounded on bounded subsets of E^* (see, e.g., [22, Proposition 1.1.11], for more details). This implies that the sequences {∇g(x_n)} _n∈ℕ, {∇g(y_n)} _n∈ℕ, and {∇g(Tx_n) : n ∈ ℕ ∪ {0}} are bounded in E^*.

In view of Proposition 4(3), we know that dom g^* = E^* and g^* is strongly coercive and uniformly convex on bounded subsets of E^*. Let s₁ = sup⁡{∥∇g(x_n)∥, ∥∇g(Tx_n)∥ : n ∈ ℕ ∪ {0}} and be the gauge of uniform convexity of the conjugate function g^*. We prove that for any w ∈ F

(88)

Let us show (88). For any given w ∈ F(T), in view of the definition of the Bregman distance (see (2)) and Lemma 6, we obtain

(89)

In view of (87), we get that

(90)

In view of (87) and (88), we conclude that

(91)

as n → ∞. From the assumption liminf⁡_n→∞α_n(1 − α_n) > 0, we get

(92)

Therefore, from the property of

we deduce that

(93)

Since ∇g^* is uniformly norm-to-norm continuous on bounded subsets of E^*, we arrive at

(94)

This implies that v ∈ F(T).

Finally, we show that . From , we conclude that

(95)

Since F ⊂ C_n for each n ∈ ℕ, we obtain

(96)

Letting n → ∞ in (96), we deduce that

(97)

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 T and extend the result to a Bregman weak relatively nonexpansive mapping, where is the set of asymptotic fixed points of the mapping T.
(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.

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Bregman f-Projection Operator with Applications to Variational Inequalities in Banach Spaces

Abstract

1. Introduction

2. Properties of Bregman Functions and Bregman Distances

3. Properties of Bregman f-Projection Operator

4. Applications to Variational Inequalities

Conflict of Interests

Acknowledgment

References

Citing Literature