The Br´ezis-Browder Theorem revisited and properties of Fitzpatrick functions of order n Liangjin Yao

Abstract In this paper, we study maximal monotonicity of linear relations (setvalued operators with linear graphs) on reflexive Banach spaces. We provide a new and simpler proof of a result due to Br´ezis-Browder which states that a monotone linear relation with closed graph is maximal monotone if and only if its adjoint is monotone. We also study Fitzpatrick functions and give an explicit formula for Fitzpatrick functions of order n for monotone symmetric linear relations. Key words: Adjoint, convex function, convex set, Fenchel conjugate, Fitzpatrick function, linear relation, maximal monotone operator, multifunction, monotone operator, set-valued operator, symmetric operator. AMS 2010 Subject Classification: 47A06, 47H05

1 Introduction Monotone operators play important roles in convex analysis and optimization [13, 16, 23, 25–27, 33, 34]. In 1978, Br´ezis-Browder gave some characterizations of a monotone operator with closed linear graph ( [14, Theorem 2]) in reflexive Banach spaces. The Br´ezis-Browder Theorem states that a monotone linear relation with closed graph is maximal monotone if and only if its adjoint is monotone if and only if its adjoint is maximal monotone, which demonstrates the connection between the monotonicity of a linear relation and that of its adjoint. In this paper we give a new and simpler proof of the hard part of the Br´ezis-Browder Theorem (Theorem 2): a monotone linear relation with closed graph is maximal monotone if its adjoint is monotone. The proof relies on a recent characterization of maximal monotonicity due to Simons and Z˘alinescu. Our proof does not require any renorming. Liangjin Yao Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada, e-mail: [email protected]

1

2

Liangjin Yao

We suppose throughout this note that X is a real reflexive Banach space with norm ∥ · ∥, that X ∗ is its continuous dual space with norm ∥ · ∥∗ and dual product ⟨·, ·⟩. We now introduce some notation. Let A : X ⇒ X ∗ be a set-valued operator or multifunction whose graph is defined by gra A := {(x, x∗ ) ∈ X × X ∗ | x∗ ∈ Ax}. The inverse operator of A, A−1 : X ∗ ⇒ X, is given by gra A−1 := {(x∗ , x) ∈ X ∗ × X | x∗ ∈ Ax}; the domain of A is dom A := {x ∈ X | Ax ̸= ∅}. The Fitzpatrick function of A (see [20]) is given by ( ) FA : (x, x∗ ) 7→ sup ⟨x, a∗ ⟩ + ⟨a, x∗ ⟩ − ⟨a, a∗ ⟩ . (1) (a,a∗ )∈gra A

For every n ∈ {2, 3, . . .}, the Fitzpatrick function of A of order n (see [1, Definition 2.2 and Proposition 2.3]) is defined by FA, n (x, x∗ ) ( := {

sup

}

(a1 ,a∗1 ),···(an−1 ,a∗n−1 ) ⊆gra A

) + ⟨a1 − x, x∗ ⟩ .

⟨x, x∗ ⟩ +

( n−2 ) ∑ ⟨ai+1 − ai , a∗i ⟩ + ⟨x − an−1 , a∗n−1 ⟩ i=1

Clearly, FA, 2 = FA . We set FA, ∞ = supn∈{2,3,··· } FA, n . If Z is a real reflexive Banach space with dual Z ∗ and a set S ⊆ Z, we denote S⊥ by S⊥ := {z∗ ∈ Z ∗ | ⟨z∗ , s⟩ = 0, ∀s ∈ S}. Then the adjoint of A, denoted by A∗ , is defined by gra A∗ := {(x, x∗ ) ∈ X × X ∗ | (x∗ , −x) ∈ (gra A)⊥ }. Note that A is said to be a linear relation if gra A is a linear subspace of X × X ∗ . (See [19] for further information on linear relations.) Recall that A is said to be monotone if for all (x, x∗ ), (y, y∗ ) ∈ gra A we have ⟨x − y, x∗ − y∗ ⟩ ≥ 0, and A is maximal monotone if A is monotone and A has no proper monotone extension (in the sense of graph inclusions). We say (x, x∗ ) ∈ X × X ∗ is monotonically related to gra A if (for every (y, y∗ ) ∈ gra A) ⟨x − y, x∗ − y∗ ⟩ ≥ 0. Recently linear relations have been become an interesting object and comprehensively studied in Monotone Operator Theory: see [1–3, 6–11, 24, 30–32]. We can now precisely describe the Br´ezis-Browder Theorem. Let A be a monotone linear relation with closed graph. Then A is maximal monotone ⇔ A∗ is maximal monotone ⇔ A∗ is monotone.

Title Suppressed Due to Excessive Length

3

The original proof of Br´ezis-Browder Theorem is based on the application of Zorn Lemma by constructing a series of finite-dimensional subspaces, which is complicated. Our goal of this paper is to give a simpler proof of Br´ezis-Browder Theorem and to derive more properties of Fitzpatrick functions of order n. The paper is organized as follows. The first main result (Theorem 2) is proved in Section 2 providing a new and simpler proof of the Br´ezis-Browder Theorem. In Section 3, some explicit formulas for Fitzpatrick functions are given. Recently, Fitzpatrick functions of order n [1] have turned out to be a useful tool in the study of n-cyclic monotonicity (see [1, 3, 4, 15]). Theorem 3 gives an explicit formula for Fitzpatrick functions of order n associated with symmetric linear relations, which generalizes and simplifies [1, Example 4.4] and [3, Example 6.4]. Our notation is standard. The notation A : X → X ∗ means that A is a single-valued mapping (with full domain) from X to X ∗ . Given a subset C of X, C is the norm closure of C. The indicator function ιC : X → ]−∞, +∞] of C is defined by { 0, if x ∈ C; x 7→ (2) +∞, otherwise. Let x ∈ X and C∗ ⊆ X ∗ . We write ⟨x,C∗ ⟩ := {⟨x, c∗ ⟩ | c∗ ∈ C∗ }. If ⟨x,C∗ ⟩ = {a} for some constant a ∈ R, then we write ⟨x,C∗ ⟩ = a for convenience. For a function f : X → ]−∞, +∞], dom f = {x ∈ X | f (x) < +∞} and f ∗ : X ∗ → [−∞, +∞] : x∗ 7→ supx∈X (⟨x, x∗ ⟩ − f (x)) is the Fenchel conjugate of f . Recall that f is said to be proper if dom f ̸= ∅. If f is convex, ∂ f : X ⇒ X ∗ : x 7→ {x∗ ∈ X ∗ | (∀y ∈ X) ⟨y − x, x∗ ⟩ + f (x) ≤ f (y)} is the subdifferential operator of f . Denote J by the duality map, i.e., the subdifferential of the function 21 ∥ · ∥2 , by [23, Example 2.26], Jx := {x∗ ∈ X ∗ | ⟨x∗ , x⟩ = ∥x∗ ∥∗ · ∥x∥, with ∥x∗ ∥∗ = ∥x∥}.

2 A new proof of the Br´ezis-Browder Theorem Fact 2.1 (Simons) (See [27, Lemma 19.7 and Section 22].) Let A : X ⇒ X ∗ be a monotone operator such that gra A is convex with gra A ̸= ∅. Then the function g : X × X ∗ → ]−∞, +∞] : (x, x∗ ) 7→ ⟨x, x∗ ⟩ + ιgra A (x, x∗ )

(3)

is proper and convex. Fact 2.2 (Simons-Z˘alinescu) (See [28, Theorem 1.2] or [26, Theorem 10.6].) Let A : X ⇒ X ∗ be monotone. Then A is maximal monotone if and only if gra A + gra(−J) = X × X ∗ .

4

Liangjin Yao

Remark 1. When J and J −1 are single-valued, Fact 2.2 yields Rockafellar’s characterization of maximal monotonicity of A. See [28, Theorem 1.3] and [27, Theorem 29.5 and Remark 29.7]. Now we state the Br´ezis-Browder Theorem. Theorem 1 (Br´ezis-Browder). (See [14, Theorem 2].) Let A : X ⇒ X ∗ be a monotone linear relation with closed graph. Then the following statements are equivalent. (The hard part is to show (iii)⇒(i)). (i) A is maximal monotone. (ii) A∗ is maximal monotone. (iii) A∗ is monotone. Proof. (i)⇒(iii): Suppose to the contrary that A∗ is not monotone. Then there exists (x0 , x0∗ ) ∈ gra A∗ such that ⟨x0 , x0∗ ⟩ < 0. Now we have ⟨−x0 − y, x0∗ − y∗ ⟩ = ⟨−x0 , x0∗ ⟩ + ⟨y, y∗ ⟩ + ⟨x0 , y∗ ⟩ + ⟨−y, x0∗ ⟩ = ⟨−x0 , x0∗ ⟩ + ⟨y, y∗ ⟩ > 0,

∀(y, y∗ ) ∈ gra A.

(4)

Thus, (−x0 , x0∗ ) is monotonically related to gra A. By maximal monotonicity of A, (−x0 , x0∗ ) ∈ gra A. Then ⟨−x0 − (−x0 ), x0∗ − x0∗ ⟩ = 0, which contradicts (4). Hence A∗ is monotone. (iii)⇒(i): See Theorem 2 below. (i)⇔(ii): Apply directly (iii)⇔(i) by using A∗∗ = A (since gra A is closed). In Theorem 2, we provide a new and simpler proof to show the hard part (iii)⇒(i) in Theorem 1. The proof was inspired by that of [34, Theorem 32.L]. Theorem 2. Let A : X ⇒ X ∗ be a monotone linear relation with closed graph. Suppose A∗ is monotone. Then A is maximal monotone. Proof. By Fact 2.2, it suffices to show that X × X ∗ ⊆ gra A + gra(−J). For this, let (x, x∗ ) ∈ X × X ∗ and we define g : X × X ∗ → ]−∞, +∞] by (y, y∗ ) 7→ 21 ∥y∗ ∥2∗ + 12 ∥y∥2 + ⟨y∗ , y⟩ + ιgra A (y − x, y∗ − x∗ ). Since gra A is closed, g is lower semicontinuous on X × X ∗ . Note that (y, y∗ ) 7→ ⟨y∗ , y⟩ + ιgra A (y − x, y∗ − x∗ ) = ⟨y∗ , y⟩ + ιgra A+(x,x∗ ) (y, y∗ ). By Fact 2.1, g is convex and coercive. According to [33, Theorem 2.5.1(ii)], g has minimizers. Suppose that (z, z∗ ) is a minimizer of g. Then (z − x, z∗ − x∗ ) ∈ gra A, that is, (x, x∗ ) ∈ gra A + (z, z∗ ).

(5)

On the other hand, since (z, z∗ ) is a minimizer of g, (0, 0) ∈ ∂ g(z, z∗ ). By a result of Rockafellar (see [18, Theorem 2.9.8] and [33, Theorem 3.2.4(ii)]), there exist (z∗0 , z0 ) ∈ ∂ (ιgra A (·−x, ·−x∗ ))(z, z∗ ) = ∂ ιgra A (z−x, z∗ −x∗ ) = (gra A)⊥ , and (v, v∗ ) ∈ X × X ∗ with v∗ ∈ Jz, z∗ ∈ Jv such that

Title Suppressed Due to Excessive Length

5

(0, 0) = (z∗ , z) + (v∗ , v) + (z∗0 , z0 ). Then (

) − (z + v), z∗ + v∗ ∈ gra A∗ .

Since A∗ is monotone, ⟨z∗ + v∗ , z + v⟩ = ⟨z∗ , z⟩ + ⟨z∗ , v⟩ + ⟨v∗ , z⟩ + ⟨v∗ , v⟩ ≤ 0.

(6)

Note that since ⟨z∗ , v⟩ = ∥z∗ ∥2∗ = ∥v∥2 , ⟨v∗ , z⟩ = ∥v∗ ∥2∗ = ∥z∥2 , by (6), we have 2 ∗ 2 ∗ 1 1 ∗ 2 1 ∗ 2 1 2 ∥z∥ + 2 ∥z ∥∗ + ⟨z , z⟩ + 2 ∥v ∥∗ + 2 ∥v∥ + ⟨v, v ⟩ ≤ 0.

Hence z∗ ∈ −Jz. By (5), (x, x∗ ) ∈ gra A + gra(−J). Thus, X × X ∗ ⊆ gra A + gra(−J). Hence A is maximal monotone. Remark 2. Haraux provides a very simple proof of Theorem 2 in Hilbert spaces in [21, Theorem 10], but the proof could not be adapted to reflexive Banach spaces.

3 Fitzpatrick functions and Fitzpatrick functions of order n Now we introduce some properties of monotone linear relations. Fact 3.1 (See [6].) Assume that A : X ⇒ X ∗ is a monotone linear relation. Then the following hold. (i) The function dom A → R : y 7→ ⟨y, Ay⟩ is convex. (ii) dom A ⊆ (A0)⊥ . For every x ∈ (A0)⊥ , the function dom A → R : y 7→ ⟨x, Ay⟩ is linear. Proof. (i): See [6, Proposition 2.3]. (ii): See [6, Proposition 2.2(i)(iii)]. Definition 1. Suppose A : X ⇒ X ∗ is a linear relation. We say A is symmetric if gra A ⊆ gra A∗ . By the definition of A∗ , we have (∀x, y ∈ dom A) ⟨x, Ay⟩ is single-valued and ⟨x, Ay⟩ = ⟨y, Ax⟩. For a monotone linear relation A : X ⇒ X ∗ (where A is not necessarily symmetric) it will be convenient to define (as in, e.g., [3]) { 1 ⟨x, Ax⟩, if x ∈ dom A; qA : x ∈ X 7→ 2 (7) +∞, otherwise. By Fact 3.1(i), qA is well defined and is at most single-valued and convex. According to the definition of qA , dom qA = dom A. Moreover, by (0, 0) ∈ gra A and A is monotone, we have that qA ≥ 0.

6

Liangjin Yao

The following generalizes a result of Phelps-Simons (see [24, Theorem 5.1]) from symmetric monotone linear operators to symmetric monotone linear relations. We write f for the lower semicontinuous hull of f . Proposition 1. Let A : X ⇒ X ∗ be a monotone symmetric linear relation. Then (i) qA is convex, and qA + ιdom A = qA . (ii) gra A ⊆ gra ∂ qA . If A is maximal monotone, then A = ∂ qA . Proof. Let x ∈ dom A. (i): Since A is monotone, qA is convex. Let y ∈ dom A. Since A is monotone, by Fact 3.1(ii), 0 ≤ 12 ⟨Ax − Ay, x − y⟩ = 12 ⟨Ay, y⟩ + 21 ⟨Ax, x⟩ − ⟨Ax, y⟩,

(8)

we have qA (y) ≥ ⟨Ax, y⟩ − qA (x). Take lower semicontinuous hull on y and then deduce that qA (y) ≥ ⟨Ax, y⟩ − qA (x). For y = x, we have qA (x) ≥ qA (x). On the other hand, qA (x) ≤ qA (x). Altogether, qA (x) = qA (x). Thus (i) holds. (ii): Let y ∈ dom A. By (8) and (i), qA (y) ≥ qA (x) + ⟨Ax, y − x⟩ = qA (x) + ⟨Ax, y − x⟩.

(9)

Since dom qA ⊆ dom qA = dom A, by (9), qA (z) ≥ qA (x)+⟨Ax, z−x⟩, ∀z ∈ dom qA . Hence Ax ⊆ ∂ qA (x). If A is maximal monotone, A = ∂ qA . Thus (ii) holds. Definition 2 (Fitzpatrick family). Let A : X ⇒ X ∗ be a maximal monotone operator. The associated Fitzpatrick family FA consists of all functions F : X × X ∗ → ]−∞, +∞] that are lower semicontinuous and convex, and that satisfy F ≥ ⟨·, ·⟩, and F = ⟨·, ·⟩ on gra A. Following [22], it will be convenient to set F | : X ∗ × X → ]−∞, +∞] : (x∗ , x) 7→ F(x, x∗ ), when F : X × X ∗ → ]−∞, +∞], and similarly for a function defined on X ∗ × X. Fact 3.2 (Fitzpatrick) (See [20, Theorem 3.10] or [17, Corollary 4.1].) Let A : X ⇒ X ∗ be a maximal monotone operator. Then for every (x, x∗ ) ∈ X × X ∗ , FA (x, x∗ ) = min{F(x, x∗ ) | F ∈ FA }

and FA∗| (x, x∗ ) = max{F(x, x∗ ) | F ∈ FA }. (10)

Proposition 2. Let A : X ⇒ X ∗ be a maximal monotone and symmetric linear relation. Then FA (x, x∗ ) = 12 qA (x) + 12 ⟨x, x∗ ⟩ + 12 q∗A (x∗ ),

∀(x, x∗ ) ∈ X × X ∗ .

Proof. Define function k : X × X ∗ → ]−∞, +∞] by (z, z∗ ) 7→ 12 qA (z) + 12 ⟨z, z∗ ⟩ + 12 q∗A (z∗ ).

Title Suppressed Due to Excessive Length

7

Claim 1: FA = k on dom A × X ∗ . Let (x, x∗ ) ∈ X × X ∗ , and suppose that x ∈ dom A. Then ( ) FA (x, x∗ ) = sup ⟨x, y∗ ⟩ + ⟨y, x∗ ⟩ − ⟨y, y∗ ⟩ (y,y∗ )∈gra A

(

= sup

) ⟨x, Ay⟩ + ⟨y, x∗ ⟩ − 2qA (y)

y∈dom A

(

1 2

qA (x) + sup

=

1 2

qA (x) + 12 sup

=

1 2

=

1 2

=

1 2

=

=

y∈dom A

) ⟨Ax, y⟩ + ⟨y, x∗ ⟩ − 12 qA (x) − 2qA (y)

y∈dom A

( ) ⟨Ax, 2y⟩ + ⟨2y, x∗ ⟩ − qA (x) − 4qA (y) (

) ⟨Ax, z⟩ + ⟨z, x∗ ⟩ − qA (x) − qA (z) z∈dom A ( ) qA (x) + 12 sup ⟨z, x∗ ⟩ − qA (z − x) z∈dom A ( ) 1 qA (x) + 2 ⟨x, x∗ ⟩ + 21 sup ⟨z − x, x∗ ⟩ − qA (z − x) qA (x) +

1 2

sup

z∈dom A ∗ 1 1 ∗ ∗ 1 2 qA (x) + 2 ⟨x, x ⟩ + 2 qA (x ) ∗

= k(x, x ) (by Proposition 1(i)). Claim 2: k is convex and proper lower semicontinuous on X × X ∗ . Since FA is convex, 12 qA + 12 ⟨·, ·⟩ + 12 q∗A is convex on dom A × X ∗ . Now we show that k is convex. Let {(a, a∗ ), (b, b∗ )} ⊆ dom k, and t ∈ ]0, 1[. Then we have {a, b} ⊆ dom qA ⊆ dom A. Thus, there exist (an ), (bn ) in dom A such that an → a, bn → b with qA (an ) → qA (a), qA (bn ) → qA (b). Since 12 qA + 12 ⟨·, ·⟩ + 12 q∗A is convex on dom A × X ∗ , we have (1 )( ) ∗ ∗ 1 1 ∗ 2 qA + 2 ⟨·, ·⟩ + 2 qA tan + (1 − t)bn ,ta + (1 − t)b ( ) ( ) ≤ t 12 qA + 12 ⟨·, ·⟩ + 12 q∗A (an , a∗ ) + (1 − t) 12 qA + 12 ⟨·, ·⟩ + 12 q∗A (bn , b∗ ). (11) Take lim inf on both sides of (11) to see that ( ) k ta + (1 − t)b,ta∗ + (1 − t)b∗ ≤ tk(a, a∗ ) + (1 − t)k(b, b∗ ). Hence k is convex on X × X ∗ . Thus, k is convex and proper lower semicontinuous. Claim 3: FA = k on X × X ∗ . To this end, we first observe that dom ∂ k∗ = gra A−1 . We have

(12)

8

Liangjin Yao

(w∗ , w) ∈ dom ∂ k∗ ⇔ (w∗ , w) ∈ dom ∂ (2k)∗ ⇔ (a, a∗ ) ∈ ∂ (2k)∗ (w∗ , w), ∃(a, a∗ ) ∈ X × X ∗ ⇔ (w∗ , w) ∈ ∂ (2k)(a, a∗ ), ∃(a, a∗ ) ∈ X × X ∗ ⇔ (w∗ − a∗ , w − a) ∈ ∂ (qA ⊕ q∗A )(a, a∗ ), ∃(a, a∗ ) ∈ X × X ∗ (by [33, Theorem 3.2.4(vi)(ii)]) ⇔ w∗ − a∗ ∈ ∂ qA (a), w − a ∈ ∂ q∗A (a∗ ), ∃(a, a∗ ) ∈ X × X ∗ ⇔ w∗ − a∗ ∈ ∂ qA (a), a∗ ∈ ∂ qA (w − a), ∃(a, a∗ ) ∈ X × X ∗ ⇔ w∗ − a∗ ∈ Aa, a∗ ∈ A(w − a), ∃(a, a∗ ) ∈ X × X ∗ (by Proposition 1(ii)) ⇔ (w, w∗ ) ∈ gra ⇔ (w∗ , w) ∈ gra A−1 . Next we observe that k∗| (z, z∗ ) = ⟨z, z∗ ⟩,

∀(z, z∗ ) ∈ gra A.

(13)

Since k(z, z∗ ) ≥ ⟨z, z∗ ⟩ and k(z, z∗ ) = ⟨z, z∗ ⟩ ⇔ qA (z) + q∗A (z∗ ) = ⟨z, z∗ ⟩ ⇔ z∗ ∈ ∂ qA (z) = Az (by Proposition 1(ii)), Fact 3.2 implies that FA ≤ k ≤ FA∗| . Hence FA ≤ k∗| ≤ FA∗| . Then by Fact 3.2, (13) holds. Now using (13), (12) and a result by J. Borwein (see [12, Theorem 1] or [33, Theorem 3.1.4(i)]), we have k = k∗∗ = (k∗ + ιdom ∂ k∗ )∗ = (⟨·, ·⟩ + ιgra A−1 )∗ = FA . Fact 3.3 (recursion) (See [4, Proposition 2.13].) Let A : X ⇒ X ∗ be monotone, and let n ∈ {2, 3, . . .}. Then ( ) FA, n+1 (x, x∗ ) = sup FA, n (a, x∗ ) + ⟨x − a, a∗ ⟩ , ∀(x, x∗ ) ∈ X × X ∗ . (a,a∗ )∈gra A

Theorem 3. Let A : X ⇒ X ∗ be a maximal monotone and symmetric linear relation, let n ∈ {2, 3, . . .}, and let (x, x∗ ) ∈ X × X ∗ . Then FA, n (x, x∗ ) = consequently, FA, n (x, x∗ ) =

∗ n−1 1 n−1 ∗ ∗ n qA (x) + n qA (x ) + n ⟨x, x ⟩,

2(n−1) 2−n ∗ ∗ n FA (x, x ) + n ⟨x, x ⟩.

FA, ∞ = qA ⊕ q∗A = 2FA − ⟨·, ·⟩.

(14)

Moreover, (15)

Proof. Let (x, x∗ ) ∈ X × X ∗ . The proof is by induction on n. If n = 2, then the result follows for Proposition 2. Now assume that (14) holds for n ≥ 2. Using Fact 3.3, we see that

Title Suppressed Due to Excessive Length

FA, n+1 (x, x∗ ) = ( =

sup

(a,a∗ )∈gra A

( sup

(a,a∗ )∈gra A

9

) FA, n (a, x∗ ) + ⟨x − a, a∗ ⟩ )

∗ ∗ n−1 ∗ ∗ 1 n−1 n qA (x ) + n qA (a) + n ⟨a, x ⟩ + ⟨x − a, a ⟩

)

(

=

n−1 ∗ ∗ n qA (x ) +

=

n−1 ∗ ∗ n qA (x ) +

=

n−1 ∗ ∗ 2n n qA (x ) + n+1

=

n−1 ∗ ∗ 2n n qA (x ) + n+1

∗ ∗ ∗ 1 ∗ n−1 2n ⟨a, a ⟩ + ⟨a, n x ⟩ + ⟨x, a ⟩ − ⟨a, a ⟩

sup

(a,a∗ )∈gra A

(

sup

(a,a∗ )∈gra A

) ∗ ⟨a, a ⟩ ⟨a, 1n x∗ ⟩ + ⟨x, a∗ ⟩ − n+1 2n (

) 1 ∗ n+1 ∗ n+1 ∗ n+1 ⟨ n+1 a, x a a, a ⟩ + ⟨x, ⟩ − ⟨ ⟩ 2n n 2n 2n 2n

(

) ⟨b, 1n x∗ ⟩ + ⟨x, b∗ ⟩ − ⟨b, b∗ ⟩

sup

(a,a∗ )∈gra A

sup

(b,b∗ )∈gra A

=

n−1 ∗ ∗ n qA (x ) + n−1 ∗ ∗ n qA (x ) + n−1 ∗ ∗ n qA (x ) +

=

∗ n 1 n ∗ ∗ n+1 qA (x ) + n+1 qA (x) + n+1 ⟨x, x ⟩,

= =

, (by Proposition 1(i))

2n 1 ∗ n+1 FA (x, n x ) ∗ n ∗ 1 ∗ n 1 n+1 qA ( n x ) + n+1 qA (x) + n+1 ⟨x , x⟩ ∗ ∗ ∗ 1 1 n (n+1)n qA (x ) + n+1 qA (x) + n+1 ⟨x , x⟩

(by Proposition 2)

∗ which is the result for n + 1. Thus, by Proposition 2, FA, n (x, x∗ ) = 2(n−1) n FA (x, x ) + 2−n ∗ n ⟨x, x ⟩. By (14), dom FA, n = dom(qA ⊕ q∗A ). (Now suppose that (x, x∗ ) )∈ dom FA, n .

By qA (x) + q∗A (x∗ ) − FA, n (x, x∗ ) =

1 n

qA (x) + q∗A (x∗ ) − ⟨x, x∗ ⟩ ≥ 0 and

FA, n (x, x∗ ) → (qA ⊕ q∗A )(x, x∗ ), n → ∞. Thus, (15) holds. Remark 3. Theorem 3 generalizes and simplifies [1, Example 4.4] and [3, Example 6.4]. See Corollary 1. Remark 4. Formula Identity (14) does not hold for nonsymmetric linear relations. See [3, Example 2.8] for an example when A is skew linear operator and (14) fails. Corollary 1. Let A : X → X ∗ be a maximal monotone and symmetric linear operator, let n ∈ {2, 3, . . .}, and let (x, x∗ ) ∈ X × X ∗ . Then FA, n (x, x∗ ) =

∗ n−1 n−1 ∗ ∗ 1 n qA (x) + n qA (x ) + n ⟨x, x ⟩,

(16)

and, FA, ∞ = qA ⊕ q∗A .

(17)

If X is a Hilbert space, then FId, n (x, x∗ ) =

2 ∗ n−1 n−1 ∗ 2 1 2n ∥x∥ + 2n ∥x ∥ + n ⟨x, x ⟩,

(18)

10

Liangjin Yao

and, FId, ∞ = 21 ∥ · ∥2 ⊕ 12 ∥ · ∥2 .

(19)

Definition 3. Let F1 , F2 : X × X ∗ → ]−∞, +∞]. Then the partial inf-convolution F1 2 F2 is the function defined on X × X ∗ by ( ) F1 2 F2 : (x, x∗ ) 7→ ∗inf ∗ F1 (x, x∗ − y∗ ) + F2 (x, y∗ ) . y ∈X

Theorem 4 (nth order Fitzpatrick function of the sum). Let A, B : X ⇒ X ∗ be maximal monotone and symmetric linear relations, and let n ∈ {2, 3, · · · }. Suppose that dom A − dom B is closed. Then FA+B, n = FA, n 2 FB, n . Moreover, FA+B, ∞ = FA, ∞ 2 FB, ∞ . Proof. By [29, Theorem 5.5] or [31], A + B is maximal monotone. Hence A + B is a maximal monotone and symmetric linear relation. Let (x, x∗ ) ∈ X × X ∗ . Then by Theorem 3, FA, n 2 FB, n (x, x∗ ) ( ) 2(n−1) ∗ ∗ ∗ ∗ ∗ ∗ 2−n 2−n = ∗inf ∗ 2(n−1) F (x, y ) + ⟨x, y ⟩ + F (x, x − y ) + ⟨x, x − y ⟩ B A n n n n y ∈X ( ) 2(n−1) ∗ = 2−n FA (x, y∗ ) + FB (x, x∗ − y∗ ) n ⟨x, x ⟩ + ∗inf ∗ n y ∈X

2(n−1) ∗ ∗ = 2−n n ⟨x, x ⟩ + n FA 2 FB (x, x ) 2(n−1) ∗ ∗ = 2−n (by n ⟨x, x ⟩ + n FA+B (x, x ), ∗ = FA+B, n (x, x ) (by Theorem 3).

[6, Theorem 5.10])

Similarly, using (15), we have FA+B, ∞ = FA, ∞ 2 FB, ∞ . Remark 5. Theorem 4 generalizes [3, Theorem 5.4]. Acknowledgements The author thanks Dr. Heinz Bauschke and Dr. Xianfu Wang for valuable discussions. The author also thanks the two anonymous referees for their careful reading and their pertinent comments.

References 1. Bartz, S., Bauschke, H.H., Borwein, J.M., Reich, S., Wang, X.: “Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative”, Nonlinear Anal., vol. 66, pp. 1198–1223 (2007). 2. Bauschke, H.H., Borwein, J.M.: “Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators”, Pacific J. Math., vol. 189, pp. 1–20 (1999). 3. Bauschke, H.H., Borwein, J.M., Wang, X.: “Fitzpatrick functions and continuous linear monotone operators”, SIAM J. Optim., vol. 18, pp. 789–809 (2007).

Title Suppressed Due to Excessive Length

11

4. Bauschke, H.H., Lucet, Y., Wang, X.: “Primal-dual symmetric antiderivatives for cyclically monotone operators”, SIAM J. Control Optim., vol. 46, pp. 2031–2051 (2007). 5. Bauschke, H.H., Wang, X.: “An explicit example of a maximal 3-cyclically monotone operator with bizarre properties”, Nonlinear Anal., vol. 69, pp. 2875–2891 (2008). 6. Bauschke, H.H., Wang, X., Yao, L.: “Monotone linear relations: maximality and Fitzpatrick functions”, J. Convex Anal., vol. 16, pp. 673–686 (2009). 7. Bauschke, H.H., Wang, X., Yao, L.: “An answer to S. Simons’ question on the maximal monotonicity of the sum of a maximal monotone linear operator and a normal cone operator”, SetValued Var. Anal., vol. 17, pp. 195-201 (2009). 8. Bauschke, H.H., Wang, X., Yao, L.: “Autoconjugate representers for linear monotone operators”, Math. Program. Ser. B, vol. 123, pp. 5-24 (2010). 9. Bauschke, H.H., Wang, X., Yao, L.: “Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter”, J. Math. Anal. Appl., vol. 370, pp. 224-241 (2010). 10. Bauschke, H.H., Wang, X., Yao, L.: “On Borwein-Wiersma Decompositions of monotone linear relations”, SIAM J. Optim., vol. 20, pp. 2636–2652 (2010). 11. Bauschke, H.H., Wang, X., Yao, L.: “On the maximal monotonicity of the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function”, to appear Proceedings of the Haifa Workshop on Optimization Theory and Related Topics. Contemp. Math., Amer. Math. Soc., Providence, RI; http://arxiv.org/abs/1001.0257v1, January (2010). 12. Borwein, J.M.: “A note on ε -subgradients and maximal monotonicity”, Pacific J. Math., vol. 103, pp. 307–314 (1982). 13. Borwein, J.M., Vanderwerff, J.D.: Convex Functions, Cambridge University Press (2010). 14. Br´ezis, H., Browder, F.E.: “Linear maximal monotone operators and singular nonlinear integral equations of Hammerstein type”, in Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic Press, pp. 31–42 (1978). 15. Bot¸, R.I., Csetnek, E.R.: “On extension results for n-cyclically monotone operators in reflexive Banach spaces”, J. Math. Anal. Appl., vol. 367, pp. 693-698 (2010). 16. Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators, Springer-Verlag (2008). 17. Burachik, R.S., Svaiter, B.F.: “Maximal monotone operators, convex functions and a special family of enlargements”, Set-Valued Anal., vol. 10, pp. 297–316 (2002). 18. Clarke, F.H.: Optimization and Nonsmooth Analysis, SIAM, Philadelphia (1990). 19. Cross, R.: Multivalued Linear Operators, Marcel Dekker, 1998. 20. Fitzpatrick, S.: “Representing monotone operators by convex functions”, in Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988), Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 20, Canberra, Australia, pp. 59–65 (1988). 21. Haraux, A.: Nonlinear Evolution Equations - Global Behavior of Solutions, Springer-Verlag Berlin Heidelberg (1981). 22. Penot, J.-P.: “The relevance of convex analysis for the study of monotonicity”, Nonlinear Anal., vol. 58, pp. 855–871 (2004). 23. Phelps, R.R.: Convex functions, Monotone Operators and Differentiability, 2nd Edition, Springer-Verlag (1993). 24. Phelps, R.R., Simons, S.: “Unbounded linear monotone operators on nonreflexive Banach spaces”, J. Convex Anal., vol. 5, pp. 303–328 (1998). 25. Rockafellar, R.T., Wets, R.J-B.: Variational Analysis, 2nd Printing, Springer-Verlag (2004). 26. Simons, S.: Minimax and Monotonicity, Springer-Verlag (1998). 27. Simons, S.: From Hahn-Banach to Monotonicity, Springer-Verlag (2008). 28. Simons, S., Z˘alinescu, C.: “A new proof for Rockafellar’s characterization of maximal monotone operators”, Proc. Amer. Math. Soc., vol 132, pp. 2969–2972 (2004). 29. Simons, S., Z˘alinescu, C.: “Fenchel duality, Fitzpatrick functions and maximal monotonicity”, J. Nonlinear Convex Anal., vol 6, pp. 1–22 (2005).

12

Liangjin Yao

30. Svaiter, B.F.: “Non-enlargeable operators and self-cancelling operators”, J. Convex Anal., vol. 17, pp. 309–320 (2010). 31. Voisei, M.D.: “The sum theorem for linear maximal monotone operators”, Math. Sci. Res. J., vol. 10, pp. 83–85 (2006). 32. Voisei, M.D., Z˘alinescu, C.: “Linear monotone subspaces of locally convex spaces”, SetValued Var. Anal., vol. 18, pp. 29–55 (2010). 33. Z˘alinescu, C.: Convex Analysis in General Vector Spaces, World Scientific Publishing (2002). 34. Zeidler, E.: Nonlinear Functional Analysis and its Application, Vol II/B Nonlinear Monotone Operators, Springer-Verlag, New York-Berlin-Heidelberg (1990).