arXiv:submit/0039715 [math.FA] 13 May 2010

The sum of a maximal monotone operator of type (FPV) and a maximal monotone operator with full domain is maximal monotone Liangjin Yao∗ May 12, 2010

Abstract The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we provide a new maximal monotonicity result for the sum of two maximal monotone operators A and B in this setting satisfying that A + Ndom B is of type (FPV) and dom A ∩ dom B ⊆ dom B. The proof relies on some results on the Fitzpatrick function.

2000 Mathematics Subject Classification: Primary 47H05; Secondary 49N15, 52A41, 90C25 Keywords: Constraint qualification, convex function, convex set, duality mapping, Fitzpatrick function, linear relation, maximal monotone operator, monotone operator, of type (FPV), subdifferential operator.

1

Introduction

Throughout this paper, we assume that X is a real Banach space with norm k · k, that X ∗ is the continuous dual of X, and that X and X ∗ are paired by h·, ·i. Let A : X ⇉ X ∗ be Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada. E-mail: [email protected]. ∗

1

∗ a set-valued operator (also known as multifunction) from X to X , i.e., for every x ∈ X, ∗ ∗ ∗ ∗ Ax ⊆ X , and let gra A = (x, x ) ∈ X × X | x ∈ Ax be the graph of A. Recall that A is monotone if

(1) ∀(x, x∗ ) ∈ gra A ∀(y, y ∗) ∈ gra A hx − y, x∗ − y ∗i ≥ 0,

and maximal monotone if A is monotone and A has no proper monotone extension (in the sense of graph inclusion). Let A : X ⇉ X ∗ be maximal monotone. We say A is of type (FPV) if for every open U ⊆ X such that U ∩ dom A 6= ∅ then x ∈ Uand (x, x∗ ) is monotonically related to gra A ∩ U × X ∗ ⇒ (x, x∗ ) ∈ gra A.

We say A is a linear relation if gra A is a linear subspace. Monotone operators have proven to be a key class of objects in modern Optimization and Analysis; see, e.g., the books [6, 7, 8, 10, 15, 16, 14, 22] and  the references therein. (We also adopt standard notation used in these books: dom A = x ∈ X | Ax 6= ∅ is the domain of A. Given a subset C of X, int C is the interior of C, and C is the closure of C. The indicator function of C, written as ιC , is defined at x ∈ X by ( 0, if x ∈ C; (2) ιC (x) := ∞, otherwise. We denote dist(x, C) by dist(x, C) := x ∈ X, the normal cone  inf c∈C kx − ck. For every operator at x is defined by NC (x) = x∗ ∈ X ∗ | suphC − x, x∗ i ≤ 0 , if x ∈ C; and NC (x) = ∅, if x ∈ / C. Given f : X → ]−∞, +∞], we set dom f = f −1 (R) and let f ∗ : X ∗ → [−∞, +∞] : x∗ 7→ supx∈X (hx, x∗ i − f (x)) Fenchel conjugate of f . If f is convex  ∗ be the and ∗ ∗ ∗ dom f 6= ∅, then ∂f : X ⇉ X : x 7→ x ∈ X | (∀y ∈ X) hy − x, x i + f (x) ≤ f (y) is the ∗ ∗ subdifferential operator of f . We denote PX :  X ×X ∗ → X by (x, x ) ∈ X ×X → x. Finally, the open unit ball in X is denoted by BX := x ∈ X | kxk < 1 , and N = {1, 2, 3, . . .}.)

Let A and B be maximal monotone operators from X to X ∗ . Clearly, the sum operator A+  ∗ ∗ ∗ ∗ ∗ B : X ⇉ X : x 7→ Ax + Bx = a + b | a ∈ Ax and b ∈ Bx is monotone. Rockafellar’s [13, Theorem 1] guarantees maximal monotonicity of A + B under the classical constraint qualification dom A ∩ int dom B 6= ∅ when X is reflexive. The most famous open problem concerns the behavior in nonreflexive Banach spaces. See Simons’ monograph [16] and [4, 5, 21] for a comprehensive account of the recent developments. Now we focus on the case that A and B satisfy that A + Ndom B is of type (FPV), and dom A ∩ dom B ⊆ dom B. We show that the sum theorem is true in this setting. We note in passing that in [18], Verona and Verona showed the sum theorem is true when A is the subdifferential operator of a proper lower semicontinuous convex function, and B is maximal monotone with full domain. In [2], it was recently shown that the sum theorem is true when A is a linear relation and B is the normal cone operator of a closed convex set. In [20], Voisei 2

showed the sum theorem is also true when A is type of (FPV) with convex domain, and B is the normal cone operator of a closed convex set. Theorem 3.4 generalizes above all and also can deduce Heisler’ result that the sum theorem is true when both operator have full domains. The remainder of this paper is organized as follows. In Section 2, we collect auxiliary results for future reference and for the reader’s convenience. The main result (Theorem 3.4) is proved in Section 3.

2

Auxiliary Results

Fact 2.1 (Rockafellar) (See [12, Theorem 3(b)], [16, Theorem 18.1], or [22, Theorem 2.8.7(iii)].) Let f, g : X → ]−∞, +∞] be proper convex functions. Assume that there exists a point x0 ∈ dom f ∩ dom g such that g is continuous at x0 . Then ∂(f + g) = ∂f + ∂g. Fact 2.2 (See [10, Theorem 2.28].) Let A : X ⇉ X ∗ be monotone with int dom A 6= ∅. Then A is locally bounded at x ∈ int dom A, i.e., there exist δ > 0 and K > 0 such that sup ky ∗k ≤ K,

∀y ∈ (x + δBX ) ∩ dom A.

y ∗ ∈Ay

Fact 2.3 (Fitzpatrick) (See [9, Corollary 3.9].) Let A : X ⇉ X ∗ be maximal monotone, and set
(3) FA : X × X ∗ → ]−∞, +∞] : (x, x∗ ) 7→ sup hx, a∗ i + ha, x∗ i − ha, a∗ i , (a,a∗ )∈gra A

which is the Fitzpatrick function associated with A. Then for every (x, x∗ ) ∈ X × X ∗ , the inequality hx, x∗ i ≤ FA (x, x∗ ) is true, and equality holds if and only if (x, x∗ ) ∈ gra A.

Fact 2.4 (See [19, Theorem 3.4 and Corollary 5.6], S or [16, Theorem 24.1(b)].) Let A, B : ∗ X ⇉ X be maximal monotone operators. Assume λ>0 λ [PX (dom FA ) − PX (dom FB )] is a closed subspace. If (4)

FA+B ≥ h·, ·i on X × X ∗ ,

then A + B is maximal monotone. Fact 2.5 (Simons) (See [16, Thereom 27.1 and Thereom 27.3].) Let A : X ⇉ X ∗ be maximal monotone with int dom A 6= ∅. Then int dom A = int [PX dom FA ], dom A = PX [dom FA ] and thus dom A is convex. 3

Now we cite some results on maximal monotone operators of type (FPV). Fact 2.6 (Simons) (See [16, Theorem 48.4(d)].) Let f : X → ]−∞, +∞] be a proper lower semicontinuous convex function. Then ∂f is of type (FPV). Fact 2.7 (Simons) (See [16, Theorem 46.1].) Let A : X ⇉ X ∗ be a maximal monotone linear relation. Then A is of type (FPV). Fact 2.8 (Simons and Verona-Verona) (See [16, Thereom 44.1] or [17].) Let A : X ⇉ X ∗ be a maximal monotone. Suppose that if C is a nonempty closed convex subset of X and dom A ∩ int C 6= ∅ then A + NC is maximal monotone. Then A is of type (FPV). Fact 2.9 (Voisei) (See [20].) Let A : X ⇉ X ∗ be maximal monotone of type (FPV) with convex domain, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C 6= ∅. Then A + NC is maximal monotone. Corollary 2.10 Let A : X ⇉ X ∗ be maximal monotone of type (FPV) with convex domain, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C 6= ∅. Then A + NC is of type (F P V ). Proof. By Fact 2.9, A + NC is maximal monotone. Let D be a nonempty closed convex subset of X, and suppose that dom(A + NC ) ∩ int D 6= ∅. Let x1 ∈ dom A ∩ int C and x2 ∈ dom(A + NC ) ∩ int D. Thus, there exists δ > 0 such that x1 + δBX ⊆ C and x2 + δBX ⊆ D. Then for small enough λ ∈ ]0, 1[, we have x2 + λ(x1 − x2 ) + 21 δBX ⊆ D. Clearly, x2 + λ(x1 − x2 ) + λδBX ⊆ C. Thus x2 + λ(x1 − x2 ) + λδ B ⊆ C ∩ D. Since dom A is 2 X convex, x2 + λ(x1 − x2 ) ∈ dom A and x2 + λ(x1 − x2 ) ∈ dom A ∩ int(C ∩ D) By Fact 2.1 , A + NC + ND = A + NC∩D . Then by Fact 2.9 (applied to A and C ∩ D), A + NC + ND = A + NC∩D is maximal monotone. Then by Fact 2.8, A + NC is of type (F P V ).  Corollary 2.11 Let A : X ⇉ X ∗ be a maximal monotone linear relation, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C 6= ∅. Then A + NC is of type (F P V ). Proof. Apply Fact 2.7 and Corollary 2.10.

3



Main Result

The following result plays a key role in Theorem 3.4 below and its first half proof follows that of [16, Theorem 44.2].

4

Proposition 3.1 Let A, B : X ⇉ X ∗ be maximal monotone with dom A ∩ int dom B 6= ∅. Assume that A + Ndom B is maximal monotone of type (FPV), and dom A ∩ dom B ⊆ dom B. Then PX [dom FA+B ] = dom A ∩ dom B. Proof. By [9, Theorem 3.4], dom A ∩ dom B = dom(A + B) ⊆ PX [dom FA+B ]. Then it suffices to show PX [dom FA+B ] ⊆ dom A ∩ dom B.

(5)

We can and do suppose that 0 ∈ dom A ∩ int dom B and (0, 0) ∈ gra B. Let z ∈ PX [dom FA+B ]. Assume that (6)

z∈ / dom A ∩ dom B

and thus α := dist(z, dom A ∩ dom B) > 0. Let y ∗ ∈ X ∗ such that ky ∗k = 1 and hz, y ∗ i ≥ 32 kzk.

(7) Let (8)

Un := [0, z] +

α B , 4n X

∀n ∈ N.

By Fact 2.5, since B is maximal monotone we have B = B + Ndom B , thus A + B = A + Ndom B + B.

(9) By assumptions,

dom A ∩ dom B ⊆ dom(A + Ndom B ) = dom A ∩ dom B ⊆ dom A ∩ dom B. Then (10)

dom A ∩ dom B = dom(A + Ndom B ).

/ gra(A + Ndom B ). By z ∈ Un and By (6) and (10), z ∈ / dom(A + Ndom B ) and thus (z, ny ∗ ) ∈ assumptions that A + Ndom B is of type (FPV), there exist (zn , zn∗ ) ∈ gra(A + Ndom B ) with zn ∈ Un such that (11)

hz − zn , zn∗ i > nhz − zn , y ∗i.

Since zn ∈ Un and zn ∈ dom A ∩ dom B by (10), and there exists λn ∈ [0, 1] such that (12)

kz − zn − λn zk = kzn − (1 − λn )zk < 41 α. 5

Then we have kz − zn k < λn kzk + 41 α. By the definition of α, α < λn kzk + 14 α. Thus, 3 α 4

(13)

< λn kzk.

By (12) and (7), hz − zn − λn z, y ∗ i ≥ −kzn − (1 − λn )zk > − 41 α.

(14) By (14), (7) and (13),

hz − zn , y ∗i > λn hz, y ∗ i − 14 α >

(15)

23 α 34

− 41 α = 41 α.

Then by (11), hz − zn , zn∗ i > 41 nα.

(16)

By (8) and (10), we have zn ∈ dom B and (17)

zn → βz,

where β ∈ [0, 1[ by (6) and (10). Then by (9) and (16), for every z ∗ ∈ X ∗ , we have FA+B (z, z ∗ ) = FA+Ndom B +B (z, z ∗ ) ≥

sup {n∈N,y ∗ ∈X ∗ }

(18)

≥

sup {n∈N,y ∗ ∈X ∗ }

We show that

[hzn , z ∗ i + hz − zn , zn∗ i + hz − zn , y ∗i − ιgra B (zn , y ∗)] 

 hzn , z ∗ i + 41 nα + hz − zn , y ∗i − ιgra B (zn , y ∗) . FA+B (z, z ∗ ) = ∞.

(19) We consider two cases. Case (1): β = 0.

By (17) and Fact 2.2 (applied to 0 ∈ int dom B), there exist N ∈ N and K > 0 such that (20)

sup ky ∗k ≤ K, y ∗ ∈Bz

n

Then by (18), we have (19) holds. Case (2): β 6= 0. 6

∀n ≥ N.

We first show lim sup sup [hz − zn , y ∗i − ιgra B (zn , y ∗)] ≥ 0.

(21)

y ∗ ∈X ∗

n→∞

Take vn∗ ∈ Bzn . We consider two subcases. Subcase 1: (vn∗ )n∈N is bounded. w*

We can suppose that vn∗ ⇁ v ∗ , where v ∗ ∈ X ∗ . Since gra B is (norm – weak star) closed, by (17), (βz, v ∗ ) ∈ gra B. By (0, 0) ∈ gra B, we have hβz, v ∗ i ≥ 0. Then hz, v ∗ i ≥ 0. We have hz − zn , vn∗ i → hz − βz, v ∗ i = (1 − β)hz, v ∗ i ≥ 0. Hence (21) holds. Subcase 2: (vn∗ )n∈N is unbounded. We can suppose that kvn∗ k → ∞. By 0 ∈ int dom B and Fact 2.2, there exist δ > 0 and M > 0 such that sup ky ∗ k ≤ M,

(22)

∀y ∈ δBX .

y ∗ ∈By

Then we have hzn − y, vn∗ − y ∗i ≥ 0, ∀y ∈ δBX , y ∗ ∈ By, n ∈ N ⇒ hzn , vn∗ i − hy, vn∗ i + hzn − y, −y ∗i ≥ 0, ∀y ∈ δBX , y ∗ ∈ By, n ∈ N ⇒ hzn , vn∗ i − hy, vn∗ i ≥ hzn − y, y ∗i, ∀y ∈ δBX , y ∗ ∈ By, n ∈ N ⇒ hzn , vn∗ i − hy, vn∗ i ≥ −(kzn k + δ)M, ∀y ∈ δBX , n ∈ N ⇒ hzn , vn∗ i ≥ hy, vn∗ i − (kzn k + δ)M, ∀y ∈ δBX , n ∈ N ⇒ hzn , vn∗ i ≥ δkvn∗ k − (kzn k + δ)M, ∀n ∈ N (23)

⇒ hzn , kvvn∗ k i ≥ δ − ∗

n

(kzn k+δ)M , ∗k kvn

∀n ∈ N.

For convenience, we suppose that ∗ w* ∗ vn ∗k ⇁ w kvn

(24)

∈ X ∗.

Then by (17) and (23), hβz, w ∗i ≥ δ.

(25) On the other hand, assume that

lim sup sup [hz − zn , y ∗ i − ιgra B (zn , y ∗ )] < −ε, n→∞

y ∗ ∈X ∗

7

where ε > 0. Then when n is large enough we have hz − zn , vn∗ i < − 2ε . Then hz − zn , kvvn∗ k i < − 2kvε∗ k . ∗

n

n

Then by (17) and (24), hz − βz, w ∗ i ≤ 0. Then hz, w ∗ i ≤ 0 which contradicts (25). Hence lim sup sup [hz − zn , y ∗ i − ιgra B (zn , y ∗ )] ≥ 0 n→∞

y ∗ ∈X ∗

Combine above two subcases, (21) holds. Then by (18),

(26)

FA+B (z, z ∗ ) = ∞.

Combine above two cases, (19) holds. This contradicts that z ∈ PX [dom FA+B ]. Hence PX [dom FA+B ] ⊆ dom A ∩ dom B and thus (5) holds. Thus PX [dom FA+B ] = dom A ∩ dom B.  Corollary 3.2 Let A : X ⇉ X ∗ be maximal monotone of type (FPV) with convex domain, and B : X ⇉ X ∗ be a maximal monotone with dom A ∩ int dom B 6= ∅. Assume dom A ∩ dom B ⊆ dom B. Then PX [dom FA+B ] = dom A ∩ dom B. Proof. Apply Corollary 2.10 and Proposition 3.1.



Corollary 3.3 Let A : X ⇉ X ∗ be a maximal monotone linear relation, and B : X ⇉ X ∗ be a maximal monotone with dom A ∩ int dom B 6= ∅. Assume dom A ∩ dom B ⊆ dom B. Then PX [dom FA+B ] = dom A ∩ dom B. Proof. Apply Corollary 2.11 and Proposition 3.1 or Fact 2.7 and Corollary 3.2.



Here is our main result. Theorem 3.4 Let A, B : X ⇉ X ∗ be maximal monotone with dom A ∩ int dom B 6= ∅. Assume that A + Ndom B is maximal monotone of type (FPV), and dom A ∩ dom B ⊆ dom B. Then A + B is maximal monotone. 8

Proof. We can and do suppose that 0 ∈ dom A ∩ int dom B and (0, 0) ∈ gra A ∩ gra B. By Fact 2.3, dom A ⊆ PX (dom FA ) and dom B ⊆ PX (dom FB ). Hence, [
(27) λ PX (dom FA ) − PX (dom FB ) = X. λ>0

Thus, by Fact 2.4, it suffices to show FA+B (z, z ∗ ) ≥ hz, z ∗ i,

(28)

∀(z, z ∗ ) ∈ X × X ∗ .

Let (z, z ∗ ) ∈ X × X ∗ , we have (29)

FA+B (z, z ∗ ) = sup [hx, z ∗ i + hz, x∗ i − hx, x∗ i + hz − x, y ∗ i − ιgra A (x, x∗ ) − ιgra B (x, y ∗ )] (x,x∗ ,y ∗ )

Assume that FA+B (z, z ∗ ) < hz, z ∗ i.

(30) Then (z, z ∗ ) ∈ dom FA+B and (31)

z ∈ dom A ∩ dom B = PX [dom FA+B ]

by Proposition 3.1. We first show (32)

FA+B (λz, λz ∗ ) ≥ λ2 hz, z ∗ i,

∀λ ∈ ]0, 1[ .

Let λ ∈ ]0, 1[. By 0 ∈ int dom B and Fact 2.5, λz ∈ int [PX dom FB ] = int [PX dom FB ] = int dom B. We consider two cases. Case (1): λz ∈ dom A. Clearly, (32) holds by putting x = λz in (29). Case (2): λz ∈ / dom A. By 0 ∈ dom A ∩ dom B and (31), λz ∈ dom A ∩ dom B. Set (33)

Un := λz + n1 BX ,

∀n ∈ N.

Since (λz, λz ∗ ) ∈ / gra(A + Ndom B ) and λz ∈ Un , there exist (bn , b∗n ) ∈ gra(A + Ndom B ) with bn ∈ Un such that (34)

hλz, b∗n i + hbn , λz ∗ i − hbn , b∗n i > λ2 hz, z ∗ i. 9

Since λz ∈ int dom B and bn → λz, by Fact 2.2, we can suppose that there exist N ∈ N and M > 0 such that (35)

sup kv ∗ k ≤ M,

bn ∈ int dom B, Bbn 6= ∅ and

∀n ≥ N.

v∗ ∈Bbn

Then Ndom B (bn ) = {0} and thus (bn , b∗n ) ∈ gra A for every n ≥ N. Thus by (29), (34) and (35), FA+B (λz, λz ∗ ) ≥ sup [hbn , λz ∗ i + hλz, b∗n i − hbn , b∗n i + hλz − bn , v ∗ i] ,

∀n ≥ N

{v∗ ∈Bbn }

≥

sup {v∗ ∈Bbn }

(36)

 2  λ hz, z ∗ i + hλz − bn , v ∗ i ,

  ≥ sup λ2 hz, z ∗ i − Mkλz − bn k , ≥ λ2 hz, z ∗ i.

∀n ≥ N

∀n ≥ N

(by (34))

(by (35))

Hence FA+B (λz, λz ∗ ) ≥ λ2 hz, z ∗ i. Combine above two cases, we have (32) holds. Since FA+B (0, 0) = 0 by (0, 0) ∈ gra A ∩ gra B. We define f : [0, 1] → R by t → FA+B (tz, tz ∗ ). Then f is continuous on [0, 1] by [22, Proposition 2.1.6]. By (32), we have (37)

FA+B (z, z ∗ ) = lim− FA+B (λz, λz ∗ ) ≥ lim− hλz, λz ∗ i = hz, z ∗ i. λ→1

λ→1

This contradicts (30). Hence (38)

FA+B (z, z ∗ ) ≥ hz, z ∗ i.

Hence (28) holds and hence A + B is maximal monotone.



Theorem 3.4 can recover many important known results: Corollary 3.5 Let f : X → ]−∞, +∞] be a proper lower semicontinuous convex function, and B : X ⇉ X ∗ be maximal monotone with dom f ∩ int dom B 6= ∅. Assume that dom ∂f ∩ dom B ⊆ dom B. Then ∂f + B is maximal monotone. Proof. By Fact 2.1, ∂f + Ndom B = ∂(f + ιdom B ). Then by Fact 2.6, ∂f + Ndom B is type of (FPV). Apply Theorem 3.4.  Corollary 3.6 Let A : X ⇉ X ∗ be maximal monotone of type (FPV), and B : X ⇉ X ∗ be maximal monotone with full domain. Then A + B is maximal monotone. 10

Proof. Since A + Ndom B = A + NX = A and thus A + Ndom B is maximal monotone of type (FPV), Theorem 3.4 implies the result as desire.  Corollary 3.7 (Verona-Verona) (See [18, Corollary 2.9(a)] or [16, Theorem 53.1].) Let f : X → ]−∞, +∞] be a proper lower semicontinuous convex function, and B : X ⇉ X ∗ be maximal monotone with full domain. Then ∂f + B is maximal monotone. Proof. Apply Corollary 3.5, or Fact 2.6 and Corollary 3.6.



Corollary 3.8 (Heisler) (See [15, Theorem 37.4].) Let A, B : X ⇉ X ∗ be maximal monotone operators with full domains. Then A + B is maximal monotone. Proof. Let C be a nonempty closed convex subset of X. By Corollary 3.7, NC +A is maximal monotone and thus A is of type (FPV) by Fact 2.8. Then Corollary 3.6 implies the result as desire.  Corollary 3.9 Let A : X ⇉ X ∗ be maximal monotone of type (FPV) with convex domain, and B : X ⇉ X ∗ be a maximal monotone with dom A ∩ int dom B 6= ∅. Assume that dom A ∩ dom B ⊆ dom B. Then A + B is maximal monotone. Proof. Apply Corollary 2.10 and Theorem 3.4.



Corollary 3.10 (Voisei) Let A : X ⇉ X ∗ be maximal monotone of type (FPV) with convex domain, let C be a nonempty closed convex subset of X, and suppose that dom A∩int C 6= ∅. Then A + NC is maximal monotone. Proof. Apply Corollary 3.9.



Corollary 3.11 Let A : X ⇉ X ∗ be a maximal monotone linear relation, and B : X ⇉ X ∗ be a maximal monotone with dom A∩int dom B 6= ∅. Assume that dom A∩dom B ⊆ dom B. Then A + B is maximal monotone. Proof. Apply Fact 2.7 and Corollary 3.9.



Corollary 3.12 (See [2].) Let A : X ⇉ X ∗ be a maximal monotone linear relation, let C be a nonempty closed convex subset of X, and suppose that dom A ∩ int C 6= ∅. Then A + NC is maximal monotone. Proof. Apply Corollary 3.11.



Corollary 3.13 Let A : X ⇉ X ∗ be maximal monotone linear relation, and B : X ⇉ X ∗ be a maximal monotone with full domain. Then A + B is maximal monotone. 11

Proof. Apply Corollary 3.11.



Remark 3.14 Theorem 3.4 generalizes Heisler’s result, [18, Corollary 2.9(a)] or [16, Theorem 53.1], [2, Theorem 3.1] and Fact 2.9. Example 3.15 Suppose that X = L1 [0, 1], let  D = x ∈ X | x is absolutely continuous, x(0) = 0, x′ ∈ X ∗ , and set

A : X ⇉ X ∗ : x 7→

(

{x′ }, if x ∈ D; ∅, otherwise.

By Phelps and Simons’ [11, Example 4.3], A is an at most single-valued maximal monotone linear relation with proper dense domain, and A is neither symmetric nor skew. Now set J to be the duality mapping, i.e, J = ∂ 12 k · k2 . Then Corollary 3.13 implies that A + J is maximal monotone. To the best of our knowledge, the maximal monotonicity of A + J in Example 3.15 cannot be deduced from any known previous results. Remark 3.16 In [3], it was shown that the sum theorem is true when A is a linear relation and B is the subdifferential operator of a proper lower semicontinuous sublinear function. To the best of our knowledge, Theorem 3.4 could not deduce this result, since the domain of the subdifferential operator of a proper lower semicontinuous sublinear function does not have to be closed. For example, see [1, Example 5.4]: Set C := {(x, y) ∈ R2 | 0 < 1/x ≤ y} and f := ι∗C . Then f is not subdifferential at any point which belongs to the boundary of its domain, except of the origin.

Acknowledgment The author thanks Dr. Heinz Bauschke and Dr. Xianfu Wang for their many valuable discussions and comments.

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