arXiv:1010.4346v1 [math.FA] 21 Oct 2010

The sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function is maximally monotone Liangjin Yao∗ October 19, 2010

Abstract The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A + ∂f provided that A is a maximally monotone linear relation, and f is a proper lower semicontinuous convex function satisfying dom A ∩ int dom ∂f #= ∅. Moreover, A + ∂f is of type (FPV). The maximal monotonicity of A + ∂f when int dom A ∩ dom ∂f #= ∅ follows from a result by Verona and Verona, which the present work complements.

2010 Mathematics Subject Classification: Primary 47A06, 47H05; Secondary 47B65, 47N10, 90C25 Keywords: Constraint qualification, convex function, convex set, Fitzpatrick function, linear relation, maximally monotone operator, monotone operator, monotone operator of type (FPV), multifunction, normal cone operator, Rockafellar’s sum theorem, set-valued operator, subdifferential operator. Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada. E-mail: [email protected]. ∗

1

1

Introduction

Throughout this paper, we assume that X is a real Banach space with norm $ · $, that X ∗ is the continuous dual of X, and that X and X ∗ are paired by %·, ·&. Let A : X ⇒ X ∗ be 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 − y, x∗ − y ∗& ≥ 0,

∀(x, x∗ ) ∈ gra A ∀(y, y ∗) ∈ gra A,

and maximally monotone if A is monotone and A has no proper monotone extension (in the sense of graph inclusion). Let A : X ⇒ X ∗ be monotone and (x, x∗ ) ∈ X × X ∗ . We say (x, x∗ ) is monotonically related to gra A if %x − y, x∗ − y ∗ & ≥ 0,

∀(y, y ∗) ∈ gra A.

Let A : X ⇒ X ∗ be maximally monotone. We say A is of type (FPV) if for every open convex set U ⊆ X such that U ∩ dom A #= ∅, the implication x ∈ U and (x, x∗ ) is monotonically related to gra A ∩ (U × X ∗ ) ⇒ (x, x∗ ) ∈ gra A holds. 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 [12, 13, 14, 18, 25, 26, 23, 38, 39]!and the references " therein. We adopt standard notation used in these books: dom A = x ∈ X | Ax #= ∅ is the domain of A. Given a subset C of X, int C is the interior of C, bdry C is the boundary, aff C is the affine hull, and C is the norm closure of C. We set C ⊥ = {x∗ ∈ X ∗ | (∀c ∈ C) %x∗ , c& = 0} and S ⊥ = {x∗∗ ∈ X ∗∗ | (∀s ∈ S) %x∗∗ , s& = 0} for a set S ⊆ X ∗ . The indicator function of C, written as ιC , is defined at x ∈ X by # 0, if x ∈ C; (2) ιC (x) = ∞, otherwise. If D ⊆ X, we set C − D = {x − y | x ∈ C,!y ∈ D}. For every x ∈ X, the "normal cone operator of C at x is defined by NC (x) = x∗ ∈ X ∗ | supc∈C %c − x, x∗ & ≤ 0 , if x ∈ C; and NC (x) = ∅, if x ∈ / C. For x, y ∈ X, we set [x, y] = {tx + (1 − t)y | 0 ≤ t ≤ 1}. Given f : X → ]−∞, +∞], we set dom f = f −1 (R) and f ∗ : X ∗ → [−∞, +∞] : x∗ 1→ supx∈X (%x, x∗ & − f (x)) ! is the Fenchel conjugate of f . If f is convex"and dom f #= ∅, then ∂f : X ⇒ X ∗ : x 1→ x∗ ∈ X ∗ | (∀y ∈ X) %y − x, x∗ & + f (x) ≤ f (y) is the subdifferential ∗ ∗ operator of f . We also ! set PX : X × X" → X : (x, x ) 1→ x. Finally, the open unit ball in X is denoted by UX = x ∈ X | $x$ < 1 , and N = {1, 2, 3, . . .}. 2

Let A and B be maximally monotone from X to X ∗ . Clearly, the sum operator ! ∗operators " ∗ ∗ ∗ A + B : X ⇒ X : x 1→ Ax + Bx = a + b | a ∈ Ax and b∗ ∈ Bx is monotone. Rockafellar’s [21, Theorem 1] guarantees maximal monotonicity of A + B under Rockafellar’s constraint qualification dom A ∩ int dom B #= ∅ when X is reflexive — this result is often referred to as “the sum theorem”. The most famous open problem concerns the maximal monotonicity of A + B in nonreflexive Banach spaces when Rockafellar’s constraint qualification holds. See Simons’ monograph [26] and [9, 10, 11, 34, 37] for a comprehensive account of some recent developments. Now we focus on the case when A is a maximally monotone linear relation, and f is a proper lower semicontinuous convex function such that dom A ∩ int dom ∂f #= ∅. We show that A + ∂f is maximally monotone. Linear relations have recently become a center of attention in Monotone Operator Theory; see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 19, 27, 28, 31, 32, 35, 36] and Cross’ book [15] for general background on linear relations. 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 3.1) is proved in Section 3.

2

Auxiliary Results

Fact 2.1 (Rockafellar) (See [20, Theorem 3(b)], [26, Theorem 18.1], or [38, 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 (Rockafellar) (See [22, Theorem A], [38, Theorem 3.2.8], [26, Theorem 18.7] or [17, Theorem 2.1]) Let f : X → ]−∞, +∞] be a proper lower semicontinuous convex function. Then ∂f is maximally monotone. Fact 2.3 (See [18, Theorem 2.28].) Let A : X ⇒ X ∗ be monotone with int dom A #= ∅. Then A is locally bounded at x ∈ int dom A, i.e., there exist δ > 0 and K > 0 such that sup $y ∗ $ ≤ K,

∀y ∈ (x + δUX ) ∩ dom A.

y ∗ ∈Ay

Fact 2.4 (Fitzpatrick) (See [16, Corollary 3.9].) Let A : X ⇒ X ∗ be maximally monotone, and set $ % (3) FA : X × X ∗ → ]−∞, +∞] : (x, x∗ ) 1→ sup %x, a∗ & + %a, x∗ & − %a, a∗ & , (a,a∗ )∈gra A

the Fitzpatrick function associated with A. Then for every (x, x∗ ) ∈ X × X ∗ , the inequality %x, x∗ & ≤ FA (x, x∗ ) is true, and the equality holds if and only if (x, x∗ ) ∈ gra A. 3

Fact 2.5 (See [33, Theorem 3.4 and Corollary 5.6], & or [26, Theorem 24.1(b)].) Let A, B : X ⇒ X ∗ be maximally monotone operators. Assume λ>0 λ [PX (dom FA ) − PX (dom FB )] is a closed subspace. If (4)

FA+B ≥ %·, ·& on X × X ∗ ,

then A + B is maximally monotone. Fact 2.6 (Simons) (See [26, Theorem 48.6(a)].) Let f : X → ]−∞, +∞] be proper, lower semicontinuous, and convex. Let (x, x∗ ) ∈ X × X ∗ and α > 0 be such that (x, x∗ ) ∈ / gra ∂f . ∗ ∗ ∗ Then for every ε > 0, there exists (y, y ) ∈ gra ∂f with y #= x and y #= x such that ' ' ' ' $x − y$ ' ' (5) ' $x∗ − y ∗$ − α' < ε and (6)

' ' ' %x − y, x∗ − y ∗& ' ' ' < ε. + 1 ' $x − y$ · $x∗ − y ∗$ '

Fact 2.7 (Simons) (See [26, Theorem 46.1].) Let A : X ⇒ X ∗ be a maximally monotone linear relation. Then A is of type (FPV). Fact 2.8 (See [18, Proposition 3.3 and Proposition 1.11].) Let f : X → ]−∞, +∞] be a lower semicontinuous convex and int dom f #= ∅. Then f is continuous on int dom f and ∂f (x) #= ∅ for every x ∈ int dom f . Fact 2.9 (See [6, Lemma 2.9].) Let A : X ⇒ X ∗ be a maximally monotone linear relation, and let z ∈ X ∩ (A0)⊥ . Then z ∈ dom A. Fact 2.10 (See [6, Lemma 2.10].) Let A : X ⇒ X ∗ be a monotone linear relation, and let f : X → ]−∞, +∞] be a proper lower semicontinuous convex function. Suppose that dom A ∩ int dom ∂f #= ∅, (z, z ∗ ) ∈ X × X ∗ is monotonically related to gra(A + ∂f ), and that z ∈ dom A. Then z ∈ dom ∂f . Fact 2.11 (Simons and Verona-Verona) (See [26, Thereom 44.1] or [29].) Let A : X ⇒ X ∗ be a maximally monotone. Suppose that for every closed convex subset C of X with dom A ∩ int C #= ∅, the operator A + NC is maximally monotone. Then A is of type (FPV). Fact 2.12 (See [5, lemma 2.5].) Let C be a nonempty closed convex subset of X such that int C #= ∅. Let c0 ∈ int C and suppose that z ∈ X " C. Then there exists λ ∈ ]0, 1[ such that λc0 + (1 − λ)z ∈ bdry C.

4

3

Main Result

Theorem 3.1 Let A : X ⇒ X ∗ be a maximally monotone linear relation, and let f : X ⇒ ]−∞, +∞] be a proper lower semicontinuous convex function with dom A ∩ int dom ∂f #= ∅. Then A + ∂f is maximally monotone. Proof. After translating the graphs if necessary, we can and do assume that 0 ∈ dom A ∩ int dom ∂f and that (0, 0) ∈ gra A ∩ gra ∂f . By Fact 2.4 and Fact 2.2, dom A ⊆ PX (dom FA ) and dom ∂f ⊆ PX (dom F∂f ). Hence, ( $ % (7) λ PX (dom FA ) − PX (dom F∂f ) = X. λ>0

Thus, by Fact 2.2 and Fact 2.5, it suffices to show that FA+∂f (z, z ∗ ) ≥ %z, z ∗ &,

(8)

∀(z, z ∗ ) ∈ X × X ∗ .

Take (z, z ∗ ) ∈ X × X ∗ . Then FA+∂f (z, z ∗ ) = sup [%x, z ∗ & + %z, x∗ & − %x, x∗ & + %z − x, y ∗ & − ιgra A (x, x∗ ) − ιgra ∂f (x, y ∗ )] .

(9)

{x,x∗ ,y ∗ }

Assume to the contrary that (10)

FA+∂f (z, z ∗ ) + λ < %z, z ∗ &,

where λ > 0. Thus by (10), (11)

(z, z ∗ ) is monotonically related to gra(A + ∂f ).

We claim that (12)

z∈ / dom A.

Indeed, if z ∈ dom A, apply (11) and Fact 2.10 to get z ∈ dom ∂f . Thus z ∈ dom A∩dom ∂f and hence FA+∂f (z, z ∗ ) ≥ %z, z ∗ & which contradicts (10). This verifies (12). By (10) and the assumption that (0, 0) ∈ gra A ∩ gra ∂f , we have sup [%0, z ∗ & + %z, A0& − %0, A0& + %z, ∂f (0)&] =

sup a∗ ∈A0,b∗ ∈∂f (0)

5

[%z, a∗ & + %z, b∗ &] < %z, z ∗ &.

Thus, because A0 is a linear subspace, z ∈ X ∩ (A0)⊥ .

(13) Then, by Fact 2.9, we have (14)

z ∈ dom A.

Combine (12) and (14), (15)

z ∈ dom A\dom A.

Set Un = z + n1 UX ,

(16)

∀n ∈ N.

By (15), (z, z ∗ ) ∈ / gra A and Un ∩ dom A #= ∅. Since z ∈ Un and A is type of (FPV) by Fact 2.7, there exist (an , a∗n ) ∈ gra A with an ∈ Un , n ∈ N such that (17)

%z, a∗n & + %an , z ∗ & − %an , a∗n & > %z, z ∗ &.

Then we have (18)

an → z.

Now we claim that (19)

z ∈ dom ∂f .

Suppose to the contrary that z #∈ dom ∂f . By the Brøndsted-Rockafellar Theorem (see [18, Theorem 3.17] or [38, Theorem 3.1.2]), dom ∂f = dom f . By 0 ∈ int dom ∂f ⊆ int dom f ⊆ int dom f , then by Fact 2.12, there exists δ ∈ ]0, 1[ such that (20)

δz ∈ bdry dom f .

Set gn : X → ]−∞, +∞] by (21)

gn = f + ι[0,an ] ,

n∈N

/ gra ∂gn . Then by Fact 2.6, Since z ∈ / dom f , z #∈ dom f ∩ [0, an ] = dom gn . Thus (z, z ∗ ) ∈ there exist βn ∈ [0, 1] and x∗n ∈ ∂gn (βn an ) with x∗n #= z ∗ and βn an #= z such that (22) (23)

$z − βn an $ ≥n $z ∗ − x∗n $ %z − βn an , z ∗ − x∗n & < − 34 . $z − βn an $ · $z ∗ − x∗n $ 6

By (18), $z − βn an $ is bounded. Then by (22), we have (24)

x∗n → z ∗ .

Since 0 ∈ int dom f , f is continuous at 0 by Fact 2.8. Then by 0 ∈ dom f ∩ dom ι[0,an ] and Fact 2.1, we have that there exist wn∗ ∈ ∂f (βn an ) and vn∗ ∈ ∂ι[0,an ] (βn an ) such that x∗n = wn∗ + vn∗ . Then by (24), (25)

wn∗ + vn∗ → z ∗ .

Since βn ∈ [0, 1], there exists a convergent subsequence of (βn )n∈N , which, for convenience, we still denote by (βn )n∈N . Then βn → β, where β ∈ [0, 1]. Then by (18), (26)

βn an → βz.

We claim that (27)

β ≤ δ < 1.

In fact, suppose to the contrary that β > δ. By (26), βz ∈ dom f . Then by 0 ∈ int dom f and [38, Theorem 1.1.2(ii)], δz = βδ βz ∈ int dom f , which contradicts (20). We can and do suppose that βn < 1 for every n ∈ N. Then by vn∗ ∈ ∂ι[0,an ] (βn an ), we have (28)

%vn∗ , an − βn an & ≤ 0.

Dividing by (1 − βn ) on both sides of the above inequality, we have (29)

%vn∗ , an & ≤ 0.

Since (0, 0) ∈ gra A, %an , a∗n & ≥ 0, ∀n ∈ N. Then by (17), we have (30) %z, βn a∗n & + %βn an , z ∗ & − βn2 %an , a∗n & ≥ %βn z, a∗n & + %βn an , z ∗ & − βn %an , a∗n & ≥ βn %z, z ∗ &. Then by (30), (31)

%z − βn an , βn a∗n & ≥ %βn z − βn an , z ∗ &.

Since gra A is a linear subspace and (an , a∗n ) ∈ gra A, (βn an , βn a∗n ) ∈ gra A. By (10), we have λ <%z − βn an , z ∗ − wn∗ − βn a∗n & = %z − βn an , z ∗ − wn∗ & + %z − βn an , −βn a∗n & < − 43 $z − βn an $ · $z ∗ − wn∗ − vn∗ $ + %z − βn an , vn∗ & + %z − βn an , −βn a∗n & (by (23)) ≤ − 43 $z − βn an $ · $z ∗ − wn∗ − vn∗ $ + %z − βn an , vn∗ & − %βn z − βn an , z ∗ & (by (31)). 7

Then λ < %z − βn an , vn∗ & − %βn z − βn an , z ∗ &.

(32)

Now we consider two cases: Case 1 : (wn∗ )n∈N is bounded. By (25), (vn∗ )n∈N is bounded. By the Banach-Alaoglu Theorem (see [24, Theorem 3.15]), there exist a weak* convergent subnet (vγ∗ )γ∈Γ of (vn∗ )n∈N , say w*

∗ vγ∗ ( v∞ ∈ X ∗.

(33)

Combine (18), (26) and (33), and pass the limit along the subnet of (32) to get that

∗ λ ≤ %z − βz, v∞ &.

(34)

By (27), divide by (1 − β) on both sides of (34) to get ∗ %z, v∞ &≥

(35)

λ 1−β

> 0.

On the other hand, by (18) and (33), passing the limit along the subnet of (29) to get that ∗ %v∞ , z& ≤ 0,

(36) which contradicts (35). Case 2 : (wn∗ )n∈N is unbounded.

Since (wn∗ )n∈N is unbounded and after passing to a subsequence if necessary, we assume that $wn∗ $ #= 0, ∀n ∈ N and that $wn∗ $ → +∞. By the Banach-Alaoglu Theorem again, there exist a weak* convergent subnet (wν∗ )ν∈I of (wn∗ )n∈N , say (37)

wν∗ w* ∗ ( w∞ ∈ X ∗ . $wν∗ $

By 0 ∈ int dom ∂f and Fact 2.3, there exist ρ > 0 and M > 0 such that (38)

∂f (y) #= ∅ and

sup $y ∗$ ≤ M, y ∗ ∈∂f (y)

8

∀y ∈ ρUX .

Then by wn∗ ∈ ∂f (βn an ), we have

(39)

%βn an − y, wn∗ − y ∗ & ≥ 0, ∀y ∈ ρUX , y ∗ ∈ ∂f (y), n ∈ N ⇒ %βn an , wn∗ & − %y, wn∗ & + %βn an − y, −y ∗& ≥ 0, ∀y ∈ ρUX , y ∗ ∈ ∂f (y), n ∈ N ⇒ %βn an , wn∗ & − %y, wn∗ & ≥ %βn an − y, y ∗&, ∀y ∈ ρUX , y ∗ ∈ ∂f (y), n ∈ N ⇒ %βn an , wn∗ & − %y, wn∗ & ≥ −($βn an $ + ρ)M, ∀y ∈ ρUX , n ∈ N (by (38)) ⇒ %βn an , wn∗ & ≥ %y, wn∗ & − ($βn an $ + ρ)M, ∀y ∈ ρUX , n ∈ N ⇒ %βn an , wn∗ & ≥ ρ$wn∗ $ − ($βn an $ + ρ)M, ∀n ∈ N ∗ ($βn an $ + ρ)M wn ⇒ %βn an , &w , ∀n ∈ N. ∗&& ≥ ρ − n $wn∗ $

Combining (26) and (37), taking the limit in (39) along the subnet, we obtain ∗ %βz, w∞ & ≥ ρ.

(40)

Then we have β #= 0 and thus β > 0. Then by (40), %z, w0∗ & ≥

(41) By (25) and

z∗ ∗& &wn

(42)

ρ β

> 0.

→ 0, we have wn∗ vn∗ + → 0. $wn∗ $ $wn∗ $

By(37), taking the weak∗ limit in (42) along the subnet, we obtain (43)

vν∗ w* ∗ ( −w∞ . $wν∗ $

Dividing by $wn∗ $ on the both sides of (32), we get that (44)

λ vn∗ %βn z − βn an , z ∗ & < %z − β a , & − . n n $wn∗ $ $wn∗ $ $wn∗ $

Combining (26), (18) and (43), taking the limit in (44) along the subnet, we obtain (45)

∗ %z − βz, −w∞ & ≥ 0.

By (27) and (45), (46)

∗ %z, −w∞ & ≥ 0,

9

which contradicts (41). Altogether z ∈ dom ∂f = dom f . Next, we show that FA+∂f (tz, tz ∗ ) ≥ t2 %z, z ∗ &,

(47)

∀t ∈ ]0, 1[ .

Let t ∈ ]0, 1[. By 0 ∈ int dom f and [38, Theorem 1.1.2(ii)], we have (48)

tz ∈ int dom f.

By Fact 2.8, (49)

tz ∈ int dom ∂f.

Set Hn = tz + n1 UX ,

(50)

∀n ∈ N.

Since dom A is a linear subspace, tz ∈ dom A\dom A by (15). Then Hn ∩ dom A #= ∅. Since (tz, tz ∗ ) ∈ / gra A and tz ∈ Hn , A is of type (FPV) by Fact 2.7, there exists (bn , b∗n ) ∈ gra A such that bn ∈ Hn and (51)

%tz, b∗n & + %bn , tz ∗ & − %bn , b∗n & > t2 %z, z ∗ &,

∀n ∈ N.

Since tz ∈ int dom ∂f and bn → tz, by Fact 2.3, there exist N ∈ N and K > 0 such that (52)

bn ∈ int dom ∂f

and

sup

$v ∗ $ ≤ K,

∀n ≥ N.

v∗ ∈∂f (bn )

Hence FA+∂f (tz, tz ∗ ) ≥ sup [%bn , tz ∗ & + %tz, b∗n & − %bn , b∗n & + %tz − bn , c∗ &] , ∀n ≥ N {c∗ ∈∂f (bn )} )2 * ≥ sup t %z, z ∗ & + %tz − bn , c∗ & , ∀n ≥ N (by (51)) {c∗ ∈∂f (bn )}

(53)

) * ≥ sup t2 %z, z ∗ & − K$tz − bn $ , ≥ t2 %z, z ∗ & (by bn → tz).

∀n ≥ N

(by (52))

Hence FA+∂f (tz, tz ∗ ) ≥ t2 %z, z ∗ &. We have verified that (47) holds. Since (0, 0) ∈ gra A ∩ gra ∂f , we obtain (∀(d, d∗ ) ∈ gra(A + ∂f )) %d, d∗& ≥ 0. Thus, FA+∂f (0, 0) = 0. Now define j : [0, 1] → R : t → FA+∂f (tz, tz ∗ ). 10

Then j is continuous on [0, 1] by (10) and [38, Proposition 2.1.6]. From (47), we obtain FA+∂f (z, z ∗ ) = lim− FA+∂f (tz, tz ∗ ) ≥ lim− %tz, tz ∗ & = %z, z ∗ &,

(54)

t→1

t→1

which contradicts (10). Hence FA+∂f (z, z ∗ ) ≥ %z, z ∗ &.

(55)

Therefore, (8) holds, and A + ∂f is maximally monotone.

"

Remark 3.2 In Theorem 3.1, when int dom A ∩ dom ∂f #= ∅, we have dom A = X since dom A is a linear subspace. Therefore, we can verify the maximal monotonicity of A + ∂f by the Verona-Verona result (see [30, Corollary 2.9(a)], [26, Theorem 53.1] or [37, Corollary 3.7]). Corollary 3.3 Let A : X ⇒ X ∗ be a maximally monotone linear relation, and f : X → ]−∞, +∞] be a proper lower semicontinuous convex function with dom A ∩ int dom ∂f #= ∅. Then A + ∂f is of type (F P V ). Proof. By Theorem 3.1, A + ∂f is maximally monotone. Let C be a nonempty closed convex subset of X, and suppose that dom(A + ∂f ) ∩ int C #= ∅. Let x1 ∈ dom A ∩ int dom ∂f and x2 ∈ dom(A + ∂f ) ∩ int C. Thus, there exists δ > 0 such that x1 + δUX ⊆ dom f and x2 +δUX ⊆ C. Then for small enough λ ∈ ]0, 1[, we have x2 +λ(x1 −x2 )+ 12 δUX ⊆ C. Clearly, x2 + λ(x1 − x2 ) + λδUX ⊆ dom f . Thus x2 + λ(x1 − x2 ) + λδ U ⊆ dom f ∩ C = dom(f + ιC ). 2 X λδ By Fact 2.8, x2 +λ(x1 −x2 )+ 2 UX ⊆ dom ∂(f +ιC ). Since dom A is convex, x2 +λ(x1 −x2 ) ∈ dom A and x2 +λ(x1 −x2 ) ∈ dom A∩int [dom ∂(f + ιC )]. By Fact 2.1 , ∂f +NC = ∂(f + ιC ). Then, by Theorem 3.1 (applied to A and f + ιC ), A + ∂f + NC = A + ∂(f + ιC ) is maximally monotone. By Fact 2.11, A + ∂f is of type (F P V ). " Corollary 3.4 (Simons) (See [26, Theorem 46.1].) Let A : X ⇒ X ∗ be a maximally monotone linear relation. Then A is of type (FPV). Proof. Let f = ιX . Then by Corollary 3.3, we have that A = A + ∂f is type of (FPV).

3.1

An example and comments

Example 3.5 Suppose that X = L1 [0, 1] with $ · $1 , let ! " D = x ∈ X | x is absolutely continuous, x(0) = 0, x( ∈ X ∗ , and set

A : X ⇒ X ∗ : x 1→

#

11

{x( }, if x ∈ D; ∅, otherwise.

"

Define f : X → ]−∞, +∞] by (56)

f (x) =

#

1 , 1−&x&21

if $x$ < 1;

+∞,

otherwise.

Clearly, X is a nonreflexive Banach space. By Phelps and Simons’ [19, Example 4.3], A is an at most single-valued maximally monotone linear relation with proper dense domain, and 1 (( A is neither symmetric nor skew. Since g(t) = 1−t 2 is convex on the [0, 1[ (by g (t) = 2(1 − t2 )−2 + 8t2 (1 − t2 )−3 ≥ 0, ∀t ∈ [0, 1[), f is convex. Clearly, f is proper lower semicontinuous, and by Fact 2.8, we have (57)

dom f = UX = int dom f = dom ∂f = int [dom ∂f ] .

Since 0 ∈ dom A ∩ int [dom ∂f ], Theorem 3.1 implies that A + ∂f is maximally monotone. To the best of our knowledge, the maximal monotonicity of A + ∂f cannot be deduced from any previously known result. Remark 3.6 To the best of our knowledge, the results in [30, 33, 34, 6, 37] can not verify the maximal monotonicity in Example 3.5. • Verona and Verona (see [30, Corollary 2.9(a)] or [26, Theorem 53.1] or [37, Corollary 3.7]) showed the following: “Let f : X → ]−∞, +∞] be proper, lower semicontinuous, and convex, let A : X ⇒ X ∗ be maximally monotone, and suppose that dom A = X. Then ∂f + A is maximally monotone.” The dom A in Example 3.5 is proper dense, hence A+∂f in Example 3.5 cannot be deduced from the Verona -Verona result. • In [33, Theorem 5.10(η)], Voisei showed that the sum theorem is true when dom A ∩ dom B is closed, dom A is convex and Rockafellar’s constraint qualification holds. In Example 3.5, dom A ∩ dom ∂f is not closed by (57). Hence we cannot apply for [33, Theorem 5.10(η)]. • In [34, Corollary 4], Voisei and Z˘alinescu showed that the sum theorem is true when ic (dom A) #= ∅,ic (dom B) #= ∅ and 0 ∈ic [dom A − dom B]. Since the dom A in Example 3.5 is a proper dense linear subspace, ic (dom A) = ∅. Thus we cannot apply for [34, Corollary 4]. (Given a set C ⊆ X, we define ic C by # i C, if aff C is closed; ic C= ∅, otherwise, where i C [38] is the intrinsic core or relative algebraic interior of C, defined by i C = {a ∈ C | ∀x ∈ aff(C − C), ∃δ > 0, ∀λ ∈ [0, δ] : a + λx ∈ C}.) 12

• In [6], it was shown that the sum theorem is true when A is a linear relation, B is the subdifferential operator of a proper lower semicontinuous sublinear function, and Rockafellar’s constraint qualification holds. Clearly, f in Example 3.5 is not sublinear. Then we cannot apply for it. Theorem 3.1 truly generalizes [6]. • In [37, Corollary 3.11], it was shown that the sum theorem is true when A is a linear relation, B is a maximally monotone operator satisfying Rockafellar’s constraint qualification and dom A ∩ dom B ⊆ dom B. In Example 3.5, since dom A is a linear subspace, we can take x0 ∈ dom A with $x0 $ = 1. Thus, by (57), we have that (58)

x0 ∈ dom A ∩ UX = dom A ∩ dom ∂f

but x0 #∈ UX = dom ∂f.

Thus dom A ∩ dom ∂f # dom ∂f and thus we cannot apply [37, Corollary 3.11] either. Open problem 3.7 Let A : X ⇒ X ∗ be a maximally monotone linear relation, and let f : X ⇒ ]−∞, +∞] be a proper lower semicontinuous convex function. Assume that ( $ % (59) λ PX (dom FA ) − PX (dom F∂f ) = X. λ>0

Is A + ∂f necessarily maximally monotone ?

References [1] H.H. Bauschke and J.M. Borwein, “Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators”, Pacific Journal of Mathematics, vol. 189, pp. 1–20, 1999. [2] H.H. Bauschke, J.M. Borwein, and X. Wang, “Fitzpatrick functions and continuous linear monotone operators”, SIAM Journal on Optimization, vol. 18, pp. 789–809, 2007. [3] H.H. Bauschke, X. Wang, and L. Yao, “Autoconjugate representers for linear monotone operators”, Mathematical Programming (Series B), vol. 123, pp. 5–24, 2010. [4] H.H. Bauschke, X. Wang, and L. Yao, “Monotone linear relations: maximality and Fitzpatrick functions”, Journal of Convex Analysis, vol. 16, pp. 673–686, 2009. [5] H.H. Bauschke, X. Wang, and L. Yao, “An answer to S. Simons’ question on the maximal monotonicity of the sum of a maximal monotone linear operator and a normal cone operator”, Set-Valued and Variational Analysis, vol. 17, pp. 195–201, 2009.

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[6] H.H. Bauschke, X. Wang, and L. Yao, “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. [7] H.H. Bauschke, X. Wang, and L. Yao, “Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter”, Journal of Mathematical Analysis and Applications, vol. 370, pp. 224-241, 2010. [8] H.H. Bauschke, X. Wang, and L. Yao, “On Borwein-Wiersma Decompositions of monotone linear relations”,SIAM Journal on Optimization, vol. 20, pp. 2636–2652, 2010. [9] J.M. Borwein, “Maximal monotonicity via convex analysis”, Journal of Convex Analysis, vol. 13, pp. 561–586, 2006. [10] J.M. Borwein, “Maximality of sums of two maximal monotone operators in general Banach space”, Proceedings of the American Mathematical Society, vol. 135, pp. 3917– 3924, 2007. [11] J.M. Borwein, “Fifty years of maximal monotonicity”, http://www.carma.newcastle.edu.au/ jb616/fifty.pdf, January 2010. [12] J.M. Borwein and J.D. Vanderwerff, Convex Functions, Cambridge University Press, 2010. [13] R.S. Burachik and A.N. Iusem, Set-Valued Mappings and Enlargements of Monotone Operators, Springer-Verlag, 2008. [14] D. Butnariu and A.N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, 2000. [15] R. Cross, Multivalued Linear Operators, Marcel Dekker, 1998. [16] S. Fitzpatrick, “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. [17] M.M. Alves and B.F. Svaiter, “A new proof for maximal monotonicity of subdifferential operators”, Journal of Convex Analysis, vol. 15, pp. 345–348, 2008. [18] R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd Edition, Springer-Verlag, 1993.

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[19] R.R. Phelps and S. Simons, “Unbounded linear monotone operators on nonreflexive Banach spaces”, Journal of Convex Analysis, vol. 5, pp. 303–328, 1998. [20] R.T. Rockafellar, “Extension of Fenchel’s duality theorem for convex functions”, Duke Mathematical Journal, vol. 33, pp. 81–89, 1966. [21] R.T. Rockafellar, “On the maximality of sums of nonlinear monotone operators”, Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970. [22] R.T. Rockafellar, “On the maximality of sums of subdifferential mappings”, Pacific Journal of Mathematics, vol. 33, pp. 209–216, 1970. [23] R.T. Rockafellar and R.J-B Wets, Variational Analysis, 3nd Printing, Springer-Verlag, 2009. [24] R. Rudin, Functinal Analysis, Second Edition, McGraw-Hill, 1991. [25] S. Simons, Minimax and Monotonicity, Springer-Verlag, 1998. [26] S. Simons, From Hahn-Banach to Monotonicity, Springer-Verlag, 2008. [27] S. Simons, “A Br´ezis-Browder theorem for SSDB spaces”; http://arxiv.org/abs/1004.4251v3, September 2010. [28] B.F. Svaiter, “Non-enlargeable operators and self-cancelling operators”, Journal of Convex Analysis, vol. 17, pp. 309–320, 2010. [29] A. Verona and M.E. Verona, “Regular maximal monotone operators”, Set-Valued Analysis, vol. 6, pp. 303–312, 1998. [30] A. Verona and M.E. Verona, “Regular maximal monotone operators and the sum theorem”, Journal of Convex Analysis, vol. 7, pp. 115–128, 2000. [31] M.D. Voisei, “A maximality theorem for the sum of maximal monotone operators in non-reflexive Banach spaces”, Mathematical Sciences Research Journal, vol. 10, pp. 36– 41, 2006. [32] M.D. Voisei, “The sum theorem for linear maximal monotone operators”, Mathematical Sciences Research Journal, vol. 10, pp. 83–85, 2006. [33] M.D. Voisei, “The sum and chain rules for maximal monotone operators”, Set-Valued and Variational Analysis, vol. 16, pp. 461–476, 2008. [34] M.D. Voisei and C. Z˘alinescu, “Maximal monotonicity criteria for the composition and the sum under weak interiority conditions”, Mathematical Programming (Series B), vol. 123, pp. 265–283, 2010. 15

[35] M.D. Voisei and C. Z˘alinescu, “Linear monotone subspaces of locally convex spaces”, Set-Valued and Variational Analysis, vol. 18, pp. 29–55, 2010. [36] L. Yao, “The Br´ezis-Browder Theorem revisited and properties of Fitzpatrick functions of order n”, to appear Fixed Point Theory for Inverse Problems in Science and Engineering (Banff 2009) , Springer-Verlag; http://arxiv.org/abs/0905.4056v1, May 2009. [37] L. Yao, “The sum of a maximal monotone operator of type (FPV) and a maximal monotone operator with full domain is maximally monotone”, submitted; http://arxiv.org/abs/1005.2247v2, August 2010. [38] C. Z˘alinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, 2002. [39] E. Zeidler, Nonlinear Functional Analysis and its Application II/B: Nonlinear Monotone Operators, Springer-Verlag, New York-Berlin-Heidelberg, 1990.

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