Fitzpatrick function for generalized monotone operators
arXiv:1801.04519v1 [math.FA] 14 Jan 2018
Mohammad Hossein Alizadeh∗ 1 ∗
Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45195-1159, Zanjan, Iran
Abstract. We define the Fitzpatrick function of a σ-monotone operator in a way similar to the original definition given by Fitzpatrick. We show that some well-known properties of Fitzpatrick function remain valid for the larger class of premonotone operators. Also, we find some conditions under which the Fitzpatrick function of a σ-monotone operator is proper, and give some results in Hilbert spaces. Keywords: Fitzpatrick function; maximal monotone operator; generalized monotone operator Mathematics Subject Classification : 47H05, 49J53, 47H04.
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1
1
Introduction
The literature on monotone and generalized monotone operators is quite extensive. These notions have been found appropriate in various branches of mathematics such as operator theory, PDE, SDE, differentiability theory of convex functions and nonlinear functional analysis. The notion of pre-monotone operators for the finite-dimensional case is introduced in [11]. The class of pre-monotone operators includes many important classes of operators such as monotone and ε-monotone operators but also contains many more: for example, if T is monotone and R is globally bounded, then T + R is premonotone. In Banach spaces, some basic properties of premonotone operators such as local boundedness, generalization of the L. Vesely theorem are studied in [3]. Also in [1] some results on the sum and difference of the σ-monotone operators, and relations between the σ-monotonicity and σ-convexity, are investigated. The Fitzpatrick function of a monotone operator was introduced in [12] by Fitzpatrick. Around 13 years later, a convex representation (Fitzpatrick function) of maximal monotone operators was rediscovered by Martinez-Legaz and Th´era [13]. Burachik and Svaiter [8] also rediscovered Fitzpatrick functions and studied the whole family of lower semicontinuous and convex functions associated with a given maximal monotone operator. The Fitzpatrick function makes a bridge between the results on convex functions and results on maximal monotone operators ( see [4, 5, 6, 14] and the references therein). The Fitzpatrick function was also used for deterministic and stochastic differential equations driven by multivalued maximal monotone operators [15]. Recently, Fitzpatrick functions (or “Fitzpatrick transforms”) were proved to be a valuable tool in the study of monotone bifunctions (see [2], [7] and [10]). Moreover, very recently a generalization of the strong Fitzpatrick inequality is studied in [9]. In this paper we introduce the Fitzpatrick function of a σ-monotone operator and we investigate some properties of this function. We hope that the Fitzpatrick function will be a tool for linking maximal σ-monotone theory to convex analysis and add to their lure. The rest of the paper is organized as follows: The next section contains some notation, and facts on σ-monotone operators, and Fitzpatrick function. In Section 3 after defining the Fitzpatrick function of a σ-monotone operator, we show that for a σ-monotone operator, the inequality FT (x, x∗ ) ≥ hx∗ , xi holds for all (x, x∗ ) ∈ X × X ∗ and we have the equality if (x, x∗ ) ∈ gr T and σ (x) = 0. Also we find some conditions under which FT is proper . In Section 4 we provide some kind of surjectivity results on Hilbert spaces. We conclude the paper by presenting some examples and a related remark.
2
Preliminaries
Suppose that X is a real Banach space, with topological dual space X ∗ . We ∗ will denote the duality pairing by h·, ·i. Given an operator T : X → 2X , its 2
domain is D (T ) = {x ∈ X : T (x) 6= ∅} , its range R (T ) = {x∗ ∈ X ∗ : ∃x ∈ X s.t. x∗ ∈ T (x)} and its graph gr T := {(x, x∗ ) ∈ X × X ∗ : x∗ ∈ T (x)} . Recall that T is said to be monotone if for every x, y ∈ D (T ) , x∗ ∈ T (x) and y ∗ ∈ T (y), hx∗ − y ∗ , x − yi ≥ 0,
and T is said to be maximal monotone if its graph is not properly included in any other monotone graph. For a monotone operator T , its Fitzpatrick function introduced in [12] is defined by FT (x, x∗ ) = sup {hx∗ , yi + hy ∗ , x − yi : (y, y ∗ ) ∈ gr T } . It is a lower semicontinuous (lsc from now on) and convex function. ∗
Proposition 1 [12] For a maximal monotone operator T : X → 2X one has FT (x, x∗ ) ≥ hx∗ , xi
∀ (x, x∗ ) ∈ X × X ∗ ,
and the equality holds if and only if (x, x∗ ) ∈ gr T. ∗
Definition 2 (i) Given an operator T : X → 2X and a map σ : D (T ) → R+ , T is said to be σ-monotone if for every x, y ∈ D(T ), x∗ ∈ T (x) and y ∗ ∈ T (y), hx∗ − y ∗ , x − yi ≥ − min{σ(x), σ(y)}kx − yk.
(1)
(ii) An operator T is called pre-monotone if it is σ-monotone for some σ : D (T ) → R+ . (iii) A σ-monotone operator T is called maximal σ-monotone, if for every operator T ′ which is σ ′ -monotone with gr T ⊆ gr T ′ and σ ′ an extension of σ, one has T = T ′ . As for the monotone case, each σ-monotone operator has a maximal σmonotone extension [11]. Definition 3 Let A be a subset of X. Given a mapping σ : A → R+ , two pairs (x, x∗ ), (y, y ∗ ) ∈ A × X ∗ are σ-monotonically related if hx∗ − y ∗ , x − yi ≥ − min{σ(x), σ(y)}kx − yk. An equivalent condition for maximal σ-monotonicity, which we use in Section 3, is stated next: ∗
Proposition 4 [3] The σ-monotone operator T : X → 2X is maximal σmonotone if and only if, for every point (x0 , x∗0 ) ∈ X × X ∗ and every extension σ ′ of σ to D (T ) ∪ {x0 } such that (x0 , x∗0 ) is σ-monotonically related to all pairs (y, y ∗ ) ∈ gr(T ), we have (x0 , x∗0 ) ∈ gr(T ). The paper [11] contains some examples of maximal σ-monotone operators. 3
3
The Fitzpatrick function of σ-monotone operators
The definition we use for the Fitzpatrick function is exactly the same as for monotone operators: ∗
Definition 5 Let X be a Banach space and T : X → 2X be a σ-monotone operator. The Fitzpatrick function associated with T is the function FT : X × X ∗ → R ∪ {+∞} defined by FT (x, x∗ ) =
sup (y,y ∗ )∈gr T
(hx∗ , yi + hy ∗ , xi − hy ∗ , yi) .
FT is norm×weak∗- lsc and convex on X × X ∗ . We need the following lemma for proving Theorem 7. Lemma 6 Suppose that T is maximal σ-monotone for some σ : D (T ) → R+ . Then for all (x, x∗ ) ∈ X × X ∗ we have inf
(y,y ∗ )∈gr T
hy ∗ − x∗ , y − xi ≤ 0.
(2)
Proof. We consider two cases: If (x, x∗ ) ∈ gr T , then obviously inf
(y,y ∗ )∈gr T
hy ∗ − x∗ , y − xi ≤ hx∗ − x∗ , x − xi = 0.
If (x, x∗ ) ∈ / gr T , then we will show that inf
(y,y ∗ )∈gr T
hy ∗ − x∗ , y − xi < 0.
(3)
Indeed, if on the contrary (3) does not hold, then for all (y, y ∗ ) ∈ gr T , hy ∗ − x∗ , y − xi ≥ 0. Consequently, for any extension σ ′ of σ on D(T ) ∪ {x} and every (y, y ∗ ) ∈ gr T , hy ∗ − x∗ , y − xi ≥ − min{σ ′ (x), σ ′ (y)} kx − yk .
By Proposition 4 this implies that (x, x∗ ) ∈ gr T , a contradiction. Thus (3) holds. So (2) holds in all cases. ∗
Theorem 7 Suppose that T : X → 2X is a maximal σ-monotone operator. Then FT (x, x∗ ) ≥ hx∗ , xi ∀ (x, x∗ ) ∈ X × X ∗ . (4) If (x, x∗ ) ∈ / gr T , then we have strict inequality in (4). If (x, x∗ ) ∈ gr T and σ (x) = 0, then the equality holds.
4
Proof. Let (x, x∗ ) ∈ X × X ∗ . It can be easily verified FT (x, x∗ ) = hx∗ , xi −
inf
(y,y ∗ )∈gr T
hy ∗ − x∗ , y − xi .
(5)
According to (2) we have inf (y,y∗ )∈gr T hy ∗ − x∗ , y − xi ≤ 0. Thus FT (x, x∗ ) ≥ hx∗ , xi . Let (x, x∗ ) ∈ / gr T , and assume in contradiction that FT (x, x∗ ) = ∗ hx , xi . From (5) we get inf (y,y∗ )∈gr T hy ∗ − x∗ , y − xi = 0. But according to (3) it is not possible. Now, suppose that x∗ ∈ T (x) and σ (x) = 0. Then for each (y, y ∗ ) ∈ gr T hx∗ − y ∗ , x − yi ≥ − min{σ(x), σ(y)}kx − yk = 0. Therefore hx∗ , yi + hy ∗ , xi − hy ∗ , yi ≤ hx∗ , xi. By taking the supremum over all (y, y ∗ ) ∈ gr T we get FT (x, x∗ ) ≤ hx∗ , xi and so FT (x, x∗ ) = hx∗ , xi . Remark 8 • Note that if FT (x, x∗ ) = hx∗ , xi, then indeed (x, x∗ ) ∈ gr T because when (x, x∗ ) ∈ / gr T by Theorem 7 we would have strict inequality. • Note that x∗ ∈ T (x) ⇔ hx∗ − y ∗ , x − yi ≥ − min{σ(x), σ(y)}kx − yk ∀ (y, y ∗ ) ∈ gr T ⇔ hx∗ , xi + min{σ(x), σ(y)}kx − yk ≥ hx∗ , yi + hy ∗ , xi − hy ∗ , yi ∀ (y, y ∗ ) ∈ gr T
⇔ hx∗ , xi + sup {min{σ(x), σ(y)}kx − yk} y∈D(T )
≥
sup (y,y ∗ )∈gr T
(hx∗ , yi + hy ∗ , xi − hy ∗ , yi)
⇔ hx∗ , xi + sup {min{σ(x), σ(y)}kx − yk} ≥ FT (x, x∗ ) . y∈D(T )
• One can easily check that if T and S are σ-monotone operators and S is an extension of T i.e., gr T ⊆ gr S, then FT ≤ FS . Notice that in some cases the Fitzpatrick function can be +∞ on X × X ∗ as the following example shows. Example 9 Suppose that X is a Hilbert space and T : X → 2X is an operator. Assume that D (T ) contains a ray L = Rz (z 6= 0) such that for each y ∈ L, {0, y0∗ } ⊆ T (y) where y0∗ is a fixed element, not identically zero on L. Then FT can not be proper. Proof. Let (x, x∗ ) ∈ X × X ∗ . First we suppose that x∗ 6= 0 on L. Then hx∗ , zi 6= 0, and we can assume without loss of generality that hx∗ , zi > 0. Then take y ∗ = 0, FT (x, x∗ ) ≥ sup hx∗ , yi ≥ lim hx∗ , nzi = +∞. n→∞
y∈L
5
If x∗ = 0, then FT (x, x∗ ) =
sup (y,y ∗ )∈gr T
(hy ∗ , x − yi)
≥ hy0∗ , xi − inf hy0∗ , yi . y∈L
Again without loss of generality we may assume hy0∗ , zi is negative and then get inf y∈L hy0∗ , yi = −∞.Thus we conclude that FT (x, x∗ ) = +∞. According to the above example if the graph of a maximal σ-monotone operator contains the lines L × {0∗ } and L × {y0∗ } and y0∗ 6= 0 on L, then the Fitzpatrick function can be everywhere +∞. Indeed, take T : R → 2R to be defined by T (x) = [0, 1] for all x ∈ R. One can see that this operator is σ-monotone with σ(x) = 1 for all x. In fact, it is maximal σ-monotone and FT ≡ +∞. According to the Theorem 7 if T is maximal σ-monotone, (x, x∗ ) in gr T and σ(x) = 0, then FT (x, x∗ ) = hx∗ , xi and so FT (x, x∗ ) < ∞. ∗ Recall that [3], given an operator T : X → 2X , we define the function σT : D (T ) → R+ ∪ {+∞} by σT (x) = inf{a ∈ R+ : hx∗ − y ∗ , x − yi ≥ −a kx − yk , ∀(y, y ∗ ) ∈ gr T, x∗ ∈ T (x)}. (6) The following proposition shows that if the equality holds in Theorem 7 for all x∗ ∈ T (x), then σT (x) = 0. ∗
Proposition 10 Suppose that T : X → 2X is σ-monotone. Let x ∈ X and for each x∗ ∈ T (x) , FT (x, x∗ ) = hx∗ , xi , then σT (x) = 0. In particular, if T is a single-valued σ-monotone, and FT (x, x∗ ) = hx∗ , xi , then σT (x) = 0. Proof. For every x∗ ∈ T (x) we have FT (x, x∗ ) = hx∗ , xi . Thus hx∗ , yi + hy ∗ , xi − hy, y ∗ i ≤ hx∗ , xi
∀ (y, y ∗ ) ∈ gr T, x∗ ∈ T (x) ,
Therefore 0 ≤ hx∗ − y ∗ , x − yi
∀ (y, y ∗ ) ∈ gr T, x∗ ∈ T (x) .
(7)
From (6) and (7) we get σT (x) = 0. According to the above proposition, for a maximal σ-monotone operator T if σT (x) 6= 0, then FT (x, x∗ ) > hx∗ , xi . In the following proposition, we find a sufficient condition, for finiteness of the Fitzpatrick function. ∗
Proposition 11 Suppose that T : X → 2X is σ-monotone and set ( ) M :=
(x, x∗ ) ∈ gr T : sup {min {σ (x) , σ (y)} kx − yk} < ∞ . y∈D(T )
Then co M ⊆ D (FT ) . 6
Proof. Assume that (x, x∗ ) ∈ M. According to Remark 8 part II, we have FT (x, x∗ ) < ∞. Thus M ⊂ D (FT ). On the other hand FT is convex so D (FT ) is also convex and so co M ⊆ D (FT ) .
4
Some results on Hilbert spaces
From now on, X is a Hilbert space. Proposition 12 Assume that T : X → 2X is σ-monotone. Suppose that R (I + T ) = X. If (y, y ∗ ) ∈ X × X is σ-monotonically related to gr T , then there exists (x, x∗ ) ∈ gr T, such that ||x∗ − y ∗ || = ||x − y|| ≤ σ (x) . Moreover, if σ (x) = 0, then (y, y ∗ ) ∈ gr T . Proof. By assumption R (I + T ) = X, so we can find x ∈ X such that y + y ∗ ∈ (I + T ) (x). This means that there exists (x, x∗ ) ∈ gr T such that y+y ∗ = x+x∗ . From here we get y − x = x∗ − y ∗ . (8) On the other hand, (y, y ∗ ) ∈ X × X is σ-monotonically related to gr T . Thus − σ(x)kx − yk ≤ hx∗ − y ∗ , x − yi.
(9)
From (8) and (9) we infer that −σ(x)kx − yk ≤ −hx − y, x − yi = −||x − y||2 . Therefore ||x − y|| ≤ σ (x) . Now if σ (x) = 0, then form the above inequality and (8) we conclude that x = y and x∗ = y ∗ . Thus (y, y ∗ ) ∈ gr T . Corollary 13 Assume that T : X → 2X is monotone. If R (I + T ) = X, then T is maximal monotone. For the proof of following fact, see [4, Corollary 15.17]. Fact 1: Suppose that f : X → R ∪ {+∞} is proper, convex, and lower semicontinuous. If f (·) + 21 ||·||2 ≥ 0, then there exists x∗ ∈ X such that f (x) +
1 1 2 2 ||x|| ≥ ||x + x∗ || . 2 2
∀x ∈ X
Proposition 14 Suppose that T : X → 2X is maximal σ-monotone. Then there exists (z, z ∗ ) ∈ X × X such that hx∗ −z ∗, x−zi ≥
1 ||z +z ∗||2 + ∗inf hx∗ − y ∗ , x − yi . 2 (y,y )∈gr T 7
∀ (x, x∗ ) ∈ X ×X
Proof. Let T be a maximal σ-monotone operator. Then by Theorem 7 FT (x, x∗ ) ≥ hx∗ , xi
∀ (x, x∗ ) ∈ X × X.
We equip X × X with the usual inner product h(x, y), (x′ , y ′ )i = hx, x′ i + hy, y ′ i. Then for each (x, x∗ ) ∈ X × X FT (x, x∗ ) +
1 1 2 2 ||(x, x∗ )|| ≥ hx∗ , xi + ||(x, x∗ )|| 2 2 1 2 = ||x + x∗ || ≥ 0. 2
Since FT : X × X → R ∪ {+∞} is proper, convex, and lower semicontinuous, by using Fact 1 we find (z ∗ , z) ∈ X × X such that for each (x, x∗ ) ∈ X × X we have FT (x, x∗ ) +
1 1 2 2 ||(x, x∗ )|| ≥ ||(x, x∗ ) + (z ∗ , z)|| 2 2 1 = || (x, x∗ ) ||2 + || (z ∗ , z) ||2 + 2 h(x, x∗ ) , (z ∗ , z)i . 2
Now using (5), we get hx∗ , xi −
inf
(y,y ∗ )∈gr T
hy ∗ − x∗ , y − xi ≥
1 || (z ∗ , z) ||2 + hz ∗ , xi + hx∗ , zi . 2
Therefore hx∗ , xi − hz ∗ , xi − hx∗ , zi ≥
inf
(y,y ∗ )∈gr T
1 hy ∗ − x∗ , y − xi + || (z ∗ , z) ||2 . 2
By adding hz ∗ , zi in the two sides of the above inequality we obtain the desired inequality. We conclude the paper by presenting two examples related to the Fitzpatrick function of a pre-monotone operator. Example 15 Consider the triangular function T : R → R defined by T (x) := max {1 − |x|, 0} . Clearly T is not monotone but it is pre-monotone by Proposi2 tion 3.4 in [11]. Then FT (x, x∗ ) = ι{0∗ } + 14 (x + 1) where ιC is the indicator function of C. Proof. Fix (x, x∗ ) ∈ R × R, by definition FT (x, x∗ ) = sup {x∗ y + y ∗ x − yy ∗ : (y, y ∗ ) ∈ gr T } = sup {x∗ y + T (y) x − yT (y)} . y∈R
Set h (y) = x∗ y + xT (y) − yT (y) . 1) If |y| ≥ 1, then T (y) = 0 and so h (y) = x∗ y. Therefore 0 if x∗ = 0, sup h (y) = +∞ if x∗ 6= 0. |y|≥1 8
2) If y ∈ [0, 1], then T (y) = 1 − y and h (y) = x∗ y + (x − y) (1 − y). Thus supy∈[0,1] h (y) = max {x, x∗ } . 3) If y ∈ [−1, 0], thenT (y) = 1 + y and h (y) = x + (x∗ + x − 1) y − y 2 . We 1 4
2
(x + 1) − x∗ . +∞ ∗ Finally we have FT (x, x ) = 2 1 (x + 1) 4 ι{0∗ } + 14 (x + 1)2 .
get supy∈[−1,0] h (y) =
if x∗ 6= 0, In other words FT (x, x∗ ) = if x∗ = 0.
1 Example 16 (Normal function) Suppose X is R and let T (x) = 1+x 2 . Then T is not monotone but it is σ-monotone (for σ = 1). Moreover, FT (x, x∗ ) = ι{0∗ } + 2 √x21+1−x . ( )
Proof. Fix (x, x∗ ) ∈ R × R. The Fitzpatrick function of T is FT (x, x∗ ) = sup {x∗ y + T (y) x − yT (y)} . y∈R
1 ∗ We define h (y) := x∗ y + (x − y) 1+y 6= 0, then 2 . It is easy to see that if x ∗ supy∈R h (y) = +∞. Now for x = 0, a direct calculation shows that the max√ imum is attend at x − x2 + 1 and so supy∈R h (y) = 2 √x21+1−x .Thus we ( ) ( ∗ +∞ if x = 6 0, conclude that FT (x, x∗ ) = supy∈R h (y) = √ 1 if x∗ = 0. 2( x2 +1−x)
Acknowledgement 17 It is my great pleasure to thanks Professor Hadjisavvas for insightful comments and various helpful discussions which led to considerable improvements throughout the earlier draft of this manuscript. Also, I would like to thank Professor Martinez-Legaz who motivated me to study the subject of this paper.
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