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## Computational & Applied Mathematics

*On-line version* ISSN 1807-0302

### Comput. Appl. Math. vol.30 no.1 São Carlos 2011

#### http://dx.doi.org/10.1590/S1807-03022011000100002

**Regularity results for semimonotone operators**

**Rolando Gárciga Otero ^{I, *}; Alfredo Iusem^{II, **}**

^{I}Instituto de Economia, Universidade Federal de Rio de Janeiro, Avenida Pasteur 250, Urca, 22290-240 Rio de Janeiro, RJ, Brazil

^{II}Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil E-mails: rgarciga@ie.ufrj.br / iusp@impa.br

**ABSTRACT**

We introduce the concept of ρ-semimonotone point-to-set operators in Hilbert spaces. This notion is symmetrical with respect to the graph of *T*, as is the case for monotonicity, but not for other related notions, like e.g. hypomonotonicity, of which our new class is a relaxation. We give a necessary condition for ρ-semimonotonicity of *T* in terms of Lispchitz continuity of *[T +* ρ^{-1}*1*]^{-1} and a sufficient condition related to expansivity of *T*. We also establish surjectivity results for maximal ρ-semimonotone operators.

**Mathematical subject classification:** 47H05,47Hxx.

**Key words:** hypomonotonicity, surjectivity, prox-regularity, semimonotonicity.

**1 Introduction**

Before introducing the class of ρ-semimonotone operators we recall the concept of monotonicity and a few of its relaxations.

**Definition 1.** Let *H* be a Hilbert space, *T : H **→* P (H) a point-to-set operator and *G (T)* its graph.

I)

Tis said to be monotone iffII)

Tis said to be maximal monotone if it is monotone and additionallyG(T) =G(T') for all monotone operatorT' : H→P (H)such thatG (T)⊂G (T^{1}).III) For ρ ∈ R

_{++},Tis said to be p-hypomonotone iffIV) For ρ ∈ R

_{++},Tis said to be maximal ρ-hypomonotone if it is ρ -hypomonotone and additionallyG (T) =G (T')for all ρ-hypomonotone operatorT' : H→ P (H)such thatG (T) ⊂G (T').V)

Tis said to be premonotone iffwhere σ :

H→R is a positive valued function defined over the whole spaceH.

Next we introduce the class of operators which are the main subject of this paper.

**Definition 2.** Let *T : H **→ P (H)* be a point-to-set operator, *G (T)* its graph and ρ ∈ (0, 1) a real number.

I)

Tis said to be ρ-semimonotone ifffor all

(x, u), (y,v)∈G (T).II)

Tis said to be maximal ρ-semimonotone if it is ρ-semimonotone and additionallyG (T)= G (T')for all ρ-semimonotone operatorT' : H→ P (H)such thatG (T)⊂G (T').

The concepts of hypomonotonicity and premonotonicity were introduced in [5] and [2] respectively. We mention that a notion of maximal premonotonicity has also been introduced in [2], but the definition is rather technical and thus we prefer to omit it.

We mention that we restrict the range of the parameter ρ to the interval (0,1) because all operators turn out to be ρ-semimonotone for ρ __>__ 1, as can be easily verified.

It is clear that monotone operators are both premonotone and ρ-hypomonotone for all ρ *>* 0, and that ρ-hypomonotone operators with ρ ∈ *(0,1/2)* are 2ρ-semimonotone. It is also elementary that *T* is ρ-hypomonotone iff *T* + ρ*I* is monotone *(I* being the identity operator in *H).*

In order to have a clearer view of the relation among these notions, it is worthwhile to look at the special case of self-adjoint linear operators in the finite dimensional case. If Λ *(A)* is the spectrum (i.e., set of eigenvalues) of the self-adjoint linear operator *A : H **→ H*, it is well known that *A* is monotone iff λ *(A) * ⊂ *[0, * ∞) and it follows easily from the comment above that *A* is ρ-hypomonotone iff λ *(A) * ⊂ *[**→* ρ, ∞). On the other hand, linear premonotone operators are just monotone. It is also elementary that *A* is *p*-semimonotone iff

with 0 *< **β(*ρ*) < **η(*ρ*)* given by (7) and (9), i.e., the eigenvualues of self-adjoint ρ-semimonotone operators can lie anywhere on the real line, excepting for an open interval around → 1/ρ contained in the negative halfline.

One of the main properties of maximal monotone operators is related to the regularization of the inclusion problem consisting of finding *x* ∈ * H* such that *b * ∈ *T (x),* with *T* monotone and *b * ∈ *H*. Such problem may have no solution, or an infinite set of solutions, but the problem *b * ∈ *(T +* λ*I) (x)* is well posed in Hadamard's sense for all λ *>* 0, meaning that there exists a unique solution, and it depends continuously on *b*. This is a consequence of Minty's Theorem (see [4]), which states that for a maximal monotone operator *T*, the operator *T +* λ*I* is onto, and its inverse is Lipschitz continuous with constant *L =* λ* ^{ -1}*, (and henceforth point-to-point), for all λ

*>*0.

When the notion of monotonicity is relaxed, one expects to preserve at least some version of Minty's result. In the case of hypomonotonicity, the fact that *T + * ρ*I* is monotone when *T* is ρ-hypomonotone easily implies that Minty's result holds for maximal ρ-hypomonotone operators whenever λ belongs to *(* ρ, ∞*),* with the Lipschitz constant of *(T +* λ*I*)^{-1}taking the value *(*λ - ρ*) ^{ -1}.*

The situation is more complicated when *T* is premonotone. Examples of premonotone operators *T* defined on the real line such that *T + * λ*I* fails to be monotone for all λ *>* 0 have been presented in [2]. Nevertheless, the following surjectivity result has been proved in [2]: when *T* is maximal premonotone and *H* is finite dimensional then *T +* λ*I* isontoforall λ *>* 0. Minty's Theorem cannot be invoked in this case, and the proof uses an existence result for equilibrium problems originally established in [3] and extended later on in [1].

Before discussing the ρ-semimonotone case, it might be illuminating to look at the surjetivity issue in the one-dimensional case. It is easy to check that *T + * λ*I* is strictly increasing when *T* is monotone and λ *>* 0, or *T* is ρ*-*hypomonotone and λ *> **ρ,* and furthermore the values of the regularized operator *T + * λ*I* go from -∞ to *+ * ∞ . The surjectivity is then an easy consequence of the maximality of the graph *G (T).* When *T* is pre-monotone, *T + * λ*I* may fail to be increasing for all λ * >* 0 (see Example 3 in [2]), but still it holds that the operator values go from -∞ to *+ * ∞, and the surjectivity is also guaranteed. This is not the case for ρ-semimonotone operators. Not only a ρ-semimonotone operator *T* defined on R may be such that *T +* λ*I* fails to be monotone for all λ *>* 0, but *T,* and even *T + * λ*I,* may happen to be strictly decreasing! (see Example 1 below). We will nevertheless manage to establish regularity of *T + * λ*I* when *T* is ρ-semimonotone and λ belongs to a certain interval *(**β (ρ), η(ρ)) * ⊂ *(0,* + ∞), with β*(**ρ), η (ρ)* as in (7), (9) below (in the case of *T* like in Example 1, the surjectivity will be a consequence of the fact that *T* is strictly decreasing). We cannot invoke Minty's result in an obvious way, since *T + * λ*I* will not in general be monotone; rather, the proof will proceed through the analysis ofthe regularity properties ofthe operator *[T + **β (ρ) I*]^{ -1} + γ (ρ) *I*, with γ (ρ) as in (8) below.

**2 Semimonotone operators**

In this section e ill establish several properties of seionotone operators. We start our analysis with some elementary ones.

**Proposition 1.** *An operator T : H **→ P(H) is ρ-semimonotone if and only if the operator T*^{ -1} *is p-semimonotone; furthermore, T is maximal p-semimonotone if and only if T*^{ -1} *is maximal **ρ-semimonotone.*

**Proof.** The result follows immediately from Definition 2, taking into account that *(x, u) * ∈ *G(T)* iff *(u, x) * ∈ *G*(T^{ -1}).

We mention that monotonicity of *T* is also equivalent to monotonicity of *T ^{ -1},* but the similar statement fails to hold for ρ-hypomonotone operators. In fact, one of the motivations behind the introduction of the class of ρ-semimonotone operators is the preservation of this symmetry property enjoyed by monotone operators.

**Proposition 2.** *If T : H **→ P(H) is p-semimonotone and α belongs to (*ρ, * 1/*ρ*) then **αT is * *-semimonotone with* *=* ρmax{α, 1/α.

**Proof.** Note first that belongs to (0, 1). Let * = **αT* and take *(x, *), *(y, ) * ∈ *G(*). By definition of , there exist *u * ∈ *T(x), v * ∈ *T(y)* such that * = **αu, = α*ν. By ρ-semimonotonicity of *T*

establishing -semimonotonicity of * = **αT.*

**Proposition 3.** *If T : H **→ P(H) is * δ*-semimonotone for some * δ ∈ (0, 1), *then T is **ρ-semimonotone for all ρ * ∈ *(* δ, 1).

**Proof.** Elementary.

**Proposition 4.** *If T : H **→ P (H) (or T ^{ -1}* :

*H*

*→ P (H)) is S-hypomonotone with*δ ∈

*(0, 1/2), then T is 2*δ

*-semimonotone. Moreover, if both T and T*δ

^{ -1}are*-hypomonotone with*δ ∈

*(0, 1) then T is*δ

*-semimonotone.*

**Proof.** Elementary.

**Remark 1.** We mention that a δ-hypomonotone operator *T* with δ __>__ 1/2, may fail to be ρ-semimonotone for all ρ, but the operator * T* is ρ-semimonotone for all ρ ∈ (0, 1*)*.

**Proposition 5.** *An operator T : H **→ P(H) is p -semimonotone if and only if*

**Proof.** Elementary.

**Proposition 6.** *If T : H **→ P(H} is maximal p-semimonotone then its graph is closed (in the strong topology).*

**Proof.** Elementary.

*2.1 The one dimensional case*

We study in this section *p*-semimonotone real valued functions, providing a simple characterization that helps in the construction of a key example and also suggests the line to follow in order to study the general case.

**Lemma 1.** *Given **ρ * ∈ (0, 1) *define * θ*(**ρ) as*

*A function f : X * ⊂ *R **→ R is ρ-semimonotone if and only if g : X → R defined by g(x) = f (x) + ρ*^{-1} *x satisfies*

*for all x, y * ∈ *X, or equivalently, g* ^{1} *= (f + **ρ*^{1}*I)* ^{1} *is Lipschitz continuous with constant* θ*(**ρ) ^{-1}.*

**Proof.** Assume that *f : X **→ R* is ρ-semimonotone and define *g(x) = f (x) + **ρ ^{-1} x*. By Definition 2, for all

*x, y*∈

*X*

or, equivalently,

Take any *x* ≠ *y* ∈ *X* and define . Then, (4) is equivalent to , i.e.,

Since for any *x * ≠ *y,*

the proof is complete.

**Example 1.** Fix ρ ∈ (0, 1) , and define *g* : R → R as . Then

for all *x* ∈ R. Thus, *g* verifies (3). Hence, the function *f : R **→ R* defined

is a ρ-semimonotone function, in view of Lemma 1. On the other hand, the function *h(x) = f (x) +* λ*x* fails to be non-decreasing for all λ ∈ R, and hence *f +* λ*I* is not monotone, so that *f* fails to be λ-hypomonotone for all λ __>__ 0. In connection with premonotonicity, note that, as an easy consequence of Definition 1 (v), if *T* is point-to-point and pre-monotone, then

for all *x* ∈ *R ^{n}\{0}.* In the one-dimensional case, (6) entails that, for a premonotone

*T, T(x)*is bounded from below on the positive half-line and bonded from above in the negative half-line. It follows that

*f*, as defined by (5), is not pre-monotone. Informally speaking, this example shows that one-dimensional semimonotone operators can be "very" decreasing, while hypomonotone or premonotone ones cannot. In a multidimensional setting, the operator

*T :*R

^{n}→ R

^{n}defined as

*T(x*) = (

_{1}, ..., x_{n}*f(x*)...,

_{1}*f (x*)), with

_{n}*f*as in (5), provides an example of a nonlinear p-semimonotone operator which fails to be both premonotone and λ-hypomonotone for all λ

__>__0.

**3 Prox-regularity properties**

The surjectivity properties of *T* + λ*I* for a ρ-semimonotone operator *T* are related to its connection with the operator *[T + **βI*]^{-1} + γ*I*, presented in the next theorem.

**Theorem 2.** *Let I be the identity operator in H. Take **ρ *∈ (0, 1) *and **β, γ, η * ∈ R_{++} *as*

i)

An operator T : H→ P( H ) is ρ-seionotone ifand only if the operator (T + βI)^{-1}+γ I is monotone.ii)

An operator T : H→ P(H} is maximal ρ -semimonotone if and only if the operator (T + βI)^{-1}+γ I is maximal monotone.

**Proof.** Consider *A : H x H **→ H x H* defined as *A(x, u) = (u - **γx, (1 + βγ)x - βu).* It is elementary that *A* is invertible, with A^{-1} *(x, u) = (u+**βx, (1 + βy)x + y u).* Let *(, ) = A(x, u)* and * = (T + **β I) ^{-1}+yI*. We claim that

*(x, u)*∈

*G ()*if and only if

*(, )*∈

*G (T).*We proceed to prove the claim:

*(x, u)*∈

*G ()*iff

*∈*

**u***(T +*

*β I)*iff

^{-1}(x) + yx*= u - yx*∈

*(T +*

*βI)*) iff

^{-1}(x*x*∈

*(T +*

*βI)() = T() + βx*iff

*= x -*

*β*∈

*T()*iff

*(, )*∈

*G (T*).

The claim is established and we proceed with the proof of (i). Consider pairs *(x, u), (y, v) * ∈ *G ()* and let *(, ) = A(x, u)* as before, and also *(, ) = A(y, v).* Observe that is monotone if and only if, for all *(x, u), (y, v)* ∈ *G (),* it holds that

using the definition of *(, ), (, )* and the formula of *A ^{-}*

^{1}in the first equality. Note that the inequality in (10) is equivalent to

using (7), (8) in the equality. In view of the claim above and the invertibility of A, *(, ), (, )* cover *G(T)* when *(x, u), (y, v)* run over *G().* Thus, we conclude from (1) that the inequality in (11) is equivalent to the ρ-semimonotonicity of *T.*

We proceed now with the proof of (ii): In view of (i), if we can add a pair *(x, u)* to *G()* while preserving the monotonicity of *,* then we can add the pair *(, ) = A(x, u)* to *G(T)* and preserve the ρ-semimonotonicity of *T*, and viceversa. It follows that the maximal monotonicity of *T* is equivalent to the maximal ρ-semimonotonicity of *T.*

**Corollary 1.** *If T : H **→ P (H) is maximal ρ-semimonotone then the operator (T + βI) ^{}^{→}^{1} + * μ

*I is onto for all*μ

*> y(*

*ρ), where y(ρ) is given by (8).*

**Proof.** By Theorem 2(ii), * = (T + **βI+ yI*, with β*(**ρ)* as in (7), is maximal monotone. Since

and μ → γ > 0, the result follows from Minty's Theorem.

**Corollary 2.** *If T : H **→ P(H} is maximal p-semimonotone then the operator T + * λ*I is onto for all * λ ∈ *(**β (ρ), η(ρ)), where β(p) and n(ρ) are given by (7) and (9) respectively.*

**Proof.** Fix β*(**ρ), γ(ρ)* and η(ρ) as in (7)-(9). Given λ ∈ *(**β,η),* define μ = (λ - β)^{-1} > 0. In view of (9), λ < η implies that μ > γ .By Corollary 1, *(T + **βI) ^{-1} + μI*is onto. Fix

*y*∈

*H*. We must exhibit some

*z*∈

*H*such that

*y*∈

*(T +*λ

*I)(z).*Since

*(T +*

*βI)*is onto, there exists

^{-1}+ μI*x*∈

*H*such that μ

*y*∈

*[(T +*

*βI*)

^{-1}

*+*

*μI*]

*(x*), or equivalently, μ

*(y - x)*∈

*(T +*

*βI )*),that is to say,

^{-1}(xDefine *z = **μ*x(y - *x*). In view of (12), , which is equivalent to

in view ofthe definition of μ. It follows from (13) that the chosen *z* is an appropriate one, thus establishing the surjectivity of *T + * λ*I*. *□*

We prove next that if *T* is ρ-semimonotone then *[T +* λ*I] ^{-1}* is point-to-point and continuous for an apropriate λ.

**Theorem 3.** *Let **β(ρ) and η(ρ) be given by (7) and (9) respectively. If T : H → P(H) is ρ-semimonotone then the operator (T + * λI)^{-1} *is Lipschitz continuous for all * λ ∈ *(**β(ρ), η(ρ)), with Lipschitz constant L (* λ*) given by*

*and henceforth point-to-point.*

**Proof.** Take *u, v * ∈ *H*, *x * ∈ *(T + * λ*I) ^{-1}(u)* and

*y*∈

*(T +*λ

*I)*We must prove that

^{-1}(v).Note that *u - * λ*x * ∈ *T(x), v - * λ*y * ∈ *T(y*), so that, applying Definition 2,

Expanding the last term in the leftmost expression of (16) and rearranging, we get

From the fact that λ ∈ *(**β, h),* it follows easily that , so that, taking *u = v* in (17), we obtain that *x = y*, and henceforth (15) holds when *u =* v. Otherwise, define

and observe that the inequality in (17) is equivalent to

Again, the fact that λ ∈ *(**β, n)* guarantees that the coefficient of ω* ^{2}* in the left hand side of (18) is positive, so that (18) holds iff ω

*>*belongs to the interval whose extrems are the two roots of the quadratic in the left hand side of (18), namely

It is not hard to check that ω* _{1} <* 0

*<*ω

*the right inequality is immediate, and the left one follows easily from the fact that λ belongs to*

_{2};*(*

*β(ρ), η(ρ)).*Since ω

*= ||x - y||/||u - v||*is positive, we conclude that (18) is equivalent to ω

__<__ω

*which is itself equivalent to (15), in view of the definition of*

_{2},*L (*λ

*),*given in (14). The fact that

*(T +*λ

*I)*

^{-1}is point-to-point is an immediate consequence of (15).

**Corollary 3.** *If T : H **→ P(H) is p-semimonotone then the operator (T*^{ -1} *+* λ)^{-1} *is Lipschitz continuous for all * λ ∈ *(**β (ρ), η(ρ)), with Lipschitz constant L (* λ*) given by* (14). *If in addition T is maximal, then T*^{ -1} *+ * λ*I is onto for all * λ ∈ *(**β(ρ),η(ρ)).*

**Proof.** The result follows from Proposition 1, Corollary 2 and Theorem 3. □

**Remark 2.** Note that lim_{}_{ρ→1}- β*(**ρ) =* lim_{}_{ρ→1}- η(ρ) = 1, and that lim_{}_{ρ→0}+ β (ρ) = 0, lim_{}_{ρ→0}+ *n(**ρ) =* + ∞, so that the "regularity window" of a ρ-semimonotone operator *T* (i.e., the interval of values of λ for which *T + * λ*I* is onto and its inverse is Lipschitz continuous), approaches the whole positive halfline when ρ approaches 0, i.e., when *T* approaches plain monotonicity, and reduces to a thin interval around 1 when ρ approaches 1 (remember that when ρ = 1 the inequality in (1) holds for any operator *T*, meaning that no "regularity window" can occur for ρ = 1).

**Remark 3.** Observe that

for all ρ ∈ *(0,* 1), so that 1, ρ and ρ* ^{-1}* always belong to the "regularity window" of a ρ-semimonotone operator

*T.*We present next the values of the Lipschitz constant

*L(*λ

*)*of

*(T +*λ

*I)*for the case in which λ takes these three special values:

^{-1}We state next that the characterization of semimonotonicity presented in Lemma 1 for the one dimensional case is a necessary condition for the general case.

**Corollary 4.** *If T : H **→ P(H) is p-semimonotone then the operator (T + ρ ^{-1}I)^{-1} is Lipschitz continuous with Lipschitz constant equal to * θ

*(*

*ρ)*θ

^{-1}, where*(*

*ρ) is given by*(2).

**Proof.** The result follows from Theorem 3 and Remark 3 with λ *= **ρ ^{-1}*.

A sufficient condition can be stated in terms of expansivity of *T*. We prove next that if *T* is expansive, with expansivity constant larger than or equal to η(ρ) as given by (9) (an assumption stronger than Lipschitz continuity of *(T + **ρ ^{-1}I)^{-1}* with Lipschitz constant equal to θ

*(*

*ρ)*then

^{-1}),*T*is ρ-semimonotone.

**Proposition 7.** *Take **ρ * ∈ (0, 1). *If T : H **→ P(H) is v-expansive with v* __>__ η(ρ), *then T is **ρ-semimonotone.*

**Proof.** Fix *u* ∈ *T(x)* and *v* ∈ *T(y*), with *x* ≠ *y.* Define . Then *t* __>__ v because *T* is v-expansive. Therefore , where *t _{2}* is the largest root of the quadratic as in the proof of Lemma 1. Thus,

for all *x * ≠ *y.* Since the inequality in (1) is trivially valid when *x = y,* the result holds.

**REFERENCES**

[1] A.N. Iusem, G. Kassay and W. Sosa, *On certain conditions for the existence of solutions of equilibrium problems.* Mathematical Programming, 116 (2009), 259-273. [ Links ]

[2] A.N. Iusem, G. Kassay and W. Sosa, *An existence result for equilibrium problems with some surjectivity consequences.* Journal of Convex Analysis, 16 (2009), 807826. [ Links ]

[3] A.N. Iusem and W. Sosa, *New existence results for equilibrium problems.* Nonlinear Analysis, 52 (2002), 621-635. [ Links ]

[4] G. Minty, *A theorem on monotone sets in Hilbert spaces.* Journal of Mathematical Analysis and Applications, 11 (1967), 434^39. [ Links ]

[5] T. Pennanen, *Local convergence of the proximal point method and multiplier methods without monotonicity.* Mathematics of Operations Research, 27 (2002), 170-191. [ Links ]

**Correspondência | Correspondence:**

Rolando Gárciga Otero

Avenida Pasteur, 250

Urca

22290-240 Rio de Janeiro, RJ, Brazil

E-mail: rgarciga@ie.ufrj.br

Received: 15/8/10.

Accepted: 05/1/11.

* Partially supported by CNPq.

** Partially supported by CPq grant no. 301280/86