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CUBO A Mathematical Journal
Vol.13, No¯ 02, (119–126). June 2011

Degree theory for the sum of VMO maps and maximal
monotone maps

Yuqing Chen

Faculty of Applied Mathematics,

Guangdong University of Technology,

Guangdong 510006, P. R. China,

email: ychen64@163.com

Donal O’Regan

Department of Mathematics,

National University of Ireland,

Galway, Ireland,

email: donal.oregan@nuigalway.ie

and

Ravi P. Agarwal

Department of Mathematical Science,

Florida Institute of Technology

Melbourne, FL, 32901, USA,

email: agarwal@fit.edu

ABSTRACT

Let Ω ⊂ Rn be an open bounded domain, f : Ω → Rn a VMO map, and T : D(T ) ⊆

Rn → Rn a maximal monotone map with D(T ) ∩ Ω 6= ∅. We construct a degree for

the sum of f + T , which can be viewed as a generalization of the degree both for VMO

maps and maximal monotone maps.



120 Yuqing Chen, Donal O’Regan & Ravi P. Agarwal CUBO
13, 2 (2011)

RESUMEN

Sea Ω ⊂ Rn un dominio abierto, f : Ω → Rn un mapa VMO, y T : D(T ) ⊆ Rn → Rn

un mapa monotono maximal con D(T ) ∩ Ω 6= ∅. Construimos un grado por la suma de

f + T , que se puede ver como una generalización de la medida, tanto para los mapas de

VMO y para los mapas monotono maximal.

Keywords and phrases: Degree theory, Maximal monotone map.

Mathematics Subject Classification: 47H11, 47H05

1. Introduction

Degree theory for continuous maps in finite dimensional spaces has a long history and has

been extensively studied. In the early 80’s of the last century a degree for some classes of non-

continuous maps was established (see [8,1,17,18] and the references therein). In 1995 and 1996, H.

Brezis and L. Nirenberg [12], [13] invented a degree theory for VMO maps; see [2-6,9-11,19,21,22].

Generally, VMO functions need not be continuous. Another important class of non-continuous

maps is the class of maximal monotone maps, and there is no relation between the VMO maps and

the maximal monotone maps. In this paper, we consider the sum of a VMO map and a maximal

monotone map, and we will define a degree theory for such a map. First we recall some definitions.

Let Ω be an open bounded domain in Rn. The class of bounded mean oscillation functions (see

[20]) are defined as

BMO(Ω) = {f : Ω → Rn is locally integrable, and |f|BMO < ∞},

where |f|BMO = supB⊂Ω
1

m(B)

∫
B

|f(x)−f|dx, f = 1
m(B)

∫
B

f(x)dx (here m(·) represents the Lebesgue

measure), and the class of vanishing mean oscillation functions (see [23]) are defined as

VMO(Ω) = {f : Ω → Rn is locally integrable, and limm(B)→ 0
1

m(B)

∫

B

|f(x) − f|dx = 0},

where B ⊂ Rn is an open ball with its closure contained in Ω. It is well known that if f ∈ VMO,

then fǫ(x) =
1

m(Bǫ(x))

∫
Bǫ(x)

f(y)dy is continuous in ǫ and x where it is defined. Let T : D(T ) ⊂

Rn → Rn be a function. If (h − g, x − y) ≥ 0 for all x, y ∈ D(T ) and h ∈ Tx, g ∈ Ty, then T is

said to be monotone. If T is monotone and T has no monotone extension in Rn, then T is said to

be maximal monotone. It is well known that T is maximal monotone iff T is monotone and T + ǫI

is surjective for all ǫ > 0. If T is maximal monotone, we use Tǫ = (T
−1 + ǫI)−1 to represent the

Yosida approximation, and Rǫ = I − ǫTǫ, the resolvent with respect to Tǫ. For maximal monotone

maps we refer the reader to [7]. Let f : Ω → Rn be a VMO map, T : D(T ) ⊆ Rn → Rn a maximal

monotone map, p ∈ Rn, and D(T ) ∩ Ω 6= ∅. Under appropriate assumptions, see (2.1) below, we

define the degree deg(f + T, Ω ∩ D(T ), p). If T = 0, this degree coincides with the degree for VMO



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Degree theory for the sum of VMO maps and maximal monotone maps 121

maps in [13], and if f = 0, then it coincides with the degree for maximal monotone maps (see

[14-16]).

2. Results

In this section, Ω ⊂ Rn is an bounded open domain, f ∈ VMO(Ω), T : D(T ) ⊆ Rn → Rn is a

maximal monotone map, p ∈ Rn, and Ω ∩ D(T ) 6= ∅. Suppose there exists an open neighborhood

U of ∂Ω in Ω and a constant β > 0 such that

1

m(Bǫ(y))

∫

Bǫ(y)

|f(x) + g − p|dx ≥ β (2.1)

for all 0 < ǫ < 1
2
d(y, ∂Ω), g ∈ Tz, z ∈ D(T ) ∩ Bǫ(y), where Bǫ(y) is an open ball centered at y

with radius ǫ such that Bǫ(y) ⊂ U, and d(y, ∂Ω) is the distance between y and ∂Ω.

We remark that if T = 0, then (2.1) was first used in [13]. If f = 0, then (2.1) is equivalent to

|g − p| ≥ β for all z ∈ D(T ) ∩ U and g ∈ Tz, and in this case Proposition 2.1 below shows that the

assumption p /∈ T (∂Ω ∩ D(T )) will guarantee (2.1) holds.

Proposition 2.1. If p /∈ T (∂Ω ∩ D(T )), then there exists d0 > 0, α0 > 0 such that d(p, Tx) ≥

d0 for all x ∈ Ω ∩ D(T ) with d(x, ∂Ω) < α0.

Proof. Suppose the conclusion is not true. There exist xn ∈ Ω ∩ D(T ), gn ∈ Txn such that

d(xn, ∂Ω) → 0, and gn − p → 0. Without loss of generality, we may assume that xn → x0 ∈ ∂Ω.

Since (gn − g, xn − x) ≥ 0 for all x ∈ D(T ), g ∈ Tx, we have

(p − g, x0 − x) ≥ 0, for all x ∈ D(T ), g ∈ Tx.

Therefore x0 ∈ ∂Ω ∩ D(T ), p ∈ Tx0, which is a contradiction.

As in [13], we define Ωǫ = {x ∈ Ω : d(x, ∂Ω) > 2ǫ} for each ǫ > 0. By definition of VMO

functions, there exists ǫ0 > 0 such that

1

m(Bǫ(x))

∫

Bǫ(x)

|f(y) − f|dy <
β

2
(2.2)

for all ǫ < ǫ0, x ∈ Ω and ǫ <
d(x,∂Ω)

2
. We may also take ǫ0 such that {x ∈ Ω : d(x, ∂Ω) ≤ 3ǫ0} ⊂ U,

where U is the same as in (2.1). Now for 0 < ǫ < ǫ0, and x ∈ ∂Ωǫ ∩ D(T ), g ∈ Tx, by (2.1) and

(2.2), we obtain

|fǫ(x) + g − p| ≥
β

2
, (2.3)

where fǫ(x) =
1

m(Bǫ(x))

∫
Bǫ(x))

f(y)dy.



122 Yuqing Chen, Donal O’Regan & Ravi P. Agarwal CUBO
13, 2 (2011)

Lemma 2.2. Suppose |fǫ(x) + g − p| ≥
β
2
, for x ∈ ∂Ωǫ ∩ D(T ), g ∈ Tx. Then there exists

λ0(ǫ) > 0 such that

p 6= fǫ(x) + Tλ(x), for all x ∈ ∂Ωǫ, λ ∈ (0, λ0(ǫ)).

Proof. If this is not true, there exist λn → 0
+, xn ∈ ∂Ωǫ with xn → x0 ∈ ∂Ωǫ, such that

fǫxn + Tλn xn = p, n ∈ {1, 2, · · · }.

Since fǫxn → fǫx0, Rλn xn = xn − λnTλn xn → x0, the maximal monotonicity of T implies that

x0 ∈ D(T ), and p − fǫx0 ∈ Tx0, which is a contradiction.

Now, assume that (2.1) holds. In view of (2.3) and Lemma 2.2, we define the degree deg(f +

T, Ω ∩ D(T ), p) by

deg(f + T, Ω ∩ D(T ), p) = limǫ→ 0+ limλ→ 0+ deg(fǫ + Tλ, Ωǫ, p). (2.4)

We claim this definition is reasonable. First, for each ǫ < ǫ0, and λ1, λ2 ∈ (0, λ0(ǫ)), since

Ttλ1+(1−t)λ2 x is continuous in (t, x) (see Corollary 2.8 in [15]) we know that {fǫ+Ttλ1+(1−t)λ2 }t∈[0,1]
is a homotopy, so

deg(fǫ + Tλ1 , Ωǫ, p) = deg(fǫ + Tλ2 , Ωǫ, p).

Now, for any ǫ ∈ (0, ǫ0), by the continuity of ft(x) in (t, x) and (2.3), there exists δ > 0 such that

|ft(x) + g − p| >
β

4
,

for |t − ǫ| ≤ δ and x ∈ ∂Ωǫ and g ∈ Tx. The same proof as in Lemma 2.2 guarantees that there

exists λ1 > 0 such that

p 6= ft(x) + Tλ(x), for all x ∈ ∂Ωǫ, |t − ǫ| ≤ δ, λ ∈ (0, λ1),

so deg(ft + Tλ, Ωǫ, p) is well defined for λ ∈ (0, λ1), and |t − ǫ| ≤ δ. By homotopy invariance, we

have

deg(ft + Tλ, Ωǫ, p) = deg(fǫ + Tλ, Ωǫ, p),

so the degree in (2.4) is well defined.

For a measurable function f : Ω → Rn, we recall that the essential range of f is defined as the

smallest closed subset essR(f) such that f(x) ∈ essR(f) a. e. x ∈ Ω (see [12]).

Proposition 2.3. If deg(f + T, Ω ∩ D(T ), p) 6= 0, then p ∈ essR(f) + T (Ω ∩ D(T )).

Proof. Suppose the conclusion is not true. Then exists r > 0 such that B(p, r) ∩ essR(f) +

T (Ω ∩ D(T )) = ∅. Set Σ = Rn \ (B(p, r) − T (Ω ∩ D(T ))). Clearly, essR(f) ⊂ Σ. Also f(x) ∈ essR(f),



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Degree theory for the sum of VMO maps and maximal monotone maps 123

a. e. x ∈ Ω, and f ∈ VMO(Ω), so we deduce that limǫ→ 0+ d(fǫ(x), Σ) = 0 uniformly. Therefore,

there exists ǫ1 ∈ (0, ǫ0) such that

|fǫ(x) − p + g| ≥
r

2
,

for all x ∈ Ω, z ∈ D(T ) ∩ Ω, g ∈ Tz, ǫ ∈ (0, ǫ1).

Thus deg(fǫ + Tλ, Ω, p) = 0 for all λ ∈ (0, λ0(ǫ)), and ǫ ∈ (0, ǫ1). Consequently, it follows

from the definition that deg(f + T, Ω ∩ D(T ), p) = 0, which is a contradiction.

Proposition 2.4. Let {ht(·)}t∈[0,1] be a family of functions in VMO(Ω), and ht(·) depends

continuously on the parameter t in the topology of BMO ∩ L1loc(Ω). Assume that there exists an

open neighborhood U of ∂Ω in Ω and a constant β > 0 such that

1

m(Bǫ(y))

∫

Bǫ(y)

|ht(x) + g − p|dx ≥ β (2.5)

for all 0 < ǫ < 1
2
d(y, ∂Ω), g ∈ Tz, z ∈ D(T ) ∩ Bǫ(y), t ∈ [0, 1], where Bǫ(y) is an open ball

centered at y with radius ǫ such that Bǫ(y) ⊂ U. Then deg(ht + T, Ω ∩ D(T ), p) does not depend

on t ∈ [0, 1].

Proof. Since ht(·) depends continuously on the parameter t in the topology of BMO∩L
1
loc(Ω),

we have

limm(B)→ 0
1

m(B)

∫

B

|ht(x) − ht| = 0, (2.6)

uniformly in t. From (2.5), (2.6), and using the same proof as in (2.3), we know that there exists

ǫ0 > 0, such that

|ht,ǫ(x) + g − p| ≥
β

2
, (2.7)

for all x ∈ ∂Ωǫ ∩ D(T ), g ∈ Tx, t ∈ [0, 1], ǫ ∈ (0, ǫ0). By using the same proof as in Lemma 2.2,

we know that there exists λ(ǫ) > 0, such that

p 6= ht,ǫ(x) + Tλx,

for all x ∈ ∂Ωǫ, t ∈ [0, 1], λ ∈ (0, λ(ǫ)). Thus deg(ht,ǫ + Tλ, Ωǫ, p) does not depend on t for each

ǫ ∈ (0, ǫ0), λ ∈ (0, λ(ǫ)). Thus deg(ht + T, Ω ∩ D(T ), p) does not depend on t ∈ [0, 1].

Corollary 2.5. Let f1, f2 ∈ VMO(Ω) satisfying (2.1). Suppose there exists 0 < β0 < β such

that
1

m(B)

∫

B

|f1(x) − f2(x)|dx < β0,

for all B ⊂ U. Then deg(f1 + T, Ω ∩ D(T ), p) = deg(f2 + T, Ω ∩ D(T ), p).

Proof. Set ht(x) = tf1(x) + (1 − t)f2(x) for t ∈ [0, 1], x ∈ Ω. Then it is easy to see that ht
depends continuous on t in the topology of BMO ∩ L1loc(Ω). Also we have

1

m(Bǫ(y))

∫

Bǫ(y)

|ht(x) + g − p|dx ≥ β − β0



124 Yuqing Chen, Donal O’Regan & Ravi P. Agarwal CUBO
13, 2 (2011)

for all 0 < ǫ < 1
2
d(y, ∂Ω), g ∈ Tz, z ∈ D(T )∩Bǫ(y), t ∈ [0, 1], where Bǫ(y) is an open ball centered

at y with radius ǫ such that Bǫ(y) ⊂ U. Therefore the conclusion follows from Proposition 2.4.

Proposition 2.6. Let Ti : D ⊆ R
n, i = 1, 2, be two maximal monotone maps. If tT1 + (1 − t)T2

is maximal monotone for each t ∈ [0, 1], and there exist an open neighborhood U of ∂Ω in Ω and

a constant β > 0 such that

1

m(Bǫ(y))

∫

Bǫ(y)

|f(x) + gt − p|dx ≥ β (2.8)

for all 0 < ǫ < 1
2
d(y, ∂Ω), gt ∈ [tT1 + (1 − t)T2]z, z ∈ D ∩Bǫ(y)), t ∈ [0, 1], where Bǫ(y) is an open

ball centered at y with radius ǫ such that Bǫ(y) ⊂ U. Then deg(f + [tT1 + (1 − t)T2], Ω ∩ D, p)

does not depend on t ∈ [0, 1].

Proof. By (2.8), using the same proof as in (2.3), we know that there exists ǫ0 > 0, such that

|fǫ(x) + gt − p| ≥
β

2
, (2.9)

for all x ∈ ∂Ωǫ ∩ D, gt ∈ tT1x + 1 − t)T2x, t ∈ [0, 1], ǫ ∈ (0, ǫ0). From (2.9), and using the same

proof as in Lemma 2.2, we know that there exists λ(ǫ) > 0, such that

p 6= fǫ(x) + T
t
λx,

for all x ∈ ∂Ωǫ, t ∈ [0, 1], λ ∈ (0, λ(ǫ)), where T
t
λ is the Yosida approximation of tT1 + (1 − t)T2.

From Lemma 2.7 in [15], we know

deg(fǫ + T
t
λ, Ωǫ, p)

does not depend on t ∈ [0, 1], λ ∈ (0, λ(ǫ)). Therefore, deg(f + [tT1 + (1 − t)T2], Ω ∩ D, p) does not

depend on t ∈ [0, 1].

Acknowledgement: The first author was supported by a NSFC grant, grant no. 10871052.

Received: April 2009. Revised: May 2010.

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	Introduction
	 Results