NONLINEAR RESOLVENTS
AND QUASILINEAR ELLIPTIC EQUATIONS
CORNELIU UDREA
This article deals with the nonlinear potential theory associated to a quasilinear
equation. In his paper, C. Dellacherie ([4]) showed that nonlinear kernels can
also have a resolvent associated to them. In this work, we construct an example
of this type; more precisely, by solving a quasilinear elliptic boundary-value problem we define a nonlinear operator on the space of essentially bounded functions,
and we associate a sub-Markovian nonlinear resolvent to it. Since the p-Laplace
boundary-value problem is an example of a quasilinear elliptic boundary-value
problem, this result generalizes the corresponding assertion obtained by the author for the p-Laplace operator.
AMS 2010 Subject Classification: Primary 31C45, Secondary 31D05, 35J62.
Key words: nonlinear potential theory, quasilinear equation, complete maximum
principle, resolvent.
1. INTRODUCTION
In this text, Ω is a nonempty, open, bounded, and connected subset of
the Euclidean space Rk , and λ is the Lebesgue measure on Ω. The Euclidean
scalar product, and its corresponding norm are denoted by h·, ·i, respectively
p
| · |. Let p be a real number which is strictly greater than 1 and p0 := p−1
be
its Hölder conjugate.
As usual Lp (Ω) denotes the space of all real valued Lebesgue measurable
functions f on Ω such that |f |p is Lebesgue integrable on Ω; for all functions
1
p
R
p
p
f in L (Ω), kf kp :=
|f | dλ . Similarly, L∞ (Ω) is the space of all realΩ
valued Lebesgue measurable functions on Ω that are essentially bounded on Ω
and
kf k∞ := inf{α ∈ (0, ∞) : |f | ≤ λ a.e. on Ω}.
We do not make any distinction between a function and its class in Lp (Ω),
so that the equalities and inequalities hold in the sense of classes or λ a.e. for
the representatives.
Furthermore, Lp (Ω; k) is the space of all Rk -valued
sLebesgue
!p measurable
k
P 2
functions f = (f1 , f2 , · · · , fk ) on Ω such that |f |p =
fj
is Lebesgue
j=1
MATH. REPORTS 15(65), 4 (2013), 511–521
512
Corneliu Udrea
integrable; for every such function f , kf kp :=
2
R
1
|f |p dλ
p
. Obviously, we
Ω
have Lp (Ω; 1) = Lp (Ω).
For each normed space (E, k · ||), E 0 denotes the spaces of all linear continuous functionals on E, and σ(E, E 0 ) is the weakest (smallest) topology of E
such that every function f in E 0 is continuous with respect to it.
p
p
The
0 normed dual
space of L (Ω)
0(respectivelyof L (Ω; k)) is identified
with Lp (Ω), k · kp0 (respectively Lp (Ω; k), k · kp0 ), and (L1 (Ω), k · k1 )0 =
(L∞ (Ω), k · k∞ ).
Let H 1,p (Ω) be the Sobolev space on Ω i.e. the completion of the space
{ϕ ∈ C ∞ (Ω) : kϕkp + k∇ϕkp < ∞} =: E
with respect to the norm
(ϕ 7→ kϕkp + k∇ϕkp =: kϕk1·p ) : E → [0, ∞).
Furthermore, H01,p (Ω) denotes the closure of Cc∞ (Ω) in the space
· k1,p ).
As usual
(H 1,p (Ω), k
0
(H −1,p (Ω), k · k−1,p0 ) = (H01,p (Ω), k · k1,p )0 , and
1,p
(Ω), k · k1,p )0 .
(H 1,p (Ω)∼ , k · k∼
1,p ) = (H
0
In this sense, the canonical pairing on H01,p (Ω) × H −1,p (Ω), and on
H 1,p (Ω) × H 1,p (Ω)∼ also, is denoted by (·, ·)1,p . The similar application on
0
0
Lp (Ω) × Lp (Ω) (respectively on Lp (Ω; k) × Lp (Ω; k)) is denoted by (·, ·)p .
The inequalities of Poincaré, and Sobolev are powerful tools in the Sobolev
spaces (see [1, 5, 7]. In this text, we use the forms of these inequalities given
in [7]. That is, there exist CP = C(k, λ(Ω)) (respectively CS = C(k, λ(Ω)) and
χ = χ(p, k) ∈ (1, ∞)) such that for all u ∈ H01,p (Ω) we have that
kukp ≤ CP k∇ukp (the Poincaré inequality)
and
kukχp ≤ CS k∇ukp (the Sobolev inequality).
Following [4, 6] or [16] we recall the basic notions of our work. We consider
T, N, {Vp : p ∈ (0, ∞)} functions from L∞ (Ω) into L∞ (Ω), V := (Vp )p∈(0,∞) ,
and I the identity map of L∞ (Ω).
Definition 1.1. (i). (a). An increasing function T is called nonlinear operator.
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Nonlinear resolvents and quasilinear elliptic equations
513
(b). If T is a Lipschitz (respectively, nonexpansive) nonlinear operator,
then T is called nonlinear bounded operator (respectively, nonlinear
sub-Markovian operator ).
(c). Assume that (I + T )(I − N ) = I = (I − N )(I + T ). Then T (respectively N ) are called the conjugate (respectively anticonjugate) operator of N
(respectively T ), and (T, N ) is called a pair of conjugated operators.
(d). We say that T satisfies the complete maximum principle iff
∀f, g ∈ L∞ (Ω), ∀α ∈ (0, ∞) : T f ≤ T g + α on {f > g} ⇒ T f ≤ T g + α,
where {f > g} := {x ∈ Ω : f (x) > g(x)}.
(ii). (a). If Vp = Vq (I + (q − p)Vp ) for all p, q ∈ (0, ∞) then V is called a
nonlinear resolvent on L∞ (Ω).
(b). Let V be a (nonlinear) resolvent on L∞ (Ω) such that pVp is a (nonlinear) sub-Markovian operator for all p ∈ (0, ∞). Then V is called a subMarkovian resolvent.
(c). Assume that for all p ∈ (0, ∞) we have
T = Vp (I + pT ), and Vp = T (I − pVp ).
Then, either T is called the initial operator of the resolvent V, or we say
that the resolvent is generated by T .
In his paper, C. Dellacherie [4] fixed a framework of the nonlinear potential theory; for a proper nonlinear operator (on the continuous functions)
he proved in that paper a Meyer type theorem, and a Hunt type theorem.
H. Maagli [10, 11] studied the semilinear perturbation of a linear resolvent
which is a nonlinear resolvent; N. Yazidi constructed some examples of the
nonlinear resolvents in [18]. The author built nonlinear resolvents associated
to the Monge-Ampère boundary-valued problem in [14], and to the p-Laplace
boundary-value problem, respectively in [17]. For general theory of the nonlinear resolvent see also [2, 6, 16].
In this direction, here we consider the quasilinear elliptic boundary-value
problem (as in [7]); according to the Leray-Lions theorem [9] (or by the Browder
theorem [3]) we obtain some similar results which generalize the corresponding assertions from [17]. Obviously, there are some similarities between the
techniques used in this work, and the ones used in [17] because the p-Laplace
equation is an example of quasilinear equation.
The main results of this work are the following ones.
(1). For a function h ∈ H 1,p (Ω) a nonlinear operator V h on L∞ (Ω) is
defined, and we shall prove that this operator is weakly continuous and satisfies
the complete maximum principle.
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Corneliu Udrea
4
(2). When p ∈ (2, ∞) we prove the existence of the anticonjugate operator
V1h of V h which is sub-Markovian; following the nonlinear technique, we show
the existence of a nonlinear sub-Markovian resolvent on L∞ (Ω) associated to
V h for this values of p.
At least two more problems related to this topic remain open.
(1). The existence of the nonlinear resolvent associated with V h for
p ∈ (1, 2).
(2). Determining the class of the excessive functions with respect to the
obtained resolvent.
2. THE ELLIPTIC QUASILINEAR EQUATION
From now on, we shall consider (similar to [7]) a function a : Ω×Rk → Rk
such that:
(QE1 ). For all ξ ∈ Rk the partial mapping a(·, ξ) : Ω → Rk is Lebesgue
measurable.
(QE2 ). The partial application a(x, ·) : Rk → Rk is continuous on Rk , λ
a.e. with respect to x ∈ Ω.
(QE3 ). There exists a positive number δ such that for all ξ ∈ Rk
|a(x, ξ)| ≤ δ|ξ|p−1
λ a.e. with respect tox ∈ Ω.
(QE4 ). For all ξ ∈ Rk we have
ha(x, ξ), ξi ≥ |ξ|p
λ a.e. with respect tox ∈ Ω.
(QE5 ). If ξ, and η are different vectors from Rk
ha(x, ξ) − a(x, η), ξ − ηi > 0 λ a.e. with respect tox ∈ Ω.
Definition 2.1. For all f ∈ Lp (Ω; k) we define λ a.e. on Ω the mapping
α(f )(x) := a(x, f (x)).
Remark 2.2. In view of Definition 2.1 and properties (QE) the following
assertions hold:
(QE10 ). For all f ∈ Lp (Ω; k) the function α(f ) is Lebesgue measurable.
(QE20 ). If {fn }n∈N? ⊂ Lp (Ω; k), is such that (fn )n is λ a.e. convergent to
f0 on Ω, then (α(fn ))n is λ a.e. convergent to α(f0 ) on Ω.
(QE30 ). For all f ∈ Lp (Ω; k) we have that
|α(f )| ≤ δ|f |p−1
λ a.e. on Ω.
(QE40 ). If f ∈ Lp (Ω; k) then
hα(f ), f i ≥ |f |p
λ a.e. on Ω.
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Nonlinear resolvents and quasilinear elliptic equations
515
(QE50 ). For all f, g ∈ Lp (Ω; k), it follows that
hα(f ) − α(g), f − gi ≥ 0
λ a.e. on Ω.
Furthemore, if hα(f ) − α(g), f − gi = 0 λ a.e. on Ω, then f = g λ a.e.
on Ω.
Lemma 2.3. (i). For all f ∈ Lp (Ω; k) we have the following assertions:
0
p
(a). α(f ) ∈ Lp (Ω; k); (b). kα(f )kp0 ≤ δkf kp−1
p ; (c). (α(f ), f )p ≥ kf kp .
0
p
p
(ii). The operator α : ((L (Ω; k), k · kp ) → (L (Ω; k), k · kp0 ) is coercive,
bounded, and nonlinear operator.
Theorem 2.4. Let us consider {fn }n∈N ⊂ Lp (Ω; k) such that
lim (fn − f0 , α(fn ) − α(f0 ))p = 0.
n→∞
We have the following properties:
(i). The set {fn }n∈N? (respectively {α(fn )}f ∈N? ) are bounded in ((Lp (Ω); k),
0
k · kp ) (respectively ((Lp (Ω); k), k · kp0 )).
(ii). The sequence (fn )n∈N? and (α(fn ))n∈N? converges weakly to f0 and
α(f0 ) respectively.
Proof. (i). (similar to [12]). In view of the conditions (QE03 ) and (QE04 )
and by Hölder’s inequality it results that
∀n ∈ N? ,
kfn kpp ≤ (fn − f0 , α(fn ) − α(f0 ))p + δkfn kp kf0 kp−1
+ δkf0 kp kfn kp−1
p
p
so that
lim sup kfn kpp
n→∞
≤ δ lim sup kfn kp · kf0 kp−1
+ δkf0 kp lim sup kfn kpp−1 .
p
n→∞
n→∞
The last inequality
shows us that (kfn kp )n∈N is a bounded sequence, hence
0
by (QE3 ), kα(fn )kp0 n∈N? is bounded.
0
(ii). By the hypothesis, and (QE
5 ) we have (hfn − f0 , α(fn ) − α(f0 )i)n∈N
1
is convergent to 0 in L (Ω), k · k1 , hence there exists (fkn )n∈N? such that
lim hfkn − f0 , α (fkn ) − α (f0 )i = 0 λ a.e. on Ω.
n→∞
From the conditions (QE) it follows that the sequences (fkn )n∈N? and
(α (fkn ))n∈N? converge λ a.e. on Ω to f0 , and α(f0 ), respectively. Moreover
(fkn )n∈N? , and (α((fkn ))n∈N? , converge weakly.
On the other hand, by Alaoglu-Bourbaki theorem the set {fn }n∈N? , respectively {α (fn )}n∈N? are both weakly relatively compact in Lp (Ω; k), re0
spectively Lp (Ω; k). By the standard arguments we obtain that f0 and α(f0 ),
are the single weakly limit points of (fn )n∈N? , and (α (fn ))n∈N? , respectively.
Hence, (fn )n∈N? and (α (fn ))n∈N? converge weakly to f0 , and α(f0 ), respectively. 516
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6
Corollary 2.5. The function
0
0
α : (Lp (Ω; k), k · kp ) → (Lp (Ω; k), σ(Lp (Ω; k), Lp (Ω; k))
is a continuous nonlinear operator.
Proof. It is obvious.
Remark 2.6. In the following section, we shall consider the quasilinear
equation
0
−div a(·, ∇u) = γ, γ ∈ H −1,p (Ω).
(i). As usual, a function u ∈ H 1,p (Ω) is called a (weak) solution of the
previous equation if for all ϕ ∈ H01,p (Ω) we have that
Z
hϕ, γi1,p = hϕ, −div a(·, ∇u)i1,p = ha(·, ∇u), ∇ϕi dλ.
Ω
(ii). Moreover, a boundary-value solution of the mentioned quasilinear
equation with data h ∈ H 1,p (Ω) is the solution of the equation
0
−div a(·, ∇u + ∇h) = γ, γ ∈ H −1,p (Ω).
(iii). If a(x, ξ) = |ξ|p−2 · ξ, for all x ∈ Ω, and ξ ∈ Rk then
a(·, ∇u) = |∇u|p−2 ∇u, and − div (|∇u|p−2 ∇u) = ∆p u
for all u ∈ H 1,p (Ω). So that, in this case, we obtain the p-Laplace operator.
3. THE INITIAL OPERATOR OF THE RESOLVENT
From now on, h is a Sobolev function from H 1,p (Ω).
Definition 3.1. For all u ∈ H 1,p (Ω), let Ah u be equal to α(∇u + ∇h) i.e.
(Ah u)(x) := a(x, ∇u(x) + ∇h(x)) λ a.e. on Ω.
Remark 3.2. (i). In view of the conditions (QE 0 ) and Lemma 2.3, we have
the following properties.
0
(QE100 ). For all u ∈ H 1,p (Ω), the function Ah u ∈ Lp (Ω; k) (and by the
0
canonical embedding Ah u ∈ H −1,p (Ω)).
(QE300 ). For all u ∈ H 1,p (Ω)
kAh uk−1,p0 ≤ kAh ukp0 ≤ δk∇u + ∇hkp−1
≤ δku + hkp−1
p
1,p .
(QE400 ). If u ∈ H01,p (Ω), then
(u, Ah u)1,p ≥ k∇u + ∇hkpp − δk∇hkp k∇u + ∇hkp−1
p .
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Nonlinear resolvents and quasilinear elliptic equations
517
(QE500 ). For all u, v ∈ H01,p (Ω), it follows that
(u − v, Ah u − Ah v)1,p ≥ 0.
Furthemore, if (u − v, Ah u − Ah v)1,p = 0 then u = v λ a.e. on Ω.
(ii). According to the previous point we have that the nonlinear operator
0
Ah : (H01,p (Ω), k · k1,p ) → (H −1,p (Ω), k · k−1,p0 )
is coercive, strictly monotone and bounded (in conformity with Corollary 2.5).
0
0
As function from (H01,p (Ω), k · k1,p ) into (H −1,p (Ω), σ(H0−1,p (Ω), H 1,p (Ω)) the
nonlinear operator Ah is continuous.
(iii). By the point (ii) the operator Ah satisfies the hypothesis of the
Leray-Lions theorem [9], or of the Browder theorem [3], hence it is a surjective
0
mapping. Therefore, for all γ ∈ H −1,p (Ω) there exists a unique function u
from H01,p (Ω) such that γ = Ah (u).
Definition 3.3. We shall denote the function u + h by Vγh . Therefore, if
h + H01,p (Ω) = {h + ϕ : ϕ ∈ H01,p (Ω)} then
0
V h : H −1,p (Ω) → h + H01,p (Ω) ,→ H 1,p (Ω).
Remark 3.4. In view of Definition 3.3, we have the following properties.
0
(i). For all γ from H −1,p (Ω), V h γ is the unique function from h+H01,p (Ω)
0
such that Ah (V h γ) = γ. Therefore, the mapping V h : H −1,p (Ω) → h+H01,p (Ω)
is the inverse map of
0
(f 7→ α(∇f )) : h + H01,p (Ω) → H −1,p (Ω), i.e. V h = (Ah )−1 .
(ii). In particular, Ah (V h γ) = γ means that for all u ∈ H01,p (Ω)
Z D
E
γ(u) = (u, γ)1,p = (u, Ah (V h γ))1,p =
a(·, ∇V h γ), ∇u dλ.
Ω
Lemma 3.5. The mapping V h : H
−1,p0
(Ω) → H 1,p (Ω) is increasing.
0
Proof (similar to [7]). Let us consider γ1 , γ2 from H −1,p (Ω) such that
γ1 ≤ γ2 , and u := inf(V h γ2 − V h γ1 , O). Then u ∈ H01,p (Ω), and
0k
on V h γ2 ≥ V h γ1 ∇u =
.
∇(V h γ2 ) − ∇(V h γ1 ) on V h γ2 ≤ V h γ1
The set F := V h γ2 ≤ V h γ1 is Lebesgue measurable, and since Ah is
monotone we have that
Z D
E
0 ≥ (u, γ2 − γ1 )1,p =
a(·, ∇V h γ2 ) − a(·, ∇V h γ1 ), ∇V h γ2 − ∇V h γ1 dλ ≥ 0.
F
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Corneliu Udrea
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Therefore ∇V h γ2 = ∇V h γ1 , λ a.e. on F , so that ∇u = 0k , λ a.e. on Ω,
which means that u = 0 λ a.e. on Ω, i.e. V h γ2 ≥ V h γ1 . Theorem 3.6 (the complete maximum principle). Let f , and g be essentially bounded functions on Ω, and c ∈ (0, ∞) such that
V hf ≤ V hg + c
λ a.e. on the set {f > g}.
then
V hf ≤ V hg + c
λ a.e. on Ω.
Proof. Let us define u := inf(V h g + c − V h f, O), E := V h g + c ≥ V h f ,
and F := V h g + c ≤ V h f . Since u ∈ H01,p (Ω), u ≤ 0, and {f > g} ⊂ E,
F ⊂ {f ≤ g} it follows
Z
Z
Z D
E
0 ≤ (f − g)udλ = (f − g)udλ =
a(·, ∇V h f ) − a(·, ∇V h g), ∇u dλ
F
Ω
Ω
Z D
E
=
a(·, ∇V h f ) − a(·, ∇V h g, ∇V h g − ∇V h f dλ ≤ 0.
F
Therefore,
D
E
a(·, ∇V h f ) − a(·, ∇V h g), ∇V h g − ∇V h f = 0
a.e. on F ⇒
∇u = 0 λ a.e. on Ω.
So that u = 0 λ a.e. on Ω i.e. V h f ≤ V h g + c on Ω.
Lemma 3.7 ([8, 15]). For all u, v ∈ H 1,p (Ω), and s ∈ R+ we define us :=
(sgn u)(|u| − s)+ , and vs := (sgn v)(|v| − s)+ Then
|us − vs | ≤ |u − v|.
V
hγ
Theorem 3.8. If γ and h are essentially bounded functions on Ω, then
is also essentially bounded.
Proof (similar to [16], see also [13] for a particular result). Let u be equal
to V h γ, s0 ∈ [khkL∞ (Ω) , ∞), s ∈ [s0 , ∞) and As := {|u| ≥ s}. In view of (QE40 )
Z
Z
p
k∇us kp ≤ ha(·, ∇us ), ∇us i dλ = ha(·, ∇u), ∇ui dλ = (us , Ah u)1,p ,
Ω
As
and
k∇us kpp
Z
≤ (us , Ah u)1,p =
Ω
1
us γdλ ≤ k1As γkp0 kus kp ≤ λ(As ) p0 kγk∞ kus kp .
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Nonlinear resolvents and quasilinear elliptic equations
519
Therefore, by the Poincaré inequality
1
0
0
k∇us kp ≤ CPp −1 λ(As ) p kγkp∞−1 ,
and by the Sobolev inequality, for all s, t ∈ [s0 , ∞) with s > t
Z
1
1
λp
(s − t)λ(As ) ≤ ( (|u| − t)χp dλ) χp ≤
As
Z
≤(
1
(|u| − t)χp dλ) χp
Z
1
1
0
0
= ( |ut |χp dλ) χp ≤ CS k∇ut kp ≤ CS CPp −1 λ(At ) p kγkp∞−1
At
Ω
1
0
0
CS CPp −1 λ(At ) p kγkp∞−1
⇒ λ(As ) ≤
λ(At )χ .
(s − t)χp
According to [8], if
0
ϕ : [s0 , ∞) → [0, ∞), ϕ(t) := λ(At ), and d := CS CPp −1 ϕ(s0 )
then λ(As,+d ) = 0, i.e. |u| ≤ s0 + d λ a.e. on Ω, hence
χ−1
χp
χp
0
2 χp−1 kγkp∞−1
kV h γkL∞ (Ω) ≤ s0 + d ⇒
0
kV h γkL∞ (Ω) ≤ khkL∞ (Ω) + CS CPp −1 λ(Ω)
χ−1
χp
χp
0
2 χp−1 kγkp∞−1 .
4. THE NONLINEAR RESOLVENT ASSOCIATED
WITH THE OPERATOR V h
In this section, we consider p ∈ (2, ∞), and we remark that since Lp (Ω)
0
is continuously embedded in Lp (Ω), we have that H 1,p (Ω) is also continuously
0
embedded in H −1,p (Ω).
in
Definition 4.1. We shall denote by J the continuous embedding of H 1,p (Ω)
and we shall consider the nonlinear operator
0
H −1,p (Ω)
0
J + Ah : H 1,p (Ω) → H −1,p (Ω), h ∈ H 1,p (Ω).
Proposition 4.2. We have the following assertions.
(i). The nonlinear operator J + Ah is strictly
bounded and
monotone,
0
0
1,p
−1,p
a continuous mapping from H (Ω, k · k1,p into H
(Ω), σ H −1,p (Ω),
H01,p (Ω) .
1,p
(ii). The function J + Ah is coercive on H0 (Ω, k · k1,p .
Proof. The assertions are consequences of Remark 3.2.(ii).
−1,p0
Corollary 4.3. For all h ∈ H 1,p (Ω), and γ ∈ H
(Ω) there exists a
unique element V1h γ from h + H01,p (Ω) such that (J + Ah )(V1h γ) = γ.
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Proof. We use again either the Leray-Lions theorem or the Browder
theorem. Remark 4.4. (i). The nonlinear operator V1h is the inverse of the function
(J + Ah )|h+H 1,p (Ω) .
0
0
(ii). Furthemore, by the definitions for all ϕ ∈ H01,p (Ω), γ ∈ H −1,p (Ω),
and h ∈ H 1,p (Ω) we have that
γ(ϕ) = (ϕ, γ)1,p = (ϕ, (J + Ah )(V1h γ))1,p = (ϕ, V1h γ)p + (ϕ, Ah (V1h γ))1,p
Z
Z D
E
h
= ϕV1 γdλ +
a(·, ∇V1h γ), ∇ϕ dλ.
Ω
Ω
(iii). Similarly to the results of Theorems 3.6 and 3.8, we can prove that
h
V1 satisfies the complete maximum principle, and if h ∈ H 1,p (Ω) ∩ L∞ (Ω),
then V1h (L∞ (Ω) ⊂ L∞ (Ω).
0
Theorem 4.5. If I−1,p0 , denotes the identity map of the space H −1,p (Ω),
then for all h ∈ H 1,p (Ω) the following assertions hold:
(i). V h (I−1,p0 − V1h ) = V1h .
(ii). V1h (I−1,p0 + V h ) = V h .
0
Proof. (i). In view of the Definitions 3.3 and 4.1, for all γ ∈ H −1,p (Ω) it
follows that
Ah (V1h γ) = γ − V1h γ = Ah V h (γ − V1h γ).
Since the functions V1h γ, and V h (γ − V1h ) are from h + H01,p (Ω) and the
nonlinear operator Ah is one to one on h + H01,p (Ω) we have that for all γ from
0
H −1,p (Ω)
V1h γ = V h (γ − V1h γ) ⇔ V h (I−1,p0 − V1h ) = V1h .
0
(ii). Similarly, for all γ ∈ H −1,p (Ω), A(V h γ) = γ so that
(J + Ah )(V h γ) = V h γ + γ, V h γ ∈ h + H01,p (Ω) ⇒
V h γ = (J + Ah )−1 (J + Ah )V h γ = (J + Ah )−1 (γ + V h γ) = V1h (γ + V h γ),
therefore,
V h = V1h (I−1,p0 + V h ).
H 1,p (Ω) ∩ L∞ (Ω),
Remark 4.6. (i). Let h be a function from
and let I be
∞
the identity map of the space L (Ω). By the preceding theorem, on the space
L∞ (Ω) we have the following identities
(I + V h )(I − V1h ) = I = (I − V1h )(I + V h )
i.e. (V h , V1h ) is a pair of conjugated nonlinear operators on L∞ (Ω).
(ii). Since V h satisfies the complete maximum principle according to
the non-linear version of Hunt’s theorem ([4] or [14] ) V1h is a sub-Markovian
nonlinear operator on L∞ (Ω).
11
Nonlinear resolvents and quasilinear elliptic equations
521
Theorem 4.7. For all h ∈ H 1,p (Ω) ∩ L∞ (Ω)there exists a nonlinear subMarkovian resolvent (Vαh )α∈(0,∞) associated to V h .
Proof. We can apply Remark 4.4. (i), and Corollary 4.3 to the operators
αJ + Ah (where α ∈ (0, ∞), and then Vαh is the inverse function of (αJ +
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Received 1 August 2013
University of Piteşti,
Department of Mathematics-Computer Science,
Str. Târgu din Vale, No.1,
110040 Piteşti, Jud.Argeş,
Romania
corneliu [email protected]