INTRODUCTION In this text, Ω is a nonempty, open, bounded, and connected subset of the Euclidean spaceRk, andλ is the Lebesgue measure on Ω

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 prob- lem we define a nonlinear operator on the space of essentially bounded functions, and we associate a sub-Markovian nonlinear resolvent to it. Since thep-Laplace boundary-value problem is an example of a quasilinear elliptic boundary-value problem, this result generalizes the corresponding assertion obtained by the au- thor for thep-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 spaceR^k, andλ is the Lebesgue measure on Ω. The Euclidean scalar product, and its corresponding norm are denoted by h·,·i, respectively

| · |. Letp be a real number which is strictly greater than 1 and p⁰ := _p−1^p be its H¨older conjugate.

As usual L^p(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 f in L^p(Ω), kfk_p :=

R

Ω

|f|^pdλ ¹

p

. Similarly, L^∞(Ω) is the space of all real- valued Lebesgue measurable functions on Ω that are essentially bounded on Ω and

kfk_∞:= inf{α∈(0,∞) :|f| ≤λ a.e. on Ω}.

We do not make any distinction between a function and its class inL^p(Ω), so that the equalities and inequalities hold in the sense of classes orλ a.e. for the representatives.

Furthermore, L^p(Ω;k) is the space of allR^k-valued Lebesgue measurable functions f = (f₁, f₂,· · · , f_k) on Ω such that |f|^p =

s k

P

j=1

f_j²

!p

is Lebesgue

MATH. REPORTS15(65),4(2013), 511–521

integrable; for every such function f, kfk_p :=

R

Ω

|f|^pdλ ¹_p

. Obviously, we have L^p(Ω; 1) =L^p(Ω).

For each normed space (E,k · ||), E⁰ denotes the spaces of all linear con- tinuous functionals onE, andσ(E, E⁰) is the weakest (smallest) topology ofE such that every function f inE⁰ is continuous with respect to it.

The normed dual space of L^p(Ω) (respectively of L^p(Ω;k)) is identified with

L^p0(Ω),k · k_p⁰

(respectively

L^p0(Ω;k),k · k_p⁰

), and (L¹(Ω),k · k₁)⁰ = (L^∞(Ω),k · k_∞).

Let H^1,p(Ω) be the Sobolev space on Ωi.e. the completion of the space {ϕ∈C^∞(Ω) :kϕk_p+k∇ϕk_p <∞}=:E

with respect to the norm

(ϕ7→ kϕk_p+k∇ϕk_p=:kϕk1·p) :E →[0,∞).

Furthermore, H₀^1,p(Ω) denotes the closure of C_c^∞(Ω) in the space (H^1,p(Ω),k · k_1,p).

As usual

(H^−1,p0(Ω),k · k_−1,p⁰) = (H₀^1,p(Ω),k · k_1,p)⁰, and (H^1,p(Ω)^∼,k · k^∼_1,p) = (H^1,p(Ω),k · k_1,p)⁰.

In this sense, the canonical pairing on H₀^1,p(Ω) ×H^−1,p0(Ω), and on H^1,p(Ω)×H^1,p(Ω)^∼ also, is denoted by (·,·)_1,p. The similar application on L^p(Ω)×L^p0(Ω) (respectively on L^p(Ω;k)×L^p0(Ω;k)) is denoted by (·,·)_p.

The inequalities of Poincar´e, 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 existC_P =C(k, λ(Ω)) (respectivelyC_S =C(k, λ(Ω)) and χ=χ(p, k)∈(1,∞)) such that for allu∈H₀^1,p(Ω) we have that

kuk_p≤C_Pk∇uk_p (the Poincar´e inequality) and

kuk_χp ≤CSk∇uk_p (the Sobolev inequality).

Following [4, 6] or [16] we recall the basic notions of our work. We consider T, N,{V_p : p ∈ (0,∞)} functions from L^∞(Ω) into L^∞(Ω), V := (V_p)_p∈(0,∞), and I the identity map of L^∞(Ω).

Definition 1.1. (i). (a). An increasing function T is called nonlinear op- erator.

(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). ThenT (respec- tively 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 V_p =V_q(I+ (q−p)V_p) for all p, q∈(0,∞) thenV is called a nonlinear resolvent on L^∞(Ω).

(b). Let V be a (nonlinear) resolvent onL^∞(Ω) such thatpV_p is a (non- linear) sub-Markovian operator for all p ∈ (0,∞). Then V is called a sub- Markovian resolvent.

(c). Assume that for all p∈(0,∞) we have

T =V_p(I+pT), and V_p=T(I−pV_p).

Then, either T is called the initial operator of the resolventV, or we say that the resolvent is generated by T.

In his paper, C. Dellacherie [4] fixed a framework of the nonlinear po- tential 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-Amp`ere boundary-valued problem in [14], and to the p-Laplace boundary-value problem, respectively in [17]. For general theory of the non- linear 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 correspond- ing 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.

(2). Whenp∈(2,∞) we prove the existence of the anticonjugate operator V₁^h ofV^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 ofp.

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 functiona: Ω×R^k→R^k such that:

(QE₁). For all ξ ∈R^k the partial mapping a(·, ξ) : Ω →R^k is Lebesgue measurable.

(QE₂). The partial application a(x,·) :R^k →R^k is continuous on R^k, λ a.e. with respect to x∈Ω.

(QE₃). There exists a positive numberδ such that for allξ∈R^k

|a(x, ξ)| ≤δ|ξ|^p−1 λ a.e. with respect tox∈Ω.

(QE₄). For allξ∈R^k we have

ha(x, ξ), ξi ≥ |ξ|^p λ a.e. with respect tox∈Ω.

(QE5). Ifξ, and η are different vectors fromR^k

ha(x, ξ)−a(x, η), ξ−ηi>0 λ a.e. with respect tox∈Ω.

Definition 2.1.For allf ∈L^p(Ω;k) we defineλa.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 allf ∈L^p(Ω;k) the functionα(f) is Lebesgue measurable.

(QE₂⁰). If{f_n}_n∈N^?⊂L^p(Ω;k), is such that (fn)nisλa.e. convergent to f₀ on Ω, then (α(f_n))_n isλa.e. convergent toα(f₀) on Ω.

(QE₃⁰). For allf ∈L^p(Ω;k) we have that

|α(f)| ≤δ|f|^p−1 λ a.e. on Ω.

(QE₄⁰). Iff ∈L^p(Ω;k) then

hα(f), fi ≥ |f|^p λ a.e. on Ω.

(QE₅⁰). For allf, g∈L^p(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 allf ∈L^p(Ω;k) we have the following assertions:

(a). α(f)∈L^p0(Ω;k); (b). kα(f)k_p⁰ ≤δkfk^p−1_p ; (c). (α(f), f)p ≥ kfk^p_p. (ii). The operator α : ((L^p(Ω;k),k · k_p) → (L^p0(Ω;k),k · k_p⁰) is coercive, bounded, and nonlinear operator.

Theorem 2.4. Let us consider {f_n}_n∈N⊂L^p(Ω;k) such that

n→∞lim(fn−f0, α(fn)−α(f0))p = 0.

We have the following properties:

(i). The set{f_n}_n∈N^?(respectively{α(f_n)}_f∈N^?) are bounded in((L^p(Ω);k), k · k_p) (respectively ((L^p0(Ω);k),k · k_p⁰)).

(ii). The sequence (f_n)n∈N^? and (α(f_n))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 H¨older’s inequality it results that

∀n∈N^?, kf_nk^p_p ≤(fn−f0, α(fn)−α(f0))p+δkf_nk_pkf₀k^p−1_p +δkf₀k_pkf_nk^p−1_p so that

lim sup

n→∞

kf_nk^p_p≤δ

lim sup

n→∞

kf_nk_p

· kf₀k^p−1_p +δkf₀k_plim sup

n→∞

kf_nk^p−1_p . The last inequality shows us that (kf_nk_p)_n∈

Nis a bounded sequence, hence by (QE₃⁰), kα(f_n)k_p⁰

n∈N^? is bounded.

(ii). By the hypothesis, and (QE₅⁰) we have (hf_n−f₀, α(f_n)−α(f₀)i)_n∈

is convergent to 0 in L¹(Ω),k · k₁ N

, hence there exists (f_kn)_n∈

N^? such that

n→∞lim hf_kn−f0, α(fkn)−α(f0)i= 0 λ a.e. on Ω.

From the conditions (QE) it follows that the sequences (f_kn)_n∈

N^? and (α(f_kn))_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 {f_n}_n∈

N^?, re- spectively {α(f_n)}_n∈

N^? are both weakly relatively compact in L^p(Ω;k), re- spectively L^p0(Ω;k). By the standard arguments we obtain that f₀ andα(f₀), are the single weakly limit points of (fn)_n∈N?, and (α(fn))_n∈N?,respectively.

Hence, (f_n)_n∈

N^? and (α(f_n))_n∈

N^? converge weakly to f₀, and α(f₀), respec- tively.

Corollary 2.5. The function

α: (L^p(Ω;k),k · k_p)→(L^p0(Ω;k), σ(L^p0(Ω;k), L^p(Ω;k)) is a continuous nonlinear operator.

Proof. It is obvious.

Remark 2.6. In the following section, we shall consider the quasilinear equation

−div a(·,∇u) =γ, γ∈H^−1,p0(Ω).

(i). As usual, a function u ∈ H^1,p(Ω) is called a (weak) solution of the previous equation if for all ϕ∈H₀^1,p(Ω) we have that

hϕ, γi_1,p=hϕ,−div a(·,∇u)i_1,p= Z

Ω

ha(·,∇u),∇ϕidλ.

(ii). Moreover, a boundary-value solution of the mentioned quasilinear equation with data h∈H^1,p(Ω) is the solution of the equation

−div a(·,∇u+∇h) =γ, γ∈H^−1,p0(Ω).

(iii). If a(x, ξ) =|ξ|^p−2·ξ, for all x∈Ω, and ξ∈R^k then a(·,∇u) =|∇u|^p−2∇u, and −div (|∇u|^p−2∇u) = ∆_pu

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 fromH^1,p(Ω).

Definition 3.1.For allu∈H^1,p(Ω), let A_hu be equal toα(∇u+∇h) i.e.

(A_hu)(x) :=a(x,∇u(x) +∇h(x)) λ a.e. on Ω.

Remark 3.2.(i). In view of the conditions (QE⁰) and Lemma 2.3, we have the following properties.

(QE₁⁰⁰). For all u ∈ H^1,p(Ω), the function A_hu ∈ L^p0(Ω;k) (and by the canonical embedding Ahu∈H^−1,p0(Ω)).

(QE₃⁰⁰). For all u∈H^1,p(Ω)

kA_huk_−1,p⁰ ≤ kA_huk_p⁰ ≤δk∇u+∇hk^p−1_p ≤δku+hk^p−1_1,p . (QE₄⁰⁰). If u∈H₀^1,p(Ω), then

(u, A_hu)_1,p≥ k∇u+∇hk^p_p−δk∇hk_pk∇u+∇hk^p−1_p .

(QE₅⁰⁰). For all u, v∈H₀^1,p(Ω), it follows that (u−v, A_hu−A_hv)1,p ≥0.

Furthemore, if (u−v, A_hu−A_hv)_1,p= 0 then u=v λa.e. on Ω.

(ii). According to the previous point we have that the nonlinear operator A_h: (H₀^1,p(Ω),k · k_1,p)→(H^−1,p0(Ω),k · k_−1,p⁰)

is coercive, strictly monotone and bounded (in conformity with Corollary 2.5).

As function from (H₀^1,p(Ω),k · k_1,p) into (H^−1,p0(Ω), σ(H₀^−1,p0(Ω), H^1,p(Ω)) the nonlinear operator A_h is continuous.

(iii). By the point (ii) the operator A_h satisfies the hypothesis of the Leray-Lions theorem [9], or of the Browder theorem [3], hence it is a surjective mapping. Therefore, for all γ ∈ H^−1,p0(Ω) there exists a unique function u from H₀^1,p(Ω) such thatγ =A_h(u).

Definition 3.3. We shall denote the function u+h by V_γ^h. Therefore, if h+H₀^1,p(Ω) ={h+ϕ:ϕ∈H₀^1,p(Ω)} then

V^h :H^−1,p0(Ω)→h+H₀^1,p(Ω),→H^1,p(Ω).

Remark 3.4.In view of Definition 3.3, we have the following properties.

(i). For allγfromH^−1,p0(Ω),V^hγ is the unique function fromh+H₀^1,p(Ω) such thatA_h(V^hγ) =γ. Therefore, the mappingV^h :H^−1,p0(Ω)→h+H₀^1,p(Ω) is the inverse map of

(f 7→α(∇f)) :h+H₀^1,p(Ω)→H^−1,p0(Ω), i.e. V^h = (Ah)⁻¹. (ii). In particular, A_h(V^hγ) =γ means that for allu∈H₀^1,p(Ω)

γ(u) = (u, γ)_1,p= (u, A_h(V^hγ))_1,p= Z

Ω

Da(·,∇V^hγ),∇uE dλ.

Lemma 3.5. The mapping V^h :H^−1,p0(Ω)→H^1,p(Ω)is increasing.

Proof (similar to [7]). Let us consider γ₁, γ₂ from H^−1,p0(Ω) such that γ₁≤γ₂, and u:= inf(V^hγ₂−V^hγ₁,O). Thenu∈H₀^1,p(Ω), and

∇u=

0_k on

V^hγ₂≥V^hγ₁

∇(V^hγ2)− ∇(V^hγ1) on

V^hγ2≤V^hγ1 . The set F :=

V^hγ₂ ≤V^hγ₁ is Lebesgue measurable, and since A_h is monotone we have that

0≥(u, γ₂−γ₁)_1,p= Z

F

Da(·,∇V^hγ₂)−a(·,∇V^hγ₁),∇V^hγ₂− ∇V^hγ₁E

dλ≥0.

Therefore ∇V^hγ₂ =∇V^hγ₁,λ a.e. on F, so that∇u= 0_k,λa.e. on Ω, which means that u= 0 λa.e. on Ω,i.e. V^hγ₂≥V^hγ₁.

Theorem 3.6 (the complete maximum principle). Let f, andg be essen- tially bounded functions on Ω, andc∈(0,∞) such that

V^hf ≤V^hg+c λa.e. on the set {f > g}.

then

V^hf ≤V^hg+c λ a.e. on Ω.

Proof. Let us defineu:= inf(V^hg+c−V^hf,O),E:=

V^hg+c≥V^hf , and F :=

V^hg+c≤V^hf . Since u ∈ H₀^1,p(Ω), u ≤ 0, and {f > g} ⊂ E, F ⊂ {f ≤g} it follows

0≤ Z

F

(f−g)udλ= Z

Ω

(f−g)udλ= Z

Ω

D

a(·,∇V^hf)−a(·,∇V^hg),∇uE dλ

= Z

F

Da(·,∇V^hf)−a(·,∇V^hg,∇V^hg− ∇V^hfE

dλ≤0.

Therefore,

Da(·,∇V^hf)−a(·,∇V^hg),∇V^hg− ∇V^hfE

= 0 a.e. onF ⇒

∇u= 0 λa.e. on Ω.

So that u= 0λa.e. on Ωi.e. V^hf ≤V^hg+con Ω.

Lemma 3.7 ([8, 15]). For allu, v∈H^1,p(Ω), and s∈R+ we defineu_s:=

(sgn u)(|u| −s)⁺, and vs:= (sgn v)(|v| −s)⁺ Then

|u_s−v_s| ≤ |u−v|.

Theorem 3.8. If γ and h are essentially bounded functions on Ω, then V^hγ is also essentially bounded.

Proof(similar to [16], see also [13] for a particular result). Letu be equal toV^hγ,s₀ ∈[khk_L^∞_(Ω),∞),s∈[s₀,∞) andA_s:={|u| ≥s}. In view of (QE₄⁰)

k∇u_sk^p_p ≤ Z

Ω

ha(·,∇u_s),∇u_sidλ= Z

As

ha(·,∇u),∇uidλ= (u_s, A_hu)_1,p,

and

k∇u_sk^p_p ≤(u_s, A_hu)_1,p= Z

Ω

u_sγdλ≤ k1_Asγk_p⁰ku_sk_p ≤λ(A_s)

1

p0kγk_∞ku_sk_p.

Therefore, by the Poincar´e inequality

k∇u_sk_p≤C_P^p0−1λ(As)^1pkγk^p_∞⁰⁻¹, and by the Sobolev inequality, for all s, t∈[s0,∞) with s > t

(s−t)λ(As)^λp1 ≤( Z

As

(|u| −t)^χpdλ)^χp1 ≤

≤( Z

At

(|u| −t)^χpdλ)^χp1 = ( Z

Ω

|u_t|^χpdλ)^χp1 ≤C_Sk∇u_tk_p ≤C_SC_P^p0−1λ(A_t)^1pkγk^p_∞⁰⁻¹

⇒λ(As)≤ C_SC_P^p0−1λ(A_t)^1pkγk^p_∞⁰⁻¹

(s−t)^χp λ(At)^χ. According to [8], if

ϕ: [s₀,∞)→[0,∞),ϕ(t) :=λ(A_t), andd:=C_SC_P^p0−1ϕ(s₀)

χ−1

χp 2^χp−1χp kγk^p_∞⁰⁻¹ thenλ(As,+d) = 0, i.e. |u| ≤s0+d λa.e. on Ω, hence

kV^hγk_L∞(Ω) ≤s₀+d⇒ kV^hγk_L∞(Ω) ≤ khk_L∞(Ω)+C_SC_P^p0−1λ(Ω)

χ−1

χp 2^χp−1χp kγk^p_∞⁰⁻¹.

4.THE NONLINEAR RESOLVENT ASSOCIATED WITH THE OPERATORV^h

In this section, we consider p ∈(2,∞), and we remark that sinceL^p(Ω) is continuously embedded inL^p0(Ω), we have thatH^1,p(Ω) is also continuously embedded in H^−1,p0(Ω).

Definition 4.1. We shall denote byJthe continuous embedding ofH^1,p(Ω) inH^−1,p0(Ω) and we shall consider the nonlinear operator

J+Ah:H^1,p(Ω)→H^−1,p0(Ω), h∈H^1,p(Ω).

Proposition 4.2. We have the following assertions.

(i). The nonlinear operator J +A_h is strictly monotone, bounded and a continuous mapping from H^1,p(Ω,k · k_1,p

into

H^−1,p0(Ω), σ

H^−1,p0(Ω), H₀^1,p(Ω)

.

(ii). The functionJ +A_h is coercive on

H₀^1,p(Ω,k · k_1,p . Proof. The assertions are consequences of Remark 3.2.(ii).

Corollary 4.3. For all h ∈H^1,p(Ω), and γ ∈H^−1,p0(Ω) there exists a unique element V₁^hγ from h+H₀^1,p(Ω)such that (J +A_h)(V₁^hγ) =γ.

Proof. We use again either the Leray-Lions theorem or the Browder theorem.

Remark 4.4. (i). The nonlinear operatorV₁^h is the inverse of the function (J+Ah)|_h+H1,p

0 (Ω).

(ii). Furthemore, by the definitions for all ϕ∈ H₀^1,p(Ω), γ ∈H^−1,p0(Ω), and h∈H^1,p(Ω) we have that

γ(ϕ) = (ϕ, γ)_1,p= (ϕ,(J+A_h)(V₁^hγ))_1,p = (ϕ, V₁^hγ)_p+ (ϕ, A_h(V₁^hγ))_1,p

= Z

Ω

ϕV₁^hγdλ+ Z

Ω

Da(·,∇V₁^hγ),∇ϕE dλ.

(iii). Similarly to the results of Theorems 3.6 and 3.8, we can prove that V₁^h satisfies the complete maximum principle, and if h ∈ H^1,p(Ω)∩L^∞(Ω), thenV₁^h(L^∞(Ω)⊂L^∞(Ω).

Theorem 4.5. IfI−1,p⁰, denotes the identity map of the space H^−1,p0(Ω), then for all h∈H^1,p(Ω) the following assertions hold:

(i). V^h(I−1,p⁰−V₁^h) =V₁^h. (ii). V₁^h(I−1,p⁰ +V^h) =V^h.

Proof. (i). In view of the Definitions 3.3 and 4.1, for allγ ∈H^−1,p0(Ω) it follows that

A_h(V₁^hγ) =γ−V₁^hγ =A_hV^h(γ−V₁^hγ).

Since the functions V₁^hγ, andV^h(γ−V₁^h) are fromh+H₀^1,p(Ω) and the nonlinear operatorA_h is one to one onh+H₀^1,p(Ω) we have that for allγ from H^−1,p0(Ω)

V₁^hγ =V^h(γ−V₁^hγ)⇔V^h(I−1,p⁰−V₁^h) =V₁^h. (ii). Similarly, for all γ ∈H^−1,p0(Ω),A(V^hγ) =γ so that

(J+A_h)(V^hγ) =V^hγ+γ, V^hγ ∈h+H₀^1,p(Ω)⇒

V^hγ = (J+Ah)⁻¹(J+Ah)V^hγ = (J+Ah)⁻¹(γ+V^hγ) =V₁^h(γ+V^hγ), therefore,

V^h=V₁^h(I−1,p⁰+V^h).

Remark 4.6. (i). Lethbe a function fromH^1,p(Ω)∩L^∞(Ω), and letI be the identity map of the spaceL^∞(Ω). By the preceding theorem, on the space L^∞(Ω) we have the following identities

(I+V^h)(I −V₁^h) =I = (I−V₁^h)(I+V^h)

i.e. (V^h, V₁^h) 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] ) V₁^h is a sub-Markovian nonlinear operator on L^∞(Ω).

Theorem 4.7. For all h∈H^1,p(Ω)∩L^∞(Ω)there exists a nonlinear sub- Markovian resolvent (V_α^h)_α∈(0,∞) associated to V^h.

Proof. We can apply Remark 4.4. (i), and Corollary 4.3 to the operators αJ +A_h (where α ∈ (0,∞), and then V_α^h is the inverse function of (αJ + A_h)|_h+H1,p(Ω) (or we can use the results from [6, 17]).

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Received 1 August 2013 University of Pite¸sti,

Department of Mathematics-Computer Science, Str. Tˆargu din Vale, No.1,

110040 Pite¸sti, Jud.Arge¸s, Romania

corneliu udrea@yahoo.com