The obtained results are then used to show the correct solvability of a mixed problem for parabolic partial differential equation with non regular boundary conditions

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

BOUNDARY-VALUE PROBLEMS FOR SECOND-ORDER DIFFERENTIAL OPERATORS WITH NONLOCAL BOUNDARY

CONDITIONS

MOHAMED DENCHE, ABDERRAHMANE MEZIANI

Abstract. In this paper, we study a second-order differential operator com- bining weighting integral boundary condition with another two-point boundary condition. Under certain conditions on the weighting functions, called regular and non regular cases, we prove that the resolvent decreases with respect to the spectral parameter inL^p(0,1), but there is no maximal decrease at infinity forp >1. Furthermore, the studied operator generates inL^p(0,1), an analytic semi group forp = 1 in the regular case, and an analytic semi group with singularities forp >1, in both cases, and forp= 1, in the non regular case only. The obtained results are then used to show the correct solvability of a mixed problem for parabolic partial differential equation with non regular boundary conditions.

1. Introduction

In spaceL^p(0,1) we consider the boundary-value problem L(u) :=u⁰⁰=f(x),

B_i(u) :=a_iu(0) +b_iu⁰(0) +c_iu(1) +d_iu⁰(1) +

Z 1

0

Ri(t)u(t)dt+ Z 1

0

Si(t)u⁰(t)dt= 0,

(1.1)

where i= 1,2 and the functionsR_i, S_i belong to C([0,1],C). To problem (1.1) in L^p(0,1) we associate the operator

Lp(u) =u⁰⁰,

with domainD(L_p) ={u∈W^2,p(0,1) :B_i(u) = 0, i= 1,2}.

Many papers and books give the full spectral theory of Birkhoff regular differen- tial operators with two point linearly independent boundary conditions, in terms of coefficients of boundary conditions. The reader should refer to [9, 13, 24, 25, 26, 33, 36, 37] and references therein. Few works were devoted to the study of a non regular situation. The case of separated non regular boundary conditions was studied by, Eberhard, Hopkins, Jakson, Keldysh, Khromov, Seifert, Stone, Ward (see Yakubov

2000Mathematics Subject Classification. 47E05, 35K20.

Key words and phrases. Green’s function; regular and non regular boundary conditions;

semi group with singularities; weighted mixed boundary conditions.

c

2007 Texas State University - San Marcos.

Submitted May 10, 2006. Published April 17, 2007.

1

and Yakubov [37] for exact references). A situation of non regular non-separated boundary conditions was considered by Benzinger [2], Denche [4, 5, 6], Eberhard and Freiling [10], Gasumov and Magerramov [16, 17], Khromov [22], Mamedov [23], Shkalikov [29], Silchenko [31], Tretter [34], Vagabov [35], Yakubov [38] and Yakubov [39].

A mathematical model with integral boundary conditions was derived by [11, 27]

in the context of optical physics. The importance of this kind of problems has been also pointed out by Samarskii [28].

In this paper, we study a problem for second order ordinary differential equation with mixed nonlocal boundary conditions combined weighting integral boundary conditions with another two point boundary conditions. The regular case was studied in the spaceL1(0,1) by Gallardo [15]. The Particular case whereSi(t) = 0, and nonregular boundary conditions is studied by Silchenko [31]. A situation of a variable coefficient ofu⁰⁰(x) in the equation has been treated in [12, 32]. The integral boundary conditions are again non regular but they assume less restrictions on the functions Ri(t) (here again Si(t) = 0, ai =bi = ci = di = 0). In particular the corresponding estimate inL2(0,1) has been established.

Following the technique in [5, 6, 14, 15, 24, 25, 26], we should bound the resolvent in the spaceL^p(0,1) by means of a suitable formula for Green’s function. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called non regular boundary conditions, we prove that the resolvent decreases with respect to the spectral parameter inL^p(0,1), but there is no maximal decreasing at infinity for p≥1. In contrast to the regular case this decreasing is maximal forp= 1 as shown in [14, 15]. Furthermore, the studied operator generates inL^p(0,1) an analytic semi group with singularities forp≥1. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with non regular non local boundary conditions.

2. Green’s Function

Letλ ∈C, u₁(x) = u₁(x, λ) andu₂(x) =u₂(x, λ) be a fundamental system of solutions of equation

L(u)−λu= 0.

Following [24] , the Green’s function of problem (1.1) is G(x, s, λ) = N(x, s, λ)

∆(λ) , (2.1)

where ∆(λ) is the characteristic determinant of the considered problem defined by

∆(λ) =

B₁(u₁) B₁(u₂) B₂(u₁) B₂(u₂)

, (2.2)

and

N(x, s, λ) =

u1(x) u2(x) g(x, s, λ) B1(u1) B1(u2) B1(g)x

B₂(u₁) B₂(u₂) B₂(g)_x

, (2.3)

forx, s∈[0,1]. The functiong(x, s, λ) is defined as follows g(x, s, λ) =±1

2

u2(s)u1(x)−u2(x)u1(s)

u⁰₁(s)u₂(s)−u₁(s)u⁰₂(s), (2.4)

where it takes the plus sign forx > s and the minus forx < s. Given an arbitrary δ∈(0,^π₂), we consider the sector

X

δ ={λ∈C;|arg(λ)| ≤ π

2 +δ, λ6= 0}.

Forλ∈P

δ, defineρas the square root of λwith positive real part. Forλ6= 0, we can consider a fundamental system of solutions of equationu⁰⁰=λu=ρ²ugiven by u1(t) =e^−ρtandu2(t) =e^ρt. In the following we are going to deduce an adequate formulae for ∆(λ) andG(x, s, λ). First of all, fori, j= 1,2, we have

B_i(u_j) =a_i+ (−1)^jρb_i+c_ie^(−1)jρ+ (−1)^jρd_ie^(−1)jρ +

Z 1

0

(Ri(t) + (−1)^jρSi(t))e^(−1)jρtdt.

So we obtain from (2.2),

∆(λ) = (a1−ρb1+c1e^−ρ−ρd1e^−ρ+ Z 1

0

(R1(t)−ρS1(t))e^−ρtdt)

×(a2+ρb2+c2e^ρ+ρd2e^ρ+ Z 1

0

(R2(t) +ρS2(t))e^ρtdt)

−(a2−ρb2+c2e^−ρ−ρd2e^−ρ+ Z 1

0

(R2(t)−ρS2(t))e^−ρtdt)

×(a1+ρb1+c1e^ρ+ρd1e^ρ+ Z 1

0

(R1(t) +ρS1(t))e^ρtdt), (2.5) andg(x, s, λ) has the form

g(x, s, λ) = (₁

4ρ(e^ρ(x−s)−e^ρ(s−x)) ifx > s,

1

4ρ(e^ρ(s−x)−e^ρ(x−s)) ifx < s.

Thus we have Bi(g)x=e^ρs

4ρ h

ai−ρbi−cie^−ρ+ρdie^−ρ +

Z s

0

(R_i(t)−ρS_i(t))e^−ρtdt+ Z 1

s

(−R_i(t) +ρS_i(t))e^−ρtdt]

+e^−ρs

4ρ [−ai−ρbi+cie^ρ+ρdie^ρ

− Z s

0

(R_i(t) +ρS_i(t))e^ρtdt+ Z 1

s

(R_i(t) +ρS_i(t))e^ρtdti ,

where i = 1,2. For x, y ∈ {ai, bi, ci, di} and F, G ∈ {R, S}, we introduce ∆xy = x1y2−x2y1,∆xF(t) =x1F2(t)−x2F1(t),∆F(t, ξ) = F1(t)F2(ξ)−F2(t)F1(ξ), and

∆_{F G}(t, ξ) =F₁(t)G₂(ξ)−F₂(t)G₁(ξ). After a long calculation, formula (2.3) can be written as

N(x, s, λ) =ϕ(x, s;λ) +ϕi(x, s, λ), (2.6) where

ϕ(x, s, λ)

= 1 2ρ

h

e^ρ(x+s){(∆_ac−ρ(∆_ad+ ∆_bc) +ρ²∆_bd)e^−ρ

− Z s

0

(∆cR(t)−ρ(∆dR(t) + ∆cS(t)) +ρ²∆dS(t))e^−ρ(t+1)dt +

Z 1

s

(∆aR(t)−ρ(∆aS(t) + ∆bR(t)) +ρ²∆bS(t))e^−ρtdt +

Z 1

s

Z s

0

(∆R(ξ, t) +ρ(∆RS(t, ξ)−∆RS(ξ, t)) +ρ²∆S(ξ, t))e^−ρ(ξ+t)dξdt}

+e^−ρ(x+s){(∆ac+ρ(∆_ad+ ∆_bc) +ρ²∆_bd)e^ρ

− Z s

0

(∆cR(t) +ρ(∆dR(t) + ∆cS(t)) +ρ²∆dS(t))e^ρ(t+1)dt +

Z 1

s

(∆aR(t) +ρ(∆bR(t) + ∆aS(t)) +ρ²∆bS(t))e^ρtdt +

Z 1

s

Z s

0

(∆R(ξ, t) +ρ(∆RS(ξ, t)−∆RS(t, ξ)) +ρ²∆S(ξ, t))e^ρ(ξ+t)dξdt}i , (2.7) and the functionϕi(x, s, λ) is given by

ϕi(x, s, λ) =

(ϕ₁(x, s, λ) ifx > s,

ϕ2(x, s, λ) ifx < s, (2.8) with

ϕ1(x, s, λ)

= 1 2ρ

he^ρ(x−s){(−∆ac+ρ(∆_ad−∆_bc) +ρ²∆_bd)e^−ρ + 2ρ∆ab+

Z s

0

(∆aR(t) +ρ(∆aS(t) + ∆bR(t))−ρ²∆bS(t))e^ρtdt +

Z s

0

(∆_cR(t) +ρ(∆_cS(t)−∆_dR(t))−ρ²∆_dS(t))e^ρ(t−1)dt

− Z 1

0

(∆_aR(t) +ρ(∆_bR(t)−∆_aS(t))−ρ²∆_bS(t))e^−ρtdt +

Z 1

0

Z s

0

(∆_R(t, ξ) +ρ(∆_RS(ξ, t)−∆_RS(t, ξ))−ρ²∆_S(t, ξ))e^ρ(ξ−t)dξdt}

+e^ρ(s−x){(−∆ac+ρ(∆bc−∆ad) +ρ²∆bd)e^ρ−2ρ∆ab

+ Z s

0

(∆_aR(t) +ρ(∆_bR(t)−∆_aS(t))−ρ²∆_bS(t))e^−ρtdt +

Z s

0

(∆cR(t) +ρ(∆dR(t)−∆cS(t))−ρ²∆dS(t))e^ρ(1−t)dt

− Z 1

0

(∆aR(t) +ρ(∆aS(t)−∆bR(t))−ρ²∆bS(t))e^ρtdt +

Z 1

0

Z s

0

(∆R(t, ξ)−ρ(∆RS(ξ, t) + ∆RS(t, ξ))−ρ²∆S(t, ξ))e^ρ(t−ξ)dξdt}i , (2.9)

and

ϕ2(x, s, λ)

= 1 2ρ

h

e^ρ(x−s){(−∆ac+ρ(∆bc−∆ad) +ρ²∆bd)e^ρ−2ρ∆ad

− Z 1

s

(∆aR(t) +ρ(∆aS(t)−∆bR(t))−ρ²∆bS(t))e^ρtdt

− Z 1

s

(∆cR(t) +ρ(∆cS(t)−∆dR(t))−ρ²∆dS(t))e^ρ(t−1)dt +

Z 1

0

(∆cR(t) +ρ(∆dR(t)−∆cS(t))−ρ²∆dS(t))e^ρ(1−t)dt +

Z 1

0

Z 1

s

(∆R(ξ, t)−ρ(∆RS(t, ξ) + ∆RS(ξ, t))−ρ²∆S(ξ, t))e^ρ(ξ−t)dξdt}

+e^ρ(s−x){(−∆ac+ρ(∆ad−∆bc) +ρ²∆bd)e^−ρ+ 2ρ∆cd

− Z 1

s

(∆aR(t) +ρ(∆bR(t)−∆aS(t))−ρ²∆bS(t))e^−ρtdt

− Z 1

s

(∆cR(t) +ρ(∆dR(t)−∆cS(t))−ρ²∆dS(t))e^ρ(1−t)dt +

Z 1

0

(∆cR(t) +ρ(∆cS(t)−∆dR(t))−ρ²∆dS(t))e^ρ(t−1)dt +

Z 1

0

Z 1

s

(∆R(ξ, t) +ρ(∆RS(t, ξ) + ∆RS(ξ, t))−ρ²∆S(ξ, t))e^ρ(t−ξ)dξdt}i (2.10) 3. Bounds on the Resolvent

Everyλ∈Csuch that ∆(λ)6= 0 belongs toρ(Lp), and the associated resolvent operatorR(λ, Lp) can be expressed as a Hilbert Schmidt operator

(λI−L_p)⁻¹f =R(λ;L_p)f =− Z 1

0

G(., s;λ)f(s)ds, f ∈L^p(0,1). (3.1) Then, for everyf ∈L^p(0,1) we estimate (3.1),

kR(λ;Lp)fkL_p(0,1)≤ sup

0≤s≤1

Z 1

0

|G(x, s;λ)|^pdx^1/p

kfkL_p(0,1), and so we need to bound

( sup

0≤s≤1

Z 1

0

|G(x, s;λ)|^pdx)^1/p= 1

|∆(λ)|( sup

0≤s≤1

Z 1

0

|N(x, s;λ)|^pdx)^1/p. (3.2) 3.1. Estimation of N(x, s, λ). We will denote by k · k the supremum norm for functions in one and two variables. Since

N(x, s, λ) =

(ϕ(x, s, λ) +ϕ₁(x, s, λ) ifx > s, ϕ(x, s, λ) +ϕ2(x, s, λ) ifx < s, it follows that

|N(x, s, λ)|^p≤2^p−1(|ϕ(x, s, λ)|^p+|ϕi(x, s, λ)|^p).

Form (2.8), we have Z 1

0

|N(x, s, λ)|^pdx

≤2^p−1Z 1 0

|ϕ(x, s, λ)|^pdx+ Z 1

s

|ϕ1(x, s, λ)|^pdx+ Z s

0

|ϕ1(x, s, λ)|^pdx .

(3.3)

From (2.7), we have Z 1

0

|ϕ(x, s, λ)|^pdx

≤ 2^2p−2 2p|ρ|^pRe(ρ)

h(|ρ|²|∆_bd|+|ρ|(|∆_ad|+|∆_bc|) +|∆_ac|)^p

×

(e^psRe(ρ)−ep(s−1) Re(ρ)) + (ep(1−s) Re(ρ)−e^−psRe(ρ)) + 1

Re(ρ)(|ρ|²k∆bSk+|ρ|(k∆aSk+k∆bRk) +k∆aRk)^p

×((e^pRe(ρ)−1)×(1−e(s−1) Re(ρ))^p+ (1−e^−pRe(ρ))×(e(1−s) Re(ρ)−1)^p) + 1

Re(ρ)(|ρ|²k∆dSk+|ρ|(k∆dRk+k∆cSk) +k∆cRk)^p

× (e^pRe(ρ)−1)

×(e(s−1) Re(ρ)−e^−Re(ρ))^p+ (1−e^−pRe(ρ))×(e^Re(ρ)−e(1−s) Re(ρ))^p

+ 1

(Re(ρ))²(k∆Rk+ 2|ρ|k∆RSk+|ρ|²k∆Sk)^p

×((e^pRe(ρ)−1)

×(e^sRe(ρ)−1)^p×(e^−sRe(ρ)−e^−Re(ρ))^p+ (1−e^−pRe(ρ))

×(e^Re(ρ)−e^sRe(ρ))^p(1−e^−sRe(ρ))^p)i , from (2.9), we have

Z 1

s

|ϕ1(x, s, λ)|^pdx

≤ 5^p−1

2p|ρ|^pRe(ρ)[(|ρ|²|∆bd|+|ρ|(|∆ad|+|∆bc|)

+|∆ac|)^p×((e^pRe(ρ)−e^−psRe(ρ)) + (e^−psRe(ρ)−e^−pRe(ρ))) + (2|ρ||∆ab|)^p

× ep(1−s) Re(ρ)−ep(s−1) Re(ρ) + ( 1

Re(ρ)(|ρ|²k∆bSk+|ρ|(k∆aSk+k∆bRk) +k∆aRk))^p×((e^−psRe(ρ)−e^−pRe(ρ))×(e(1+s) Re(ρ)−1)^p+ (e^pRe(ρ)

−e^psRe(ρ))×(1−e−(1+s) Re(ρ))^p) + ( 1

Re(ρ)(|ρ|²k∆dSk+|ρ|(k∆dRk +k∆cSk) +k∆cRk))^p×((e^−psRe(ρ)−e^−pRe(ρ))×(ep(s+1) Re(ρ)−e^pRe(ρ))^p + (e^pRe(ρ)−e^psRe(ρ))×(e^−Re(ρ)−e−(1+s) Re(ρ))^p) + ( 1

(Re(ρ))²(k∆Rk + 2|ρ|k∆RSk+|ρ|²k∆Sk))^p×((e^−psRe(ρ)−e^−pRe(ρ))×(e^Re(ρ)−1)^p

×(e^sRe(ρ)−1)^p+ (e^pRe(ρ)−e^psRe(ρ))×(1−e^−Re(ρ))^p×(1−e^−sRe(ρ))^p)i .

From (2.10), we have Z s

0

|ϕ2(x, s, λ)|^pdx

≤ 5^p−1 2p|ρ|^pRe(ρ)

h(|ρ|²|∆bd|+|ρ|(|∆ad|+|∆bc|)

+|∆ac|)^p×((e^pRe(ρ)−ep(1−s) Re(ρ)) + (ep(s−1) Re(ρ)−e^−pRe(ρ))) + (2|ρ||∆cd|)^p×(e^psRe(ρ)−e^−psRe(ρ)) + ( 1

Re(ρ)(|ρ|²k∆bSk+|ρ|(k∆aSk +k∆bRk) +k∆aRk))^p×((e^psRe(ρ)−1)×(e(1−s) Re(ρ)−1)^p

+ (1−e^−psRe(ρ))×(1−e(s−1) Re(ρ))^p) + ( 1

Re(ρ)(|ρ|²k∆_dSk+|ρ|(k∆_dRk +k∆_cSk) +k∆_cRk))^p×((e^psRe(ρ)−1)×(e(1−s) Re(ρ)−e^−Re(ρ))^p + (1−e^−psRe(ρ))×(e^Re(ρ)−e(s−1) Re(ρ))^p) + ( 1

(Re(ρ))²(k∆Rk+ 2|ρ|k∆RSk +|ρ|²k∆Sk))^p×((e^psRe(ρ)−1)×(e(1−s) Re(ρ)−1)^p×(1−e^−Re(ρ))^p + (1−e^−psRe(ρ))×(e^Re(ρ)−1)^p×(1−e(s−1) Re(ρ))^p)i

. So that

Z 1

0

|N(x, s, λ)|^pdx

≤ 2^p−1×5^p−1

2p|ρ|^pRe(ρ)[(|ρ|²|∆bd|+|ρ|(|∆ad|+|∆bc|) +|∆ac|)^p

×2(e^pRe(ρ)−e^−pRe(ρ)) + (2|ρ||∆ab|)^p×(ep(1−s) Re(ρ)−ep(s−1) Re(ρ)) + (2|ρ||∆cd|)^p×(e^psRe(ρ)−e^−psRe(ρ)) + ( 1

Re(ρ)(|ρ|²k∆bSk+|ρ|(k∆aSk +k∆bRk) +k∆aRk))^p×((e^pRe(ρ)−1)×(1−e(s−1) Re(ρ))^p

+ (1−e^−pRe(ρ))(e(1−s) Re(ρ)−1)^p+ (e^−psRe(ρ)−e^−pRe(ρ))×(e(1+s) Re(ρ)−1)^p + (e^pRe(ρ)−e^psRe(ρ))(1−e−(1+s) Re(ρ))^p+ (e^psRe(ρ)−1)×(e(1−s) Re(ρ)−1)^p + (1−e^−psRe(ρ))(1−e(s−1) Re(ρ))^p) + ( 1

Re(ρ)(|ρ|²k∆dSk+|ρ|(k∆dRk

+k∆cSk) +k∆cRk))^p((e^pRe(ρ)−1)×(e(s−1) Re(ρ)−e^−Re(ρ))^p+ (1−e^−pRe(ρ))

×(e^Re(ρ)−e(1−s) Re(ρ))^p+ (e^−psRe(ρ)−e^−pRe(ρ))×(ep(s+1) Re(ρ)−e^pRe(ρ))^p + (e^pRe(ρ)−e^psRe(ρ))×(e^−Re(ρ)−e−(1+s) Re(ρ))^p+ (e^psRe(ρ)−1)

×(e(1−s) Re(ρ)−e^−Re(ρ))^p+ (1−e^−psRe(ρ))×(e^Re(ρ)−e(s−1) Re(ρ))^p) + ( 1

(Re(ρ))²(k∆Rk+ 2|ρ|k∆RSk+|ρ|²k∆Sk))^p×((e^pRe(ρ)−1)

×(e^sRe(ρ)−1)^p×(e^−sRe(ρ)−e^−Re(ρ))^p+ (1−e^−pRe(ρ))×(e^Re(ρ)−e^sRe(ρ))^p

×(1−e^−sRe(ρ))^p+ (e^−psRe(ρ)−e^−pRe(ρ))×(e^Re(ρ)−1)^p×(e^sRe(ρ)−1)^p + (e^pRe(ρ)−e^psRe(ρ))×(1−e^−Re(ρ))^p×(1−e^−sRe(ρ))^p+ (e^psRe(ρ)−1)

×(e(1−s) Re(ρ)−1)^p×(1−e^−Re(ρ))^p+ (1−e^−psRe(ρ))×(e^Re(ρ)−1)^p

×(1−e(s−1) Re(ρ))^p)i . Since Re(ρ)>0, we obtain

sup

0≤s≤1

Z 1

0

|N(x, s, λ)|^pdx1/p

≤ 2^(2−2p)×5^(1−p1) p^1p|ρ|(Re(ρ))^1/pe^Re(ρ)h

|ρ|²|∆bd|

+|ρ|(|∆ad|+|∆bc|+|∆cd|) +|∆ac|+ 3

Re(ρ)(|ρ|²k∆bSk+|ρ|(k∆aSk +k∆bRk) +k∆aRk) + 3

Re(ρ)(|ρ|²k∆dSk+|ρ|(k∆dRk+k∆cSk) +k∆cRk) + 3

(Re(ρ))²(k∆Rk+ 2|ρ|k∆RSk+|ρ|²k∆Sk)i

. (3.4)

From the above inequality, (3.2) and (3.3), we obtain kR(λ, Lp)k ≤ 2^(2−2p)×5^(1−1p)

|∆(ρ²)||ρ|(Re(ρ))^1/pp^1/pe^Re(ρ)h

(|ρ|²|∆bd|+|ρ|(|∆ad|+|∆bc| +|∆ab|+|∆cd|) +|∆ac|+ 3

Re(ρ)(|ρ|²(k∆bSk+k∆dSk) +|ρ|(k∆dRk+k∆cSk+k∆aSk+k∆bRk) +k∆cRk+k∆aRk)

+ 3

(Re(ρ))²(k∆Rk+ 2|ρ|k∆RSk+|ρ|²k∆Sk)i ,

forρ∈P

δ

2 ={ρ∈C:|argρ| ≤ ^π₄ +^δ₂, ρ6= 0}, we have (Re(ρ))⁻¹< ¹

|ρ|cos(^π₄+^δ₂). Then

kR(λ, Lp)k

≤ 2^(2−2p)×5^(1−1p)

|∆(ρ²)||ρ|^1+1p(cos(^π₄ +^δ₂))^1/pp^1p e^Re(ρ)h

(|ρ|²|∆_bd|+|ρ|(|∆_ad| +|∆bc|+|∆bc|+|∆ab|+|∆cd|) +|∆ac|+ 3

|ρ|cos(^π₄ +^δ₂)(|ρ|²(k∆bSk+k∆dSk) +|ρ|(k∆dRk+k∆cSk+k∆aSk+k∆bRk) +k∆cRk+k∆aRk)

+ 3

(|ρ|cos(^π₄ +^δ₂))²(k∆Rk+ 2|ρ|k∆RSk+|ρ|²k∆Sk)i .

Finally, we obtain, forλ=ρ²∈P

δ, kR(λ, Lp)k

≤ c

|∆(ρ²)||ρ|^1+1pe^Re(ρ)h

(|ρ|²|∆_bd|+|ρ|(|∆_ab|+|∆_ad|

+|∆bc|+|∆cd|+k∆bSk+k∆dSk) +|∆ac|+k∆aSk+k∆bRk+k∆dRk +k∆cSk+k∆Sk+ 1

|ρ|(k∆aRk+k∆cRk+ 2k∆RSk) +k∆_Rk

|ρ|² i

≤c H(ρ)

|ρ|^1+p1, (3.5)

where

H(ρ) = e^Re(ρ)

|∆(ρ²)|

h(|ρ|²|∆bd|+|ρ|(|∆ab|+|∆ad|

+|∆bc|+|∆cd|+k∆bSk+k∆dSk) +|∆ac|+k∆aSk+k∆bRk+k∆dRk +k∆_cSk+k∆_Sk+ 1

|ρ|(k∆_aRk+k∆_cRk+ 2k∆_RSk) +k∆_Rk

|ρ|² i

, (3.6) and

c= 2^(2−2p)×5^(1−p1)

p^1/p max 1

(cos(^δ₂))^1/p, 3

(cos(^δ₂))^1+1p, 3 (cos(^δ₂))^2+1p

.

The following step is to analyze the functionH(ρ) in order to determine the cases for which it is bounded in the sectorP

δ/2.

3.2. Estimation of the characteristic determinant, regular case. The next step is to determine the cases for which |∆(λ)| remains bounded below. It will then be necessary to bound |∆(λ)| appropriately. However, formula (2.5) is not useful for this purpose, it will be then necessary to make some additional regularity hypotheses on the functionsR_iandS_i, and so we assume that the functionsR_iand S_i,i= 1,2, are inC¹([0,1],C). Integrating twice by parts in (2.5), we obtain

∆(λ) =e^ρh

−ρ²∆_bd+ρ(∆_ad−∆_bc+ ∆_dS(0)−∆_bS(1)) + ∆_ac + ∆aS(1) + ∆cS(0)−∆bR(1)−∆dR(0) + ∆S(1,0) +1

ρ(∆aR(1)

−∆cR(0) + ∆RS(0,1) + ∆RS(1,0)) + 1

ρ²(∆R(0,1)) + Φ(ρ)i ,

(3.7)

where

Φ(ρ) = 2e^−ρ[ρ(∆_ab+ ∆_cd+ ∆_dS(0)−∆_bS(1)) +1

ρ(∆_cR(1)

−∆aR(0)−∆RS(0,0)−∆RS(1,1))]

×e^−2ρ[ρ²∆bd+ρ(∆ad−∆bc+ ∆dS(0)−∆bS(1))−∆ac−∆aS(1)

−∆_cS(0) + ∆_bR(1) + ∆_dR(0)−∆_S(1,0) + 1

ρ(∆_aR(1)−∆_cR(0) + ∆_RS(0,1) + ∆_RS(1,0)) + 1

ρ²(∆_R(1,0))]

−1 ρ[

Z 1

0

(∆aR⁰(t) +ρ(∆aS⁰(t)−∆bR⁰(t))−ρ²∆bS⁰(t))e^ρ(t−1)dt +

Z 1

0

(∆aR⁰(t) +ρ(∆bR⁰(t)−∆aS⁰(t))−ρ²∆bS⁰(t))e^−ρ(t+1)dt +

Z 1

0

(∆cR⁰(t) +ρ(∆cS⁰(t)−∆dR⁰(t))−ρ²∆dS⁰(t))e^ρ(t−2)dt +

Z 1

0

(∆cR⁰(t) +ρ(∆dR⁰(t)−∆cS⁰(t))−ρ²∆dS⁰(t))e^−ρtdt]

+ 1 ρ²[

Z 1

0

(∆R⁰R(t,0)−ρ(∆R⁰S(t,0) + ∆RS⁰(0, t))−ρ²∆S⁰S(t,0))e^ρ(t−1)dt +

Z 1

0

(∆R⁰R(0, t)−ρ(∆R⁰S(0, t) + ∆RS⁰(t,0)) +ρ²∆S⁰S(1, t))e^−ρ(t+1)dt +

Z 1

0

(∆R⁰R(1, t) +ρ(∆R⁰S(1, t) + ∆RS⁰(t,1)) +ρ²∆S⁰S(t,1))e^ρ(t−2)dt +

Z 1

0

(∆R⁰R(t,1) +ρ(∆R⁰S(t,1) + ∆RS⁰(1, t)) +ρ²∆S⁰S(1, t))e^−ρtdt +

Z 1

0

Z 1

0

(∆R⁰(ξ, t) +ρ(∆R⁰S⁰(t, ξ) + ∆R⁰S⁰(ξ, t))

−ρ²∆S⁰(ξt))e^{ρ(ξ−t−1)}dξdt].

Suppose first that ∆_bd 6= 0. From (3.7) we can write the characteristic determinant in the form

∆(λ) =ρ²∆_bde^ρ(−1 +F(ρ))

for a certain functionF(ρ), It is not difficult to see that a constantr0can be chosen in order that|F(ρ)| ≤ ¹₂ for|ρ|> r0. Thus, forρ∈r0 +P

δ

2, we have

|∆(λ)| ≥ |ρ|²|∆bd|e^Re(ρ)(1− |F(ρ)|)≥ 1

2|ρ|²|∆bd|e^Re(ρ). Finally, from (3.6) we obtain

H(ρ)≤ 2

|∆bd|

h|∆_bd|+ 1 r0

(|∆_ab|+|∆_ad|+|∆_bc|+|∆_cd|+k∆_bSk+k∆_dSk) + 1

r²₀(|∆ac|+k∆aSk+k∆bRk+k∆dRk+k∆cSk+k∆Sk) + 1

r³₀(k∆aRk+k∆cRk+ 2k∆RSk) +k∆Rk r₀⁴

i≡H0,

which proves thatH(ρ) is bounded by a constantH₀in the sector r₀ +P

δ 2. The other cases can be treated in a similar way [5, 6, 13, 15], and we do not include them here for lack of space. After doing the complete analysis of cases, we obtain thatH(ρ) is bounded by a constantH0>0 in a sector of the formr0 +P

δ 2, only in the following five cases

(1) ∆bd 6= 0

(2) ∆bd = 0 and ∆ad−∆bc−∆bS(1) + ∆dS(0)6= 0

(3) ∆ab= ∆ad= ∆bc= ∆bd= ∆cd= 0, ∆bS≡0, ∆dS≡0 and ∆ac+∆aS(1)+

∆cS(0)−∆bR(1)−∆dR(0) + ∆S(1,0)6= 0

(4) ∆ab = ∆ac = ∆ad = ∆bc = ∆bd = ∆cd = 0, ∆bR ≡0,∆dR ≡ 0,∆aS ≡ 0,∆bS≡0, ∆cS≡0, ∆dS≡0, ∆S≡0 and ∆aR(1)−∆cR(0) + ∆RS(0,1) +

∆RS(1,0)6= 0

(5) ai=bi=ci=di= 0, Si ≡0,i= 1,2 and ∆R(0,1)6= 0.

Definition 3.1. Suppose that Ri, Si ∈C¹([0,1],C),i= 1,2. The boundary con- ditions in (1.1) are called regular if they verify one of the conditions above.

The above arguments prove the following theorem.

Theorem 3.2. If the boundary value conditions in (1.1) are regular, then P

δ ⊂ ρ(Lp)for sufficiently large|λ| and there existsc >0 such that

kR(λ, Lp)k ≤ c

|λ|^12+2p1 ,|λ| →+∞.

Remark 3.3. From theorem 3.2 results that the operatorLp forp6=∞, generates an analytic semi group with singularities [30] of typeA(^p−1_p+1,^3p−1_p+1).

Remark 3.4. Forp= 1, the decrease of the resolvent is maximal and the operator L1 generates an analytic semi group [15].

3.3. Estimation of the characteristic determinant, non regular case. As in the regular case Formula (3.7) is not useful for the estimation of the charac- teristic determinant, it will be then necessary to make some additional hypotheses on the functions RiandSi, and so we assume that the functions Ri andSi are in C²([0,1],C),i= 1,2. Integrating twice by parts in (3.7), we obtain

∆(λ) =e^ρh

−ρ²∆bd+ρ(∆ad−∆bc+ ∆dS(0)−∆bS(1)) + ∆ac

+ ∆aS(1) + ∆cS(0)−∆bR(1)−∆dR(0) + ∆S(1,0) + ∆bS⁰(1) + ∆dS⁰(0) + 1

ρ(∆aR(1)−∆cR(0) + ∆bR⁰(1)−∆aS⁰(1) + ∆cS⁰(0)

−∆dR⁰(0) + ∆RS(0,1) + ∆RS(1,0) + ∆SS⁰(1,0) + ∆SS⁰(0,1)) + 1

ρ²(∆R(0,1)−∆RS⁰(0,1) + ∆RS⁰(1,0)−∆R⁰S(1,0) + ∆R⁰S(0,1)

−∆dR⁰(1)−∆cR⁰(0)) + 1

ρ³(∆R⁰R(1,0) + ∆R⁰R(0,1)) + Φ(ρ)], (3.8) where

Φ(ρ) = 2e^−ρh

ρ(∆ab+ ∆cd+ ∆bS(0)−∆dS(1)) +1

ρ(∆cR(1)

−∆_aR(0) + ∆_dR⁰(1)−∆_cS⁰(1) + ∆_aS⁰(0)−∆_bR⁰(0)−∆_RS(0,0)

−∆RS(1,1) + ∆S⁰S(1,1) + ∆S⁰S(0,0)) + 1

ρ³(∆RR⁰(1,1) + ∆RR⁰(0,0))]

+e^−2ρ[ρ²∆bd+ρ(∆ad−∆bc+ ∆dS(0)−∆bS(1))−∆ac−∆aS(1)

−∆_cS(0) + ∆_bR(1) + ∆_dR(0)−∆_S(1,0)−∆_bS⁰(1)−∆_dS⁰(0) +1

ρ(∆_aR(1)−∆_cR(0) + ∆_bR⁰(1)−∆_aS⁰(1) + ∆_cS⁰(0)−∆_dR⁰(0) + ∆RS(0,1) + ∆RS(1,0) + ∆SS⁰(1,0) + ∆SS⁰(0,1))

+ 1

ρ²(∆_R(0,1)−∆_RS⁰(0,1) + ∆_RS⁰(1,0)−∆_R⁰_S(1,0) + ∆_R⁰_S(0,1)

−∆_dR⁰(1)−∆_cR⁰(0)) + 1

ρ³(∆_R⁰_R(1,0) + ∆_R⁰_R(0,1))]

+ 1 ρ²[

Z 1

0

(∆aR⁰⁰(t) +ρ(∆aS⁰⁰(t)−∆bR⁰⁰(t))−ρ²∆bS⁰⁰(t))e^ρ(t−1)dt +

Z 1

0

(∆cR⁰⁰(t) +ρ(∆cS⁰⁰(t)−∆dR⁰⁰(t))−ρ²∆dS⁰⁰(t))e^ρ(t−2)dt

− Z 1

0

(∆aR⁰⁰(t) +ρ(∆bR⁰⁰(t)−∆aS⁰⁰(t))−ρ²∆bS⁰⁰(t))e^−ρ(t+1)dt

− Z 1

0

(∆cR⁰⁰(t) +ρ(∆dR⁰⁰(t)−∆cS⁰⁰(t))−ρ²∆dS⁰⁰(t))e^−ρtdt +{(

Z 1

0

(R⁰₂(t)−ρS₂⁰(t))e^−ρtdt)×( Z 1

0

(R⁰₁(t) +ρS₁⁰(t))e^ρ(t−1)dt)

−( Z 1

0

(R⁰₁(t)−ρS₁⁰(t))e^−ρtdt)×( Z 1

0

(R₂⁰(t) +ρS₂⁰(t))e^ρ(t−1)dt)}i + 1

ρ³ h−

Z 1

0

(∆RR⁰⁰(1, t) +ρ(∆RS⁰⁰(1, t) + ∆R⁰⁰S(t,1)) +ρ²∆S⁰⁰S(t,1))e^ρ(t−2)dt

+ Z 1

0

(∆RR⁰⁰(0, t) +ρ(∆S⁰⁰R(t,0) + ∆SR⁰⁰(0, t)) +ρ²∆S⁰⁰S(t,0))e^−ρ(t+1)dt +

Z 1

0

(∆R⁰⁰R(t,1) +ρ(∆R⁰⁰S(1, t) + ∆RS⁰⁰(1, t)) +ρ²∆SS⁰⁰(1, t))e^−ρtdt

− Z 1

0

(∆R⁰⁰R(t,0) +ρ(∆R⁰⁰S(t,0) + ∆RS⁰⁰(0, t)) +ρ²∆SS⁰⁰(0, t))e^ρ(t−1)dti . After a straightforward calculation, we obtain the following inequality valid for ρ∈P

δ

2, with|ρ|sufficiently large,

|Φ(ρ)|

≤2e^−Re(ρ)h

|ρ|(|∆ab|+|∆cd|+k∆bSk+k∆dSk) + 1

|ρ|(k∆cRk+k∆aRk+k∆dR⁰k+k∆cS⁰k+k∆aS⁰k+k∆bR⁰k + 2k∆SRk+ 2k∆S⁰Sk) 1

|ρ|³(k∆RR⁰k+k∆RR⁰k)

+e^−Re(ρ)[|ρ|²|∆bd|+|ρ|(|∆ad|+|∆bc|+k∆dSk+k∆bSk) +|∆ac|+k∆bRk+k∆aSk+k∆dRk+k∆cSk+k∆dS⁰k+k∆bS⁰k +k∆_Sk+ 1

|ρ|(k∆_aRk+k∆_cRk+k∆_bR⁰k+k∆_aS⁰k+k∆_cS⁰k +k∆dR⁰k+ 2k∆RS⁰k+ 2k∆S⁰Sk) + 1

|ρ|²(k∆aR⁰k+k∆cR⁰k +k∆Rk+ 2k∆RS⁰k+ 2k∆R⁰Sk+ 2

|ρ|³k∆R⁰Rki

+ 1

|ρ|²Re(ρ)[(k∆|aR⁰⁰|+|ρ|(k∆aS⁰⁰k+k∆bR⁰⁰k) +|ρ|²k∆bS⁰⁰k)×(1−e^−Re(ρ)) + (k∆_cR⁰⁰k+|ρ|(k∆_cS⁰⁰k+k∆_dR⁰⁰k) +|ρ|²k∆_dS⁰⁰k)×(e^−Re(ρ)−e^{−2 Re(ρ)}) + (k∆aR⁰⁰k+|ρ|(k∆aS⁰⁰k+k∆bR⁰⁰k) +|ρ|²k∆bS⁰⁰k)×(e^−Re(ρ)−e^{−2 Re(ρ)}) + (k∆cR⁰⁰k+|ρ|(k∆cS⁰⁰k+k∆dR⁰⁰k) +|ρ|²k∆dS⁰⁰k)×(1−e^−Re(ρ))

+ 1

|ρ|²(Re(ρ))²(k∆R⁰k+ 2|ρk|∆R⁰S⁰k+|ρ|²k∆S⁰k)×(1−e^−Re(ρ))²+ 2

|ρ|³Re(ρ)

×(k∆R⁰⁰Rk+|ρ|(k∆R⁰⁰Sk+k∆RS⁰⁰k) +|ρ|²k∆SS⁰⁰k)×(e^−Re(ρ)−e^{−2 Re(ρ)})

+ 2

|ρ|³Re(ρ)(k∆R⁰⁰Rk+|ρ|(k∆R⁰⁰Sk+k∆RS⁰⁰k) +|ρ|²k∆SS⁰⁰k)×(1−e^−Re(ρ)).

Then

|Φ(ρ)| ≤ 4

|ρ|²(cos(^π₄+^δ₂))²

h|ρ|(|∆ab|+|∆cd|+k∆bSk+k∆dSk)

+ 1

|ρ|(k∆cRk+k∆aRk+k∆dR⁰k+k∆cS⁰k+k∆aS⁰k+k∆bR⁰k + 2k∆SRk+ 2k∆S⁰Sk) 1

|ρ|³(k∆RR⁰k+k∆RR⁰k)

+ 1

2|ρ|²(cos(^π₄ +^δ₂))²[|ρ|²|∆bd|+|ρ|(|∆ad|+|∆bc|+k∆dSk+k∆bSk) +|∆_ac|+k∆_bRk+k∆_aSk+k∆_dRk+k∆_cSk+k∆_dS⁰k+k∆_bS⁰k +k∆Sk+ 1

|ρ|(k∆aRk+k∆cRk+k∆bR⁰k+k∆aS⁰k+k∆cS⁰k +k∆dR⁰k+ 2k∆RS⁰k+ 2k∆S⁰Sk) + 1

|ρ|²(k∆aR⁰k+k∆cR⁰k +k∆Rk+ 2k∆RS⁰k+ 2k∆R⁰Sk+ 2

|ρ|³k∆R⁰Rk]

+ 2

|ρ|³cos(^π₄+^δ₂)[(k∆_aR⁰⁰k+|ρ|(k∆_aS⁰⁰k+k∆_bR⁰⁰k) +|ρ|²k∆_bS⁰⁰k) + (k∆cR⁰⁰k+|ρ|(k∆cS⁰⁰k+k∆dR⁰⁰k) +|ρ|²k∆dS⁰⁰k)i

+ 1

|ρ|⁴(cos(^π₄ +^δ₂))²(k∆R⁰k+ 2|ρk|∆R⁰S⁰k+|ρ|²k∆S⁰k)

+ 4

|ρ|⁴cos(^π₄+^δ₂)(k∆R⁰⁰Rk+|ρ|(k∆R⁰⁰Sk+k∆RS⁰⁰k) +|ρ|²k∆SS⁰⁰k).

(3.9) Where we have used that Re(ρ)>|ρ|cos(^π₄+^δ₂), 1−e^−Re(ρ)<1, 1−e^{−2 Re(ρ)}<1,

|ρ|²e^−Re(ρ) ≤ 2(cos(^π₄ + ^δ₂))⁻² and 2|ρ|²e^{−2 Re(ρ)} ≤ (cos(^π₄ + ^δ₂))⁻². There are several cases to analyze depending on the functionsRiand Si,i= 1,2.

Case 1. Suppose that ∆_bd = 0, ∆_ad−∆_bc+ ∆_dS(0)−∆_bS(1) = 0, max(|∆ab|,|∆ad|,|∆bc|,|∆cd|,k∆bSk,k∆dSk)6= 0

andk1= ∆ac+∆aS(1)+∆cS(0)−∆bR(1)−∆dR(0)+∆S(1,0)+∆bS⁰(1)+∆dS⁰(0)6= 0 From (3.3), we have for|ρ|sufficiently large

|∆(λ)| ≥e^Re(ρ)h

|∆ac+ ∆aS(1) + ∆cS(0)−∆bR(1)−∆dR(0) + ∆S(1,0) + ∆_bS⁰(1) + ∆_dS⁰(0)| − 1

|ρ||∆aR(1)−∆_cR(0) + ∆_bR⁰(1)−∆_aS⁰(1) + ∆cS⁰(0)−∆dR⁰(0) + ∆RS(0,1) + ∆RS(1,0) + ∆SS⁰(1,0) + ∆SS⁰(0,1)|

− 1

|ρ|²|∆R(0,1)−∆_RS⁰(0,1) + ∆_RS⁰(1,0)−∆_R⁰_S(1,0) + ∆_R⁰_S(0,1)

−∆_dR⁰(1)−∆_cR⁰(0)| − 1

|ρ|³|∆R⁰R(1,0) + ∆_R⁰_R(0,1)| −Φ(ρ)i . From (3.9) we deduce forρ∈P

δ

2,|ρ| ≥r0>0.

|Φ(ρ)| ≤ c(r₀)

|ρ| . Then, we have

|∆(λ)| ≥e^Re(ρ)[|∆ac+ ∆_aS(1) + ∆_cS(0)−∆_bR(1)−∆_dR(0) + ∆_S(1,0) + ∆_bS⁰(1) + ∆_dS⁰(0)| − 1

|ρ||∆_aR(1)−∆_cR(0) + ∆_bR⁰(1)−∆_aS⁰(1) + ∆_cS⁰(0)−∆_dR⁰(0) + ∆_RS(0,1) + ∆_RS(1,0) + ∆_SS⁰(1,0) + ∆_SS⁰(0,1)|

− 1

|ρ|²|∆_R(0,1)−∆_RS⁰(0,1) + ∆_RS⁰(1,0)−∆_R⁰_S(1,0) + ∆_R⁰_S(0,1)

−∆dR⁰(1)−∆cR⁰(0)| − 1

|ρ|³|∆R⁰R(1,0) + ∆R⁰R(0,1)| −c(r₀)

|ρ|

i , we can now chooser₀>0, such that

1

r₀|∆aR(1)−∆cR(0) + ∆bR⁰(1)−∆aS⁰(1) + ∆cS⁰(0)−∆dR⁰(0) + ∆_RS(0,1) + ∆_RS(1,0) + ∆_SS⁰(1,0) + ∆_SS⁰(0,1)|+ 1

r0²

|∆_R(0,1)

−∆_RS⁰(0,1) + ∆_RS⁰(1,0)−∆_R⁰_S(1,0) + ∆_R⁰_S(0,1)−∆_dR⁰(1)−∆_cR⁰(0)|

+ 1

r³₀|∆R⁰R(1,0) + ∆R⁰R(0,1)|+c(r0)

|ρ|

≤ 1

2|∆ac+ ∆aS(1) + ∆cS(0)

−∆_bR(1)−∆_dR(0) + ∆_S(1,0) + ∆_bS⁰(1) + ∆_dS⁰(0)|, then, forρ∈P

δ

2,|ρ| ≥r0>0, we get

|∆(ρ)| ≥ e^Re(ρ) 2 |k1|.

From (3.5), we deduce the following bound, valid for every|argρ| ≤ ^π₄ +^δ₂,ρ6= 0, kR(λ, Lp)k ≤ cH(ρ)

|ρ|^1+1p ≤ 1

|ρ|^1/p ×cH(ρ)

|ρ| , where

H(ρ)

|ρ| = e^Re(ρ)

|∆(ρ²)||ρ|[|ρ|(|∆ab|+|∆ad|+|∆bc|+|∆cd|+k∆bSk +k∆dSk) + (|∆ac|+k∆aSk+k∆bRk+k∆dRk+k∆cSk +k∆Sk) + 1

|ρ|(k∆aRk+k∆cRk+ 2k∆RSk) +k∆_Rk

|ρ|² ]

= e^Re(ρ)

|∆(ρ²)|[(|∆ab|+|∆ad|+|∆bc|+|∆cd|+k∆bSk+k∆dSk) + 1

|ρ|(|∆ac|+k∆aSk+k∆bRk+k∆dRk+k∆cSk+k∆Sk)