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volume 5, issue 2, article 38, 2004.

Received 15 October, 2003;

accepted 16 April, 2004.

Communicated by:A.G. Ramm

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Journal of Inequalities in Pure and Applied Mathematics

BOUNDARY VALUE PROBLEM FOR SECOND-ORDER DIFFERENTIAL OPERATORS WITH MIXED NONLOCAL BOUNDARY CONDITIONS

M. DENCHE AND A. KOURTA

Laboratoire Equations Differentielles, Departement de Mathematiques,

Faculte des Sciences, Universite Mentouri, 25000 Constantine, Algeria.

EMail:denech@wissal.dz

c

2000Victoria University ISSN (electronic): 1443-5756 147-03

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Boundary Value Problem for Second-Order Differential Operators with Mixed Nonlocal

Boundary Conditions M. Denche and A. Kourta

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J. Ineq. Pure and Appl. Math. 5(2) Art. 38, 2004

http://jipam.vu.edu.au

Abstract

In this paper, we study a second order differential operator with mixed nonlocal boundary conditions combined weighting integral boundary condition with another two point boundary condition. 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 forp≥1. 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 boundary conditions.

2000 Mathematics Subject Classification:47E05, 35K20.

Key words: Green’s Function, Non Regular Boundary Conditions, Regular Boundary Conditions, Semi group with Singularities.

The authors are grateful to the editor and the referees for their valuable comments and helpful suggestions which have much improved the presentation of the paper.

1. Introduction

In the spaceL^p(0,1)we consider the boundary value problem

(1.1)







L(u) =u⁰⁰ =f(x),

B₁(u) =a₀u(0) +b₀u⁰(0) +c₀u(1) +d₀u⁰(1) = 0, B₂(u) =R1

0 R(t)u(t)dt+R1

0 S(t)u⁰(t)dt= 0,

where the functionsR, S ∈C([0,1],C). We associate to problem (1.1) in space L^p(0,1)the operator:

L_p(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 differential operators with two point linearly independent boundary conditions, in terms of coefficients of boundary conditions. The reader should refer to [7, 10, 20, 21, 22, 28, 31, 33] and references therein. Few works have been devoted to the study of a non regular situation. The case of separated non regular boundary conditions was studied by W. Eberhard, J.W. Hopkins, D. Jak- son, M.V. Keldysh, A.P. Khromov, G. Seifert, M.H. Stone, L.E. Ward (see S.

Yakubov and Y. Yakubov [33] for exact references). A situation of non regular non-separated boundary conditions was considered by H. E. Benzinger [2], M.

Denche [4], W. Eberhard and G. Freiling [8], M.G. Gasumov and A.M. Mager- ramov [12, 13], A.P. Khromov [18], Yu. A. Mamedov [19], A.A. Shkalikov [24], Yu. T. Silchenko [26], C. Tretter [29], A.I. Vagabov [30], S. Yakubov [32]

and Y. Yakubov [34].

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A mathematical model with integral boundary conditions was derived by [9,23] in the context of optical physics. The importance of this kind of problem has been also pointed out by Samarskii [27].

In this paper, we study a problem for second order ordinary differential equations with mixed nonlocal boundary conditions combined with weighted integral boundary conditions and another two point boundary condition. Following the technique in [11, 20, 21, 22], we should bound the resolvent in the space L^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 in L^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= 1as shown in [11]. 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.

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2. Green’s Function

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

L(u)−λu = 0.

Following [20], the Green’s function of problem (1.1) is given by

(2.1) G(x, s;λ) = N(x, s;λ)

∆(λ) ,

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

(2.2) ∆(λ) =

B₁(u₁) B₁(u₂) B₂(u₁) B₂(u₂)

and

(2.3) 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

forx, s∈[0,1].The functiong(x, s, λ)is defined as follows (2.4) g(x, s;λ) =±1

2

u₁(x)u₂(s)−u₁(s)u₂(x) u⁰₁(s)u₂(s)−u₁(s)u⁰₂(s), where it takes the plus sign forx > sand the minus sign forx < s.

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Given an arbitraryδ∈ ^π₂, π

, we consider the sector Σ_δ ={λ∈C,|argλ| ≤δ, λ6= 0}. Forλ∈P

δ,defineρas the square root ofλwith positive real part. Forλ6= 0, we can consider a fundamental system of solutions of the equation u⁰⁰ =λu = ρ²ugiven byu1(t) = e^−ρtandu2(t) =e^ρt.

In the following we are going to deduce an adequate formulae for∆(λ)and G(x, s;λ).First of all, forj = 1,2,we have

B₁(u_j) =a₀+ (−1)^jb₀ρ+c₀e⁽⁻¹⁾^j^ρ+ (−1)^jd₀ρe⁽⁻¹⁾^j^ρ, B₂(u_j) =

Z 1 0

R(t)e⁽⁻¹⁾^j^ρtdt+ (−1)^jρ Z 1

0

S(t)e⁽⁻¹⁾^j^ρtdt.

so we obtain from (2.2)

(2.5) ∆(λ) = a₀−b₀ρ+c₀e^−ρ−d₀ρe^−ρ Z 1

0

(R(t) +ρS(t))e^ρt dt

−(a₀+b₀ρ+c₀e^ρ+d₀ρe^ρ) Z 1

0

(R(t)−ρS(t))e^−ρtdt

, 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.

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Thus we have

B₁(g) = a₀−b₀ρ−c₀e^−ρ+d₀ρe^−ρe^ρs 4ρ + (−a₀−b₀ρ+c₀e^ρ+d₀ρe^ρ)e^−ρs

4ρ , B₂(g) = e^ρs

4ρ Z s

0

(R(t)−ρS(t))e^−ρtdt+ Z 1

s

(−R(t) +ρS(t))e^−ρtdt

+ e^−ρs 4ρ

− Z s

0

(R(t) +ρS(t))e^ρtdt+ Z 1

s

(R(t) +ρS(t))e^ρtdt

. After a long calculation, formula (2.3) can be written as

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

(2.7) ϕ(x, s;λ) = 1 2ρ

Z 1 0

(a₀+ρb₀) (R(t) +ρS(t))e^ρtdt

− e^ρ(c₀+ρd₀) Z s

0

e^−ρ(x+s) +

Z 1 s

(a₀−ρb₀) (R(t)−ρS(t))e^−ρtdt

− e^ρ(c₀−ρd₀) Z s

0

(R(t)−ρS(t))e^−ρtdt

e^ρ(x+s)

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and the functionϕ_i(x, s;λ)is given by

(2.8) ϕ_i(x, s;λ) =







ϕ1(x, s;λ) ifx > s, ϕ₂(x, s;λ) ifx < s, with

(2.9) ϕ₁(x, s;λ)

= e^ρ(s−x) 2ρ

Z s 0

(a₀+ρb₀+c₀e^ρ+ρe^ρd₀) (R(t)−ρS(t))e^−ρtdt

− Z 1

s

(a₀−ρb₀) (R(t) +ρS(t))e^ρtdt

+e^ρ(x−s) 2ρ

Z s 0

a₀−ρb₀+c₀e^−ρ−ρe^−ρd₀

− Z 1

0

(a₀+ρb₀) (R(t)−ρS(t))e^−ρtdt

, and

(2.10) ϕ₂(x, s;λ)

= 1 2ρ

− Z 1

s

(a0+ρb0+c0e^ρ+ρe^ρd0) (R(t)−ρS(t))e^−ρtdt +

Z 1 0

(c₀−ρd₀)e^−ρ(R(t) +ρS(t))e^ρtdt

e^ρ(s−x)

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− Z 1

s

a₀−ρb₀+c₀e^−ρ−ρe^−ρd₀

− Z 1

0

(c₀+ρd₀)e^ρ(R(t)−ρS(t))e^−ρtdt

e^ρ(x−s)

.

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3. Bounds on the Resolvent

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

(3.1) (λI−L_p)⁻¹f =R(λ, L_p)f =− Z 1

0

G(·, s;λ)f(s)ds, f ∈L^p(0,1).

Then, for everyf ∈L^p(0,1)we estimate (3.1) kR(λ;L_p)fk_Lp(0,1) ≤

sup

0≤s≤1

Z 1 0

|G(x, s;λ)|^pdx ¹_p

kfk_Lp(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

.

3.1. Estimation of N (x, s; λ)

We will denote byk·kthe supremum norm which is defined bykRk= sup

0≤s≤1

|R(s)|.

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 ; then

(3.2) kN(x, s;λ)k_Lp ≤ kϕ(x, s;λ)k_Lp+kϕi(x, s;λ)k_Lp,

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from (2.8), we have (3.3) kϕ_i(x, s;λ)k_Lp

≤2^1/p

(Z s 0

|ϕ2(x, s;λ)|^pdx ¹_p

+ Z 1

s

|ϕ1(x, s;λ)|^pdx ¹_p)

. From (2.7) we have

|ϕ(x, s;λ)| ≤ e^{−(x+s) Re}^ρ 2|ρ|

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) Z 1

s

e^tRe^ρdt + (kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|)e^Re^ρ

Z s 0

e^t^Re^ρdt

+e^{(x+s) Re}^ρ 2|ρ|

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) Z 1

s

e^−t^Re^ρdt + e⁻^Re^ρ(kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|)

Z s 0

e^−t^Re^ρdt

then

|ϕ(x, s;λ)| ≤ e^{−(x+s) Re}^ρ 2|ρ|Reρ

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) e^Re^ρ−e^s^Re^ρ + (kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) e^{(s+1) Re}^ρ−e^Re^ρ

+ e^{(x+s) Re}^ρ 2|ρ|Reρ

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) e^−s^Re^ρ−e⁻^Re^ρ + (kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) e⁻^Re^ρ−e^{−(s+1) Re}^ρ

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and Z 1

0

|ϕ(x, s;λ)|^pdx

≤ 2^pR1

0 e^−px^Re^ρdx (|ρ|Reρ)^p

h(kRk+|ρ| kSk)^p(|a₀|+|ρ| |b₀|)^p e^{(1−s) Re}^ρ−1p

+ (kRk+|ρ| kSk)^p× (|c0|+|ρ| |d0|)^p e^Re^ρ−e^{(1−s) Re}^ρ^pi +2^pR1

0 e^pxRe^ρdx

(|ρ|Reρ)^p [(kRk+|ρ| kSk)^p (|a₀|+|ρ| |b₀|)^p 1−e^{(s−1) Re}^ρ^p + (kRk+|ρ| kSk)^p(|c₀|+|ρ| |d₀|)^p e^{(s−1) Re}^ρ−e^{−1 Re}^ρpi

, after calculation we obtain

Z 1 0

|ϕ(x, s;λ)|^pdx ¹_p

≤ 2¹⁺²^pe^Re^ρ

|ρ|(Reρ)¹⁺^p¹ p^1/p

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|)

e^−s^Re^ρ−e⁻^Re^ρ

× 1−e^−p^Re^ρ¹_p

+ 1−e^−p^Re^ρ_p¹

1−e^{(s−1) Re}^ρi

+(kRk+|ρ| kSk)

×(|c0|+|ρ| |d0|)h

1−e^−p^Re^ρ¹_p

1−e^−s^Re^ρ + 1−e^−p^Re^ρ¹_p

e^{(s−1) Re}^ρ−e⁻^Re^ρii

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asReρ >0so

(3.4) sup

0≤s≤1

Z 1 0

|ϕ(x, s;λ)|^pdx ¹_p

≤ 2¹⁺²^pe^Re^ρ(kRk+|ρ| kSk) [(|a₀|+|c₀|) +|ρ|(|b₀|+|d₀|)]

|ρ|(Reρ)¹⁺¹^p p^1/p

.

From (2.9) we have

|ϕ₁(x, s;λ)| ≤ e^{(s−x) Re}^ρ 2|ρ|

(|a₀|+|ρ| |b₀|) +e^Re^ρ(|c₀|+|ρ| |d₀|)

×(kRk+|ρ| kSk) Z s

0

e^−t^Re^ρdt + (|a₀|+|ρ| |b₀|) (kRk+|ρ| kSk)

Z 1 0

e^tRe^ρdt

+ e^{(x−s) Re}^ρ 2|ρ|

(kRk+|ρ| kSk)

[(|a₀|+|ρ| |b₀|) +e^Re^ρ(|c0|+|ρ| |d0|)]

Z s 0

e^t^Re^ρdt + (|a₀|+|ρ| |b₀|) (kRk+|ρ| kSk)

Z 1 0

e^−tRe^ρdt

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then

|ϕ1(x, s;λ)| ≤ e^{(s−x) Re}^ρ 2|ρ|Reρ

(|a0|+|ρ| |b0|) (kRk+|ρ| kSk) e^Re^ρ−e^−s^Re^ρ + (|c₀|+|ρ| |d₀|) (kRk+|ρ| kSk) e^Re^ρ−e^{(1−s) Re}^ρ

+e^{(x−s) Re}^ρ 2|ρ|Reρ

(kRk+|ρ| kSk) (|a₀|+|ρ| |b₀|) e^s^Re^ρ−e⁻^Re^ρ + (|c₀|+|ρ| |d₀|) (kRk+|ρ| kSk) e^{(s−1) Re}^ρ−e⁻^Re^ρ and

Z 1 s

|ϕ₁(x, s;λ)|^pdx

≤ 2^pR1

s e^−px^Re^ρdx

|ρ|^p(Reρ)^p

h(|a₀|+|ρ| |b₀|)^p(kRk+|ρ| kSk)^p e^{(1+s) Re}^ρ−1^p + (|c₀|+|ρ| |d₀|)^p(kRk+|ρ| kSk)^p e^{(1+s) Re}^ρ−e^Re^ρpi

+2^pR1

s e^pxRe^ρdx

|ρ|^p(Reρ)^p h

(|a0|+|ρ| |b0|)^p(kRk+|ρ| kSk)^p 1−e^{−(1+s) Re}^ρ^p + (|c₀|+|ρ| |d₀|)^p(kRk+|ρ| kSk)^p e⁻^Re^ρ−e^{−(1+s) Re}^ρp

, this yields

Z 1 s

|ϕ₁(x, s;λ)|^pdx≤ 2^p+1e^p^Re^ρ

|ρ|^p(Reρ)^p+1[(|a₀|+|ρ| |b₀|)^p(kRk+|ρ| kSk)^p

× 1−e^{−(1+s) Re}^ρ^p

1−e^{p(s−1) Re}^ρ

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+ (|c₀|+|ρ| |d₀|)^p(kRk+|ρ| kSk)^p 1−e^−s^Re^ρp

1−e^{p(s−1) Re}^ρ asReρ >0we obtain

(3.5) sup

0≤s≤1

Z 1 s

|ϕ₁(x, s;λ)|^pdx ¹_p

≤ 2¹⁺²^pe^Re^ρ(kRk+|ρ| kSk) [(|a₀|+|c₀|) +|ρ|(|b₀|+|d₀|)]

|ρ|(Reρ)¹⁺¹^p p^1/p

. From (2.10) we have

|ϕ₂(x, s;λ)|

≤ e^{(x−s) Re}^ρ 2|ρ|

(kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) Z 1

0

e^{(1−t) Re}^ρdt + (kRk+|ρ| kSk)

(|a₀|+|ρ| |b₀|) +e⁻^Re^ρ(|c₀|+|ρ| |d₀|) Z 1

0

e^tRe^ρdt

+ e^{(s−x) Re}^ρ

2|ρ| [(|c₀|+|ρ| |d₀|) (kRk+|ρ| kSk) Z 1

0

e^{(t−1) Re}^ρdt +

(|a₀|+|ρ| |b₀|) +e^Re^ρ(|c₀|+|ρ| |d₀|)

(kRk+|ρ| kSk) Z 1

s

e^−t^Re^ρdt

then

|ϕ₂(x, s;λ)| ≤ e^x^Re^ρ 2|ρ|Reρ

(kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) e^{(1−s) Re}^ρ−e⁻^Re^ρ + (|a0|+|ρ| |b0|) (kRk+|ρ| kSk) e^{(1−s) Re}^ρ−1

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+ e^−x^Re^ρ 2|ρ|Reρ

(kRk+|ρ| kSk) (|c₀|+|ρ| |d₀|) e^Re^ρ−e^{(s−1) Re}^ρ + (|a₀|+|ρ| |b₀|) (kRk+|ρ| kSk) 1−e^{(s−1) Re}^ρ

. So

Z s 0

|ϕ₂(x, s;λ)|^pdx

≤ 2^p+1e^p^Re^ρ p|ρ|^p(Reρ)^p+1

h(kRk+|ρ| kSk)^p(|c₀|+|ρ| |d₀|)^p 1−e^{(s−2) Re}^ρp

× 1−e^−ps^Re^ρ

+ (|a0|+|ρ| |b0|)^p

× (kRk+|ρ| kSk)^p 1−e^−ps^Re^ρ

1−e^{(s−1) Re}^ρ^pi asReρ >0we obtain

(3.6) sup

0≤s≤1

Z s 0

|ϕ₂(x, s;λ)|^pdx _p¹

≤ 2¹⁺²^pe^Re^ρ(kRk+|ρ| kSk) [(|a0|+|c0|) +|ρ|(|b0|+|d0|)]

|ρ|(Reρ)¹⁺¹^p p^1/p

. From (3.4), (3.5) and (3.6), we obtain

kR(λ, L_p)k

≤ 2¹⁺²^pe^Re^ρ(kRk+|ρ| kSk) [(|a0|+|c0|) +|ρ|(|b0|+|d0|)]

|4(λ)| |ρ|(Reρ)¹⁺¹^pp^1/p

2¹⁺²^p + 1

,

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forρ∈Σδ

2 =

ρ∈C:|argρ| ≤ ^δ₂, ρ6= 0 , we have(Reρ)⁻¹ < ¹

|ρ|cos(^δ₂), then kR(λ, L_p)k

≤

2¹⁺²^p

2¹⁺²^p + 1

e^Re^ρ(kRk+|ρ| kSk) [(|a₀|+|c₀|) +|ρ|(|b₀|+|d₀|)]

|4(λ)| |ρ|

|ρ|¹⁺¹^p

cos₂^δ1+¹_p

p^1/p

.

Finally, we obtain forλ=ρ² ∈Σ_δ (3.7) kR(λ, L_p)k_Lp

≤ c e^Re^ρ

|4(λ)| |ρ|¹⁺¹^p

kRk

|ρ| +kSk

[(|a₀|+|c₀|) +|ρ|(|b₀|+|d₀|)], where

c=

2¹⁺²^p

2¹⁺²^p + 1 p^1/p cos^δ₂1+¹_p .

3.2. Estimation of the Characteristic Determinant

The next step is to determine the cases for which |∆(λ)| remains bounded be- low. 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 ad- ditional regularity hypotheses on the functionsRandS, and so we assume that the functionsRandS are inC²([0,1],C).

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Integrating twice by parts in (2.5), we obtain (3.8) ∆(λ) = e^ρh

ρ(d₀S(0)−b₀S(1)) + (d₀S⁰(0) +b₀S⁰(1) +a₀S(1)

−b0R(1)−d0R(0) +c0S(0)) + 1

ρ(a0R(1)−c0R(0) +b0R⁰(1)

−a₀S⁰(1)−d₀R⁰(0) +c₀S⁰(0)) + 1

ρ²(−a₀R⁰(1)−c₀R⁰(0)) + Φ(ρ)

, where

(3.9) Φ(ρ) =

(a₀−ρb₀)e^−ρ+ (c₀−ρd₀)e^−2ρ Z 1

0

R⁰⁰(t)

ρ² − S⁰⁰(t) ρ

e^ρtdt +

(a₀+ρb₀)e^−ρ+ (c₀+ρd₀) Z 1

0

R⁰⁰(t)

ρ² − S⁰⁰(t) ρ

e^−ρtdt +e^−ρ

1 ρ

−b₀R⁰(0) +d₀R⁰(1)−c₀S⁰(1) +a₀S⁰(0) +c₀R(1)−a₀R(0) +ρ(b₀S(0)−d₀S(1))] + 2e^−2ρ[(a₀+ρb₀)(R(1)−ρS(1))

−(c₀−ρd₀)(R(0) +ρS(0)) + 1

ρ²(a₀+ρb₀)(R⁰(1)−ρS⁰(1)) +1

ρ²(c₀−ρd₀)(R⁰(0) +ρS⁰(0))

. After a straightforward calculation, we obtain the following inequality valid for

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ρ∈Σ^δ

2,with|ρ|sufficiently large,

|Φ(ρ)|

≤

(|a₀|+|ρ| |b₀|)e⁻^Re^ρ+ (|c₀|+|ρ| |d₀|)e^{−2 Re}^ρ

×

kR⁰⁰k

|ρ|² +kS⁰⁰k

|ρ|

e^Re^ρ−1 Reρ

+

(|a₀|+|ρ| |b₀|)e⁻^Re^ρ+ (|c₀|+|ρ| |d₀|)

×

kR⁰⁰k

|ρ|² +kS⁰⁰k

|ρ|

1−e⁻^Re^ρ Reρ

+ 2e⁻^Re^ρ 1

|ρ|

−b₀R⁰(0) +d₀R⁰(1)−c₀S⁰(1) +a₀S⁰(0) +c₀R(1)−a₀R(0) + |ρ| |b₀S(0)−d₀S(1)|] +e^{−2 Re}^ρ

1

|ρ|(|a₀|+|ρ| |b₀|) (kRk+|ρ| kSk) + 1

|ρ|(|c0|+|ρ| |d0|) (kRk+|ρ| kSk) + (|a0|+|ρ| |b0|)

R⁰

+|ρ|

S⁰

|ρ|²

!

+ (|c₀|+|ρ| |d₀|)

R⁰

+|ρ|

S⁰

|ρ|²

!#

.

Then

|Φ(ρ)| ≤ 2

|ρ|cos₂^δ

(|a₀|+|c₀|)

|ρ| + (|b₀|+|d₀|)

kR⁰⁰k

|ρ| +kS⁰⁰k

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+ 1

|ρ|cos^δ₂ 1

|ρ|²

−b₀R⁰(0) +d₀R⁰(1)−c₀S⁰(1) + +a₀S⁰(0) +c₀R(1)−a₀R(0)

|b₀S(0)−d₀S(1)|i

+ 1

|ρ|(cos^δ₂)²

|a0|

|ρ| +|b₀| kRk

|ρ| +kSk

+ |c₀|

|ρ| +|d₀| kRk

|ρ| +kSk

+ |a0|

|ρ| +|b₀|

R⁰

|ρ|² + S⁰

|ρ|

!

+ |c₀|

|ρ| +|d₀|

R⁰

|ρ|² + S⁰

|ρ|

!#

(3.10) .

Where we have used thatRe(ρ)≥ |ρ|cos(^δ₂),(1−e⁻^Re^ρ)<1, e⁻^Re^ρ≤ 2

|ρ|²(cos^δ₂)²ande^{−2 Re}^ρ≤ 1

2|ρ|²(cos^δ₂)², e⁻^Re^ρ<1.

There are several cases to analyze depending on the functionsRandS.

• Case 1.

Suppose thatS 6= 0,d0 6= 0, b0 6= 0, d0S(0)−b0S(1) = 0and

k1 =d0S⁰(0) +b0S⁰(1) +a0S(1)−b0R(1)−d0R(0) +c0S(0)6= 0.

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From (3.8), we have for|ρ|sufficiently large

|4(λ)|

≥e^Re^ρh

d₀S⁰(0) +b₀S⁰(1) +a₀S(1)−b₀R(1)−d₀R(0) +c₀S(0)

− 1

|ρ|

a₀R(1)−c₀R(0) +b₀R⁰(1)−a₀S⁰(1)−d₀R⁰(0) +c₀S⁰(0)

− 1

|ρ|²

a₀R⁰(1) +c₀R⁰(0)

− |Φ(ρ)|

. From (3.10) we deduce forρ∈Σ^δ

2,|ρ| ≥r₀ >0.

|Φ(ρ)| ≤ c(r₀)

|ρ| . Then, we have

|4(λ)|

≥e^Re^ρh

d₀S⁰(0) +b₀S⁰(1) +a₀S(1)−b₀R(1)−d₀R(0) +c₀S(0)

− 1

|ρ|

a₀R(1)−c₀R(0) +b₀R⁰(1)−a₀S⁰(1)−d₀R⁰(0) +c₀S⁰(0)

− 1

|ρ|²

a₀R⁰(1) +c₀R⁰(0)

− c(r₀)

|ρ|

.

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we can now chooser₀ >0, such that 1

r0

a₀R(1)−c₀R(0) +b₀R⁰(1)−a₀S⁰(1)−d₀R⁰(0) +c₀S⁰(0) + 1

r₀²

a₀R⁰(1) + c₀R⁰(0)

+ c(r₀)

|ρ|

≤ 1 2

d₀S⁰(0) +b₀S⁰(1) +a₀S(1)−b₀R(1)−d₀R(0) +c₀S(0) , then, forρ∈Σ^δ

2,|ρ| ≥r0 >0, we get

|4(λ)| ≥ e^Re^ρ 2 |k₁|.

From (3.7), we deduce the following bound, valid for every|argρ| ≤ ^δ₂, ρ6= 0 kR(λ, L_p)k ≤ 2c

|ρ|¹^p|k₁|

|a₀|+|c₀|

|ρ|

+ (|b₀|+|d₀|) kRk

|ρ| +kSk

, then

kR(λ, Lp)k ≤ c₁

|λ|^2p¹ , as|λ| −→+∞,where

c₁ = 2ckSk(|b₀|+|d₀|)

|k₁| .

• Case 2.

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Ifb₀ =d₀ = 0, S 6= 0anda₀S(1) +c₀S(0) = 0,with

k₂ =a₀R(1)−c₀R(0)−a₀S⁰(1) +c₀S⁰(0) 6= 0, we have the following bound, valid forλ∈Σ_δand|λ|big enough,

kR(λ, L_p)k ≤ 2c(|a₀|+|c₀|)

|ρ|¹^p |k₂|

kRk

|ρ| +kSk

, then

kR(λ, L_p)k ≤ c₁

|λ|^2p¹ , as|λ| −→+∞,where

c₁ = 2c(|a₀|+|c₀|)kSk

|k₂| .

• Case 3.

IfS = 0,b₀ 6= 0,d₀ 6= 0andb₀R(1) +d₀R(0) = 0,with k₃ =a₀R(1)−c₀R(0) +b₀R⁰(1)−d₀R⁰(0)6= 0.

Similarly, we get

kR(λ, L_p)k ≤ 2ckRk

|ρ|¹⁺¹^p|k₃|

(|a₀|+|c₀|)

|ρ| + (|b₀|+|d₀|)

,

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then, we have

kR(λ, L_p)k ≤ c₁

|λ|^2p¹ , as|λ| −→+∞,where

c₁ = 2ckRk(|b₀|+|d₀|)

|k₃| .

• Case 4.

Ifb0 =d0 = 0,S = 0anda0R(1)−c0R(0) = 0with k₄ =a₀R⁰(1)−c₀R⁰(0)6= 0.

Again in this case, we have

kR(λ, Lp)k ≤ 2c(|a0|+|c0|)kRk

|ρ|¹^p|k₄| , then

kR(λ, L_p)k ≤ c1

|λ|^2p¹ , as|λ| −→+∞,where

c₁ = ckRk(|a₀|+|c₀|)

|k₄| .

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Definition 3.1. The boundary value conditions in (1.1) are called non regular if the functions R, S ∈ C²([0,1],C) and if and only if one of the following conditions holds

1-d₀S(0)−b₀(1) = 0, b₀ 6= 0, d₀ 6= 0, S 6= 0and

d₀S⁰(0) +b₀S⁰(1) +a₀S(1)−b₀R(1)−d₀R(0) +c₀S(0) 6= 0 2-b₀ =d₀ = 0,kSk 6= 0, a₀S(1) +c₀S(0) = 0with

a₀R(1)−c₀R(0)−a₀S⁰(1) +c₀S⁰(0)6= 0 3-S = 0,b₀ 6= 0, d₀ 6= 0, b₀R(1) +₀R(0)6= 0with

a₀R(1)−c₀R(0) +b₀R⁰(1)−d₀R⁰(0)6= 0 4-b₀ =d₀ = 0,S = 0,a₀R(1)−c₀R(0) = 0with

a0R⁰(1)−c0R⁰(0)6= 0.

This proves the following theorem

Theorem 3.1. If the boundary value conditions in (1.1) are non regular, then Σ_δ ⊂ρ(L_p)for sufficiently large|λ|and there existsc >0such that

kR(λ, L_p)k ≤ c

|λ|^2p¹ .

Remark 3.1. From Theorem3.1it follows that the operatorL_p,forp6=∞, gen- erates an analytic semigroup with singularities [25] of typeA(2p−1, 4p−1).

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3.3. Application

In the following, we apply the above obtained results to the study of a class of a mixed problem for a parabolic equation with an weighted integral boundary condition combined with another two point boundary condition of the form

(3.11)











∂u(t, x)

∂t −a∂²u(t, x)

∂x² =f(t, x)

L₁(u) = a₀u(0, t) +b₀u⁰(0, t) +c₀u(1, t) +d₀u⁰(1, t) = 0, L2(u) = R1

0 R(ξ)u(t, ξ)dξ+R1

0 S(ξ)u⁰(t, ξ)dξ = 0, u(0, x) =u₀(x),

where(t, ξ)∈[0, T]×[0,1]. Boundary value problems for parabolic equations with integral boundary conditions are studied by [1, 3, 5, 6, 14, 15, 16, 17, 35] using various methods. For instance, the potential method in [3] and [17], Fourier method in [1, 14, 15,16] and the energy inequalities method has been used in [5, 6, 35]. In our case, we apply the method of operator differential equation. The study of the problem is then reduced to a cauchy problem for a parabolic abstract differential equation, where the operator coefficients has been previously studied. For this purpose, letE,E₁, andE₂be Banach spaces.

Introduce two Banach spaces C_µ((0, T], E)

= (

f /f ∈C((0, T], E), kfk= sup

t∈(0,T]

kt^µf(t)k<∞ )

, µ≥0,

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C_µ^γ((0, T], E) = (

f /f ∈C((0, T], E), kfk= sup

t∈(0,T]

kt^µf(t)k + sup

0<t<t+h≤T

kf(t+h)−f(t)kh^−γt^µ<∞

,

µ≥0, γ ∈(0,1], and a linear space

C¹((0, T], E1, E2) =

f /f ∈C((0, T], E1)∩C¹((0, T], E2) , E1 ⊂E2, where C((0, T], E) andC¹((0, T], E)are spaces of continuous and continu- ously differentiable, respectively, vector-function from (0, T] into E. We denote, for a linear operatorAin a Banach spaceE, by

E(A) = n

u/u∈D(A), kuk_E(A) = kAuk²+kuk²¹₂o , and

C¹((0, T], E(A), E) =n

f /f ∈C((0, T], E(A)), f⁰ ∈C((0, T], E))o . Let us derive a theorem which was proved by various methods in [25] and [31, 33]. Consider, in a Banach spaceE, the Cauchy problem

(3.12)

( u⁰(t) = Au(t) +f(t), t∈[0, T], u(0) =u0,

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where Ais, generally speaking, unbounded linear operator inE, u₀ is a given element of E, f(t) is a given vector-function and u(t) is an unknown vector- function inE.

Theorem 3.2. Let the following conditions be satisfied:

1. Ais a closed linear operator in a Banach spaceEand for someβ∈(0,1], α >0

kR(λ, A)k ≤C|λ|^−β, |argλ| ≤ π

2 +α, |λ| → ∞;

2. f ∈C_µ^γ((0, T], E)for someγ ∈(1−β,1], µ∈[0, β) ; 3. u₀ ∈D(A).

Then the Cauchy problem (3.12) has a unique solution u∈C([0, T], E)∩C¹((0, T], E(A), E) and for the solutionuthe following estimates hold

ku(t)k ≤C

kAu₀k+ku₀k+kfk_C

µ((0,t],E)

, t∈(0, T],

u⁰(t)

+kAu(t)k

≤C

t^β−1(kAu₀k+ku₀k) +t^β−µ−1kfk_C^γ

µ((0,t],E)

, t∈(0, T].