Journal of Inequalities in Pure and
Applied Mathematics
BOUNDARY VALUE PROBLEM FOR SECOND-ORDER
DIFFERENTIAL OPERATORS WITH MIXED NONLOCAL
BOUNDARY CONDITIONS
volume 5, issue 2, article 38,
2004.
M. DENCHE AND A. KOURTA
Laboratoire Equations Differentielles,
Departement de Mathematiques,
Faculte des Sciences, Universite Mentouri,
25000 Constantine, Algeria.
EMail: denech@wissal.dz
Received 15 October, 2003;
accepted 16 April, 2004.
Communicated by: A.G. Ramm
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
147-03
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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 in Lp (0, 1), but there is no maximal decreasing at infinity
for p ≥ 1. Furthermore, the studied operator generates in Lp (0, 1) an analytic
semi group with singularities for p ≥ 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.
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
2000 Mathematics Subject Classification: 47E05, 35K20.
Key words: Green’s Function, Non Regular Boundary Conditions, Regular Boundary
Conditions, Semi group with Singularities.
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Contents
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.
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Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bounds on the Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Estimation of N (x, s; λ) . . . . . . . . . . . . . . . . . . . . . . .
3.2
Estimation of the Characteristic Determinant . . . . . . .
3.3
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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5
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1.
Introduction
In the space Lp (0, 1) we consider the boundary value problem

00
 L (u) = u = f (x) ,
0
0
B1 (u) = a0 u (0) + b0 u (0) + c0 u (1) + d0 u (1) = 0,
(1.1)
R1
R1

0
B2 (u) = 0 R (t) u (t) dt + 0 S (t) u (t) dt = 0,
where the functions R, S ∈ C([0, 1] , C). We associate to problem (1.1) in space
Lp (0, 1) the operator:
Lp (u) = u00 ,
with domain D(Lp ) = {u ∈ W 2,p (0, 1) : Bi (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. Jakson, 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. Magerramov [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].
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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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
Lp (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 Lp (0, 1), but there is no
maximal decreasing at infinity for p ≥ 1. In contrast to the regular case this
decreasing is maximal for p = 1 as shown in [11]. Furthermore, the studied operator generates in Lp (0, 1) an analytic semi group with singularities for p ≥ 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.
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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2.
Green’s Function
Let λ ∈ C, u1 (x) = u1 (x, λ ) and u2 (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
B1 (u1 ) B1 (u2 ) (2.2)
∆(λ) = B2 (u1 ) B2 (u2 ) u1 (x) u2 (x) g(x, s, λ)
B1 (g)x
N (x, s; λ) = B1 (u1 ) B1 (u2 )
B2 (u1 ) B2 (u2 )
B2 (g)x
for x, s ∈ [0, 1].The function g(x, s, λ) is defined as follows
(2.4)
M. Denche and A. Kourta
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and
(2.3)
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
g(x, s; λ) = ±
1 u1 (x)u2 (s) − u1 (s)u2 (x)
,
2 u01 (s)u2 (s) − u1 (s)u02 (s)
where it takes the plus sign for x > s and the minus sign for x < s.
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Given an arbitrary δ ∈
π
,π
2
, we consider the sector
Σδ = {λ ∈ C, |arg λ| ≤ δ, λ 6= 0} .
P
For λ ∈ δ , define ρ as the square root of λ with positive real part. For λ 6= 0,
we can consider a fundamental system of solutions of the equation u00 = λu =
ρ2 u given by u1 (t) = e−ρt and u2 (t) = eρt .
In the following we are going to deduce an adequate formulae for ∆(λ) and
G(x, s; λ). First of all, for j = 1, 2, we have
(−1)j ρ
j
j
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
(−1)j ρ
B1 (uj ) = a0 + (−1) b0 ρ + c0 e
+ (−1) d0 ρe
,
Z 1
Z 1
j
j
B2 (uj ) =
R(t)e(−1) ρt dt + (−1)j ρ
S(t)e(−1) ρt dt.
0
M. Denche and A. Kourta
0
so we obtain from (2.2)
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(2.5) ∆(λ) = a0 − b0 ρ + c0 e−ρ − d0 ρe−ρ
ρ
ρ
1
Z
Z0
− (a0 + b0 ρ + c0 e + d0 ρe )
ρt
(R(t) + ρS(t))e dt
1
(R(t) − ρS(t))e
0
dt ,
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and g(x, s; λ) has the form
g(x, s; λ) =
−ρt
Contents

1 ρ(x−s)


(e
− eρ(s−x) ) if x > s,

 4ρ


1 ρ(s−x)


(e
− eρ(x−s) ) if x < s.
4ρ
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Thus we have
B1 (g) = a0 − b0 ρ − c0 e−ρ + d0 ρe−ρ
eρs
4ρ
e−ρs
+ (−a0 − b0 ρ + c0 eρ + d0 ρeρ )
,
4ρ
Z s
Z 1
eρs
−ρt
−ρt
B2 (g) =
(R(t) − ρS(t))e dt +
(−R(t) + ρS(t))e dt
4ρ
0
s
Z s
Z 1
e−ρs
ρt
ρt
+
−
(R(t) + ρS(t))e dt +
(R(t) + ρS(t))e dt .
4ρ
0
s
After a long calculation, formula (2.3) can be written as
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
N (x, s; λ) = ϕ(x, s; λ) + ϕi (x, s; λ),
(2.6)
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where
Contents
1
2ρ
Z
1
(a0 + ρb0 ) (R (t) + ρS (t)) eρt dt
0
Z s
ρ
ρt
− e (c0 + ρd0 )
(R (t) + ρS (t)) e dt e−ρ(x+s)
0
Z 1
+
(a0 − ρb0 ) (R (t) − ρS (t)) e−ρt dt
s
Z s
ρ
−ρt
ρ(x+s)
− e (c0 − ρd0 )
(R (t) − ρS (t)) e dt e
(2.7) ϕ(x, s; λ) =
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and the function ϕi (x, s; λ) is given by

 ϕ1 (x, s; λ) if x > s,
(2.8)
ϕi (x, s; λ) =

ϕ2 (x, s; λ) if x < s,
with
(2.9) ϕ1 (x, s; λ)
Z s
eρ(s−x)
=
(a0 + ρb0 + c0 eρ + ρeρ d0 ) (R (t) − ρS (t)) e−ρt dt
2ρ
0
Z 1
ρt
−
(a0 − ρb0 ) (R (t) + ρS (t)) e dt
s
Z s
eρ(x−s)
+
a0 − ρb0 + c0 e−ρ − ρe−ρ d0 (R (t) + ρS (t)) eρt dt
2ρ
0
Z 1
−ρt
−
(a0 + ρb0 ) (R (t) − ρS (t)) e dt ,
0
and
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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(2.10) ϕ2 (x, s; λ)
Z 1
1
−
(a0 + ρb0 + c0 eρ + ρeρ d0 ) (R (t) − ρS (t)) e−ρt dt
=
2ρ
s
Z 1
−ρ
ρt
+
(c0 − ρd0 ) e (R (t) + ρS (t)) e dt eρ(s−x)
0
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Z
−
s
1
a0 − ρb0 + c0 e−ρ − ρe−ρ d0 (R (t) + ρS (t)) eρt dt
Z 1
ρ
−ρt
ρ(x−s)
−
(c0 + ρd0 ) e (R (t) − ρS (t)) e dt e
.
0
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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3.
Bounds on the Resolvent
Every λ ∈ C such that ∆(λ) 6= 0 belongs to ρ(Lp ), and the associated resolvent
operator R(λ, Lp ) can be expressed as a Hilbert Schmidt operator
Z 1
−1
(3.1) (λI − Lp ) f = R(λ, Lp )f = −
G(·, s; λ)f (s)ds, f ∈ Lp (0, 1).
0
Then, for every f ∈ Lp (0, 1) we estimate (3.1)
kR(λ; Lp )f kLp (0,1) ≤
Z
sup
0≤s≤1
and so we need to bound
p1
Z 1
p
sup
|G(x, s; λ)| dx
=
0≤s≤1
3.1.
0
1
|G(x, s; λ)|p dx
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
p1
kf kLp (0,1) ,
0
M. Denche and A. Kourta
1
|∆(λ)|
Z
p
|N (x, s; λ)| dx
sup
0≤s≤1
p1
1
.
0
Contents
Estimation of N (x, s; λ)
We will denote by k·k the supremum norm which is defined by kRk = sup |R (s)|.
0≤s≤1
Since
(
N (x, s; λ) =
ϕ (x, s; λ) + ϕ1 (x, s; λ) if x > s
ϕ (x, s; λ) + ϕ2 (x, s; λ) if x < s
;
then
(3.2)
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kN (x, s; λ)kLp ≤ kϕ (x, s; λ)kLp + kϕi (x, s; λ)kLp ,
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from (2.8), we have
(3.3)
kϕi (x, s; λ)kLp
(Z
p1 Z
s
p
1/p
≤2
|ϕ2 (x, s; λ)| dx +
0
1
|ϕ1 (x, s; λ)|p dx
p1 )
.
s
From (2.7) we have
Z 1
e−(x+s) Re ρ
|ϕ (x, s; λ)| ≤
(kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |)
et Re ρ dt
2 |ρ|
s
Z s
Re ρ
t Re ρ
+ (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e
e
dt
0
Z 1
e(x+s) Re ρ
(kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |)
e−t Re ρ dt
+
2 |ρ|
Z ss
− Re ρ
−t Re ρ
+e
(kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |)
e
dt
0
then
e−(x+s) Re ρ (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) eRe ρ − es Re ρ
2 |ρ| Re ρ
+ (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e(s+1) Re ρ − eRe ρ
e(x+s) Re ρ (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) e−s Re ρ − e− Re ρ
+
2 |ρ| Re ρ
+ (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e− Re ρ − e−(s+1) Re ρ
|ϕ (x, s; λ)| ≤
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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and
Z
1
|ϕ (x, s; λ)|p dx
0
R1
p
2p 0 e−px Re ρ dx h
≤
(kRk + |ρ| kSk)p (|a0 | + |ρ| |b0 |)p e(1−s) Re ρ − 1
p
(|ρ| Re ρ)
i
p
p
Re ρ
(1−s) Re ρ p
+ (kRk + |ρ| kSk) × (|c0 | + |ρ| |d0 |) e
−e
R
1
p
2p 0 epx Re ρ dx
+
[(kRk + |ρ| kSk)p (|a0 | + |ρ| |b0 |)p 1 − e(s−1) Re ρ
p
(|ρ| Re ρ)
p i
+ (kRk + |ρ| kSk)p (|c0 | + |ρ| |d0 |)p e(s−1) Re ρ − e−1 Re ρ
,
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
after calculation we obtain
Title Page
Z
1
|ϕ (x, s; λ)|p dx
p1
Contents
0
1+ p2 Re ρ
≤
2
e
1
|ρ| (Re ρ)1+ p p1/p
(kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) e−s Re ρ − e− Re ρ
× 1 − e−p Re ρ
p1
p1
i
1 − e(s−1) Re ρ +(kRk + |ρ| kSk)
h
1
× (|c0 | + |ρ| |d0 |) 1 − e−p Re ρ p 1 − e−s Re ρ
1
ii
+ 1 − e−p Re ρ p e(s−1) Re ρ − e− Re ρ
+ 1 − e−p Re ρ
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as Re ρ > 0 so
Z
(3.4)
p
p1
|ϕ (x, s; λ)| dx
sup
0≤s≤1
1
0
2
≤
21+ p eRe ρ (kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)]
1
|ρ| (Re ρ)1+ p p1/p
From (2.9) we have
e(s−x) Re ρ |ϕ1 (x, s; λ)| ≤
(|a0 | + |ρ| |b0 |) + eRe ρ (|c0 | + |ρ| |d0 |)
2 |ρ|
Z s
× (kRk + |ρ| kSk)
e−t Re ρ dt
0
Z 1
t Re ρ
+ (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk)
e
dt
0
e(x−s) Re ρ
+
(kRk + |ρ| kSk) [(|a0 | + |ρ| |b0 |)
2 |ρ|
Z s
Re ρ
+e
(|c0 | + |ρ| |d0 |)]
et Re ρ dt
0
Z 1
−t Re ρ
+ (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk)
e
dt
0
.
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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then
e(s−x) Re ρ |ϕ1 (x, s; λ)| ≤
(|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) eRe ρ − e−s Re ρ
2 |ρ| Re ρ
+ (|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk) eRe ρ − e(1−s) Re ρ
e(x−s) Re ρ (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) es Re ρ − e− Re ρ
+
2 |ρ| Re ρ
+ (|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk) e(s−1) Re ρ − e− Re ρ
and
Z
1
|ϕ1 (x, s; λ)|p dx
s
R1
p
2p s e−px Re ρ dx h
≤
(|a0 | + |ρ| |b0 |)p (kRk + |ρ| kSk)p e(1+s) Re ρ − 1
p
p
|ρ| (Re ρ)
p i
+ (|c0 | + |ρ| |d0 |)p (kRk + |ρ| kSk)p e(1+s) Re ρ − eRe ρ
R1
p
2p s epx Re ρ dx h
(|a0 | + |ρ| |b0 |)p (kRk + |ρ| kSk)p 1 − e−(1+s) Re ρ
+
p
p
|ρ| (Re ρ)
p
+ (|c0 | + |ρ| |d0 |)p (kRk + |ρ| kSk)p e− Re ρ − e−(1+s) Re ρ ,
this yields
Z 1
|ϕ1 (x, s; λ)|p dx ≤
s
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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2p+1 ep Re ρ
p
p
p
p+1 [(|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk)
|ρ| (Re ρ)
p
× 1 − e−(1+s) Re ρ
1 − ep(s−1) Re ρ
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+ (|c0 | + |ρ| |d0 |)p (kRk + |ρ| kSk)p 1 − e−s Re ρ
p
1 − ep(s−1) Re ρ
as Re ρ > 0 we obtain
Z
(3.5)
p
p1
|ϕ1 (x, s; λ)| dx
sup
0≤s≤1
1
s
2
≤
21+ p eRe ρ (kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)]
1
|ρ| (Re ρ)1+ p p1/p
.
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
From (2.10) we have
|ϕ2 (x, s; λ)|
Z 1
e(x−s) Re ρ
≤
(kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |)
e(1−t) Re ρ dt
2 |ρ|
0
Z
1 t Re ρ
− Re ρ
+ (kRk + |ρ| kSk) (|a0 | + |ρ| |b0 |) + e
(|c0 | + |ρ| |d0 |)
e
dt
0
Z 1
e(s−x) Re ρ
+
[(|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk)
e(t−1) Re ρ dt
2 |ρ|
0
Z 1
Re ρ
−t Re ρ
+ (|a0 | + |ρ| |b0 |) + e
(|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk)
e
dt
M. Denche and A. Kourta
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s
then
ex Re ρ (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) e(1−s) Re ρ − e− Re ρ
|ϕ2 (x, s; λ)| ≤
2 |ρ| Re ρ
+ (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) e(1−s) Re ρ − 1
Close
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Page 15 of 36
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+
So
Z
e−x Re ρ (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |) eRe ρ − e(s−1) Re ρ
2 |ρ| Re ρ
+ (|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk) 1 − e(s−1) Re ρ .
s
|ϕ2 (x, s; λ)|p dx
0
p
2p+1 ep Re ρ h
p
p
1 − e(s−2) Re ρ
p
p+1 (kRk + |ρ| kSk) (|c0 | + |ρ| |d0 |)
p |ρ| (Re ρ)
× 1 − e−ps Re ρ + (|a0 | + |ρ| |b0 |)p
i
p
−ps Re ρ
(s−1) Re ρ p
× (kRk + |ρ| kSk) 1 − e
1−e
≤
as Re ρ > 0 we obtain
Z
sup
(3.6)
0≤s≤1
s
|ϕ2 (x, s; λ)|p dx
1+ p2 Re ρ
2
Contents
e
(kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)]
1
|ρ| (Re ρ)1+ p p1/p
.
≤
II
I
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kR (λ, Lp )k
2
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From (3.4), (3.5) and (3.6), we obtain
1+ p2
M. Denche and A. Kourta
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p1
0
≤
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
Quit
eRe ρ (kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)] 1+ p1
|4 (λ)| |ρ| (Re ρ)
1+ p2
2
+1 ,
Page 16 of 36
p1/p
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for ρ ∈ Σ δ = ρ ∈ C : |arg ρ| ≤ 2δ , ρ 6= 0 , we have (Re ρ)−1 <
2
1
,
|ρ| cos( 2δ )
then
kR (λ, Lp )k
1+ p2
1+ p2
2
2
+ 1 eRe ρ (kRk + |ρ| kSk) [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)]
.
≤
1+ p1
1
|4 (λ)| |ρ| |ρ|1+ p cos 2δ
p1/p
Finally, we obtain for λ = ρ2 ∈ Σδ
(3.7)
kR (λ, Lp )kLp
≤
c eRe ρ
1+ p1
|4 (λ)| |ρ|
kRk
+ kSk [(|a0 | + |c0 |) + |ρ| (|b0 | + |d0 |)] ,
|ρ|
M. Denche and A. Kourta
Title Page
where
1+ p2
2
c=
p1/p
3.2.
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
2
+1
1+ p1 .
cos 2δ
1+ p2
Estimation of the Characteristic Determinant
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 functions R and S, and so we assume that
the functions R and S are in C 2 ([0, 1] , C).
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Integrating twice by parts in (2.5), we obtain
h
0
0
(3.8) ∆(λ) = eρ ρ(d0 S(0) − b0 S(1)) + (d0 S (0) + b0 S (1) + a0 S (1)
1
0
−b0 R (1) − d0 R (0) + c0 S (0)) + (a0 R (1) − c0 R (0) + b0 R (1)
ρ
1
0
0
0
0
0
−a0 S (1) − d0 R (0) + c0 S (0)) + 2 (−a0 R (1) − c0 R (0)) + Φ(ρ) ,
ρ
where
Z 1 R00 (t) S 00 (t) ρt
−ρ
−2ρ
(3.9) Φ(ρ) = (a0 − ρb0 )e + (c0 − ρd0 )e
−
e dt
ρ2
ρ
0
Z 1 R00 (t) S 00 (t) −ρt
−ρ
+ (a0 + ρb0 )e + (c0 + ρd0 )
−
e dt
ρ2
ρ
0
0
−ρ 1
+e
−b0 R0 (0) + d0 R0 (1) − c0 S 0 (1) + a0 S (0) + c0 R(1) − a0 R(0)
ρ
+ ρ(b0 S(0) − d0 S(1))] + 2e−2ρ [(a0 + ρb0 )(R(1) − ρS(1))
1
0
0
− (c0 − ρd0 )(R(0) + ρS(0)) + 2 (a0 + ρb0 )(R (1) − ρS (1))
ρ
1
0
0
+ 2 (c0 − ρd0 )(R (0) + ρS (0)) .
ρ
After a straightforward calculation, we obtain the following inequality valid for
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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ρ ∈ Σ δ ,with |ρ| sufficiently large,
2
|Φ(ρ)|
≤ (|a0 | + |ρ| |b0 |) e− Re ρ + (|c0 | + |ρ| |d0 |) e−2 Re ρ
00
Re ρ
kR k kS 00 k
e
−1
×
+
|ρ|
Re ρ
|ρ|2
− Re ρ
+ (|a0 | + |ρ| |b0 |) e
+ (|c0 | + |ρ| |d0 |)
00
kR k kS 00 k
1 − e− Re ρ
+
×
|ρ|
Re ρ
|ρ|2
1 0
0
0
0
−b
R
(0)
+
d
R
(1)
−
c
S
(1)
+
a
S
(0)
+
c
R(1)
−
a
R(0)
+ 2e− Re ρ
0
0
0
0
0
0
|ρ|
1
+ |ρ| |b0 S(0) − d0 S(1)|] + e−2 Re ρ
(|a0 | + |ρ| |b0 |) (kRk + |ρ| kSk)
|ρ|
0
!
R + |ρ| S 0 1
+
(|c0 | + |ρ| |d0 |) (kRk + |ρ| kSk) + (|a0 | + |ρ| |b0 |)
|ρ|
|ρ|2
0
!#
R + |ρ| S 0 + (|c0 | + |ρ| |d0 |)
.
|ρ|2
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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Then
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|Φ(ρ)| ≤
2
(|a0 | + |c0 |)
+ (|b0 | + |d0 |)
δ
|ρ|
|ρ| cos 2
00
kR k
+ kS 00 k
|ρ|
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1
1 0
0
0
+
2 −b0 R (0) + d0 R (1) − c0 S (1)
δ
|ρ| cos 2 |ρ|
i
0
+ +a0 S (0) + c0 R(1) − a0 R(0) |b0 S(0) − d0 S(1)|
1
|a0 |
kRk
+
+ |b0 |
+ kSk
|ρ|
|ρ|
|ρ| (cos 2δ )2
kRk
|c0 |
+
+ |d0 |
+ kSk
|ρ|
|ρ|
0!
0
S R
|a0 |
+
+ |b0 |
2 +
|ρ|
|ρ|
|ρ|
0 !#
0
R S |c0 |
+
+ |d0 |
+
.
|ρ|
|ρ|
|ρ|2
(3.10)
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
Title Page
Where we have used that Re(ρ) ≥
e− Re ρ ≤
|ρ| cos( 2δ ),
− Re ρ
(1 − e
) < 1,
2
1
ande−2 Re ρ ≤
, e− Re ρ < 1.
2
δ 2
|ρ| (cos 2 )
2 |ρ| (cos 2δ )2
2
There are several cases to analyze depending on the functions R and S.
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• Case 1.
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Suppose that S 6= 0, d0 6= 0, b0 6= 0, d0 S (0) − b0 S (1) = 0 and
0
Page 20 of 36
0
k1 = d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) 6= 0.
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From (3.8), we have for |ρ| sufficiently large
|4 (λ)|
h
0
0
≥e
d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0)
1 0
0
0
0
−
a0 R (1) − c0 R (0) + b0 R (1) − a0 S (1) − d0 R (0) + c0 S (0)
|ρ|
1 0
0
− 2 a0 R (1) + c0 R (0) − |Φ(ρ)| .
|ρ|
Re ρ
From (3.10) we deduce for ρ ∈ Σ δ , |ρ| ≥ r0 > 0.
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
2
M. Denche and A. Kourta
c(r0 )
|Φ(ρ)| ≤
.
|ρ|
Then, we have
|4 (λ)|
h
0
0
Re ρ ≥e
d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0)
1 0
0
0
0
−
a
R
(1)
−
c
R
(0)
+
b
R
(1)
−
a
S
(1)
−
d
R
(0)
+
c
S
(0)
0
0
0
0
0
0
|ρ|
c(r ) 1 0
0
0
.
− 2 a0 R (1) + c0 R (0) −
|ρ|
|ρ|
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we can now choose r0 > 0, such that
1 0
0
0
0
a0 R (1) − c0 R (0) + b0 R (1) − a0 S (1) − d0 R (0) + c0 S (0)
r0
c(r )
1 0
0
0
+ 2 a0 R (1) + c0 R (0) +
r0
|ρ|
1
0
0
≤ d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) ,
2
then, for ρ ∈ Σ δ , |ρ| ≥ r0 > 0, we get
2
|4 (λ)| ≥
eRe ρ
|k1 | .
2
From (3.7), we deduce the following bound, valid for every |arg ρ| ≤ 2δ , ρ 6= 0
2c
|a0 | + |c0 |
kRk
kR (λ, Lp )k ≤
+ (|b0 | + |d0 |)
+ kSk ,
1
|ρ|
|ρ|
|ρ| p |k1 |
then
kR (λ, Lp )k ≤
c1
1
,
|λ| 2p
as |λ| −→ +∞, where
M. Denche and A. Kourta
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c1 =
• Case 2.
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
2c kSk (|b0 | + |d0 |)
.
|k1 |
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If b0 = d0 = 0, S 6= 0 and a0 S (1) + c0 S (0) = 0, with
0
0
k2 = a0 R (1) − c0 R (0) − a0 S (1) + c0 S (0) 6= 0,
we have the following bound, valid for λ ∈ Σδ and |λ| big enough,
2c (|a0 | + |c0 |) kRk
kR (λ, Lp )k ≤
+ kSk ,
1
|ρ|
|ρ| p |k2 |
then
kR (λ, Lp )k ≤
c1
1
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
,
|λ| 2p
as |λ| −→ +∞, where
M. Denche and A. Kourta
c1 =
2c (|a0 | + |c0 |) kSk
.
|k2 |
Title Page
Contents
• Case 3.
If S = 0, b0 6= 0, d0 6= 0 and b0 R(1) + d0 R (0) = 0, with
0
0
k3 = a0 R (1) − c0 R (0) + b0 R (1) − d0 R (0) 6= 0.
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Similarly, we get
Quit
kR (λ, Lp )k ≤
2c kRk
1+ p1
|ρ|
|k3 |
(|a0 | + |c0 |)
+ (|b0 | + |d0 |) ,
|ρ|
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then, we have
kR (λ, Lp )k ≤
c1
1
,
|λ| 2p
as |λ| −→ +∞, where
c1 =
2c kRk (|b0 | + |d0 |)
.
|k3 |
• Case 4.
If b0 = d0 = 0, S = 0 and a0 R (1) − c0 R (0) = 0 with
0
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
0
k4 = a0 R (1) − c0 R (0) 6= 0.
M. Denche and A. Kourta
Again in this case, we have
Title Page
kR (λ, Lp )k ≤
2c (|a0 | + |c0 |) kRk
1
p
|ρ| |k4 |
then
kR (λ, Lp )k ≤
c1
1
,
|λ| 2p
,
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as |λ| −→ +∞, where
c kRk (|a0 | + |c0 |)
c1 =
.
|k4 |
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Definition 3.1. The boundary value conditions in (1.1) are called non regular
if the functions R, S ∈ C 2 ([0, 1] , C) and if and only if one of the following
conditions holds
1-d0 S (0) − b0 (1) = 0, b0 6= 0, d0 6= 0, S 6= 0 and
0
0
d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) 6= 0
2- b0 = d0 = 0, kSk =
6 0, a0 S (1) + c0 S (0) = 0 with
0
0
a0 R (1) − c0 R (0) − a0 S (1) + c0 S (0) 6= 0
3-S = 0, b0 6= 0, d0 6= 0, b0 R(1) +0 R (0) 6= 0 with
0
0
a0 R (1) − c0 R (0) + b0 R (1) − d0 R (0) 6= 0
4-b0 = d0 = 0, S = 0, a0 R (1) − c0 R (0) = 0 with
0
0
a0 R (1) − c0 R (0) 6= 0.
This proves the following theorem
Theorem 3.1. If the boundary value conditions in (1.1) are non regular, then
Σδ ⊂ ρ(Lp ) for sufficiently large |λ| and there exists c > 0 such that
kR (λ, Lp )k ≤
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c
|λ|
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
1
2p
.
Remark 3.1. From Theorem 3.1 it follows that the operator Lp , for p 6= ∞, generates an analytic semigroup with singularities [25] of type A (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

∂ 2 u(t, x)
∂u(t, x)


−
a
= f (t, x)


∂t
∂x2



 L (u) = a u (0, t) + b u0 (0, t) + c u (1, t) + d u0 (1, t) = 0,
1
0
0
0
0
(3.11)
R
R

0
1
1


L2 (u) = 0 R(ξ) u(t, ξ) dξ + 0 S(ξ) u (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, let E, E1 , and E2 be Banach spaces.
Introduce two Banach spaces
Cµ ((0, T ] , E)
(
=
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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)
f /f ∈ C ((0, T ] , E) , kf k = sup ktµ f (t)k < ∞ , µ ≥ 0,
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(
Cµγ ((0, T ] , E) =
f /f ∈ C ((0, T ] , E) , kf k = sup ktµ f (t)k
t∈(0,T ]
+
sup
−γ µ
kf (t + h) − f (t)k h t < ∞ ,
0<t<t+h≤T
µ ≥ 0, γ ∈ (0, 1] ,
and a linear space
C 1 ((0, T ] , E1 , E2 ) = f /f ∈ C ((0, T ] , E1 ) ∩ C 1 ((0, T ] , E2 ) , E1 ⊂ E2 ,
where C ((0, T ] , E) and C 1 ((0, T ] , E) are spaces of continuous and continuously differentiable, respectively, vector-function from (0, T ] into E. We denote, for a linear operator A in a Banach space E, by
n
1 o
E(A) = u/u ∈ D(A), kukE(A) = kAuk2 + kuk2 2 ,
and
n
o
0
C 1 ((0, T ] , E(A), E) = f /f ∈ C ((0, T ] , E(A)) , f ∈ C ((0, T ] , E)) .
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
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Let us derive a theorem which was proved by various methods in [25] and [31,
33]. Consider, in a Banach space E, the Cauchy problem
( 0
u (t) = Au(t) + f (t), t ∈ [0, T ] ,
(3.12)
u(0) = u0 ,
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where A is, generally speaking, unbounded linear operator in E, u0 is a given
element of E, f (t) is a given vector-function and u(t) is an unknown vectorfunction in E.
Theorem 3.2. Let the following conditions be satisfied:
1. A is a closed linear operator in a Banach space E and for some β ∈ (0, 1] ,
α>0
π
kR (λ, A)k ≤ C |λ|−β ,
|arg λ| ≤ + α, |λ| → ∞;
2
2. f ∈ Cµγ ((0, T ] , E) for some γ ∈ (1 − β, 1] , µ ∈ [0, β) ;
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
3. u0 ∈ D(A).
Then the Cauchy problem (3.12) has a unique solution
Title Page
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u ∈ C ([0, T ] , E) ∩ C 1 ((0, T ] , E(A), E)
and for the solution u the following estimates hold
ku(t)k ≤ C kAu0 k + ku0 k + kf kCµ ((0,t],E) , t ∈ (0, T ] ,
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0 u (t) + kAu(t)k
≤ C tβ−1 (kAu0 k + ku0 k) + tβ−µ−1 kf kCµγ ((0,t],E) , t ∈ (0, T ] .
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As a result of this we get the following theorem
Theorem 3.3. Let the following conditions be satisfied
1. a 6= 0, |arg a| <
π
,
2
2. the functions R(t), S(t) ∈ C 2 ([0, 1] , C) and one of the following conditions is satisfied
d0 S (0) − b0 S (1) = 0, b0 6= 0, d0 6= 0, and S 6= 0
0
0
d0 S (0) + b0 S (1) + a0 S (1) − b0 R (1) − d0 R (0) + c0 S (0) 6= 0,
or b0 = 0, d0 = 0, S 6= 0 , a0 S (1) + c0 S (0) = 0 with
0
Boundary Value Problem for
Second-Order Differential
Operators with Mixed Nonlocal
Boundary Conditions
M. Denche and A. Kourta
0
a0 R (1) − c0 R (0) − a0 S (1) + c0 S (0) 6= 0,
Title Page
or b0 6= 0, d0 6= 0, S = 0, b0 R(1) + d0 R (0) = 0 with
0
0
a0 R (1) − c0 R (0) + b0 R (1) − d0 R (0) 6= 0,
or b0 = d0 = 0, S = 0, a0 R (1) − c0 R (0) = 0 with
0
0
a0 R (1) − c0 R (0) 6= 0.
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h
1
1
3. f ∈ Cµγ ((0, T ] , Lq (0, 1)) for some γ ∈ 1 − 2q
, 1 and some µ ∈ 0, 2q
,
4. u0 ∈ Wq2 (0, 1) , Li u = 0, i = 1, 2 .
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Then the problem (3.11) has a unique solution
u ∈ C ((0, T ] , Lq (0, 1)) ∩ C 1 (0, T ] , Wq2 (0, 1), Lq (0, 1)
and for this solution we have the estimates:
(3.13) ku(t, ·)kLq (0,1) ≤ c ku0 kWq2 (0,1) + kf kCµ ((0,t],Lq (0,1)) , t ∈ (0, T ] ,
(3.14)
ku00 (t, ·)kLq (0,1) + ku0 (t, ·)kLq (0,1)
1
1
≤ c t 2q −1 ku0 kWq2 (0,1) + t 2q −1−µ kf kCµγ ((0,t],Lq (0,1)) , t ∈ (0, T ] .
Proof. We consider in the space Lq (0, 1) 1 ≤ q < ∞, the operator A defined by
A(u) = au00 (x), D(A) = u ∈ Wq2 (0, 1), Li (u) = 0, i = 1, 2 .
Then problem (3.11) can be written as
( 0
u (t) = Au(t) + f (t),
u(0) = u0 ,
where u(t) = u(t, ·), f (t) = f (t, ·), and u0 = u0 (·) are functions with values in
the Banach space Lq (0, 1). From Theorem 3.1 we conclude that kR(λ, A)k ≤
1
π
c |λ|− 2q , for |arg λ| ≤ + α, as |λ| → ∞.
2
Then, from Theorem 3.2 the problem (3.11) has a unique solution
u ∈ C ((0, T ] , Lq (0, 1)) ∩ C 1 (0, T ] , Wq2 (0, 1), Lq (0, 1)
Boundary Value Problem for
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and we have the following estimates
(3.15) ku(t, .)kLq (0,1) ≤ c kAu0 kLq (0,1) + ku0 kLq (0,1) + kf kCµ ((0,t],Lq (0,1)) ,
0 (3.16) u (t)
+ kAu(t, ·)kLq (0,1)
Lq (0,1)
1 1
≤ c t 2q −1 kAu0 kLq (0,1) + ku0 kLq (0,1) + t 2q −1−µ kf kCµγ ((0,t],Lq (0,1)) ,
where t ∈ [0, T ], from (3.15) we get
ku(t, ·)kLq (0,1) ≤ c ku000 kLq (0,1) + ku0 kLq (0,1) + kf kCµ ((0,t],Lq (0,1))
≤ c ku0 kWq2 (0,1) + kf kCµ ((0,t],Lq (0,1)) , t ∈ [0, T ] .
and from (3.16) we get
0 + ku00 (t, ·)kLq (0,1)
u (t) q
L (0,1)
1 1
−1
−1−µ
2q
2q
≤c t
kAu0 kLq (0,1) + ku0 kLq (0,1) + t
kf kCµγ ((0,t],Lq (0,1))
1
1
≤ c t 2q −1 ku0 kWq2 (0,1) + t 2q −1−µ kf kCµγ ((0,t],Lq (0,1)) , t ∈ [0, T ] .
Boundary Value Problem for
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which gives the desired result.
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