NONLOCAL BOUNDARY VALUE-PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

ADJM ISSN 1539-854X

AFRICAN DIASPORA JOURNALOF MATHEMATICS

(A publication of Nova Science Publishers)www.african-j-math.orgVolume 5 (2006), no. 1, pp. 1-11

NONLOCAL BOUNDARY VALUE-PROBLEM FORPARTIAL DIFFERENTIAL EQUATIONS

WITH VARIABLE COEFFICIENTS

E. KENGNE

Abstract. This work is about nonlocal boundary-value problem forpartial di�erential equations with variable with respect to t and x coef-�cients in a rectangular domain and discusses existence and uniquenessof solutions of the problem under consideration. To investigate the wellposedness of the problem, we prove metric statements related to lowerbound of small denominator appearing in the course of solution of theproblem.

1. IntroductionThe problems with nonlocal boundary conditions in the time variable for partial

di�erential equations are, in general, Hadamard ill-posed [5], [8], [22] in the scalesof Sobolev spaces of functions periodic in spatial variables. Uniqueness of a solu-tion to such problems in some cases depends on the Diophantine properties of theircoe�cients and parameters [5], [22], while solvability and smoothness of the solu-tions are connected with the problem of small denominators [1], [9]. In recent years,problems with nonlocal boundary conditions for partial di�erential equations anddi�erential operator equations have been studied by many authors. In particularregular problems without small denominators or with conditions under which smalldenominators are not equal to zero, which guarantees the solvability of the prob-lem, were mainly investigated [5], [6], [10]�[12], [15]�[17], [19], [24]. However, in thegeneral case, nonlocal boundary-value problems for general di�erential operatorswith partial derivatives are ill posed, and the problem of their solvability is oftenconnected with the problem of small denominators. For some classes of equations

AMS Subject Classi�cation: 35G15, 34B10.Keywords: Nonlocal boundary value-problem; Uniqueness; Well-posed problems; Problems of

small denominator; Nonlocal boundary conditions; ill-posed problems.Copyright c© 2006 Nova Science Publishers.

1

2 E. KENGNE

and systems of equations, nonregular cases of problems with nonlocal two-pointconditions with respect to a selected variable t were investigated in works of V.M.Borok and her students [3], [4], [7], [13], [14], [18].

In the present paper, we extend the investigations carried out in [14] to partialdi�erential equations with variable with respect to t and x coe�cients. We study aproblem with nonlocal conditions with respect to the variablet and Dirichlet-typeconditions with respect to x for factorized operators with partial derivatives.

2. Formulation of problem and notationsIn the domain

∏= (0, l) × (0, T ) of the variables (x, t) we consider the partial

di�erential equation

(2.1) ∂

∂t

[∂

∂t− P

(t,

d

dx

)]u = f(x, t),

where P(t, d

dx

)= a(t)

[q(x)− d

dx

(p(x) d

dx

)]+ b(t). Here a(t) and b(t) are complex-

valued functions, a, b ∈ C1 ([0, T ]) , p, q ∈ C1 ([0, l]) , p(x) ≥ p0 = const > 0 andq(x) ≥ 0, for every x ∈ [0, l]. We need to �nd a solution to equation (2.1) satisfyingthe conditions

(2.2)(

1 α2 −µ α4

β1 1 −µβ1 −µ

)

u(x, 0)ut(x, 0)u(x, T )ut(x, T )

=

(ϕ1(x)ϕ2(x)

), x ∈ [0, l],

and(2.3) u(0, t) = u(l, t) = 0,

where µ, α2, α4 and β1 are four complex numbers such thatµ 6= 0, 1− µ− β1α2 −β1α4 6= 0 and rank

(1 α2 −µ α4

β1 1 −µβ1 −µ

)= 2, and ϕ1 and ϕ2 are two given

complex-valued functions.The eigenvalue problem corresponding to problem (2.1)�(2.3) reads

(2.4) d

dx

(p(x)

d

dx

)X + (λ− q(x))X = 0, X(0) = X(l) = 0.

Let Λ = {λn, n ∈ N} and X = {Xn(x), n ∈ N} be the system of eigenvalues andthe system of eigenfunctions of problem (2.4), respectively. According to [21] thesystem of eigenfunctions of problem (2.4) is complete and orthonormal inL2 (0, l)(in what follows, we assume that it is normalized),λn > 0, and λi 6= λj when i 6= j.Furthermore, Xn ∈ C2 ([0, l]) , n ∈ N and the following estimates are true [21], [20]:(2.5) c1n

2 ≤ λn ≤ c2n2, 0 < c1 ≤ c2, n ∈ N,

(2.6) maxx∈[0,l]

∣∣∣X(j)n (x)

∣∣∣ ≤ c3jλj/2n , j = 0, 1, 2, n ∈ N.

It follows from (2.5) that limn→+∞

λn = +∞.

In what follows we denote by C(1,2)(∏)

the Banach space of functions w(x, t)with the norm

‖w‖C(1,2)(Q) =

∑

2k+m≤2

max(x,t)∈Q

∣∣∣∣∂k+mw(x, t)

∂tk∂xm

∣∣∣∣ ,

A NONLOCAL BOUNDARY VALUE-PROBLEM 3

Bk, k > 0, is the space of functions ϕ ∈ L2 (0, l),

ϕ(x) =+∞∑n=1

ϕnXn(x)

with the norm

‖ϕ‖Bk=

+∞∑n=1

|ϕn| exp (kλn) ,

and C (Bk, [0, T ]) is the space of functions

ψ(x, t) =+∞∑n=1

ψn(t)Xn(x)

de�ned in∏

, continuous in t, and belonging to the space Bk for every �xed t ∈ [0,T ] with the norm

‖ψ‖C(Bk, [0, T ]) =+∞∑n=1

maxt∈[0, T ]

|ψn(t)| exp (kλn) .

For every complex function or constantα, we write α = αr + iαi, which means thatαr = Re(α) and αi = Im(α).

3. On the uniqueness of solutions of problem (2.1)�(2.3)We seek a solution of problem (2.1)�(2.3) in the form

(3.1) u(x, t) =+∞∑n=1

Yn(t)Xn(x).

If series (3.1) and the series obtained by the termwise di�erentiation of series (3.1)with respect to x up to the order two inclusive converge uniformly in the domain∏

, then function u(x, t) de�ned by (3.1) satis�es the boundary conditions (2.3). Inorder to determine Yn(t), we assume that the boundary functionsϕ1(x) and ϕ2(x)and the free term f(x, t) can be expanded into Fourier series:

(3.2) ϕs(x) =+∞∑n=1

ϕs,nXn(x), s = 1, 2, f(x, t) =+∞∑n=1

fn(t)Xn(x),

where

(3.3) ϕs,n =∫ l

0

ϕs(x)Xn(x)dx, fn(t) =∫ l

0

f(x, t)Xn(x)dx, n ∈ N, s = 1, 2.

If we insert (3.1) and (3.2) into (2.1) and (2.2) we obtain for determination ofYn(t)the following Cauchy problem for an ordinary di�erential equation:

(3.4) d

dt

(d

dt− λna− b

)Yn = fn(t),

(3.5)(

1 α2 −µ α4

β1 1 −µβ1 −µ

)

Yn(0)Y ′

n(0)Yn(T )Y ′

n(T )

=

(ϕ1,n

ϕ2,n

).

4 E. KENGNE

A fundamental system of solutions of the homogeneous equation corresponding toequation (3.4) reads

{Yn1(t) = exp

(∫ t

0

(λna(τ) + b(τ)) dτ

),

Yn2(t) =∫ t

0

exp(∫ t

0

(λna(τ) + b(τ)) dτ −∫ y

0

(λna(τ) + b(τ)) dτ

)dy

}.

In what follows we restrict ourselves to the case where the coe�cients of equation(2.1) satisfy the conditions(3.6) a(0) = a(T ) = 0; b(0) = b(T ) = 0.

Under conditions (3.6) the characteristic determinant [21] of the homogeneous prob-lem(3.7)

d

dt

(d

dt− λna− b

)Yn = 0,

(1 α2 −µ α4

β1 1 −µβ1 −µ

)

Yn(0)Y ′

n(0)Yn(T )Y ′

n(T )

=

(00

)

reads(3.8) ∆(λn) = (1− µ− β1α2 − β1α4) (1− µ exp (λnA + B)) ,

whereA =

∫ T

0

a(τ)dτ, B =∫ T

0

b(τ)dτ.

since 1− µ− β1α2 − β1α4 6= 0, ∆(λn) = 0 if and only, if 1− µ exp (λnA + B) = 0.

1− µ exp (λnA + B) = 0 ⇔ µ−1 = exp (λnA + B) = 0⇔ − ln |µ| − i arg µ = λnA + B + 2πim, m ∈ Z⇔ ln |µ|+ λnAr + Br = 0

and 2π arg µ + λnAi + Bi = 2πm, m ∈ .ZThus in order that ∆(λn) 6= 0 under condition 1 − µ − β1α2 − β1α4 6= 0, it isnecessary and su�cient that at least one of the following conditions holds

1) ln |µ|+ λnAr + Br 6= 0

2) arg µ + λnAi + Bi 6= 2πm, m ∈ Z.

Theorem 3.1. For the uniqueness of solutions of problem (2.1)�(2.3), (3.6) inspace C(1,2)

(∏), it is necessary and su�cient that

(3.9) 1− µ exp (λnA + B) 6= 0, for every λn ∈ Λ.

Proof.Necessity.Assume that for certainλn0 ∈ Λ, we have 1−µ exp (λn0A + B) =0. Then ∆(λn0) = 0 and there exist nontrivial solutions

Yn0(t) = C1 exp(∫ t

0

(λn0a(τ) + b(τ)) dτ

), C1 6= 0

of homogeneous problem (3.7). In this case the homogeneous problem associatedto problem (2.1)�(2.3) (f(x, t) ≡ 0, ϕ1(x) ≡ 0, and ϕ2(x) ≡ 0) admits nontrivialsolutions of the form

u(x, t) = Yn0(t)Xn0(x)


and a solution of the inhomogeneous problem (2.1)�(2.3), if it exists, is not unique.Su�ciency. Let condition (3.9) be satis�ed. Let u1(x, t) and u2(x, t) be two

solutions of problem (2.1)�(2.3) in space C(1,2)(∏)

. Then function u(x, t) =

u1(x, t) − u2(x, t) belongs to space C(1,2)(∏)

and is a solution of homogeneousproblem associated to problem (2.1)�(2.3) and can be expanded in Fourier series

(3.10) u(x, t) =+∞∑n=1

un(t)Xn(x).

We can also expand∂

∂t

[∂

∂t− P

(t,

d

dx

)]u

and(

1 α2 −µ α4

β1 1 −µβ1 −µ

)

u(x, 0)ut(x, 0)u(x, T )ut(x, T )

into Fourier series in the system of functionsXn(x), and these series coincide with

∂

∂t

[∂

∂t− P

(t,

d

dx

)] +∞∑n=1

un(t)Xn(x)

and(

1 α2 −µ α4

β1 1 −µβ1 −µ

)

∑+∞n=1 un(0)Xn(x)∑+∞n=1 u′n(0)Xn(x)∑+∞n=1 un(T )Xn(x)∑+∞n=1 u′n(T )Xn(x)

,

respectively. Using Parseval equality for the functions ∂∂t

[∂∂t − P

(t, d

dx

)]u and

(1 α2 −µ α4

β1 1 −µβ1 −µ

)

u(x, 0)ut(x, 0)u(x, T )ut(x, T )

, we obtain that for every n ∈ N, un(t) is

a solution of problem (3.7) under conditions (3.6). This solution can be expressedin the form

un(t) = Cn1Yn1(t) + Cn2Yn2(t),

where (Cn1, Cn2) is a solution of a linear homogeneous algebraic system with de-terminant ∆(λn) 6= 0. Therefore Cn1 = Cn2 = 0, and un(t) ≡ 0, for every n ∈ N.From the Parseval equality for the function u(x, t) and continuity of this functionin

∏, we conclude that u(x, t) ≡ 0, and this means that u1(x, t) ≡ u2(x, t). We

have then proved that the solution of problem (2.1)�(2.3) in spaceC(1,2)(∏)

, ifit exists, is unique. Theorem 3.1 is proven.

4. On the existence of solutions of problem (2.1)�(2.3)In this section, we assume that the solution of problem (2.1)�(2.3), (3.6) in

space C(1,2)(∏)

, if it exists, is unique. According to the above theorem, condition

6 E. KENGNE

(3.9) is satis�ed. Then for every n ∈ N, the solution of problem (3.4)�(3.6) reads(4.1) Yn(t) = Cn1Yn1(t) + Cn2Yn2(t) + Y ∗

n (t),

where

(4.2)Y ∗

n (t) = exp(∫ t

0(λna(τ) + b(τ)) dτ

) ∫ t

0fn(z)[∫ z

0exp

(− ∫ y

0(λna(τ) + b(τ)) dτ − 2

∫ z


)dy

− ∫ t

0exp

(− ∫ y

0(λna(τ) + b(τ)) dτ − 2

∫ z


)dy

]dz,

Yn1(t) and Yn2(t) are the above de�ned functions, and coe�cients Cn1 and Cn2

satisfy the system

(1 α2 −µ α4

β1 1 −µβ1 −µ

)

Cn1Yn1(0) + Cn2Yn2(0) + Y ∗n (0)

Cn1Y′n1(0) + Cn2Y

′n2(0) + Y ∗′

n (0)Cn1Yn1(T ) + Cn2Yn2(T ) + Y ∗

n (T )Cn1Y

′n1(T ) + Cn2Y

′n2(T ) + Y ∗′

n (T )

=(

ϕ1,n

ϕ2,n

),

that is,

(4.3)

[1− µ exp (λnA + B)] Cn1 − ϕ1,n − µY ∗n (T ) + α4Y

∗′n (T )

+ [α2 + α4 − µ exp (λnA + B))× ∫ T

0exp

(− ∫ y


)dy

]Cn2 = 0,

[β1 + exp (λnA + B)] Cn1 − ϕ2,n − µβ1Y∗n (T )− µY ∗′

n (T )+ [1− µ− µβ1 exp (λnA + B))× ∫ T

0exp

(− ∫ y


)dy

]Cn2 = 0.

The determinant of system (4.3) coincides with∆(λn). Therefore(4.4)Cn1 =

1∆(λn)

[(ϕ1,n + µ exp (λnA + B)

∫ T

0fn(z)δT (z)dz + α4T exp (− (λnA + B))

× ∫ T

0fn(z) exp (2C(z)) dz

)×

(1− µ− µβ1 exp (λnA + B))

∫ T

0exp (C(y)) dy

)

−(α2 + α4 − µ exp (λnA + B))

∫ T

0exp (C(y)) dy

)

×(ϕ2,n + µβ1 exp (λnA + B)

∫ T

0fn(z)δT (z)dz − µT exp (− (λnA + B))

× ∫ T


)]

Cn2 =1

∆(λn)

[(ϕ2,n + µβ1 exp (λnA + B)

∫ T

0fn(z)δT (z)dz − µT exp (− (λnA + B))

× ∫ T


)× (1− µ exp (λnA + B))− (β1 + exp (λnA + B))(

ϕ1,n + µ exp (λnA + B)∫ T

0fn(z)δT (z)dz) + α4T exp (− (λnA + B))

× ∫ T


)],

where

(4.5) C(ξ) = − ∫ ξ

0(λna(τ) + b(τ)) dτ,

δT (z) =∫ z

0exp (C(y) + 2C(z)) dy − ∫ T

0exp (C(y) + 2C(z)) dy.


Inserting (4.1) into (3.1), a formal solution of problem (2.1)�(2.3), (3.6) reads

(4.6) u(x, t) =+∞∑n=1

(Cn1Yn1(t) + Cn2Yn2(t) + Y ∗n (t))Xn(x).

In general, series (4.6) is divergent because the moduli of the nonzero expression(see (3.8) and (4.4)) ∆(λn) = 1−µ exp (λnA + B) can be arbitrarily small for in�n-itely many numbers λn ∈ Λ. Therefore, the question of the existence of a solutionof problem (2.1)�(2.3), (3.6) is connected with the problem of small denominator.Lemma 4.1. Let F (λn) be a bounded sequence of real numbers. Then, for almostall (with respect to the Lebesgue measure in R ) numbers a ∈ R, the followinginequality holds for all (except �nitely many) pairs (λn, m) λn ∈ Λ, m ∈ Z:

(4.7)∣∣∣∣F (λk)− am

λn

∣∣∣∣ ≥ λ−3/2−εn , 0 < ε < 1.

Proof. To establish inequality (4.7) it is su�cient to prove that for almost all(with respect to the Lebesque measure inR) numbers a ∈ R, the inequality

(4.8)∣∣∣∣F (λk)− am

λn

∣∣∣∣ < λ−3/2−εn , 0 < ε < 1

admits no more that a �nite number of solutions(λn, m) ,with m ∈ Z and λn ∈ Λ.To prove the inequality (4.8) we use the procedure of the proof of lemma 2.4 in [24,Chap. 1]. Let us denote by K the set of real numbers a ∈ [d, 2d] (d ∈ R) for whichthe inequality (4.8) has in�nitely many solutions (λn, m), λn ∈ Λ and m ∈ Z. Letus �x λn = λn∗ ∈ Λ, which corresponds to n∗ ∈ N. The corresponding values of mfor which inequality (4.8) holds belong to the interval

(4.9) λn∗

a

(F (λk)− λ

−3/2−εn∗

)< m <

λn∗

a

(F (λk) + λ

−3/2−εn∗

).

Let m = m0 ∈ Z satisfying inequality (4.9). It is obvious that for some positiveconstant Ca that depends on a and does not depend on n∗ we have

∣∣m0∣∣ < Caλn∗ .

Then the measure of the setK(n∗,m0) of real numbers a for which inequality (4.8)holds for �xed n = n∗ and m = m0 satis�es the inequality

(4.10) mes K(λn∗ ,m0) <

2

|m0|λ1/2+εn∗

.

It follows from (4.9) and (4.10) that for the measure of the set K(λn∗) of realnumbers a for which the inequality (4.8) for a �xed λn = λn∗ admits solutions inintegers m 6= 0, the following estimate is satis�ed

mes K(λn∗) <

E(Ca|λn∗ |)+1∑

m0=−E(Ca|λn∗ |)−1

2

|m0| |λn∗ |1/2+ε≤ C1a ln λn∗

λ1/2+εn∗

.

If we sum this last estimate for all λn we obtain

mes K < C1a

+∞∑n=1

ln λn

λ1/2+εn

.

From the convergence of the series at the right-hand side of the last inequality andlemma 2.1 in [24, Chap. 1], we conclude that the measure of setK = [d, 2d] of realnumbers a for which inequality (4.8) admits in�nitely many solutions in integersm

8 E. KENGNE

is equal to zero. Therefore we establish that for almost alla ∈ [d, 2d], estimate (4.7)is true for all (except �nitely many)λn ∈ Λ. Because the real axis (−∞, +∞) canbe covered by �nitely many segments of the form [d, 2d] (e.g.,

(−∞, +∞) =

(+∞⋃

l=−∞

[−2l, − 2l−1])∪

(+∞⋃

l=−∞

[2l−1, 2l

])

),

Theorem 4.2. If condition (3.9) is satis�ed then there exist α ≥ 0 and C > 0 sothat the following estimate holds for all (except �nitely many)n ∈ N :

(4.11) |1− µ exp (λnA + B)| ≥ Cλ−αn .

Proof. Estimate (4.11) is evident ifA = 0. SupposeA 6= 0 and let µ = exp (ρ + iϕ) ,ρ ∈ R and ϕ = arg µ ∈ [0, 2π). Then 1−µ exp (λnA + B) = 1−exp ((Arλn + Br + ρ)+i

(Aiλn + Bi + ϕ

)). It is proven in [2] that for every polynomialP (iσ) with com-

plex coe�cients, if 1 − exp (P (iσ)) = 0 admits no more than a �nite number ofsolutions, then there exist C > 0 and r ∈ R so that the following estimate is true:

(4.12) |1− exp (P (iσ))| ≥ C (1 + |σ|)r, ∀σ ∈ R so that |σ| ≥ σ0.

In (4.12) σ0 is a positive number so that 1− exp (P (iσ)) 6= 0 for every |σ| ≥ σ0. Inother words, all the zeros of1−exp (P (iσ)) = 0 are in the interval (−σ0, σ0) . In par-ticular for the �rst degree polynomialP (iσ) = (Arσ + Br + ρ)+i

(Aiσ + Bi + ϕ

)=

Aσ + B + ρ + iϕ, (4.8) is satis�ed as soon as |Ar|+ |Br + ρ| > 0 or Ar = Ai = 0.Therefore from (4.12) we obtain for some constantC > 0

|1− µ exp (λnA + B)| =∣∣1− exp

((Arλn + Br + ρ) + i

(Aiλn + Bi + ϕ

))∣∣ ≥ C,

and estimate (4.11) is established.Let now

(4.13) |Ar|+ |Br + ρ| = 0 and Ai 6= 0.

In this case equation 1− exp (P (iσ)) = 0 with P (iσ) = Aσ + B + ρ + iϕ admits anin�nite number of solutions. Now we have

1− µ exp (λnA + B) = 1− exp(i(Aiλn + Bi + ϕ

)),

and 1− µ exp (λnA + B) 6= 0 if and only, if

Aiλn + Bi + ϕ /∈ 2πZ, for every n ∈ N.

∣∣1− exp(i(Aiλn + Bi + ϕ

))∣∣ = 2

√sin2 (Aiλn + Bi + ϕ)

2

= 2∣∣∣∣sin

(ϕ

2+

Aiλn + Bi

2

)∣∣∣∣ .

For every n ∈ N there exists an integer m = m(n) so that

ϕ + Aiλn + Bi − 2mπ ∈ (−π, π)

we complete the proof of lemma 4.1.


and ∣∣∣∣sin(

ϕ

2+

Aiλn + Bi

2

)∣∣∣∣ =∣∣∣∣sin

(ϕ

2+

Aiλn + Bi

2−mπ

)∣∣∣∣

≥ 2∣∣∣∣

ϕ

2π+

Aiλn + Bi

2π−m

∣∣∣∣

=1π

∣∣Bi + Aiλn + ϕ− 2πm∣∣

Thus∣∣1− exp

(i(Aiλn + Bi + ϕ

))∣∣ ≥ 4λn

∣∣Ai∣∣∣∣∣∣Aiλn + Bi + ϕ

2πAiλn− 1

Ai

m

λn

∣∣∣∣ .

BecauseΛ = {λn}+∞n=1 is a sequence satisfying limn→+∞

λn = +∞, F (λn) = Aiλn+Bi+ϕ2πAiλn

is a bounded sequence. According to lemma 4.1 in which we takea = 1/Ai, we have∣∣1− exp

(i(Aiλn + Bi + ϕ

))∣∣ ≥ 4λn

∣∣Ai∣∣∣∣∣∣Aiλn + Bi + ϕ

2πAiλn− 1

Ai

m

λn

∣∣∣∣≥ 4λn

∣∣Ai∣∣ λ−3/2−ε

n

= 4∣∣Ai

∣∣ λ−1/2−εn = Cλ−α

n ,

withα = 1/2 + ε, 0 < ε < 1,

for almost all (with respect to the Lebesgue measure in R) numbers Ai and all(except �nitely many) λn ∈ Λ. Theorem 4.2 is completely proven.Theorem 4.3. Suppose that condition (3.9) is satis�ed and at least one of condi-tions (4.13) is not satis�ed. Then if ϕq ∈ Bδ1 , q = 1, 2, f ∈ C (Bδ2 , [0, T ]) , withδ1 > ε1 and δ2 > ε2, there always exists a unique solution of problem (2.1)�(2.3),(3.6) that belongs to the space C(1,2)

(∏)and continuously depends on f(x, t) and

ϕq, q = 1, 2.

Theorem 4.4. Suppose that condition (3.9) is satis�ed and each of conditions(4.13) is satis�ed. Then if ϕq ∈ Bδ1 , q = 1, 2, f ∈ C (Bδ2 , [0, T ]) , with δ1 > α1

and δ2 > α2, for almost all (with respect to the Lebesgue measure inR) values ofAi, there exists a unique solution of problem (2.1)�(2.3), (3.6) that belongs to thespace C(1,2)

(∏)and continuously depends on f(x, t) and ϕq, q = 1, 2.

Proof. If the conditions of theorem 4.3 and theorem 4.4 are satis�ed then ac-cording to theorem 4.2, there existα ≥ 0 and C > 0 so that the following estimateholds for all (except �nitely many) integern ∈ N:

(4.14) |1− µ exp (λnA + B)| ≥ Cλ−αn .

Under the conditions of theorems 4.3 and 4.4, by using estimates (2.5) and (2.6),the expressions for Yn1(t) and Yn2(t), and relations (4.2), (4.4), (4.6) and (4.14),we have

(4.15)‖u‖

C(1,2)(Q) ≤∑+∞

n=1 |Cn1Yn1(t) + Cn2Yn2(t) + Y ∗n (t)| |Xn(x)|

≤ C ′1∑+∞

n=1 λ3α−1n (|ϕ1n|+ |ϕ2n|) exp (ε1λn)

+C ′2∑+∞

n=1 λ2+α maxt∈[0, T ]

|fn(t)| exp (ε2λn) ,

10 E. KENGNE

where C ′1 and C ′2 are two positive constants. Knowing that for every p ≥ 0 andε > 0, there exists a positive constant C ′3 = C(p, ε) and a real h ≥ 0 so thathp ≤ C ′3 exp (εh) , we can reduce (4.15) to the form

‖u‖C(1,2)(Q) ≤ C ′4

{+∞∑n=1

|ϕ1n| exp ((ε1 + ε1) λn) ++∞∑n=1

|ϕ2n| exp ((ε1 + ε1) λn)

}

+C ′5

+∞∑n=1

maxt∈[0, T ]

|fn(t)| exp ((ε2 + ε2)λn)

= C ′6[‖ϕ1‖Bε1+ε1

+ ‖ϕ2‖Bε1+ε1

]+ C ′7 ‖f‖C(Bε2+ε2 , [0, T ]) < +∞,

where C ′6 and C ′7 are positive constants, εj = δj − εj , j = 1, 2. This last estimatecompletes the proof of theorem 4.3 and theorem 4.4.

AcknowledgmentsThis work was made possible thanks to the CDE/IMU grant.

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Department of Mathematics and Computer Sciences, Faculty of ScienceUniversity of DschangP.O. Box 4509, Douala, Republic of CameroonE-mail address: [email protected]

NONLOCAL BOUNDARY VALUE-PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

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