DEMONSTRATIO MATHEMATICA
Vol. XLVII
No 1
2014
Le Thi Phuong Ngoc, Huynh Thi Hoang Dung, Pham Hong
Danh, Nguyen Thanh Long
LINEAR APPROXIMATION AND ASYMPTOTIC
EXPANSION ASSOCIATED WITH THE SYSTEM OF
NONLINEAR FUNCTIONAL EQUATIONS
Abstract. This paper is devoted to the study of the following perturbed system of
nonlinear functional equations
ˆ
˙

ż Xijk pxq
m ÿ
n „
ÿ
(E) fi pxq “
εaijk Ψ x, fj pRijk pxqq,
fj ptqdt ` bijk fj pSijk pxqq ` gi pxq,
0
k“1 j“1
x P Ω “ r´b, bs, i “ 1, . . . , n, where εis a small parameter, aijk , bijk are the given real
constants, Rijk , Sijk , Xijk : Ω Ñ Ω, gi : Ω Ñ R, Ψ : Ω ˆ R2 Ñ R are the given continuous
functions and fi : Ω Ñ R are unknown functions. First, by using the Banach fixed point
theorem, we find sufficient conditions for the unique existence and stability of a solution
of (E). Next, in the case of Ψ P C 2 pΩ ˆ R2 ; Rq, we investigate the quadratic convergence
of (E). Finally, in the case of Ψ P C N pΩ ˆ R2 ; Rq and ε sufficiently small, we establish an
asymptotic expansion of the solution of (E) up to order N ` 1 in ε. In order to illustrate
the results obtained, some examples are also given.
1. Introduction
In this paper, we consider the following system of nonlinear functional
equations
(1.1)
fi pxq
ˆ
˙

ż Xijk pxq
m
n
ÿ ÿ„
“
εaijk Ψ x, fj pRijk pxqq,
fj ptqdt `bijk fj pSijk pxqq `gi pxq,
k“1 j“1
0
i “ 1, . . . , n, x P Ω “ r´b, bs, where aijk , bijk are the given real constants;
2010 Mathematics Subject Classification: 39B72, 45F10.
Key words and phrases: system of nonlinear functional equations, converges quadratically, perturbed problem, asymptotic expansion.
104
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
Rijk , Sijk , Xijk : Ω Ñ Ω, gi : Ω Ñ R, Ψ : Ω ˆ R2 Ñ R are the given
continuous functions and fi : Ω Ñ R are unknown functions, ε is a small
parameter.
The existence of solutions for functional integral equations of the form
(1.1) have been extensively studied by many authors via various techniques
in functional analysis, topology and fixed point theory, such as using the Banach fixed point theorem, fixed point theorems of Krasnoselskii type, the
Darbo fixed point theorem, the nonlinear alternative of Leray–Schauder
and using the technique of the measure of noncompactness. There are
many interesting results of solvability, asymptotic stability and some properties of solutions; for example, we refer to the [1], [4]–[17] and references
therein.
It is well known that, integral equations and functional integral equations
as above have attracted great interest in the field of nonlinear analysis not
only because of their mathematical context but also because of their applications in various fields of science and technology, in engineering, mechanics,
physics, economics, . . . . For the details of such applications, see for example,
C. Corduneanu [2], K. Deimling [3].
In [17], system (1.1) is studied with m “ n “ 2, Ψ “ 0 and Sijk binomials
of first degree. The solution is approximated by a uniformly convergent
recurrent sequence and it is stable with respect to the functions gi .
In [7], [8], [10], the existence and uniqueness of a solution of the functional
equation
f pxq “ apx, f pSpxqqq,
in the functional space BCra, bs, have been studied.
In [11]–[14], special cases of (1.1) have been studied corresponding the
following form
fi pxq “
m ÿ
n
ÿ
aijk px, fj pSijk pxqqq ` gi pxq,
k“1 j“1
i “ 1, . . . , n, x P I Ă R, where I is a bounded or unbounded interval. By
using the Banach fixed point theorem, the authors have established the existence, uniqueness and stability of the solution of (1.1) with respect to the
functions gi . Furthermore, the quadratic convergence and an asymptotic expansion of solutions are also investigated.
Applying a fixed point theorem of Krasnosel’skii type and giving the
suitable assumptions, Dhage and Ntouyas [4], Purnaras [16] obtained some
results on the existence of solutions to the following nonlinear functional
integral equation
Linear approximation and asymptotic expansion associated. . .
ż σptq
ż µptq
vpt, sqgps, xpηpsqqqds,
kpt, sqf ps, xpθpsqqqds `
xptq “ qptq `
105
0
0
t P r0, 1s,
where 0 ≤ µptq ≤ t; 0 ≤ σptq ≤ t; 0 ≤ θptq ≤ t; 0 ≤ ηptq ≤ t, for all t P r0, 1s.
Purnaras also showed that the technique used in [16] can be applied to
yield existence results for the following equation
ż µptq
xptq “ qptq `
kpt, sqf ps, xpθpsqqqds
αptq
ż λptq
ż σpsq
ˆ
p
kpt, sqF s, xpνpsqq,
`
βptq
˙
k0 s, v, xpηpvqqqdv ds, t P r0, 1s.
0
Recently, using the technique of the measure of noncompactness and the
Darbo fixed point theorem, Z. Liu et al. [9] have proved the existence and
asymptotic stability of solutions for the equation
ˆ
˙
żt
xptq “ f t, xptq,
upt, s, xpapsqq, xpbpsqqq ds , t P R` .
0
Motivated by the above mentioned works, we introduce and investigate
the more general nonlinear functional integral equation of the form (1.1).
This paper consists of five sections. In section 2, by using the Banach
fixed point theorem, we find sufficient conditions for the unique existence
and stability of a solution of (1.1). In section 3, in the case of Ψ P C 2 pΩ ˆ
R2 ; Rq, we investigate the quadratic convergence of (1.1). In the case of
Ψ P C N pΩ ˆ R2 ; Rq and ε sufficiently small, an asymptotic expansion of the
solution of (1.1) up to order N ` 1 in ε is established in section 4. We end
the paper with illustrated examples.
The results obtained here relatively generalize the ones in [1], [4]–[17].
2. The theorems on existence, uniqueness and stability of solutions
With Ω “ r´b, bs, we denote by X “ CpΩ; Rn q the Banach space of
functions f : Ω Ñ Rn continuous on Ω with respect to the norm
n
ÿ
}f }X “ sup
|fi pxq| , f “ pf1 , . . . , fn q P X.
xPΩ i“1
For any non-negative integer r, we put
!
)
pkq
C r pΩ; Rn q “ f P CpΩ; Rn q : fi P CpΩ; Rq, 0 ≤ k ≤ r, 1 ≤ i ≤ n .
It is clear that C r pΩ; Rn q is the Banach space with respect to the norm
n ˇ
ˇ
ÿ
ˇ pkq ˇ
}f }r “ max sup
ˇfi pxqˇ .
0≤k≤r xPΩ
i“1
106
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
We write the system (1.1) in the form of an operational equation in X
as follows
f “ εAf ` Bf ` g,
(2.1)
where
f “ pf1 , . . . , fn q, Af “ ppAf q1 , . . . , pAf qn q, Bf “ ppBf q1 , . . . , pBf qn q,
with
$
ˆ
˙
ż Xijk pxq
m ÿ
n
ÿ
’
’
pAf qi pxq “
aijk Ψ x, fj pRijk pxqq,
fj ptqdt ,
’
’
&
0
j“1
k“1
m ÿ
n
ÿ
’
’
’
’
pBf
q
pxq
“
bijk fj pSijk pxqq, x P Ω, i “ 1, 2, . . . , n.
i
%
k“1 j“1
We set the following notions
n ÿ
m
ÿ
max |αijk | ,
}rαijk s} “
i“1 k“1
1≤j≤n
for any set rαijk s “ tαijk P R : i, j “ 1, . . . , n; k “ 1, . . . , mu,
n ÿ
m
ÿ
}rFijk s} “
i“1 k“1
max }Fijk }8 ,
1≤j≤n
for any set rFijk s “ tFijk P CpΩ; Rq : i, j “ 1, . . . , n; k “ 1, . . . , mu, where
the symbol }¨}8 denotes the supremum norm on CpΩ; Rq; and the following
assumptions
All the functions Rijk , Sijk , Xijk : Ω Ñ Ω are continuous,
g P X,
}rbijk s} ă 1,
Ψ : Ω ˆ R2 Ñ R satisfying the following condition: @M ą 0, DC1 pM q
ą 0: |Ψpx, y1 , z1 q ´ Ψpx, y2 , z2 q| ≤ C1 pM q p|y1 ´ y2 | ` |z1 ´ z2 |q for all
px, y1 , z1 q, px, y2 , z2 q P Ω ˆ r´M, M s ˆ r´bM, bM s,
M p1´}rbijk s}q
2}g}
,
pH5 q M ą 1´ rb X s and 0 ă ε0 ă 2rp1`bqM C pM q`nM
} ijk }
1
0 s}raijk s}
where
pH1 q
pH2 q
pH3 q
pH4 q
(2.2)
M0 “ sup t|Ψpx, 0, 0q| : x P Ωu .
Given M ą 0, we put
KM “ tf P X : }f }X ≤ M u.
The following lemmas are useful to establish our main results, the proof
are not difficult so we omit it.
Linear approximation and asymptotic expansion associated. . .
107
Lemma 2.1. Let pH1 q and pH3 q hold. Then the linear operator I ´ B :
X Ñ X is invertible and
›
›
1
›pI ´ Bq´1 › ≤
.
1 ´ }rbijk s}
By Lemma 2.1, we rewrite the functional equations system (2.1) as follows
f “ pI ´ Bq´1 pεAf ` gq ” T f.
(2.3)
Lemma 2.2. Let pH1 q, pH3 q, pH4 q hold. Then, for every M ą 0 we have
(i) ›}Af }X ≤ }ra
› ijk s} rp1 ` bq C1 pM q }f }X ›` nM0›s , @f P KM ;
(ii) ›Af ´ Af¯›X ≤ p1 ` bq C1 pM q }raijk s} ›f ´ f¯›X , @f, f¯ P KM .
Then, we have the following theorem.
Theorem 2.3. Let pH1 q–pH5 q hold. Then, for every ε, with |ε| ≤ ε0 , the
system (2.3) has a unique solution f P KM .
Proof. It is evident that T : X Ñ X. Considering f, f¯ P KM , by Lemmas
2.1 and 2.2, we easily verify that
1
rε0 }raijk s} pp1 ` bq M C1 pM q ` nM0 q ` }g}X s ;
1 ´ }rbijk s}
›
›
›
ε0 p1 ` bq C1 pM q }raijk s} ››
(2.5) ›T f ´ T f¯›X ≤
f ´ f¯›X .
1 ´ }rbijk s}
(2.4) }T f }X ≤
Notice that, from pH3 q–pH5 q we have
(2.6)
1
rε0 }raijk s} pp1 ` bq M C1 pM q ` nM0 q ` }g}X s ≤ M.
1 ´ }rbijk s}
It follows from (2.4)–(2.6), that T : KM Ñ KM is a contraction mapping.
Then, using Banach fixed point theorem, there exists a unique function f P
KM such that f “ T f.
Remark 2.4. Theorem 2.3 gives a consecutive approximate algorithm
f pνq “ T f pν´1q , ν “ 1, 2, . . . , where f p0q P X is given.
Then the sequence tf pνq u converges in X to the solution f of (2.3) and
we have the error estimation
›
›
›
σ ν ›› p0q
› pνq
›
›
›f ´ f › ≤
›T f ´ f p0q › for all ν P N,
1´σ
X
X
ε0 p1 ` bq C1 pM q }raijk s}
where σ “
ă 1.
1 ´ }rbijk s}
108
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
3. The second order algorithm
In this part, let Ψ P C 1 pΩ ˆ R2 ; Rq.
First, using the approximation
¯
´
BΨ
px, upν´1q , v pν´1q q upνq ´ upν´1q
By
´
¯
BΨ
px, upν´1q , v pν´1q q v pνq ´ v pν´1q ,
`
Bz
şX pxq pνq
pνq
where upνq “ fj pRijk pxqq, v pνq “ 0 ijk fj ptqdt, we obtain the following
algorithm for system (1.1)
Ψpx, upνq , v pνq q – Ψpx, upν´1q , v pν´1q q `
(3.1)
pνq
fi pxq “ ε
m
ÿ
n
m ÿ
ÿ
´
¯
pνq
aijk Ψ Wijk pxq
k“1 j“1
n
ÿ
¯
BΨ ´ pνq ¯ ´ pνq
pν´1q
pRijk pxqq
Wijk pxq fj pRijk pxqq ´ fj
By
k“1 j“1
ˆż Xijk pxq ´
m ÿ
n
¯ ˙
ÿ
BΨ ´ pνq ¯
pν´1q
pνq
ptq dt
fj ptq ´ fj
`ε
aijk
Wijk pxq
Bz
0
k“1 j“1
`ε
m ÿ
n
ÿ
`
aijk
pνq
bijk fj pSijk pxqq ` gi pxq,
k“1 j“1
for all x P Ω, 1 ≤ i ≤ n, and ν “ 1, 2, . . . where
˙
ˆ
ż Xijk pxq
pν´1q
pν´1q
pνq
pRijk pxqq,
fj
ptqdt ,
(3.2)
Wijk pxq “ x, fj
0
p0q
p0q
and f p0q “ pf1 , . . . , fn q P KM is given.
Rewrite (3.1) as a linear system of functional equations
(3.3)
pνq
fi pxq “ pBf pνq qi pxq ` ε
m ÿ
n
ÿ
pνq
pνq
αijk pxqfj pRijk pxqq
k“1 j“1
`ε
m ÿ
n
ÿ
k“1 j“1
pνq
βijk pxq
ż Xijk pxq
0
pνq
pνq
fj ptqdt ` gi pxq,
pνq
pνq
pνq
for x P Ω, i “ 1, 2, . . . , n and ν “ 1, 2, . . . with αijk pxq, βijk pxq and gi pxq
depending on f pν´1q as follows
BΨ ´ pνq ¯ pνq
BΨ ´ pνq ¯
pνq
(3.4)
αijk pxq “ aijk
Wijk pxq , βijk pxq “ aijk
Wijk pxq ,
By
Bz
Linear approximation and asymptotic expansion associated. . .
109
and
(3.5)
pνq
gi pxq “ gi pxq ` εpAf pν´1q qi pxq ´ ε
m ÿ
n
ÿ
pνq
pν´1q
αijk pxqfj
pRijk pxqq
k“1 j“1
´ε
m ÿ
n
ÿ
pνq
βijk pxq
ż Xijk pxq
k“1 j“1
pν´1q
fj
0
ptqdt.
Then, we have the following.
Theorem 3.1. Let pH1 q–pH3 q hold and let Ψ P C 1 pΩˆR2 ; Rq. If f pν´1q P X
satisfies
›
›
›¯
´›
› pνq ›
› pνq ›
γν “ }rbijk s} ` |ε| ›rαijk s› ` b ›rβijk s› ă 1,
there exists a unique function f pνq P X being solution of system (3.3)–(3.5).
Proof. We write system (3.3)–(3.5) in the form of an operational equation
in X “ CpΩ; Rn q
f pνq “ Tν f pνq ,
where
pTν f qi pxq “ pBf qi pxq ` ε
n
m ÿ
ÿ
pνq
αijk pxqfj pRijk pxqq
k“1 j“1
`ε
n
m ÿ
ÿ
k“1 j“1
pνq
βijk pxq
ż Xijk pxq
0
pνq
fj ptqdt ` gi pxq,
for x P Ω, i “ 1, 2, . . . , n and f “ pf1 , . . . , fn q P X.
It is easy to check that Tν : X Ñ X and
›
›
›
›
›Tν f ´ Tν f¯› ≤ γν ›f ´ f¯›
for all f, f¯ P X.
X
X
Using the Banach fixed point theorem, there exists a unique function
f pνq P X being a solution of system (3.3)–(3.5).
Next, we make the following hypotheses:
pH6 q Ψ P C 2 pΩ “ˆ R2 ; Rq,
‰
1
1
2
0
pH7 q ε0 }raijk s} nM
M ` p1 ` bqM1 ` 2 p1 ` bq M2 M ≤ 1´}rbijk s}´ M }g}X ,
where M0 is given by (2.2) and
ˇ ˇ
ˇ˙
*
"ˆˇ
$
ˇ BΨ ˇ ˇ BΨ ˇ
’
ˇ
ˇ
ˇ
ˇ
’
& M1 “ sup
ˇ By ˇ ` ˇ Bz ˇ px, y, zq : px, y, zq P A˚ ,
"ˆˇ 2 ˇ ˇ 2 ˇ ˇ 2 ˇ˙
*
ˇB Ψˇ ˇ B Ψ ˇ ˇB Ψˇ
’
’
ˇ`ˇ
ˇ`ˇ
ˇ px, y, zq : px, y, zq P A˚ ,
ˇ
% M2 “ sup
ˇ By 2 ˇ ˇ ByBz ˇ ˇ Bz 2 ˇ
with A˚ “ tpx, y, zq : x P Ω, |y| ≤ M, |z| ≤ bM u .
110
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
Theorem 3.2. Let pH1 q–pH3 q, pH6 q, pH7 q hold, let f be the solution of
system (1.1) and the sequence tf pνq u be defined by algorithm (3.3)–(3.5).
›
›
piq If ›f p0q ›X ≤ M, then
›
›2
›
›
›
›
› pνq
›
(3.6)
›f ´ f › ≤ βM ›f pν´1q ´ f › , @ν “ 1, 2, . . .
X
X
where
p1 ` bq2 M2 }raijk s}
ą 0.
1 ´ }rbijk s} ´ ε0 p1 ` bq M1 }raijk s}
›
›
piiq If the first term f p0q sufficiently near f such that βM ›f p0q ´ f ›X ă 1,
then the sequence tf pνq u converges quadratically to f and furthermore
›
›
›
› ¯2ν
1 ´
› pνq
›
›
›
(3.8)
βM ›f p0q ´ f ›
, @ν “ 1, 2, . . .
›f ´ f › ≤
βM
X
X
›
›
Proof. First, we verify that if ›f p0q ›X ≤ M, then
›
›
› pνq ›
›f › ≤ M, @ν “ 1, 2, . . .
βM “
(3.7)
1
2 ε0
X
Indeed, supposing
› pν´1q ›
› ≤ M,
›f
X
(3.9)
we deduce from (3.3) that
›
›
›
›¯ ›
›
›
›
›
›
´
›
›
›
› pνq › ›
›
›
› pνq ›
(3.10) ›f pνq › ≤ }rbijk s} ` |ε| ›rαijk s› ` |ε| b ›rβijk s› ›f pνq › ` ›g pνq › .
X
X
X
On the other hand, we have
ˇ
ˇ
$ˇ
ˇ
ˇ BΨ ´ pνq ¯ˇ
ˇ
ˇ
pνq
’
ˇ
’
& ˇαijk pxqˇ ≤ |aijk | ˇ By Wijk pxq ˇˇ ≤ M1 |aijk | ,
ˇ
ˇ
(3.11)
ˇ
ˇ
ˇ BΨ ´ pνq ¯ˇ
’
ˇ
pνq
’
ˇ
% ˇˇβijk
pxqˇ ≤ |aijk | ˇ
Wijk pxq ˇˇ ≤ M1 |aijk | .
Bz
Hence, we deduce from (3.10), (3.11) that
›
›
›
›
›
›
› pνq ›
›
›
›
›
›f › ≤ r}rbijk s} ` ε0 p1 ` bq M1 }raijk s}s ›f pνq › ` ›g pνq › .
X
X
Note that pH7 q implies }rbijk s} ` ε0 p1 ` bq M1 }raijk s} ă 1, so
› pνq ›
›
›
›g ›
› pνq ›
X
(3.12)
.
›f › ≤
1 ´ }rbijk s} ´ ε0 p1 ` bq M1 }raijk s}
X
›
›
Now, we need an estimate on the term ›g pνq ›X .
X
Linear approximation and asymptotic expansion associated. . .
111
From (3.4) and (3.5), we obtain
(3.13)
pνq
gi pxq
“ gi pxq ` ε
m ÿ
n
ÿ
„ ´
¯
pνq
aijk Ψ Wijk pxq
k“1 j“1

ż
BΨ ´ pνq ¯ Xijk pxq pν´1q
BΨ ´ pνq ¯ pν´1q
Wijk pxq fj
pRijk pxqq ´
Wijk pxq
fj
ptqdt .
´
By
Bz
0
On the other hand,
of the function
¯ Ψpx, 0, 0q at
´ the Taylor’s expansion
şXijk pxq pν´1q
pνq
pν´1q
ptqdt , up to order 2,
fj
pRijk pxqq, 0
the point Wijk pxq “ x, fj
leads to
¯ BΨ ´
´
¯
pνq
pν´1q
pνq
Ψpx, 0, 0q “ Ψ Wijk pxq ´
pRijk pxqq
Wijk pxq fj
By
ż
¯2
BΨ ´ pνq ¯ Xijk pxq pν´1q
1 B 2 Ψ ´ pνq ¯ ´ pν´1q
pR
pxqq
´
ptqdt `
pxq
f
fj
W̄
Wijk pxq
ijk
j
ijk
Bz
2 By 2
0
ż
Xijk pxq
B 2 Ψ ´ pνq ¯ pν´1q
pν´1q
`
ptqdt
pRijk pxqq
fj
W̄ijk pxq fj
ByBz
0
˜ż
¸2
Xijk pxq
1 B 2 Ψ ´ pνq ¯
pν´1q
`
W̄ijk pxq
fj
ptqdt ,
2 Bz 2
0
where
pνq
W̄ijk pxq
˙
ˆ
ż Xijk pxq
pν´1q
pν´1q
ptqdt ,
pRijk pxqq, ´θijk
fj
“ x, ´θijk fj
0
0 ă θijk ă 1.
Therefore
ˇ ´
¯
´
¯
ˇ
ˇΨ W pνq pxq ´ BΨ W pνq pxq f pν´1q pRijk pxqq
(3.14)
j
ijk
ijk
ˇ
By
ˇ
ż
ˇ
BΨ ´ pνq ¯ Xijk pxq pν´1q
´
Wijk pxq
fj
ptqdtˇˇ
Bz
0
ˇ
´
¯
´
¯
2
2
ˇ
1B Ψ
pνq
pν´1q
“ ˇˇΨpx, 0, 0q ´
W̄ijk pxq fj
pRijk pxqq
2 By 2
ż Xijk pxq
B 2 Ψ ´ pνq ¯ pν´1q
pν´1q
´
W̄ijk pxq fj
pRijk pxqq
fj
ptqdt
ByBz
0
ˆż Xijk pxq
˙2 ˇ
ˇ
1 B 2 Ψ ´ pνq ¯
pν´1q
W̄ijk pxq
fj
ptqdt ˇˇ
´
2
2 Bz
0
112
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
ˇ
ˇ
ż Xijk pxq
´
¯2
ˇ pν´1q
ˇ
1
pν´1q
pν´1q
ˇ
≤ M0 ` M2 fj
pRijk pxqq ` M2 ˇfj
fj
pRijk pxqq
ptqdtˇˇ
2
0
ˆż Xijk pxq
˙2
1
pν´1q
fj
ptqdt
` M2
2
0
ˇ ˇż X pxq
ˇ2
„ˇ
ˇ pν´1q
ˇ ˇ ijk
ˇ
1
pν´1q
ˇ
ˇ
ˇ
≤ M0 ` M2 ˇfj
pRijk pxqqˇ ` ˇ
fj
ptqdtˇˇ .
2
0
It follows from (3.13), (3.14) that

„
›
›
1
› pνq ›
2
2
(3.15)
›g › ≤ }g}X ` ε0 }raijk s} nM0 ` p1 ` bq M2 M .
2
X
Hence, from (3.12), (3.15) and pH7 q, we obtain
ı
”
2
1
2
›
›
}g}
p1
`
bq
}ra
`
ε
M
M
s}
nM
`
0
2
0
ijk
X
2
› pνq ›
≤ M.
›f › ≤
1 ´ }rbijk s} ´ ε0 p1 ` bq M1 }raijk s}
X
›
›
Now, we shall estimate ›f ´ f pνq ›X .
Put epνq “ f ´ f pνq , we obtain from (1.1) and (3.1) the system
pνq
pνq
ei pxq “ fi pxq ´ fi pxq
n
m ÿ
ÿ
pνq
pνq
pνq
αijk pxqej pRijk pxqq
“ pBe qi pxq ` ε
(3.16)
k“1 j“1
`ε
`ε
n
m ÿ
ÿ
k“1 j“1
m ÿ
n
ÿ
pνq
βijk pxq
ż Xijk pxq
0
pνq
ej ptqdt
„
´
¯
pνq
aijk Ψ pWijk pxqq ´ Ψ Wijk pxq
k“1 j“1
´

ż
BΨ ´ pνq ¯ pν´1q
BΨ ´ pνq ¯ Xijk pxq pν´1q
Wijk pxq ej
pRijk pxqq ´
Wijk pxq
ej
ptqdt ,
By
Bz
0
pνq
where Wijk pxq is given by (3.2) and
˜
Wijk pxq “
ż Xijk pxq
x, fj pRijk pxqq,
¸
fj ptqdt .
0
´
¯
şZ
Using Taylor’s expansion of the function Ψ x, fj pY q, 0 fj ptqdt at the
şZ pν´1q
pν´1q
point px, fj
pY q, 0 fj
ptqdtq, up to order 2, we obtain
Linear approximation and asymptotic expansion associated. . .
(3.17)
113
ˆ
˙
ˆ
˙
żZ
żZ
pν´1q
pν´1q
Ψ x, fj pY q,
fj ptqdt “ Ψ x, fj
pY q,
fj
ptqdt
0
0
ˆ
˙
żZ
BΨ
pν´1q
pν´1q
pν´1q
x, fj
pY q,
fj
ptqdt ej
pY q
`
By
0
˙ż Z
ˆ
żZ
BΨ
pν´1q
pν´1q
pν´1q
fj
ptqdt
ej
ptqdt
x, fj
pY q,
`
Bz
0
0
ˇ2
¯ˇ
1 B 2 Ψ ´ pνq
ˇ
ˇ pν´1q
`
pY
q
px,
Y,
Zq
e
ω
ˇ
ˇ
j
j
2 By 2
żZ
¯
B 2 Ψ ´ pνq
pν´1q
pν´1q
ptqdt
ej
`
pY q
ωj px, Y, Zq ej
ByBz
0
˙2
¯ ˆż Z
1 B 2 Ψ ´ pνq
pν´1q
`
ptqdt ,
ej
ωj px, Y, Zq
2 Bz 2
0
where
pνq
ωj px, Y, Zq
ˆ
żZ”
ı ˙
pν´1q
pν´1q
pν´1q
pν´1q
ptq dt ,
ptq ` θj ej
pY q,
fj
pY q ` θj ej
“ x, fj
0
0 ă θj ă 1.
pν´1q
Substituting (3.17) into (3.16) where the arguments of fj , fj
pνq
ωj
pν´1q
, ej
appearing in (3.17) are replaced by Y “ Rijk pxq, Z “ Xijk pxq, we get
(3.18)
pνq
ei pxq
“ pBe
pνq
qi pxq ` ε
m ÿ
n
ÿ
pνq
pνq
αijk pxqej pRijk pxqq
k“1 j“1
`ε
m ÿ
n
ÿ
k“1 j“1
m ÿ
n
ÿ
pνq
βijk pxq
ż Xijk pxq
0
pνq
ej ptqdt
ˇ2
1 B 2 Ψ ´ pνq ¯ ˇˇ pν´1q
ˇ
ω
pxq
e
pR
pxqq
ˇ
ˇ
ijk
j
ijk
2
2 By
k“1 j“1
ż Xijk pxq
B 2 Ψ ´ pνq ¯ pν´1q
pν´1q
`
pRijk pxqq
ptqdt
ωijk pxq ej
ej
ByBz
0
ˇż
ˇ2 
ˇ
1 B 2 Ψ ´ pνq ¯ ˇˇ Xijk pxq pν´1q
`
ωijk pxq ˇ
ej
ptqdtˇˇ ,
2
2 Bz
0
`ε
pνq
pνq
„
aijk
where ωijk pxq “ ωj px, Rijk pxq, Xijk pxqq.
,
114
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
Combining (3.9), (3.11), (3.18), the result is
ˇ
m
n ÿ
ˇ
ˇ n ˇ
›
›
›
›
ÿ
ˇ
ˇ pνq ˇ ÿ ˇˇ pνq
› pνq ›
› pνq ›
ˇ
max sup ˇαijk pxqˇ
e
pR
pxqq
›e › ≤ ›Be › `ε0
ijk
j
ˇ
ˇ
X
X
i“1 k“1
1≤j≤n xPΩ
j“1
ˇ
ˇ
n ˇż Xijk pxq
ˇ
ˇÿ
ˇ
ˇ pνq ˇ
ˇ
ˇ
pνq
ej ptqdtˇ
`ε0
max sup ˇβijk pxqˇ
ˇ
ˇ
ˇ
1≤j≤n xPΩ
i“1 k“1
j“1 0
ˇ˙2
n ÿ
m
n ˆˇ
ˇ ˇˇż Xijk pxq
ÿ
ÿ
ˇ
1
ˇ ˇ
ˇ pν´1q
pν´1q
` ε 0 M2
pRijk pxqqˇ` ˇ
ptqdtˇˇ
ej
max |aijk | sup
ˇej
1≤j≤n
2
xPΩ j“1
0
i“1 k“1
›
›
›
›
≤ r}rbijk s}`ε0 p1`bq M1 }raijk s}s ›epνq ›
X
›
›2
1
›
›
` ε0 p1`bq2 M2 }raijk s} ›epν´1q › .
2
X
Consequently
2
›
›
›
›
1
› pνq ›
› pν´1q ›2
2 ε0 p1 ` bq M2 }raijk s}
(3.19)
›e › ≤
›e
›
1 ´ }rbijk s} ´ ε0 p1 ` bq M1 }raijk s}
X
X
›
›2
›
›
” βM ›epν´1q › .
n ÿ
m
ÿ
X
Hence, we obtain (3.6) by (3.7) and (3.19). Finally, from (3.6), (3.8)
follows.
Remark 3.3. If we choose µ0 sufficient large such that
›
›
›
› σ µ0
›
›
›
›
βM ›g pµ0 q ´ f › ≤ βM ›T g p0q ´ g p0q ›
ă 1,
X
X 1´σ
and›choose f›p0q “ g pµ0 q , then first term f p0q sufficiently near f such that
βM ›f p0q ´ f ›X ă 1.
4. Asymptotic expansion of solutions
In this part, we assume that the functions Rijk , Sijk , Xijk , g, Ψ and the
real numbers aijk , bijk , M satisfy the assumptions pH1 q–pH5 q, respectively.
We use the following notation
ˆ
˙
ż Xijk pxq
Ψrfj s “ Ψ x, fj pRijk pxqq,
fj ptqdt .
0
Now, we assume that
pH8 q Ψ P C N pΩ ˆ R2 ; Rq.
We consider the perturbed system (2.1), where ε is a small parameter |ε| ≤ ε0 . Let us consider the finite sequence of functions tf rrs u, r “
Linear approximation and asymptotic expansion associated. . .
115
0, 1, . . . , N, f rrs P KM (with suitable constants M ą 0, ε0 ą 0q defined as
follows:
f rrs “ pI ´ Bq´1 P rrs , r “ 0, 1, . . . , N,
(4.1)
where
´
¯
rrs
P rrs “ P1 , . . . , Pnrrs , r “ 0, 1, . . . , N,
and
P r0s “ g.
With r “ 1:
r1s
Pi
“ pAf
r0s
m ÿ
n
ÿ
qi pxq “
r0s
aijk πj rΨs,
k“1 j“1
where
$
¯
´
r0s
r0s
r0s
r0s
’
’
pX
pxqq
,
pR
pxqq,
Jf
s
“
Ψ
x,
f
rΨs
“
Ψrf
π
ijk
ijk
& j
j
j
j
ż Xijk pxq
’
r0s
r0s
’
fj ptqdt.
% Jfj pXijk pxqq “
(4.2)
0
With r “ 2:
r2s
Pi
m ÿ
n
ÿ
“
r1s
aijk πj rΨs,
k“1 j“1
where
r1s
r0s
r1s
r0s
r1s
(4.3) πj rΨs “ πj rD2 Ψsfj ` πj rD3 ΨsJfj , D2 Ψ “
BΨ
BΨ
, D3 Ψ “
.
By
Bz
For 2 ≤ r ≤ N,
rrs
Pi
m ÿ
n
ÿ
“
rr´1s
aijk πj
rΨs,
k“1 j“1
rrs
where, πj rΨs, 0 ≤ r ≤ N ´ 1 defined by the recurrence formulas
rrs
πj rΨs “
(4.4)
r´1
ÿ
)
r ´ s ! rss
rr´ss
rss
rr´ss
πj rD2 Ψsfj
` πj rD3 ΨsJfj
.
r
s“0
rrs
rrs
We also note that πj rΨs is the first-order function with respect to fj ,
rrs
Jfj . In fact,
rrs
r0s
rrs
r0s
rrs
πj rΨs “ πj rD2 Ψsfj ` πj rD3 ΨsJfj ` terms depending on
rss
rss
rss
rss
pj, r, πj rD2 Ψs, πj rD3 Ψs, fj , Jfj q, s “ 1, . . . , r ´ 1.
116
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
Put
h“f
r0s
N
ÿ
`
f rrs εr ” f r0s ` U,
r“1
then
v “ fε ´
N
ÿ
f rrs εr ” fε ´ h
r“0
satisfies the system
(4.5)
pI ´ Bqv “ εrApv ` hq ´ Ahs ` Eε ,
where
Eε “ εrApf r0s ` U q ´ Apf r0s qs ´
N
ÿ
P rrs εr .
r“2
Then, we have the following lemmas.
rrs
Lemma 4.1. The functions πj rΨs, 0 ≤ r ≤ N ´ 1 as above are defined
by the following formulas:
ˇ
ˇ
1 Br
rrs
πj rΨs “
Ψrhj sˇˇ
, 0 ≤ r ≤ N ´ 1.
r
r! Bε
ε“0
Proof. (i) It is easy to see that
ˇ
ˇ
1 B0
Ψrhj sˇˇ
“ Ψrhj s|ε“0
0
0! Bε
ε“0
¯
´
r0s
r0s
r0s
r0s
“ Ψ x, fj pRijk pxqq, Jfj pXijk pxqq “ Ψrfj s “ πj rΨs.
With r “ 1, we shall show that
r1s
πj rΨs
(4.6)
ˇ
ˇ
1 B
“
Ψrhj sˇˇ
.
1! Bε
ε“0
We have
B
B
B
Ψrhj s “ D2 Ψrhj s hj ` D3 Ψrhj s Jhj .
Bε
Bε
Bε
On the other hand, from the formulas
(4.7)
hj “
N
ÿ
rrs
fj εr ,
r“0
N
N
ÿ
ÿ
B
rrs r´1 B
rrs
hj “
rfj ε , Jhj “
rJfj εr´1 ,
Bε
Bε
r“1
r“1
we have
(4.8)
hj |ε“0 “
r0s
fj ,
ˇ
ˇ
ˇ
B ˇˇ
r1s B
r1s
hj ˇ
“ fj , Jhj ˇˇ
“ Jfj .
Bε ε“0
Bε
ε“0
Linear approximation and asymptotic expansion associated. . .
117
Hence, it follows from (4.7), (4.8) that
ˇ
ˇ
ˇ
ˇ
ˇ
B
1 B
B ˇˇ
ˇ
` D3 Ψrhj s Jhj ˇˇ
Ψrhj sˇ
“ D2 Ψrhj s hj ˇ
1! Bε
Bε ε“0
Bε
ε“0
ε“0
r0s
r1s
r0s
r1s
“ D2 Ψrfj sfj ` D3 Ψrfj sJfj
r0s
r1s
r0s
r1s
“ πj rD2 Ψsfj ` πj rD3 ΨsJfj
r1s
“ πj rΨs.
Thus, (4.6) holds.
rss
Suppose that we have defined the functions πj rΨs, 0 ≤ s ≤ r ´ 1 from
formulas (4.2), (4.3) and (4.4). Therefore, it follows from (4.7) that
ˆ
˙
„

B r´1
B
B r´1
B
B
Br
Ψrhj s “ r´1
Ψrhj s “ r´1 D2 Ψrhj s hj ` D3 Ψrhj s Jhj
Bεr
Bε
Bε
Bε
Bε
Bε
„

r´1
ÿ
Bs
B r´s
Bs
B r´s
s
“
Cr´1
D2 Ψrhj s r´s hj ` s D3 Ψrhj s r´s Jhj .
Bεs
Bε
Bε
Bε
s“0
We also note that
ˇ
ˇ
s
ˇ
B s ˇˇ
rss B
rss
hj ˇ
“ s!fj , s Jhj ˇˇ
“ s!Jfj , 0 ≤ s ≤ r.
s
Bε
Bε
ε“0
ε“0
Hence
ˇ
ˇ
ˇ
„ s
r´1
ˇ
ˇ
1 Br
1 ÿ s
B r´s ˇˇ
B
ˇ
ˇ
“
Ψrhj sˇ
C
D2 Ψrhj sˇ
h
r´s j ˇ
r! Bεr
r! s“0 r´1 Bεs
ε“0
ε“0 Bε
ε“0
ˇ
ˇ 
r´s
ˇ
ˇ
Bs
B
`
D3 Ψrhj sˇˇ
Jhj ˇˇ
s
r´s
Bε
ε“0 Bε
ε“0
r´1
ÿ
1
rss
rr´ss
s
“
Cr´1
s!πj rD2 Ψspr ´ sq!fj
r! s“0
r´1
ÿ
`
rss
rr´ss
s
Cr´1
s!πj rD3 Ψspr ´ sq!Jfj
s“0
r´1
ÿ
“
”
ı
1
rss
rr´ss
rss
rr´ss
s
Cr´1
s!pr ´ sq! πj rD2 Ψsfj
` πj rD3 ΨsJfj
r! s“0
r´1
ÿ
“
ı
r ´ s ” rss
rr´ss
rss
rr´ss
rrs
πj rD2 Ψsfj
` πj rD3 ΨsJfj
“ πj rΨs.
r
s“0
Lemma 4.1 is proved completely.
118
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
p1q
Lemma 4.2. Let pH1 q–pH5 q, pH8 q hold.› Then
› there exists a constant C̄N
rrs
›
depending only on N , }raijk s}, }rbijk s} , f ›X , 0 ≤ r ≤ N such that
}Eε }X ≤ C̄N |ε|N `1 .
p1q
Proof. In the case of N “ 1, the proof of Lemma 4.2 is easy, hence we omit
the details, so we only prove with N ≥ 2. We have
´
(4.9)
Apf r0s ` U q ´ Apf r0s q
m ÿ
n
ÿ
¯
i
pxq “
ı
”
r0s
r0s
aijk Ψrfj ` Uj s ´ Ψrfj s ,
k“1 j“1
in which
¯
´
r0s
r0s
r0s
Ψrfj s “ Ψ x, fj pRijk pxqq, Jfj pXijk pxqq ,
¯
´
r0s
r0s
r0s
Ψrfj ` Uj s “ Ψ x, fj pRijk pxqq ` Uj pRijk pxqq, Jpfj ` Uj qpXijk pxqq .
By using Maclaurin’s expansion of the function Ψrhj s round the point
ε “ 0 up to order N, we obtain
r0s
r0s
r0s
Ψrhj s ´ Ψrfj s “ Ψrfj ` Uj s ´ Ψrfj s
ˇ
Nÿ
´1
ˇ
1 Br
εN rN s
r
ˇ
“
Ψrh
s
ε
`
R rΨs
j
ˇ
r! Bεr
N! j
ε“0
r“1
(4.10)
Nÿ
´1
“
rrs
πj rΨsεr `
r“1
εN rN s
R rΨ, θ1 s,
N! j
rrs
rN s
where dj rΨs, 0 ≤ r ≤ N ´1 are defined by (4.2), (4.3) and (4.4); Rj rΨ, θ1 s
is defined as follows
ˇ
ˇ
BN
rN s
(4.11)
Rj rΨ, θ1 s “ N Ψrhj sˇˇ
,
Bε
ε“θ1 ε
with 0 ă θ1 ă 1.
r0s
r0s
Substituting Ψrfj ` Uj s ´ Ψrfj s in (4.10) into (4.9), we obtain after
some rearrangements in order of ε that
(4.12)
Eεi “ εpApf r0s ` U q ´ Apf r0s qqi ´
N
ÿ
rrs
Pi εr
r“2
“ε
m
ÿ
n
ÿ
k“1 j“1
N
”
ı ÿ
r0s
r0s
rrs
aijk Ψrfj ` Uj s ´ Ψrfj s ´
P i εr
r“2
Linear approximation and asymptotic expansion associated. . .
Nÿ
´1 ÿ
m
n
ÿ
“
rrs
aijk πj rΨsεr`1 `
r“1 k“1 j“1
Nÿ
´1
“
rr`1s r`1
Pi
ε
`
r“1
“
119
m n
N
ÿ
εN `1 ÿ ÿ
rN s
rrs
aijk Rj rΨ, θ1 s ´
P i εr
N ! k“1 j“1
r“2
m n
N
ÿ
εN `1 ÿ ÿ
rN s
rrs
aijk Rj rΨ, θ1 s ´
P i εr
N ! k“1 j“1
r“2
m n
εN `1 ÿ ÿ
rN s
aijk Rj rΨ, θ1 s.
N ! k“1 j“1
By the boundedness of the functions f rrs , r “ 0, 1, 2, . . . , N, f rrs P KM ,
it implies from (4.11), (4.12) that
}Eε }X “ |ε|N `1 }RN rΦ, εs}X ≤ C̄N |ε|N `1 .
p1q
Lemma 4.2 is proved completely.
Theorem 4.3. Let pH1 q–pH5 q, pH8 q hold. Then there exists a constant
ε1 ą 0 such that, for every ε P R, with |ε| ≤ ε1 , the system (2.3) has a
unique solution fε P KM satisfying the asymptotic estimation up to order
N ` 1 as follows
›
›
N
ÿ
›
›
2
p1q
rrs r ›
›fε ´
C̄N |ε|N `1 ,
f ε › ≤
›
1 ´ }rbijk s}
X
r“0
where the functions f rrs , r “ 0, 1, . . . , N are defined by (4.1).
Proof. From (4.5), Lemmas 2.1 and 4.2 we have
›
›
(4.13)
}v}X ≤ ›pI ´ Bq´1 › p|ε| }Apv ` hq ´ Ah}X ` }Eε }X q
¯
´
1
p1q
≤
ε1 }Apv ` hq ´ Ah}X ` C̄N |ε|N `1 .
1 ´ }rbijk s}
On the other hand
}v ` h}X “ }fε }X ≤ N1 , }h}X
N ›
›
ÿ
› rrs ›
≤
›f ›
r“0
N
ÿ
(4.14)
}Jv ` Jh}X “ }Jfε }X ≤ N2 , }Jh}X ≤
X
›
›
› rrs ›
›Jf ›
r“0
It follows from (4.14) that
(4.15)
where
” N̄1 ,
}Apv ` hq ´ Ah}X ≤ C2 pM q }raijk s} }v}X ,
X
” N̄2 .
120
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
C2 pM q
ˇ ˇ
ˇ
"ˇ
*
ˇ BΨ
ˇ ˇ BΨ
ˇ
ˇ
ˇ
ˇ
ˇ
“ sup ˇ
px, y, zqˇ ` ˇ
px, y, zqˇ : x P Ω, |y| ≤ N1 ` N̄1 , |z| ≤ N2 ` N̄2 .
By
Bz
From (4.13), (4.15), we obtain
´
¯
1
p1q
ε1 C2 pM q }raijk s} }v}X ` C̄N |ε|N `1 .
}v}X ≤
1 ´ }rbijk s}
Choosing 0 ă ε1 ă ε0 , such that
(4.16)
ε1 C2 pM q }raijk s}
1
≤ 1{2.
1 ´ }rbijk s}
So, (4.16) leads to
}v}X ≤
2
p1q
C̄ |ε|N `1 ,
1 ´ }rbijk s} N
or
›
›
N
ÿ
›
›
rrs r ›
›fε ´
f
ε
›
›
r“0
X
≤
2
p1q
C̄N |ε|N `1 .
1 ´ }rbijk s}
Theorem 4.3 is proved completely.
5. Examples
Let us give two following illustrated examples for the results obtained as
above.
5.1. Example 1.
Let us consider system (1.1) with n “ 2, m “ 1, Ψ px, y, zq “ cos y sin z :
$
ˆ ˆ
˙˙
ˆż x
˙
x
2
’
’ f1 pxq “ εa11 cos f1
`
sin
f1 ptqdt
’
’
’
3 3
0
’
’
˜ż 3
¸
’
ˆ ˆ
˙˙
’
x
’
x
2
’
’
’
` εa12 cos f2
´
sin
f2 ptqdt
’
’
3 3
’
0
’
’
’
’
2x 1
x 1
’
’
` b11 f1 p
` q ` b12 f2 p ´ q ` g1 pxq,
’
’
’
3
3
2˜ 2
&
¸
ˆ ˆ
˙˙
ż x5
(5.1)
x 3
’
f2 pxq “ εa21 cos f1
`
sin
f1 ptqdt
’
’
4 4
’
0
’
’
ˆ ˆ
˙˙
ˆż x
˙
’
’
’
x 3
’
’
` εa22 cos f2
´
sin
f2 ptqdt
’
’
4 4
’
0
’
ˆ
˙
ˆ
˙
’
’
’
x 1
2x 1
’
’
` b21 f1
`
` b22 f2
´
` g2 pxq,
’
’
2 2
3
3
’
’
’
%
x P Ω “ r´1, 1s,
Linear approximation and asymptotic expansion associated. . .
121
where ε ą 0 is small enough; aijk ” aij , bijk ” bij are constants and all the
functions g1 , g2 : Ω Ñ R; Rijk ” Rij , Sijk ” Sij , Xijk ” Xij : Ω Ñ Ω are
continuous defined respectively as follows
$
’
a P R;
’
& ij
bij P R such that
ř
’
’
% }rbij s} “ 2i“1 max |bij | “ max |b1j | ` max |b2j | ă 1;
1≤j≤2
1≤j≤2
1≤j≤2
$
’
g1 , g2 P CpΩ;
’
’
ff
« Rq;
ff «
’
’
2
x
2
x
’
`
´
R
pxq
R
pxq
’
11
12
’
’
rRij pxqs “
“ x3 33 x3 33 ;
’
’
R
pxq
R
pxq
’
21
22
4 ` 4
4 ´ 4
’
&
«
ff «
ff
2x
1
x
1
S11 pxq S12 pxq
`
´
2
2 ;
’
rSij pxqs “
“ x3 13 2x
’
1
’
’
S
pxq
S
pxq
`
´
21
22
’
2
2
3
3
’
«
ff «
ff
’
’
’
3
’
X11 pxq X12 pxq
x x
’
’
rX
pxqs
“
“
.
’
ij
%
X21 pxq X22 pxq
x5 x
It is obvious that pH1 q–pH5 q hold with M ą
2}g}X
1´}rbij s}
and 0 ă ε0 ă
1´}rbij s}
4}raij s} .
So we conclude that, for every ε with |ε| ă ε0 , equation (5.1) has
a unique solution f P KM .
On the other hand, because Ψ px, y, zq “ Ψ py, zq “ cos y sin z, pH6 q and
2}g}
1´}rb s}´
1
}g}
ij
X
M
pH8 q are also satisfied. Therefore, if M ą 1´}rbijXs} , 0 ă ε0 ă 2p2`3M
q}raij s}
and ε ą 0 is small enough, then we obtain the results as in Theorems 3.2
and 4.3.
5.2. Example 2.
Consider system (1.1) with n “ m “ 2, Ψ px, y, zq “ Φ pzq , Φ P C 1 pRq :
$
ˆż x
˙
ˆż x3
˙
ˆ
˙
’
x`1
’
’
f1 pxq “ εa11 Φ
f1 ptqdt ` εa12 Φ
f2 ptqdt ` b111 f1
’
’
2
’
0
0
’
ˆ
˙
’
’
’
2x ` 1
&
` b112 f1 pcos πxq ` b122 f2
` g1 pxq,
(5.2)
3
’
ˆ
˙
ˆ
˙
’
’
x
´
1
2x
´
1
’
’
f2 pxq “ b211 f1
` b221 f2 psin πxq ` b222 f2
` g2 pxq,
’
’
’
2
3
’
’
%
x P Ω “ r´1, 1s,
where ε ą 0 is small enough; aijk ” aij , bijk are constants and all the
functions g1 , g2 : Ω Ñ R; Rijk ” Rij , Sijk , Xijk ” Xij : Ω Ñ Ω are
122
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
continuous defined respectively as follows
$
’
’ aij P R «such thatff «
ff
’
’
’
a
a
a
a
’
11
12
11
12
’
’
raij s “
“
;
’
’
a21 a22
0
0
’
’
’
’
’
’
& bijk P R «such that
ff «
ff
b111 b121 b112 b122
b111
0
b112 b122
’
rbijk s “
“
,
’
’
b211 b221 b212 b222
b211 b221
0
b222
’
’
’
’
2 ÿ
2
’
ÿ
’
’
’
}rb
s}
“
max |bijk |
ijk
’
’
1≤j≤2
’
’
i“1
k“1
’
%
“ |b111 | ` |b222 | ` max t|b112 | , |b122 |u ` max t|b211 | , |b221 |u ă 1;
$
’
’
’ g1 , g2 P CpΩ;
« Rq;
ff
’
’
’
S
pxq
S
pxq
S
pxq
S
pxq
’
111
121
112
122
’
’
rSijk pxqs “
’
’
S211 pxq S221 pxq S212 pxq S222 pxq
’
’
&
ff
«
2x`1
x`1
0
cos
πx
2
3
’
;
“ x´1
’
2x´1
’
’
sin
πx
0
’
2
3
’
«
ff «
ff
’
’
’
3
’
X
pxq
X
pxq
x
x
11
12
’
’
’
% rXij pxqs “ X pxq X pxq “ 0 0 .
21
22
It is also clear to see that pH1 q ´ pH5 q hold with M ą
0 ă ε0 ă
M p1´}rbijk s}q
«
ff .
2}g}X
1´}rbijk s}
and
So, for every ε such that |ε| ă ε0 ,
4}raij s} M sup |Φ1 pzq|`|Φp0q|
|z|≤M
equation (5.2) has a unique solution f P KM .
2}g}
Furthermore, if Φ P C 2 pRq or Φ P C N pRq, M ą 1´ rb X s and
} ijk }

„
ˇ
ˇ
ˇ
ˇ
|Φ p0q|
` sup ˇΦ1 pzqˇ ` M sup ˇΦ2 pzqˇ
0 ă 2ε0 }raij s}
M
|z|≤M
|z|≤M
1
}g}X
M
and ε ą 0 is small enough, the results of Theorems 3.2 and 4.3 are obtained.
ă 1 ´ }rbijk s} ´
Acknowledgements. The authors wish to express their sincere thanks
to the referees for their valuable comments and important remarks. The
comments uncovered several weaknesses in the presentation of the paper
and helped us to clarify it. The authors are also extremely grateful for the
Linear approximation and asymptotic expansion associated. . .
123
support given by Vietnam’s National Foundation for Science and Technology
Development (NAFOSTED) under Project 101.01-2012.12.
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Le Thi Phuong Ngoc
NHATRANG EDUCATIONAL COLLEGE
01 Nguyen Chanh Str.
NHATRANG CITY, VIETNAM
E-mail: ngocltp@gmail.com, ngoc1966@gmail.com
124
L. T. P. Ngoc, H. T. H. Dung, P. H. Danh, N. T. Long
Huynh Thi Hoang Dung
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF ARCHITECTURE OF HO CHI MINH CITY
196 Pasteur Str., Dist. 3
HO CHI MINH CITY, VIETNAM
E-mail: dunghth1980@gmail.com
Pham Hong Danh
DEPARTMENT OF MATHEMATICS
STATISTICS AND INFORMATICS
UNIVERSITY OF ECONOMICS OF HO CHI MINH CITY
59C Nguyen Dinh Chieu Str., Dist. 3
HO CHI MINH CITY, VIETNAM
E-mail: hongdanh282@gmail.com
Nguyen Thanh Long DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
UNIVERSITY OF NATURAL SCIENCE
VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY
227 Nguyen Van Cu Str., Dist. 5
HO CHI MINH CITY, VIETNAM
E-mail: longnt1@yahoo.com, longnt2@gmail.com
Received March 8, 2012; revised version December 12, 2012.