Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 203, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
RAPID CONVERGENCE OF APPROXIMATE SOLUTIONS FOR
SINGULAR DIFFERENTIAL SYSTEMS
PEIGUANG WANG, XIANG LIU
Abstract. In this article we show the existence and approximation of solutions for a class of singular differential system with initial value condition. We
present a generalized quasilinearization method for obtain monotone iterative
sequences of approximate solutions converging uniformly to a solution at a
rate higher than quadratic.
1. Introduction
In 1974, Rosenbrock [17] introduced the concept of singular systems, which is
more complicated than the ordinary ones, and its qualitative analysis involve greater
difficulty than those of the ordinary systems. A systematic development of the
basic theory of the linear singular systems has been provided by Campbell [5, 6].
The results of qualitative properties for nonlinear singular differential systems can
be found in [9, 12, 18, 19, 20, 22, 23]. However, we noticed that most of the
previous results focused on stability problems. In fact, the convergence of the
solution has an important function in the development of qualitative theory, and
the rapid convergence of solutions is also very important in practical applications.
Generalized quasilinearization is an efficient method for constructing approximate solutions of nonlinear problems. Bellman and Kalaba [2], Lakshmikantham
and Vatsala [10] gave a systematic development of the method to ordinary differential equations. Up till now, only some rapid convergence results have been
found on ordinary differential equations, for initial value problems [3, 11, 13, 14],
for boundary value problem [4, 15, 21]. The applications of the method of quasilinearization in singular differential systems are rare [1, 7, 8, 16]. For example, in
[1], the authors investigated the uniform and quadratic convergence of singular differential systems. We do not find any results on the rapid convergence of solutions
for singular systems.
In this paper, we discuss the existence and uniqueness of solutions and give rapid
convergence result of solutions for singular differential systems by using the method
of generalized quasilinearization and the order relation of lower and upper solutions.
2010 Mathematics Subject Classification. 34A09, 34A45.
Key words and phrases. Singular system; generalized quasilinearization; rapid convergence.
c
2015
Texas State University - San Marcos.
Submitted February 12, 2015. Published August 10, 2015.
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P. WANG, X. LIU
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2. Preliminaries and Lemmas
We consider the following initial value problem for singular differential system
Ax0 = f (t, x),
t ∈ J,
(2.1)
x(0) = x0 .
where A is a singular n × n matrix, x ∈ Rn , f ∈ C(J × Rn , Rn ), J = [0, a], a > 0
is a fixed constant.
Definition 2.1. The function α0 ∈ C 1 (J, Rn ) is called a lower solution of (2.1), if
the following inequalities are satisfied:
Aα00 ≤ f (t, α0 ),
t ∈ J,
(2.2)
α0 (0) ≤ x0 .
Definition 2.2. The function β0 ∈ C 1 (J, Rn ) is called an upper solution of IVP
(2.1), if the following inequalities are satisfied:
Aβ00 ≥ f (t, β0 ),
t ∈ J,
(2.3)
β0 (0) ≥ x0 .
In our further investigations we will need some results on linear singular differential inequalities and linear singular differential systems. Consider the singular
differential inequality
Ax0 + M (t)x ≤ 0,
x(0) ≤ 0,
t ∈ J,
(2.4)
where A, M (t) are n × n matrices, A is singular and M (t) is continuous for t ∈ J.
Lemma 2.3 ([1]). Assume that
(A1) There exists a constant λ such that, L(t) = [λA + M (t)]−1 exists and  =
AL(t) is a constant matrix.
(A2) There exists a nonsingular matrix T such that T −1 , (LT )−1 exist and T −1 ,
(LT ), (LT )−1 ≥ 0, satisfying
C 0
I − λC 0
T −1 ÂT =
, T −1 [I − λÂ]T = 1
,
0 0
0
I2
where C is a diagonal matrix with C −1 ≥ 0.
Then x(0) ≤ 0 implies x(t) ≤ 0 for t ∈ J.
For the singular linear initial value problem
Ax0 + M (t)x = g(t),
x(0) = y0 ,
(2.5)
we have the following result.
Lemma 2.4 ([6]). Assume that condition (A1) holds, index(A) = 1 and
(A3) y0 satisfies (I − ÂÂD )(y0 − w(0)) = 0, where w(t) = M̂ D g(t), M̂ =
M (t)L(t).
Then the unique solution y(t) of
Ây 0 + M̂ y = g(t),
y(0) = y0 .
(2.6)
is given by
y(t) = e
−ÂD M̂ t
D
ÂÂ y0 + e
−ÂD M̂ t
Z
0
t
D
eÂ
M̂ s
ÂD g(s)ds + (I − ÂÂD )M̂ D g(t),
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RAPID CONVERGENCE OF APPROXIMATE SOLUTIONS
3
where the notation ÂD , M̂ D mean the Drazin inverse of the matrices Â, M̂ respectively. Furthermore, we note that x(t) = L(t)y(t), then we obtain the solution x(t)
of (2.5).
To prove our main result, we need the following comparison result.
Lemma 2.5. Assume that the conditions (A1), (A2) hold, and
(A4) The functions α0 , β0 ∈ C 1 (J, Rn ) are lower and upper solutions of (2.1),
fx exists and are continuous.
Then α0 (0) ≤ β0 (0) implies that α0 (t) ≤ β0 (t) on J.
Proof. It can be noted from the condition (A4) that
Aα00 − Aβ00 ≤ f (t, α0 ) − f (t, β0 )
Z 1
=
fx (t, σα0 + (1 − σ)β0 )dσ (α0 − β0 ).
0
Taking
M (t) = −
Z
1
fx (t, σα0 + (1 − σ)β0 )dσ ,
0
we have
A(α0 − β0 )0 + M (t)(α0 − β0 ) ≤ 0.
Noting that α0 (0) ≤ β0 (0), therefore, by Lemma 2.3, we have α0 (t) ≤ β0 (t) on
J.
The following existence result is needed for our main result. For convenience we
define the set
S(α0 , β0 ) = {u ∈ C(J, Rn ) : α0 (t) ≤ u(t) ≤ β0 (t), t ∈ J}.
Lemma 2.6. Assume that the conditions (A1)–(A3) hold, and
(A5) The functions α0 , β0 ∈ C 1 (J, Rn ) are lower and upper solutions of (2.1)
with α0 ≤ β0 on J.
(A6) The function f ∈ C(S(α0 , β0 ), Rn ) satisfies the inequality
f (t, x) − f (t, y) ≥ −M0 (x − y)
for x ≥ y, M (t0 ) = M0 , and t0 ∈ J.
Then (2.1) has a solution x(t) that satisfies α0 (t) ≤ x(t) ≤ β0 (t) on J.
Proof. Let αn+1 and βn+1 be the solutions of the singular linear systems
0
Aαn+1
= f (t, αn ) − M0 (αn+1 − αn ),
t ∈ J,
αn+1 (0) = x0 ,
and
0
Aβn+1
= f (t, βn ) − M0 (βn+1 − βn ),
βn+1 (0) = x0 ,
t ∈ J,
(2.7)
(2.8)
where αn+1 and βn+1 exist because of Lemma 2.4. According to the iterative
schemes (2.7) and (2.8), we obtain the sequences {αn (t)} and {βn (t)} which were
generated by the initial conditions α0 (t) and β0 (t), respectively.
Let n = 0, we first show that α0 (t) ≤ α1 (t) ≤ β1 (t) ≤ β0 (t) on J.
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P. WANG, X. LIU
EJDE-2015/203
For this purpose, setting p(t) = α0 (t) − α1 (t). Using the condition (A5), we
obtain
Ap0 = A(α0 − α1 )0
≤ f (t, α0 ) − [f (t, α0 ) − M0 (α1 − α0 )]
= −M0 p,
t ∈ J.
Noting that p(0) ≤ 0, by Lemma 2.3, we have p(t) ≤ 0, that is, α0 (t) ≤ α1 (t) on J.
similarly, letting p(t) = β1 (t) − β0 (t), we can show that β1 (t) ≤ β0 (t) on J.
To prove α1 (t) ≤ β1 (t). Setting p(t) = α1 (t) − β1 (t) so that p(0) = 0. From the
condition (A6), we have
Ap0 = A(α1 − β1 )0
= f (t, α0 ) − M0 (α1 − α0 ) − [f (t, β0 ) − M0 (β1 − β0 )]
≤ M0 (β0 − α0 ) − M0 (α1 − α0 ) + M0 (β1 − β0 )
= −M0 p,
t ∈ J.
As before, this implies that α1 (t) ≤ β1 (t) on J. Thus, we conclude that
α0 (t) ≤ α1 (t) ≤ β1 (t) ≤ β0 (t), t ∈ J.
The process can be continued successively to obtain that
α0 (t) ≤ α1 (t) ≤ . . . αn (t) ≤ βn (t) ≤ · · · ≤ β1 (t) ≤ β0 (t),
t ∈ J.
It is easy to see that the sequence {αn (t)} is uniformly bounded and equicontinuous,
employing the Ascoli-Arzela Theorem, the nondecreasing sequence {αn (t)} has a
pointwise limit x(t) that satisfies α0 (t) ≤ x(t) ≤ β0 (t). A passage to the limit based
on the Dominated Convergence Theorem shows that x(t) is a solution of
Ax0 = f (t, x) − M0 (x − x),
t ∈ J,
x(0) = x0 ;
that is, x(t) is a solution of (2.1). Thus, we conclude that there exists a solution
x(t) of IVP (2.1) satisfies α0 (t) ≤ x(t) ≤ β0 (t) on J. The proof is complete.
3. Main results
In this section we prove the convergence of the sequence of successive approximations is of order k ≥ 2.
Theorem 3.1. Assume that the conditions (A1)–(A3), (A6) hold, and
(A7) The functions α0 , β0 ∈ C 1 (J, Rn ) are lower and upper solutions of (2.1)
with α0 ≤ β0 on J.
i
(A8) The Frechet derivatives ∂ f∂x(t,x)
(i = 0, 1, 2, . . . , k) exist and are continuous
i
satisfying f (t, x) + M xk is (k − 1)-hyperconvex and f (t, x) − N xk is (k − 1)k
k
xk )
xk )
hyperconcave in x; that is, ∂ (f (t,x)+M
≥ 0 and ∂ (f (t,x)−N
≤ 0,
∂xk
∂xk
where the n × n matrices M , N > 0, k ≥ 1.
Then there exist monotone sequences {αn (t)}, {βn (t)} which convergence is of order
k, that is, there exist constant matrices λ1 and µ1 > 0 such that for the solution
x(t) of (2.1) in S(α0 , β0 ), the inequalities
max |x(t) − αn+1 (t)| ≤ λ1 max |x(t) − αn (t)|k ,
t∈J
t∈J
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RAPID CONVERGENCE OF APPROXIMATE SOLUTIONS
5
max |βn+1 (t) − x(t)| ≤ µ1 max |βn (t) − x(t)|k
t∈J
t∈J
hold, where
max |u(t)| = (max |u1 (t)|, . . . , max |un (t)|)T ,
t∈J
k
k
t∈J
t∈J
k T
|u| = (|u1 | , . . . , |un | ) for any function u ∈ C(J, Rn ).
Proof. Firstly, for t ∈ J, applying Taylor mean value theorem yields
f (t, x) =
k−1
X
∂ i f (t, y) (x − y)i
∂xi
i!
i=0
Z 1
∂ k f (t, σx + (1 − σ)y) (x − y)k
dσ
,
+
(1 − σ)k−1
∂xk
(k − 1)!
0
(3.1)
0
with α0 ≤ x, y ≤ β0 , ∂∂xf0 = f . Furthermore, for α0 ≤ y ≤ x ≤ β0 , in view of the
condition (A8), we have
f (t, x) ≥
k−1
X
i=0
∂ i f (t, y) (x − y)i
− M (x − y)k ≡ F (t, x, y),
∂xi
i!
(3.2)
and F (t, x, x) = f (t, x), where xi = (xi1 , xi2 , . . . , xin )T , x ∈ Rn , i = 0, 1, 2, . . . , k.
Similarly, for a giving t ∈ J, α0 ≤ x ≤ y ≤ β0 , we obtain
(P
k−1 ∂ i f (t,y) (x−y)i
− M (x − y)k , k = 2n + 1.
i=0
∂xi
i!
f (t, x) ≤ Pk−1
i
i
∂ f (t,y) (x−y)
(3.3)
+ N (x − y)k , k = 2n.
i=0
∂xi
i!
≡ G(t, x, y)
and G(t, x, x) = f (t, x).
Consider the singular differential system
Ax0 =
k−1
X
i=0
∂ i f (t, α0 ) (x − α0 )i
− M (x − α0 )k ≡ F (t, x, α0 ),
∂xi
i!
t ∈ J,
(3.4)
x(0) = x0 .
We will show that α0 and β0 are lower and upper solutions of (3.4) respectively.
The condition (A7) and the inequality (3.2) imply
Aα00 ≤ f (t, α0 ) = F (t, α0 , α0 ),
t ∈ J,
α0 (0) ≤ x0 ,
and
Aβ00 ≥ f (t, β0 ) ≥ F (t, β0 , α0 ),
t ∈ J,
β0 (0) ≥ x0 .
Hence by Lemma 2.6, there exists a solution α1 (t) of (3.4) such that α0 (t) ≤
α1 (t) ≤ β0 (t) on J. Furthermore, we can prove that α1 (t) is the unique solution of
(3.4). For this purpose, we assume x1 (t) and x2 (t) are two solutions of (3.4) and
α0 ≤ x2 ≤ x1 ≤ β0 holds. Then, from (3.4), and noting that x1 (0) − x2 (0) = 0, and
Ai − B i = (A − B)
i−1
X
j=0
Ai−1−j B j ,
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P. WANG, X. LIU
EJDE-2015/203
we have
Ax01 − Ax02 = F (t, x1 , α0 ) − F (t, x2 , α0 )
=
k−1
X
i=0
−
∂ i f (t, α0 ) (x1 − α0 )i
− M (x1 − α0 )k
∂xi
i!
k−1
X
i=0
∂ i f (t, α0 ) (x2 − α0 )i
+ M (x2 − α0 )k
∂xi
i!
i−1
n k−1
o
X ∂ i f (t, α0 ) 1 X
i−1−j
j
=
(x
−
α
)
(x1 − x2 )
(x
−
α
)
1
0
2
0
∂xi
i! j=0
i=1
−M
k−1
X
(x1 − α0 )k−1−j (x2 − α0 )j (x1 − x2 )
j=0
i−1
n k−1
X ∂ i f (t, α0 ) 1 X
(x1 − α0 )i−1−j (x2 − α0 )j (x1 − x2 )
≤
i
∂x
i! j=0
i=1
≤ L1 (x1 − x2 ),
where
k−1
X
i=1
i−1
∂ i f (t, α0 ) 1 X
(x1 − α0 )i−1−j (x2 − α0 )j ≤ L1 ,
∂xi
i! j=0
M (t̄) = −L1 , t̄ ∈ J. Using Lemma 2.3, we can get x1 (t) ≤ x2 (t). Then we have
x1 (t) ≡ x2 (t), that is, α1 (t) is the unique solution of (3.4).
Furthermore, we consider the singular differential system
(P
k−1 ∂ i f (t,β0 ) (y−β0 )i
− M (y − β0 )k , k = 2n + 1,
0
i=0
∂xi
i!
Ay = Pk−1
i
i
∂ f (t,β0 ) (y−β0 )
+ N (y − β0 )k , k = 2n,
i=0
∂xi
i!
(3.5)
≡ G(t, y, β0 ), t ∈ J,
y(0) = x0 .
Now we show that α0 and β0 are lower and upper solutions of (3.5) respectively.
For this purpose, the condition (A7) and the inequality (3.3) yield
Aα00 ≤ f (t, α0 ) ≤ G(t, α0 , β0 ),
t ∈ J,
α0 (0) ≤ x0 ,
and
Aβ00 ≥ f (t, β0 ) = G(t, β0 , β0 ),
t ∈ J,
β0 (0) ≥ x0 .
Thus from Lemma 2.6, we see that there exists a solution β1 (t) of (3.5) such
that α0 (t) ≤ β1 (t) ≤ β0 (t) on J. Similarly, we can prove that β1 (t) is the unique
solution of (3.5). To do this, we talk about two cases. If k = 2n + 1, the uniqueness
of the solution of (3.5) is the same as (3.4). If k = 2n, taking two solutions x1 , x2
of (3.5) such that α0 ≤ x2 ≤ x1 ≤ β0 . Then, we obtain
Ax01 − Ax02 = G(t, x1 , β0 ) − G(t, x2 , β0 )
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RAPID CONVERGENCE OF APPROXIMATE SOLUTIONS
=
7
i−1
n k−1
o
X ∂ i f (t, β0 ) 1 X
i−1−j
j
(x1 − x2 )
(x
−
β
)
(x
−
β
)
1
0
2
0
∂xi
i! j=0
i=1
+N
k−1
X
(x1 − β0 )k−1−j (x2 − β0 )j (x1 − x2 )
j=0
≤
n k−1
X
i=1
i−1
o
∂ i f (t, β0 ) 1 X
(x1 − β0 )i−1−j (x2 − β0 )j (x1 − x2 )
i
∂x
i! j=0
≤ L1 (x1 − x2 ).
Noting that x1 (0) − x2 (0) = 0, by Lemma 2.3, we obtain x1 (t) ≤ x2 (t). Thus, (3.5)
has a unique solution.
Next, we prove that α1 (t) ≤ β1 (t) on J. From (3.2), we obtain
Aα10 = F (t, α1 , α0 ) ≤ f (t, α1 ),
t ∈ J,
α1 (0) = x0 .
Proceeding as before, one can obtain that β1 (t) is an upper solution of (2.1). It
then follows from Lemma 2.5 that α1 (t) ≤ β1 (t) on J. Consequently,
α0 (t) ≤ α1 (t) ≤ β1 (t) ≤ β0 (t),
t ∈ J.
Continuing this process by induction, we obtain two monotone sequences {αn (t)}
and {βn (t)} satisfying
α0 (t) ≤ α1 (t) ≤ · · · ≤ αn (t) ≤ βn (t) ≤ · · · ≤ β1 (t) ≤ β0 (t),
t ∈ J.
Let αn , βn be lower and upper solutions of (2.1) respectively with αn ≤ βn on J.
Then we consider the singular differential system
Ax0 =
k−1
X
i=0
∂ i f (t, αn ) (x − αn )i
− M (x − αn )k ≡ F (t, x, αn ),
∂xi
i!
t ∈ J,
(3.6)
x(0) = x0 .
In this case, we can show easily that αn and βn are lower and upper solutions of
(3.6). Therefore, by Lemma 2.6, there exists a solution αn+1 (t) of (3.6) such that
αn (t) ≤ αn+1 (t) ≤ βn (t) on J. The uniqueness of αn+1 (t) is analogous to α1 (t), we
omit the details.
Next, consider the singular differential system
(P
k−1 ∂ i f (t,βn ) (y−βn )i
− M (y − βn )k , k = 2n + 1,
0
i=0
∂xi
i!
Ay = Pk−1
i
i
∂ f (t,βn ) (y−βn )
+ N (y − βn )k , k = 2n,
i=0
∂xi
i!
(3.7)
≡ G(t, y, βn ), t ∈ J,
y(0) = x0 .
Similarly, we can show that αn and βn are lower and upper solutions of (3.7),
respectively. Consequently, by Lemma 2.6, we obtain a solution βn+1 (t) of (3.7)
exists such that αn (t) ≤ βn+1 (t) ≤ βn (t) on J. Furthermore, we can show that
αn+1 (t) ≤ βn+1 (t) on J. By induction, we have that for all n,
α0 (t) ≤ α1 (t) ≤ · · · ≤ αn (t) ≤ βn (t) ≤ · · · ≤ β1 (t) ≤ β0 (t), t ∈ J.
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P. WANG, X. LIU
EJDE-2015/203
Employing the Ascoli-Arzela Theorem, it can be shown that they have pointwise
limits ρ(t) and r(t). Taking the limit as n → ∞, we obtain
lim αn (t) = ρ(t) ≤ r(t) = lim βn (t).
n→∞
n→∞
We can show easily that ρ(t) and r(t) are solutions of (2.1).
Finally, we show that the order of convergence is k ≥ 2. For that, let x(t) be a
solution of (2.1) in S(α0 , β0 ), we define
en (t) = x(t) − αn (t), an (t) = αn+1 (t) − αn (t), t ∈ J,
so that, en ≥ 0, an ≥ 0, en (0) = 0, an (0) = 0. From the equality (3.1), we have
Ax0 =
k−1
X
∂ i f (t, αn ) (x − αn )i
∂xi
i!
i=0
Z 1
∂ k f (t, σx + (1 − σ)αn ) (x − αn )k
+
(1 − σ)k−1
dσ
.
∂xk
(k − 1)!
0
On the other hand, by (3.6), we have
0
Aαn+1
=
k−1
X
i=0
∂ i f (t, αn ) (αn+1 − αn )i
− M (αn+1 − αn )k .
∂xi
i!
Therefore,
Ae0n+1 =
k−1
X
∂ i f (t, αn ) (x − αn )i
∂xi
i!
i=0
Z
1
∂ k f (t, σx + (1 − σ)αn ) (x − αn )k
+
(1 − σ)k−1
dσ
∂xk
(k − 1)!
0
−
k−1
X
i=0
≤
k−1
X
i=1
∂ i f (t, αn ) (αn+1 − αn )i
+ M (αn+1 − αn )k
∂xi
i!
∂ i f (t, αn ) (ein − ain )
+ N ekn + M akn
∂xi
i!
i−1
n k−1
o
X ∂ i f (t, αn ) 1 X
i−1−j j
k
≤
e
a
n
n en+1 + cen
i
∂x
i!
i=1
j=0
≤ L1 en+1 + cekn ,
where c = N + M , an ≤ en . Furthermore, we have
Ae0n+1 ≤ −M (t̄)en+1 + cekn .
Lemma 2.3 implies en+1 (t) ≤ x(t) on J, where x(t) is the solution of
Ax0 + M (t̄)x = cekn , x(0) = 0.
Thus, using the expression of x(t) in Lemma 2.4, we obtain
Z t
D
−1 −ÂD M̂ t
x(t) = [λA + M (t̄)] e
e M̂ s ÂD cekn (s)ds + (I − ÂÂD )M̂ D cekn (t) ,
0
we arrive at after taking suitable estimates
max |x(t) − αn+1 (t)| ≤ λ1 max |x(t) − αn (t)|k ,
t∈J
t∈J
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RAPID CONVERGENCE OF APPROXIMATE SOLUTIONS
9
where λ1 is an appropriate positive matrix.
Similarly, we define
gn (t) = x(t) − βn (t), bn (t) = βn+1 (t) − βn (t), t ∈ J,
so that, en (t) ≤ 0, bn (t) ≤ 0, gn (0) = 0, bn (0) = 0. In view of (3.1), we have
0
Ax =
k−1
X
∂ i f (t, βn ) (x − βn )i
∂xi
i!
i=0
Z
1
∂ k f (t, σx + (1 − σ)βn ) (x − βn )k
+
(1 − σ)k−1
dσ
.
∂xk
(k − 1)!
0
Furthermore, by (3.7), we obtain
(P
k−1 ∂ i f (t,βn ) (βn+1 −βn )i
− M (βn+1 − βn )k ,
0
i=0
∂xi
i!
Aβn+1 = Pk−1
∂ i f (t,βn ) (βn+1 −βn )i
+ N (βn+1 − βn )k ,
i=0
∂xi
i!
k = 2n + 1,
k = 2n.
Hence, if k = 2n + 1, we obtain
0
− Agn+1
0
= Aβn+1
− Ax0
=
≤
k−1
X ∂ i f (t, βn ) (x − βn )i
∂ i f (t, βn ) (βn+1 − βn )i
k
−
M
(β
−
β
)
−
n+1
n
∂xi
i!
∂xi
i!
i=0
i=0
Z 1
k
k
∂ f (t, σx + (1 − σ)βn )
(x − βn )
−
(1 − σ)k−1
dσ
k
∂x
(k − 1)!
0
k−1
X
k−1
X
i=1
∂ i f (t, βn ) bin − gni
− M bkn + N (−gn )k
∂xi
i!
i−1
n k−1
o
X ∂ i f (t, βn ) 1 X
i−1−j j
b
g
≤
(−gn+1 ) + (M + N )(−gn )k
n
n
i
∂x
i!
i=1
j=0
≤ L1 (−gn+1 ) + (M + N )(−gn )k .
Furthermore, we obtain
A(−gn+1 )0 ≤ −M (t̄)(−gn+1 ) + (M + N )(−gn )k .
According to Lemma 2.3, we have −gn+1 (t) ≤ x(t) on J, where x(t) is the solution
of
Ax0 + M (t̄)x = (M + N )(−gn )k , x(0) = 0.
Thus, using the expression of x(t) in Lemma 2.4 and taking suitable estimates, we
obtain
max |βn+1 (t) − x(t)| ≤ µ1 max |βn (t) − x(t)|k ,
t∈J
t∈J
where µ1 is a positive matrix.
If k = 2n, we obtain
0
− Agn+1
=
k−1
X
i=0
k−1
X ∂ i f (t, βn ) (x − βn )i
∂ i f (t, βn ) (βn+1 − βn )i
k
+
N
(β
−
β
)
−
n+1
n
∂xi
i!
∂xi
i!
i=0
10
P. WANG, X. LIU
−
Z
1
(1 − σ)k−1
0
≤
k−1
X
i=1
EJDE-2015/203
∂ k f (t, σx + (1 − σ)βn ) (x − βn )k
dσ
∂xk
(k − 1)!
∂ i f (t, βn ) bin − gni
+ N bkn + M gnk
∂xi
i!
i−1
n k−1
o
X ∂ i f (t, βn ) 1 X
i−1−j j
k
≤
b
g
n
n (−gn+1 ) + (M + N )gn
i
∂x
i!
j=0
i=1
≤ L1 (−gn+1 ) + (M + N )gnk .
Then, we obtain
A(−gn+1 )0 ≤ −M (t̄)(−gn+1 ) + (M + N )gnk .
Now applying Lemma 2.3, we have −gn+1 (t) ≤ x(t) on J, where x(t) is the solution
of
Ax0 + M (t̄)x = (M + N )gnk , x(0) = 0.
Analogous to the above discussion, after taking suitable estimates, we obtain
max |βn+1 (t) − x(t)| ≤ µ1 max |βn (t) − x(t)|k .
t∈J
t∈J
The proof is complete.
The following corollaries are immediate results of Theorem 3.1.
Corollary 3.2. Assume that conditions (A1)–(A3), (A6), (A7) hold, and
i
(i = 0, 1, 2, 3) exist and are continuous, also
(A9) The Frechet derivatives ∂ f∂x(t,x)
i
f (t, x) + M x3 is (2)-hyperconvex and f (t, x) − N x3 is (2)-hyperconcave in
x; that is,
∂ 3 (f (t, x) − N x3 )
∂ 3 (f (t, x) + M x3 )
≥ 0,
≤ 0,
3
∂x
∂x3
where the n × n matrices M , N > 0.
Then there exist monotone sequences {αn (t)}, {βn (t)} which converge uniformly to
the solution of (2.1) and the convergence is cubic.
Corollary 3.3. Assume that conditions (A1)–(A3), (A6), (A7) hold, and
i
(A10) The Frechet derivatives ∂ f∂x(t,x)
(i = 0, 1, 2, 3, 4) exist and are continui
ous satisfying f (t, x) + M x4 is (3)-hyperconvex and f (t, x) − N x4 is (3)4
4
x4 )
x4 )
hyperconcave in x, that is, ∂ (f (t,x)+M
≥ 0 and ∂ (f (t,x)−N
≤ 0, where
∂x4
∂x4
the n × n matrices M , N > 0.
Then there exist monotone sequences {αn (t)}, {βn (t)} which converge uniformly to
the solution of (2.1) and the convergence is quartic.
Remark 3.4. If f (t, x) is (k − 1)-hyperconvex or (k − 1)-hyperconcave in x, by the
method of quasilinearization, we also can obtain the convergence of the monotone
sequences is of order k (k is odd or even).
3.1. Acknowledgments. This research is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei
Province of China (A2013201232). The authors would like to thank the reviewers
for their valuable suggestions and comments.
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RAPID CONVERGENCE OF APPROXIMATE SOLUTIONS
11
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Peiguang Wang
College of Electronic and Information Engineering, Hebei University, Baoding 071002,
China
E-mail address: pgwang@hbu.edu.cn
12
P. WANG, X. LIU
EJDE-2015/203
Xiang Liu
College of Mathematics and Information Science, Hebei University, Baoding 071002,
China
E-mail address: 1065265114@qq.com