Electronic Journal of Differential Equations, Vol. 2002(2002), No. 90, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Generalized quasilinearization method for a
second order three point boundary-value
problem with nonlinear boundary conditions
∗
Bashir Ahmad, Rahmat Ali Khan, & Paul W. Eloe
Abstract
The generalized quasilinearization technique is applied to obtain a
monotone sequence of iterates converging uniformly and quadratically to
a solution of three point boundary value problem for second order differential equations with nonlinear boundary conditions. Also, we improve
the convergence of the sequence of iterates by establishing a convergence
of order k.
1
Introduction
The method of quasilinearization pioneered by Bellman and Kalaba [1] and
generalized by Lakshmikantham [8, 9] has been applied to a variety of problems
[2, 10, 11, 12, 13, 16].
Multipoint boundary value problems for second order differential equations
have also been receiving considerable attention recently. Kiguradze and Lomtatidze [7] and Lomtatidze [14, 15] have studied closely related problems. Gupta
et.al. [4, 5, 6] have studied problems related to three point boundary value
problems. More recently, Paul Eloe and Yang Gao [3] discussed the method of
quasilinearization for a three point boundary value problem. In this paper, we
develop the method of generalized quasilinearization for a three point boundary
value problem involving nonlinear boundary conditions and obtain a monotone
sequence of approximate solutions converging uniformly and quadratically to a
solution of the problem. Also, we have discussed the convergence of order k.
∗ Mathematics Subject Classifications: 34B10, 34B15.
Key words: Generalized quasilinearization, boundary value problem,
nonlinear boundary conditions, quadratic and rapid convergence.
c
2002
Southwest Texas State University.
Submitted March 14, 2002. Published October 22, 2002.
1
2
Generalized quasilinearization method
EJDE–2002/90
Basic Results
Consider the three point boundary value problem with nonlinear boundary conditions
x00 = f (t, x(t)), t ∈ [0, 1] = J
(1.1)
1
x(0) = a, x(1) = g(x( )),
2
where f ∈ C[J × R, R] and g : R → R is continuous. Let G(t, s) denote the
Green’s function for the conjugate or Dirichlet boundary value problem and is
given by
(
t(s − 1), 0 ≤ t < s ≤ 1
G(t, s) =
s(t − 1), 0 ≤ s < t ≤ 1.
We note that G(t, s) < 0 on (0, 1) × (0, 1). If x(t) is the solution of (1.1) and
(1.2), then
1
x(t) = a(1 − t) + g(x( ))t +
2
1
Z
G(t, s)f (s, x(s))ds.
(1.2)
0
Let α, β ∈ C 2 [0, 1]. We say that α is a lower solution of the BVP (1.1), if
α00 ≥ f (t, α),
α(0) ≤ a,
t ∈ [0, 1]
1
α(1) ≤ g(α( )),
2
and β be an upper solution of the BVP (1.1), if
β 00 ≤ f (t, β),
β(0) ≥ a,
t ∈ [0, 1]
1
β(1) ≥ g(β( )).
2
Now, we state the following theorems without proof [3].
Theorem 1.1 Assume that f is continuous with fx > 0 on [0, 1] × R and g is
continuous with 0 ≤ g 0 < 1 on R. Let β and α be the upper and lower solutions
of the BVP (1.1) respectively. Then α(t) ≤ β(t), t ∈ [0, 1].
Theorem 1.2 (Method of upper and lower solutions) Assume that f is
continuous on [0, 1]×R and g is continuous on R satisfying 0 ≤ g 0 < 1. Further,
we assume that there exists an upper solution β and a lower solution α of the
BVP (1.1) such that α(t) ≤ β(t), t ∈ [0, 1]. Then there exists a solution x of
the BVP (1.1) such that
α(t) ≤ x ≤ β(t),
t ∈ [0, 1].
EJDE–2002/90
2
Bashir Ahmad, Rahmat Ali Khan, & Paul W. Eloe
3
Main Result
Theorem 2.1 (Generalized quasilinearization method)
(A1 ) f, fx are continuous on [0, 1] × R and fxx exists on [0, 1] × R. Further,
fx > 0 and fxx + φxx ≤ 0, where φ, φx are continuous on [0, 1] × R and
φxx ≤ 0.
(A2 ) g, g 0 are continuous on R and g 00 exists and 0 ≤ g 0 < 1, g 00 (x) ≥ 0, x ∈ R.
(A3 ) α and β are lower and upper solutions of the BVP (1.1) respectively.
Then there exists a monotone sequence {wn } of solutions converging quadratically to the unique solution x of the BVP (1.1).
Proof.
Define F : [0, 1] × R → R as
F (t, x) = f (t, x) + φ(t, x).
Then, in view of (A1 ), we note that F , Fx are continuous on [0, 1] × R, and Fxx
exists such that
Fxx (t, x) ≤ 0.
(2.1)
Using the mean value theorem and the assumptions (A1 ) and (A2 ), we obtain
f (t, x) ≤ F (t, y) + Fx (t, y)(x − y) − φ(t, x),
g(x) ≥ g(y) + g 0 (y)(x − y),
(2.2)
(2.3)
where x, y ∈ R such that x ≥ y and t ∈ [0, 1]. Here, we remark that (2.2) and
(2.3) are also valid independent of the requirement x ≥ y. Define the functions
∗
F (t, x, y) and h(x, y) as
∗
F (t, x, y) = F (t, y) + Fx (t, y)(x − y) − φ(t, x),
h(x, y) = g(y) + g 0 (y)(x − y).
We observe that
f (t, x) = min F ∗ (t, x, y).
(2.4)
F x (t, x, y) =Fx (t, y) − φx (t, x) ≥ Fx (t, x) − φx (t, x)
=fx (t, x) > 0,
(2.5)
y
Further
∗
∗
implies that F (t, x, y) is increasing in x for each fixed (t, y) ∈ [0, 1]×R. Similarly
g(x) = max h(x, y),
(2.6)
0 ≤ h0 (x, y) < 1.
(2.7)
y
4
Generalized quasilinearization method
EJDE–2002/90
Now, set α = w0 , and consider the three point BVP
∗
x00 =F (t, x(t), w0 (t)),
x(0) = a,
t ∈ [0, 1] = J
1
1
x(1) = h(x( ), w0 ( )).
2
2
(2.8)
Using (A3 ) together with (2.4) and (2.6), we have
∗
w000 ≥ f (t, w0 ) =F (t, w0 , w0 ), t ∈ [0, 1]
1
1
1
w0 (0) ≤ a, w0 (1) ≤ g(w0 ( )) = h(w0 ( ), w0 ( )),
2
2
2
and
∗
β 00 ≤ f (t, β) ≤F (t, β, w0 ), t ∈ [0, 1]
1
1
1
β(0) ≥ a, β(1) ≥ g(β( )) ≥ h(β( ), w0 ( )),
2
2
2
which imply that w0 and β are lower and upper solutions of the BVP (2.8)
respectively. In view of (2.5) (2.7) and the fact that w0 and β are lower and
upper solutions of the BVP (2.8) respectively, it follows by Theorems 1.1 and
1.2 that there exists a unique solution w1 of the BVP (2.8) such that
w0 (t) ≤ w1 (t) ≤ β(t),
t ∈ [0, 1].
Now, consider the BVP
∗
x00 =F (t, x(t), w1 (t)),
t ∈ [0, 1] = J
1
1
x(0) = a, x(1) = h(x( ), w1 ( )).
2
2
(2.9)
Again, using (A3 ), (2.4) and (2.6), we find that w1 and β are lower and upper
solutions of (2.9) respectively, that is,
∗
∗
w100 =F (t, w1 , w0 ) ≥F (t, w1 , w1 ), t ∈ [0, 1]
1
1
1
1
w1 (0) = a, w1 (1) = h(w1 ( ), w0 ( )) ≤ h(w1 ( ), w1 ( )),
2
2
2
2
and
∗
β 00 ≤ f (t, β) ≤F (t, β, w1 ), t ∈ [0, 1]
1
1
1
β(0) ≥ a, β(1) ≥ g(β( )) ≥ h(β( ), w1 ( )).
2
2
2
Hence, by Theorems 1.1 and 1.2, there exists a unique solution w2 of (2.9) such
that
w1 (t) ≤ w2 (t) ≤ β(t), t ∈ [0, 1].
EJDE–2002/90
Bashir Ahmad, Rahmat Ali Khan, & Paul W. Eloe
5
Continuing this process successively, we obtain a monotone sequence {wn } of
solutions satisfying
w0 (t) ≤ w1 (t) ≤ · · · ≤ wn (t) ≤ β(t),
t ∈ [0, 1],
where each element wn of the sequence is a solution of the BVP
∗
x00 =F (t, x(t), wn−1 (t)),
x(0) = a,
t ∈ [0, 1] = J
1
1
x(1) = h(x( ), wn−1 ( )),
2
2
and
1
1
wn (t) = a(1 − t) + h(wn ( ), wn−1 ( ))t +
2
2
Z
1
∗
G(t, s) F (s, wn , wn−1 )ds. (2.10)
0
Employing the fact that [0, 1] is compact and the monotone convergence is
pointwise, it follows that the convergence of the sequence is uniform. If x(t) is
the limit point of the sequence, then passing onto the limit n → ∞, (2.10) gives
Z 1
∗
1
1
x(t) = a(1 − t) + h(x( ), x( ))t +
G(t, s) F (s, x(s), x(s))ds
2
2
0
Z 1
1
= a(1 − t) + g(x( ))t +
G(t, s)f (s, x(s))ds.
2
0
Thus, x(t) is the solution of the BVP (1.1). Now, we show that the convergence
of the sequence is quadratic. For that, set
en (t) = x(t) − wn (t),
t ∈ [0, 1].
Observe that
en (t) ≥ 0, en (0) = 0,
1
1
1
en (1) = g(x( )) − h(wn ( ), wn−1 ( )).
2
2
2
Using the mean value theorem repeatedly, (A1 ) and the nonincreasing property
of Fx , we have
e00n+1 (t)
00
= x00 (t) − wn+1
(t)
= f (t, x) − [F (t, wn ) + Fx (t, wn )(wn+1 − wn ) − φ(t, wn+1 )]
= Fx (t, c1 )(x − wn ) − Fx (t, wn )(x − wn ) + Fx (t, wn )(x − wn+1 )
−φx (t, c2 )(x − wn+1 )
= (Fxx (t, c3 )(c1 − wn )(x − wn ) + (Fx (t, wn ) − φx (t, c2 ))(x − wn+1 )
≥ Fxx (t, c3 )(x − wn )2 + (Fx (t, c2 ) − φx (t, c2 ))(x − wn+1 )
= Fxx (t, c3 )(en )2 + fx (t, c2 )en+1
≥ Fxx (t, c2 )(en )2 ≥ −M k en k2 ,
6
Generalized quasilinearization method
EJDE–2002/90
where M is a bound on Fxx (t, x) for t ∈ [0, 1], wn < c3 < c1 < x(t), wn+1 <
c2 < x(t), and || · || denotes the supremum norm on C[0, 1]. Thus, we have
Z 1
1
1
1
en+1 (t) = [g(x( )) − h(wn+1 ( ), wn ( ))]t +
G(t, s)e00n+1 (s)ds
2
2
2
0
1
1
1
1
1
≤ [g(x( )) − g(wn ( )) − g 0 (wn ( ))(wn+1 ( ) − wn ( ))]t
2
2
2
2
2
Z 1
+
G(t, s)M ken k2 ds
0
1
1
1
1
1
≤ [g (co )(x( ) − wn ( )) − g 0 (wn ( ))(wn+1 ( ) − wn ( ))]t
2
2
2
2
2
Z 1
+M ken k2
|G(t, s)| ds
0
0
1
1
1
1
[g (c1 )(co − wn ( ))(x( ) − wn ( )) + g 0 (wn ( ))en+1 ]t
2
2
2
2
+M1 ken k2
1
≤ [g 00 (c1 )e2n (t) + g 0 (wn ( ))en+1 ]t + M1 k en k2 ,
2
=
00
where wn ( 12 ) < c1 < co < x( 12 ). Taking the maximum over the interval [0, 1],
we get
ken+1 k ≤ M2 ken k2 + λken+1 k + M1 ken k2 .
Solving algebraically, we get
M3
ken k2 ,
1−λ
R1
where, |g 0 | ≤ λ < 1, M1 provides a bound on M 0 | G(t, s) | ds, M2 provides
a bound for |g 00 | on [wn ( 12 ), x( 12 )] and M3 = M1 + M2 . This establishes the
quadratic convergence.
ken+1 k ≤
3
Rapid Convergence
Theorem 3.1 Assume that
(B1 )
∂i
∂xi f (t, x)
(i = 0, 1, 2, . . . k) are continuous on [0, 1] × R satisfying
∂i
f (t, x) ≥ 0, (i = 0, 1, 2, . . . k − 1)
∂xi
∂k
(f (t, x) + φ(t, x)) ≤ 0,
∂xk
i
∂
where ∂x
i φ(t, x) (i = 0, 1, 2, . . . k) are continuous and
some function φ(t, x).
∂k
φ(t, x)
∂xk
< 0 for
EJDE–2002/90
Bashir Ahmad, Rahmat Ali Khan, & Paul W. Eloe
7
(B2 ) α, β ∈ C 2 [J, R] are lower and upper solutions of the BVP (1.1).
(B3 )
di
dxi g(x)
(i = 0, 1, 2, . . . k) are continuous on R satisfying
di
M
g(x) <
,
dxi
(β − α)i−1
0≤
with 0 < M <
1
3
dk
g(x)
dxk
and
≥ 0.
Then there exists a monotone sequence of solutions {wn } that converge to the
unique solution, x, of the BVP (1.1) with the order of convergence k ≥ 2.
Proof.
Define F : [0, 1] × R → R as
F (t, x) = f (t, x) + φ(t, x).
Using (B1 ), (B3 ) and the generalized mean value theorem, we obtain
f (t, x) ≤
k−1
X
i=0
∂i
(x − y)i
F
(t,
y)
− φ(t, x),
∂xi
i!
g(x) ≥
k−1
X
i=0
di
(x − y)i
g(y)
.
i
dx
i!
Define
∗∗
F (t, x, y) =
k−1
X
i=0
∂i
(x − y)i
F
(t,
y)
− φ(t, x),
∂xi
i!
(3.1)
and
∗
h (x, y) =
k−1
X
i=0
(x − y)i
di
g(y)
.
dxi
i!
(3.2)
∗
∗∗
Observe that F (t, x, y) and h (x, y) are continuous and further
∗∗
f (t, x) = min F (t, x, y),
y
∗
g(x) = max h (x, y).
y
(3.3)
(3.4)
Using generalized mean value theorem, (3.1) can be written as
∗∗
F (t, x, y) =
k−1
X
i=0
∂i
(x − y)i
∂k
(x − y)k
f
(t,
y)
−
φ(t,
ξ)
.
∂xi
i!
∂xk
k!
(3.5)
Differentiating (3.5) and using (B1 ), we get
∗∗
F x (t, x, y) >
k−1
X
i=0
∂i
(x − y)i−1
f
(t,
y)
≥ 0,
∂xi
(i − 1)!
(3.6)
8
Generalized quasilinearization method
EJDE–2002/90
∗∗
which implies that F (t, x, y) is increasing in x for each (t, y) ∈ [0, 1] × R.
Similarly, differentiation of (3.2), in view of (B3 ), yields
∗
0
h (x, y) =
k−1
X
i=0
(x − y)i−1
di
g(y)
.
dxi
(i − 1)!
∗
Clearly h 0 (x, y) ≥ 0 and
∗
0
h (x, y)
=
k−1
X
≤
k−1
X
i=0
i=0
k−1
X di
(x − y)i−1
(β − α)i−1
di
g(y)
≤
g(y)
i
i
dx
(i − 1)!
dx
(i − 1)!
i=0
k−2
X 1
M
1
< M (1 +
) = M (3 − k−3 )
k−1
(i − 1)!
2
2
i=0
< 3M < 1.
Now, set α = w0 , and consider the linear BVP
∗∗
x00 =F (t, x(t), w0 (t)),
x(0) = a,
t ∈ [0, 1] = J
∗
1
1
x(1) =h (x( ), w0 ( )).
2
2
(3.7)
Using (B2 ), (3.3) and (3.4), we find that
∗∗
w000 ≥ f (t, w0 ) =F (t, w0 , w0 ), t ∈ [0, 1]
∗
1
1
1
w0 (0) ≤ a, w0 (1) ≤ g(w0 ( )) =h (w0 ( ), w0 ( )),
2
2
2
and
∗∗
β 00 ≤ f (t, β) ≤F (t, β, w0 ), t ∈ [0, 1]
∗
1
1
1
β(0) ≥ a, β(1) ≥ g(β( )) ≥h (β( ), w0 ( )),
2
2
2
imply that w0 and β are lower and upper solutions of the BVP (3.7) respectively.
It follows by Theorems 1.1 and 1.2 that there exists a unique solution w1 of the
BVP (3.7) such that
w0 (t) ≤ w1 (t) ≤ β(t),
t ∈ [0, 1].
Continuing this process successively, we obtain a monotone sequence {wn } of
solutions satisfying
w0 (t) ≤ w1 (t) ≤ w2 (t) ≤ · · · ≤ wn (t) ≤ β(t),
t ∈ [0, 1],
where each element wn of the sequence is a solution of the BVP
∗∗
x00 =F (t, x(t), wn−1 (t)),
x(0) = a,
t ∈ [0, 1] = J
1
1
x(1) =h (x( ), wn−1 ( )),
2
2
∗
EJDE–2002/90
Bashir Ahmad, Rahmat Ali Khan, & Paul W. Eloe
9
and is given by
∗
1
1
wn (t) = a(1 − t)+ h (wn ( ), wn−1 ( ))t +
2
2
Z
1
∗∗
G(t, s) F (s, wn , wn−1 )ds. (3.8)
0
Again, using the standard arguments employed in the last section, it follows
that
Z 1
∗
∗∗
1
1
= a(1 − t)+ h (x( ), x( ))t +
G(t, s) F (s, x(s), x(s))ds
2
2
0
Z 1
1
= a(1 − t) + g(x( ))t +
G(t, s)f (s, x(s))ds.
2
0
x(t)
Hence x(t) is the solution of the BVP (1.1). Now, we show that the convergence
of the sequence of iterates is of order k ≥ 2. For that, we set
en (t) = x(t) − wn (t),
an (t) = wn+1 (t) − wn (t),
t ∈ [0, 1].
Note that en (t) ≥ 0, an (t) ≥ 0, en (t) − an (t) = en+1 (t), and
en (0) = 0,
1
1
1
en (1) = g(x( )) − h(wn ( ), wn−1 ( )).
2
2
2
Also, en (t) ≥ an (t) and hence by induction ekn (t) ≥ akn (t). Using the generalized
mean value theorem, we have
e00n+1 (t)
00
= x00 − wn+1
h k−1
X ∂i
(x − wn )i
∂k
(x − wn )k i
=
f
(t,
w
)
+
f
(t,
ξ)
n
∂xi
i!
∂xk
k!
i=0
h k−1
X ∂i
(wn+1 − wn )i
∂k
(wn+1 − wn )k i
−
f
(t,
w
)
−
φ(t,
ξ)
n
∂xi
i!
∂xk
k!
i=0
=
k−1
X
i=0
≥
∂i
(ei − ain )
∂k
(en )k
∂k
(an )k
f (t, wn ) n
+ k f (t, ξ)
+ k φ(t, ξ)
i
∂x
i!
∂x
k!
∂x
k!
k−1
X
i=0
k
≥
k−1
∂i
1 X j i−1−j ∂k
∂k
(en )k
f (t, wn )
en an
en+1 + ( k f (t, ξ) + k φ(t, ξ))
i
∂x
i! i=0
∂x
∂x
k!
∂
(en )k
F (t, ξ)
≥ −M ken kk ,
k
∂x
k!
(3.9)
10
Generalized quasilinearization method
1 ∂k
k! ∂xk F (t, ξ)
where M is a bound on
have
EJDE–2002/90
for t ∈ [0, 1]. Thus, in view of (3.9), we
en+1 (t)
Z 1
1
1 1
g(x( )) − h(wn+1 ( ), wn ( )) t +
G(t, s)e00n+1 (t)ds
2
2
2
0
Z 1
1
1 1
|G(t, s)| ds
≤ g(x( )) − h(wn+1 ( ), wn ( )) t + M ken kk
2
2
2
0
h k−1
X di
1 (x( 12 ) − wn ( 12 ))i
dk
1 (x( 12 ) − wn ( 12 ))k
g(w
(
))
+
g(ξ(
))
=
n
dxi
2
i!
dxk
2
k!
i=0
=
−
k−1
X
i=0
di
1 (wn+1 ( 12 ) − wn ( 12 ))i i
g(w
(
))
t + M1 ken kk
n
dxi
2
i!
h k−1
X di
1 (ein ( 12 ) − an ( 12 ))i
dk
1 (en ( 12 ))k i
=
g(w
(
))
+
g(ξ(
))
t + M1 ken kk
n
i
k
dx
2
i!
dx
2
k!
i=0
k−1
h k−1
X di
1 1 X j 1 i−1−j 1
1
=
g(wn ( ))
en ( )an
( ))en+1 ( )
i
dx
2
i!
2
2
2
i=0
i=0
dk
1 (en ( 12 ))k i
+ k g(ξ( ))
t + M1 ken kk
(3.10)
dx
2
k!
i−1
h k−1
X
M
1 X i−1−j 1 j 1 i
1
≤
en
( )an ( ) en+1 ( ) + M2 ken kk + M1 ken kk .
i−1
(β − α)
i! j=0
2
2
2
i=1
Letting
Pn (t) =
k−1
X
i=1
i−1
M
1 X i−1−j 1 j 1
e
( )an ( ),
i−1
(β − α)
i! j=0 n
2
2
we observe that
lim Pn (t) = lim
n→∞
n→∞
k−1
X
i=1
i−1
M
1 X i−1−j 1 j 1
1
en
( )an ( ) = M < .
i−1
(β − α)
i! j=0
2
2
3
Therefore, we can choose λ < 1/3 and n0 ∈ N such that for n ≥ n0 , we have
Pn (t) < λ and consequently (3.10) becomes
ken+1 k < λken+1 k + M3 ken kk .
Solving algebraically, we obtain
M3
ken kk ,
1−λ
R1
where M3 = M1 +M2 , M1 provides bound for M 0 |G(t, s)|ds, and M2 provides
bound for
dk
1 1
g(ξ( )) .
k
dx
2 k!
ken+1 k ≤
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11
References
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Bashir Ahmad
Department of Mathematics, King Abdulaziz University,
Jeddah-21589, Saudi Arabia
e-mail:bashir qau@yahoo.com
Rahmat Ali Khan
Department of Mathematics, Quaid-i-Azam University,
Islamabad, Pakistan
e-mail:rahmat alipk@yahoo.com
Paul W. Eloe
Department of Mathematics, University of Dayton,
Dayton, OH 45469-2316, USA
e-mail: Paul.Eloe@notes.udayton.edu