General Mathematics Vol. 12, No. 4 (2004), 23–28
Smooth dependence on parameters for some
functional-differential equations
Olaru Ion Marian
Abstract
I this paper we study the smooth dependence on parameters for
the following equation:
x0 (t, λ) = f (t, x(t, λ), x(g(t), λ), λ), t ∈ [a, b]
by using theorem of fibre contraction.
2000 Mathematical Subject Classification: 34K10, 47H10
Keywords: Picard operators, weakly Picard operators, fixed points, data
dependence
1
Introduction
The purpose of this paper is to study the following problem:
(1)
x0 (t, λ) = f (t, x(t, λ), x(g(t), λ), λ), t ∈ [a, b]
(2)
x(t, λ) = ϕ(t, λ), t ∈ [a1 , a], λ ∈ J
where
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24
Olaru Ion Marian
(H0 ) a1 ≤ a < b, J ⊂ R, a compact interval.
(H1 ) g ∈ (C[a, b], [a1 , b]), ϕ ∈ C 1 ([a1 , a] × J), f ∈ C 1 ([a, b] × R2n × J).
¯¯
¯¯
¯¯ ∂f
¯¯
(H2 ) Exists Lf > 0 such that ¯¯
(t, u1 , u2 , λ)¯¯ ≤ Lf , for all t ∈ [a, b],
∂ui
Rn
n
ui ∈ R , i = 1, 2, λ ∈ J.
(H3 )2Lf (b − a) < 1
We note X = C([a1 , b] × J).
The problem (1) + (2) is equivalent with

t

 ϕ(a, λ) + R f (s, x(s, λ), x(g(s), λ))ds, t ∈ [a, b]
(3)
x(t, λ) =
a

 ϕ(t, λ),
t ∈ [a1 , a]
We consider the following operator: B : X −→ X,

t

 ϕ(a, λ) + R f (s, x(s, λ), x(g(s), λ))ds, t ∈ [a, b]
B(x)(t, λ) =
a

 ϕ(t, λ)
, t ∈ [a1 , a]
We need the theorem of fiber contractions(see [3]).
Theorem 1.Let (X, d) and (Y, ρ) be two metric space and
A : X × Y → X × Y, A(x, y) = (B(x), C(x, y)).
We suppose that
(i)(Y, ρ) is a complete metric space .
(ii) The operator B : X → X is weakly Picard operator.
(iii)There exists a ∈ [0, 1) such that C(x, ·) is an a-contraction,for all
x ∈ X.
(iv)If (x∗ , y ∗ ) ∈ FA then C(·, y ∗ ) is continuous in x∗ .
Then A is weakly Picard operator.If B is Picard operator then A is a
Picard operator.
Smooth dependence on parameters ...
2
25
Main results
Differentiability with respect the parameters was studied in [2],[4].
For x ∈ C[a1 , b] × J we have:
|x|C =
max
(t,λ)∈[a1 ,b]×J
||f (t, λ)||Rn .
(X, | · |C ) has a structure of Banach space.
Proposition 1. We suppose that the conditions (H0 ), (H1 ), (H2 ), (H3 ), are
satisfied. Then:
i) the problem (1) + (2) has an unique solutions x∗ ∈ X
ii) x∗ (t, ·) ∈ C 1 (J), for all t ∈ [a1 , b1 ].
Proof. From
||B(x)(t, λ) − B(y)(t, λ)||Rn ≤
Zt
||f (s, x(s, λ), x(g(s), λ), λ) − f (s, y(s, λ), y(g(s), λ), λ)||Rn ds ≤
≤
a
Zt
||x(s, λ) − y(s, λ)||Rn + ||x(g(s), λ) − y(g(s), λ)||Rn ds ≤
≤ Lf
a
≤ 2Lf (b − a)||x − y||C
we have that the B is Picard operator.It follow that there exists a unique
solution x∗ (t, λ) ∈ X.
We have

t

 ϕ(a, λ) + R f (s, x∗ (s, λ), x∗ (g(s), λ), λ)ds, t ∈ [a, b]
(4) x∗ (t, λ) =
a

 ϕ(t, λ)
, t ∈ [a1 , a]
26
Olaru Ion Marian
We consider the operator C : X × X → X.

Rt ∂f
∂ϕ


(a,
λ)
+
((s, x(s), λ), x(g(s), λ), λ) · y(s, λ)ds+


∂λ
∂u1

a



Rt

 + ∂f (s, x(s, λ), x(g(s), λ), λ)y(g(s), λ)ds+
∂u2
C(x, y)(t, λ) =
a

t
R ∂f



+
(s, x(s, λ), x(g(s), λ), λ)ds, t ∈ [a, b]


∂λ

a


 ∂ϕ
(t, λ), t ∈ [a1 , a]
∂λ
We show that C(x, ·) : X → X is a contraction.
||C(x, y)(t, λ) − C(x, z)(t, λ)||Rn ≤
Zt
(||y(s, λ) − z(s, λ)||Rn + ||y(g(s), λ) − z(g(s), λ)||Rn ) ds ≤
≤ Lf
a
≤ 2Lf (b − a)||y − z||C
From fiber contractions theorem we have that the operator A : X ×X →
X × X,
A(x, y) = (B(x), C(x, y))
is a Picard operator.
So, the sequences
xn+1 = B(xn ), n ∈ N
yn+1 = C(xn , yn ), n ∈ N
converges uniformly (with respect to t ∈ [a1 , b], λ ∈ J) to (x∗ , y ∗ ) ∈ FA , for
all x0 , y0 ∈ X.
If we take x0 = 0, y0 = ∂x0 = 0, then
∂λ

t

 ϕ(a, λ) + R f (s, 0, 0, λ)ds, t ∈ [a, b]
x1 (t, λ) = B(x0 )(t, λ) =
a

 ϕ(t, λ)
, t ∈ [a1 , a]
Smooth dependence on parameters ...
y1 (t, λ) = C(x0 , y0 )(t, λ) =

Rt

 ∂ϕ (a, λ) + ∂f (s, 0, 0, λ)ds, t ∈ [a, b]
∂x1
∂λ
∂λ
=
=
(t, λ).
a

∂λ
 ∂ϕ (t, λ), t ∈ [a , a]
1
∂λ
We suppose that yn (t, λ) = ∂xn (t, λ).
∂λ
∂xn+1
We show that yn+1 (t, λ) =
(t, λ).
∂λ
xn+1 (t, λ) = B(xn )(t, λ) =

t

 ϕ(a, λ) + R f (s, xn (s, λ), xn (g(s), λ))ds, t ∈ [a, b]
=
a

 ϕ(t, λ)
, t ∈ [a1 , a]
yn+1 (t, λ) = C(xn , yn )(t, λ) =

Rt ∂f
∂ϕ


(a,
λ)
+
(s, xn (s, λ), xn (g(s), λ), λ)yn (s, λ)ds+


∂λ
∂u1

a



Rt

 + ∂f (s, xn (s, λ), xn (g(s), λ), λ)yn (s, λ)ds+
∂u2
=
=
a

t
R

∂f



 + a ∂λ (s, xn (s, λ), xn (g(s), λ), λ)ds, t ∈ [a, b]



 ∂ϕ
(t, λ)
, t ∈ [a1 , a]
∂λ

Rt ∂f
∂ϕ


(a,
λ)
+
(s, xn (s, λ), xn (g(s), λ), λ) ∂xn (s, λ)ds+


∂λ
∂u
∂λ
1

a


t

R

 + ∂f (s, xn (s, λ), xn (g(s), λ), λ) ∂xn (s, λ)ds+
∂u2
∂λ
=
=
a

t
R

∂f


+
(s, xn (s, λ), xn (g(s), λ), λ)ds, t ∈ [a, b]


∂λ

a


 ∂ϕ
(t, λ), t ∈ [a, a1 ]
∂λ
∂xn+1
(t, λ).
=
∂λ
Thus,
xn −→ x∗ as n → ∞,
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Olaru Ion Marian
∂xn
−→ y ∗ as n → ∞.
∂λ
∗
∗
These imply that there exists ∂x and ∂x = y ∗ .
∂λ
∂λ
References
[1] L. Ladeira,J. K. Hale, Differentiability with respect to delays, J. Differential equations, 92, 1991,14-26.
[2] I. A. Rus, Functional-Differential equation of mixed point, via weakly
Picard operator Seminar on mixed point theory Cluj-Napoca, vol. 3,
2002,335-345.
[3] I. A. Rus, Weakly Picard operators and applications, Seminar on fixed
point theory Cluj-Napoca, vol. 2, 2001,41-57.
[4] J.Sotomayor,Smooth dependence of solutions of differential equations
on initial data:a simple proof, Bul. Soc. Bras. Mat., 1973,55-59
“Lucian Blaga” University of Sibiu
Faculty of Sciences
Department of Mathematics
Str. Dr. I. Raţiu, no. 5-7
550012 - Sibiu, Romania
E-mail: olaruim@yahoo.com