General Mathematics Vol. 14, No. 1 (2006), 31–40
Cg asymptotic equivalence for some
functional equation of type Voltera1
Marian Olaru
and
Vasilica Olaru
Abstract
In this paper , by using the notion of ϕ− contraction, we study
the Cg asymptotic equivalence for the solutions of the equations
x0 (t) = A(t)x(t) and y 0 (t) = A(t)y(t) + f (t, yt ),where f (t, ·) is a
Voltera operator.
2000 Mathematical Subject Classification: 34K10, 47H10
Keywords: Cg asymptotic equivalence ,Voltera operator, ϕ− contraction.
1
Introduction
Let Cg be the Banach space of continuous functions defined on
Rt0 = [t0 , ∞), t0 ≥ 0 which satisfied the condition :
(1)
|u(t)| = O(g(t)), t −→ ∞,
1
Received January 17, 2006
Accepted for publication (in revised form) February 25, 2006
31
32
M. Olaru and V. Olaru
where g is a continuous and positive function defined on Rt0 ,| · | is the
Euclidean norm in Rn .
We define the norm on Cg by relation:
|u(t)|
.
t∈Rt0 g(t)
|u|Cg = sup
(2)
We note by ut the restriction of function u at [t0 , t]. For ut ∈ C([t0 , t], Rn )
we define
(3)
kut k = sup |u(s)|
s∈[t0 ,t]
On [1] it is presented the following lemma:
Lemma 1.1.Let g be a nondecreasing, positive function defined on R+ and
x ∈ Cg . Then:
kxt k
|x|Cg = sup
t ∈ R+ g(t)
Next we consider the equations :
(4)
x0 = A(t)x
(5)
y 0 = A(t)y + f (t, yt ),
where for t ≥ t0 the application ψ −→ f (t, ψ) is an application from
C([t0 , t], Rn ) to Rn that satisfies some conditions that assure the existence
of the equation (5), conditions that are to be explained below .
Definition 1.1.[1] We say that the equations (4) and (5) are Cg -asymptotic
equivalence if for all solution x ∈ Cg of equation (4) corresponding a unique
solution y of equation (5) such that :
(6)
lim
t−→∞
|x(t) − y(t)|
= 0.
g(t)
Cg asymptotic equivalence for some functional equation of type Voltera 33
Through the following definitions we shall further present the notion of
comparison function and ϕ− contraction:
Definition 1.2.[2],[3] ϕ : R+ −→ R+ is a strict comparison function if ϕ
satisfies the following:
i) ϕ is continuous.
ii)ϕ is monotone increasing.
iii) ϕn (t) −→ 0, for all t > 0.
iv) t-ϕ(t) −→ ∞,for t −→ ∞.
Let (X, d) be an metric space and f : X −→ X be an operator.
Definition 1.3.[2],[3] The operator f is called a strict ϕ-contraction if:
(i) ϕ is a strict comparison function.
(ii)d(f (x), f (y)) ≤ ϕ(d(x, y)), for all x, y ∈ X.
We shall make the following hypothesis :
(H) We supose that there exists a comparison function ϕ which satisfies
condition
(7)
ϕ(λ · r) ≤ λ · ϕ(r),
for all r ≥ 0 and λ ≥ 1
An example of such a function is shown in the next figure:
ϕ : R+ −→ R+ , ϕ(t) =
r
r+1
On [2] I.A Rus obtains the following result:
Theorem 1.1.Let (X, d) be a complete metric space and f : X −→ X a
ϕ-contraction.Then f , is a Picard operator.
34
2
M. Olaru and V. Olaru
Main result
Theorem 2.1.Let X(t) be a fundamental matrix of equation (4).We suppose that :
(i) There exists the projectors P1 ,P2 and a constant K > 0 such that
Zt
(8)
Z∞
|X(t)P1 X
t0
−1
q
|X(t)P2 X
(s)| ds +
−1
1q
(s)| ds ≤ K,
q
t
for t ≥ t0 , q > 1;
(ii) The application t −→ f (t, yt ) is continuous for all y ∈ Cg .
(iii) There exists ϕ : R+ → R+ , a comparison function which satisfies the
hypothesis (H) , and λ a continuous,nonnegative function defined on
Rt0 such that
(9)
|f (t, yt ) − f (t, yt )| ≤ λ(t)ϕ(kyt − yt k),
for all t ≥ t0 ,y ∈ Cg
(iv)
1
∞
p
Z
p
(λ(s)g(s)) ds
∈ Cg , p > 1


(10)
t
∞
Z
(11)

t0
1
∞
1
Z
p

p
|f (s, 0)|p ds
< ∞,
(λ(s))p
< ∞, p > 1.



t0
Cg asymptotic equivalence for some functional equation of type Voltera 35
Then for all solution x ∈ Cg of equation (4) there exists a unique solution
y(t) of equation (5).
If we replace the condition (10) with
(12)
∞
1
Z
p
p
(λ(s)g(s)) ds
= o(t), t −→ ∞ ,


t
then the equations (4)and (5) are Cg -asymptotic equivalence.
Proof. The function g being nondecreasing and positive we cant suppose
that g ≥ 1, because Cg = Ckg for all k > 0.(for more details see [1])
On Cg we define the operator T by relation:
Zt
Z∞
X(t)P1 X −1 (s)f (s, ys )ds −
T (y)(t) = x(t) +
t0
X(t)P2 X −1 (s)f (s, ys )ds
t
Let x ∈ Cg be a solution for the equation(4).Then |x(t)| ≤ A · g(t), for
all t ≥ t0 .We prove that T (Cg ) ⊆ Cg .Let be y ∈ Cg . Then
Zt
|X(t)P1 X −1 (s)| · |f (s, ys )|ds+
|T (y)(t)| ≤ |x(t)| +
t0
Z∞
|X(t)P2 X −1 (s)| · |f (s, ys )|ds ≤
+
t
Zt
|X(t)P1 X −1 (s)| · |f (s, ys ) − f (s, 0)|ds+
≤ Ag(t) +
t0
Zt
|X(t)P1 X −1 (s)| · |f (s, 0)|ds+
+
t0
36
M. Olaru and V. Olaru
Z∞
|X(t)P2 X −1 (s)| · |f (s, ys ) − f (s, 0)|ds+
+
t
Z∞
|X(t)P2 X −1 (s)| · |f (s, 0)|ds ≤
+
t
∞
Z
≤ Ag(t) +

1
q
|X(t)P2 X −1 (s)|q ds

∞
Z
·

t
 t
Z
+

t
1
q
|X(t)P1 X −1 (s)|q ds

 t
Z
·

t0
∞
Z
+

+

1
p
|f (s, 0)|p ds
+

t0
1
q
|X(t)P2 X −1 (s)|q ds

∞
1
Z

p
·
(λ(s)ϕ(kys k))p ds
+


t
 t
Z
1
p
|f (s, 0)|p ds
+

t
1
q
|X(t)P1 X −1 (s)|q ds

 t
Z
·
t0

(λ(s))ϕ(kys k)p ds
t0
 t
Z
≤ Ag(t) + K ·

1
p
p
|f (s, 0)| ds
+

t0
∞
Z
+K ·

|f (s, 0)|p ds
1
p

+
t
 t
Z
+K · ϕ(|y|Cg ) ·

t0
(λ(s)g(s))p ds
1
p

+
1
p

≤
Cg asymptotic equivalence for some functional equation of type Voltera 37
∞
1
Z

p
+K · ϕ(|y|Cg ) ·
(λ(s)g(s))p ds
≤


t
≤ M · g(t),
where
1
∞
Z
p

M = A + K · ϕ(|y|Cg ) ·
(λ(s))p ds
+


t0
∞
Z
+K · ϕ(|y|Cg ) · B1 + 2K ·

1
p
.
|f (s, 0)|p ds

t0
Next we consider y, y ∈ Cg . We prove that, the operator T is a ϕ-contraction.
Zt
|X(t)P1 X −1 (s)| · |f (s, ys ) − f (s, y s )|ds+
kT (y)(t) − T (y)(t)k ≤
t0
Z∞
|X(t)P2 X −1 (s)| · |f (s, ys ) − f (s, y s )|ds ≤
+




Z t

t

1  ∞
1 


 p Z
p 
p
p
|f (s, ys ) − f (s, y s )| ds
≤
≤ K·
+
|f (s, ys ) − f (s, y s )| ds




 




t
 t0


1 
1






 Zt

p 


 p Z∞
p
p
≤ K · ϕ(|y − y|Cg )
(λ(s)g(s)) ds
+
≤
(λ(s)g(s)) ds



 





t

 t0




∞
1




Z
p

p
≤K·
+ B1 ϕ(|y − y|Cg ) · g(t).
(λ(s)) ds








 t0

38
M. Olaru and V. Olaru


1


(
)



 R∞
p
p
+ B1 < 1
We choose t0 ≥ 0 such that K ·
(λ(s)) ds





 t0
Let be x an solution of equation (4) and y the unique solution of the
equation(5) that corresponds to x. Then
Zt
|X(t)P1 X −1 (s)| · |f (s, ys )|ds+
|y(t) − x(t)| ≤
t0
Z∞
|X(t)P2 X −1 (s)| · |f (s, ys )|ds ≤
+
t
Zt
Z∞
|X(t)P1 X −1 (s)|·|f (s, ys )|ds+
≤
t0
|X(t)P2 X −1 (s)|·|f (s, ys )−f (s, 0)|ds+
t
Z∞
|X(t)P2 X −1 (s)| · |f (s, 0)|ds = I1 + I2 .
+
t
Z∞
Z∞
|X(t)P2 X −1 (s)|·|f (s, ys )−f (s, 0)|ds+
I2 =
t
|X(t)P2 X −1 (s)|·|f (s, 0)|ds
t
∞
1
∞
1
Z
Z

p

p
+K
|f (s, 0)|p ds
.
≤ Kϕ(|y|Cg )
(λ(s)g(s))p ds




t
t1
If t ≥ t1 ≥ t0 then
1
∞
Z
p

(λ(s)g(s))p ds
<


6Kϕ(|y|Cg )
t
∞
Z

t1
|f (s, 0)|p ds
1
p

<
.
6K
Cg asymptotic equivalence for some functional equation of type Voltera 39
ε
Then I2 ≤ g(t)
3
Zt
|X(t)P1 X −1 (s)| · |f (s, ys )|ds ≤
I1 =
t0
Zt1
Zt
≤ |X(t)P1 |
|X
−1
|X(t)P1 X −1 (s)|·|f (s, ys )−f (s, 0)|ds+
(s)f (s, ys )|ds+
t0
t1
Zt
+
|f (s, 0)|ds
t1
Zt1
|X −1 (s)f (s, ys )|ds+
≤ |X(t)P1 |





Z t
+K ·




 t1
t0

1  t
1 


 p Z
p 
p
p
≤
(λ(s)ϕ(kys k)) ds
+
|f (s, 0)|


 


t1

Zt1
|X −1 (s)f (s, ys )|ds+
≤ |X(t)P1 |
t0
 t
Z
+K · ϕ(|y|Cg )

1
 t
1
Z
p

p
(λ(s)g(s))p ds
+K ·
|f (s, 0)|p
≤



t1
Zt1
≤ |X(t)P1 |
t1
∞
1
Z
p

(λ(s))p ds
|X −1 (s)f (s, ys )|ds + K · ϕ(|y|Cg )g(t)
+


t1
t0
∞
Z
+K ·

t1
|f (s, 0)|p
1
p

.
40
M. Olaru and V. Olaru
Using Lemma 1.1 we obtain that |X(t)P1 |
2ε
t ≥ t2 ≥ t1 . Then I1 < g(t)
3
Rt1
t0
ε
|X −1 (s)f (s, ys )|ds < , for all
3
References
[1] C.Corduneanu,Sur certaines equations fonctioneneles de Voltera, Funkcialaj Ekvacioj,9(1966),pp517-573.
[2] I.A.Rus, Generalized contractions, Seminar on fixed point theory, No
3, 1983, 1-130.
[3] I.A. Rus,S Muresan,Data dependence of the fixed points set of weakly
Picard operators, Studia Univ.”Babes-Bolyai”, Mathematica, Volume
XLIII, Number 1, March 1998,pp79-83.
Department of Mathematics
Faculty of Sciences
“Lucian Blaga” University of Sibiu
Str. Dr. I. Raţiu nr. 5-7,
550012 - Sibiu, România,
E-mail: olaruim@yahoo.com
olaru2@yahoo.com