Ann. Funct. Anal. 2 (2011), no. 2, 34–41
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL: www.emis.de/journals/AFA/
CLASSIFICATION OF POSITIVE SOLUTIONS OF NONLINEAR
SYSTEMS OF VOLTERRA INTEGRAL EQUATIONS
YOUSSEF N. RAFFOUL
Communicated by K. Ciesielski
Abstract. We give asymptotic classification of the positive solutions of a class
of two-dimensional nonlinear Volterra integro-differential equations. Also, we
furnish necessary and sufficient conditions for the existence of such positive
solutions.
1. Introduction
In this paper we consider the two-dimensional nonlinear Volterra integro-differen
tial equations
(
Rt
x0 (t) = h(t)x(t) + t0 a(t, s)f (y(s))ds,
Rt
(1.1)
y 0 (t) = p(t)y(t) + t0 b(t, s)g(x(s))ds, 0 ≤ t0 ≤ s ≤ t < ∞.
The functions a(t, s) and b(t, s) are positive and continuous functions for 0 ≤
t0 ≤ s ≤ t < ∞. The functions f and g are real-valued continuous in y and x,
respectively, and increasing on the real line R. Also, the coefficients h and p are
positive and continuous functions for t ≥ 0 and satisfy the condition
Z ∞
Z ∞
h(t)dt < ∞ and
p(t)dt < ∞.
(1.2)
0
0
Throughout this paper we assume that the functions f and g are well behaved
so that the solutions of system (1.1) exist on the interval [0, ∞). Also, we assume
that f (x) > 0, g(x) > 0 for x 6= 0.
Definition 1.1. A pair of real-valued functions (x, y) is said to be
Date: Received: 31 October 2010; Revised: 13 May 2011; Accepted: 17 June 2011.
2010 Mathematics Subject Classification. Primary 34K20; Secondary 45J05, 45D05.
Key words and phrases. Volterra integro-differential equations, positive solutions, classification, Systems, fixed point theorems.
34
POSITIVE SOLUTIONS OF NONLINEAR SYSTEMS
35
(1) a solution of system (1.1) if it satisfies (1.1) for t ≥ t0 ;
(2) eventually positive if both x(t) and y(t) are eventually positive;
(3) nonoscillatory if both x(t) and y(t) are either eventually positive or eventually negative.
There are many papers written on the subject of oscillatory and nonoscillatory
behavior of solutions in differential equations. For such topics we refer the interested reader to [1], [2], [5]-[7] and the reference therein. Recently, Li [6] and Li
and Cheng [7] studied the class of two-dimensional nonlinear differential systems
of the form
0
x (t) = a(t)f (y(t)),
y 0 (t) = b(t)g(x(t)),
under similar assumptions. They provided a classification scheme for positive
solutions of the above system. They also provided conditions for the existence of
solutions with designated asymptotic properties. However, no study has been devoted to systems of Volterra integro-differential equations. In [8] the author used
the notion of Lyapunov functional and obtain results concerning the boundedness
of solutions of Volterra integro-differential equations with unbounded perturbation.
Systems of the form of (1.1) are used for continuous risk models where the risk
process is a variation/extension of the classical compound Poisson process, see
for example Dickson and dos Reis (1997) [2] and Kluppelberg and Stadtmuller
(1998) [4].
In this paper, we classify positive solutions of (1.1) according to their limiting behaviors and then provide sufficient and/or necessary conditions for their
existence.
To simplify our notation we let
Z t Z u
A(t) =
a(u, s)ds du,
t0
and
t0
Z t Z
u
B(t) =
b(u, s)ds du.
t0
t0
Let
A(∞) = lim A(t) and B(∞) = lim B(t).
t→∞
−t−s
Note that if a(t, s) = e
t→∞
, then
1
A(∞) = .
2
We will discuss each of the following cases:
(i) A(∞) = ∞, and B(∞) = ∞;
(ii) A(∞) = ∞, and B(∞) < ∞;
(iii) A(∞) < ∞, and B(∞) = ∞;
(iv) A(∞) < ∞, and B(∞) < ∞.
Let C be the set of all continuous functions and define
Ω = {(x, y) ∈ C([0, ∞), R) × C([0, ∞), R) : x, y are eventually positive }.
36
Y.N. RAFFOUL
Let t0 ≥ 0 with x(t0 ) = Rx0 and y(t0 ) = y0 . By multiplying both sides of the first
− t h(s)ds
equation in (1.1) by e t0
and then integrating from u = t0 to u = t we
arrive at the variation of parameters formula
R
Z t Z u
Rt
t
h(s)ds
x(t) = x0 e t0
+
a(u, s)f (y(s))ds e u h(s)ds du.
(1.3)
t0
t0
In a similar fashion, we obtain from the second equation of (1.1),
R
Z t Z u
Rt
t
p(s)ds
y(t) = y0 e t0
+
b(u, s)g(x(s))ds e u p(s)ds du.
t0
(1.4)
t0
It is clear from (1.3) and (1.4) that x and y are positive provided that x0 , y0 ≥ 0.
Moreover, since h, p > 0, then from (1.1) we have x0 , y 0 > 0. Now, for some
positive constants α and β, we define the set
K(α, β) = {(x, y) ∈ Ω : lim x(t) = α, lim y(t) = β}.
t→∞
t→∞
Note that α and β maybe considered to be infinite.
2. Classification of Solutions and Existence
In this section, we should classify positive solutions of (1.1) according to their
limiting behaviors and then provide necessary and sufficient conditions for their
existence in the cases (ii), (iii) and (iv). Our results are based on the application
of Knaster’s fixed point theorem, which we state below.
Knaster’s Fixed Point Theorem Let X be a partially ordered Banach space
with ordering ≤. Let M be a subset of X with the following properties: The infimum of M belongs to M and every nonempty subset of M has a supermum which
belongs to M . Let T : M → M be an increasing mapping, i.e., x ≤ y implies
T x ≤ T y. Then T has a fixed point in M .
Theorem 2.1. Any solution (x, y) ∈ Ω of (1.1)) belongs to one of the following
subsets
K(α, β), K(α, ∞), K(∞, β) and K(∞, ∞).
Proof. Since (x, y) ∈ Ω, we have x0 , y 0 > 0 for t ≥ t0 . Thus x and y are increasing.
Hence,
lim x(t) = α > 0 or lim x(t) = ∞,
t→∞
t→∞
and
lim y(t) = β > 0 or lim y(t) = ∞.
t→∞
t→∞
In the following we state four theorems. Each theorem is related to one of the
above mentioned cases.
Theorem 2.2. Suppose (1.2), A(∞) = ∞ and B(∞) = ∞ hold. Then any
solution (x, y) ∈ Ω of (1.1) belongs to the set K(∞, ∞).
POSITIVE SOLUTIONS OF NONLINEAR SYSTEMS
37
Proof. Let (x, y) ∈ Ω be a solution of system (1.1). Then x0 (t), y 0 (t) > 0 for
t ≥ t0 . Thus x and y are increasing. As a consequence of this and the fact that
f is increasing, an integration of (1.1) yields
Z t
Z t Z u
x(t) = x0 +
h(s)x(s)ds +
a(u, s)f (y(s))ds du
t0
t0
t0
Z t Z u
≥ f (y0 )
a(u, s)ds du → ∞, for t → ∞,
t0
t0
and
t
Z
u
Z t Z
p(s)y(s)ds +
b(u, s)g(x(s))ds du
t0
t0
Z t Z u
≥ g(x0 )
b(u, s)ds du → ∞, for t → ∞.
y(t) = y0 +
t0
t0
t0
Showing that x(t) → ∞ and y(t) → ∞, as t → ∞.
Theorem 2.3. Suppose (1.2), A(∞) = ∞ and B(∞) < ∞ hold. Then there
exists a solution (x, y) ∈ Ω of (1.1) that belongs to the set K(∞, β) if and only if
Z t Z
u
lim
t→∞
Z s b(u, s)g
t0
Rs
e
t0
v
h(τ )dτ
t0
Z
v
a(v, k)f (c)dk dv ds du < ∞, (2.1)
t0
for some positive constant c.
Proof. Let (x, y) ∈ Ω be a solution of system (1.1). Then x0 (t), y 0 (t) > 0 for
t ≥ t0 . Thus x and y are increasing and there exists a positive constant β > 0
such that y0 ≤ y(t) ≤ β for t ≥ t0 . From (1.3) we have
R
Z t Z u
Rt
t
t0 h(s)ds
x(t) = x0 e
+
a(u, s)f (y(s))ds e u h(s)ds du
t0
t0
R
Z t Z u
t
≥
a(u, s)f (y0 )ds e u h(s)ds du,
t0
t0
and
Z
t
Z t Z
β ≥ y(t) = y0 +
p(s)y(s)ds +
t0
t0
Z s R
Z t Z u
Z
s
h(τ )dτ
v
≥
b(u, s)g
e
t0
t0
t0
u
b(u, s)g (x(s)) ds du
a(v, k)f (y0 )dk dv ds du.
t0
v
t0
By taking the limit at infinity in the above inequality we obtain (2.1).
Conversely, suppose that (2.1) holds. First notice that for t ≥ t0 , the second
equation of (1.1) can be written as
Z t
Z t Z u
y(t) = y0 +
p(s)y(s)ds +
b(u, s)g(x(s))ds du.
t0
t0
t0
38
Y.N. RAFFOUL
Next, we can choose a large enough T so that
Z s R
Z ∞ Z u
Z v
s
h(τ
)dτ
b(u, s)g
ev
a(v, k)dk dv ds du ≤
T
t0
t0
t0
and
∞
Z
T
c
,
4f (c)
1
p(s)ds ≤ .
4
Let X be the Banach space of all bounded real-valued functions y(t) on [T, ∞)
with the norm
||y|| = sup |y(t)|
t≥T
and with the usual pointwise ordering ≤. Define a subset ω of X by
c
ω = {y(t) ∈ X : ≤ y(t) ≤ c, t ≥ T }.
2
It is clear that for any subset B of ω, inf B ∈ ω and sup B ∈ ω. Define the
operator E : ω → X by
Z t
c
(Ey)(t) =
+
p(s)y(s)ds
2
T
Z s R
Z t Z u
Z v
s
h(τ )dτ
v
+
b(u, s)g
e
a(v, k)f (y(k))dk dv ds du,
T
t0
t0
t0
y ∈ ω.
We claim that E maps ω into ω. To see this we let y ∈ ω. Then
Z t
c/2 ≤ (Ey)(t) = c/2 +
p(s)y(s)ds
T
Z s R
Z t Z u
Z v
s
h(τ )dτ
v
+
b(u, s)g
e
a(v, k)f (y(k))dk dv ds du
T
t0
t0
t0
Z ∞
≤ c/2 + c
p(s)ds
T
Z s R
Z ∞ Z u
Z v
s
h(τ
)dτ
+ f (c)
b(u, s)g
ev
a(v, k)dk dv ds du
T
t0
t0
t0
c c
≤ c/2 + + = c.
4 4
Since E is increasing, the mapping E satisfies the hypothesis of Knaster’s fixed
point theorem and hence we conclude that there exists y in ω such that y = Ey.
Set
R
Z t Z u
t
x(t) =
a(s)f (y(s))ds e u h(s)ds du,
t0
t0
then
0
Z
t
x (t) = h(t)x(t) +
a(s)f (y(s))ds,
t0
POSITIVE SOLUTIONS OF NONLINEAR SYSTEMS
39
and
u
Z t Z
x(t) =
Rt
a(u, s)f (y(s))ds e
t0
t0
Z t Z u
≥ f (y0 )
a(u, s)ds du.
t0
u
h(s)ds
du
t0
In view of A(∞) = ∞, we have
lim x(t) = ∞.
t→∞
On the other hand,
Z t
c
y(t) = +
p(s)y(s)ds
2
T
Z s R
Z t Z u
Z
s
h(τ )dτ
v
+
b(u, s)g
e
T
t0
t0
v
a(v, k)f (y(k))dk dv ds du,
t0
from which we obtain,
lim y(t) = β,
t→∞
where β is a constant. Hence, (x, y) is a positive solution of (1.1) which belongs
to K(∞, β).
The proof of the next theorem follows along the lines of the proof of Theorem
2.3 and hence we omit.
Theorem 2.4. Suppose (1.2), A(∞) < ∞ and B(∞) = ∞ hold. Then there
exists a solution (x, y) ∈ Ω of (1.1) that belongs to the set K(α, ∞) if and only if
Z s R
Z t Z u
Z v
s
p(τ
)dτ
lim
a(u, s)f
ev
b(v, k)g(c)dk dv ds du < ∞,
t→∞
t0
t0
t0
t0
for some positive constant c.
Theorem 2.5. Suppose (1.2) hold. Then any solution (x, y) ∈ Ω of (1.1) that
belongs to the set K(α, β) if and only A(∞) < ∞ and B(∞) < ∞.
Proof. Let (x, y) be a solution in Ω with limt→∞ x(t) → α > 0 and limt→∞ y(t) →
β > 0. Then, there exists a T ≥ t0 and two positive constants; namely, c1 and c2
such that c1 ≤ x(t) ≤ α, c2 ≤ y(t) ≤ β for t ≥ T ≥ t0 . From system (1.1) we
have for t ≥ T that
Z t
Z t Z u
x(t) = x(T ) +
h(s)x(s)ds +
a(u, s)f (y(s))ds du,
(2.2)
T
T
t0
and
Z
y(t) = y(T ) +
t
Z t Z
u
p(s)y(s)ds +
T
b(u, s)g(x(s))ds du.
T
t0
(2.3)
40
Y.N. RAFFOUL
Thus
t
Z
Z t Z
u
c1 ≤ x(t) = x(T ) +
h(s)x(s)ds +
Z ∞
Z
≤ x(T ) + α
h(s)ds +
T
T
T
∞
t
Z 0u
T
a(u, s)f (y(s))ds du,
a(u, s)f (β)ds du < ∞,
t0
and
Z
t
Z t Z
u
c2 ≤ y(t) = y(T ) +
p(s)y(s)ds +
Z ∞
Z
≤ y(T ) + β
p(s)ds +
T
T
T
∞
t
Z 0u
T
b(u, s)g(x(s))ds du,
b(u, s)g(α)ds du < ∞.
t0
Conversely, suppose that A(∞) < ∞ and B(∞) < ∞. First notice that for
t ≥ t0 , the first equation of (1.1) can be written as
Z t
Z t Z u
x(t) = x0 +
h(s)x(s)ds +
a(u, s)f (y(s))ds du.
t0
t0
t0
In a similar fashion, we obtain from the second equation of (1.1),
Z t
Z t Z u
y(t) = y0 +
p(s)y(s)ds +
b(u, s)g(x(s))ds du.
t0
t0
t0
Next, we can choose a T large enough so that
Z ∞ Z u
a(u, s)ds du ≤
T
t0
d
,
4f (c)
and
Z
∞
p(s)ds ≤
T
1
.
4d
Let X be the Banach space of all bounded real-valued functions (x, y) on [T, ∞)
with the norm
||(x, y)|| = max{sup |x(t)|, sup |y(t)|}
t≥T
t≥T
and with the usual pointwise ordering ≤. Define a subset ω of X by
ω = {(x, y) ∈ X :
d
c
≤ x(t) ≤ d, ≤ y(t) ≤ c, t ≥ T }.
2
2
It is clear that any subset B of ω, inf B ∈ ω and sup B ∈ ω. Define the operator
E : ω → X by
E
x
y
(t) =
d
2
c
2
 R

R t R u
t
h(s)x(s)ds + T t0 a(u, s)f (y(s))ds du
T
,
+ Rt
R t R u
p(s)y(s)ds
+
b(u,
s)g(x(s))ds
du
T
T
t0
POSITIVE SOLUTIONS OF NONLINEAR SYSTEMS
41
(x, y) ∈ ω. We claim that E maps ω into ω. To see this we let x ∈ ω. Then
Z t
Z t Z u
d/2 ≤ (Ex)(t) = d/2 +
p(s)x(s)ds +
a(u, s)f (y(s))ds du
T
T
t0
Z ∞
Z ∞ Z u
≤ d/2 + d
p(s)ds + f (c)
a(u, s)ds du
T
T
t0
d d
≤ d/2 + + = d.
4 4
Showing that, for y ∈ ω, c/2 ≤ Ey ≤ c is similar and hence we omit it. Since
E is increasing, the mapping E satisfies the hypothesis of Knaster’s fixed point
theorem and hence we conclude that there exists (x, y) in ω such that (x, y) =
E(x, y). That is,
Z t
Z t Z u
d
x(t) = +
h(s)x(s)ds +
a(u, s)f (y(s))ds du,
2
T
T
t0
Z t
Z t Z u
c
y(t) = +
p(s)y(s)ds +
b(u, s)g(x(s))ds du
2
T
T
t0
from which we obtain,
lim x(t) = α and lim y(t) = β,
t→∞
t→∞
where α and β are positive constants. Hence, (x, y) is a positive solution of (1.1)
which belongs to K(α, β).
Remark It is easy to extended the results of this paper to systems of more
than two-dimensional Volterra integro-differential equations.
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Department of Mathematics, University of Dayton, Dayton, OH 45469-2316,
USA.
E-mail address: youssef.raffoul@notes.udayton.edu