Memoirs on Differential Equations and Mathematical Physics
Volume 57, 2012, 123–162
Tomoyuki Tanigawa
GENERALIZED REGULARLY
VARYING SOLUTIONS OF SECOND ORDER
NONLINEAR DIFFERENTIAL EQUATIONS
WITH DEVIATING ARGUMENTS
Dedicated to 80th birthday anniversary of Professor Kusano Takaŝi
Abstract. The sharp sufficient conditions of the existence of generalized regularly varying solutions (in the sense of Karamata) of differential
equations of the type
n h
¡
¢0 X
¡
¢
¡
¢i
p(t)ϕ(x0 (t)) ±
qi (t)ϕ x(gi (t)) + ri (t)ϕ x(hi (t)) = 0
i=1
are established. Here, p, qi , ri : [a, ∞) → (0, ∞) are continuous functions,
gi , hi : [a, +∞) → R are continuous and increasing functions such that
gi (t) < t, hi (t) > t for t ≥ a, lim gi (t) = ∞ and ϕ(ξ) ≡ |ξ|α sgn ξ, α > 0.
t→∞
2010 Mathematics Subject Classification. 34C11, 26A12.
Key words and phrases. Second order nonlinear differential equation
with deviating arguments, generalized regularly varying solution, asymptotic behavior of a solution.
Ł
Ø
æØ . Æ Æ Œ Ł
æŁ Æ øª Ł Æ ØºŒ
º
¡
Ø
Œ
Œ º ƺ æŁ
º
æØ º
Æ
æ-
n h
¢0 X
¡
¢
¡
¢i
p(t)ϕ(x0 (t)) ±
qi (t)ϕ x(gi (t)) + ri (t)ϕ x(hi (t)) = 0
i=1
Æ
Œø Łæ
Œ ºŁ
ª , Æ ø p, qi , ri : [a, ∞) →
(0, ∞) æßıª
æŒ ø
, gi , hi : [a, +∞) → R
æßıª
Æ Æ æŒ ø
, ºØ gi (t) < t, hi (t) > t, ºø t ≥ a, lim gi (t) =
∞, ºŁº ϕ(ξ) ≡ |ξ|α sgn ξ, α > 0.
t→∞
Generalized Regularly Varying Solutions . . .
125
1. Introduction
The equation to be studied in this paper is
n h
¡
¢
¡
¢i
¡
¢0 X
qi (t)ϕ x(gi (t)) + ri (t)ϕ x(hi (t)) = 0
p(t)ϕ(x0 (t)) ±
¡
(A± )
i=1
¢
ϕ(ξ) = |ξ|α sgn ξ, α > 0, ξ ∈ R ,
where p, qi , ri : [a, ∞) → (0, ∞) are continuous functions, gi , hi are continuous and increasing functions with gi (t) < t, hi (t) > t and lim gi (t) = ∞
t→∞
for i = 1, 2, . . . , n. In what follows we always assume that the function p(t)
satisfies
Z∞
dt
(1.1)
1 = ∞.
p(t) α
a
It is shown in the monograph [8] that the class of regularly varying functions in the sense of Karamata is a well-suited framework for the asymptotic
analysis of nonoscillatory solutions of the second order linear differential
equation of the form
x00 (t) = q(t)x(t), q(t) > 0.
The study of asymptotic analysis of nonoscillatory solutions of functional
differential equations with deviating arguments in the framework of regularly varying functions (called Karamata functions) was first attempted by
Kusano and Marić [5], [6]. They established a sharp condition for the existence of a slowly varying solution of the second order functional differential
equation with retarded argument of the form
x00 (t) = q(t)x(g(t)),
and the following functional differential equation of the form
£
¤
x00 (t) ± q(t)x(g(t)) + r(t)x(h(t)) = 0,
(1.2)
(1.3)
where q, r : [a, ∞) → (0, ∞) are continuous functions, g, h are continuous
and increasing with g(t) < t, h(t) > t for t = a, lim g(t) = ∞.
t→∞
It is well known that there is the qualitative similarity between linear differential equations and half-linear differential equations (see the book Došlý
and Řehák [2]). Therefore, in our previous papers [4], [7] we proved how
useful the regularly varying functions were for the study of nonoscillation
and asymptotic analysis of the half-linear differential equation involving
nonlinear Sturm–Liouville type differential operator of the form
¡
¢0
p(t)ϕ(x0 (t)) ± f (t)ϕ(x(t)) = 0, p(t) > 0,
(B± )
and the half-linear functional differential equation with both retarded and
advanced arguments of the form
¡
¢0 h
¡
¢
¡
¢i
(1.4)
ϕ(x0 (t)) ± q(t)ϕ x(g(t)) + r(t)ϕ x(h(t)) = 0,
126
Tomoyuki Tanigawa
where f : [a, ∞) → (0, ∞) is a continuous function, p, g, h are just as in
the above equations.
Theorem A (J. Jaroš, T. Kusano and T. Tanigawa [4]). Suppose
that (1.1) holds. The equations (B± ) have a normalized slowly varying
solution with respect to P (t) and a normalized regularly varying solution of
index 1 with respect to P (t) if and only if
Z∞
α
lim P (t)
f (s) ds = 0,
t→∞
(1.5)
t
where the function P (t) is defined by
Zt
P (t) =
ds
1
a
p(s) α
.
(1.6)
Theorem B (J. Manojlović and T. Tanigawa [7]). Suppose that
lim
t→∞
g(t)
h(t)
= 1 and lim
=1
t→∞ t
t
hold. Then the equations (1.4) have a slowly varying solution and a regularly
varying solution of index 1 if and only if
Z∞
lim t
α
Z∞
α
q(s) ds = lim t
t→∞
r(s) ds = 0.
t→∞
t
t
The objective of this paper is to establish a sharp condition of the existence of a normalized slowly varying solution with respect to P (t) and a
normalized regularly varying solution of index 1 with respect to P (t) of the
equation (A± ). Our main result is the following
Theorem 1.1. Suppose that
lim
P (gi (t))
= 1 for i = 1, 2, . . . , n
P (t)
(1.7)
lim
P (hi (t))
= 1 for i = 1, 2, . . . , n
P (t)
(1.8)
t→∞
and
t→∞
hold. The equation (A± ) possesses a normalized slowly varying solution
with respect to P (t) and a normalized regularly varying solution of index 1
with respect to P (t) if and only if
Z∞
Z∞
α
lim P (t) qi (s) ds = lim P (t) ri (s) ds = 0 for i = 1, 2, . . . , n.
α
t→∞
t→∞
t
t
(1.9)
Generalized Regularly Varying Solutions . . .
127
This paper is organized as follows. In Section 2 we briefly recall the definitions and properties of the slowly varying and regularly varying functions
of index ρ with respect to P (t) which are called the generalized regularly
varying functions introduced by Jaroš and Kusano [3]. Explicit expressions
for the normalized slowly varying solution with respect to P (t) and the
normalized regularly varying solution of index 1 with respect to P (t) of
the equations (B± ) obtained in [4] do not meet our need for application
to the functional differential equations (A± ), and thus we present a modified proof of Theorem A in Section 3. The proof of Theorem 1.1 which is
based on Theorems A and B will be presented in Section 4. Some examples
illustrating our result will also be presented in Section 5.
2. Definitions and Properties of the Generalized Regularly
Varying Functions
For the reader’s convenience we first state the definitions and some basic
properties of the regularly varying functions and then refer to the generalized regularly varying functions. The generalized regularly varying functions
are introduced for the first time by Jaroš and Kusano [3] in order to gain
useful information about an asymptotic behavior of nonoscillatory solutions
for the self-adjoint differential equations of the form
¡
¢0
p(t)x0 (t) + f (t)x(t) = 0.
The definitions and properties of regularly varying functions:
Definition 2.1. A measurable function f : [a, ∞) → (0, ∞) is said to be
a regularly varying of index ρ if it satisfies
f (λt)
= λρ for any λ > 0, ρ ∈ R.
t→∞ f (t)
lim
Proposition 2.1 (Representation Theorem). A measurable function f :
[a, ∞) → (0, ∞) is regularly varying of index ρ if and only if it can be written
in the form
½ Zt
¾
δ(s)
f (t) = c(t) exp
ds , t = t0 ,
s
t0
for some t0 > a, where c(t) and δ(t) are measurable functions such that
lim c(t) = c ∈ (0, ∞) and lim δ(t) = ρ.
t→∞
t→∞
The totality of regularly varying functions of index ρ is denoted by RV(ρ).
The symbol SV is used to denote RV(0) and a member of SV = RV(0) is
referred to as a slowly varying function. If f (t) ∈ RV(ρ), then f (t) = tρ L(t)
for some L(t) ∈ SV. Therefore, the class of slowly varying functions is of
128
Tomoyuki Tanigawa
fundamental importance in the theory of regular variation. In addition to
the functions tending to positive constants as t → ∞, the following functions
½Y
¾
N
N
n log t o
Y
mi
ni
,
(logi t) (mi ∈ R), exp
(logi t)
(0 < ni < 1), exp
log2 t
i=1
i=1
where log1 t = log t and logk t = log logk−1 t for k = 2, 3, . . . , N , also belong
to the set of slowly varying functions.
Proposition 2.2. Let L(t) be any slowly varying function. Then, for
any γ > 0,
lim tγ L(t) = ∞ and lim t−γ L(t) = 0.
t→∞
t→∞
Proposition 2.3 (Karamata’s integration theorem). Let L(t) ∈ SV.
Then
(i) if γ > −1,
Zt
sγ L(s) ds ∼
a
tγ+1
L(t), as t → ∞;
γ+1
(ii) if γ < −1,
Z∞
sγ L(s) ds ∼ −
t
tγ+1
L(t), as t → ∞.
γ+1
Here and hereafter the notation ϕ(t) ∼ ψ(t) as t → ∞ is used to mean
the asymptotic equivalence of ϕ(t) and ψ(t): lim ψ(t)/ϕ(t) = 1.
t→∞
For an excellent explanation of the theory of regularly varying functions
the reader is referred to the book [1].
The definitions and properties of generalized regularly varying
functions:
Definition 2.2. A measurable function f : [a, ∞) → (0, ∞) is said to be
slowly varying with respect to P (t) if the function f ◦P (t)−1 is slowly varying
in the sense of Karamata, where the function P (t) is defined by (1.6) and
P (t)−1 denotes the inverse function of P (t). The totality of slowly varying
functions with respect to P (t) is denoted by SVP .
Definition 2.3. A measurable function g : [a, ∞) → (0, ∞) is said to
be regularly varying function of index ρ with respect to P (t) if the function
g ◦ P (t)−1 is regularly varying of index ρ in the sense of Karamata. The set
of all regularly varying functions of index ρ with respect to P (t) is denoted
by RVP (ρ).
Of fundamental importance is the following representation theorem for
the generalized slowly and regularly varying functions, which is an immediate consequence of Proposition 2.1.
Generalized Regularly Varying Solutions . . .
129
Proposition 2.4.
(i) A function f (t) is slowly varying with respect to P (t) if and only if
it can be expressed in the form
½ Zt
¾
δ(s)
(2.1)
f (t) = c(t) exp
ds , t = t0
1
p(s) α P (s)
t0
for some t0 > a, where c(t) and δ(t) are measurable functions such
that
lim c(t) = c ∈ (0, ∞) and lim δ(t) = 0.
t→∞
t→∞
(ii) A function g(t) is regularly varying of index ρ with respect to P (t)
if and only if it has the representation
½ Zt
¾
δ(s)
g(t) = c(t) exp
ds , t = t0
(2.2)
1
p(s) α P (s)
t0
for some t0 > a, where c(t) and δ(t) are measurable functions such
that
lim c(t) = c ∈ (0, ∞) and lim δ(t) = ρ.
t→∞
t→∞
If the function c(t) in (2.1) (or (2.2)) is identically a constant on [t0 , ∞),
then the function f (t) (or g(t)) is called normalized slowly varying (or normalized regularly varying of index ρ) with respect to P (t). The totality of
such functions is denoted by n-SVP (or n-RVP ).
It is easy to see that if g(t) ∈ RVP (ρ) (n-RVP (ρ)), then g(t) = P (t)ρ f (t)
for some f (t) ∈ SVP (or n-SVP ).
Proposition 2.5. Let f (t) ∈ SVP . Then, for any γ > 0,
lim P (t)γ f (t) = ∞ and lim P (t)−γ f (t) = 0.
t→∞
t→∞
(2.3)
The Karamata’s integration theorem is generalized in the following manner.
Proposition 2.6 (The generalized Karamata’s integration theorem). Let
f (t) ∈ n-SVP . Then
(i) If γ > −1,
Zt
P (s)γ
p(s)
t0
(ii) If γ < −1,
R∞
1
α
f (s) ds ∼
P (t)γ+1
f (t) as t → ∞;
γ+1
(2.4)
1
P (t)γ f (t)/p(t) α dt < ∞ and
t0
Z∞
t
P (s)γ
p(s)
1
α
f (s) ds ∼ −
P (t)γ+1
f (t) as t → ∞.
γ+1
(2.5)
130
Tomoyuki Tanigawa
3. The Existence of Generalized Regularly Varying Solution
of Self-Adjoint Differential Equation without Deviating
Arguments
R∞
Theorem 3.1. Put F (t) = P (t)α f (s) ds, Fb(t) = sup F (s),
t
s=t
1
F+ (t, w) = |1 + F (t) − w|1+ α
and
³
1´
w − 1,
+ 1+
α
(3.1)
³
1
1´
w − |1 + F (t) − w|1+ α .
F− (t, w) = 1 + 1 +
(3.2)
α
(i) The equation (B+ ) possesses a n-SVP solution x(t) having the expression
½ Zt ³
x(t) = exp
t0
¾
v(s) + F (s) ´ α1
ds
, t = t0
p(s)P (s)α
(3.3)
for some t0 > a, in which v(t) satisfies
Z∞
1
(v(s) + F (s))1+ α
α
v(t) = αP (t)
1
p(s) α P (s)α+1
t
ds, t = t0
(3.4)
and
0 5 v(t) 5 Fb(t0 ) for t = t0
(3.5)
if and only if (1.5) holds.
(ii) The equation (B+ ) possesses a n-RVP (1) solution x(t) having the
expression
½ Zt ³
x(t) = exp
t1
¾
1 + F (s) − w(s) ´ α1
ds , t = t1
p(s)P (s)α
(3.6)
for some t1 > a, in which w(t) satisfies
w(t) =
and
α
P (t)
Z∞
F+ (s, w(s)) ds, t = t1
(3.7)
t
q
0 5 w(t) 5 Fb(t1 ) for t = t1
(3.8)
if and only if (1.5) holds.
(iii) The equation (B− ) possesses a n-SVP solution x(t) having the expression
½ Zt ³
x(t) = exp
t0
¾
v(s) − F (s) ´ α1 ∗
ds
, t = t0
p(s)P (s)α
(3.9)
Generalized Regularly Varying Solutions . . .
131
for some t0 > a, in which v(t) satisfies
Z∞
v(t) = αP (t)α
1
|v(s) − F (s)|1+ α
1
p(s) α P (s)α+1
t
ds, t = t0
(3.10)
and (3.5) if and only if (1.5) holds. Here, the meaning of the asterisk notation is defined by ξ γ∗ = |ξ|γ sgn ξ, γ > 0, ξ ∈ R.
(iv) The equation (B− ) possesses a n-RVP (1) solution x(t) having the
expression
½ Zt ³
x(t) = exp
t1
¾
1 − F (s) + w(s) ´ α1
ds
, t = t1
p(s)P (s)α
(3.11)
for some t1 > a, in which w(t) satisfies
α
w(t) =
P (t)
Z∞
F− (s, w(s)) ds, t = t1
(3.12)
t
and (3.8) if and only if (1.5) holds.
Our purpose in this section is to give a proof of the above Theorem 3.1.
The following lemma will be needed for our purpose.
Lemma 3.1.
(i) If x(t), a nonoscillatory solution of (B± ), is not zero on [a, ∞),
then the function u(t) = p(t)ϕ(x0 (t)/x(t)) satisfies the generalized
Riccati equation
1
u0 (t) + α
|u(t)|1+ α
1
p(t) α
± f (t) = 0, t = a.
(C± )
(ii) If u(t) is a solution of (C± ), then the function
½ Zt ³
x(t) = exp
a
u(s) ´ α1 ∗
ds
p(s)
¾
is a nonoscillatory solution of (B± ) on [a, ∞).
Proof of Theorem 3.1. Since the idea of the proof of Theorem 3.1 for the
equation (B− ) is similar to the way of proving the equation (B+ ), we restrict
our attention to the proof for equation (B+ ).
(The “only if” part): Let x(t) be a positive solution of (B+ ) belonging
to n-SVP or n-RVP (1), respectively. Then, by the representation theorem,
½ Zt
x(t) = exp
δ(s)
1
t0
p(s) α P (s)
¾
ds , t = t0 ,
132
Tomoyuki Tanigawa
for some t0 > a, where lim δ(t) = 0 or 1 according as x(t) ∈ n-SVP or
t→∞
x(t) ∈ n-RVP (1). Since the function
³ x0 (t) ´
³ δ(t) ´
u(t) = p(t)ϕ
=ϕ
x(t)
P (t)
satisfies the generalized Riccati equation (C+ ) and u(t) → 0 as t → ∞, we
obtain
Z∞
Z∞
1
|P (s)α u(s)|1+ α
u(t) = α
ds + f (s) ds
1
p(s) α P (s)α+1
t
t
or
Z∞
α
P (t) u(t) = αP (t)
α
1
|P (s)α u(s)|1+ α
1
p(s) α P (s)α+1
t
ds+
Z∞
+ P (t)α
f (s) ds, t = t0 .
(3.13)
t
Letting t → ∞ in (3.13), we easily conclude that (1.5) holds in either case
of P (t)α u(t) → 0 or P (t)α u(t) → 1 as t → ∞.
(The “if” part) Suppose that (1.5) holds.
(The existence of a n-SVP solution of (B+ )): Choose t0 > max{a, 1} so
large that
n
1
1o
φ = (2Fb(t0 )) α max 2, 1 +
< 1,
(3.14)
α
and define the set of continuous functions V and the integral operators F
by
n
o
V = v ∈ C0 [t0 , ∞) : 0 5 v(t) 5 Fb(t0 ), t = t0
(3.15)
and
Z∞
α
Fv(t) = αP (t)
1
(v(s) + F (s))1+ α
1
p(s) α P (s)α+1
t
ds, t = t0 ,
(3.16)
where C0 [t0 , ∞) denotes the Banach space consisting of all continuous functions on [t0 , ∞) and tend to 0 as t → ∞ and equipped with the norm
kvk0 = sup |v(t)|. It can be verified that F is a contraction mapping on V .
t=t0
In fact, using (3.14), we see that v ∈ V implies lim Fv(t) = 0 and
t→∞
Z∞
ds
1
Fv(t) 5 α(2Fb(t0 ))1+ α P (t)α
1
t
p(s) α P (s)α+1
1
= (2Fb(t0 ))1+ α 5 Fb(t0 ),
Generalized Regularly Varying Solutions . . .
133
and that v1 , v2 ∈ V implies
¯
¯
¯1+ 1 ¯¯
1
¯
¯ |v1 (t) + F (t)|1+ α − ¯v2 (t) + F (t)¯ α ¯ 5
³
1
1´ b
5 1+
(2F (t0 )) α |v1 (t) − v2 (t)| 5 φ|v1 (t) − v2 (t)|, t = t0 ,
α
which ensures that F is a contraction mapping. Therefore, there exists a
unique element v0 ∈ V such that v0 = Fv0 , that is,
Z∞
1
(v0 (s) + F (s))1+ α
α
ds, t = t0 .
v0 (t) = αP (t)
1
p(s) α P (s)α+1
t
Obviously, v0 (t) satisfies the integral equation
³ v (t) ´0 (v(t) + F (t))1+ α1
0
+
= 0, t = t0 .
1
P (t)α
p(t) α P (t)α+1
(3.17)
By virtue of the function v0 (t) we define the function
½ Zt ³
x0 (t) = exp
t0
¾
v0 (s) + F (s) ´ α1
ds , t = t0 .
p(s)P (s)α
Since the function u(t) = v0 (t)+F (t)/P (t)α satisfies the generalized Riccati
equation (C+ ) associated with (B+ ) which is easily seen to be equivalent to
(3.17), x0 (t) is a solution of the differential equation (B+ ).
(The existence of a n-RVP (1) solution of (B+ )): We will construct a
n-RVP (1) solution of (B+ ). Let us consider the function
½ Zt ³
x(t) = exp
t1
¾
1 + F (s) − w(s) ´ α1 ∗
ds , t = t1
p(s)P (s)α
(3.18)
for some t1 > a to be determined later. According to (ii) of Lemma 3.1, the
function x(t) is a solution of (B+ ) on [t1 , ∞) if w(t) is chosen in such way
that u(t) = 1 + F (t) − w(t)/P (t)α satisfies the generalized Riccati equation
(C+ ) on [t1 , ∞). Then the differential equation for w(t) is derived:
£
1 ¤
α
α
w0 (t) −
w(t) +
1 − |1 + F (t) − w(t)|1+ α = 0. (3.19)
1
1
p(t) α P (t)
p(t) α P (t)
We rewrite (3.19) as
(P (t)w(t))0 −
α
1
p(t) α
F+ (t, w(t)) = 0,
(3.20)
where F+ (t, w(t)) is defined with (3.1). It is convenient to express F+ (t, w) as
F+ (t, w) = G(t, w) + H(t, w) + k(t),
(3.21)
with G(t, w), H(t, w) and k(t) defined, respectively, by
³
1
1
1
1´
(1+F (t)) α w −(1+F (t))1+ α , (3.22)
G(t, w) = |1+F (t)−w|1+ α + 1+
α
134
Tomoyuki Tanigawa
³
o
1
1´n
H(t, w) = 1 +
1 − (1 + F (t)) α w,
α
(3.23)
and
1
k(t) = (1 + F (t))1+ α − 1.
(3.24)
Since F (t) → 0 as t → ∞ by hypothesis, we can choose t1 > max{a, 1} such
that
q
³
1´
[K + L + α] Fb(t1 ) 5 1,
1+
(3.25)
α
where K and L are positive constants such that
K=
³ 4 ´1− α1
and L = 1 if α > 1;
3
³ 3 ´ α1 −1
³ 5 ´ α1 −1
K=
and L =
if α 5 1.
2
4
(3.26)
Noting
q that since 1 + 1/α > 1 and K + L + α = 2, we have in view of (3.25)
that Fb(t1 ) 5 1/2 and F (t) 5 1/4 for all t = t1 . It is easily shown that,
using the mean value theorem and L’Hospital rule, the following inequalities
hold for (3.22), (3.23) and (3.24):
¯ ∂G(t, w) ¯
1³
1´
¯
¯
1+
K|w|,
(3.27)
¯5
¯
∂w
α
α
¯ ∂H(t, w) ¯
1´
1³
¯
¯
1+
L F (t),
(3.28)
¯
¯5
∂w
α
α
³
´
1
1
1+
L w2 ,
(3.29)
|G(t, w)| 5
α
α
³
1
1
|H(t, w)| 5
1 + )L F (t)|w|,
(3.30)
α
α
and
³
1´
|k(t)| 5 1 +
F (t)
α
for t = t1 and for |w| 5 1/4.
Consider the set W ⊂ C0 [t1 , ∞) defined by
q
n
o
W = w ∈ C0 [t1 , ∞) : |w(t)| 5 Fb(t1 ) , t = t1
(3.31)
(3.32)
and define the integral operator G : W → C0 [t1 , ∞) by
α
Gw(t) =
P (t)
Zt
F+ (s, w(s))
1
t1
p(s) α
ds, t = t1 ,
(3.33)
where F+ (t, w) is given by (3.1). Then, it can be shown that G is a contraction mapping on W . In fact, if w ∈ W , then, by means of (3.29)–(3.31)
Generalized Regularly Varying Solutions . . .
135
and (3.25), we can see that
α
|Gw(t)| 5
P (t)
Zt
t1
1
p(s)
1
α
³
1´ 1
5 1+
α P (t)
h
i
|G(s, w)| + |H(s, w)| + |k(s)| ds 5
Zt
1
1
t1
p(s) α
h
i
Lw(s)2 + LF (s)|w(s)| + αF (s) ds 5
³
i
3
1 ´h b
LF (t1 ) + LFb(t1 ) 2 + αFb(t1 ) =
5 1+
α
q
³
i
h
1´b
= 1+
F (t1 ) L + L Fb(t1 ) + α 5
q α ³
q
q
1´
5 Fb(t1 ) 1 +
[K + L + α] Fb(t1 ) 5 Fb (t1 ) , t = t1 .
α
Since F+ (t, w(t)) → 0 as t → ∞, we obtain lim Gw(t) = 0. Thus, it follows
t→∞
that Gw ∈ W , and hence G maps W into itself. Moreover, if w1 , w2 ∈ W ,
then, using (3.27) and (3.28), we obtain
¯
¯
¯Gw1 (t) − Gw2 (t)¯ 5
Zt
×
t1
1
1
p(s) α
α
×
P (t)
h¯
¯ ¯
¯i
¯G(s, w1 (s))−G(s, w2 (s))¯ + ¯H(s, w1 (s))−H(s, w2 (s))¯ ds 5
1 ´h
K
5 1+
α
³
q
i
Fb(t1 ) + L Fb(t1 ) kw1 − w2 k0 5
q
³
1´
5 1+
[K + L] Fb (t1 ) kw1 − w2 k0 ,
α
which implies that
q
´
³
°
°
°Gw1 − Gw2 ° 5 1 + 1 [K + L] Fb(t1 ) kw1 − w2 k0 .
0
α
In view of (3.25) this shows that G is a contraction mapping on W . Therefore, the contraction mapping principle ensures the existence of a unique
fixed element w1 ∈ W such that w1 = Gw1 , which is equivalent to the
integral equation
w1 (t) =
α
P (t)
Zt
F+ (s, w1 (s))
1
t1
p(s) α
ds, t = t1 .
(3.34)
Differentiation of (3.34) shows that w1 (t) satisfies the differential equation
(3.20), and substitution of this w1 (t) into (3.6) gives rise to a solution x(t)
of the half-linear differential equation (B+ ) defined on [t1 , ∞). Furthermore, since lim w1 (t) = 0, it follows from the representation theorem that
t→∞
136
Tomoyuki Tanigawa
x(t) ∈ n-RVP (1). This completes the proof of Theorem 3.1 for the equation (B+ ).
¤
Remark 3.1. Consider another half-linear differential equation
¡
¢0
p(t)ϕ(x0 (t)) + fe(t)ϕ(x(t)) = 0,
e +)
(B
where fe(t) is a positive continuous function such that
fe(t) = f (t), t = a
and
Z∞
fe(s) ds = 0.
α
lim P (t)
t→∞
t
We take t0 > max{a, 1} so large that
Z∞
n
1
1o
α
e
e
α
< 1 where F (t) = P (t)
fe(s) ds.
(2F (t0 )) max 2, 1 +
α
t
Then, by means of Theorem 3.1, both x0 (t) and x
e0 (t) are given, respectively,
by (3.3) and
½ Zt ³
x
e0 (t) = exp
t0
¾
ve0 (s) + Fe(s) ´ α1
ds , t = t0 ,
p(s)P (s)α
where ve0 (t) is a solution of the integral equation
Z∞
1
(e
v0 (s) + Fe(s))1+ α
ve0 (t) = α P (t)α
ds, t = t0 .
1
p(s) α P (s)α+1
t
We here compare x0 (t) with x
e0 (t). From the proof of Theorem 3.1, v0 (t)
and ve0 (t) are the fixed points of the contraction mapping F and Fe given,
respectively, by (3.16) and
Z∞
1
(e
v (s) + Fe(s))1+ α
α
Feve(t) = α P (t)
ds, t = t0 .
1
p(s) α p(s)α+1
t
Noting that v0 (t) and ve0 (t) are the limit points of uniform convergence on
[t0 , ∞) of the sequences defined by
vn+1 (t) = Fvn (t), t = t0 , n = 1, 2, . . . , v1 (t) = 0
and
ven+1 (t) = Feven (t), t = t0 , n = 1, 2, . . . , ve1 (t) = 0.
We conclude that ve0 (t) = v0 (t), t = t0 , which implies that x
e0 (t) = x0 (t) for
t = t0 .
Generalized Regularly Varying Solutions . . .
137
4. The Existence of Generalized Regularly Varying Solution
of Self-Adjoint Functional Differential Equation with
Deviating Arguments
In this section we first present the proof of Theorem 1.1 for equation
(A+ ) and then give the proof for the equation (A− ).
4.1. The proof of Theorem 1.1 for the equation (A+ ). (The “only
if” part) Suppose that there exists a positive solution x1 (t) ∈ n-SVP or
x2 (t) ∈ n-RVP (1) of (A+ ). The equation (A+ ) can be written as the halflinear differential equation without retarded and advanced arguments
n
¡
¢0 X
£
¤
p(t)ϕ(x0 (t)) +
qx,gi (t) + rx,hi (t) ϕ(x(t)) = 0,
(4.1)
i=1
where
³ x(g (t)) ´
³ x(h (t)) ´
i
i
qx,gi (t) = qi (t)ϕ
and rx,hi (t) = ri (t)ϕ
,
x(t)
x(t)
i = 1, 2, . . . , n.
Here, applying Theorem 3.1, we see that
Z∞ X
n
£
¤
α
lim P (t)
qx,gi (s) + rx,hi (s) ds = 0
t→∞
i=1
t
or
α
lim P (t)
Z∞ X
n
t→∞
t
α
qx,gi (s) ds = lim P (t)
Z∞ X
n
t→∞
i=1
rx,hi (s) ds = 0.
i=1
t
By the representation theorem, xj (t), j = 1, 2 can be expressed as
½ Zt
xj (t) = exp
δj (s)
1
t0
p(s) α P (s)
¾
ds , j = 1, 2
for some t0 > a, where δj (t) satisfies
(
0
lim δj (t) =
t→∞
1
(j = 1)
(j = 2).
The solutions xj (t), j = 1, 2 satisfy
xj (gi (t))
= exp
xj (t)
½
and
xj (hi (t))
= exp
xj (t)
Zt
−
δj (s)
1
gi (t)
½ hZi (t)
p(s) α P (s)
δj (s)
1
t
p(s) α P (s)
¾
ds , t = t1 ,
¾
ds , t = t1 ,
(4.2)
138
Tomoyuki Tanigawa
respectively, where t1 is such that gi (t1 ) = t0 , i = 1, 2, . . . , n. Then, using
the properties of δj (t), (1.7) and (1.8), we see that
Zt
|δj (s)|
1
gi (t)
p(s) α P (s)
ds 5 sup |δj (s)| · log
s=gi (t)
P (t)
→ 0 as t → ∞
P (gi (t))
and
hZi (t)
|δj (s)|
1
α
t
p(s) P (s)
ds 5 sup |δj (s)| · log
s=t
P (hi (t))
→ 0 as t → ∞.
P (t)
Thus, it follows that
lim
t→∞
xj (gi (t))
xj (hi (t))
= lim
= 1, i = 1, 2, . . . , n, j = 1, 2.
t→∞
xj (t)
xj (t)
(4.3)
Consequently, from (4.3) we find that (1.9) holds.
(The “if” part)
(The existence of a n-SVP solution of (A+ )): ©Suppose that
ª (1.9) is satisfied. Choose t0 > a so large that t∗ = min
inf gi (t) > max{a, 1},
i=1,2,...,n
t=t0
¾1
½ X
n
n
¤ α
£
1o
αb
b
max 2, 1 +
2
Qi (t0 ) + 2 Ri (t0 )
<1
α
i=1
(4.4)
and
n
³ X
£
¤´ α1
P (hi (t))
2
Qi (t0 ) + 2α Ri (t0 )
log
5 log 2, t = t0 ,
P (t)
i=1
b i (t) and R
bi (t) for i = 1, 2 . . . , n are defined by
where Qi (t), Ri (t), Q
Z∞
α
b i (t) = sup Qi (s)
Qi (t) = P (t)
qi (s) ds, Q
(4.6)
s=t
t
and
(4.5)
Z∞
bi (t) = sup Ri (s).
ri (s) ds, R
α
Ri (t) = P (t)
t
(4.7)
s=t
Let Ξ denote the set of all positive continuous nondecreasing functions
ξ(t) on [t∗ , ∞) satisfying
ξ(t) = 1 for t∗ 5 t 5 t0 ;
n
( Z∞ µ v0 (s) + P [Qi (s) + 2α Ri (s)] ¶ 1
α
i=1
ξ(t) 5 exp
t0
p(s)P (s)α
ξ(hi (t))
5 2 for t = t0
ξ(t)
(4.8)
)
ds
for t = t0 ;
(4.9)
(4.10)
Generalized Regularly Varying Solutions . . .
139
for i = 1, 2, . . . , n, where v0 (t) satisfies the following integral equation:
n
P
1
[Qi (s) + 2α Ri (s)])1+ α
Z∞ (v0 (s) +
i=1
v0 (t) = αP (t)α
ds, t = t0 . (4.11)
1
p(s) α P (s)α+1
t
We note that the function
n
( Zt µ v0 (s) + P [Qi (s) + 2α Ri (s)] ¶ 1
α
i=1
X0 (t) = exp
ds , t = t0
p(s)P (s)α
t0
)
is a solution of the half-linear differential equation
n
¡
¢0 X
£
¤
p(t)ϕ(x0 (t)) +
qi (t) + 2α ri (t) ϕ(x(t)) = 0,
(4.12)
(4.13)
i=1
since the function
v0 (t) +
u(t) =
n
P
i=1
[Qi (t) + 2α Ri (t)]
P (t)α
satisfies the generalized Riccati equation
1
n
X
£
¤
|u(t)|1+ α
u0 (t) + α
+
qi (t) + 2α ri (t) = 0.
1
p(t) α
i=1
Since v0 (t)+
n
P
(4.14)
(4.15)
[Qi (t)+2α Ri (t)] → 0 as t → ∞, X0 (t) is a normalized slowly
i=1
varying function with respect to P (t) by the representation theorem. It is
obvious that Ξ is a nonvoid closed and convex subset of the locally convex
space C[t0 , ∞) of all continuous functions on [t0 , ∞) equipped with the
metric topology of uniform convergence on compact subintervals of [t0 , ∞).
For any ξ ∈ Ξ, we define qξ,gi (t) and rξ,hi (t) by
³ ξ(g (t)) ´
³ ξ(h (t)) ´
i
i
qξ,gi (t) = qi (t)ϕ
and rξ,hi (t) = ri (t)ϕ
,
(4.16)
ξ(t)
ξ(t)
respectively. Taking into account (4.10), we have
n
n
n
n
X
X
X
X
qξ,gi (t) 5
qi (t),
rξ.hi (t) 5 2α
ri (t),
i=1
i=1
i=1
and accordingly,
n
n
n
n
X
X
X
X
Qξ,gi (t) 5
Qi (t),
Rξ,hi (t) 5 2α
Ri (t)
i=1
i=1
i=1
(4.18)
i=1
where Qξ,gi (t) and Rξ,hi (t) are defined by
Z∞
Z∞
α
α
Qξ,gi (t) = P (t)
qξ,gi (s) ds, Rξ,hi (t) = P (t)
rξ,hi (s) ds.
t
(4.17)
i=1
t
(4.19)
140
Tomoyuki Tanigawa
Consequently, it follows from (4.4) that
½ X
n h
i ¾ α1
n
1o
αb
b
2
Qξ,gi (t0 ) + 2 Rξ,hi (t0 )
max 2, 1 +
< 1,
α
i=1
(4.20)
b ξ,g (t) = sup Qξ,g (t) and R
bξ,h (t) = sup Rξ,h (s). Thus, Theowhere Q
i
i
i
i
s=t
s=t
rem 3.1 implies that for any ξ ∈ Ξ the half-linear differential equation
n
¡
¢0 X
£
¤
p(t)ϕ(x0 (t)) +
qx,gi (t) + rx,hi (t) ϕ(x(t)) = 0
(4.21)
i=1
has a n-SVP solution
n
( Zt µ vξ (s) + P [Qξ,g (s) + Rξ,h (s)] ¶ 1
i
i
α
i=1
Xξ (t) = exp
p(s)P (s)α
t0
)
ds , t = t0 , (4.22)
where vξ (t) is a solution of the integral equation
n
¡
¢1+ α1
P
[Qξ,gi (s) + Rξ,hi (s)]
Z∞ vξ (s) +
i=1
vξ (t) = αP (t)α
ds, t = t0 , (4.23)
1
p(s) α P (s)α+1
t
and satisfies
0 5 vξ (t) 5
n
n
X
¤
£
¤ X
£
bi (t0 ) for t = t0 .
b ξ,g (t0 )+ R
bξ,h (t0 ) 5
b i (t0 )+2α R
Q
Q
i
i
i=1
i=1
Let us now define the mapping Φ which assigns to each ξ ∈ Ξ the function
given by
Φξ(t) = 1 for t∗ 5 t 5 t0 , Φξ(t) = Xξ (t) for t = t0 .
(4.24)
To apply the Schauder–Tychonoff fixed point theorem to Φ we will show
that Φ is a continuous mapping which sends Ξ into a relatively compact
subset of Ξ.
(i) Φ maps Ξ into itself. Let ξ ∈ Ξ. Then
n
( Zt µ vξ (s) + P [Qξ,g (s) + Rξ,h (s)] ¶ 1 )
i
i
α
i=1
Φξ(t) = Xξ (t) = exp
ds 5
α
p(s)P (s)
t0
n
( Zt µ vξ (s) + P [Qi (s) + 2α Ri (s)] ¶ 1
α
i=1
5 exp
ds
p(s)P (s)α
t0
( Zt µ v0 (s)+ P [Qi (s)+2α Ri (s)] ¶ 1
n
i=1
5 exp
t0
p(s)P (s)α
)
5
)
α
ds , t = t0 ,
Generalized Regularly Varying Solutions . . .
141
where we make use of the fact that vξ (t) 5 v0 (t), t = t0 for all ξ ∈ Ξ (cf.
n
P
b i (t0 ) + 2α R
bi (t0 )], using (4.5),
Remark 3.1). Furthermore, since vξ (t) 5
[Q
i=1
we see that
Φ(ξ(hi (t)))
= exp
Φ(ξ(t))
n
( hZi (t)µ vξ (s) + P [Qξ,g (s) + Rξ,h (s)] ¶ 1
i
i
α
i=1
ds
p(s)P (s)α
t
)
(µ n
¶ 1 hZi (t)
X£
¤ α
αb
b
5 exp
2
Qi (t0 ) + 2 Ri (t0 )
ds
5
)
=
1
p(s) α P (s)
t
(µ n
)
i¶ α1
Xh
P
(h
(t))
i
α
b i (t0 )+2 R
bi (t0 )
= exp
2
Q
log
5 2, t = t0 .
P (t)
i=1
i=1
This shows that Φξ ∈ Ξ, that is, Φ is a self-map on Ξ.
(ii) Φ(Ξ) is relatively compact in C[t∗ , ∞). Since Φ maps Ξ into itself,
that is, Φ(Ξ) ⊂ Ξ, Φ(Ξ) is locally uniformly bounded on [t∗ , ∞), and since
ξ ∈ Ξ implies
05
d
d
Φξ(t) =
Xξ (t) =
dt
dt
n
( Zt µ vξ (s) + P [Qξ,g (s) + Rξ,h (s)] ¶ 1 )
i
i
α
i=1
= exp
ds ×
α
p(s)P (s)
t0
µ vξ (t) +
×
n
P
[Qξ,gi (t) + Rξ,hi (t)] ¶ α1
i=1
5
p(t)P (t)α
(µ n
)
¶ α1
X
P
(t)
α
bi (t0 )]
b i (t0 ) + 2 R
5 exp
[Q
log
×
P (t0 )
i=1
µ2
×
n
P
b i (t0 ) + 2α R
bi (t0 )] ¶ 1
[Q
α
i=1
p(t)P (t)α
,
Φ(Ξ) is locally equi-continuous on [t∗ , ∞). From the Arzela–Ascoli lemma
it then follows that Φ(Ξ) is relatively compact in C[t∗ , ∞).
(iii) Φ is a continuous mapping. Let {ξm (t)} be a sequence of functions
in Ξ converging to δ(t) uniformly on the compact subintervals of [t∗ , ∞).
To prove the continuity of Φ, we have to prove that {Φξm (t)} converges
to Φδ(t) uniformly on compact subintervals in [t∗ , ∞). Applying the mean
142
Tomoyuki Tanigawa
value theorem, for t = t∗ we obtain
¯
¯
¯Φξm (t) − Φδ(t)¯ = |Xξm (t) − Xδ (t)| =
n
¯
( Zt µ vξ (s) + P [Qξ ,g (s) + Rξ ,h (s)] ¶ 1 )
¯
m
m i
m
i
α
¯
i=1
= ¯ exp
ds −
α
¯
p(s)P (s)
t0
n
( Zt µ vδ (s) + P [Qδ,g (s) + Rδ,h (s)] ¶ 1
i
i
α
i=1
− exp
p(s)P (s)α
t0
n
( Zt µ v0 (s) + P [Q
b i (s) + 2α R
bi (s)] ¶ 1
α
i=1
5 exp
Zt
×
t0
)
ds ×
p(s)P (s)α
t0
)¯
¯
¯
ds ¯
¯
n
¯µ v (s) + P [Q
1
ξm
ξm ,gi (s) + Rξm ,hi (s)] ¶ α
¯
1 ¯
i=1
−
1 ¯
P (s)α
p(s) α ¯
µ vδ (s) +
−
n
P
[Qδ,gi (s) + Rδ,hi (s)] ¶ α1 ¯¯
¯
¯ ds.
¯
P (s)α
i=1
By means of the inequality |xλ − y λ | 5 |x − y|λ for x, y ∈ R+ and 0 < λ < 1,
we find that the integrand of the last integral in the previous inequality is
bounded from above by the function
n
¯µ v (t) + P [Q
1
ξm ,gi (t) + Rξm ,hi (t)] ¶ α
¯ ξm
¯
i=1
−
¯
¯
P (t)α
n
P
µ vδ (t) +
[Qδ,gi (t) + Rδ,hi (t)] ¶ α1 ¯¯
¯
i=1
−
¯5
¯
P (t)α
n
n
P
P
à |vξ (t)−vδ (t)|+ |Qξ ,g (t)−Qδ,g (t)|+ |Rξ ,h (t)−Rδ,h (t)| ! α1
m
m i
i
m
i
i
i=1
i=1
5
α
P (t)
if α > 1.
Similarity, using the mean value theorem, we find that
n
¯µ v (t) + P [Q
1
ξm ,gi (t) + Rξm ,hi (t)] ¶ α
¯ ξm
¯
i=1
−
¯
¯
P (t)α
n
P
µ vδ (t) +
[Qδ,gi (t) + Rδ,hi (t)] ¶ α1 ¯¯
¯
i=1
−
¯5
¯
P (t)α
Generalized Regularly Varying Solutions . . .
|vξm (t)−vδ (t)|+
n
P
i=1
5 C1
143
|Qξm ,gi (t)−Qδ,gi (t)|+
n
P
i=1
|Rξm ,hi (t)−Rδ,hi (t)|
P (t)α
if α 5 1,
b i (t0 ) and R
bi (t0 ). Accordingly,
where C1 is a constant depending only on α, Q
the continuity of Φ is guaranteed if we prove that the two sequences
n
P
|vξm (t)−vδ (t)|
,
P (t)α
i=1
|Qξm ,gi (t)−Qδ,gi (t)|+
n
P
|Rξm ,hi (t)−Rδ,hi (t)|
i=1
P (t)α
(4.25)
converge to 0 on any compact subinterval of [t∗ , ∞). In fact, it can be shown
more strongly that they converge to 0 uniformly on [t∗ , ∞). The uniform
convergence of the second sequence in (4.25) follows from the Lebesgue
dominated convergence theorem applied to the inequality
n
P
i=1
|Qξm ,gi (t) − Qδ,gi (t)| +
5
Z∞ " X
n
n
P
i=1
P (t)α
|Rξm ,hi (t) − Rδ,hi (t)|
5
¯ ³
³ δ(g (s)) ´¯¯
¯ ξm (gi (s)) ´
i
¯+
qi (s)¯¯ϕ
−ϕ
¯
ξ
(s)
δ(s)
m
i=1
t
+
n
X
i=1
#
¯ ³
³ δ(h (s)) ´¯¯
¯ ξm (hi (s)) ´
i
¯ ds
−ϕ
ri (s)¯¯ϕ
¯
ξm (s)
δ(s)
for t = t0 . To examine the first sequence in (4.25) we proceed as follows.
Using (4.23) and the mean value theorem, we obtain
Z∞
|vξm (t) − vδ (t)|
1
5α
×
1
α
P (t)
p(s) α P (s)α+1
t
¯
n
¯³
´1+ α1
X
¯
× ¯ vξm (s) +
−
[Qξm ,gi (s) + Rξm ,hi (s)]
¯
i=1
³
− vδ (s) +
n
X
i=1
¯
´1+ α1 ¯
¯
[Qδ,gi (s) + Rδ,hi (s)]
¯ ds.
¯
Therefore, we have
· Z∞
5 ατ1
1
1
t
p(s) α
|vξm (t) − vδ (t)|
5
P (t)α
¸
Z∞
|vξm (s) − vδ (s)|
1
Sm,n (s)
ds
+
ds
,
1
P (s)α+1
p(s) α P (s)α+1
t
(4.26)
144
Tomoyuki Tanigawa
where τ1 is a positive constant defined by
½ n
¾ α1
³
1´ X b
bi (t0 )]
τ1 = 1 +
2
[Qi (t0 ) + 2α R
α
i=1
(4.27)
and Sm,n (t) is defined by
n h
X
¯
¯ ¯
¯i
¯Qξm ,gi (t) − Qδ,gi (t)¯ + ¯Rξm ,hi (t) − Rδ,hi (t)¯ .
Sm,n (t) =
i=1
Note that τ1 < 1 by (4.27). Letting
Z∞
|vξm (s) − vδ (s)|
Zm (t) =
ds,
1
p(s) α P (s)α+1
(4.28)
t
we derive from (4.26) the following differential inequality for zm (t):
Z∞
¡
¢0
P (t)ατ1 −1
Sm,n (s)
ατ1
P (t) Zm (t) = −ατ1
ds.
(4.29)
1
1
p(t) α
p(s) α P (s)α+1
t
Noting that P (t)ατ1 Zm (t) → ∞ and that the right-hand side of (4.29) is
integrated over [t, ∞), we obtain
Z∞
1
Sm,n (s)
Zm (t) 5
ds, t = t0 .
(4.30)
1
ατ
1
P (t)
p(s) α P (s)1+α−ατ1
t
Combining (4.26) with (4.30), we have
|vξm (t) − vδ (t)|
5
P (t)α
·
5 ατ1
1
P (t)ατ1
5
Z∞
Sm,n (s)
1
t
p(s) α P (s)1+α−ατ1
ατ1
P (t)ατ1
Z∞
ds +
Sm,n (s)
1
t
Sm,n (s)
1
t
Z∞
p(s) α P (s)1+α−ατ1
p(s) α P (s)α+1
¸
ds 5
ds, t = t0 .
This shows that |uξm (t) − vδ (t)|/P (t)α converges to 0 uniformly on [t∗ , ∞).
We therefore conclude that the mapping Φ defined by (4.24) is continuous
in the topology of C[t∗ , ∞). Thus, all the hypotheses of the Schauder–
Tychonoff fixed point theorem are fulfilled, and hence there exists ξ0 (t) ∈ Ξ
satisfying the half-linear functional differential equation
n
£
¤
¡
¢0 X
qξ0 ,gi (t) + rξ0 ,hi (t) ϕ(ξ0 (t)) = 0, t = t0 ,
p(t)ϕ(ξ00 (t)) +
i=1
which is rewritten as
n h
¡
¢
¡
¢i
¡
¢0 X
qi (t)ϕ ξ0 (gi (t)) + ri (t)ϕ ξ0 (hi (t)) = 0, t = t0 .
p(t)ϕ(ξ00 (t)) +
i=1
Generalized Regularly Varying Solutions . . .
145
This implies that the equation (A+ ) has a n-SVP solution ξ0 (t) existing on
[t0 , ∞).
(The existence of a n-RVP (1) solution of (A+ )): Next, we will be concerned with the construction of a n-RVP (1) solution of equation
© (A+ ) under
ª
the condition (1.9). Choose t1 > a so large that t∗ = min
inf gi (t) >
i=1,2,...,n
t=t1
max{a, 1},
v
u n
uX £
¤
1´
b i (t1 ) + 2α R
bi (t1 ) 5 1, t = t1
1+
[K + L + α]t
Q
α
i=1
(4.31)
¶1
n
¤ α
3 X£b
P (hi (t))
αb
+
5 log 2, t = t1 .
Qi (t1 ) + 2 Ri (t1 )
log
2 i=1
P (t)
(4.32)
³
and
µ
Let H denote the set of all continuous nondecreasing functions η(t) on [t∗ , ∞)
satisfying
η(t) = 1 for t∗ 5 t 5 t1 ;
n
( Zt µ 3 + P [Qi (s) + 2α Ri (s)] ¶ 1
2
α
i=1
1 5 η(t) 5 exp
)
ds
p(s)P (s)α
t1
(4.33)
for t = t1 ; (4.34)
η(hi (t))
5 2 for t = t1 , i = 1, 2, . . . , n.
η(t)
(4.35)
For any η ∈ H we consider the differential equation
n
¡
¢0 X
0
p(t)ϕ(x (t)) +
[qη,gi (t) + rη,hi (t)]ϕ(x(t)) = 0, t = t1 ,
(4.36)
i=1
where
qη,gi (t) = qi (t)ϕ
³ η(g (t)) ´
³ η(h (t)) ´
i
i
and rη,hi (t) = ri (t)ϕ
, i = 1, 2, . . . , n.
η(t)
η(t)
Since η(gi (t))/η(t) 5 1 and η(hi (t))/η(t) 5 2, we have
qη,gi (t) 5 qi (t) and rη,hi (t) 5 2α ri (t), t = t1 for i = 1, 2, . . . , n,
so that
Z∞
Z∞
α
Qη,gi (t) := P (t) qη,gi (s) ds 5 P (t) qi (s) ds = Qi (t), t = t1 ,
α
t
Z∞
Rη,hi (t) := P (t)
(4.37)
t
Z∞
rη,hi (s) ds 5 2 P (t) ri (s) ds = 2α Ri (t), t = t1 . (4.38)
α
α
t
α
t
146
Tomoyuki Tanigawa
Accordingly, from (4.31) we have
v
u n
³
´
uX £
¤
1
b η,g (t1 ) + R
bη,h (t1 ) 5 1,
Q
1+
[K + L + α]t
i
i
α
i=1
(4.39)
b η,g (t) = sup Qη,g (s) and R
bη,h (t) = sup Rη,h (s). Moreover, we
where Q
i
i
i
i
s=t
s=t
1
α
notice that from K + L + α > 2 and 1 + > 1 follows
v
u n
uX £
¤
t
b η,g (t1 ) + R
bη,h (t1 ) 5 1 .
Q
i
i
2
i=1
(4.40)
This enables us to apply Theorem 3.1 and thus we conclude that half-linear
differential equation (4.36) has a n-RVP (1) solution of the form
n
( Zt µ 1 + P [Qη,g (s) + Rη,h (s)] − wη (s) ¶ 1 )
i
i
α
i=1
Xη (t) = exp
ds , t = t1 ,
α
p(s)P (s)
t1
(4.41)
where wη (t) is a solution of the integral equation
Z∞
α
Fη (s, wη (s))
wη (t) =
ds, t = t1 ,
(4.42)
1
P (t)
p(s) α
t
s
n
P
b η,g (t1 ) + R
bη,h (t1 )] for t = t1 . Furthermore, it
satisfying |wη (t)| 5
[Q
i
i
i=1
follows from (4.40) that |wη (t)| 5 1/2 for t = t1 . Here Fη (t, wη (t)) is
¯
¯1+ α1 ³
n
¯ X
¯
£
¤
1´
+ 1+
Fη (t, wη ) = ¯¯1+
wη −1, t = t1 .
Qη,gi (t)+Rη,hi (t) −wη ¯¯
α
i=1
Denote by Ψ the mapping which assigns to each η ∈ H the function Ψη(t)
defined by
Ψη(t) = 1 for t∗ 5 t 5 t1 , Ψη(t) = Xη (t) for t = t1 .
(4.43)
(i) Ψ is a self-map on H. For any η ∈ H from (4.37) and (4.38), for t = t1
we find that
¯
¯
n
X
¯
¯
£
¤
¯1 +
¯5
Q
(t)
+
R
(t)
−
w
(t)
ξ,gi
ξ,hi
η
¯
¯
i=1
51+
n
X
£
¤
Qi (t) + 2α Ri (t) + |wη (t)| 5
i=1
5
n
¤
3 X£
Qi (t) + 2α Ri (t) ,
+
2 i=1
Generalized Regularly Varying Solutions . . .
147
or accordingly,
n
( Zt µ 3 + P [Qi (s) + 2α Ri (s)] ¶ 1
2
α
i=1
Xη (t) 5 exp
ds , t = t1 .
p(s)P (s)α
t1
)
Moreover, we have
Ψη(hi (t))
= exp
Ψη(t)
n
( hZi (t) (1 + P [Qη,g (s) + Rη,h (s)] − wη (s)) α1
i
i
i=1
ds
1
p(s) α P (s)
t
n
1
( hZi (t) ¡ 3 + P [Qi (s) + 2α Ri (s)]¢ α
2
i=1
5 exp
1
p(s) α P (s)
t
)
5
)
ds
5
(µ
)
¶ α1 hZi (t)
n
¤
3 X£b
ds
bi (s)
5 exp
+
Qi (s) + 2α R
5
1
2 i=1
p(s) α P (s)
t
(µ
)
¶1
n
¤ α
3 X£b
P (hi (t))
αb
5 exp
log
+
Qi (s)+2 Ri (s)
5 2, t = t1 .
2 i=1
P (t)
(ii) Ψ(H) is relatively compact in C[t∗ , ∞). This is a consequence of the
inclusion Ψ(H) ⊂ H and the following inequality holding for any η ∈ H:
05
d
d
Ψη(t) =
Xη (t) =
dt
dt
n
P
µ1 +
[Qη,gi (t) + Rη,hi (t)] − wη (t) ¶ α1
i=1
p(t)P (t)α
n
P
bi (t1 )] ¶ 1
b i (t1 ) + 2α R
µ 32 +
[Q
α
5
i=1
(µ
× exp
p(t)P (t)α
Xη (t) 5
×
)
¶ α1
n
¤
3 X£b
P
(t)
bi (t1 )
+
Qi (t1 ) + 2α R
log
.
2 i=1
P (t1 )
(iii) Ψ is continuous in the topology of C[t∗ , ∞). Let {ηn } be a sequence
in H converging to θ ∈ H, which amounts to supposing that the sequence
{ηn (t)} converges to θ(t) uniformly on the compact subintervals of [t∗ , ∞).
We will show that {Ψηn (t)} converges to Ψθ(t) uniformly on the compact
subintervals [t∗ , ∞). In order to simplify notation, for arbitrary η ∈ H we
denote
n
P
1+
[Qη,gi (t) + Rη,hi (t)] − wη (t)
i=1
, t = t1 .
(4.44)
Vη (t) =
P (t)α
148
Tomoyuki Tanigawa
In view of (4.41), we have
¯
¯ ¯
¯
¯Ψηm (t) − Ψθ(t)¯ = ¯Xηm (t) − Xθ (t)¯ =
¯
½ Zt ³
¾
½ Zt ³
¾¯¯
¯
Vηm (s) ´ α1
Vθ (s) ´ α1
¯
¯
= ¯ exp
ds − exp
ds ¯ 5
¯
¯
p(s)
p(s)
t1
t1
( Zt µ 3 +
2
5 exp
n
P
[Qi (s) + 2α Ri (s)] ¶ α1
i=1
ds ×
p(s)P (s)α
t1
Zt
×
t1
1
p(s)
1
α
)
¯
¯
1
1 ¯
¯
¯(Vηm (s)) α − (Vθ (s)) α ¯ ds.
As in the previous part of the proof, we can verify that the integrand of the
last integral is bounded from the above by
à P |Qη
n
i=1
m ,gi
(t)−Qθ,gi (t)|+
n
P
i=1
|Rηm ,hi (t)−Rθ,hi (t)|+|wηm (t)−wθ (t)| ! α1
P (t)α
if α > 1,
n
P
C2
i=1
|Qηm ,gi (t)−Qθ,gi (t)|+
n
P
i=1
|Rηm ,hi (t)−Rθ,hi (t)|+|wηm (t)−wθ (t)|
P (t)α
if α 5 1,
b i (t1 ) and R
bi (t1 ). Accordingly,
where C2 is a constant depending only on α, Q
it suffices to prove the uniform convergence to 0 on the compact subintervals
of the two sequences
|wηm (t) − wθ (t)|
πm,n (t)
and
,
P (t)α
P (t)α
where
πm,n (t) =
n
X
¯
¯Qη
m ,gi
i=1
n
¯ X
¯
¯
¯Rη ,h (t) − Rθ,h (t)¯.
(t) − Qθ,gi (t)¯ +
m
i
i
i=1
The uniform convergence of the sequence πm,n (t)/P (t)α is an immediate
consequence of the Lebesgue dominated convergence theorem. Therefore,
let us examine the sequence |wηm (t) − wθ (t)|/P (t)α . Applying the mean
value theorem to Fηm (t, wηm (t)) and Fθ (t, wθ (t)) in |wηm (t) − wθ (t)|/P (t)α ,
Generalized Regularly Varying Solutions . . .
149
we obtain for t = t1
´
¯
¯ ³
¯Fηm (t, wηm (t)) − Fθ (t, wθ (t))¯ 5 1 + 1 |wηm (t) − wθ (t)|+
α
¯¯
¯1+ α1
n
¯¯
X
¯
£
¤
¯¯
+ ¯ ¯1 +
Qηm ,gi (t) + Rηm ,hi (t) − wηm (t)¯¯
−
¯
i=1
¯
¯1+ α1 ¯¯
n
X
¯
¯
£
¤
¯
− ¯¯1 +
Qθ,gi (t) + Rθ,hi (t) − wθ (t)¯¯
¯5
¯
i=1
³
³
1´
1´
5 1+
(1 + τ2 )|wηm (t) − wθ (t)| + 1 +
τ2 πn,m (t),
α
α
b i (t1 ) and R
bi (t1 ).
where τ2 is a positive constant depending only on α, Q
α
Consequently, the sequence |wηm (t) − wθ (t)|/P (t) implies
|wηm (t) − wθ (t)|
(α + 1)(1 + τ2 )
5
P (t)α
P (t)α+1
+
Zt
(α + 1)τ2
P (t)α+1
Zt
|wηm (s) − wθ (s)|
ds+
P (s)α
t1
πm,n (s)
1
p(s) α
t1
ds, t = t1 .
(4.45)
Putting for simplicity
Zt
Wm (t) =
t1
|wηm (s) − wθ (s)|
ds,
P (s)α
(4.46)
we transform (4.45) into
¡
¢0
P (t)−(α+1)(1+τ2 ) Wm (t) 5
Zt
(α + 1)τ2
πm,n (s) ds, t = t1 ,
1
p(t) α P (t)(α+1)(1+τ2 )+1
t1
which, after integration over [t1 , t], yields
τ2
Wm (t) 5
P (t)(α+1)(1+τ2 )
1 + τ2
Zt
t1
πm,n (s)
ds, t = t1 .
P (s)(α+1)(1+τ2 )
Combining (4.45) with (4.47), we have
(α + 1)τ2
|wηm (t) − wθ (t)|
5
P (t)α
P (t)−(α+1)τ2
+
(α + 1)τ2
P (t)α+1
Zt
Zt
t1
πm,n (s)
ds+
P (s)(α+1)(1+τ2 )
πm,n (s)
1
t1
p(s) α
ds, t = t1 .
(4.47)
150
Tomoyuki Tanigawa
This ensures the desired convergence of the sequence |wηm (t)−wθ (t)|/P (t)α ,
whence the continuity of the mapping Ψ has been assured. Thus, all the
hypotheses of the Schauder–Tychonoff fixed point theorem are fulfilled, and
so there exists η0 ∈ H such that η0 = Ψη0 . Since η0 (t) = Xη0 (t) for t = t1 ,
η0 (t) satisfies the differential equation
¡
n
¢0 X
£
¤
p(t)ϕ(η00 (t)) +
qη0 ,gi (t) + rη0 ,hi (t) ϕ(η0 (t)) = 0, t = t1
i=1
or
¡
n h
¢0 X
¡
¢
¡
¢i
p(t)ϕ(η00 (t)) +
qi (t)ϕ η0 (gi (t)) + ri (t)ϕ η0 (hi (t)) = 0, t = t1 .
i=1
Therefore, η0 (t) is a desired n-RVP (1) solution of the functional differential
equation (A+ ) on [t1 , ∞).
The proof of Theorem 1.1 for the equation (A− ).
(The existence of a n-SVP solution of ©(A− )): Suppose
that (1.9) holds.
ª
Choose t0 > a so large that t∗ = min
inf gi (t) > max{a, 1} and such
i=1,2,...,n
t=t0
that
n
n
o
³ X
¤´ 1
£ α
b i (t0 ) + R
bi (t0 ) α max 2, 1 + 1 < 1,
2
2 Q
α
i=1
n
³ X
¤´ 1
£ α
b i (t0 ) + R
bi (t0 ) α log
2
2 Q
i=1
P (t)
< log 2,
P (gi (t))
(4.48)
(4.49)
b i (t) and R
bi (t) are defined by (4.6)
are satisfied for all t = t0 , where Q
and (4.7).
Let M denote the set of all positive continuous nonincreasing functions
µ(t) on [t∗ , ∞) with the properties
µ(t) = 1 for t∗ 5 t 5 t0 ;
n
¡
¢1
( Zt 2 P [2α Q
b i (s) + R
bi (s)] α )
i=1
µ(t) = exp −
ds for t = t0 ;
1
p(s) α P (s)
(4.50)
(4.51)
t0
µ(gi (t))
5 2 for t = t0 , i = 1, 2, . . . , n.
µ(t)
(4.52)
We here consider the following differential equation:
¡
n
¢0 X
£
¤
p(t)ϕ(x0 (t)) =
qµ,gi (t) + rµ,hi (t) ϕ(x(t))
(4.53)
i=1
where, for arbitrary µ ∈ M, the functions qµ,gi (t) and rµ,hi (t) are defined
by (4.2). In view of Theorem 3.1, for each µ ∈ M, the equation (4.53) has
Generalized Regularly Varying Solutions . . .
151
a n-SVP solution Xµ (t) having the representation
n
( Zt µ rµ (s)− P [Qµ,g (s)+Rµ,h (s)] ¶ 1
)
i
i
α∗
i=1
Xµ (t) = exp
ds , t = t0 , (4.54)
p(s)P (s)α
t0
where rµ (t) is a solution of the integral equation
n
P
1
[Qµ,gi (s) + Rµ,hi (s)]|1+ α
Z∞ |rµ (s) −
i=1
rµ (t) = αP (t)α
ds, t = t0 , (4.55)
1
p(s) α P (s)α+1
t
satisfying the inequality
0 5 rµ (t) 5
n
X
£
¤
b µ,g (t0 ) + R
bµ,h (t0 ) 5
Q
i
i
i=1
5
n
X
£ α
¤
b i (t0 ) + R
bi (t0 ) , t ≥ t0 .
2 Q
(4.56)
i=1
b µ,g (t) = sup Qµ,g (s)
Here Qµ,gi (t) and Rµ,hi (t) are defined by (4.19) and Q
i
i
s=t
bµ,h (t) = sup Rµ,h (s). Furthermore, using the decreasing nature of
and R
i
i
s=t
µ(t), we have
qµ,gi (t) 5 2α qi (t) and rµ,hi (t) 5 ri (t), t = t0 , i = 1, 2, . . . , n,
accordingly,
n
n
X
£
¤ X
£ α
¤
Qµ,gi (t) + Rµ,hi (t) 5
2 Qi (t) + Ri (t) 5
i=1
i=1
5
n
X
£ α
¤
b i (t0 ) + R
bi (t0 ) , t = t0 .
2 Q
(4.57)
i=1
Let us now define H to be the mapping which assigns to each µ ∈ M the
function Hµ given by
Hµ(t) = 1 for t∗ 5 t 5 t0 , Hµ(t) = Xµ (t) for t = t0 .
(4.58)
Proceeding as in the proof of the existence of n-SVP solution of (A+ ), it can
be proved that H maps M into a relatively compact subset of M with the
help of the Schauder–Tychonoff fixed point theorem, so that there exists a
µ0 ∈ M such that
µ0 (t) = Xµ0 (t) =
n
( Zt µ rµ (s) − P [Qµ ,g (s) + Rµ ,h (s)] ¶ 1
0
0 i
0
i
α∗
i=1
= exp
t0
p(s)P (s)α
)
ds , t = t0 .
152
Tomoyuki Tanigawa
This means that µ0 (t) is a solution satisfying the functional differential
equation
¡
¢0
p(t)ϕ(µ00 (t))
n
X
£
¤
qµ0 ,gi (t) + rµ0 ,hi (t) ϕ(µ0 (t)), t = t0
=
i=1
or consequently,
n h
¡
¢0 X
¡
¢
¡
¢i
p(t)ϕ(µ00 (t)) =
qi (t)ϕ µ0 (gi (t)) + ri (t)ϕ µ0 (hi (t)) , t = t0 .
i=1
Therefore, we conclude that the equation (A− ) has a n-SVP solution.
(The existence of a n-RVP (1) solution of (A− )):
© Suppose
ª that (1.9) is
satisfied. Choose t1 > a so large that t∗ = min
inf gi (t) > max{a, 1}
i=1,2,...,n
t=t1
v
u n
³
´
uX £
¤
1 e e
b i (t1 ) + 2α R
bi (t1 ) 5 1
1+
[K + L + α]t
Q
α
i=1
and
v
) α1
u n
uX
P (hi (t))
α
t
b
1+
log
[Qi (t1 ) + 2 Ri (t1 )]
5 log 2,
P (t)
i=1
(4.59)
(
(4.60)
b i (t) and R
bi (t) are defined by (4.6) and
where the functions Qi (t), Ri (t), Q
(4.7), while
³ ´ 1
³ ´1− 1
4 1− α


α
 4

if α > 1
if α > 1
3
e =
e
3
K
.
(4.61)
and
L
=
³ 3 ´ α1 −1




1
if α 5 1
if α 5 1
2
Let K define the set of all positive continuous nondecreasing functions ν(t)
on [t∗ , ∞) satisfying
ν(t) = 1 for t∗ 5 t 5 t1 ;
½ Zt ³
¾
1 + ρ(s) ´ α1
1 5 ν(t) 5 exp
ds
for t = t1 ;
p(s)P (s)α
(4.62)
(4.63)
t1
ν(hi (t))
5 2 for t = t1 , i = 1, 2, . . . , n,
ν(t)
(4.64)
where ρ(t) is a solution of the integral equation
n
¤
1´X£b
bi (t1 ) 1
Qi (t1 ) + 2α R
ρ(t) = 1 +
α i=1
P (t)
³
Zt e
e + α]
[Lρ(s) + L
1
t1
p(s) α
ds. (4.65)
Generalized Regularly Varying Solutions . . .
153
In order to verify that ρ(t) is a solution of (4.65), we now consider the
integral operator R defined by
n
³
¤
1´X£b
bi (t1 ) ×
Rρ(t) = 1 +
Qi (t1 ) + 2α R
α i=1
1
×
P (t)
Zt
t1
1
p(s)
1
α
£
¤
e
e + α ds (4.66)
Lρ(s)
+L
on the set
(
P=
v
)
u n
uX £
¤
α
t
b
b
ρ ∈ C0 [t1 , ∞) : 0 5 ρ(t) 5
Qi (t1 ) + 2 Ri (t1 ) , t = t1 .
i=1
It is easy to see that R sends P into itself and satisfies
n
´
°
° X
£
¤³
e 1 − ρ2 k0 , ρ1 , ρ2 ∈ R.
b i (t1 ) + 2α R
bi (t1 ) 1 + 1 Lkρ
°Rρ1 − Rρ2 ° 5
Q
α
i=1
Therefore, there exists a unique fixed point of R which solves the integral
equation (4.65).
Consider a family of half-linear differential equations
n
¡
¢0 X
£
¤
p(t)ϕ(x0 (t)) =
qν,gi (t) + rν,hi (t) ϕ(x(t)), t = t1 ,
(4.67)
i=1
where, for any ν ∈ K, the functions qν,gi (t) and rν,hi (t) are defined by
³ ν(g (t)) ´
³ ν(h (t)) ´
i
i
qν,gi (t) = qi (t)ϕ
and rν,hi (t) = ri (t)ϕ
.
ν(t)
ν(t)
b ν,g (t) =
Then, we define Qν,gi (t), Rν,hi (t) for every ν ∈ K by (4.19) and Q
i
bν,h (t) = sup Rν,h (t). It follows from Theorem 3.1 that for
sup Qν,gi (t), R
i
i
s=t
s=t
each ν ∈ K, the equation (4.67) has a n-RVP (1) solution Xν (t) expressed
in the form
n
( Zt µ 1 − P [Qν,g (s) + Rν,h (s)] + wν (s) ¶ 1 )
i
i
α
i=1
Xν (t) = exp
ds , (4.68)
α
p(s)P (s)
t1
t = t1 ,
where wν (t) is a solution of the integral equation
α
wν (t) =
P (t)
Zt e
Fν (s, wν (s))
1
t1
p(s) α
ds, t = t1
(4.69)
154
Tomoyuki Tanigawa
and
n
¯
¯1+ α1
³
X
£
¤
1´
¯
¯
Feν (t, wν ) = 1 + 1 +
wν − ¯1 −
Qν,gi (t) + Rν,hi (t) + wν ¯
. (4.70)
α
i=1
We notice that for some fixed ν ∈ K, wν (t) is a fixed point of the contraction
mapping Fν defined by
α
Fν wν (t) =
P (t)
Zt e
Fν (s, wν (s))
1
p(s) α
t1
ds, t = t1 ,
(4.71)
which satisfies
v
u n
uX £
¤
b ν,g (t1 ) + R
bν,h (t1 ) , t = t1 .
|wν (t)| 5 t
Q
i
i
(4.72)
i=1
Furthermore, using the increasing nature of ν(t), we obtain
qν,gi (t) 5 qi (t), rν,hi (t) 5 2α ri (t) for t = t1 , ν ∈ K,
or consequently,
n
X
£
¤
Qν,gi (t) + Rν,hi (t) 5
i=1
5
n
n
X
¤
£
¤ X
£
bi (t1 ) , t = t1 .
b i (t1 ) + 2α R
Qi (t) + 2α Ri (t) 5
Q
i=1
(4.73)
i=1
We will show that for every ν ∈ K,
|wν (t)| 5 ρ(t), t = t1 .
(4.74)
To this end, it is convenient to express Feν (t, wν ) as
e ν (t, wν ) + H
e ν (t, wν ) + e
Feν (t, wν ) = G
kν (t),
e ν (t, wν ), H
e ν (t, wν ) and e
where G
kν (t) are defined, respectively, by
n
³
X
£
¤´1+ α1
e ν (t, wν ) = 1 −
G
+
Qν,gi (t) + Rν,hi (t)
i=1
³
n
X
£
¤´1+ α1
1 ´³
1−
Qν,gi (t) + Rν,hi (t)
wν −
α
i=1
n
¯1+ α1
¯
X
£
¤
¯
¯
Qν,gi (t) + Rν,hi (t) + wν ¯
− ¯1 −
,
+ 1+
i=1
¾
n
´½
³
X
£
¤´ α1
e ν (t, wν ) = 1 + 1
Qν,gi (t) + Rν,hi (t)
H
1− 1−
wν ,
α
i=1
³
Generalized Regularly Varying Solutions . . .
and
155
n
³
X
£
¤´1+ α1
e
kν (t) = 1 − 1 −
Qν,gi (t) + Rν,hi (t)
.
i=1
Using the mean value theorem, we find that for some θ ∈ (0, 1) the inequalities hold:
¯1
n
³
´¯¯
³X
¯
¯
£
¤´¯ α −1
e ν (t, wν (t))¯ 5 1 1+ 1 ¯1−(1−θ)
¯H
¯
×
Q
(t)+R
(t)
ν,gi
ν,hi
¯
α
α ¯
i=1
n
X
£
¤
×
Qν,gi (t) + Rν,hi (t) |wν (t)|
i=1
n
¤
1³
1 ´e X £ b
bi (t1 ) |wν (t)|
5
1+
L
Qi (t1 ) + 2α R
α
α
i=1
and
(4.75)
¯
¯
¯e
kν (t)¯ 5
¯
¯1 n
n
X
£
¤
£
¤¯ α X
1 ´¯¯
5 1+
Qν,gi (t)+Rν,hi (t) 5
1−(1−θ)
Qν,gi (t)+Rν,hi (t) ¯¯
¯
α
i=1
i=1
³
n
³
¤
1´X£b
bi (t1 ) , t = t1 .
Qi (t1 ) + 2α R
5 1+
α i=1
(4.76)
Moreover, by means of the mean value theorem and L’Hospital rule, it
follows that
³
´
¯
¯
e ν2 (t), t = t1 .
e ν (t, wν (t))¯ 5 1 1 + 1 Lw
¯G
(4.77)
α
α
Let ν ∈ K be fixed. Recalling that ρ and wν are the fixed point of the
contraction mappings R and Fν defined by (4.66) and (4.71), we see that ρ
and wν are constructed, respectively, as the limits as n → ∞ of the sequences
{ρn = Rρn−1 , n = 1, 2, . . . , with ρ0 = 0} and {wn = Fν wn−1 , n =
1, 2, . . . , n, with w0 = 0}. First we note that for t = t1 ,
|w1 (t)| = Fν w0 (t) =
=
α
P (t)
Zt
1
1
p(s) α
t1
α+1
5
P (t)
Zt
t1
5 (α + 1)
·
¸
n
¯
X
£
¤¯¯1+ α1
¯
1 − ¯1 +
ds 5
Qν,gi (s) + Rν,hi (s) ¯
1
p(s)
n
X
i=1
1
α
i=1
n
X
£
¤
Qν,gi (s) + Rν,hi (s) ds 5
i=1
¤
£
b i (t1 )(t) + 2α R
bi (t1 ) 5
Q
´
¤³
£
b i (t1 )(t) + 2α R
bi (t1 ) 1 + 1 [L
e + α] = ρ1 (t), t = t1 .
5 Q
α
156
Tomoyuki Tanigawa
Then, assuming that |wn (t)| 5 ρn (t), t = t1 , for some n ∈ N and using
(4.75), (4.76) and (4.77), we have
|wn+1 (t)| = Fν wn (t) =
α
=
P (t)
Zt
p(s)
t1
α+1
5
P (t)
1
Zt
1
α
h¯
¯ ¯
¯ ¯
¯i
e ν (s, wn (s))¯ + ¯H
e ν (s, wn (s))¯ + ¯e
¯G
kν (s)¯ ds 5
1
1
p(s) α
t1
·e
n
eX
£
¤
L 2
L
b i (t1 ) + 2α R
bi (t1 ) |wn (s)|+
wν (s) +
Q
α
α i=1
+
¸
n
X
£
¤
b i (t1 ) + 2α R
bi (t1 ) ds 5
Q
i=1
³
n
¤
1´X£b
bi (t1 ) 1
5 1+
Qi (t1 )+2α R
α i=1
P (t)
Zt e e
[L + Lρn (s) + α]
1
t1
p(s) α
ds = ρn+1 (t).
Therefore, inductive arguments ensure the validity of (4.74).
We define by M the mapping which assigns to each ν ∈ K the function
Hν(t) defined by
Mν(t) = 1 for t∗ 5 t 5 t1 , Mν(t) = Xν (t) for t = t1 .
(i) M is a self-map on K, since it readily follows from (4.60) and 0 5 ρ(t) 5
s
n
P
b i (t) + 2α R
bi (t1 )] , t = t1 that
[Q
i=1
½ Zt ³
1 5 Xν (t) 5 exp
t1
¾
1 + ρ(s) ´ α1
, t = t1 for any ν ∈ K
p(s)P (s)α
and
Mν(hi (t))
=
Mν(t)
n
( hZi (t)µ 1 − P [Qν,g (s) + Rν,h (s)] + wν (s) ¶ 1
i
i
α∗
i=1
= exp
p(s)P (s)α
t
( hZi (t)
³
5 exp
(µ
5 exp
t
)
ds
5
)
1 + ρ(s) ´ α1
ds 5
p(s)P (s)α
v
)
u n
¶ α1
uX
P
(h
(t))
i
b i (t1 ) + 2α R
bi (t1 )]
5 2, t = t1 .
1 + t [Q
log
P (t)
i=1
Generalized Regularly Varying Solutions . . .
157
(ii) M(K) is relatively compact in C[t∗ , ∞). The inclusion M(K) ⊂ K
shows that M(K) is locally uniformly bounded on [t∗ , ∞). Since
05
d
d
Mν(t) =
Xν (t) =
dt
dt
n
P
µ1 −
[Qν,gi (t) + Rν,hi (t)] + wν (t) ¶ α1
i=1
=
Xν (t) 5
p(t)P (t)α
s
n
o1
n
P
b i (t1 ) + 2α R
bi (t1 )] α
1+
[Q
i=1
5
×
1
p(t) α P (t)
v
(µ
)
u n
¶ α1
uX
P
(t)
b i (t1 ) + 2α R
bi (t1 )]
× exp
1 + t [Q
log
,
P (t1 )
i=1
we conclude that M(K) is locally equi-continuous on [t∗ , ∞).
(iii) M is continuous in the topology of C[t∗ , ∞). Let {νm (t)} be a sequence
in K converging to δ(t) uniformly on compact subintervals of [t∗ , ∞). We
have to prove that {Mνm (t)} converges to Mδ(t) uniformly on any compact
subintervals of [t∗ , ∞). In order to simplify the notation, for arbitrary ν ∈ K
we define
1−
n
P
i=1
Zν (t) =
[qν,gi (t) + Rν,hi (t)] + wν (t)
, t = t1 .
P (t)α
(4.78)
Then, using (4.68) and the mean value theorem, we get
¯
¯ ¯
¯
¯Mνm (t) − Mδ(t)¯ = ¯Xνm (t) − Xδ (t)¯ =
¯
¾
½ Zt ³
¾¯¯
½ Zt ³
¯
Zδ (s) ´ α1
Zνm (s) ´ α1
¯
¯
= ¯ exp
ds − exp
ds ¯ 5
¯
¯
p(s)
p(s)
t1
t1
½ Zt ³
5 exp
t1
¾ Zt
1
1
|(Zνm (s)) α − (Zδ (s)) α |
1 + ρ(s) ´ α1
ds
ds.
1
p(s)P (s)α
p(s) α
t1
As in the proof of the existence of a n-RVP (1) solution of the equation (A+ ),
we can show that the integrand of the last integral is bounded from above
by the functions
µ
n
P
i=1
|Qνm ,gi (t)−Qδ,gi (t)|+
n
P
i=1
|Rνm ,hi (t)−Rδ,hi (t)|+|wνm (t)−wδ (t)| ¶ α1
P (t)α
if α > 1,
158
Tomoyuki Tanigawa
n
P
C3
i=1
|Qνm ,gi (t)−Qδ,gi (t)|+
n
P
i=1
|Rνm ,hi (t)−Rδ,hi (t)|+ |wνm (t)−wδ (t)|
P (t)α
if α 5 1,
where C3 is a positive constant depending only on α and ρ(t1 ). Therefore, it
suffices to prove the uniform convergence to 0 on the compact subintervals
of the two sequences
|wνm (t) − wδ (t)|
and
P (t)α
n
n
P
P
|Qνm ,gi (t) − Qδ,gi (t)| +
|Rνm ,hi (t) − Rδ,hi (t)|
i=1
i=1
P (t)α
(4.79)
Sen,m (t)
=
.
P (t)α
The second sequence in (4.79) can be dealt with exactly as in the case of
n-RVP (1) solution of the equation (A+ ). In order to prove the uniform
convergence of the first sequence in (4.79), we consider Feνm (t, wνm ) and
Feδ (t, wδ ) defined by (4.70). Applying the mean value theorem, for t = t1
we get
´¯
¯
¯ ³
¯
¯Feν (t, wν (t)) − Feδ (t, wδ (t))¯ 5 1 + 1 ¯wν (t) − wδ (t)¯+
m
m
m
α
¯¯
n
¯1+ α1
X
¯¯
£
¤
¯
¯
+ ¯¯1 −
−
Qνm ,gi (t) + Rνm ,hi (t) + wνm (t)¯
i=1
n
¯
¯1+ α1 ¯¯
X
£
¯
¯
¯5
− ¯1 −
Qδ,gi (t) + Rδ,hi (t)] + wδ (t)¯
¯
i=1
³
¯ ³
¯
¯o
1 ´¯¯
1 ´ ne
5 1+
wνm (t)−wδ (t)¯ + 1+
τ3 Sm,n (t)+ ¯wνm (t)−wδ (t)¯ =
α
α
³
¯
¯ ³
1´
1´ e
= 1+
(1 + τ3 )¯wνm (t) − wδ (t)¯ + 1 +
τ3 Sm,n (t),
α
α
b i (t1 ) and R
bi (t1 ).
where τ3 is a positive constant depending only on α, Q
Consequently, the first sequence in (4.79) implies that
|wνm (t) − wδ (t)|
(α + 1)(1 + τ3 )
5
P (t)α
P (t)(α+1)
Zt
|wνm (s) − wδ (s)|
1
t1
p(s) α
Zt e
(α + 1)τ3
Sm,n (s)
+
ds, t = t1 .
1
P (t)α+1
p(s) α
t1
Putting for brevity
Zt
fm (t) =
W
|wνm (s) − wδ (s)|
1
t1
p(s) α
ds,
ds+
(4.80)
Generalized Regularly Varying Solutions . . .
159
fm (t):
we derive the following differential inequality for W
¢
¡
fm (t) 0 5
P (t)−(α+1)(1+τ3 ) W
5
Zt e
Sm,n (s)
(α + 1)τ3
1
1
p(t) α P (t)(α+1)(τ3 +1)+1
t1
p(s) α
ds, t = t1 .
(4.81)
Integrating (4.81) from t1 to t, we obtain
1
fm (t) 5 τ3
W
(α+1)(τ
3 +1)
τ3 +1 P (t)
Zt
Sem,n (s)
1
α
t1
p(s) P (s)(α+1)(τ3 +1)
ds, t = t1 . (4.82)
Using (4.80) and (4.82), we conclude that
|wνm (t) − wδ (t)|
(α + 1)τ3
5
α
P (t)
P (t)(α+1)(τ3 +2)
(α + 1)τ3
+
P (t)α+1
Zt
Sem,n (s)
1
α
t1
p(s) P (s)(α+1)(τ3 +1)
Zt e
Sm,n (s)
1
t1
p(s) α
ds+
ds, t = t1 ,
whence it follows that the sequence |wνm (t) − wδ (t)|/P (t)α converges to
0 uniformly on [t1 , ∞). Therefore, we have proved that the mapping M
is continuous in the topology of C[t∗ , ∞). Thus, applying the Schauder–
Tychonoff fixed point theorem, M has a fixed point ν0 in K. Since ν0 =
Xν0 (t) for t = t1 , ν0 (t) satisfies the functional differential equation
¡
n h
¢0 X
¡
¢
¡
¢i
p(t)ϕ(ν00 (t)) =
qi (t)ϕ ν0 (gi (t)) +ri (t)ϕ ν0 (hi (t)) , t = t1 . (4.83)
i=1
It is obvious that ν0 (t) is a n-RVP (1) solution of the equation (A− ). This
completes the proof of Theorem 1.1.
5. Examples
We here present four examples illustrating application of Theorem 1.1 to
the functional differential equations of the type (A+ ) and (A− ), respectively.
We begin with two examples of the existence of n-SVP and n-RVP (1) solutions of the type (A+ ) with the case i = 1.
Example 5.1. Consider the following functional differential equation
with both retarded and advanced arguments
³ ³
³ ³
¡ −αt
¢0
1 ´´
1 ´´
+ r(t)ϕ x t +
= 0, (5.1)
e ϕ(x0 (t)) + q(t)ϕ x t −
log t
log t
t = e,
160
Tomoyuki Tanigawa
where the functions q(t) and r(t) are given by
·
¸
α ³
λ ´α−1
λ(λ − 1)
λ
λ
q(t) = αt α 1 +
1−
+
+
×
2e t
log t
t log t
t(log t)2
log t
½
¾−αλ
1
³
log(1 − t log
1 ´−α
t)
× 1−
1+
t log t
log t
and
·
¸
λ
λ(λ − 1)
λ ´α−1
λ
r(t) = αt α 1 +
1−
+
+
×
2e t
log t
t log t
t(log t)2
log t
½
¾−αλ
1
³
log(1 + t log
1 ´−α
t)
1+
× 1+
t log t
log t
α
³
for λ being a positive constant. The function p(t) = e−αt satisfies (1.1) and
the function P (t) given by (1.6) is P (t) ∼ et . Moreover, the functions
g(t) = t −
1
1
and h(t) = t +
log t
log t
(5.2)
satisfy the conditions (1.7) and (1.8). The condition (1.9) is satisfied for
this equation, since
Z∞
t
α
q(s) ds ∼ α αt and
2t e
Z∞
h(s) ds ∼
t
α
2tα eαt
as t → ∞.
Therefore, equation (5.1) has a n-SVet solution x(t) by Theorem 1.1. One
such solution is x(t) = t(log t)λ .
Example 5.2. Consider the following functional differential equation:
¡α
¢0
¡
¡
1 ¢
1 ¢
t ϕ(x0 (t)) + q(t)ϕ x(te− t ) + r(t)ϕ x(te t ) = 0, t = ee ,
(5.3)
where the functions q(t) and r(t) are given by
³
αµ
µ ´α−1 ³
µ + 1´
q(t) =
1−
1−
×
α+1
2t(log t)
log2 t
log2 t
log2 t
½
¾αµ
1
³
log(1 − t log
1 ´−α
t)
× 1−
1+
t log t
log2 t
and
r(t) =
³
µ ´α−1 ³
µ + 1´
αµ
1
−
1
−
×
2t(log t)α+1 log2 t
log2 t
log2 t
½
¾αµ
1
³
log(1 + t log
1 ´−α
t)
× 1+
1+
,
t log t
log2 t
respectively, and µ is a positive constant. The function p(t) = tα satisfies
(1.1) and the function P (t) reduces to P (t) ∼ log t, while the functions
Generalized Regularly Varying Solutions . . .
1
161
1
g(t) = te− t and h(t) = te t satisfy the conditions (1.7) and (1.8). Moreover,
since
Z∞
Z∞
αµ
αµ
q(s) ds ∼
and
h(s) ds ∼
2t(log t)α+1 log2 t
2t(log t)α+1 log2 t
t
t
as t → ∞,
the condition (1.9) is satisfied and thus, the equation (5.3) possesses a nRVlog t solution by Theorem 1.1. One such solution is log t/(log2 t)µ .
Next, two examples illustrating application of Theorem 1.1 to the functional differential equation of the type (A− ) with the case i = 1 will be
presented below.
Example 5.3. We consider the functional differential equation with both
retarded and advanced arguments
³ ³
³ ³
¡ −αt
¢0
1 ´´
1 ´´
e ϕ(x0 (t)) = q(t)ϕ x t −
+ r(t)ϕ x t +
, t = e, (5.4)
log t
log t
where the functions q(t) and r(t) are given by
α ³
λ ´α−1
q(t) = α αt 1 −
×
2t e
log t
·³
¸
2 ´³
λ ´
λ ³
λ ´
λ
× 1+
1−
+
1−
+
×
t
log t
t log t
log t
t(log t)2
½
¾−αλ
1
³
log(1 − t log
1 ´α
t)
× 1−
1+
,
t log t
log t
α ³
λ ´α−1
r(t) = α αt 1 −
×
2t e
log t
·³
¸
2 ´³
λ ´
λ ³
λ ´
λ
× 1+
1−
+
1−
+
×
t
log t
t log t
log t
t(log t)2
½
¾−αλ
1
³
log(1 + t log
1 ´α
t)
× 1+
1+
t log t
log t
for λ being a positive constant. As in Example 5.1, it could be shown
without difficulty that all conditions of Theorem 1.1 are satisfied, so that
the equation (5.4) has a n-SVet solution x(t) by Theorem 1.1. One such
solution is (log t)λ /t.
Example 5.4. Consider the following functional differential equation:
¡α
¢0
¡
¡
1 ¢
1 ¢
t ϕ(x0 (t)) = q(t)ϕ x(te− t ) + r(t)ϕ x(te t ) , t = ee ,
(5.5)
where the functions q(t) and r(t) are given by
³
µ ´α−1 ³
µ − 1´
αµ
1
+
1
+
×
q(t) =
2t(log t)α+1 log2 t
log2 t
log2 t
162
Tomoyuki Tanigawa
½
¾−αµ
1
log(1 − t log
1 ´−α
t)
× 1−
1+
t log t
log2 t
³
and
³
µ ´α−1 ³
µ − 1´
αµ
1+
1+
×
α+1
2t(log t)
log2 t
log2 t
log2 t
½
¾−αµ
1
³
log(1 + t log
1 ´−α
t)
× 1+
1+
,
t log t
log2 t
respectively, and µ is a positive constant. As in Example 5.2, it can be
verified that all conditions of Theorem 1.1 are satisfied. Therefore, the
equation (5.5) possesses a n-RVlog t solution x(t). One such solution is
x(t) = log t(log2 t)µ .
r(t) =
Acknowledgement
This research was supported by Grant-in-Aid for Scientific Research (C)
(No. 23540218), the Ministry of Education, Culture, Sports, Science and
Technology, Japan.
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787.
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7. J. Manojlović and T. Tanigawa, Regularly varying solutions of half-linear differential equations with retarded and advanced arguments. (Submitted for publication)
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(Received 16.06.2012)
Author’s address:
Kumamoto University, Department of Mathematics, Faculty of Education, 2 – 40 – 1, Kurokami, Chuo-ku, Kumamoto, 860 – 8555, Japan.
e-mail: tanigawa@educ.kumamoto-u.ac.jp