Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 313725, 14 pages
doi:10.1155/2012/313725
Research Article
Nonoscillatory Solutions of Second-Order
Differential Equations without Monotonicity
Assumptions
Lianwen Wang and Rhonda McKee
Department of Mathematics and Computer Science, University of Central Missouri, Warrensburg,
MO 64093, USA
Correspondence should be addressed to Lianwen Wang, lwang@ucmo.edu
Received 25 March 2012; Accepted 7 June 2012
Academic Editor: Chong Lin
Copyright q 2012 L. Wang and R. McKee. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The continuability, boundedness, monotonicity, and asymptotic properties of nonoscillatory
solutions for a class of second-order nonlinear differential equations pthxtfx t qtgxt are discussed without monotonicity assumption for function g. It is proved that all
solutions can be extended to infinity, are eventually monotonic, and can be classified into disjoint
classes that are fully characterized in terms of several integral conditions. Moreover, necessary
and sufficient conditions for the existence of solutions in each class and for the boundedness of all
solutions are established.
1. Introduction
This paper studies the continuability, boundedness, monotonicity, and asymptotic properties
of nonoscillatory solutions for a class of second-order nonlinear differential equations
pthxtf x t qtgxt,
t ≥ a.
1.1
Some special cases of 1.1 such as half-linear equation
ptΦp x t qtΦp xt,
1.2
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where Φp r |r|p−2 r, p > 1, is the so-called p-Laplacian operator, Emden-Fowler equation
ptΦp x t qtΦβ xt,
1.3
pthxtx t qtgxt
1.4
and differential equation
have been extensively discussed in the literature; see, for example, 1–15 and references cited
therein. Equation 1.1 with general nonlinear function fr is investigated in 16–18. It is
worth to point out that gr is assumed to be monotonic in most cited papers, but 2, 6
explain that this assumption does not hold in some applications. The aim of this paper is
to investigate the continuability, boundedness, monotonicity, and asymptotic properties of
nonoscillatory solutions of 1.1 without monotonic assumption for g. Some techniques and
ideas have been used by the authors in 17.
By solution of 1.1, we mean a differentiable function x such that pthxtfx t
is differentiable and satisfies 1.1 on the maximum existence interval a, αx , αx ≤ ∞. A
solution x of 1.1 is said to be eventually monotonic if there exists a tx ≥ a such that x
is monotonic on tx , αx . In this paper, we consider only solutions that are not eventually
identically equal to zero.
Throughout the paper, we always assume that
H pt, qt : a, ∞ → R are continuous and pt > 0 and qt > 0;
hr : R → R is continuous and hr > 0;
gr : R → R is continuous and rgr > 0 for r /
0;
fr : R → R is continuous, increasing, and rfr > 0 for r /
0.
H1 There exists a constant M1 > 0 such that
−1
f uv ≤ M1 f −1 uf −1 v,
∀u, v ∈ R.
1.5
Remark 1.1. H1 holds for p-Laplacian operator; indeed,
f −1 uv f −1 uf −1 v.
1.6
However, there are nonlinear functions f that satisfy H1 but not 1.6; see 17.
The paper is organized as follows: Section 1 briefly addresses the background and
the motivation of the paper. Continuability, classification, and boundedness of solutions are
discussed in Section 2. Sections 3 and 4 deal with the existence of class A and class B solutions,
respectively. Finally, several remarks are provided in Section 5 to compare our results with
existing ones.
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3
2. Continuability, Classification, and Boundedness of Solutions
In this section we discuss continuability, classification, and boundedness of solutions of 1.1.
First of all, we cite a result from 17 that will be used later on.
Lemma 2.1. If x is a solution of 1.1 with maximal existence interval a, αx , αx ≤ ∞, then x is
eventually monotonic. Moreover, if x is bounded on all finite subinterval of a, αx , then αx ∞.
Remark 2.2. From Lemma 2.1 all solutions of 1.1 except eventually trivial solutions can be
classified into two classes
A x is defined on a, αx : xtx t > 0 in a left neighborhood of αx ,
B x is defined on a, ∞ : xtx t < 0 for t ≥ a .
2.1
Next theorem establishes the continuability for all solutions of 1.1, in other words,
all solutions can be extended to a, ∞.
Theorem 2.3. Assume the following assmputions hold.
H2 There exists a real number m > 0 and a continuous function Gr : R → R such that Gr
is increasing and |gr| ≤ |Gr| for |r| ≥ m, and rGr > 0 for r / 0;
H3 There exists a real number r0 > 0 such that
∞
r0
dr
∞,
f −1 zr
−r0
−∞
dr
−∞,
f −1 zr
2.2
where zr Gr/hr.
Then all solutions of 1.1 can be extended to a, ∞.
Proof. The proof is similar to that of Theorem 2.3 17. We point out that as in the proof of
Theorem 2.3 17, for a class A solution x, we have
f x t ≤
xt
xt1 Gxt
pthxt
pdhxdfx d
Gxd
dr
≤ M12 f −1 k
−1
f zr
t
t1
f −1
1
ps
s
t
qsds ,
d
2.3
qσdσ ds.
d
Remark 2.4. The function gr r sin r is not monotonic. Clearly, |gr| ≤ 2|r|, so g is
bounded by an increasing function Gr 2r. Therefore, the existing results which require
the monotonic condition for g would not apply, but Theorem 2.3 does.
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From Remark 2.2 and Theorem 2.3, all solutions of 1.1 can be classified further into
four disjoint classes
Ab x ∈ A : lim |xt| < ∞ ,
t→∞
A∞ x ∈ A : lim |xt| ∞ ,
t→∞
2.4
0 ,
Bb x ∈ B : lim xt /
t→∞
B0 x ∈ B : lim xt 0 .
t→∞
We will show that the existence of solutions in each class and the boundedness of all
solutions are fully characterized by means of convergence or divergence of the following
integrals:
J1 ∞
f
−1
a
J2 J3 J4 J5 1
pt
t
qsds dt,
a
t
1
−
f
qsds dt,
pt a
a
∞
∞
1
f −1
qsds dt,
pt t
a
∞
∞
1
−1
f
qsds dt,
−
pt t
a
∞
1
−1
dt.
f
pt
a
∞
−1
2.5
Theorem 2.5. Let (H2) and (H3) hold. Then all positive (negative) solutions of 1.1 are bounded if
and only if J1 < ∞J2 > −∞.
Proof. We consider positive solutions only since the case of negative solutions can be handled
similarly.
Necessity. Let x be a positive bounded class A solution. Then xt > 0 and x t > 0
for t ≥ b > a and limt → ∞ xt l ∈ 0, ∞. By the Extreme Value Theorem, we have L1 :
minxb≤r≤l gr > 0. Hence
pthxtf x t pbhxbf x b t
b
qsgxsds ≥ L1
t
qsds.
b
2.6
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5
Since x is continuous and bounded and hr is continuous, then hxt is bounded. Let
hxt ≤ K for t ∈ a, ∞. Then
Kptf x t ≥ pthxtf x t ≥ L1
qsds,
b
t
K 1
f x t ≥
L1
pt
t
2.7
qsds.
b
By H1, we have
f
−1
1
pt
t
≤ f −1
qsds
b
K f x t
L1
K
x t.
L1
2.8
K
l − xb < ∞.
L1
2.9
≤ M1 f −1
Integrating from b to t and letting t → ∞, we have
J1 ∞
f
−1
b
1
pt
t
qsds dt ≤ M1 f
b
−1
Sufficiency. We will prove by contradiction. Let x be a unbounded class A solution.
Then xt > 0 and x t > 0 on b, ∞, and there exists a real number d ≥ b such that xt ≥ m
for d ≤ t < ∞. Similar to the proof of Theorem 2.3, we have
xt
xt1 dr
≤ M12 f −1 k
f −1 zr
t
f
−1
t1
1
ps
s
qσdσ ds.
2.10
d
Letting t → ∞ and noting that x∞ ∞, we have
∞
xt1 dr
≤ M12 f −1 k
f −1 zr
∞
t1
f −1
1
ps
s
b
qσdσ ds ≤ M12 f −1 kJ1 < ∞,
2.11
a contradiction to H3. Therefore, x is bounded.
Corollary 2.6. Let (H2) and (H3) hold. If 1.1 has a positive (negative) bounded class A solution,
then all positive (negative) solutions are bounded. On the other hand, if 1.1 has an unbounded
positive (negative) class A solution, then all positive (negative) solutions are unbounded.
3. Class A Solutions
In this section, we consider the existence of class Ab and class A∞ solutions of 1.1. The
necessary and sufficient conditions for the existence of class Ab solutions and the sufficient
conditions for the existence of class A∞ solutions are provided.
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Journal of Applied Mathematics
Theorem 3.1. Equation 1.1 has both positive and negative class A solutions.
Proof. Similar to the proof of Theorem 3.1 in 17.
Theorem 3.2. Equation 1.1 has a positive (negative) Ab solution if and only if J1 < ∞ J2 > −∞.
Proof. Necessity. Without loss of generality, we assume that x is a positive Ab solution. In
this case, there exists a b ≥ a such that xt > 0 and x t > 0 for t ≥ b. Note that x∞ :
limt → ∞ xt < ∞, we have
m1 :
min
gr > 0,
xb≤r≤x∞
c1 : max hxt ≤
b≤t<∞
3.1
hr < ∞.
max
xb≤r≤x∞
Then
pthxtf x t pbhxbf x b t
qsgxsds ≥ m1
b
t
qsds,
3.2
b
and hence
1
pt
t
qsds ≤
b
1
hxtf x t .
m1
3.3
Taking f −1 on both sides and applying H1 imply that
f
−1
1
pt
t
≤f
qsds
−1
b
c1
x t.
m1
3.4
c1
x∞ − xb < ∞.
m1
3.5
1
hxtf x t
m1
≤ M1 f
−1
Therefore
J1 ∞
f
−1
b
1
pt
t
qsds ds ≤ M1 f −1
b
Sufficiency. Define
m2 max gr > 0,
c2 min hr > 0.
3.6
1≤r≤2
1≤r≤2
Since J1 < ∞, we may select a d ≥ a such that
∞
f
d
−1
1
pt
t
d
qsds dt ≤
1
M1
f −1 m
2 /c2 .
3.7
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Let CBd, ∞ be the Banach space of all bounded and continuous functions defined on d, ∞
endowed with the supremum norm, and let X {x ∈ CBd, ∞ : 1 ≤ xt ≤ 2, t ≥ d}. Clearly,
X is a bounded convex subset of CBd, ∞. Define a mapping F1 : X → CBd, ∞ by
F1 xt 1 t
f
d
−1
1
pshxs
s
qσgxσdσ ds.
3.8
d
In order to apply Schauder’s fixed-point theorem to show that F1 has a fixed point in X, we
need to prove that F1 maps into X and is continuous, and F1 X is precompact in CBd, ∞.
Let x ∈ X. Considering 3.7, we have
s
t
m2
1
−1
f
qσdσ ds
1 ≤ F1 xt ≤ 1 M1 f
c2
ps d
d
t
∞
1
−1 m2
−1
≤ 1 M1 f
f
qsds dt ≤ 2.
c2
pt d
d
−1
3.9
Hence, F1 maps X into X.
Now, we show that if xn , x∗ ∈ X and xn −x∗ → 0 as n → ∞, then F1 xn −F1 x∗ → 0.
Indeed, for any fixed s ∈ d, ∞, since xn s → x∗ s as n → ∞, we have
s
−1
1
f
qσgx
σdσ
n
pshxn s d
s
1
∗
−→ 0 as n −→ ∞.
qσgx
−f −1
σdσ
∗
pshx s d
3.10
Note that
s
s
−1
1
1
−1
∗
f
qσgxn σdσ − f
qσgx σdσ pshxn s d
pshx∗ s d
s
−1
1
≤ f
qσgxn σdσ pshxn s d
3.11
s
−1
1
∗
f
qσgx σdσ pshx∗ s d
s
m2
1
≤ 2M1 f −1
qσdσ : Fs,
f −1
c2
ps d
and that
∞
d
Fsds ∞
2M1 f
d
−1
s
m2
1
−1
−1 m2
f
J1 .
qσdσ ds 2M1 f
c2
ps d
c2
3.12
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Journal of Applied Mathematics
By Lebesgue’s dominated convergence theorem and considering 3.11 and 3.12 we have
F1 xn − F1 x∗ t ≤ sup f −1
s
1
qσgxn σdσ
pshxn s d
b≤t<∞ d
s
1
∗
ds
− f −1
qσgx
σdσ
∗
pshx s d
∞ s
−1
1
≤
qσgxn σdσ
f
pshxn s d
d
s
1
−1
∗
qσgx σdσ ds −→ 0
−f
pshx∗ s d
3.13
as n → ∞. Therefore, F1 is continuous in X.
Finally, we show the precompactness of F1 X in CBd, ∞, which means that for any
sequence xn ∈ X, F1 xn has a convergent subsequence in CBd, ∞. This can be proved by
showing that F1 xn has a convergent subsequence in Cb1 , b2 for any compact subinterval
b1 , b2 of b, ∞ as well as the diagonal rule. In fact, F1 xn is uniformly bounded on b1 , b2 .
Since
F1 xn t f
−1
1
pthxn t
t
≤ M1 f
qsgxn sds
d
−1
t
1
m2
−1
f
qtds .
c2
pt d
3.14
By the Mean Value Theorem, we have
|F1 xn t1 − F1 xn t2 |
t
1
−1 m2
−1
F1 xn ξt1 − t2 ≤ M1 f
qsds |t1 − t2 |.
max f
c2 b1 ≤t≤b2
pt d
3.15
Then F1 xn is uniformly bounded and equicontinuous in Cb1 , b2 . So F1 xn has a convergent
subsequence in Cb1 , b2 by Arzelà-Ascoli Theorem.
Now all conditions of Schauder’s fixed-point theorem are satisfied, so F1 has a fixed
point x in X, that is,
xt 1 t
f
d
−1
1
pshxs
s
qσgxσdσ ds.
3.16
d
It is easy to verify that pthxtfx t qtgxt. Hence, x is a positive Ab solution
of 1.1. The proof is complete.
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Theorem 3.3. Let (H2) and (H3) hold. Then
a A∞ ∅ if and only if J1 < ∞ and J2 > −∞.
b Equation 1.1 has a positive (negative) A∞ solution if J1 ∞ J2 −∞.
Proof. By Theorem 2.5 all solutions of 1.1 are bounded if and only if J1 < ∞ and J2 > −∞,
so part a follows.
If J1 ∞, there is no positive Ab solution of 1.1 from Theorem 3.2. Therefore,
Theorem 3.1 guarantees the existence of a positive A∞ solution of 1.1. Similarly, there exists
a negative A∞ solution of 1.1 if J2 −∞.
4. Class B Solutions
In this section the existence of class B, Bb , and B0 solutions are discussed. We assume that
0 and x a x1 .
1.1 has a unique solution for any initial conditions xa x0 /
Theorem 4.1. Assume the following assumptions hold.
H2a There exists a continuous function Gr : R → R such that G is increasing, rGr > 0 for
r/
0 and |gr| ≤ |Gr|;
H4 There exists r0 > 0 such that
±r0
0
dr
∞.
f −1 zr
4.1
Then 1.1 has
a both positive and negative solutions in class B;
b no solution which is eventually identically equal to zero.
Proof. a We prove that class B has a positive solution, the case of having a negative solution
is similar. Assume x0 > 0. The solution of 1.1 with initial conditions xa x0 and x a c,
denoted by xt : xt, c, has the form
xt x0 t
f −1
a
pahx0 fc
1
pshxs
pshxs
s
qσgxσdσ ds.
4.2
a
Define two sets U and L as
U c ∈ R : there exists some t ≥ a such that x t, c > 0 ,
L c ∈ R : there exists some t ≥ a such that x t, c < 0 .
4.3
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Then U ∩ L ∅. Clearly, U /
∅. We claim that U is open. Indeed, if c0 ∈ U, there exists t > a
such that x t, c0 > 0. For any c ∈ R, we have
p t h x t, c0 f x t, c0 − p t h x t, c f x t, c
pahx0 fc0 − pahx0 fc t
qs gxs, c0 − gxs, c ds.
4.4
a
Since 1.1 has a unique solution for any initial conditions xa /
0, x a, this solution is
continuously dependent on initial data. If c → c0 , we have gxs, c − gxs, c0 → 0
uniformly for s on a, t . Hence, x t, c > 0 for all c that are close to c0 , this proves the
openness of U.
Next we show that L /
∅. Define
M2 : min hr > 0,
M3 : min pt > 0.
0≤r≤x0
Let
a≤t≤a
1
⎛
M2 M3 f −1 −x0 − Gx0 c < f −1 ⎝
pahx0 a
1
a
4.5
⎞
qsds
⎠ < 0.
4.6
If there exists b ∈ a, a 1 such that xb, c < 0, then c ∈ L and L /
∅. Otherwise, xt, c ≥ 0 on
a, a 1. In this case, we claim x t, c < 0 on a, a 1. If this is not true, since x a, c c < 0,
there exists t1 ∈ a, a 1 such that x t1 , c 0 and x t, c < 0 for t ∈ a, t1 . Taking into
account 4.6 we have
0 pthxt1 , cf x t1 , c
t1
pahx0 fc qsgxs, cds
a
≤ pahx0 fc Gx0 4.7
a
1
qsds < 0.
a
This is a contradiction and hence x t, c < 0 on a, a 1. Notice that
t
pahx0 fc
1
xa 1, c x0 f
qsgxsds dt
pthxt
pthxt a
a
⎛
⎞
a
1
a
1
pahx
qsds
Gx
fc
0
0
a
⎠dt < 0,
≤ x0 f −1 ⎝
M2 M3
a
a
1
−1
4.8
we know c ∈ L. Clearly, L is open, then R − U ∪ L /
∅. Take c ∈ R − U ∪ L, xt, c is
a nonincreasing nonnegative solution on a, ∞. We will show that xt, c > 0 on a, ∞. If
not, there exists t0 > a such that xt0 0 and xt 0 for t ≥ t0 and x t0 0. Note that
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11
for t ∈ a, t0 we have
x t f
1
−
pthxt
−1
≥f
−1
Gxt
−
pthxt
t0
qsgxsds
t
t0
−1
≥ M1 f zxtf
4.9
qsds
t
1
−
pt
−1
t0
qsds .
t
Dividing both sides by f −1 zxt and integrating from a to t0 , we have
t0
a
x t
dt ≥ M1
f −1 zxt
t0
f
1
−
pt
−1
a
t0
qsds dt.
4.10
t
That is
x0
0
1
dr ≤ −M1
−1
f zr
t0
f
1
−
pt
−1
a
t0
qsds dt < ∞,
4.11
t
a contradiction to H4. Therefore, xt > 0 for t ≥ a and x ∈ B.
The proof of part b follows from the end part of the proof of part a.
Theorem 4.2. Equation 1.1 has a positive (negative) Bb solution if and only if J4 > −∞ J3 < ∞.
Proof. Necessity. We assume that x is a positive Bb solution. The case of negative Bb solution
is similar. In this case, we have xt > 0 and x t < 0 for t ≥ a. Let
m1 min
c1 gr > 0,
x∞≤r≤xa
max
hr > 0
x∞≤r≤xa
4.12
and note that pthxtfx t < 0, pthxtfx t > 0. Then
lim pthxtf x t B ≤ 0.
t→∞
4.13
Integrating both sides of 1.1 from t to ∞ implies that
∞
m1
t
qsds ≤
∞
t
qsgxsds B − pthxtf x t
≤ −pthxtf x t .
4.14
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Journal of Applied Mathematics
Hence,
∞
1
c1
f −1 −
x t.
qsds ≥ M1 f −1
pt t
m1
4.15
Again, integrating both sides of the above inequality we have
J4 ∞
f
−1
a
1
−
pt
∞
t
c1
−1
qsds dt ≥ M1 f
x∞ − xa > −∞.
m1
4.16
Sufficiency. Let
m2 max gr > 0,
1≤r≤2
c2 min hr > 0.
1≤r≤2
4.17
Since J4 > −∞ we choose d > a such that
∞
f
−1
d
1
−
pt
∞
qsds dt ≥ −
t
1
.
M1 f −1 m2 /c2 4.18
Let X and CBd, ∞ as defined in Theorem 3.2. Define F2 : X → CBd, ∞ by
F2 xt 1 −
∞
f −1 −
t
1
pshxs
∞
qσgxσdσ ds.
4.19
s
For any x ∈ X, we have
1 ≤ F2 xt ≤ 1 −
∞
M1 f −1
d
∞
m2
1
qσdσ ds ≤ 2.
f −1 −
c2
ps s
4.20
This proves that F2 maps X into X. Similar to the proof of Theorem 3.2, we are able to show
that F2 is continuous in X, and F2 X is precompact in CBd, ∞. Then F2 has a fixed-point
xt in X by Schauder’s fixed-point theorem, that is,
xt 1 −
∞
t
f −1 −
1
pshxs
∞
qσgxσdσ ds.
4.21
s
It is easy to verify that xt is a positive Bb solution of 1.1. The proof is complete.
Theorem 4.3. Let (H2a) and (H4) hold and let J5 ∞. Then 1.1 has a positive (negative) B0
solution if and only if J4 −∞ (J4 ∞).
Proof. We prove the assertion for positive solutions without loss of generality.
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13
Necessity. Assume xt is a positive B0 solution. Then xt > 0 and x t < 0 for t ≥ a,
x∞ 0, and limt → ∞ pthxtfx t L ∈ −∞, 0. We claim L 0. In fact, if L < 0, since
pthxtfx t is negative and increasing on a, ∞, then pthxtfx t ≤ L and
x t ≤ f −1
L
c1 pt
≤ M1 f −1
L
1
,
f −1
c1
pt
4.22
where c1 max0≤r≤xa hr > 0.
Integrating both sides from a to ∞ and noting that x∞ 0, we have
xa ≥ −M1 f
−1
L
c1
∞
f −1
a
1
dt,
pt
4.23
a contradiction to J5 ∞ and hence L 0.
Integrating both sides of 1.1 from t to ∞ we have
pthxtf x t −
∞
qsgxsds.
4.24
t
Then
x t f
−
−1
1
pthxt
∞
qsgxsds
t
∞
1
−1
−1
≥ M1 f zxtf
−
qsds .
pt t
4.25
Hence,
∞
x t
1
−1
f
qsds
.
−
≥
M
1
pt t
f −1 zxt
4.26
Integrating both sides of the above inequality from a to ∞ implies that
xa
0
dr
≤ −M1
f −1 zr
∞
a
∞
1
f −1 −
qsds dt.
pt t
4.27
Therefore, J4 −∞ from H4.
Sufficiency. By Theorem 4.1 1.1 has a positive class B solution x, either x ∈ Bb or
/ Bb from Theorem 4.2. So x ∈ B0 . The proof is
x ∈ B0 . Note that J4 −∞ implies that x ∈
complete.
5. Remarks
In this section, we present several remarks about comparison of our results with the existing
ones in the literature.
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Journal of Applied Mathematics
Theorems 2.3 and 2.5 improve 10, Theorem 1 since H3 reduces to iii of 10 if
fr r and the differentiability of p· and h· is not required. Theorems 2.3, 2.5, and
4.2 complement and generalize 2, Theorem 8. Moreover, under H2, Theorems 2.3, 2.5,
and 4.2 improve 2, Theorem 8 since H3 improves 22 of 2; see the discussion in
16. Theorem 2.5 generalizes 13, Theorem 3.9. Theorems 2.3, 3.1, and 4.2 generalize 16,
Theorem 1. Theorem 3.2 generalizes 2, Theorem 3, 16, Theorem 3, and 18, Theorem 2.1.
Theorem 3.3 generalizes 18, Theorem 2.2. Theorem 4.1 generalizes 13, Theorem 2.1 and
improves 3, Theorem 6 under H2a since hp in 3 is replaced by a weaker condition
H4. Theorem 4.2 generalizes 2, Theorem 1, 16, Theorem 5, and 18, Theorem 3.1.
Theorem 4.3 generalize 16, Theorem 6 and 18, Theorem 3.2.
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