Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 247071, 11 pages
doi:10.1155/2008/247071
Research Article
Existence Theorems of Periodic Solutions for
Second-Order Nonlinear Difference Equations
Xiaochun Cai1 and Jianshe Yu2
1
2
College of Statistics, Hunan University, Changsha, Hunan 410079, China
College of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, China
Correspondence should be addressed to Xiaochun Cai, cxchn8@hnu.cn
Received 14 August 2007; Accepted 14 November 2007
Recommended by Patricia J. Y. Wong
The authors consider the second-order nonlinear difference equation of the type Δpn Δxn−1 δ qn xnδ fn, xn , n ∈ Z, using critical point theory, and they obtain some new results on the existence
of periodic solutions.
Copyright q 2008 X. Cai and J. Yu. 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.
1. Introduction
We denote by N, Z, R the set of all natural numbers, integers, and real numbers, respectively.
For a, b ∈ Z, define Za {a, a 1, . . . }, Za, b {a, a 1, . . . , b} when a ≤ b.
Consider the nonlinear second-order difference equation
δ Δ pn Δxn−1
qn xnδ f n, xn ,
n ∈ Z,
1.1
where the forward difference operator Δ is defined by the equation Δxn xn1 − xn and
Δ2 xn−1 ΔΔxn−1 Δxn − Δxn−1 .
1.2
In 1.1, the given real sequences {pn }, {qn } satisfy pnT pn > 0, qnT qn for any n ∈ Z,
f : Z × R → R is continuous in the second variable, and fn T, z fn, z for a given positive
integer T and for all n, z ∈ Z × R. −1δ −1, δ > 0, and δ is the ratio of odd positive integers.
By a solution of 1.1, we mean a real sequence x {xn }, n ∈ Z, satisfying 1.1.
In 1, 2, the qualitative behavior of linear difference equations of type
Δpn Δxn qn xn 0
1.3
2
Advances in Difference Equations
has been investigated. In 3, the nonlinear difference equation
Δpn Δxn−1 qn xn fn, xn 1.4
has been considered. However, results on periodic solutions of nonlinear difference equations
are very scarce in the literature, see 4, 5. In particular, in 6, by critical point method, the
existence of periodic and subharmonic solutions of equation
Δ2 xn−1 f n, xn 0, n ∈ Z,
1.5
has been studied. Other interesting contributions can be found in some recent papers 7–11
and in references contained therein. It is interesting to study second-order nonlinear difference
equations 1.1 because they are discrete analogues of differential equation
ptϕu ft, u 0.
1.6
In addition, they do have physical applications in the study of nuclear physics, gas aerodynamics, infiltrating medium theory, and plasma physics as evidenced in 12, 13.
The main purpose here is to develop a new approach to the above problem by using
critical point method and to obtain some sufficient conditions for the existence of periodic
solutions of 1.1.
Let X be a real Hilbert space, I ∈ C1 X, R, which implies that I is continuously Fréchet
differentiable functional defined on X. I is said to be satisfying Palais-Smale condition P-S
condition if any sequence {Iun } is bounded, and I un → 0 as n → ∞ possesses a convergent subsequence in X. Let Bρ be the open ball in X with radius ρ and centered at 0, and let
∂Bρ denote its boundary.
Lemma 1.1 mountain pass lemma, see 14. Let X be a real Hilbert space, and assume that I ∈
C1 X, R satisfies the P-S condition and the following conditions:
I1 there exist constants ρ > 0 and a > 0 such that Ix ≥ a for all x ∈ ∂Bρ , where Bρ {x ∈ X :
xX < ρ};
I2 I0 ≤ 0 and there exists x0 ∈ Bρ such that Ix0 ≤ 0.
Then c inf h∈Γ sup s∈0,1 Ihs is a positive critical value of I, where
Γ h ∈ C 0, 1, X : h0 0, h1 x0 .
1.7
Lemma 1.2 saddle point theorem, see 14, 15. Let X be a real Banach space, X X1 ⊕ X2 , where
X1 / {0} and is finite dimensional. Suppose I ∈ C1 X, R satisfies the P-S condition and
I3 there exist constants σ, ρ > 0 such that I|∂Bρ X1 ≤ σ;
I4 there is e ∈ Bp X1 and a constant ω > σ such that I|eX2 ≥ ω.
Then I possesses a critical value c ≥ ω and
c inf
max
Ihu,
h∈Γ u∈Bρ
where Γ {h ∈ CBρ
X1, X|h|∂Bρ X1 id}.
X1
1.8
X. Cai and J. Yu
3
2. Preliminaries
In this section, we are going to establish the corresponding variational framework for 1.1.
Let Ω be the set of sequences
x xn n∈Z . . . , x−n , . . . , x−1 , x0 , x1 , . . . , xn , . . . ,
2.1
that is,
Ω x xn : xn ∈ R, n ∈ Z .
2.2
For any x, y ∈ Ω, a, b ∈ R, ax by is defined by
∞
ax by : axn byn n−∞ .
2.3
Then Ω is a vector space. For given positive integer T, ET is defined as a subspace of Ω by
ET x xn ∈ Ω : xnT xn , n ∈ Z .
2.4
Clearly, ET is isomorphic to RT , and can be equipped with inner product
x, y T
xi y i ,
∀x, y ∈ ET ,
2.5
i1
by which the norm · can be induced by
1/2
T
2
xi
,
x :
∀x ∈ ET .
2.6
i1
It is obvious that ET with the inner product defined by 2.5 is a finite-dimensional Hilbert
space and linearly homeomorphic to RT . Define the functional J on ET as follows:
Jx T
T
T
1 1 pn Δxn−1 δ1 −
qn xnδ1 Fn, xn ,
δ 1 n1
δ 1 n1
n1
∀x ∈ ET ,
2.7
z
where Ft, z 0 ft, sds. Clearly, J ∈ C1 ET , R, and for any x {xn }n∈Z ∈ ET , by using
x0 xT , x1 xT 1 , we can compute the partial derivative as
∂J
−Δpn Δxn−1 δ − qn xnδ fn, xn , n ∈ Z1, T .
∂xn
Thus x {xn }n∈Z is a critical point of J on ET i.e., J x 0 if and only if
Δpn Δxn−1 δ qn xnδ fn, xn ,
n ∈ Z1, T .
2.8
2.9
By the periodicity of xn and fn, z in the first variable n, we have reduced the existence
of periodic solutions of 1.1 to that of critical points of J on ET . In other words, the functional J is just the variational framework of 1.1. For convenience, we identify x ∈ ET with
x x1 , x2 , . . . , xT T . Denote W {x1 , x2 , . . . , xT T ∈ ET : xi ≡ v, v ∈ R, i ∈ Z1, T } and
W⊥ Y such that ET Y ⊕ W. Denote other norm ·r on ET as follows see, e.g., 16:
xr Ti1 |xi |r 1/r , for all x ∈ ET and r > 1. Clearly, x2 x. Due to ·r1 and ·r2
being equivalent when r1 , r2 > 1, there exist constants c1 , c2 , c3 , and c4 such that c2 ≥ c1 > 0,
c4 ≥ c3 > 0, and
for all x ∈ ET , δ > 0 and β > 1.
c1 x ≤ xδ1 ≤ c2 x,
2.10
c3 x ≤ xβ ≤ c4 x,
2.11
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Advances in Difference Equations
3. Main results
In this section, we will prove our main results by using critical point theorem. First, we prove
two lemmas which are useful in the proof of theorems.
Lemma 3.1. Assume that the following conditions are satisfied:
F1 there exist constants a1 > 0, a2 > 0, and β > δ 1 such that
z
fn, sds ≤ −a1 |z|β a2 ,
∀z ∈ R;
3.1
0
F2 qn ≤ 0,
∀n ∈ Z.
3.2
Then the functional
Jx T
T
T
1 1 pn Δxn−1 δ1 −
qn xnδ1 Fn, xn δ 1 n1
δ 1 n1
n1
3.3
satisfies P-S condition.
Proof. For any sequence {xl } ⊂ ET , with Jxl being bounded and J xl → 0 as
l → ∞, there exists a positive constant M such that |Jxl | ≤ M. Thus, by F1 ,
−M ≤ Jxl ≤
T T
l
l δ1
1 l δ1
l pn xn − xn−1
− qn xn
F n, xn
δ 1 n1
n1
T
T
T
δ1 l δ1 1 1 l
l δ1
l −
pn 2δ1 xn
xn−1
qn xn
F n, xn
δ 1 n1
δ 1 n1
n1
T
T
T l δ1
l δ1
1 2δ1 l β
pn pn1 xn
−
qn xn
− a1 xn a2 T
≤
δ 1 n1
δ 1 n1
n1
3.4
T
β
δ1
l δ1
1 − a1 xl β a2 T.
2 pn pn1 − qn xn
δ 1 n1
Set
A0 max 2δ1 pn pn1 − qn .
n∈Z1,T 3.5
Then A0 > 0. Also, by the above inequality, we have
−M ≤ Jxl ≤
A0 xl δ1 − a1 xl β a2 T.
δ1
β
δ1
3.6
X. Cai and J. Yu
5
In view of
δ1/β
T T xnl δ1 ≤ T β−δ−1/β
xnl β
,
n1
3.7
n1
we have
l β
x ≥ T δ1−β/δ1 xl β .
β
δ1
3.8
A0 xl δ1 − a1 T δ1−β/δ1 xl β a2 T.
δ1
δ1
δ1
3.9
Then we get
−M ≤ Jxl ≤
Therefore, for any l ∈ N,
β
A0 xl δ1 ≤ M a2 T.
a1 T δ1−β/δ1 xl δ1 −
δ1
δ1
3.10
Since β > δ 1, the above inequality implies that {xl } is a bounded sequence in ET . Thus {xl }
possesses convergent subsequences, and the proof is complete.
Theorem 3.2. Suppose that F1 and following conditions hold:
F3 for each n ∈ Z,
fn, z
0;
z→0
zδ
lim
3.11
F4 qn < 0,
∀n ∈ Z1, T .
3.12
Then there exist at least two nontrivial T -periodic solutions for 1.1.
Proof. We will use Lemma 1.1 to prove Theorem 3.2. First, by Lemma 3.1, J satisfies P-S condition. Next, we will prove that conditions I1 and I2 hold. In fact, by F3 , there exists ρ > 0
such that for any |z| < ρ and n ∈ Z1, T ,
|Fn, z| ≤ −
qmax
zδ1 ,
2δ 1
3.13
where qmax maxn∈Z1,T qn < 0. Thus for any x ∈ ET , x ≤ ρ for all n ∈ Z1, T , we have
Jx ≥ −
T
T
qmax qmax xnδ1 xδ1
δ 1 n1
2δ 1 n1 n
−
qmax
xδ1
δ1
2δ 1
≥−
qmax δ1
c xδ1
2 .
2δ 1 1
3.14
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Advances in Difference Equations
Taking a −c1δ1 qmax /2δ 1ρδ1 , we have
Jx|∂Bρ ≥ a > 0,
3.15
and the assumption I1 is verified. Clearly, J0 0. For any given w ∈ ET with w 1 and a
constant α > 0,
Jαw ≤
≤
T
T
1 pn αwn − αwn−1 δ1 − qn αwn δ1 Fn, αwn δ 1 n1
n1
T
T
1 pn 2αδ1 − qn αδ1 − a1 |αwn |β a2 T
δ 1 n1
n1
3.16
T
1
2δ1 pn − qn αδ1 αδ1 − a1 T 2−β/2 αβ a2 T
δ 1 n1
−→ − ∞,
α −→ ∞.
Thus we can easily choose a sufficiently large α such that α > ρ and for x αw ∈ ET , Jx < 0.
Therefore, by Lemma 1.1, there exists at least one critical value c ≥ a > 0. We suppose that x is
a critical point corresponding to c, that is, Jx
c, and J x
0. By a similar argument to the
proof of Lemma 3.1, for any x ∈ ET , there exists x ∈ ET such that J x
cmax . Clearly, x /
0. If
x /
x,
and the proof is complete; otherwise, x x and c cmax . By Lemma 1.1,
c inf sup J hs ,
3.17
h∈Γ s∈0,1
where Γ {h ∈ C0, 1, ET | h0 0, h1 x}. Then for any h ∈ Γ, cmax max s∈0,1 Jhs.
By the continuity of Jhs in s, J0 ≤ 0 and Jx < 0 show that there exists some s0 ∈
0, 1 such that Jhs0 cmax . If we choose h1 , h2 ∈ Γ such that the intersection {h1 s |
s ∈ 0, 1} {h2 s | s ∈ 0, 1} is empty, then there exist s1 , s2 ∈ 0, 1 such that Jh1 s1 Jh2 s2 cmax . Thus we obtain two different critical points x1 h1 s1 , x2 h2 s2 of J in
ET . In this case, in fact, we may obtain at least two nontrivial critical points which correspond
to the critical value cmax . The proof of Theorem 3.2 is complete. When fn, xn ≡ hn , we have
the following results.
Theorem 3.3. Assume that the following conditions hold:
G1 qn < 0,
∀n ∈ Z1, T ;
3.18
G2 1
c1δ1
T
n1
δ1/2
h2n
T
n1
δ1
T
δ1/2
−qn < pmin λ2
− qmax
hn
,
n1
3.19
X. Cai and J. Yu
7
where pmin min n∈Z1,T pn , qmax max n∈Z1,T qn , c1 is a constant in 2.10, and λ2 is the minimal
positive eigenvalue of the matrix
⎞
⎛
2 −1 0 · · · 0 −1
⎜ −1 2 −1 · · · 0 0 ⎟
⎟
⎜
⎜ 0 −1 2 · · · 0 0 ⎟
⎟
⎜
A⎜
3.20
⎟ .
⎜· · · · · · · · · · · · · · · · · · ⎟
⎟
⎜
⎝ 0 0 0 · · · 2 −1 ⎠
−1 0 0 · · · −1 2 T ×T
Then equation
δ
Δ pn Δxn−1 qn xnδ hn ,
n ∈ Z,
3.21
possesses at least one T -periodic solution.
First, we proved the following lemma.
Lemma 3.4. Assume that G1 holds, then the functional
Jx T
T
T
δ1
1 1 pn Δxn−1
−
qn xnδ1 hn xn
δ 1 n1
δ 1 n1
n1
3.22
satisfies P-S condition on ET .
Proof. For any sequence {xl } ⊂ ET with Jxl being bounded and J xl → 0 as n → ∞,
there exists a positive constant M such that |Jxl | ≤ M. In view of G3 and
1/2 1/2
T
T
T
l 2
l
|hn xn | ≤
h2n
,
3.23
xn
n1
we have
M ≥ Jxl ≥−
n1
n1
T T
T
l δ1
l δ1 1 1 l
pn Δxn−1
−
qn xn
hn xn
δ 1 n1
δ 1 n1
n1
T
T l δ1 1 hn xnl qn xn
−
δ 1 n1
n1
T
l δ1
1
−
xn
≥−
qmax
δ1
n1
qmax xl δ1 −
−
δ1
δ1
T
T
1/2 h2n
n1
1/2
h2n
T
l 2
xn
1/2
3.24
n1
l x n1
δ1
qmax δ1 ≥−
c1 xl −
δ1
T
1/2
h2n
l x .
n1
By δ 1 > 1, the above inequality implies that {xl } is a bounded sequence in ET . Thus {xl }
possesses a convergent subsequence, and the proof of Lemma 3.4 is complete. Now we prove
Theorem 3.3 by the saddle point theorem.
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Advances in Difference Equations
Proof of Theorem 3.3. For any w z, z, . . . , zT ∈ W, we have
Jw −
T
T
1 qn zδ1 hn z.
δ 1 n1
n1
3.25
1/δ
1/δ
Take z Tn1 hn / Tn1 qn and ρ w T 1/2 | Tn1 hn / Tn1 qn | , then
δ1/δ
T
h
n
n1
δ
Jw 1/δ .
δ 1 T
n1 qn Set
σ
δ
δ1
T
n1 hn
δ1/δ
,
3.27
Y.
3.28
1/δ
T
n1 qn then we have
Jw σ,
3.26
∀w ∈ ∂Bρ
On the other hand, for any x ∈ Y, we have
Jx ≥
T
T
T
δ1
1 1 pn Δxn−1
−
qn xnδ1 hn xn
δ 1 n1
δ 1 n1
n1
T
T
T
δ1 qmax pmin −
xnδ1 −
|hn xn |
Δxn−1
δ 1 n1
δ 1 n1
n1
1/2
δ1/2
T
T
pmin δ1 qmax
δ1
2
2
≥
Δxn−1 −
hn
x
c
xδ1 −
δ1 1
δ1
n1
n1
3.29
T
pmin δ1 T
qmax
c1 x Axδ1/2 −
xδ1
|hn xn |,
δ1 −
δ1
δ1
n1
where xT x1 , x2 , . . . , xT .
Clearly, λ1 0 is an eigenvalue of the matrix A and ξ v, v, . . . , vT ∈ ET is an eigenvector of A corresponding to 0, where v /
0, v ∈ R. Let λ2 , λ3 , . . . , λT be the other eigenvalues
of A. By matrix theory, we have λj > 0 for all j ∈ Z2, T . Without loss of generality, we may
assume that 0 λ1 < λ2 ≤ · · · ≤ λT , then for any x ∈ Y,
1/2
T
pmin δ1 δ1/2
qmax
δ1
δ1
2
c λ
xδ1 −
Jx ≥
x −
hn
x
δ1 1 2
δ1
n1
1/2
T
pmin δ1 δ1/2 qmax δ1
δ1
2
c λ
c
x −
−
hn
x
δ1 1 2
δ1 1
n1
δ
≥−
δ1
T
n1
1/2 h2n
T
2
n1 hn
δ1/2
pmin c1δ1 λ2
1/2
− qmax c1δ1
1/δ
,
3.30
X. Cai and J. Yu
9
as one finds by minimizing with respect to x. That is
δ1/2δ
T
2
h
1/c1 δ1/δ
n
n1
δ
Jx ≥ −
1/δ .
δ1 δ1/2
pmin λ2
− qmax
Set
δ
w0 −
δ1
T
3.31
δ1/2δ
1/c1 δ1/δ
1/δ ,
δ1/2
pmin λ2
− qmax
2
n1 hn
3.32
then by G2 , we have
Jx ≥ w0 > σ,
∀x ∈ Y.
3.33
This implies that the assumption of saddle point theorem is satisfied. Thus there exists at least
one critical point of J on ET , and the proof is complete. When qn > 0, we have the following
result.
Theorem 3.5. Assume that the following conditions are satisfied:
G3 2δ1 pn pn1 < qn , qn > 0 for all n ∈ Z1, T ;
δ1/2δ T
1/δ
δ1/δ
G4 Tn1 h2n n1 qn C1δ1 < −A0 Tn1 hn ,
where A0 maxn∈Z1,T 2δ1 pn pn1 − qn .
Then 3.21 possesses at least one T -periodic solution.
Before proving Theorem 3.5, first, we prove the following result.
Lemma 3.6. Assume that G3 holds, then Jx defined by 3.22 satisfies P-S condition.
Proof. For any sequence {xl } ∈ ET with Jxl being bounded and J xl → 0 as
n → ∞, there exists a positive constant M such that |Jxl | ≤ M.
Thus
−M ≤ J xl ≤
≤
T
T
T
l δ1
l δ1 1 1 l
pn Δxn−1
−
qn xn
hn xn
δ 1 n1
δ 1 n1
n1
T
T
T l δ1
l δ1 2δ1 1 hn xnl −
qn xn
pn pn1 xn
δ 1 n1
δ 1 n1
n1
T
δ1 l δ1
1 ≤
2
pn pn1 − qn xn
δ 1 n1
≤
δ1
1
A0 xl δ1 δ1
δ1
A0 δ1 c2 xl ≤
δ1
T
1/2
h2n
l x n1
T
n1
1/2
h2n
l x .
T
n1
1/2
h2n
l x 3.34
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Advances in Difference Equations
That is,
−c2δ1
A0 xl δ1 −
δ1
T
1/2
h2n
l x ≤ M,
∀n ∈ N.
3.35
n1
By δ 1 > 1, the above inequality implies that {xl } is a bounded sequence in ET . Thus {xl }
possesses convergent subsequences, and the proof is complete.
Proof of Theorem 3.5. For any w z, z, . . . , zT ∈ W, we have
Jω −
T
T
1 qn zδ1 hn z.
δ 1 n1
n1
3.36
1/δ
Take z Tn1 hn / Tn1 qn , ρ w T 1/2 | Tn1 hn / Tn1 | , then
Jw δ
δ1
T
n1 hn
δ1/δ
1/δ
T
n1 qn ,
∀w ∈ ∂Bρ
W.
3.37
Set
σ
then Jw σ for all w ∈ ∂Bρ
δ
δ1
T
n1 hn
δ1/δ
| Tn1 qn |1/δ
,
3.38
W. On the other hand, for any x ∈ Y, we have
1/2
T
T
1 δ1
δ1
2
Jx ≤
2 pn pn1 − qn xn hn
x
δ 1 n1
n1
1/2
T
A0 δ1
h2n
x
≤
c2 xδ1 δ1
n1
δ1/2δ
!
!
T
1 1/δ 1 δ1/δ δ
2
≤ −
hn
.
δ 1 A0
c2
n1
3.39
δ1/2δ
Set w0 −δ/δ 11/A0 1/δ 1/c2 δ1/δ Tn1 h2n , then Jx ≤ w0 < σ. Thus −Jx
satisfies the assumption of saddle point theorem, that is, there exists at least one critical point
of J on ET . This completes the proof of Theorem 3.5.
Acknowledgment
This project is supported by specialized research fund for the doctoral program of higher education, Grant no. 20020532014.
X. Cai and J. Yu
11
References
1 C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, and Riccati Equations, vol. 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic, Dordrecht, The Netherlands, 1996.
2 S. S. Cheng, H. J. Li, and W. T. Patula, “Bounded and zero convergent solutions of second-order difference equations,” Journal of Mathematical Analysis and Applications, vol. 141, no. 2, pp. 463–483, 1989.
3 T. Peil and A. Peterson, “Criteria for C-disfocality of a selfadjoint vector difference equation,” Journal
of Mathematical Analysis and Applications, vol. 179, no. 2, pp. 512–524, 1993.
4 F. Dannan, S. Elaydi, and P. Liu, “Periodic solutions of difference equations,” Journal of Difference Equations and Applications, vol. 6, no. 2, pp. 203–232, 2000.
5 S. Elaydi and S. Zhang, “Stability and periodicity of difference equations with finite delay,” Funkcialaj
Ekvacioj, vol. 37, no. 3, pp. 401–413, 1994.
6 Z. M. Guo and J. S. Yu, “The existence of periodic and subharmonic solutions for second-order superlinear difference equations,” Science in China Series A, vol. 3, pp. 226–235, 2003.
7 R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition,
2000.
8 R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, vol. 404 of Mathematics and Its
Applications, Kluwer Academic, Dordrecht, The Netherlands, 1997.
9 M. Cecchi, Z. Došlá, and M. Marini, “Positive decreasing solutions of quasi-linear difference equations,” Computers & Mathematics with Applications, vol. 42, no. 10-11, pp. 1401–1410, 2001.
10 P. J. Y. Wong and R. P. Agarwal, “Oscillations and nonoscillations of half-linear difference equations
generated by deviating arguments,” Computers & Mathematics with Applications, vol. 36, no. 10–12, pp.
11–26, 1998, Advances in difference equations, II.
11 P. J. Y. Wong and R. P. Agarwal, “Oscillation and monotone solutions of second order quasilinear
difference equations,” Funkcialaj Ekvacioj, vol. 39, no. 3, pp. 491–517, 1996.
12 M. Cecchi, M. Marini, and G. Villari, “On the monotonicity property for a certain class of second order
differential equations,” Journal of Differential Equations, vol. 82, no. 1, pp. 15–27, 1989.
13 M. Marini, “On nonoscillatory solutions of a second-order nonlinear differential equation,” Bollettino
della Unione Matematica Italiana, vol. 3, no. 1, pp. 189–202, 1984.
14 P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,
vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence,
RI, USA, 1986.
15 J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical
Sciences, Springer, New York, NY, USA, 1989.
16 K. C. Chang and Y. Q. Lin, Functional Analysis, Peking University Press, Beijing, China, 1986.