Existence and multiple solutions to a discrete fourth order
Transcript Of Existence and multiple solutions to a discrete fourth order
Liu et al. Advances in Difference Equations https://doi.org/10.1186/s13662-018-1879-1
(2018) 2018:427
RESEARCH
Open Access
Existence and multiple solutions to a discrete fourth order boundary value problem
Xia Liu1,2*, Tao Zhou3 and Haiping Shi4,5
*Correspondence: [email protected] 1College of Continuing Education and Open College, Guangdong University of Foreign Studies, Guangzhou, China 2Science College, Hunan Agricultural University, Changsha, China Full list of author information is available at the end of the article
Abstract In this article, we study a discrete fourth order boundary value problem. By making use of variational methods and critical point theory, we obtain some criteria for the existence and multiple solutions. Moreover, two examples are included to illustrate the applicability of the main results.
MSC: 39A10; 34B05; 58E05; 65L10
Keywords: Boundary value problems; Existence and multiple solutions; Fourth order; Critical point theory; Discrete
1 Introduction and statement of the main results In this article, we are interested in the existence and multiple solutions to the discrete fourth order nonlinear equation
4un–2 – (rn–1 un–1) = f (n, un), n ∈ Z[1, k],
(1.1)
with boundary value conditions
iu–1 = iuk–1, i = 0, 1, 2, 3,
(1.2)
where jun = ( j–1un) (j = 2, 3, 4), 0un = un, un = un+1 – un, f (s, u) ∈ C(R2, R), rn > 0 is real-valued for each n ∈ Z[0, k], r0 = rk, k ≥ 1 is an integer. Here, Z denotes the sets of integers, R denotes the sets of real numbers, N denotes the sets of natural numbers. Given a ≤ b in Z, let Z[a, b] := Z ∩ [a, b]. Let u* denote the transpose of a vector u.
Boundary value problem (1.1) with (1.2) can be regarded as being a discrete analogue of the fourth order differential equation
u(4)(s) – r(s)u (s) = f s, u(s) , s ∈ (0, 1),
(1.3)
with boundary value conditions
u(i)(0) = u(i)(1), i = 0, 1, 2, 3.
(1.4)
© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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(1.3) includes the following equation:
u(4)(s) = f s, u(s) , s ∈ R,
(1.5)
which is used to describe the stationary states of the deflection of an elastic beam [29]. Differential equations similar to (1.3) and special cases of it have been studied using a number of different methods in the literature, we refer the reader to papers [1, 2, 11–14, 24, 25] and the references contained therein.
Difference equations [1–10, 15–20, 22, 26–28, 30, 31] appear in numerous settings and forms, both as a fundamental tool in the discrete analogue of a differential equation and as a useful model for several economical and population problems.
If f (n, un) = qnun, Peterson and Ridenhour [22] considered the fourth order difference equation
4un–2 + qnun = 0, n ∈ [a + 2, a + 3, . . . , b + 2],
(1.6)
and gave some conditions on qn that ensure (1.6) is (2,2)-disconjugate on [a, b + 4] utilizing an appropriately defined quadratic form.
Making use of the symmetric mountain pass lemma, Chen and Tang [5] established some existence criteria to guarantee the fourth order difference system
4un–2 + qnun = f (n, un+1, un, un–1), n ∈ Z
(1.7)
has infinitely many homoclinic orbits. In [16], the existence, multiplicity, and nonexistence results of nontrivial solutions for
discrete nonlinear fourth order boundary value problems
4un–2 + η 2un–1 – ξ un = λf (n, un), n ∈ Z[a + 1, b + 1],
with ua = 2ua–1 = 0,
ub+2 = 2ub+1 = 0,
are obtained. The methods used here are based on the critical point theory and monotone operator theory.
Positive solutions of the following fourth order nonlinear difference equations with a deviating argument
an bn cn( un)γ β α + dnuλn+τ = 0
(1.8)
are investigated. Došlá, Krejčová, and Marini [8] introduced for (1.8) the notions of a minimal solution and a maximal solution, and gave necessary and sufficient conditions for their existence. Some relationships with nonoscillatory solutions, which have a different growth at infinity, were presented as well.
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Graef, Kong, and Wang [10] studied the discrete fourth order periodic boundary value problem with a parameter
⎧ ⎨ 4un–2 –
(pn–1
un–1) + qnun = λf (n, un),
⎩ iu–1 = iuN–1, i = 0, 1, 2, 3.
n ∈ Z(1, N),
By using variational methods and the mountain pass lemma, sufficient conditions are found under which the above problem has at least two nontrivial solutions.
In 2015, Liu, Zhang, and Shi [19] considered the following fourth order nonlinear difference equation:
2 pn–2 2un–2 – (qn–1 un–1) = f (n, un+1, un, un–1), n ∈ Z(1, k),
with boundary value conditions
u–1 = u0 = 0, uk+1 = uk+2 = 0.
Using the critical point theory, the authors established various sets of sufficient conditions for the existence and nonexistence of solutions for the Dirichlet boundary value problem and gave some new results.
By using the invariant set of descending flow and variational method, Long and Chen [20] in 2018 established the existence of multiple solutions to a class of second order discrete Neumann boundary value problem
⎧ ⎨– (pn–1 un–1) + qnun = kf (n, un), ⎩ u0 = uN .
n ∈ Z(1, N),
The solutions included sign-changing solutions, positive solutions, and negative solutions. Moreover, an example was given to illustrate our results.
In the last few years, variational methods and critical point theory have been used to study the existence and multiple solutions of discrete boundary value problems. In this article, we utilize this approach to obtain some sufficient conditions for the existence and multiple solutions to the boundary value problem (BVP for short) (1.1) with (1.2). What is more, two examples are included to illustrate the applicability of the main results.
Throughout this article, assume that there is a function F(s, u) such that
u
F(s, u) = f (s, t) dt
0
for any (s, u) ∈ R2. Our main results are the following theorems.
Theorem 1.1 Assume that the function F(s, u) ≥ 0 satisfies the following assumptions:
(F1)
There
exist
two
constants
δ1
>
0
and
a1
∈
(0,
λmin 2
)
such
that
F(s, u) ≤ a1u2, ∀s ∈ R2, |u| ≤ δ1.
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(F2)
There
exist
two
constants
a2
∈
(
λmax 2
,
+∞)
and
a3
>
0
such
that
F(s, u) ≥ a2u2 – a3, ∀(s, u) ∈ R2,
where λmin and λmax are constants which can be referred to (2.4) and (2.5). Then BVP (1.1) with (1.2) admits at least three solutions which are a trivial solution and two nontrivial solutions.
Remark 1.1 In [10], the authors considered the discrete fourth order periodic boundary value problem with a parameter
⎧ ⎨ 4un–2 –
(pn–1
un–1) + qnun = λf (n, un),
⎩ iu–1 = iuN–1, i = 0, 1, 2, 3.
n ∈ Z(1, N),
(1.9)
The following hypotheses are satisfied in [10]:
(H1) pn > 0 for n ∈ Z(0, N + 1) and qn > 0 for n ∈ Z(0, N);
(H2)
lim|n|→∞
|F (n,u)| |u|2
=
0
for
n
∈
Z(0, N);
(H3)
lim sup|n|→∞
|F (n,u)| |u|2
≤
0
for
n
∈
Z(0, N);
(H4) there exists ω ∈ U such that
N n=1
F
(n,
ωn
)
>
0.
Note that (F2) of Theorem 1.1 does not satisfy (H2). At least two nontrivial solutions of
(1.9) are obtained by the mountain pass lemma in [10]. However, in our paper, we employ
a linking theorem to obtain at least two nontrivial solutions. Furthermore, our conditions
on the nonlinear term are weaker than [10].
Theorem 1.2 Assume that the function F(s, u) ≥ 0 satisfies the following assumptions: (F3) lim|u|→0 F(us2,u) = 0, ∀(s, u) ∈ R2. (F4) There exist three constants a4 > 0, γ > 2, and a5 > 0 such that
F(s, u) ≥ a4|u|γ – a5, ∀(s, u) ∈ R2.
Then BVP (1.1) with (1.2) admits at least three solutions which are a trivial solution and two nontrivial solutions.
Theorem 1.3 Assume that the function F(s, u) ≥ 0, (F1) and (F2) and the following assumptions are satisfied:
(f ) f (s, –u) = –f (s, u), ∀(s, u) ∈ R2.
Then BVP (1.1) with (1.2) admits at least q distinct pairs of nontrivial solutions, where q is the dimension of Q which can be referred to (2.6).
Remark 1.2 In [9], the authors considered the fourth order nonlinear difference equation
⎧ ⎨ 4un–2 –
(pn–1
un–1) + qnun = f (n, un),
⎩ iu–1 = iuN–1, i = 0, 1, 2, 3.
n ∈ Z(1, N),
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Define
f0
=
lim infu→0 minn∈Z(1,N)
f (n,u) u
and
f∞
=
lim supu→0 maxn∈Z(1,N)
f (nu,u) .
The following hypotheses are satisfied in [9]:
(H1) pn ≥ 0 and qn ≥ 0 for n ∈ Z(1, N) and there exists η with η < q such that f ∞ ≤ η,
where q = minn∈Z(1,N) qn;
(H2) f (n, u) is odd in u, i.e., f (n, –u) = –f (n, u) for (n, u) ∈ Z(1, N) × R;
(H3) there exists m ∈ {1, . . . , N} such that f0 > λm.
Note that (F1) of Theorem 1.3 does not satisfy (H3). Furthermore, our conditions on the
nonlinear term are weaker than [9].
If f (n, un) = τnψ(un), (1.1) reduces to the following fourth order nonlinear equation:
4un–2 – (rn–1 un–1) = τnψ (un), n ∈ Z[1, k],
(1.10)
where ψ ∈ C(R, R), τn > 0 is real-valued for each n ∈ Z[1, k]. Therefore, we can easily obtain the following results.
Theorem 1.4 Assume that the following assumptions are satisfied: (Ψ1) There exists a function Ψ (u) ∈ C1(R, R) with Ψ (u) ≥ 0 such that
Ψ (u) = ψ(u).
(Ψ2)
There
exist
two
constants
δ2
>
0
and
a6
∈
(0,
λmin 2
)
such
that
Ψ (u) ≤ a6u2, ∀s ∈ R2, |u| ≤ δ2.
(Ψ3)
There
exist
two
constants
a7
∈
(
λmax 2
,
+∞)
and
a8
>
0
such
that
Ψ (u) ≥ a7u2 – a8, ∀(s, u) ∈ R2,
where λmin and λmax are constants which can be referred to (2.4) and (2.5). Then BVP (1.10) with (1.2) admits at least three solutions which are a trivial solution and two nontrivial solutions.
Corollary 1.1 Assume that (Ψ1), (Ψ2), (Ψ3) and the following assumption are satisfied.
(ψ) ψ(–u) = –ψ(u), ∀u ∈ R.
Then BVP (1.10) with (1.2) admits at least q distinct pairs of nontrivial solutions, where q is the dimension of Q which can be referred to (2.6).
2 Variational framework In this section, we shall establish the corresponding variational framework for BVP (1.1) with (1.2) which will be of fundamental importance in proving our main results.
In order to apply the critical point theory, we define a k-dimensional Hilbert space U by
U := u : Z[–1, k + 2] → R | iu–1 = iuk–1, i = 0, 1, 2, 3 ,
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and equip it with the inner product
k
(u, v) := ujvj,
j=1
∀u, v ∈ U,
and the induced norm
u :=
1
k
2
u2j ,
j=1
∀u ∈ U.
Remark 2.1 It is obvious that
u–1 = uk–1, u0 = uk, u1 = uk+1, u2 = uk+2, ∀u ∈ U.
(2.1)
As a matter of fact, U is isomorphic to Rk. Throughout this article, when we say u = (u1, u2, . . . , uk) ∈ Rk, we always imply that u can be extended to a vector in U so that (2.1) holds.
Define a functional J on U by
1k J(u) :=
2 n=1
k
2un–2 2 + rn–1(
n=1
k
un–1)2 – F(n, un).
n=1
After a careful computation, we have
∂J =
∂ un
4un–2 –
(rn–1 un–1) – f (n, un),
n ∈ Z[1, k].
Therefore, J (u) = 0 if and only if
(2.2)
4un–2 – (rn–1 un–1) = f (n, un), n ∈ Z[1, k].
Consequently, we reduce the problem of finding a solution of BVP (1.1) with (1.2) to that of seeking a critical point of the functional J on U. Denote the k × k matrices S and R.
For k = 1, let S = R = (0). For k = 2, let
8 –8
S=
,
–8 8
and
R = r0 + r1 –r0 – r1 . –r0 – r1 r0 + r1
For k = 3, let ⎛ 6 –3
S = ⎜⎝–3 6 –3 –3
⎞ –3 –3⎟⎠ . 6
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For k = 4, let
⎛
⎞
6 –4 2 –4
⎜⎜–4 6 –4 2 ⎟⎟
S = ⎜⎝ 2 –4 6 –4⎟⎠ .
–4 2 –4 6
For k ≥ 5, let
⎛
⎞
6 –4 1 0 0 · · · 0 0 1 –4
⎜⎜–4 6 –4 1 0 · · · 0 0 0 1 ⎟⎟
⎜⎜⎜ 1 –4 6 –4 1 · · · 0 0 0 0 ⎟⎟⎟
⎜⎜ 0 1 –4 6 –4 · · · 0 0 0 0 ⎟⎟
⎜⎜ 0 0 1 –4 6 · · · 0 0 0 0 ⎟⎟
S = ⎜⎜⎜· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·⎟⎟⎟ .
⎜⎜ 0 0 0 0 0 · · · 6 –4 1 0 ⎟⎟
⎜⎜ ⎜
0
0
0
0
0
· · · –4
6
–4
1
⎟⎟ ⎟
⎜⎝ 1 0 0 0 0 · · · 1 –4 6 –4⎟⎠
–4 1 0 0 0 · · · 0 1 –4 6
For k ≥ 3, let
⎛
r0 + r1 –r1
0 ···
⎜⎜ –r1 r1 + r2 –r2 · · ·
⎜⎜ 0
–r r + r · · ·
R = ⎜⎜ ⎜
···
2
23
··· ··· ···
⎜⎝ 0
0
0 ···
⎞
–r0
0 ⎟⎟
0 ⎟⎟
···
⎟⎟ . ⎟
–rk–1 ⎟⎠
–r0
0
0 · · · rk–1 + r0
Let M := S + R. We rewrite J(u) as
J(u) = 1 u∗Mu – k F(n, un). 2 n=1
(2.3)
It is easy to see that 0 is an eigenvalue of M, (1, 1, . . . , 1)∗ is an eigenvector associated with 0. M is semi-positive definite. Let λ1, λ2, . . . , λk be the eigenvalues of M.
Set
λmin = min{λj | λj = 0, j = 1, 2, . . . , k},
(2.4)
and
λmax = max{λj | λj = 0, j = 1, 2, . . . , k}.
(2.5)
Let P = {(c, c, . . . , c)∗ ∈ U | c ∈ R}, then P is an invariant subspace of U. Denote Q by
U = P ⊕ Q.
(2.6)
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3 Some basic lemmas Assume that U is a real Banach space and J ∈ C1(U, R). As usual, J is said to satisfy the Palais–Smale condition if every sequence {u(j)} ⊂ U such that {J(u(j))} is bounded and J (u(j)) → 0 (j → ∞) has a convergent subsequence. The sequence {u(j)} is called a Palais– Smale sequence.
Let U be a real Banach space. Define the symbol Bρ(u) as the open ball in U about u of radius ρ, ∂Bρ(u) as its boundary, and B¯ ρ(u) as its closure.
Lemma 3.1 (Linking theorem [21, 23]) Let U be a real Banach space, U = U1 ⊕ U2, where U1 is finite dimensional. Suppose that J ∈ C1(U, R) satisfies the Palais–Smale condition and the following:
(J1) There are positive constants c and ρ such that J|∂Bρ(0)∩U2 ≥ c. (J2) There are μ ∈ ∂B1(0) ∩ U2 and a positive constant cˆ ≥ ρ such that J|∂Ω ≤ 0, where
Ω = (B¯ cˆ(0) ∩ U1) ⊕ {sμ | 0 < s < cˆ}. Then J possesses a critical value c0 ≥ c, where
c0 = inf sup J d(u) ,
d∈Υ u∈Ω
and Υ = {d ∈ C(Ω¯ , U) | d|∂Ω = id}, where id denotes the identity operator.
Lemma 3.2 (Clark theorem [21]) Let U be a real Banach space, J ∈ C1(U, R), with J being even, bounded from below and satisfying the Palais–Smale condition. Assume J(0) = 0, there is a set Γ ⊂ U such that Γ is homeomorphic to Sk–1 (k – 1 dimension unit sphere) by an odd map, and supΓ J < 0. Then J has at least k distinct pairs of nonzero critical points.
Lemma 3.3 Assume that (r) and (F1)–(F3) are satisfied. Then the functional J satisfies the Palais–Smale condition.
Proof Let {u(j)}j∈N ⊂ U be such that {J(u(j))}j∈N is bounded and J (u(j)) → 0 as j → ∞. Then there is a constant A > 0 such that
–A ≤ J u(j) ≤ A, ∀j ∈ N.
From (F2) and (2.3), for any {u(j)}j∈N ⊂ U, we have
–A ≤ J u(j)
1 =
u(j)
∗Mu(j) –
k
F n, u(j)
2 n=1
≤ λmax u(j) 2 – k 2
a2 u(j) 2 – a3
n=1
= λmax – a2 2
u(j) 2 + a3k.
Then
a2 – λmax 2
u(j) 2 ≤ A + a3k.
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It
comes
from
a2
∈
(
λmax 2
,
+∞)
that
we
can
find
a
constant
B
>
0
such
that,
for
any
j
∈
N,
u(j) ≤ B. Thus, we know that the sequence {u(j)}j∈N is bounded in the k dimensional
space U. Therefore, the Palais–Smale condition holds.
4 Proofs of theorems
Proof of Theorem 1.1 Obviously, F(n, 0) = 0 and f (n, 0) = 0 for any n ∈ Z[1, k] via (F1) and (F2). Hence, u = 0 is a trivial solution of BVP (1.1) with (1.2).
It comes from Lemma 3.3 that J(u) is bounded from above in U. Let
J¯ = sup J(u).
u∈U
Therefore, there exists a sequence {u(j)} on U such that
J¯ = lim J u(j) .
j→∞
What is more, from the proof of Lemma 3.3, we have
J(u) ≤ λmax – a2 2
u 2 + a3k,
∀u ∈ U.
(4.1)
This implies that lim u →+∞ J(u) = –∞. Thus, {u(j)} is bounded. Then {u(j)} has a convergent subsequence defined by {u(jn)}. Set
u¯ = lim u(jn).
n→+∞
Due to the continuity of J(u) in u, there must be a point u¯ ∈ U, J(u¯ ) = J¯. Clearly, u¯ ∈ U is a critical point of J(u).
From (F1), for any u ∈ Q, u ≤ δ1, we have
J(u) = 1 u∗Mu – k F(n, un) 2 n=1
≥ λmin 2
k
u 2 – a1 u2n
n=1
≥ λmin – a1 u 2. 2
Denote
c = λm2in – a1 δ12.
We have
J(u) ≥ c, ∀u ∈ Q ∩ ∂Bδ1 (0).
(4.2)
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Thus, there are constants c > 0 and δ1 > 0 such that J|∂Bδ1 (0)∩Q ≥ c. Assumption (J1) of the linking theorem is satisfied.
In view of Mu = 0, for all u ∈ P, we have
J(u) = 12 u∗Mu – k F(n, un) = – k F(n, un) ≤ 0.
n=1
n=1
Hence, u¯ ∈/ P and the critical point u¯ of J(u) corresponding to the critical value J¯ is a nontrivial solution of BVP (1.1) with (1.2).
In the light of Lemmas 3.1 and 3.3, it is sufficient to verify condition (J2). Choose α ∈ ∂B1(0) ∩ Q, for any β ∈ P and s ∈ R, let u = sα + β. By (F2), we have
J(u) = 1 (sα + β)∗M(sα + β) – k F(n, sαn + βn) 2 n=1
≤ 12 (sα)∗M(sα) – k a2(sαn + βn)2 – a3
n=1
≤ λm2axs2 – a2 k (sαn + βn)2 + a3k
n=1
= λmax – a2 s2 – a2 β 2 + a3k 2
≤ –a2 β 2 + a3k.
Consequently, there is some positive constant χ > δ1 such that
J(u) ≤ 0, ∀u ∈ ∂Ω,
where Ω = (B¯ χ (0) ∩ Q) ⊕ {sα | 0 < s < χ }. Applying the linking theorem, J(u) has a critical value c0 ≥ c > 0, where
c0 = inf sup J d(u) ,
d∈Υ u∈Ω
and Υ = {d ∈ C(Ω¯ , U) | d|∂Ω = id}. Similar to the proof of Theorem 1.1 in [4], we can prove that BVP (1.1) with (1.2) admits
at least three solutions, and so we omit it.
Remark 4.1 Note that (F3) implies (F1). Similar to the above argument, we can also prove Theorem 1.2. For simplicity, we omit its proof.
Proof of Theorem 1.3 Obviously J ∈ C1(U, R), J is even, and J(0) = 0. From Lemma 3.3, J satisfies the Palais–Smale condition. By the proof of Theorem 1.1, we have that J is bounded from below. On account of Lemma 3.2, it is sufficient to find a set Γ and an odd map such that Γ is homeomorphic to Sq–1 by an odd map.
Choose
Γ = ∂Bδ1 (0) ∩ Q.
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RESEARCH
Open Access
Existence and multiple solutions to a discrete fourth order boundary value problem
Xia Liu1,2*, Tao Zhou3 and Haiping Shi4,5
*Correspondence: [email protected] 1College of Continuing Education and Open College, Guangdong University of Foreign Studies, Guangzhou, China 2Science College, Hunan Agricultural University, Changsha, China Full list of author information is available at the end of the article
Abstract In this article, we study a discrete fourth order boundary value problem. By making use of variational methods and critical point theory, we obtain some criteria for the existence and multiple solutions. Moreover, two examples are included to illustrate the applicability of the main results.
MSC: 39A10; 34B05; 58E05; 65L10
Keywords: Boundary value problems; Existence and multiple solutions; Fourth order; Critical point theory; Discrete
1 Introduction and statement of the main results In this article, we are interested in the existence and multiple solutions to the discrete fourth order nonlinear equation
4un–2 – (rn–1 un–1) = f (n, un), n ∈ Z[1, k],
(1.1)
with boundary value conditions
iu–1 = iuk–1, i = 0, 1, 2, 3,
(1.2)
where jun = ( j–1un) (j = 2, 3, 4), 0un = un, un = un+1 – un, f (s, u) ∈ C(R2, R), rn > 0 is real-valued for each n ∈ Z[0, k], r0 = rk, k ≥ 1 is an integer. Here, Z denotes the sets of integers, R denotes the sets of real numbers, N denotes the sets of natural numbers. Given a ≤ b in Z, let Z[a, b] := Z ∩ [a, b]. Let u* denote the transpose of a vector u.
Boundary value problem (1.1) with (1.2) can be regarded as being a discrete analogue of the fourth order differential equation
u(4)(s) – r(s)u (s) = f s, u(s) , s ∈ (0, 1),
(1.3)
with boundary value conditions
u(i)(0) = u(i)(1), i = 0, 1, 2, 3.
(1.4)
© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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(1.3) includes the following equation:
u(4)(s) = f s, u(s) , s ∈ R,
(1.5)
which is used to describe the stationary states of the deflection of an elastic beam [29]. Differential equations similar to (1.3) and special cases of it have been studied using a number of different methods in the literature, we refer the reader to papers [1, 2, 11–14, 24, 25] and the references contained therein.
Difference equations [1–10, 15–20, 22, 26–28, 30, 31] appear in numerous settings and forms, both as a fundamental tool in the discrete analogue of a differential equation and as a useful model for several economical and population problems.
If f (n, un) = qnun, Peterson and Ridenhour [22] considered the fourth order difference equation
4un–2 + qnun = 0, n ∈ [a + 2, a + 3, . . . , b + 2],
(1.6)
and gave some conditions on qn that ensure (1.6) is (2,2)-disconjugate on [a, b + 4] utilizing an appropriately defined quadratic form.
Making use of the symmetric mountain pass lemma, Chen and Tang [5] established some existence criteria to guarantee the fourth order difference system
4un–2 + qnun = f (n, un+1, un, un–1), n ∈ Z
(1.7)
has infinitely many homoclinic orbits. In [16], the existence, multiplicity, and nonexistence results of nontrivial solutions for
discrete nonlinear fourth order boundary value problems
4un–2 + η 2un–1 – ξ un = λf (n, un), n ∈ Z[a + 1, b + 1],
with ua = 2ua–1 = 0,
ub+2 = 2ub+1 = 0,
are obtained. The methods used here are based on the critical point theory and monotone operator theory.
Positive solutions of the following fourth order nonlinear difference equations with a deviating argument
an bn cn( un)γ β α + dnuλn+τ = 0
(1.8)
are investigated. Došlá, Krejčová, and Marini [8] introduced for (1.8) the notions of a minimal solution and a maximal solution, and gave necessary and sufficient conditions for their existence. Some relationships with nonoscillatory solutions, which have a different growth at infinity, were presented as well.
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Graef, Kong, and Wang [10] studied the discrete fourth order periodic boundary value problem with a parameter
⎧ ⎨ 4un–2 –
(pn–1
un–1) + qnun = λf (n, un),
⎩ iu–1 = iuN–1, i = 0, 1, 2, 3.
n ∈ Z(1, N),
By using variational methods and the mountain pass lemma, sufficient conditions are found under which the above problem has at least two nontrivial solutions.
In 2015, Liu, Zhang, and Shi [19] considered the following fourth order nonlinear difference equation:
2 pn–2 2un–2 – (qn–1 un–1) = f (n, un+1, un, un–1), n ∈ Z(1, k),
with boundary value conditions
u–1 = u0 = 0, uk+1 = uk+2 = 0.
Using the critical point theory, the authors established various sets of sufficient conditions for the existence and nonexistence of solutions for the Dirichlet boundary value problem and gave some new results.
By using the invariant set of descending flow and variational method, Long and Chen [20] in 2018 established the existence of multiple solutions to a class of second order discrete Neumann boundary value problem
⎧ ⎨– (pn–1 un–1) + qnun = kf (n, un), ⎩ u0 = uN .
n ∈ Z(1, N),
The solutions included sign-changing solutions, positive solutions, and negative solutions. Moreover, an example was given to illustrate our results.
In the last few years, variational methods and critical point theory have been used to study the existence and multiple solutions of discrete boundary value problems. In this article, we utilize this approach to obtain some sufficient conditions for the existence and multiple solutions to the boundary value problem (BVP for short) (1.1) with (1.2). What is more, two examples are included to illustrate the applicability of the main results.
Throughout this article, assume that there is a function F(s, u) such that
u
F(s, u) = f (s, t) dt
0
for any (s, u) ∈ R2. Our main results are the following theorems.
Theorem 1.1 Assume that the function F(s, u) ≥ 0 satisfies the following assumptions:
(F1)
There
exist
two
constants
δ1
>
0
and
a1
∈
(0,
λmin 2
)
such
that
F(s, u) ≤ a1u2, ∀s ∈ R2, |u| ≤ δ1.
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(F2)
There
exist
two
constants
a2
∈
(
λmax 2
,
+∞)
and
a3
>
0
such
that
F(s, u) ≥ a2u2 – a3, ∀(s, u) ∈ R2,
where λmin and λmax are constants which can be referred to (2.4) and (2.5). Then BVP (1.1) with (1.2) admits at least three solutions which are a trivial solution and two nontrivial solutions.
Remark 1.1 In [10], the authors considered the discrete fourth order periodic boundary value problem with a parameter
⎧ ⎨ 4un–2 –
(pn–1
un–1) + qnun = λf (n, un),
⎩ iu–1 = iuN–1, i = 0, 1, 2, 3.
n ∈ Z(1, N),
(1.9)
The following hypotheses are satisfied in [10]:
(H1) pn > 0 for n ∈ Z(0, N + 1) and qn > 0 for n ∈ Z(0, N);
(H2)
lim|n|→∞
|F (n,u)| |u|2
=
0
for
n
∈
Z(0, N);
(H3)
lim sup|n|→∞
|F (n,u)| |u|2
≤
0
for
n
∈
Z(0, N);
(H4) there exists ω ∈ U such that
N n=1
F
(n,
ωn
)
>
0.
Note that (F2) of Theorem 1.1 does not satisfy (H2). At least two nontrivial solutions of
(1.9) are obtained by the mountain pass lemma in [10]. However, in our paper, we employ
a linking theorem to obtain at least two nontrivial solutions. Furthermore, our conditions
on the nonlinear term are weaker than [10].
Theorem 1.2 Assume that the function F(s, u) ≥ 0 satisfies the following assumptions: (F3) lim|u|→0 F(us2,u) = 0, ∀(s, u) ∈ R2. (F4) There exist three constants a4 > 0, γ > 2, and a5 > 0 such that
F(s, u) ≥ a4|u|γ – a5, ∀(s, u) ∈ R2.
Then BVP (1.1) with (1.2) admits at least three solutions which are a trivial solution and two nontrivial solutions.
Theorem 1.3 Assume that the function F(s, u) ≥ 0, (F1) and (F2) and the following assumptions are satisfied:
(f ) f (s, –u) = –f (s, u), ∀(s, u) ∈ R2.
Then BVP (1.1) with (1.2) admits at least q distinct pairs of nontrivial solutions, where q is the dimension of Q which can be referred to (2.6).
Remark 1.2 In [9], the authors considered the fourth order nonlinear difference equation
⎧ ⎨ 4un–2 –
(pn–1
un–1) + qnun = f (n, un),
⎩ iu–1 = iuN–1, i = 0, 1, 2, 3.
n ∈ Z(1, N),
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Define
f0
=
lim infu→0 minn∈Z(1,N)
f (n,u) u
and
f∞
=
lim supu→0 maxn∈Z(1,N)
f (nu,u) .
The following hypotheses are satisfied in [9]:
(H1) pn ≥ 0 and qn ≥ 0 for n ∈ Z(1, N) and there exists η with η < q such that f ∞ ≤ η,
where q = minn∈Z(1,N) qn;
(H2) f (n, u) is odd in u, i.e., f (n, –u) = –f (n, u) for (n, u) ∈ Z(1, N) × R;
(H3) there exists m ∈ {1, . . . , N} such that f0 > λm.
Note that (F1) of Theorem 1.3 does not satisfy (H3). Furthermore, our conditions on the
nonlinear term are weaker than [9].
If f (n, un) = τnψ(un), (1.1) reduces to the following fourth order nonlinear equation:
4un–2 – (rn–1 un–1) = τnψ (un), n ∈ Z[1, k],
(1.10)
where ψ ∈ C(R, R), τn > 0 is real-valued for each n ∈ Z[1, k]. Therefore, we can easily obtain the following results.
Theorem 1.4 Assume that the following assumptions are satisfied: (Ψ1) There exists a function Ψ (u) ∈ C1(R, R) with Ψ (u) ≥ 0 such that
Ψ (u) = ψ(u).
(Ψ2)
There
exist
two
constants
δ2
>
0
and
a6
∈
(0,
λmin 2
)
such
that
Ψ (u) ≤ a6u2, ∀s ∈ R2, |u| ≤ δ2.
(Ψ3)
There
exist
two
constants
a7
∈
(
λmax 2
,
+∞)
and
a8
>
0
such
that
Ψ (u) ≥ a7u2 – a8, ∀(s, u) ∈ R2,
where λmin and λmax are constants which can be referred to (2.4) and (2.5). Then BVP (1.10) with (1.2) admits at least three solutions which are a trivial solution and two nontrivial solutions.
Corollary 1.1 Assume that (Ψ1), (Ψ2), (Ψ3) and the following assumption are satisfied.
(ψ) ψ(–u) = –ψ(u), ∀u ∈ R.
Then BVP (1.10) with (1.2) admits at least q distinct pairs of nontrivial solutions, where q is the dimension of Q which can be referred to (2.6).
2 Variational framework In this section, we shall establish the corresponding variational framework for BVP (1.1) with (1.2) which will be of fundamental importance in proving our main results.
In order to apply the critical point theory, we define a k-dimensional Hilbert space U by
U := u : Z[–1, k + 2] → R | iu–1 = iuk–1, i = 0, 1, 2, 3 ,
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and equip it with the inner product
k
(u, v) := ujvj,
j=1
∀u, v ∈ U,
and the induced norm
u :=
1
k
2
u2j ,
j=1
∀u ∈ U.
Remark 2.1 It is obvious that
u–1 = uk–1, u0 = uk, u1 = uk+1, u2 = uk+2, ∀u ∈ U.
(2.1)
As a matter of fact, U is isomorphic to Rk. Throughout this article, when we say u = (u1, u2, . . . , uk) ∈ Rk, we always imply that u can be extended to a vector in U so that (2.1) holds.
Define a functional J on U by
1k J(u) :=
2 n=1
k
2un–2 2 + rn–1(
n=1
k
un–1)2 – F(n, un).
n=1
After a careful computation, we have
∂J =
∂ un
4un–2 –
(rn–1 un–1) – f (n, un),
n ∈ Z[1, k].
Therefore, J (u) = 0 if and only if
(2.2)
4un–2 – (rn–1 un–1) = f (n, un), n ∈ Z[1, k].
Consequently, we reduce the problem of finding a solution of BVP (1.1) with (1.2) to that of seeking a critical point of the functional J on U. Denote the k × k matrices S and R.
For k = 1, let S = R = (0). For k = 2, let
8 –8
S=
,
–8 8
and
R = r0 + r1 –r0 – r1 . –r0 – r1 r0 + r1
For k = 3, let ⎛ 6 –3
S = ⎜⎝–3 6 –3 –3
⎞ –3 –3⎟⎠ . 6
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For k = 4, let
⎛
⎞
6 –4 2 –4
⎜⎜–4 6 –4 2 ⎟⎟
S = ⎜⎝ 2 –4 6 –4⎟⎠ .
–4 2 –4 6
For k ≥ 5, let
⎛
⎞
6 –4 1 0 0 · · · 0 0 1 –4
⎜⎜–4 6 –4 1 0 · · · 0 0 0 1 ⎟⎟
⎜⎜⎜ 1 –4 6 –4 1 · · · 0 0 0 0 ⎟⎟⎟
⎜⎜ 0 1 –4 6 –4 · · · 0 0 0 0 ⎟⎟
⎜⎜ 0 0 1 –4 6 · · · 0 0 0 0 ⎟⎟
S = ⎜⎜⎜· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·⎟⎟⎟ .
⎜⎜ 0 0 0 0 0 · · · 6 –4 1 0 ⎟⎟
⎜⎜ ⎜
0
0
0
0
0
· · · –4
6
–4
1
⎟⎟ ⎟
⎜⎝ 1 0 0 0 0 · · · 1 –4 6 –4⎟⎠
–4 1 0 0 0 · · · 0 1 –4 6
For k ≥ 3, let
⎛
r0 + r1 –r1
0 ···
⎜⎜ –r1 r1 + r2 –r2 · · ·
⎜⎜ 0
–r r + r · · ·
R = ⎜⎜ ⎜
···
2
23
··· ··· ···
⎜⎝ 0
0
0 ···
⎞
–r0
0 ⎟⎟
0 ⎟⎟
···
⎟⎟ . ⎟
–rk–1 ⎟⎠
–r0
0
0 · · · rk–1 + r0
Let M := S + R. We rewrite J(u) as
J(u) = 1 u∗Mu – k F(n, un). 2 n=1
(2.3)
It is easy to see that 0 is an eigenvalue of M, (1, 1, . . . , 1)∗ is an eigenvector associated with 0. M is semi-positive definite. Let λ1, λ2, . . . , λk be the eigenvalues of M.
Set
λmin = min{λj | λj = 0, j = 1, 2, . . . , k},
(2.4)
and
λmax = max{λj | λj = 0, j = 1, 2, . . . , k}.
(2.5)
Let P = {(c, c, . . . , c)∗ ∈ U | c ∈ R}, then P is an invariant subspace of U. Denote Q by
U = P ⊕ Q.
(2.6)
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3 Some basic lemmas Assume that U is a real Banach space and J ∈ C1(U, R). As usual, J is said to satisfy the Palais–Smale condition if every sequence {u(j)} ⊂ U such that {J(u(j))} is bounded and J (u(j)) → 0 (j → ∞) has a convergent subsequence. The sequence {u(j)} is called a Palais– Smale sequence.
Let U be a real Banach space. Define the symbol Bρ(u) as the open ball in U about u of radius ρ, ∂Bρ(u) as its boundary, and B¯ ρ(u) as its closure.
Lemma 3.1 (Linking theorem [21, 23]) Let U be a real Banach space, U = U1 ⊕ U2, where U1 is finite dimensional. Suppose that J ∈ C1(U, R) satisfies the Palais–Smale condition and the following:
(J1) There are positive constants c and ρ such that J|∂Bρ(0)∩U2 ≥ c. (J2) There are μ ∈ ∂B1(0) ∩ U2 and a positive constant cˆ ≥ ρ such that J|∂Ω ≤ 0, where
Ω = (B¯ cˆ(0) ∩ U1) ⊕ {sμ | 0 < s < cˆ}. Then J possesses a critical value c0 ≥ c, where
c0 = inf sup J d(u) ,
d∈Υ u∈Ω
and Υ = {d ∈ C(Ω¯ , U) | d|∂Ω = id}, where id denotes the identity operator.
Lemma 3.2 (Clark theorem [21]) Let U be a real Banach space, J ∈ C1(U, R), with J being even, bounded from below and satisfying the Palais–Smale condition. Assume J(0) = 0, there is a set Γ ⊂ U such that Γ is homeomorphic to Sk–1 (k – 1 dimension unit sphere) by an odd map, and supΓ J < 0. Then J has at least k distinct pairs of nonzero critical points.
Lemma 3.3 Assume that (r) and (F1)–(F3) are satisfied. Then the functional J satisfies the Palais–Smale condition.
Proof Let {u(j)}j∈N ⊂ U be such that {J(u(j))}j∈N is bounded and J (u(j)) → 0 as j → ∞. Then there is a constant A > 0 such that
–A ≤ J u(j) ≤ A, ∀j ∈ N.
From (F2) and (2.3), for any {u(j)}j∈N ⊂ U, we have
–A ≤ J u(j)
1 =
u(j)
∗Mu(j) –
k
F n, u(j)
2 n=1
≤ λmax u(j) 2 – k 2
a2 u(j) 2 – a3
n=1
= λmax – a2 2
u(j) 2 + a3k.
Then
a2 – λmax 2
u(j) 2 ≤ A + a3k.
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It
comes
from
a2
∈
(
λmax 2
,
+∞)
that
we
can
find
a
constant
B
>
0
such
that,
for
any
j
∈
N,
u(j) ≤ B. Thus, we know that the sequence {u(j)}j∈N is bounded in the k dimensional
space U. Therefore, the Palais–Smale condition holds.
4 Proofs of theorems
Proof of Theorem 1.1 Obviously, F(n, 0) = 0 and f (n, 0) = 0 for any n ∈ Z[1, k] via (F1) and (F2). Hence, u = 0 is a trivial solution of BVP (1.1) with (1.2).
It comes from Lemma 3.3 that J(u) is bounded from above in U. Let
J¯ = sup J(u).
u∈U
Therefore, there exists a sequence {u(j)} on U such that
J¯ = lim J u(j) .
j→∞
What is more, from the proof of Lemma 3.3, we have
J(u) ≤ λmax – a2 2
u 2 + a3k,
∀u ∈ U.
(4.1)
This implies that lim u →+∞ J(u) = –∞. Thus, {u(j)} is bounded. Then {u(j)} has a convergent subsequence defined by {u(jn)}. Set
u¯ = lim u(jn).
n→+∞
Due to the continuity of J(u) in u, there must be a point u¯ ∈ U, J(u¯ ) = J¯. Clearly, u¯ ∈ U is a critical point of J(u).
From (F1), for any u ∈ Q, u ≤ δ1, we have
J(u) = 1 u∗Mu – k F(n, un) 2 n=1
≥ λmin 2
k
u 2 – a1 u2n
n=1
≥ λmin – a1 u 2. 2
Denote
c = λm2in – a1 δ12.
We have
J(u) ≥ c, ∀u ∈ Q ∩ ∂Bδ1 (0).
(4.2)
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Thus, there are constants c > 0 and δ1 > 0 such that J|∂Bδ1 (0)∩Q ≥ c. Assumption (J1) of the linking theorem is satisfied.
In view of Mu = 0, for all u ∈ P, we have
J(u) = 12 u∗Mu – k F(n, un) = – k F(n, un) ≤ 0.
n=1
n=1
Hence, u¯ ∈/ P and the critical point u¯ of J(u) corresponding to the critical value J¯ is a nontrivial solution of BVP (1.1) with (1.2).
In the light of Lemmas 3.1 and 3.3, it is sufficient to verify condition (J2). Choose α ∈ ∂B1(0) ∩ Q, for any β ∈ P and s ∈ R, let u = sα + β. By (F2), we have
J(u) = 1 (sα + β)∗M(sα + β) – k F(n, sαn + βn) 2 n=1
≤ 12 (sα)∗M(sα) – k a2(sαn + βn)2 – a3
n=1
≤ λm2axs2 – a2 k (sαn + βn)2 + a3k
n=1
= λmax – a2 s2 – a2 β 2 + a3k 2
≤ –a2 β 2 + a3k.
Consequently, there is some positive constant χ > δ1 such that
J(u) ≤ 0, ∀u ∈ ∂Ω,
where Ω = (B¯ χ (0) ∩ Q) ⊕ {sα | 0 < s < χ }. Applying the linking theorem, J(u) has a critical value c0 ≥ c > 0, where
c0 = inf sup J d(u) ,
d∈Υ u∈Ω
and Υ = {d ∈ C(Ω¯ , U) | d|∂Ω = id}. Similar to the proof of Theorem 1.1 in [4], we can prove that BVP (1.1) with (1.2) admits
at least three solutions, and so we omit it.
Remark 4.1 Note that (F3) implies (F1). Similar to the above argument, we can also prove Theorem 1.2. For simplicity, we omit its proof.
Proof of Theorem 1.3 Obviously J ∈ C1(U, R), J is even, and J(0) = 0. From Lemma 3.3, J satisfies the Palais–Smale condition. By the proof of Theorem 1.1, we have that J is bounded from below. On account of Lemma 3.2, it is sufficient to find a set Γ and an odd map such that Γ is homeomorphic to Sq–1 by an odd map.
Choose
Γ = ∂Bδ1 (0) ∩ Q.