Nonexistence and existence results for a class of fourth-order difference Neumann boundary value problems

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ScienceDirect Indagationes Mathematicae xx (xxxx) xxx–xxx www.elsevier.com/locate/indag

Review

Nonexistence and existence results for a class of fourth-order difference Neumann boundary value problems✩ Xia Liu a,b,∗ , Yuanbiao Zhang c , Haiping Shi d

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a Oriental Science and Technology College, Hunan Agricultural University, Changsha 410128, China b Science College, Hunan Agricultural University, Changsha 410128, China c Packaging Engineering Institute, Jinan University, Zhuhai 519070, China d Modern Business and Management Department, Guangdong Construction Vocational Technology Institute,

Guangzhou 510450, China Received 2 December 2013; accepted 27 May 2014 Communicated by S.J. van Strien

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Abstract In this paper, a class of fourth-order nonlinear difference equations are considered. By making use of the critical point method, we establish various sets of sufficient conditions for the nonexistence and existence of solutions for Neumann boundary value problems and give some new results. Our results successfully complement the existing ones. c 2014 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). ⃝

Keywords: Nonexistence and existence; Neumann boundary value problems; Fourth-order; Mountain Pass Lemma; Discrete variational theory

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✩ This project is supported by Specialized Research Fund for the Doctoral Program of Higher Eduction of China (No. 20114410110002), National Natural Science Foundation of China (No. 11171078), Natural Science Foundation of Guangdong Province (No. S2013010014460), Science and Research Program of Hunan Provincial Science and Technology Department (No. 2012FJ4109) and Scientific Research Fund of Hunan Provincial Education Department (No. 12C0170). ∗ Corresponding author at: Oriental Science and Technology College, Hunan Agricultural University, Changsha 410128, China. Tel.: +86 13787799383. E-mail address: [email protected] (X. Liu).

http://dx.doi.org/10.1016/j.indag.2014.05.001 c 2014 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). 0019-3577/⃝

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Contents 1. 2. 3.

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Introduction...................................................................................................................... 2 Variational structure and some lemmas ................................................................................ 5 Proof of the main results .................................................................................................... 7 3.1. Proof of Theorem 1.1 .............................................................................................. 7 3.2. Proof of Theorem 1.2 .............................................................................................. 8 3.3. Proof of Theorem 1.3 .............................................................................................. 9 3.4. Proof of Theorem 1.4 .............................................................................................. 10 Examples ......................................................................................................................... 10 References ....................................................................................................................... 11

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1. Introduction Below N, Z and R denote the sets of all natural numbers, integers and real numbers respectively. k is a positive integer. For any a, b ∈ Z, define Z(a) = {a, a+1, . . .}, Z(a, b) = {a, a+1, . . . , b} when a ≤ b. ∆ is the forward difference operator defined by 1u n = u n+1 − u n , and ∆2 u n = ∆(1u n ). Besides, ∗ denotes the transpose of a vector. The present paper considers the fourth-order nonlinear difference equation     ∆2 pn−1 ∆2 u n−2 − ∆ qn (1u n−1 )δ + rn u δn = f (n, u n ), n ∈ Z(1, k), (1.1)

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with boundary value conditions

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1u −1 = 1u 0 = 0,

1u k = 1u k+1 = 0,

(1.2)

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where pn is nonzero and real valued for each n ∈ Z(0, k + 1), {qn }n∈Z(1,k+1) and {rn }n∈Z(1,k) are real sequences, δ is the ratio of odd positive integers, f ∈ C(R2 , R). Recently, difference equations have attracted the interest of many researchers since they provide a natural description of several discrete models. Such discrete models are often investigated in various fields of science and technology such as computer science, economics, neural network, ecology, cybernetics, biological systems, optimal control, and population dynamics. These studies cover many of the branches of difference equations, such as stability, attractivity, periodicity, oscillation, and boundary value problem, see [9,21,19,20,25,27,29,35,43] and the references therein. We may think of (1.1) with (1.2) as being a discrete analogue of the following fourth-order nonlinear differential equation  ′′  ′ p(t)u ′′ (t) − q(t)u ′ (t) = f (t, u(t)), t ∈ [a, b], (1.3)

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with boundary value conditions

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u(a) = u ′ (a) = 0,

u(b) = u ′ (b) = 0.

(1.4)

Eq. (1.3) includes the following equation u (4) (t) = f (t, u(t)),

t ∈ R,

(1.5)

which is used to describe the bending of an elastic beam; see, for example, [1,6,18,22,24,28,40] and references therein. Owing to its importance in physics, many methods are applied to study fourth-order boundary value problems by many authors.

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In recent years, the study of boundary value problems for differential equations develops at relatively rapid rate. By using various methods and techniques, such as fixed point theory, topological degree theory, coincidence degree theory, a series of existence results of nontrivial solutions for differential equations have been obtained in literatures, we refer to [3–5,8,23,38]. And critical point theory is also an important tool to deal with problems on differential equations [12,16,17, 30,34,44]. Only since 2003, critical point theory has been employed to establish sufficient conditions on the existence of periodic solutions of difference equations. By using the critical point theory, Guo and Yu [21,19,20] and Shi et al. [36] have successfully proved the existence of periodic solutions of second-order nonlinear difference equations. We also refer to [41,42] for the discrete boundary value problems. Compared to first-order or second-order difference equations, the study of higher-order equations, and in particular, fourth-order equations, has received considerably less attention (see, for example, [10,11,15,32,33,37,39] and the references contained therein). Yan, Liu [39] in 1997 and Thandapani, Arockiasamy [37] in 2001 studied the following fourth-order difference equation of form,   ∆2 pn ∆2 u n + f (n, u n ) = 0, n ∈ Z, (1.6) and obtained criteria for the oscillation and nonoscillation of solutions for Eq. (1.6). In 2005, Cai, Yu and Guo [7] obtained some criteria for the existence of periodic solutions of the fourth-order difference equation   ∆2 pn−2 ∆2 u n−2 + f (n, u n ) = 0, n ∈ Z. (1.7) In 1995, Peterson and Ridenhour considered the disconjugacy of Eq. (1.7) when pn ≡ 1 and f (n, u n ) = qn u n (see [32]). The boundary value problem (BVP) for determining the existence of solutions of difference equations has been a very active area of research in the last twenty years, and for surveys of recent results, we refer the reader to the monographs by Agarwal et al. [2,13,14,26,35]. We, in this paper, use the critical point theory to give some sufficient conditions of the nonexistence and existence of solutions for the BVP (1.1) with (1.2). We shall study the superlinear and sublinear cases. The main idea in this paper is to transfer the existence of the BVP (1.1) with (1.2) into the existence of the critical points of some functional. The proof is based on the notable Mountain Pass Lemma in combination with variational technique. The purpose of this paper is two-folded. On one hand, we shall further demonstrate the powerfulness of critical point theory in the study of solutions for boundary value problems of difference equations. On the other hand, we shall complement existing ones. The motivation for the present work stems from the recent paper in [12]. Let p¯ = max{ pn : n ∈ Z(1, k)}, q¯ = max{qn : n ∈ Z(2, k)}, r¯ = max{rn : n ∈ Z(1, k)},

p = min{ pn : n ∈ Z(1, k)}, q = min{qn : n ∈ Z(2, k)}, r = min{rn : n ∈ Z(1, k)}.

Our main results are as follows. Theorem 1.1. Assume that the following hypotheses are satisfied: ( p) for any n ∈ Z(1, k), pn < 0; (q) for any n ∈ Z(2, k), qn ≤ 0; (r ) for any n ∈ Z(1, k), rn ≤ 0;

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(F1 ) there exists a functional F(n, v) ∈ C 1 (Z × R, R) with F(0, ·) = 0 such that ∂ F(n, v) = f (n, v), ∀n ∈ Z(1, k); ∂v (F2 ) there exists a constant M0 > 0 such that for all (n, v) ∈ Z(1, k) × R | f (n, v)| ≤ M0 . Then the BVP (1.1) with (1.2) possesses at least one solution. Remark 1.1. Assumption (F2 ) implies that there exists a constant M1 > 0 such that (F2′ ) |F(n, v)| ≤ M1 + M0 |v|, ∀(n, v) ∈ Z(1, k) × R. Theorem 1.2. Suppose that (F1 ) and the following hypotheses are satisfied: ( p ′ ) for any n ∈ Z(1, k), pn > 0; (q ′ ) for any n ∈ Z(2, k), qn ≥ 0; (r ′ ) for any n ∈ Z(1, k), rn ≥ 0; (F3 ) there exists a functional F(n, v) ∈ C 1 (Z × R, R) such that F(n, v) = 0, ∀n ∈ Z(1, k); lim v→0 |v| (F4 ) there exists a constant β > max{2, δ + 1} such that for any n ∈ Z(1, k), 0 < v f (n, v) < β F(n, v),

∀v ̸= 0.

Then the BVP (1.1) with (1.2) possesses at least two nontrivial solutions. Remark 1.2. Assumption (F4 ) implies that there exist constants a1 > 0 and a2 > 0 such that (F4′ ) F(n, v) > a1 |v|β − a2 , ∀n ∈ Z(1, k). Theorem 1.3. Suppose that ( p ′ ), (q ′ ), (r ′ ), (F1 ) and the following assumption are satisfied: (F5 ) there exist constants R > 0 and 1 < α < 2 such that for n ∈ Z(1, k) and |v| ≥ R, 0 < f (n, v) ≤ α F(n, v). Then the BVP (1.1) with (1.2) possesses at least one solution. Remark 1.3. Assumption (F5 ) implies that for each n ∈ Z(1, k) there exist constants a3 > 0 and a4 > 0 such that (F5′ ) F(n, v) ≤ a3 |v|α + a4 , ∀(n, v) ∈ Z(1, k) × R. Theorem 1.4. Suppose that ( p), (q), (r ), (F1 ) and the following assumption are satisfied: (F6 ) v f (n, v) > 0, for v ̸= 0, ∀n ∈ Z(1, k). Then the BVP (1.1) with (1.2) has no nontrivial solutions. Remark 1.4. In the existing literature, results on the nonexistence of solutions of discrete boundary value problems are very scarce. Hence, Theorem 1.4 complements existing ones. The remaining of this paper is organized as follows. First, in Section 2, we shall establish the variational framework for the BVP (1.1) with (1.2) and transfer the problem of the existence of the BVP (1.1) with (1.2) into that of the existence of critical points of the corresponding functional. Some related fundamental results will also be recalled. Then, in Section 3, we shall complete the proof of the results by using the critical point method. Finally, in Section 4, we shall give three examples to illustrate the main results. For the basic knowledge of variational methods, the reader is referred to [30,31,34,44].

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2. Variational structure and some lemmas

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In order to apply the critical point theory, we shall establish the corresponding variational framework for the BVP (1.1) with (1.2) and give some lemmas which will be of fundamental importance in proving our main results. First, we state some basic notations. Let Rk be the real Euclidean space with dimension k. Define the inner product on Rk as follows: ⟨u, v⟩ =

k 

u jvj,

∀u, v ∈ Rk ,

(2.1)

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j=1

by which the norm ∥ · ∥ can be induced by ∥u∥ =

 k 



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,

u 2j

∀u ∈ Rk .

(2.2)

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j=1

On the other hand, we define the norm ∥ · ∥r on Rk as follows: ∥u∥r =

 k 

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 r1 r

|u j |

,

(2.3)

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j=1

for all u ∈ Rk and r > 1. Since ∥u∥r and ∥u∥2 are equivalent, there exist constants c1 , c2 such that c2 ≥ c1 > 0, and c1 ∥u∥2 ≤ ∥u∥r ≤ c2 ∥u∥2 ,

∀u ∈ Rk . )∗

Clearly, ∥u∥ = ∥u∥2 . For any u = (u 1 , u 2 , . . . , u k ∈ k > 2, consider the functional J defined on Rk as follows: J (u) =

Rk ,

(2.4)

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for the BVP (1.1)–(1.2), with

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k−2 k−1 k  2 1 1  1  pn+1 ∆2 u n + qn+1 (1u n )δ+1 + rn u δ+1 n 2 n=1 δ + 1 n=1 δ + 1 n=1

−

k 

F(n, u n ) +

n=1

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1 1 p1 (1u 1 )2 + pk (1u k−1 )2 , 2 2

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(2.5)

where

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∂ F(n, v) = f (n, v), ∂v

1u −1 = 1u 0 = 0,

1u k = 1u k+1 = 0.

Clearly, J ∈ C 1 (Rk , R) and for any u = {u n }kn=1 = (u 1 , u 2 , . . . , u k )∗ , by using 1u −1 = 1u 0 = 0, 1u k = 1u k+1 = 0, we can compute the partial derivative as     ∂J = ∆2 pn−1 ∆2 u n−2 − ∆ qn (1u n−1 )δ + rn u δn − f (n, u n ), ∀n ∈ Z(1, k). ∂u n Thus, u is a critical point of J on Rk if and only if     ∆2 pn−1 ∆2 u n−2 − ∆ qn (1u n−1 )δ + rn u δn = f (n, u n ),

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∀n ∈ Z(1, k).

We reduce the existence of the BVP (1.1) with (1.2) to the existence of critical points of J on Rk . That is, the functional J is just the variational framework of the BVP (1.1) with (1.2).

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Remark 2.1. In the case k = 1 and k = 2 are trivial, and we omit their proofs. Let P and Q be the k × k  6 −4 1 −4 6 −4   1 −4 6  1 −4 0 P= · · · · · · · · ·  0 0 0 0 0 0 0 0 0  2 −1 0 −1 2 −1   0 −1 2 Q= · · · · · · · · · 0 0 0 0 0 0

matrices defined by 0 0 ··· 0 1 0 ··· 0 −4 1 · · · 0 6 −4 · · · 0 ··· ··· ··· ··· 0 0 ··· 6 0 0 · · · −4 0 0 ··· 1 ··· 0 0 ··· 0 0  ··· 0 0 . · · · · · · · · · · · · 2 −1 · · · −1 2

0 < λ1 ≤ λ2 ≤ · · · ≤ λk ,

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0 < λ˜ 1 ≤ λ˜ 2 ≤ · · · ≤ λ˜ k .

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(2.6) (2.7) C 1 (E, R),

Let E be a real Banach space, J ∈ i.e., J is a continuously Fr´echet-differentiable functional defined on E. J is said to satisfy the Palais–Smale condition       (P.S.  condition for short) if any sequence u (l) ⊂ E for which J u (l) is bounded and J ′ u (l) → 0 (l → ∞) possesses a convergent subsequence in E. Let Bρ denote the open ball in E about 0 of radius ρ and let ∂ Bρ denote its boundary. Lemma 2.1 (Mountain Pass Lemma [34]). Let E be a real Banach space and J ∈ C 1 (E, R) satisfy the P.S. condition. If J (0) = 0 and (J1 ) there exist constants ρ, a > 0 such that J |∂ Bρ ≥ a, and (J2 ) there exists e ∈ E \ Bρ such that J (e) ≤ 0. Then J possesses a critical value c ≥ a given by c = inf max J (g(s)), g∈Γ s∈[0,1]

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0 0  0  0 , · · ·  1 −4 6

Clearly, P and Q are positive definite. Let λ1 , λ2 , . . . , λk be the eigenvalues of P, λ˜ 1 , λ˜ 2 , . . . , λ˜ k be the eigenvalues of Q. Applying matrix theory, we know λ j > 0, λ˜ j > 0, j = 1, 2, . . . , k. Without loss of generality, we may assume that

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0 0 0 0 ··· −4 6 −4

(2.8)

where Γ = {g ∈ C([0, 1], E)|g(0) = 0, g(1) = e}.

(2.9)

Lemma 2.2. Suppose that ( p ′ ), (q ′ ), (r ′ ), (F1 ), (F3 ) and (F4 ) are satisfied. Then the functional J satisfies the P.S. condition.    Proof. Let u (l) ∈ Rk , l ∈ Z(1) be such that J u (l) is bounded. Then there exists a positive constant M2 such that   −M2 ≤ J u (l) ≤ M2 , ∀l ∈ N.

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By (F4′ ), we have

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k−2 k−1   1  2  δ+1 1  (l) −M2 ≤ J u (l) = pn+1 ∆2 u (l) + q 1u n+1 n n 2 n=1 δ + 1 n=1 k k  δ+1    1   1  1  (l) 2 (l) 2 (l) + rn u (l) − F n, u + p 1u + pk 1u k−1 1 n n 1 δ + 1 n=1 2 2 n=1   1 δ+1 δ+1   k−2  k−1  2 qc 2 2  ¯ p¯  (l) (l) (l) ≤ + 2 u − 2u n+1 + u (l) u − u (l) n n  2 n=1 n+2 δ + 1  n=1 n+1 k     2  r¯ c2δ+1   (l) δ+1  (l) β   − a1 u  u n  + a2 k + p¯ u (l)  δ+1 n=1 δ+1  ∗  δ+1 r¯ cδ+1  δ+1 qc ¯ p¯  (l) ∗   (l) (l) (l) 2 2 ≤ + 2 u (l)  u Pu + u Qu 2 δ+1 δ+1  β  2    β − a1 c1 u (l)  + a2 k + p¯ u (l)  δ+1     δ+1    ¯ 2δ+1 λ˜ k 2  λk  (l) 2 qc  (l) δ+1 r¯ c2  (l) δ+1 ≤ + 1 p¯ u  + + u  u  2 δ+1 δ+1   β  (l) β − a1 c1 u  + a2 k,   (l) (l) (l) ∗ = u1 , u2 , . . . , uk , u (l) ∈ Rk . That is,

+

where u (l)

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δ+1

   δ+1   β  λ   ¯ δ+1 λ˜ k 2  k  (l)   2 qc  (l) δ+1 r¯ c2  (l) δ+1 + 1 p¯ u (l)  − 2 − u  − u  u  2 δ+1 δ+1 ≤ M2 + a2 k.

β a1 c1

Since β > max{2, δ + 1}, there exists a constant M3 > 0 such that    (l)  u  ≤ M3 , ∀l ∈ N.     Therefore, u (l) is bounded on Rk . As a consequence, u (l) possesses a convergence subsequence in Rk . Thus the P.S. condition is verified.  3. Proof of the main results

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In this section, we shall prove our main results by using the critical point theory.

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3.1. Proof of Theorem 1.1

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Proof. By (F2′ ), for any u = (u 1 , u 2 , . . . , u k )∗ ∈ Rk , we have

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J (u) =

k−2 k−1 k  2 1 1  1  pn+1 ∆2 u n + qn+1 (1u n )δ+1 + rn u δ+1 n 2 n=1 δ + 1 n=1 δ + 1 n=1

−

k  n=1

F(n, u n ) +

1 1 p1 (1u 1 )2 + pk (1u k−1 )2 2 2

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X. Liu et al. / Indagationes Mathematicae xx (xxxx) xxx–xxx k−2 k  p¯  |u n | + M1 k (u n+2 − 2u n+1 + u n )2 + M0 2 n=1 n=1 √ p¯ ≤ u ∗ Pu + M0 k∥u∥ + M1 k 2 √ pλ ¯ 1 ≤ ∥u∥2 + M0 k∥u∥ + M1 k 2 → −∞ as ∥u∥ → +∞.

≤

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The above inequality means that −J (u) is coercive. By the continuity of J (u), J attains its maximum at some point, and we denote it u, ˇ that is,   J (u) ˇ = max J (u)|u ∈ Rk .

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Clearly, uˇ is a critical point of the functional J . This completes the proof of Theorem 1.1.

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3.2. Proof of Theorem 1.2

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Proof. By (F3 ), for any ϵ = |F(n, v)| ≤

pλ1 4

|v|,

pλ1 4

∀n ∈ Z(1, k),

J (u) =

k−2 k−1 k  2 1 1  1  pn+1 ∆2 u n + qn+1 (1u n )δ+1 + rn u δ+1 n 2 n=1 δ + 1 n=1 δ + 1 n=1

−

k 

F(n, u n ) +

n=1

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(see (2.6)), there exists ρ > 0, such that

for |v| ≤ ρ. For any u = (u 1 , u 2 , . . . , u k )∗ ∈ Rk and ∥u∥ ≤ ρ, we have |u n | ≤ ρ, n ∈ Z(1, k). For any n ∈ Z(1, k),

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≥

k−2 p

2 p

1 1 p1 (1u 1 )2 + pk (1u k−1 )2 2 2

(u n+2 − 2u n+1 + u n )2 −

n=1

k pλ1 

4

|u n |

n=1

pλ1

∥u∥2 4 pλ1 ≥ ∥u∥2 − ∥u∥2 2 4 pλ1 = ∥u∥2 , 4 where u = (u 1 , u 2 , . . . , u k )∗ , u ∈ Rk . pλ1 Take a = 4 ρ 2 > 0. Therefore, ≥

u ∗ Pu −

2 pλ1

J (u) ≥ a > 0,

∀u ∈ ∂ Bρ .

At the same time, we have also proved that there exist constants a > 0 and ρ > 0 such that J |∂ Bρ ≥ a. That is to say, J satisfies the condition (J1 ) of the Mountain Pass Lemma. For our setting, clearly J (0) = 0. In order to exploit the Mountain Pass Lemma in critical point theory, we need to verify all other conditions of the Mountain Pass Lemma. By Lemma 2.2, J satisfies the P.S. condition. So it suffices to verify the condition (J2 ).

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From the proof of the P.S. condition in Lemma 2.2, we know 

J (u) ≤

δ+1 2

 qc ¯ δ+1 λ˜ k λk + 1 p¯ ∥u∥2 + 2 2 δ+1

∥u∥δ+1 +

r¯ c2δ+1 β ∥u∥δ+1 − a1 c1 ∥u∥β + a2 k. δ+1

Since β > max{2, δ + 1}, we can choose u¯ large enough to ensure that J (u) ¯ < 0. By the Mountain Pass Lemma, J possesses a critical value c ≥ a > 0, where c = inf sup J (h(s)),

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h∈Γ s∈[0,1]

and

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Γ = {h ∈ C([0, 1], R ) | h(0) = 0, h(1) = u}. ¯ k

Let u˜ ∈ be a critical point associated to the critical value c of J , i.e., J (u) ˜ = c. Similar to the proof of the P.S. condition, we know that there exists uˆ ∈ Rk such that Rk

J (u) ˆ = cmax = max J (h(s)).

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s∈[0,1]

Clearly, uˆ ̸= 0. If u˜ ̸= u, ˆ then the conclusion of Theorem 1.2 holds. Otherwise, u˜ = u. ˆ Then c = J (u) ˜ = cmax = maxs∈[0,1] J (h(s)). That is, sup J (u) = inf sup J (h(s)).

11 12

13

h∈Γ s∈[0,1]

u∈Rk

Therefore,

14

cmax = max J (h(s)), s∈[0,1]

∀h ∈ Γ .

By the continuity of J (h(s)) with respect to s, J (0) = 0 and J (u) ¯ < 0 imply that there exists s0 ∈ (0, 1) such that J (h(s0 )) = cmax .

J (h 1 (s1 )) = J (h 2 (s2 )) = cmax . Thus, we get two different critical points of J on Rk denoted by u = h 1 (s1 ),

u = h 2 (s2 ). 2

The above argument implies that the BVP (1.1) with (1.2) possesses at least two nontrivial solutions. The proof of Theorem 1.2 is finished.  3.3. Proof of Theorem 1.3

k−2 k−1 k  2 1  1  1 pn+1 ∆2 u n + qn+1 (1u n )δ+1 + rn u δ+1 n 2 n=1 δ + 1 n=1 δ + 1 n=1

−

k  n=1

17

F(n, u n ) +

19 20

21

22

23

24 25

26

Proof. We only need to find at least one critical point of the functional J defined as in (2.5). By (F5′ ), for any u = (u 1 , u 2 , . . . , u k )∗ ∈ Rk , we have J (u) =

16

18

Choose h 1 , h 2 ∈ Γ such that {h 1 (s) | s ∈ (0, 1)} ∩ {h 2 (s) | s ∈ (0, 1)} is empty, then there exists s1 , s2 ∈ (0, 1) such that

1

15

1 1 p1 (1u 1 )2 + pk (1u k−1 )2 2 2

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≥

1

k−2 p

2

(u n+2 − 2u n+1 + u n )2 − a3

n=1

k 

|u n |α − a4 k

n=1

  α1 α k  p ∗ |u n |α  − a4 k = u Pu − a3  2 n=1   12 α k  pλ1 ∥u∥2 − a3 c2α  ≥ u 2n  − a4 k 2 n=1

2

3

pλ1

∥u∥2 − a3 c2α ∥u∥α − a4 k 2 → +∞ as ∥u∥ → +∞. ≥

4

5

8

By the continuity of J , we know from the above inequality that there exist lower bounds of values of the functional. And this means that J attains its minimal value at some point which is just the critical point of J with the finite norm. 

9

3.4. Proof of Theorem 1.4

6 7

10 11

12

13

14

Proof. Assume, for the sake of contradiction, that the BVP (1.1) with (1.2) has a nontrivial solution. Then J has a nonzero critical point u ⋆ . Since     ∂J = ∆2 pn−1 ∆2 u n−2 − ∆ qn (1u n−1 )δ + rn u δn − f (n, u n ), ∂u n we get k 

f (n, u ⋆n )u ⋆n =

n=1

k       δ   δ  ⋆ ∆2 pn−1 ∆2 u ⋆n−2 − ∆ qn 1u ⋆n−1 + rn u ⋆n un n=1

=

15

k−2 

k−1 k  2   δ+1   δ+1 + pn+1 ∆2 u ⋆n + qn+1 1u ⋆n rn u ⋆n

n=1

18

 ⋆ 2

+ p1 1u 1

16

17

n=1



+ pk 1u ⋆k−1 

n=1

2

≤ 0.

(3.1)

On the other hand, it follows from (F6 ) that k 

f (n, u ⋆n )u ⋆n > 0.

(3.2)

n=1 19

This contradicts (3.1) and hence the proof is complete.

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4. Examples

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As an application of Theorems 1.2–1.4, we give three examples to illustrate our main results. Example 4.1. For n ∈ Z(1, k), assume that   ∆4 u n−2 − ∆ 8n (1u n−1 )δ + 6n u δn = βϕ(n)u n |u n |β−2 ,

(4.1)

X. Liu et al. / Indagationes Mathematicae xx (xxxx) xxx–xxx

11

with boundary value conditions (1.2), δ is the ratio of odd positive integers, β > max{2, δ +1}, ϕ is continuously differentiable and ϕ(n) > 0, n ∈ Z(1, k) with ϕ(0) = 0. We have qn = 8n ,

pn ≡ 1,

rn = 6n ,

f (n, v) = βϕ(n)v|v|β−2

1 2

3

and

4

F(n, v) = ϕ(n)|v|β .

5

It is easy to verify all the assumptions of Theorem 1.2 are satisfied and then the BVP (4.1) with (1.2) possesses at least two nontrivial solutions. Example 4.2. For n ∈ Z(1, k), assume that     ∆2 7n−1 ∆2 u n−2 − ∆ 2n (1u n−1 )δ + 5n u δn = αψ(n)u n |u n |α−2 ,

qn = 2n ,

r n = 5n ,

7

8

(4.2)

with boundary value conditions (1.2), δ is the ratio of odd positive integers, 1 < α < 2, ψ is continuously differentiable and ψ(n) > 0, n ∈ Z(1, k) with ψ(0) = 0. We have pn = 7n ,

6

f (n, v) = αψ(n)v|v|α−2

9

10 11

12

and

13

F(n, v) = ψ(n)|v|α .

14

It is easy to verify all the assumptions of Theorem 1.3 are satisfied and then the BVP (4.2) with (1.2) possesses at least one solution. Example 4.3. For n ∈ Z(1, k), assume that

qn = −3n ,

rn = −9n ,

16

17

  6 1 − ∆4 u n−2 + ∆ 3n (1u n−1 )δ − 9n u δn = u n5 , 5 with boundary value conditions (1.2), δ is the ratio of odd positive integers. We have pn ≡ −1,

15

f (n, v) =

(4.3)

6 1 v5 5

and

18

19

20

21

6 5

F(n, v) = v . It is easy to verify all the assumptions of Theorem 1.4 are satisfied and then the BVP (4.3) with (1.2) has no nontrivial solutions. References [1] A.R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl. 116 (2) (1986) 415–426. [2] R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York, 2000. [3] C.D. Ahlbrandt, Dominant and recessive solutions of symmetric three term recurrences, J. Differential Equations 107 (2) (1994) 238–258. [4] V. Anuradha, C. Maya, R. Shivaji, Positive solutions for a class of nonlinear boundary value problems with Neumann–Robin boundary conditions, J. Math. Anal. Appl. 236 (1) (1999) 94–124. [5] D. Arcoya, Positive solutions for semilinear Dirichlet problems in an annulus, J. Differential Equations 94 (2) (1991) 217–227.

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