Uniqueness and Multiplicity of Solutions for a Second-Order Discrete Boundary Value Problem with a Parameter

Abstract

This paper is concerned with the existence of unique and multiple solutions to the boundary value problem of a second-order difference equation with a parameter, which is a complement of the work by J. S. Yu and Z. M. Guo in 2006.

1. Introduction and Preliminaries

In this paper, we consider the existence, uniqueness, and multiplicity of solutions for a second-order discrete boundary value problem𝑝(𝑛+1)𝑢(𝑛+1)+𝑐(𝑛)𝑢(𝑛)+𝑝(𝑛)𝑢(𝑛−1)=𝜆𝑓(𝑛,𝑢(𝑛)),𝑛∈𝑍(1,𝑘),𝑢(0)+𝛼𝑢(1)=𝐴,𝑢(𝑘+1)+𝛽𝑢(𝑘)=𝐵, (1.1) where 𝜆∈ℝ is a parameter. Our technique is based on critical point theory, which is successfully used to deal with the existence of solutions for discrete problems (see [1–9]), especially in [7, 9]. Similarly to [7], we denote by ℕ,ℤ, and ℝ the sets of all natural numbers, integers, and real numbers, respectively. For 𝑎,𝑏∈ℤ, 𝑍(𝑎)={𝑎,𝑎+1,…}, 𝑍(𝑎,𝑏)={𝑎,𝑎+1,…,𝑏} when 𝑎≤𝑏. We assume that 𝑝(𝑛) is nonzero and real-valued for each 𝑛∈𝑍(1,𝑘), 𝑐(𝑛) is real-valued for each 𝑛∈𝑍(1,𝑘), and 𝑓(𝑛,𝑢) is real-valued for each (𝑛,𝑢)∈𝑍(1,𝑘)×ℝ and continuous in 𝑢. Let ℝ^𝑘 be the real Euclidean space with dimension 𝑘. For any 𝑢,𝑣∈ℝ^𝑘, ‖𝑢‖ and (𝑢,𝑣), denote the usual norm and inner product in ℝ^𝑘, respectively.

Consider the functional defined on ℝ^𝑘,1𝐽(𝑢)=2(𝑀𝑢,𝑢)+(𝜂,𝑢)−𝜆𝐹(𝑢),𝑢=(𝑢(1),𝑢(2),…,𝑢(𝑘))^𝑇∈ℝ^𝑘, (1.2)where (⋅)^T is the transpose of a vector in ℝ^𝑘,,0⋮0,𝑀=𝑐(1)−𝛼𝑝(1)𝑝(2)0⋯00𝑝(2)𝑐(2)𝑝(3)⋯000𝑝(3)𝑐(3)⋯00⋯⋯⋯000⋯𝑐(𝑘−1)𝑝(𝑘)000⋯𝑝(𝑘)𝑐(𝑘)−𝛽𝑝(𝑘+1)𝜂=𝑝(1)𝐴𝑝(𝑘+1)𝐵𝐹(𝑢)=∑^𝑘_𝑗=1∫₀^𝑢(𝑗)𝑓(𝑗,𝑠)𝑑𝑠. (1.3)

It is easy to see that 𝐽(𝑢) is Fréchet differentiable with Fréchet derivative𝐽^′(𝑢)=𝑀𝑢+𝜂−𝜆𝑓(𝑢), (1.4)where 𝑓(𝑢)=(𝑓(1,𝑢(1)),𝑓(2,𝑢(2)),…,𝑓(𝑘,𝑢(𝑘)))^𝑇, and there is a one-to-one correspondence between the critical point of functional 𝐽 and the solution of BVP (1.1). Furthermore, 𝑢=(𝑢(1),𝑢(2),…,𝑢(𝑘))^𝑇 is a critical point of 𝐽 if and only if {𝑢(𝑡)}^𝑘+1_𝑡=0=(𝑢(0),𝑢(1),…,𝑢(𝑘+1))^𝑇 is a solution of (1.1), where 𝑢(0)=𝐴−𝛼𝑢(1),𝑢(𝑘+1)=𝐵−𝛽𝑢(𝑘) [7].

Recently, Yu and Guo [7] studied the BVP,𝑝(𝑛+1)𝑢(𝑛+1)+𝑐(𝑛)𝑢(𝑛)+𝑝(𝑛)𝑢(𝑛−1)=𝑓(𝑛,𝑢(𝑛)),𝑢(0)+𝛼𝑢(1)=𝐴,𝑢(𝑘+1)+𝛽𝑢(𝑘)=𝐵. (1.5)They obtained some existence results for (1.5). One of the main results is as follows.

Theorem (𝐴₀). Suppose that 𝑓(𝑛,𝑧) satisfies the following assumption:
(f3) there exist constants 𝑎₁>0, 𝑎₂>0, 𝑅>0, and 𝛽>2 such that^𝑘_𝑗=10^𝑢(𝑗)𝑓(𝑗,𝑠)𝑑𝑠≥𝑎₁‖𝑢‖^𝛽−𝑎₂for𝑢=(𝑢(1),𝑢(2),…,𝑢(𝑘))^𝑇∈ℝ^𝑘,‖𝑢‖≥𝑅. (1.6)Then BVP (1.5) has at least one solution.

Equation (1.6) shows that 𝐹(𝑢)=∑^𝑘_𝑗=1∫₀^𝑢(𝑗)𝑓(𝑗,𝑠)𝑑𝑠>0 for ‖𝑢‖ large enough. Since 𝐹(𝑢) may be negative, then the conclusion of Theorem 1.1 cannot be drawn, which motivates us to consider (1.1). Note that if we take 𝜆=−1 in (1.1), then 𝐹(𝑢)<0. Under the similar condition to (1.6) when 𝜆<0, we not only obtain the existence of solutions for (1.1), but also the multiplicity.

Let 𝐸 be a Banach space with a direct sum decomposition 𝐸=𝑋⊕𝑌. The functional 𝐼∈𝐶¹(𝐸,ℝ) has a local linking at 0 if for some 𝜌>0,𝐼(𝑢)≤0,𝑢∈𝑋, ‖𝑢‖≤𝜌, and 𝐼(𝑢)≥0,𝑢∈𝑌, ‖𝑢‖≤𝜌. The functional 𝐼∈𝐶¹(𝐸,ℝ) is said to satisfy the (PS) condition if any sequence 𝑥_𝑛⊂𝑋 for which 𝐼(𝑥_𝑛) is bounded, and 𝐼^′(𝑥_𝑛)→0(𝑛→∞) possesses a convergent subsequence in 𝐸.

Theorem 1 A (see[10]). Let 𝐸 be a Banach space. Suppose that 𝐼∈𝐶¹(𝐸,ℝ) satisfies the (PS) condition and has a local linking at 0. Assume that 𝐼 is bounded below and inf_𝐸𝐼<0. Then 𝐼 has at least two nontrivial critical points.

Theorem 1 B (see[6, 11]). Let 𝑋 be a real Banach space, 𝐼∈𝐶¹(𝑋,ℝ) with 𝐼 even, bounded from below, and satisfying (PS) condition. Suppose 𝐼(0)=0. There is a set 𝐾⊂𝑋 such that 𝐾 is homeomorphic to a unit sphere 𝑆^𝑛−1 in ℝ^𝑛 (𝑛∈ℕ) by an odd map, and sup_𝐾𝐼<0. Then 𝐼 possesses at least 𝑛 distinct pairs of critical points.

2. Main Results

Following conditions will be useful to prove our main results.

(H1)There exist numbers 𝛼₁>2 and 𝑎₁>0 such that𝐹(𝑢)≥𝑎₁‖𝑢‖^𝛼1for𝑢=(𝑢(1),𝑢(2),…,𝑢(𝑘))^𝑇∈ℝ^𝑘. (2.1)(H2)lim_𝑢→0(𝐹(𝑢)/‖𝑢‖²)=0for𝑢=(𝑢(1),𝑢(2),…,𝑢(𝑘))^𝑇∈ℝ^𝑘.

Theorem 2.1. Suppose that 𝑀 is positive definite, 𝑓(𝑗,0)=0, 𝑓(𝑗,𝑢)𝑢<0 for 𝑗∈𝑍(1,𝑘) and 𝑢≠0, and that 𝑝(1)𝐴=𝑝(𝑘+1)𝐵=0. Then (1.1) has only trivial solution for 𝜆>0.

Proof. Note that(𝐽^′(𝑢),𝑢)=(𝑀𝑢,𝑢)+(𝜂,𝑢)−𝜆(𝑓(𝑢),𝑢)≥𝜆₁‖𝑢‖²−𝜆^𝑘_𝑗=1𝑓(𝑗,𝑢(𝑗))𝑢(𝑗)≥𝜆₁‖𝑢‖²>0 (2.2)for 𝑢≠0, where 𝜆₁ is the least eigenvalue of 𝑀, which means that the Nahari manifold is empty. Thus (1.1) has only trivial solution.

Theorem 2.2. Suppose that (H1) and (H2) hold, 𝑝(1)𝐴=𝑝(𝑘+1)𝐵=0 and 𝑀 is neither positive definite nor negative definite. Then (1.1) has at least two nontrivial solutions for 𝜆<0.

Proof. We will prove that the functional 𝐽(𝑢) satisfies all conditions of Theorem A by two steps.
Step 1. 𝐽 is bounded from below and satisfies (PS) condition. Let 𝜆_−𝑙,𝜆_−𝑙+1,…,𝜆₋₁,𝜆₁,𝜆₂,…,𝜆_𝑚 denote all the eigenvalues of 𝑀, where 𝜆_−𝑙≤𝜆_−𝑙+1≤⋯≤𝜆₋₁<0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑚 and 𝑙+𝑚=𝑘. For any 𝑗∈𝑍(−𝑙,−1)⋃𝑍(1,𝑚), set 𝜉_𝑗 to be an eigenvector of 𝑀 corresponding to the eigenvalue 𝜆_𝑗, 𝑗=−𝑙,−𝑙+1,…,−1,1,2,…,𝑚, such that(𝜉_𝑖,𝜉_𝑗)=0for𝑖≠𝑗,1for𝑖=𝑗. (2.3)
Let 𝑋 and 𝑌 be subspaces of ℝ^𝑘 defined by𝑋={𝑥∈ℝ^𝑘∶𝑥=∑^𝑚_𝑗=1𝑥_𝑗𝜉_𝑗,𝑥_𝑗∈ℝ,𝑗∈𝑍(1,𝑚)},𝑌={𝑦∈ℝ^𝑘∶𝑦=∑⁻¹_𝑗=−𝑙𝑦_𝑗𝜉_𝑗,𝑦_𝑗∈ℝ,𝑗∈𝑍(−𝑙,−1)}, (2.4)respectively. Then ℝ^𝑘 has the direct sum decomposition ℝ^𝑘=𝑋⊕𝑌. In view of (H1), we have1𝐽(𝑢)≥2𝜆_−𝑙‖𝑢‖²−𝜆𝑎₁‖𝑢‖^𝛼11≥−2|𝜆_−𝑙|(1−2/𝛼₁)(|𝜆_−𝑙|/(−𝛼₁𝜆𝑎₁))^{2/(𝛼1−2)}. (2.5)The second inequality follows from the elementary inequality −𝑎𝑥²+𝑏𝑥^𝑞≥−𝑎(1−2/𝑞)((2𝑎)/(𝑞𝑏))^2/(𝑞−2), where 𝑎>0,𝑏>0,𝑥>0, and 𝑞>2, which can be easily obtained by the fact that the function ℎ(𝑥)=−𝑎𝑥²+𝑏𝑥^𝑞(𝑎,𝑏>0,𝑥≥0) attains its minimum at ((2𝑎)/(𝑞𝑏))^1/(𝑞−2). Thus 𝐽(𝑢) is bounded from below.
Equation (2.5) shows that 𝐽(𝑢) is coercive, so we can obtain that any (PS) sequence must be bounded in ℝ^𝑘 and, by a standard argument, has a convergent subsequence.
Step 2. 𝐽 has a local linking at 0. Indeed, by (H2) for given 𝜆<0 and sufficiently small 𝜀>0 such that −𝜆𝜀+(1/2)𝜆₋₁<0, there exists 𝑟>0 small enough such that for ‖𝑢‖<𝑟,𝐹(𝑢)<𝜀‖𝑢‖² (2.6)holds. Then for 𝑦∈𝑌 such that 0<‖𝑦‖<𝑟, we have1𝐽(𝑦)≤2𝜆₋₁‖𝑦‖²−𝜆𝜀‖𝑦‖²1=(2𝜆₋₁−𝜆𝜀)‖𝑦‖²<0. (2.7)
On the other hand, for 𝑥∈𝑋 with ‖𝑥‖<𝑟, we have1𝐽(𝑥)≥2𝜆₁‖𝑥‖²−𝜆𝑎₁‖𝑥‖^𝛼1≥0. (2.8)The application of Theorem A finishes our proof.

Remark 2.3. By the above proof, we see that replacing (H1) with (f3), and by adding the condition that 𝐹(𝑢)≥0 for 𝑢∈ℝ^𝑘, Theorem 2.2 still holds, where (f3) is the same as in Theorem 1.1. Indeed, we have1𝐽(𝑢)≥2𝜆_−𝑙𝑅²−1for‖𝑢‖≤𝑅,2|𝜆_−𝑙|(1−2/𝛽)(|𝜆_−𝑙|/(−𝛽𝜆𝑎₁))^2/(𝛽−2)+𝜆𝑎₂for‖𝑢‖>𝑅, (2.9)which means that 𝐽(𝑢) is bounded from below. The fact that 𝐽(𝑢) has local linking at 0 may be verified similarly.

If we further impose some condition on 𝑓(𝑢) and matrix 𝑀, then the following result can be derived.

Theorem 2.4. Suppose (H1) and (H2) hold, 𝑝(1)𝐴=𝑝(𝑘+1)𝐵=0,𝑓(𝑢) is odd in 𝑢, that is, 𝑓(𝑗,−𝑢)=−𝑓(𝑗,𝑢) for (𝑗,𝑢)∈𝑍(1,𝑘)×ℝ, and that 𝑀 is neither positive definite nor negative definite and has 𝑙 distinct negative eigenvalues. Then (1.1) has at least 𝑙 distinct pairs of solutions for 𝜆<0.

Proof. By the proof of Theorem 2.2, 𝐽(𝑢) is bounded from below and satisfies (PS) condition. In addition, 𝐽(0)=0, 𝐽(𝑢) is even. Consider the subset 𝐾 of ℝ^𝑘:𝐾={𝑦∈𝑌∶𝑦=⁻¹_𝑗=−𝑙𝑦_𝑗𝜉_𝑗,⁻¹_𝑗=−𝑙𝑦²_𝑗=𝜌²,𝑦_𝑗∈ℝ,𝑗∈𝑍(−𝑙,−1)}, (2.10)where 𝜌 is a positive number small enough to be determined later, 𝑌 is defined by Theorem 2.2 similarly. Define the mapping 𝑇∶𝐾→𝑆^𝑙−1 by𝑇(𝑦)=𝑇(𝑦=⁻¹_𝑗=−𝑙𝑦_𝑗𝜉_𝑗𝑦)=(−^−𝑙𝜌𝑦,−^−𝑙+1𝜌𝑦,…,−⁻¹𝜌), (2.11)where 𝑆^𝑙−1 is a unit sphere in ℝ^𝑙. Then 𝑇 is a homeomorphism between 𝐾 and 𝑆^𝑙−1, and 𝐾 is a subset of the finite dimensional space 𝑌 equipped with the Euclidian norm. We can choose 𝜌>0 and 𝜀>0 small enough such that (1/2)𝜆₋₁−𝜆𝜀<0 for 𝑦∈𝐾, and then we have1𝐽(𝑦)≤2𝜆₋₁‖𝑦‖²1−𝜆𝐹(𝑢)≤2𝜆₋₁𝜌²−𝜆𝜀𝜌²1≤(2𝜆₋₁−𝜆𝜀)𝜌². (2.12)For above 𝜌>0, we havesup_𝐾𝐽<0, (2.13)which together with Theorem B concludes the proof.

Remark 2.5. The condition that 𝐽(𝑢) is bounded from below is crucial to prove both Theorems 2.2 and 2.4. As in Remark 2.3, if we replace (H1) by (f3) and the condition that 𝐹(𝑢)≥0 for 𝑢∈ℝ^𝑘, then Theorem 2.4 is also true.

Acknowledgments

The authors would like to thank Professor B. G. Zhang and referee's valuable suggestions and comments. This research was partially supported by the NNSF (10571115) of China, China Postdoctoral Science Foundation (20070421106), the Key Project of Chinese Ministry of Education, and the Foundation of Chinese Nationalities Committee (07QH04).