Existence and Multiplicity of Solutions for Discrete Nonlinear Two-Point Boundary Value Problems - Jianmin Guo; Caixia Guo (Discrete Dynamics in Nature and Society)
By using Morse theory, the critical point theory, and the character of 𝐾1/2, we consider the existence and multiplicity results of solutions to the following discrete nonlinear two-point boundary value problem −Δ2𝑥(𝑘−1)=𝑓(𝑘,𝑥(𝑘)),𝑘∈ℤ(1,𝑇) subject to 𝑥(0)=0=Δ𝑥(𝑇), where 𝑇 is a positive integer, ℤ(1,𝑇)={1,2,…,𝑇},Δ is the forward difference operator defined by Δ𝑥(𝑘)=𝑥(𝑘+1)−𝑥(𝑘), and 𝑓∶ℤ(1,𝑇)×ℝ→ℝ is continuous. In argument, Morse inequalities play an important role.