Existence and Multiplicity of Solutions for Discrete Nonlinear Two-Point Boundary Value Problems

Abstract EN

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.