Abstract
In this paper, we study the existence and multiplicity of nontrivial solutions of a second-order discrete boundary value problem with resonance and sublinear or superlinear nonlinearity. The main methods are based on the Morse theory and the minimax methods. In addition, some examples are given to illustrate our results.
Highlights
Let Z and R be the sets of integers and real numbers, respectively
We note that these results were usually obtained by means of critical point theory, for example, the existence of ground state solutions [1], homoclinic orbits [2,3,4,5], the boundary value problem, and periodic solutions [6,7,8]
We study the existence and multiplicity of nontrivial solutions for problem (1) at resonance by means of the interaction between the nonlinearity and the spectrum of the symmetric matrix P + Q, where P + Q denotes a matrix whose elements are given by p(k) and q(k) for k ∈ Z(1, T)
Summary
Let Z and R be the sets of integers and real numbers, respectively. For a, b ∈ Z, Z(a, b) denotes the discrete interval {a, a + 1, . . . , b} if a ≤ b. We consider the existence and multiplicity of nontrivial solutions for the following discrete Dirichlet boundary value problem: Δ[p(k)Δu(k − 1)] + q(k)u(k) + f(k, u(k)) 0, k ∈ Z(1, T), . The existence of solutions for nonlinear difference equations has been widely studied by many authors We note that these results were usually obtained by means of critical point theory, for example, the existence of ground state solutions [1], homoclinic orbits [2,3,4,5], the boundary value problem, and periodic solutions [6,7,8]. Yu and Guo in [12] first used the critical point theory to study the following discrete boundary value problem:.
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