A note on the multiplicity of solutions for a second-order difference equation with a parameter (Q426451)

The author is motivated by the paper of \textit{L. Gao} [Appl. Math. Comput. 216, No. 5, 1592--1598 (2010; Zbl 1205.39002)], where the solvability of the discrete boundary value problem (BVP) NEWLINE\[NEWLINE \Delta (p_{k-1} \Delta x_{k-1})+q_kx_k+\lambda f(k,x_k)=0 NEWLINE\]NEWLINE with the periodic boundary condition \(x_0=x_N\), \(p_0\Delta x_0= p_N\Delta x_N\) is investigated via Clark's variational principle. He observes that certain arguments used in the proofs of some results of that paper are not correct and suggests their improvement. The main result of the paper are two existence criteria presenting conditions under which the investigated BVP possesses at least \(N\) distinct solutions.