Boundedness of solutions for a class of second-order differential equations Original Research Article

In this paper we study the boundedness of solutions for the second-order differential equation (Fp(x′)+ψ(x′))′+αFp(x+)−βFp(x−)=φ(t,x),(Fp(x′)+ψ(x′))′+αFp(x+)−βFp(x−)=φ(t,x), Turn MathJax on where Fp(s)=|s|p−2sFp(s)=|s|p−2s, p>1p>1 and αα, ββ are strictly positive constants satisfying a resonant relation View the MathML sourceπα1/psinπ/p+πβ1/psinπ/p=2πn with nn being a positive integer, and φ(t,x)φ(t,x) is a 2π2π-periodic function in tt. There exists a function W(θ)W(θ) such that if it is of constant sign, under some increasing restrictions on ψ(x′)ψ(x′) and φ(t,x)φ(t,x), by a small twist theorem we prove that all the solutions of the above equation are bounded in the sense of Lagrangian stability.