On the boundedness and oscillation of solutions to \((m(t)x^ 1)^ 1+a(t)b(x)=0\)
Conditions are given for the solutions of \((m(t)x')'+a(t)b(x)=0\) to be bounded, oscillatory and stable. Typically, the boundedness result is as follows. Suppose that a(t) and m(t) are \(C^ 1[0,\infty)\), and that \(a(t)\geq a_ 0\geq 0\) and \(m(t)\geq m_ 0\geq 0\) for some positive constants \(a_ 0\) and \(m_ 0\). Suppose also that m'(t)\(\leq 0\), a'(t)\(\leq 0\) and that \(b(x)\in C(-\infty,+\infty)\). If \(\lim_{| x| \to \infty}\int^{x}_{0}b(u)du=+\infty,\) then \(| x|\) and \(| x'|\) are bounded as \(t\to \infty\).
scientific article; zbMATH DE number 4000422