On periodic boundary value problem for the equation \(u^{\prime\prime} = f(t,u,u^{\prime})\) with one-sided growth restrictions on \(f\)
The boundary value problem \[ u''(t)= f(t,u(t), u'(t)),\quad u(a)= u(b),\;u'(a)= u'(b),\tag{P} \] is considered, where \(f: [a,b]\times \mathbb{R}^2\to \mathbb{R}\) satisfies the local Carathéodory conditions. The authors give new sufficient conditions for the solvability of (P) formulated by upper and lower functions and one-sided growth restrictions on the function \(f\) with respect to the third argument. These conditions guarantee the existence of upper and lower solutions to (P) as well. The proofs of existence results are based on new a priori estimates on solutions to second-order differential inequalities with periodic boundary conditions.
scientific article; zbMATH DE number 1741846
scientific article