A monotone method for constructing extremal solutions to second order periodic boundary value problems (Q2724134)

The authors study the existence of solutions to the second-order periodic boundary value problem NEWLINE\[NEWLINEu''(t)=f(t,u(t),u'(t)), \;t \in [0, 2 \pi], \quad u(0)=u(2\pi), \;u'(0)=u'(2\pi),NEWLINE\]NEWLINE where \(f\) is a Carathédory function. NEWLINENEWLINENEWLINEAssuming the existence of a pair of lower and upper solutions \(\alpha\) and \(\beta\), given in the reverse order, i.e. \(\alpha \geq \beta\), the authors prove the existence of extremal solutions lying between \(\alpha \) and \(\beta\). The required hypotheses on \(f\) are: NEWLINENEWLINENEWLINE(H\(_1\)) For given \(\alpha, \beta \in C[0,2\pi]\) with \(\beta(t) \leq \alpha(t)\) on \([0,2\pi]\) there exists \(0 < L < |M |\) such that NEWLINE\[NEWLINE(M-L)(v_2-v_1) \leq f(t,u,v_1) - f(t,u,v_2) \leq (M+L)(v_2-v_1)NEWLINE\]NEWLINE for a.e. \(t \in [0,2 \pi]\) whenever \(\beta(t) \leq u \leq \alpha(t)\), \(v_2 \geq v_1\). NEWLINENEWLINENEWLINE(H\(_2\)) The inequality \(f(t,u_2,v) - f(t,u_1,v) \geq 0\) holds for a.e. \(t \in [0, 2 \pi]\) whenever \(\beta(t) \leq u_1 \leq u_2 \leq \alpha(t)\), \(v \in {\mathbb{R}}\). NEWLINENEWLINENEWLINEThe main tool used in the proof is the following maximum-minimum principle: If \(0 \leq L < |M |\), \(y'(t) + M y(t)+L |y(t) |\geq 0\) for a.e. \(t \in [0,2 \pi],\) and \( y(0) \geq y(2 \pi) \), then \(M y(t) \geq 0\).