Maximum and anti-maximum principles for the general operator of second order with variable coefficients. (Q1855925)

The authors study positivity or negativity properties of the general linear second-order operator \[ L(p,q)u(t)=u''(t)+p(t)u'(t)+q(t)u(t), \quad t\in J=[0,R], \] where \(p, q\in L_1(J)\) are given and Neumann boundary conditions are considered. As a consequence of the results obtained for the Neumann problem, the authors deduce analogous results for the operator \(L(p,q)\) with other type of boundary conditions, for example the mixed, Dirichlet or periodic ones. Using the lower and upper solutions method together with the above results, the authors get the existence and approximation of extremal solutions to the nonlinear equation \[ u''(t)+p(t)u'(t)+f(t,u(t))=0 \] with different boundary conditions.