Maximum and anti-maximum principles for the general operator of second order with variable coefficients. (English) Zbl 1037.34014

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.


34B15 Nonlinear boundary value problems for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
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