Implicit nonlinear discontinuous functional boundary value \(\phi\)-Laplacian problems: Extremality results.
The authors consider nonlinear functional equations of the form \[ L_0 u(t)= f(t,u,L_0 u)\tag{\(*\)} \] on the interval \([a,b]\) with nonlinear boundary conditions of the form \[ L_1(u)= B_1(u,L_1(u)),\quad L_2(u(a), u(b))= 0, \] where \(L_0\) and \(L_1\) are defined by \[ L_0 u(t)=- {d\over dt} \varphi(t, u(t), u'(t))- g(t, u,u(t), u'(t)) \] and \[ L_1(u)= L_1(u(a), u(b),u'(a), u'(b),u), \] while \(f\), \(B_1\), \(L_2\), \(\varphi\) and \(g\) are given functions. \(u\) appearing in expressions of \(f\), \(B_1\), \(g\) and \(L_1\) refers to functionals on \(u(t)\). The goal is to show that under suitable conditions on \(f\), \(B_1\), \(L_2\), \(\varphi\) and \(g\), there are extremal solutions to \((*)\) in a sector formed by a pair of lower and upper solutions \(\alpha(t)\) and \(\beta(t)\), namely: \(\alpha(t)\leq u(t)\leq \beta(t)\). To this end, they consider first the case when (\(f\equiv 0\), \(B_1\equiv 0\)) and prove three existence theorems. Then they generalize the results to a more general case.
scientific article; zbMATH DE number 1861107