On the lower and upper solution method for higher order functional boundary value problems (Q2853199)

The authors study the \(n\)th-order differential equation NEWLINE\[NEWLINE -(\phi(u^{(n-1)}(x)))'=f(x,u(x),\dots,u^{(n-1)}(x)), \quad x\in(0,1), NEWLINE\]NEWLINE where \(\phi:\mathbb R\to \mathbb R\) is an increasing homeomorphism such that \(\phi(0)=0\) and \(f: [0,1]\times\mathbb R^n\to\mathbb R\) is an \(L^1\)-Caratheodory function, together with the boundary conditions NEWLINE\[NEWLINEg_i(u,\dots,u^{(n-2)},u^{(i)}(1))=0,\, i=0,\dots,n-3;NEWLINE\]NEWLINE NEWLINE\[NEWLINEg_{n-2}(u,\dots,u^{(n-2)},u^{(n-2)}(0),u^{(n-1)}(0))=0; g_{n-1}(u,\dots,u^{(n-2)},u^{(n-2)}(1),u^{(n-1)}(1))=0.NEWLINE\]NEWLINE Here, \(g_i\), \(i=0,\dots,n-1\), are continuous functions satisfying certain monotonicity assumptions.NEWLINENEWLINE Sufficient conditions for the existence of solutions, location sets for the solution and its derivatives up to order \((n-1)\) are given.