Focal boundary value problems, via Sperner's lemma (Q2782949)

The author considers the \(n\)th-order differential equation NEWLINE\[NEWLINEx^{(n)}(t)=f(t,x(t)), \quad t\in [0,1],NEWLINE\]NEWLINE satisfying the multipoint conditions \(x^{(j)}(0)=0\), \(j\in [0,n_1-1]\), \(x^{(l_i)}+j(a_i)=0\) for \( j\in [0,n_i-1]\), \(i=2,3,\dots ,k.\) The nonlinearity \(f\) is assumed to satisfy \(f(t,x)>0\) for nonnegative \(x\) and \(t\in [0,1]\). Moreover, the solution \( x(t)\) satisfies \(x^{(i)}(t)>0\) for \(0<t\leq 1.\) The principal tool is a result from combinatorial topology, Sperner's lemma.