A note on multiple solutions for \((p,n-p)\) focal and \((n,p)\) problems (Q2763904)

The existence of at least two nonnegative solutions to the \((p,n-p)\) focal problem NEWLINE\[NEWLINE(-1)^{n-p}y^{(n)}(t) = \varphi(t)f(t,y(t)), \quad t\in(0,1);NEWLINE\]NEWLINE NEWLINE\[NEWLINE y^{(i)}(0)=0, \quad i=0,1,\dots,p-1;NEWLINE\]NEWLINE NEWLINE\[NEWLINE y^{(i)}(1)=0, \quad i=p,p+1,\dots,n-1;NEWLINE\]NEWLINE is studied, where \(n\geq 2\), \(1\leq p <n\), \(\varphi \in L_1([0,1])\) is continuous on \((0,1)\), \(f: [0,1]\times [0,\infty)\to[0,\infty)\) is continuous, strictly positive on \([0,1]\times (0,\infty)\), and such that, for all \((t,u) \in [0,1]\times (0,\infty)\), one has \(f(t,u)\leq w(u)\) with some nonnegative \(w\) continuous and nondecreasing on \([0,\infty)\). Assuming certain additional conditions on \(f\) and \(\varphi\), the authors use the Krasnoselskii fixed-point theorem in a cone to prove the existence of ``twin nonnegative solutions.'' NEWLINENEWLINENEWLINESimilar results are obtained also for nonsingular \((n,p)\) problems.