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Existence of triple positive solutions for \((n,p)\) and \((p,n)\) boundary value problems. (English) Zbl 1085.34512
Summary: We first consider the \((n,p)\) boundary value problem \(-y^{(n)}=f(y(t))\), \(0\leq t\leq 1\), \(y^{(i)}(0)=0\), \(0\leq i\leq n-2\), \(y^{(p)}(1)=0\), where \(1\leq p\leq n-1\) is fixed and \(f:\mathbb R\to [0,\infty)\) is continuous. Best possible upper and lower bounds are obtained on the integral of Green’s function associated with this problem. These bounds are then used in conjunction with growth assumptions on \(f\) to apply the Leggett-Williams fixed-point theorem to obtain the existence of at least three positive solutions. We conclude by stating the analogous results for \((p,n)\) boundary value problems.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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