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Positive solutions of higher-order periodic boundary value problems. (English) Zbl 1062.34021

Summary: This paper deals with the existence of positive solutions for \(n\)th-order periodic boundary value problems \[ L_nu(t)=f \bigl(t,u(t)\bigr),\;0\leq t\leq 2\pi,\quad u^{(i)}(0)=u^{(i)} (2\pi),\;i=0,1,\dots,n-1, \] where \(L_nu(t)=u^{(n)}(t)+\sum^{n-1}_{i=0} a_iu^{(i)}(t)\) is an \(n\)th-order linear differential operator and \(f:[0,2\pi]\times\mathbb{R}^+\to\mathbb{R}^+\) is continuous. We obtain a sufficient condition that the operator \(L_n\) satisfies the maximum principle in the periodic boundary condition. Using this maximum principle and fixed-point index theory in cones, we obtain existence results for positive solutions.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

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