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Global structure of positive solution set for a class of second-order nonlinear periodic boundary value problems. (Chinese. English summary) Zbl 1463.34078

Summary: By using the interval bifurcation theory and topological degree theory, the author studies a class of second-order nonlinear periodic boundary value problems \[ \begin{cases} -u''+q(t)u = \lambda g(t)f(u),\, t \in (0, 2\pi), \\ u(0) = u(2\pi),\, u'(0) = u'(2\pi), \end{cases} \] and gives the global structure of the positive solution set of the problem, where \(\lambda\) is a positive parameter, \(q \in C([0, 2\pi], [0, \infty))\) and \(q\) is not always 0, \(f \in C([0, \infty), [0, \infty))\), \(g \in C([0, 2\pi], [0, \infty))\) and there exists \({t_0} \in [0, 2\pi]\) such that \(g(t_0) > 1\).

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34C23 Bifurcation theory for ordinary differential equations
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