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Global structure of positive solutions for superlinear \(2m\)th-boundary value problems. (English) Zbl 1224.34034

Summary: We consider boundary value problems of the type \[ (-1)^{m}u^{(2m)}(t)=\lambda a(t)f(u(t)),\;0<t<1, \]
\[ u^{(2i)}(0)=u^{(2i)}(1)=0,\;i=0,1,2,\cdots ,m-1 , \] where \(a\in C([0,1], [0,\infty ))\) and \(a(t_0)>0\) for some \(t_0\in [0,1]\), \(f\in C([0,\infty ),[0,\infty ))\) and \(f(s)>0\) for \(s>0\), and \(f_0=\infty \), where \(f_0=\lim _{s\rightarrow 0^+}f(s)/s\). We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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