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Bifurcation of solutions of nonlinear Sturm-Liouville problems. (English) Zbl 1088.34518

Summary: A global bifurcation theorem for the following nonlinear Sturm-Liouville problem is given \[ u''(t)= -h(\lambda, t,u(t), u'(t)),\quad\text{a.e. on }(0,1), \]
\[ u(0)\cos\eta- u'(0)\sin\eta= 0,\tag{\(*\)} \]
\[ u(1)\cos\zeta+ u'(1)\sin\zeta= 0\quad\text{with }\eta,\zeta\in \Biggl[0,{\pi\over 2}\Biggr]. \] Moreover, we give various versions of existence theorems for boundary value problems \[ u''(t)= -g(t,u(t), u'(t)),\quad \text{a.e. on }(0,1), \]
\[ u(0)\cos\eta- u'(0)\sin g\eta= 0,\tag{\(**\)} \]
\[ u(1)\cos\zeta+ u'(1)\sin g\zeta= 0. \] The main idea of these proofs is studying properties of an unbounded connected subset of the set of all nontrivial solutions of the nonlinear spectral problem \((*)\), associated with the boundary value problem \((**)\), in such a way that \(h(1,\cdot,\cdot,\cdot)= g\).

MSC:

34C23 Bifurcation theory for ordinary differential equations
34B24 Sturm-Liouville theory
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