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On the existence of minimal and maximal solutions of discontinuous functional Sturm-Liouville boundary value problems. (English) Zbl 1108.34020
The authors study the existence of minimal and maximal solutions of the discontinuous functional Sturm-Liouville boundary value problems
\[ -\frac{d}{dt}(\mu (t)u^{\prime }(t))=\lambda g(t,u,u(t),u^{\prime }(t))\text{ a.e. in } J=[t_{0},t_{1}],\tag{P} \]
\[ a_{0}u(t_{0})-b_{0}u^{\prime }(t_{0})=c_{0},\quad a_{1}u(t_{1})+b_{1}u^{\prime }(t_{1})=c_{1}, \]
where \(\lambda ,\) \(a_{j},\) \(b_{j}\) \(\in \mathbb{R}^{+},c_{j}\in \mathbb{R}\) for \(j=0,1,\mu \in C(J,(0,\infty ))\) and \(g:\) \(J\times C(J)\times \mathbb{ R\times R\rightarrow }\mathbb{R}\) is a given function. First, they give an existence result on \((P)\) where the second argument of \( g \) is a fixed function in \(C(J).\) Then, they study the dependence of the solution set of \((P)\) on the fixed function. By a fixed-point result for multifunctions, they give existence results for minimal and maximal solutions of \((P).\)

MSC:
34B24 Sturm-Liouville theory
34B15 Nonlinear boundary value problems for ordinary differential equations
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