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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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