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Existence of multiple positive solutions for one-dimensional \(p\)-Laplacian. (English) Zbl 1098.34017
Consider the following multipoint boundary value problem for the one-dimensional p-Laplacian
\[ \left( \phi _{p}\left( u^{\prime }\right) \right) ^{\prime }+f(t,u)=0, \;t\in \left( 0,1\right), \]
subject to the boundary conditions
\[ \phi _{p}\left( u^{\prime }\left( 0\right) \right) =\sum\limits_{i=1}^{n-2}a_{i}\phi _{p}\left( u^{\prime }\left( \xi _{i}\right) \right) , \quad u(1)=\sum\limits_{i=1}^{n-2}b_{i}u(\xi _{i})\text{ with }\phi _{p}(s)=\left| s\right| ^{p-2}s,\;p>1, \]
\[ 0<\xi _{1}<\xi _{2}<\dots <\xi _{n-2}<1,\;a_{i},\;b_{i} \]
are nonnegative with \(\;0<\sum\limits_{i=1}^{n-2}a_{i}<1\) and \(\sum\limits_{i=1}^{n-2}b_{i}<1.\) The authors provide conditions that are sufficient for obtaining multiple positive solutions of the problem above. They use a fixed-point theorem for operators on a cone.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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