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Existence of triple positive solutions for multi-point boundary value problems with a one dimensional \(p\)-Laplacian. (English) Zbl 1134.34017

Summary: We consider the existence of multiple positive solutions for multi-point boundary value problems with a one dimensional \(p\)-Laplacian
\[ (\varphi_p(u'(t)))'+ q(t)f(t,u(t),u'(t))=0, \quad 0<t<1, \]
\[ u(0)= \sum_{i=1}^{m-2} \alpha_iu(\xi_i), \qquad u(1)= \sum_{i=1}^{m-2} \beta_iu(\xi_i), \]
where \(\varphi_p(s)= |s|^{p-2}s\), \(p>1\), \(\alpha_i\leq\beta_i\) \((1\leq i\leq m-2)\in [0,\infty)\), \(\sum_{i=1}^{m-2} \alpha_i, \sum_{i=1}^{m-2} \beta_i\in (0,1)\), \(0<\xi_1< \xi_2<\cdots< \xi_{m-2}<1\). By the fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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