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Keywords:
boundary-value problems; positive solutions; fixed-point theorem; cone
boundary-value problems; positive solutions; fixed-point theorem; cone
Summary:
The paper deals with the existence of multiple positive solutions for the boundary value problem $$ \begin{cases} (\varphi (p(t)u^{(n-1)})(t))' + a(t)f(t, u(t), u'(t), \ldots , u^{(n-2)}(t)) = 0, \quad \ 0 < t < 1, \\ u^{(i)}(0) = 0, \quad i = 0, 1, \ldots , n - 3,\\ u^{(n-2)}(0) = \sum _{i=1}^{m-2}\alpha _iu^{(n-2)}(\xi _i),\quad u^{(n-1)}(1) = 0, \end{cases} $$ where $\varphi \colon \Bbb R \rightarrow \Bbb R$ is an increasing homeomorphism and a positive homomorphism with $\varphi (0) = 0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.
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