Existence and iteration of positive solutions for a nonlinear multi-point boundary value problem with a \(p\)-Laplacian operator. (English) Zbl 1245.34029
Summary: We are interested in the existence of positive solutions for the following nonlinear multi-point boundary value problem with a \(p\)-Laplacian operator:
\[
(\varphi_p(u'(t)))'+q(t)f(t,u(t),u'(t))=0,\;t\in (0,1)
\]
\[ \varphi_p(u(0))-\sum^{m-2}_{i=1}\alpha_i\varphi_p (u'(\xi_i))=0,\;\varphi_p(u(1))+\sum^{m-2}_{i=1}\beta_i\varphi_p (u'(\xi_i))=0, \] where \(\varphi_p(s)\) is a \(p\)-Laplacian operator, that is, \(\varphi_p(s)=|s|^{p-2}s\), \(p>1\); \(\xi_i\in(0,1)\) with \(0<\xi_1<\xi_2 <\xi_3<\cdots<\xi_{m-2}<1\).
By using the monotone iterative technique, we obtain not only the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solution.
\[ \varphi_p(u(0))-\sum^{m-2}_{i=1}\alpha_i\varphi_p (u'(\xi_i))=0,\;\varphi_p(u(1))+\sum^{m-2}_{i=1}\beta_i\varphi_p (u'(\xi_i))=0, \] where \(\varphi_p(s)\) is a \(p\)-Laplacian operator, that is, \(\varphi_p(s)=|s|^{p-2}s\), \(p>1\); \(\xi_i\in(0,1)\) with \(0<\xi_1<\xi_2 <\xi_3<\cdots<\xi_{m-2}<1\).
By using the monotone iterative technique, we obtain not only the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solution.
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
