The existence of countably many positive solutions for nonlinear singular \(m\)-point boundary value problems. (English) Zbl 1136.34027
Summary: We study the existence of countably many positive solutions for the nonlinear singular boundary value problem
\[ (\varphi(u'))'+a(t)f(u(t))=0,\quad 0<t<1, \]
subject to the boundary value conditions:
\[ u(0)=\sum^{m-2}_{i=1}\alpha_iu(\xi_i), \quad \varphi(u'(1))= \sum^{m-2}_{i=1}\beta_i\varphi(u'(\xi_i)), \]
where \(\phi\mathbb :R\to\mathbb R\) is an increasing homeomorphism and positive homomorphism and \(\varphi(0)=0\), \(\xi_i\in (0,1)\) with \(0<\xi_1<\xi_2<\cdots<\xi_{m-2}<1\) and \(\alpha_i\), \(\beta_i\) satisfy \(\alpha_i,\beta_i\in [0,+\infty]\), \(0<\sum^{m-2}_{i=1}\alpha_i<1\), \(0<\sum^{m-2}_{i=1}\beta_i<1\), \(f\in C([0,+\infty),[0,+\infty))\), \(a:[0,1]\to [0,+\infty)\) and has countably many singularities in \([0,\frac12)\). We show that there exist countably many positive solutions by using the fixed-point index theory and a new fixed-point theorem in cones.
\[ (\varphi(u'))'+a(t)f(u(t))=0,\quad 0<t<1, \]
subject to the boundary value conditions:
\[ u(0)=\sum^{m-2}_{i=1}\alpha_iu(\xi_i), \quad \varphi(u'(1))= \sum^{m-2}_{i=1}\beta_i\varphi(u'(\xi_i)), \]
where \(\phi\mathbb :R\to\mathbb R\) is an increasing homeomorphism and positive homomorphism and \(\varphi(0)=0\), \(\xi_i\in (0,1)\) with \(0<\xi_1<\xi_2<\cdots<\xi_{m-2}<1\) and \(\alpha_i\), \(\beta_i\) satisfy \(\alpha_i,\beta_i\in [0,+\infty]\), \(0<\sum^{m-2}_{i=1}\alpha_i<1\), \(0<\sum^{m-2}_{i=1}\beta_i<1\), \(f\in C([0,+\infty),[0,+\infty))\), \(a:[0,1]\to [0,+\infty)\) and has countably many singularities in \([0,\frac12)\). We show that there exist countably many positive solutions by using the fixed-point index theory and a new fixed-point theorem in cones.
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
Keywords:
singularity; multiple positive solutions; multiple boundary value problems; fixed-point theorem; coneReferences:
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