Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems
The paper presents existence principles for the nonlocal boundary-value problem $$ (\phi (u^(p-1)))' = g(t, u,...,u^{(p-1)}), \alpha_k(u)=0, 1 \leq k \leq p-1$$ where $p\geq2,\quad \phi: {\mathbb R}\rightarrow{\mathbb R}$ is an increasing and odd homeomorphism, $g$ is a Caratheodory functi...
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| Date: | 2008 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2008
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3153 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | The paper presents existence principles for the nonlocal boundary-value problem
$$ (\phi (u^(p-1)))' = g(t, u,...,u^{(p-1)}), \alpha_k(u)=0, 1 \leq k \leq p-1$$
where $p\geq2,\quad \phi: {\mathbb R}\rightarrow{\mathbb R}$
is an increasing and odd homeomorphism, $g$ is a Caratheodory function which is either regular or has singularities in its space variables
and $\alpha_k: C^{p-1}[0,T]\rightarrow{\mathbb R}$ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems
$(-1)^n(\phi(u^{(2n-1)}))' = f (t,u,...,u^{(2n-1)}),\quad u^{(2k)}(0) = 0,\quad$
$a_ku^{(2k)}(T) + b_k u^{(2k+1)}(T)=0,\quad 0\leq k\leq n-1$ is given. |
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