| Title:
|
Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions (English) |
| Author:
|
Mawhin, Jean |
| Author:
|
Szymańska-Dębowska, Katarzyna |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
141 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
239-259 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A couple ($\sigma ,\tau $) of lower and upper slopes for the resonant second order boundary value problem $$ x'' = f(t,x,x'), \quad x(0) = 0,\quad x'(1) = \int _0^1 x'(s) {\rm d}g(s), $$ with $g$ increasing on $[0,1]$ such that $\int _0^1 dg = 1$, is a couple of functions $\sigma , \tau \in C^1([0,1])$ such that $\sigma (t) \leq \tau (t)$ for all $t \in [0,1]$, \begin {gather} \sigma '(t) \geq f(t,x,\sigma (t)), \quad \sigma (1) \leq \int _0^1 \sigma (s) {\rm d}g(s),\nonumber \\ \tau '(t) \leq f(t,x,\tau (t)), \quad \tau (1) \geq \int _0^1 \tau (s) {\rm d}g(s),\nonumber \end {gather} in the stripe $\int _0^t\sigma (s) {\rm d}s \leq x \leq \int _0^t \tau (s) {\rm d}s$ and $t \in [0,1]$. It is proved that the existence of such a couple $(\sigma ,\tau )$ implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained. (English) |
| Keyword:
|
nonlocal boundary value problem |
| Keyword:
|
lower solution |
| Keyword:
|
upper solution |
| Keyword:
|
lower slope |
| Keyword:
|
upper slope |
| Keyword:
|
Leray-Schauder degree |
| MSC:
|
34B10 |
| MSC:
|
34B15 |
| MSC:
|
47H11 |
| idZBL:
|
Zbl 06587864 |
| idMR:
|
MR3499786 |
| DOI:
|
10.21136/MB.2016.17 |
| . |
| Date available:
|
2016-05-19T09:09:16Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145714 |
| . |
| Reference:
|
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