Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions
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 $\sigma,...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics of the Czech Academy of Science
2016-07-01
|
| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/141/2/mb141_2_8.pdf |
| Summary: | A couple ($\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 $\sigma(t) łeq\tau(t)$ for all $t \in[0,1]$, \begin{gather} \sigma'(t) \geq f(t,x,\sigma(t)), \quad\sigma(1) łeq\int_0^1 \sigma(s) {\rm d}g(s),\nonumber
\tau'(t) łeq f(t,x,\tau(t)), \quad\tau(1) \geq\int_0^1 \tau(s) {\rm d}g(s),\nonumber\end{gather} in the stripe $\int_0^t\sigma(s) {\rm d}s łeq x łeq\int_0^t \tau(s) {\rm d}s$ 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. |
|---|---|
| ISSN: | 0862-7959 2464-7136 |