Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions

Mawhin, Jean, and Szymańska-Dębowska, Katarzyna. "Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions." Mathematica Bohemica 141.2 (2016): 239-259. <http://eudml.org/doc/276990>.

@article{Mawhin2016,
abstract = {A couple ($\sigma ,\tau $) of lower and upper slopes for the resonant second order boundary value problem \[ x^\{\prime \prime \} = f(t,x,x^\{\prime \}), \quad x(0) = 0,\quad x^\{\prime \}(1) = \int \_0^1 x^\{\prime \}(s) \{\rm d\}g(s), \] 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 $\sigma (t) \le \tau (t)$ for all $t \in [0,1]$, \begin\{gather\} \sigma ^\{\prime \}(t) \ge f(t,x,\sigma (t)), \quad \sigma (1) \le \int \_0^1 \sigma (s) \{\rm d\}g(s),\nonumber \\ \tau ^\{\prime \}(t) \le f(t,x,\tau (t)), \quad \tau (1) \ge \int \_0^1 \tau (s) \{\rm d\}g(s),\nonumber \end\{gather\} in the stripe $\int _0^t\sigma (s) \{\rm d\}s \le x \le \int _0^t \tau (s) \{\rm d\}s$ 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.},
author = {Mawhin, Jean, Szymańska-Dębowska, Katarzyna},
journal = {Mathematica Bohemica},
keywords = {nonlocal boundary value problem; lower solution; upper solution; lower slope; upper slope; Leray-Schauder degree},
language = {eng},
number = {2},
pages = {239-259},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions},
url = {http://eudml.org/doc/276990},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Mawhin, Jean
AU - Szymańska-Dębowska, Katarzyna
TI - Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 2
SP - 239
EP - 259
AB - A couple ($\sigma ,\tau $) of lower and upper slopes for the resonant second order boundary value problem \[ x^{\prime \prime } = f(t,x,x^{\prime }), \quad x(0) = 0,\quad x^{\prime }(1) = \int _0^1 x^{\prime }(s) {\rm d}g(s), \] 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 $\sigma (t) \le \tau (t)$ for all $t \in [0,1]$, \begin{gather} \sigma ^{\prime }(t) \ge f(t,x,\sigma (t)), \quad \sigma (1) \le \int _0^1 \sigma (s) {\rm d}g(s),\nonumber \\ \tau ^{\prime }(t) \le f(t,x,\tau (t)), \quad \tau (1) \ge \int _0^1 \tau (s) {\rm d}g(s),\nonumber \end{gather} in the stripe $\int _0^t\sigma (s) {\rm d}s \le x \le \int _0^t \tau (s) {\rm d}s$ 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.
LA - eng
KW - nonlocal boundary value problem; lower solution; upper solution; lower slope; upper slope; Leray-Schauder degree
UR - http://eudml.org/doc/276990
ER -