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 $\sigma,...
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2016-07-01
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| Series: | Mathematica Bohemica |
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| Online Access: | http://mb.math.cas.cz/full/141/2/mb141_2_8.pdf |
| _version_ | 1818953165682245632 |
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| author | Jean Mawhin Katarzyna Szymańska-Dębowska |
| author_facet | Jean Mawhin Katarzyna Szymańska-Dębowska |
| author_sort | Jean Mawhin |
| collection | DOAJ |
| description | A couple ($\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. |
| first_indexed | 2024-12-20T10:01:56Z |
| format | Article |
| id | doaj.art-9be10971784a477984573465abd7a09b |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-12-20T10:01:56Z |
| publishDate | 2016-07-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-9be10971784a477984573465abd7a09b2022-12-21T19:44:20ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362016-07-01141223925910.21136/MB.2016.17MB.2016.17Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditionsJean MawhinKatarzyna Szymańska-DębowskaA 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.http://mb.math.cas.cz/full/141/2/mb141_2_8.pdf nonlocal boundary value problem lower solution upper solution lower slope upper slope Leray-Schauder degree |
| spellingShingle | Jean Mawhin Katarzyna Szymańska-Dębowska Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions Mathematica Bohemica nonlocal boundary value problem lower solution upper solution lower slope upper slope Leray-Schauder degree |
| title | Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions |
| title_full | Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions |
| title_fullStr | Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions |
| title_full_unstemmed | Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions |
| title_short | Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions |
| title_sort | couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions |
| topic | nonlocal boundary value problem lower solution upper solution lower slope upper slope Leray-Schauder degree |
| url | http://mb.math.cas.cz/full/141/2/mb141_2_8.pdf |
| work_keys_str_mv | AT jeanmawhin couplesoflowerandupperslopesandresonantsecondorderordinarydifferentialequationswithnonlocalboundaryconditions AT katarzynaszymanskadebowska couplesoflowerandupperslopesandresonantsecondorderordinarydifferentialequationswithnonlocalboundaryconditions |