Existence of solutions of a second order equation defined on unbounded intervals with integral conditions on the boundary
In this paper we shall use the upper and lower solutions method to prove the existence of at least one solution for the second order equation defined on unbounded intervals with integral conditions on the boundary: \begin{equation*} u^{\prime \prime }\left( t\right) -m^{2}u\left( t\right) +f( t,e^{-...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universidad de Santiago de Compostela (USC) |
| Repositorio: | Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela |
| Idioma: | inglés |
| OAI Identifier: | oai:minerva.usc.gal:10347/37954 |
| Acceso en línea: | https://hdl.handle.net/10347/37954 |
| Access Level: | acceso abierto |
| Palabra clave: | Boundary value problems Integral boundary conditions Upper and lower solutions method Existence of solution 120219 Ecuaciones diferenciales ordinarias |
| Sumario: | In this paper we shall use the upper and lower solutions method to prove the existence of at least one solution for the second order equation defined on unbounded intervals with integral conditions on the boundary: \begin{equation*} u^{\prime \prime }\left( t\right) -m^{2}u\left( t\right) +f( t,e^{-mt}u\left( t\right) ,e^{-mt}\,u^{\prime }\left( t\right)) =0,\quad \mbox{for all}\;t\in % \left[ 0,+\infty \right) , \label{1.1} \end{equation*} \begin{equation*} \label{1.2} u\left( 0\right) -\frac{1}{m}u^{\prime }\left( 0\right) =\int\limits_{0}^{+\infty }e^{-2ms}u\left( s\right) ds,\underset{% t\rightarrow +\infty }{\lim }{\left\{e^{-mt}u\left( t\right) \right\}} =B, \end{equation*}% where $m>0,m\neq \frac{1}{6},B\in \mathbb{R}$ and $f:\left[ 0,+\infty \right) \times \mathbb{R}^{2}\rightarrow \mathbb{R} $ is a continuous function satisfying a suitable locally $L^1$ bounded condition and a kind of Nagumo's condition with respect to the first derivative. |
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