Constant sign Green's function for simply supported beam equation
The aim of this paper consists on the study of the following fourth-order operator: \begin{equation}\label{Ec::T4} T[M]\,u(t)\equiv u^{(4)}(t)+p_1(t)\,u'''(t)+p_2(t)\,u''(t)+M\,u(t)\,,\ t\in I \equiv [a,b]\,, \end{equation} coupled with the two point boundary conditions: \be...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2017 |
| 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:10347/45439 |
| Acceso en línea: | https://hdl.handle.net/10347/45439 |
| Access Level: | acceso abierto |
| Palabra clave: | Green's function Spectral Charaterization Beam equation 1202 Análisis y análisis funcional |
| Sumario: | The aim of this paper consists on the study of the following fourth-order operator: \begin{equation}\label{Ec::T4} T[M]\,u(t)\equiv u^{(4)}(t)+p_1(t)\,u'''(t)+p_2(t)\,u''(t)+M\,u(t)\,,\ t\in I \equiv [a,b]\,, \end{equation} coupled with the two point boundary conditions: \begin{equation}\label{Ec::cf} u(a)=u(b)=u''(a)=u''(b)=0\,. \end{equation} So, we define the following space: \begin{equation}\label{Ec::esp} X=\left\lbrace u\in C^4(I)\quad\mid\quad u\text{ satisfies boundary conditions \eqref{Ec::cf}}\right\rbrace \,. \end{equation} Here $p_1\in C^3(I)$ and $p_2\in C^2(I)$. By assuming that the second order linear differential equation \begin{equation}\label{Ec::2or} L_2\, u(t)\equiv u''(t)+p_1(t)\,u'(t)+p_2(t)\,u(t)=0\,,\quad t\in I, \end{equation} is disconjugate on $I$, we characterize the parameter's set where the Green's function related to operator $T[M]$ in $X$ is of constant sign on $I \times I$. Such characterization is equivalent to the strongly inverse positive (negative) character of operator $T[M]$ on $X$ and comes from the first eigenvalues of operator $T[0]$ on suitable spaces. |
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