The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions
In this paper, first we consider the existence of a positive solution for the nonlinear fractional differential equation boundary--value problem \begin{eqnarray*} &&^C\D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\quad 2<\alpha\leq 3,\\ &&u'(0)=u''(0)=0, \quad u(1...
| 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:10347/45460 |
| Acceso en línea: | https://hdl.handle.net/10347/45460 |
| Access Level: | acceso abierto |
| Palabra clave: | Fractional differential equation Integral boundary value condition Positive solution Green’s function Fixed-point theorem 1202 Análisis y análisis funcional |
| Sumario: | In this paper, first we consider the existence of a positive solution for the nonlinear fractional differential equation boundary--value problem \begin{eqnarray*} &&^C\D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\quad 2<\alpha\leq 3,\\ &&u'(0)=u''(0)=0, \quad u(1)=\lambda\int_0^1u(s)\id s, \end{eqnarray*} where $0\leq\lambda<1$, and $^C\D^\alpha$ is the Caputo's differential operator of order $\alpha$, and $f:[0,1]\times[0,\infty)\rightarrow[0,\infty)$ is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary $\alpha$, $1\leq n<\alpha\leq n+1$: Problem 1: \begin{eqnarray*} &&^C\D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\\ &&u^{(i)}(0)=0,\; 0\leq i\leq n,\; i\neq k, \quad u(1)=\lambda\int_0^1u(s)\id s, \end{eqnarray*} where $k\in\{0, 1, \dots, n-1\}$, $0\leq\lambda<k+1$; Problem 2: \begin{eqnarray*} &&D^{\alpha}u(t)+f(t,u(t))=0, \quad 0< t<1,\label{problemn2}\\ &&u^{(i)}(0)=0,\; 0\leq i\leq n-1,\; \quad u(1)=\lambda\int_0^1u(s)\id s,\label{uslovin2} \end{eqnarray*} where $0\leq\lambda \leq \alpha$ and $D^{\alpha}$ is the Riemann--Liouville fractional derivative of order $\alpha$. For these problems we give existence results, which improve recent results in the literature. |
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