On positive solutions for a \(m\)-point fractional boundary value problem on an infinite interval. (English) Zbl 1479.34010
Summary: In this paper, by using a recent fixed point theorem, we study the existence and uniqueness of positive solutions for the following \(m\)-point fractional boundary value problem on an infinite interval
\[
\begin{cases} D_{0^+}^{\alpha}x(t)+f(t,x(t))=0, & \quad 0<t<\infty,\\
x(0)=x'(0)=0, & \quad D_{0^+}^{\alpha -1}x(+\infty )= \sum _{i=1}^{m-2}\beta _ix(\xi _i), \end{cases}
\]
where \(2<\alpha <3\), \(D_{0^+}^{\alpha }\) is the standard Riemann-Liouville fractional derivative,
\[ D_{0^+}^{\alpha -1}x(+\infty )= \lim _{t\rightarrow \infty }D_{0^+}^{\alpha -1}x(t),\\ 0<\xi _1<\xi _2<\cdots<\xi _{m-2}<\infty \quad \text{and} \quad \beta _i\ge 0 \quad \text{for} \quad i=1,2,\dots ,m-2. \]
Moreover, we present an example illustrating our results.
\[ D_{0^+}^{\alpha -1}x(+\infty )= \lim _{t\rightarrow \infty }D_{0^+}^{\alpha -1}x(t),\\ 0<\xi _1<\xi _2<\cdots<\xi _{m-2}<\infty \quad \text{and} \quad \beta _i\ge 0 \quad \text{for} \quad i=1,2,\dots ,m-2. \]
Moreover, we present an example illustrating our results.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 47H10 | Fixed-point theorems |
References:
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