文章基本信息

标题：Existence of solutions for a system of singular sum fractional q -differential equations via quantum calculus
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作者：Mohammad Esmael Samei
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2020
卷号：2020
期号：1
页码：1-23
DOI：10.1186/s13662-019-2480-y
出版社：Hindawi Publishing Corporation
摘要：In this study, we discuss the existence of positive solutions for the system of m-singular sum fractional q-differential equations$$ \begin{gathered} D_{q}^{\alpha_{i}} x_{i} + g_{i} \bigl(t, x_), \ldots, x_{m}, D_{q}^{\gamma _)} x_), \ldots, D_{q}^{\gamma_{m}} x_{m} \bigr) \\ \quad{} +h_{i} \bigl(t, x_), \ldots, x_{m}, D_{q}^{\gamma_)} x_), \ldots, D_{q}^{\gamma_{m}} x_{m} \bigr)=0 \end{gathered} $$ with boundary conditions $x_{i}(0) = x_{i}' (1) = 0$ and $x_{i}^{(k)}(t) = 0$ whenever $t=0$, here $2\leq k \leq n-1$, where $n= [\alpha_{i}]+ 1$, $\alpha_{i} \geq2$, $\gamma_{i} \in(0,1)$, $D_{q}^{\alpha}$ is the Caputo fractional q-derivative of order α, here $q \in(0,1)$, function $g_{i}$ is of Carathéodory type, $h_{i}$ satisfy the Lipschitz condition and $g_{i} (t , x_), \ldots, x_{2m})$ is singular at $t=0$, for $1 \leq i \leq m$. By means of Krasnoselskii’s fixed point theorem, the Arzelà-Ascoli theorem, Lebesgue dominated theorem and some norms, the existence of positive solutions is obtained. Also, we give an example to illustrate the primary effects.
关键词：Existence of solutions ; Caputo q -derivative ; Singularity ; Fractional q -differential equations ;