Advances in Difference Equations ( IF 4.1 ) Pub Date : 2020-09-04 , DOI: 10.1186/s13662-020-02918-0 Anwar Ali , Muhammad Sarwar , Mian Bahadur Zada , Thabet Abdeljawad
In this manuscript, we obtain sufficient conditions required for the existence of solution to the following coupled system of nonlinear fractional order differential equations:
$$ \begin{gathered} D^{\gamma}\omega(\ell)= \mathcal{F} \bigl( \ell,\omega(\lambda\ell), \upsilon(\lambda\ell) \bigr), \\ D^{\delta}\upsilon(\ell)=\mathcal{\overline{F}} \bigl(\ell,\omega ( \lambda\ell), \upsilon(\lambda\ell) \bigr), \end{gathered} $$with fractional integral boundary conditions
$$ \begin{gathered} \mathfrak{a}_{1}\omega(0)- \mathfrak{b}_{1}\omega(\eta)-\mathfrak {c}_{1}\omega(1)= \frac{1}{\varGamma(\gamma)} \int_{0}^{1}(1-\rho )^{\gamma-1} \phi \bigl( \rho, \omega(\rho) \bigr)\, d\rho\quad\text{and} \\ \mathfrak{a}_{2}\upsilon(0)-\mathfrak{b}_{2} \upsilon (\xi)-\mathfrak{c}_{2}\upsilon(1)=\frac{1}{\varGamma(\delta)} \int _{0}^{1}(1-\rho)^{\delta-1} \psi \bigl( \rho, \upsilon(\rho) \bigr) \,d\rho, \end{gathered} $$where \(\ell\in\mathfrak{Z}=[0,1]\), \(\gamma, \delta\in(0,1]\), \(0<\lambda<1\), D denotes the Caputo fractional derivative (in short CFD), \(\mathcal{F}, \mathcal{\overline{F}}: \mathfrak{Z}\times \mathfrak{R}\times\mathfrak{R} \rightarrow\mathfrak{R}\) and \(\phi , \psi:\mathfrak{Z}\times\mathfrak{R}\rightarrow\mathfrak{R}\) are continuous functions. The parameters η, ξ are such that \(0<\eta, \xi<1\), and \(\mathfrak{a}_{i}, \mathfrak{b}_{i}, \mathfrak {c}_{i}\) (\(i=1, 2\)) are real numbers with \(\mathfrak{a}_{i}\neq\mathfrak {b}_{i}+\mathfrak{c}_{i}\) (\(i=1, 2\)). Using topological degree theory, sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, an appropriate example is presented in the end.
中文翻译：
拓扑度理论的比例时滞微分方程耦合系统解的存在唯一性。
在此手稿中，我们获得了以下非线性分数阶微分方程耦合系统解存在的必要条件：
$$ \ begin {gathered} D ^ {\ gamma} \ omega（\ ell）= \ mathcal {F} \ bigl（\ ell，\ omega（\ lambda \ ell），\ upsilon（\ lambda \ ell）\ bigr ），\\ D ^ {\ delta} \ upsilon（\ ell）= \ mathcal {\ overline {F}}} \ bigl（\ ell，\ omega（\ lambda \ ell），\ upsilon（\ lambda \ ell）\更大），\ end {聚集} $$具有分数积分边界条件
$$ \ begin {gathered} \ mathfrak {a} _ {1} \ omega（0）-\ mathfrak {b} _ {1} \ omega（\ eta）-\ mathfrak {c} _ {1} \ omega（ 1）= \ frac {1} {\ varGamma（\ gamma）} \ int_ {0} ^ {1}（1- \ rho）^ {\ gamma-1} \ phi \ bigl（\ rho，\ omega（\ rho）\ bigr）\，d \ rho \ quad \ text {and} \\ \ mathfrak {a} _ {2} \ upsilon（0）-\ mathfrak {b} _ {2} \ upsilon（\ xi）- \ mathfrak {c} _ {2} \ upsilon（1）= \ frac {1} {\ varGamma（\ delta）} \ int _ {0} ^ {1}（1- \ rho）^ {\ delta-1 } \ psi \ bigl（\ rho，\ upsilon（\ rho）\ bigr）\，d \ rho，\ end {gathered} $$其中\（\ ell \ in \ mathfrak {Z} = [0,1] \），\（\ gamma，\ delta \ in（0,1] \），\（0 <\ lambda <1 \），D表示Caputo分数导数（简称CFD），\（\ mathcal {F}，\ mathcal {\ overline {F}}}：\ mathfrak {Z} \ times \ mathfrak {R} \ times \ mathfrak {R} \ rightarrow \ mathfrak {R} \）和\（\ phi，\ psi：\ mathfrak {Z} \ times \ mathfrak {R} \ rightarrow \ mathfrak {R} \）是连续函数。参数η，ξ使得\ （0 <\ eta，\ xi <1 \）和\（\ mathfrak {a} _ {i}，\ mathfrak {b} _ {i}，\ mathfrak {c} _ {i} \）（\（ i = 1，2 \））是具有\（\ mathfrak {a} _ {i} \ neq \ mathfrak {b} _ {i} + \ mathfrak {c} _ {i} \）（\（i = 1，2 \））。使用拓扑度理论，可以为有关问题的至少一个唯一解的存在建立足够的结果。为了验证结果的有效性，最后给出了一个适当的例子。
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