Existence and uniqueness for fractional neutral differential equations with infinite delay. (English) Zbl 1177.34084
Summary: We consider the Cauchy initial value problem of fractional neutral functional differential equations with infinite delay of the form
\[ D^qg(t,x_t)=f(t,x_t),\quad t\in [t_0,\infty),\tag{1} \]
\[ x_{t_0}=\varphi,\;(t_0,\varphi)\in [0,\infty)\times \Omega,\tag{2} \]
where \(D^q\) is Caputo’s fractional derivative of order \(0 < q < 1\), \(\Omega\) is an open subset of \(B\) and \(g,f : [t_0,\infty)\times \Omega\to \mathbb{R}^n\) are given functionals satisfying some assumptions. Various criteria on existence and uniqueness are obtained.
\[ D^qg(t,x_t)=f(t,x_t),\quad t\in [t_0,\infty),\tag{1} \]
\[ x_{t_0}=\varphi,\;(t_0,\varphi)\in [0,\infty)\times \Omega,\tag{2} \]
where \(D^q\) is Caputo’s fractional derivative of order \(0 < q < 1\), \(\Omega\) is an open subset of \(B\) and \(g,f : [t_0,\infty)\times \Omega\to \mathbb{R}^n\) are given functionals satisfying some assumptions. Various criteria on existence and uniqueness are obtained.
MSC:
| 34K05 | General theory of functional-differential equations |
| 26A33 | Fractional derivatives and integrals |
| 34K40 | Neutral functional-differential equations |
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