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Existence and uniqueness of solutions to fractional semilinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions. (English) Zbl 1179.45012
Summary: We study the fractional semilinear mixed Volterra-Fredholm integrodifferential equation
\[ \frac{d^\alpha x(t)}{dt^\alpha}=Ax(t)+f\left(t,x(t),\int^t_{t_0} k(t,s,x(s))\,ds,\;\int^T_{t_0} h(t,s,x(s))\,ds\right), \]
where \(t\in [t_0, T]\), \(t_0\geq 0\), \(0 < \alpha < 1\), and \(f\) is a given function. We prove the existence and uniqueness of solutions to this equation, with a nonlocal condition.

MSC:
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
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