# zbMATH — the first resource for mathematics

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
Full Text: