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Zbl 1235.34210
Wang, JinRong; Yan, X.; Zhang, X.-H.; Wang, T.-M.; Li, X.-Z.
A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions.
(English)
[J] Opusc. Math. 31, No. 1, 119-135 (2011). ISSN 1232-9274

Summary: We discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: $$\multline D^q_tx(t)= Ax(t)+\int^t_0B(t-s)x(s)ds+t^nf(t,x(t)),\quad t\in [0,T],\ n\in Z^+,\\ q\in(0,1],\ x(0)=g(x)+x_0.\endmultline$$ Some sufficient conditions for the existence of a mild solution are given. The main tools are the resolvent operator and fixed point theorems due to Banach's fixed point theorem, Krasnoselskii's fixed point theorem and Schaefer's fixed point theorem. Finally, an example is given.
MSC 2000:
*34K37
34K30 Functional-differential equations in abstract spaces
45J99

Keywords: integrodifferential equations; fractional derivative; nonlocal conditions; resolvent operator and their norm continuity; fixed point theorem; mild solutions

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