TY  - JOUR
ID  - Wang2011
LB  - Wang2011
AU  - Wang, JinRong
AU  - Yan, X.
AU  - Zhang, X.-H.
AU  - Wang, T.-M.
AU  - Li, X.-Z.
TI  - A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions
ST  - A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2011
VL  - 31
IS  - 1
SP  - 119
EP  - 135
DO  - http://dx.doi.org/10.7494/OpMath.2011.31.1.119
UR  - http://dx.doi.org/10.7494/OpMath.2011.31.1.119
AB  - In this paper, we discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: \[\begin{aligned}D_{t}^{q}x(t)=Ax(t)+\int\limits_{0}^{t}B(t-s)x(s)ds+t^{n}f\left(t,x(t)\right),&\;t\in [0,T],\;n\in Z^{+},\\&q\in(0,1],\;x(0)=g(x)+x_{0}.\end{aligned}\] Some sufficient conditions for the existence of mild solutions for the above system are given. The main tools are the resolvent operators and fixed point theorems due to Banach's fixed point theorem, Krasnoselskii's fixed point theorem and Schaefer's fixed point theorem. At last, an example is given for demonstration.
KW  - integrodifferential equations
KW  - fractional derivative
KW  - nonlocal conditions
KW  - resolvent operator and their norm continuity
KW  - fixed point theorem
KW  - mild solutions
SN  - 1232-9274
ER  -