Opuscula Math. 31, no. 1 (2011), 119-135
http://dx.doi.org/10.7494/OpMath.2011.31.1.119

Opuscula Mathematica

# A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions

JinRong Wang
X. Yan
X.-H. Zhang
T.-M. Wang
X.-Z. Li

Abstract. 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.

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

Mathematics Subject Classification: 34G20, 45J05.

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• JinRong Wang
• Guizhou University, Department of Mathematics, Guiyang, Guizhou 550025, P.R. China
• X. Yan
• Guizhou University, Department of Mathematics, Guiyang, Guizhou 550025, P.R. China
• X.-H. Zhang
• Guizhou University, Department of Mathematics, Guiyang, Guizhou 550025, P.R. China
• T.-M. Wang
• Guizhou University, Department of Mathematics, Guiyang, Guizhou 550025, P.R. China
• X.-Z. Li
• Guizhou University, Department of Mathematics, Guiyang, Guizhou 550025, P.R. China
• Revised: 2010-06-01.
• Accepted: 2010-06-01. 