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Zbl 1120.26003
Ye, Haiping; Gao, Jianming; Ding, Yongsheng
A generalized Gronwall inequality and its application to a fractional differential equation.
(English)
[J] J. Math. Anal. Appl. 328, No. 2, 1075-1081 (2007). ISSN 0022-247X

The authors prove a statement about a Gronwall type inequality in the following form. Let $\beta >0, a(t)$ be a nonnegative function locally integrable on $[0,T), \ T\le\infty$, and $g(t)$ be a nonnegative nondecreasing bounded continuous function on $[0,T)$. Suppose that $u(t)$ is nonnegative and locally integrable on $[0,T)$ and satisfies the inequality $$u(t)\le a(t)+g(t)\int\limits_0^t(t-s)^{\beta-1}u(s)ds$$ on this interval. Then $$u(t)\le a(t)+\int\limits_0^t\left[\sum\limits_{n=1}^\infty \frac{(g(t)\Gamma(\beta))^n}{\Gamma(n\beta)}(t-s)^{n\beta-1} a(s)\right]ds.$$ An application of this statement is given to a fractional differential equation $D^\alpha y(t)= f(t,y(t))$ with the Cauchy type condition $\left. D^{\alpha-1}y(t)\right|_{t=0}=\eta$, where $0<\alpha<1$.
[Stefan G. Samko (Faro)]
MSC 2000:
*26A33 Fractional derivatives and integrals (real functions)
26D15 Inequalities for sums, series and integrals of real functions

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