## A generalized Gronwall inequality and its application to a fractional differential equation.(English)Zbl 1120.26003

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 $$[0,T), \;T\leq\infty$$, 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 $u(t)\leq a(t)+g(t)\int\limits_0^t(t-s)^{\beta-1}u(s)ds$ on this interval. Then $u(t)\leq a(t)+\int\limits_0^t\left[\sum\limits_{n=1}^\infty \frac{(g(t)\Gamma(\beta))^n}{\Gamma(n\beta)}(t-s)^{n\beta-1} a(s)\right]ds.$ 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 $$0<\alpha<1$$.

### MSC:

 26A33 Fractional derivatives and integrals 26D15 Inequalities for sums, series and integrals
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### References:

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