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 \(0<\alpha<1\).


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