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Some new weakly singular integral inequalities and their applications to fractional differential equations. (English) Zbl 1337.26022

The authors give some explicit bounds to some new nonlinear Herry-Gronwall-type retarded integral inequalities with weakly singular integral kernel of the form \[ u^{p}(t) \leq a(t) + \int^{t}_{t_{0}}(t-s)^{\beta-1)}b(s)u^{q}(s)ds+\int^{t}_{t_{0}}(t-s)^{\beta-1)}c(s)u^{l}(s-r)ds, \;t\in I, \]
\[ u(t) \leq \phi(t),\; t\in [t_{0}-r, t_{0}), \] and Gronwall-Bellman type integral inequalities with nonlinear weakly singular integral kernel of the form \[ u^{p}(t) \leq a(t) + b(t)\int^{t}_{0}(t-s)^{\beta-1)}c(s)u^{m}(s)ds+d(t)\int^{t}_{0}(t^{\alpha}-s^{\alpha})^{\beta-1)}s^{\gamma-1}f(s)u^{q}(s)ds \] are derived and proved. Some consequences of the obtained results are pointed out while some examples to illustrate the applications of the results obtained are also given.

MSC:

26A33 Fractional derivatives and integrals
26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
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