Xu, Run; Meng, Fanwei Some new weakly singular integral inequalities and their applications to fractional differential equations. (English) Zbl 1337.26022 J. Inequal. Appl. 2016, Paper No. 78, 16 p. (2016). 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. Reviewer: James Adedayo Oguntuase (Abeokuta) Cited in 44 Documents MSC: 26A33 Fractional derivatives and integrals 26D10 Inequalities involving derivatives and differential and integral operators 26D15 Inequalities for sums, series and integrals Keywords:inequality; weakly singular integral kernel; fractional differential and integral equation; quantitative analysis PDFBibTeX XMLCite \textit{R. Xu} and \textit{F. Meng}, J. Inequal. Appl. 2016, Paper No. 78, 16 p. (2016; Zbl 1337.26022) Full Text: DOI References: [1] Denton, Z, Vatsala, AS: Fractional integral inequalities and applications. Comput. Math. Appl. 59, 1087-1094 (2010) · Zbl 1189.26044 [2] Ye, H, Gao, J, Ding, Y: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075-1081 (2007) · Zbl 1120.26003 [3] Medved̆, M: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. 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