Abstract
In this paper, we investigate some nonlinear integral inequalities with weakly singular kernel which can be used as tools in deriving boundedness of the solutions of certain fractional differential equations and integral equations. Our results generalize and improve some results in the literature. Besides, we give some applications for some fractional differential equations involving the Riemann-Liouville derivative and the Caputo derivative, respectively.
Highlights
In the study of the qualitative and quantitative properties of solutions of some fractional differential equations, many inequalities with singular kernels have been developed
Which came from the study of a global existence and an exponential decay result for a parabolic Cauchy problem by Henry [ ]
Ma and Pečarić [ ] used the modification of Medveď’s method [ ] to study some new weakly singular integral inequality of Henry’s type: t up(t) ≤ a(t) + b(t) tα – sα β– sγ – f (s)uq(s) ds, t >,. Used it to study the boundedness of certain fractional differential equations with the Caputo fractional derivatives and integral equations involving the Erdélyi-Kober fractional integrals
Summary
In the study of the qualitative and quantitative properties of solutions of some fractional differential equations, many inequalities with singular kernels have been developed (for example, see [ – ] and the references therein). In these inequalities, Medveď [ ] discussed the following useful integral inequality:. We discuss the more general integral inequality with weakly singular kernel:. N) are bounded and monotonically increasing and provide a handy tool to derive the boundedness of the solutions of certain fractional differential equations and integral equations.
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