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On the inequalities of integro-differential fractional equations. (English) Zbl 1033.45007

The authors present integro-differential inequalities results of equations \[ u^{(\alpha)}(t)=g(t,u(t))+\int_{t_{0}}^{t}H(t,s(t),u(s))\,ds, \quad t\in J \;\alpha\in \mathbb R, \;\;0<\alpha\leq 1, \] with the initial condition \[ u^{(\alpha-1)}(t_{0})=u_{0}, \] where \(\mathbb R\) is the set of real numbers, \(J=[t_{0},t_{0}+a], \;a>0, \;g\in C(J\times \mathbb R^{n},\mathbb R^{n})\) and \(H\in C(J\times J\mathbb R^{n},\mathbb R^{n})\). \(\mathbb R^{n}\) denotes the real \(n\)-dimensional Euclidean space.

MSC:

45J05 Integro-ordinary differential equations
26D10 Inequalities involving derivatives and differential and integral operators
45G10 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
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