## A note on the fractional calculus in Banach spaces.(English)Zbl 1086.45004

Let $$E$$ be a Banach space (in some of the results additionally required to be reflexive). The authors consider the abstract fractional integral equation $$x(t) = g(t) + \lambda I^\alpha [f(\cdot, x(\cdot))](t)$$ with unknown solution $$x:[0,1] \to E$$ and given functions $$f : [0,1]\times E \to E$$ and $$g :[0,1] \to E$$. Existence results for the solution of this equation are given, where $$I^\alpha$$ is the fractional integral of weak Riemann, Pettis or Bochner type of order $$\alpha > 0$$.

### MSC:

 45N05 Abstract integral equations, integral equations in abstract spaces 26A33 Fractional derivatives and integrals 47G10 Integral operators 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 45G10 Other nonlinear integral equations
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