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Weak solution for fractional order integral equations in reflexive Banach spaces. (English) Zbl 1111.26011
Summary: We define the fractional order Pettis-integral operator in reflexive Banach spaces and we investigate the properties of such operator. A fixed point theorem is used to establish an existence result for the nonlinear Pettis-fractional order integral equation of the following type $x(t)=g(t)+\lambda \, I^{\alpha }f\bigl (t,x(t)\bigr )\,, \qquad t\in [0,1]\,,\;\;0<\alpha <1\,.$ Moreover, the existence of a solution for the Cauchy problem $\frac {d\!x}{d\!t} = f\bigl (t,\operatorname {D}^{\beta }\!x(t)\bigr ), \qquad t\in [0,1]\,,\^^M\;0< \beta <1\,,\;\;x(0) = x_0\,,$ is proved.

##### MSC:
 26A33 Fractional derivatives and integrals 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 47G10 Integral operators
##### Keywords:
fractional calculus; pseudo solution; Cauchy problem
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##### References:
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