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

Salem, Hussein A. H.; El-Sayed, A. M. A.; Moustafa, O. L.

Stud. Sci. Math. Hung. 42, No. 2, 115-130 (2005).

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 \(\alpha > 0\).

Reviewer: Kai Diethelm (Braunschweig)

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

Keywords:

fractional calculus; Cauchy problem; Banach space; abstract fractinal integral equations