Solving fractional evolution problem in Colombeau algebra by mean generalized fixed point

Subject Areas : Ordinary differential equations

M. Elomari ¹ , S. Melliani ² , A. Taqbibt ³ , S. Chadli ⁴

1 - Department of Mathematics, Faculty of Sciences and Technics, Beni-Mellal, Morocco
2 - Department of Mathematics, Faculty of Sciences and Technics, Beni-Mellal, Morocco
3 - Department of Mathematics, Faculty of Sciences and Technics, Beni-Mellal, Morocco
4 - Department of Mathematics, Faculty of Sciences and Technics, Beni-Mellal, Morocco

Received: 2018-07-13 Accepted : 2018-09-29 Published : 2019-02-01

Keywords: Colombeau algebra, locally convexe space, generalized semigroup, generalized fixed point,

Abstract :

‎The present paper is devoted to the existence and uniqueness result of the fractional evolution equation $D^{q}_c u(t)=g(t,u(t))=Au(t)+f(t)$‎ ‎for the real $q\in (0,1)$ with the initial value $u(0)=u_{0}\in\tilde{\R}$‎, ‎where $\tilde{\R}$ is the set of all generalized real numbers and $A$ is an operator defined from $\mathcal G$ into itself‎. Here the Caputo fractional derivative $D^{q}_c$ is used instead of the usual derivative. The introduction of locally convex spaces is to use their topology in order to define generalized semigroups and generalized fixed points, then to show our requested result.

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