Solving fractional evolution problem in Colombeau algebra by mean generalized fixed point
محورهای موضوعی : Ordinary differential equationsM. 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
کلید واژه: Colombeau algebra, locally convexe space, generalized semigroup, generalized fixed point,
چکیده مقاله :
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.
[1] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Inter. J. Math. & Math. Sci. 29 (9) (2002), 531-536.
[2] J. F. Colombeau, Elementary Introduction to New Generalized Function, North Holland, Amsterdam, 1985.
[3] J. F. Colombeau, New Generalized Function and Multiplication of Distribution, North Holland, Amsterdam, 1984.
[4] M. Grosser, M. Kunzinger, M. Oberguggenberger, R. Steinbauer, Geometric Theory of Generalized Functions with Applications to General Relativity, Mathematics and its Applications 537, Dordrecht, 2001.
[5] R. Hermann, M. Oberguggenberger, Ordinary differential equations and generalized functions, in: Nonlinear Theory of Generalized Functions, Chapman & Hall, 1999.
[6] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V, Netherlands, 2006.
[7] J. A. Marti, Fixed points in algebras of generalized functions and applications, HAL Id: hal-01231272.
[8] R. Metzler, J. Klafter, The random walks guide to anomalous diffusion: a fractional dynamics approach, Physics Reports. 339 (2000), 1-77.
[9] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics, 1992.
[10] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Bull. Amer. Math. Soc. (N.S.) 12 (1985), —.
