Electron. J. Diff. Equ., Vol. 2015 (2015), No. 168, pp. 1-16.

Existence and uniqueness of mild solutions for fractional semilinear differential equations

Bambang Hendriya Guswanto, Takashi Suzuki

In this article, we study the existence and uniqueness of a local mild solution for a class of semilinear differential equations involving the Caputo fractional time derivative of order $\alpha$ $(0<\alpha<1)$ and, in the linear part, a sectorial linear operator A. We put some conditions on a nonlinear term f and an initial data $u_0$ in terms of the fractional power of A. By applying Banach's Fixed Point Theorem, we obtain a unique local mild solution with smoothing effects, estimates, and a behavior at t close to 0. An example as an application of our results is also given.

Submitted April 2, 2015. Published June 18, 2015.
Math Subject Classifications: 34A08, 34A12.
Key Words: Fractional semilinear differential equation; sectorial operator; Caputo fractional derivative; fractional power; mild solution.

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Bambang Hendriya Guswanto
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Jenderal Soedirman University (UNSOED)
Purwokerto, Indonesia
email: bambanghg_unsoed@yahoo.com; bambang.guswanto@unsoed.ac.id
Takashi Suzuki
Division of Mathematical Science
Department of Systems Innovation
Graduate School of Engineering Science
Osaka University, Osaka, Japan
email: suzuki@sigmath.es.osaka-u.ac.jp

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