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2013 Pseudo Asymptotic Solutions of Fractional Order Semilinear Equations
Carlos Lizama , Edgardo Alvarez-Pardo
Banach J. Math. Anal. 7(2): 42-52 (2013). DOI: 10.15352/bjma/1363784222

Abstract

Using a generalization of the semigroup theory of linear operators, we prove existence and uniqueness of mild solutions for the semilinear fractional order differential equation $${D}^{\alpha+1}_t u(t) + \mu {D}_t^{\beta} u(t) - Au(t) = f(t,u(t)), t\in (0,\infty), \alpha \in (0,\infty), \alpha \leq \beta \leq 1, \, \mu \geq 0, $$ with the property that the solution can be written as $u=f+h$ where $f$ belongs to the space of periodic (resp. almost periodic, compact almost automorphic, almost automorphic) functions and $h$ belongs to the space $ P_0(\mathbb{R}_{+},X):= \{ \phi\in BC(\mathbb{R}_{+},X) \, :\,\, \lim_{T \to \infty}\frac{1}{T} \int_{0}^{T}||\phi(s)||ds=0 \}$. Moreover, this decomposition is unique.

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Carlos Lizama . Edgardo Alvarez-Pardo . "Pseudo Asymptotic Solutions of Fractional Order Semilinear Equations." Banach J. Math. Anal. 7 (2) 42 - 52, 2013. https://doi.org/10.15352/bjma/1363784222

Information

Published: 2013
First available in Project Euclid: 20 March 2013

zbMATH: 1275.47092
MathSciNet: MR3039938
Digital Object Identifier: 10.15352/bjma/1363784222

Subjects:
Primary: 47D06
Secondary: 34A08 , 35R11 , 45N05

Keywords: generalized semigroup theory , pseudo asymptotic solutions , sectorial operators , two-term time fractional derivative

Rights: Copyright © 2013 Tusi Mathematical Research Group

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Vol.7 • No. 2 • 2013
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