## Banach Journal of Mathematical Analysis

### Pseudo Asymptotic Solutions of Fractional Order Semilinear Equations

#### 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.

#### Article information

Source
Banach J. Math. Anal., Volume 7, Number 2 (2013), 42 -52 .

Dates
First available in Project Euclid: 20 March 2013

https://projecteuclid.org/euclid.bjma/1363784222

Digital Object Identifier
doi:10.15352/bjma/1363784222

Mathematical Reviews number (MathSciNet)
MR3039938

Zentralblatt MATH identifier
1275.47092

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

#### Citation

Alvarez-Pardo , Edgardo; Lizama , Carlos. Pseudo Asymptotic Solutions of Fractional Order Semilinear Equations. Banach J. Math. Anal. 7 (2013), no. 2, 42 --52. doi:10.15352/bjma/1363784222. https://projecteuclid.org/euclid.bjma/1363784222

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