Electron. J. Differential Equations, Vol. 2018 (2018), No. 27, pp. 1-15.

Composition and convolution theorems for mu-Stepanov pseudo almost periodic functions and applications to fractional integro-differential equations

Edgardo Alvarez

Abstract:
In this article we establish new convolution and composition theorems for $\mu$-Stepanov pseudo almost periodic functions. We prove that the space of vector-valued mu-Stepanov pseudo almost periodic functions is a Banach space. As an application, we prove the existence and uniqueness of mu-pseudo almost periodic mild solutions for the fractional integro-differential equation
$$ D^\alpha u(t)=Au(t)+\int_{-\infty}^t a(t-s)Au(s)\,ds+f(t,u(t)), $$
where A generates an $\alpha$-resolvent family $\{S_\alpha(t)\}_{t\geq 0}$ on a Banach space X, $a\in L^1_{\rm loc}(\mathbb{R}_+)$, $\alpha>0$, the fractional derivative is understood in the sense of Weyl and the nonlinearity f is a mu-Stepanov pseudo almost periodic function.

Submitted September 12, 2016. Published January 18, 2018.
Math Subject Classifications: 45D05, 34A12, 45N05.
Key Words: mu-Stepanov pseudo almost periodic; mild solutions,; fractional integro-differential equations; composition; convolution.

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Edgardo Alvarez
Universidad del Norte
Departamento de Matemáticas y Estadística
Barranquilla, Colombia
email: edgalp@yahoo.com, ealvareze@uninorte.edu.co

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