Electron. J. Differential Equations, Vol. 2016 (2016), No. 270, pp. 1-12.

Weighted pseudo almost automorphic and S-asymptotically omega-periodic solutions to fractional difference-differential equations

Edgardo Alvarez, Carlos Lizama

Abstract:
We study weighted pseudo almost automorphic solutions for the nonlinear fractional difference equation
$$ \Delta^{\alpha}u(n)=Au(n+1)+f(n, u(n)),\quad n\in \mathbb{Z}, $$
for $0<\alpha \leq 1$, where A is the generator of an $\alpha$-resolvent sequence $\{S_{\alpha}(n)\}_{n\in\mathbb{N}_0}$ in $\mathcal{B}(X)$. We prove the existence and uniqueness of a weighted pseudo almost automorphic solution assuming that f(.,.) is weighted almost automorphic in the first variable and satisfies a Lipschitz (local and global) type condition in the second variable. An analogous result is also proved for $\mathcal{S}$-asymptotically $\omega$-periodic solutions.

Submitted July 5, 2016. Published October 7, 2016.
Math Subject Classifications: 32N05, 65Q10, 47B39.
Key Words: Weyl-like fractional difference; fractional difference equation; weighted pseudo almost automorphic sequence; alpha-resolvent sequences of operators

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Edgardo Alvarez
Universidad del Norte
Departamento de Matemáticas y Estadística
Barranquilla, Colombia
email: ealvareze@uninorte.edu.co
Carlos Lizama
Universidad de Santiago de Chile
Facultad de Ciencia
Departamento de Matemática y Ciencia de la Computacióon
Las Sophoras 173, Estación Central
Santiago, Chile
email: carlos.lizama@usach.cl

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