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

• Main Authors: Edgardo Alvarez, Carlos Lizama
• Format: Article
• Language: English
• Published: Texas State University 2016-10-01
• Series: Electronic Journal of Differential Equations
• Subjects: Weyl-like fractional difference; fractional difference equation; weighted pseudo almost automorphic sequence; alpha-resolvent sequences of operators
• Online Access: http://ejde.math.txstate.edu/Volumes/2016/270/abstr.html

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.