Almost automorphic mild solutions to fractional partial difference-differential equations

Abstract

We study existence and uniqueness of almost automorphic solutions for nonlinear partial difference-differential equations modeled in abstract form as(*)

for where is the generator of a -semigroup defined on a Banach space , denote fractional difference in Weyl-like sense and satisfies Lipchitz conditions of global and local type. We introduce the notion of -resolvent sequence and we prove that a mild solution of corresponds to a fixed point of

We show that such mild solution is strong in case of the forcing term belongs to an appropriate weighted Lebesgue space of sequences. Application to a model of population of cells is given.

Keywords:

• Weyl-like fractional difference
• almost automorphic sequence
• -semigroups
• fractional difference equations
• mild solution

AMS Subject Classifications:

• 35R11
• 34A08
• 43A60
• 47D06

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

C. Lizama has been partially supported by DICYT, Universidad de Santiago de Chile; Project CONICYT-PIA ACT1112 Stochastic Analysis Research Network; FONDECYT 1140258 and Ministerio de Educación CEI Iberus (Spain). L. Abadias has been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragón.