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We introduce the concept of pseudo S-asymptotically periodic functions and study some of the qualitative properties of functions of this type. In addition, we discuss the existence of pseudo S-asymptotically periodic mild solutions for abstract neutral functional differential equations. Some applications involving ordinary and partial differential equations with delay are presented. 2010 Mathematics subject classification: primary 35B15; secondary 34K40, 34K30, 47D06. Denote by PSAP w,p (Z) thedoi:10.1017/s0004972712000950 fatcat:ygw4a6jzpfcadcnwwcbfnu4sg4