Communications on Applied Nonlinear Analysis Volume 25(2018), Number 3, 52 - 76 On the Generalized Bi-f -harmonic Map Equations on Singly Warped Product Manifolds Hadj Mazouzi University of Mostaganem (UMAB) Department of Mathematics B.P.227,27000, Mostaganem, Algeria hadjmazouzi@yahoo.fr Hichem El Hendi University of Bechar Department of Mathematics 08000, Bechar, Algeria elhendi.h@hotmail.fr Lakehal Belarbi University of Mostaganem (UMAB) Department of Mathematics Laboratory of Pure and Applied Mathematics B.P.227,27000, Mostaganem, Algeria lakehalbelarbi@gmail.com Communicated by Ram U Verma (Received May 26, 2018; Revised Version Accepted June 13, 2018) www.internationalpubls.com Abstract In this paper, we present some new properties for bi-f -harmonic map between gener- alized warped product manifolds. AMS Subject Classification: 53A45, 53C20 Key words: Harmonic map, Bi-f -harmonic map, Bi-harmonic map, Generalized warped product manifolds. 1 Introduction Bi-harmonic maps are critical points of bi-energy functional defined one the space of smooth maps between Riemannian manifolds, introduced by Eells and Sampson in 1964, with is a 52