In this paper, we establish some results regarding the existence of solution on L ₁ spaces to a nonlinear boundary value problem originally proposed by Rotenberg (J. Theo. Biol. 1983; 103:181–199) to model the growth of cell population. Our strategy consists in establishing new variants of fixed point theorems in general Banach spaces. These topological results can be used to resolve some open problems posed by Latrach and Jeribi (Nonlinear Anal. T.M.A. 1999; 36:843–862).