On the solutions for a nonlinear boundary value problem modeling a proliferating cell population with inherited cycle length

This paper deals with existence results for a nonlinear boundary value problem derived from a model introduced by Lebowitz and Rubinow (1974) describing a proliferating cell population. Cells are distinguished by age aa and cycle length ll. The cycle length is viewed as an inherited property determined at birth. The boundary condition models the process of cell division of mother cells and the inheritance of cycle length by daughter cells. In our framework, daughter cells and mother cells are related by a general reproduction rule which covers all known biological ones. In this work, the cycle length ll is allowed to be infinite, that is, l∈[l1,+∞)l∈[l1,+∞). This hypothesis introduces some mathematical difficulties which are overcome by using domination arguments (in the lattice sense) and recent fixed point theorems involving continuous weakly compact operators on non reflexive Banach spaces.