ABSTRACT
This chapter deals with some open problems carried from [99, 115, 116, 117,
118] concerning the existence of solutions on L1 spaces to nonlinear boundary
value problems derived from three models. The first one deals with nonlinear
one-dimensional stationary transport equations arising in the kinetic theory
of gas where we must describe the interaction of gas molecules with solid walls
bounding the region where the gas follows. The second one, introduced by J.
L. Lebowitz and S. I. Rubinow [121] in 1974, models microbial populations by
age and cycle length formalism. The third one, introduced by M. Rotenberg
[142] in 1983, describes the growth of a cell population. These three models can
be transformed into a fixed point problem which has two types of equations.
The first type involves a nonlinear weakly compact operator on L1 spaces. The
second type deals with two nonlinear operators depending on the parameter
λ, say, ψ = A1(λ)ψ + A2(λ)ψ where A1(λ) is a weakly compact operator on
L1 spaces and A2(λ) is a (strict) contraction mapping for a large enough Reλ.
Consequently, Schauder’s (resp. Krasnosel’skii’s) fixed point theorem [149]
cannot be used in the first (resp. second) type of equation. This is essentially
due to the lack of compactness.
