We begin by considering various
kinds of nonlinear operators in a Banach lattice X, i.e., a Banach space which has
a compatible lattice structure. With adequate definitions we are able to
develop a theory parallel to the theory of nonlinear equations of evolutions in a
general Banach space, as carried out by Komura, Kato, Browder and others.
Existence and uniqueness theorems about solutions of the equation of evolution
du(t)∕dt = −Au(t) are developed under conditions on the space X and the
operator A. Given a solution u(t) to du(i)∕dt = −Au(t) with initial condition
u(0) = v, where v lies in the domain of A, a semi-group U(t) is defined by
U(t)v = u(t),t ≧ 0.
