To properly characterize our system, we claim that a function (x:[sigma,tau]rightarrowmathbb{X}) is a normalized piecewise continuous function on ([sigma,tau]) if x is piecewise continuous and left continuous on ((sigma,tau]).
To describe appropriately our problems we say that a function (x: [mu,tau]rightarrow H) is a normalized piecewise continuous function on ([mu,tau]) if x is piecewise continuous and continuous on ([mu,tau]).
To describe appropriately our system (1), we say that the function u : → X is a normalized piecewise continuous function on if u is piecewise continuous and left continuous on ( σ, τ ]. We denote by P C ( , X ) the space formed by the normalized piecewise continuous functions from into X.
We say that a function x : → H is a normalized piecewise continuous function on if x is piecewise continuous and leftcontinuous on ( ν, τ ]. We denote by P C ( ; H ) the space formed by the normalized piecewisecontinuous stochastic processes from { x ( t ) : t ∈ }.
Using the fact that (U t, s))t ≥ sis a evolution family of operators and assuming the conditions on f, g and the family of operator I i, i = 1,..., m, it is not difficult see that t → Γ i (t), t ∈ [0, b] is a normalized piecewise continuous function for all i = 1,..., m.
Therefore (Z') is a normalized semigroup.
Let moreover be a piecewise continuous function, denoting the space of piecewise continuous functions with values in.
A good fit would be a piecewise approximation.
The demographic function is then constrained to be a piecewise constant function with exactly i distinct levels.
It is noted that has the following features: is a piecewise quadratic polynomial; is piecewise continuously differentiable; If is positive semidefinite, then is a piecewise convex quadratic function.
