A characterization of well-posedness for abstract Cauchy problems with finite delay

作者：Lizama Carlos^*; Poblete Felipe

来源：Journal of Mathematical Analysis and Applications, 2018, 457(1): 410-435.

DOI：10.1016/j.jmaa.2017.08.023

• C-0-semigroups
• Finite delay
• Cauchy problem
• Functional equations
• Well posedness

Let A be a closed operator defined on a Banach space X and F be a bounded operator denned on a appropriate phase space. In this paper, we characterize the well-posedness of the first order abstract Cauchy problem with finite delay,
{u'(t) = Au(t) + Fu (t) , t >0;
u(0) = x;
u(t) = phi(t),-r< t< 0
solely in terms of a strongly continuous one-parameter family {G(t)}(t >= 0) of bounded linear operators that satisfy the functional equation
G(t + s)x = G(t)G(s)x + integral(0)(-r) G(t + m)[SG(s +.)x](m)dm
for all t, s >0, x is an element of X. In case F = 0 this property reduces to the characterization of well-posedness for the first order abstract Cauchy problem in terms of the functional equation that satisfy the C-0-semigroup generated by A.

• 出版日期2018-1-1