×

Perturbation theory for dual semigroups. III: Nonlinear Lipschitz continuous perturbations in the sunreflexive. (English) Zbl 0675.47036

Volterra integrodifferential equations in Banach spaces and applications, Proc. Conf., Trento/Italy 1987, Pitman Res. Notes Math. Ser. 190, 67-89 (1989).
[For the entire collection see Zbl 0664.00018.]
This paper is a continuation of earlier work of the authors [Math. Ann. 277, 709–725 (1987; Zbl 0634.47039)] and [Proc. R. Soc. Edinb., Sect. A 109, No. 1/2, 145–172 (1988; Zbl 0661.47015)]. They have developed a theory of dual semigroups in the case where the underlying Banach space is not reflexive. The importance lies in the scope of applications for semigroups on \(C_ 0(X)\) (space of continuous functions on the locally compact space \(X\)) or \(L^ 1(X,\mu)\).
Here the authors treat the problem of existence, uniqueness, and asymptotic behaviour of solutions of the semilinear integral equation \[ u(t)=T(t)x+\int^{t}_{0}T_ t^{\odot '}(t-s)F(u(s))\,ds \] where \(F\) is a globally Lipschitz-continuous function from \(X\) into \(X^{\odot_{'}}\) and \((T_ 0(t))\) is the given strongly continuous semigroup for which \(X\) is its “sun dual”.
As pointed out in the last remark many delay equations and many age dependent population equations are covered by this type of integral equation.

MSC:

47D03 Groups and semigroups of linear operators
47H20 Semigroups of nonlinear operators
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
34G20 Nonlinear differential equations in abstract spaces