LOCAL MILD SOLUTIONS FOR SOME NONLINEAR INTEGRODIFFERENTIAL EQUATIONS

摘要

where ?A is the infinitesimal generator of an analytic semigroup (T(t)) t≥0 of linear operators on X, x 0 ∈ X, T > 0, 0 < q < 1. Here, f : [0,T] × X × X ?→ X is a nonlinear function, and g : C([0,T],X) ?→ X, g(x 0 ) = 0, k ∈ C ? [0,T] × [0,T] ? , K Q = sup t∈[0,Q] R t 0 |k(t,τ)|dτ < ∞ for Q ∈ (0,T], and lim Q→0 K Q = 0. In this paper, the derivative D q is understood in the Riemann Liouville sense. The use of nonlocal condition is not a new fact. Indeed, in several recent papers, its presence has been pointed out (see for instance [3], [4], [11], [12]). As indicated by many authors, the fact to utilize nonlocal conditions has its interest in physics because it allows the solution to have a better effect than the solution of the classical Cauchy problem with initial datum x(0) = x 0 . For instance, the authors used g(x) = P p i=1 c i x(t i ) where c i , i = 1,2,..., p are given constants and 0 < t 1 < t 2 < ... < t p ≤ T to describe the diffusion phenomenon of a small amount in a transparent tube. In this case, the Cauchy problem permits the additional measurements at t i , i = 1,2,..., p.