Existence of Solutions and Controllability of Nonlinear Integrodifferential Systems in Banach Spaces

where 0 ≤ t0 < t1 < ··· < tp ≤ a, a > 0, −A is the infinitesimal generator of aC0-semigroup in a Banach space X , u0 ∈ X , and f : [0,a]×X → X , g : [0,a]p ×X → X are given functions. Subsequently he has investigated the same type of problem for different kinds of evolution equations in Banach spaces [10, 11, 12, 13, 14]. Ntouyas and Tsamatos [31] have established the global existence of solutions of semilinear evolution equations with nonlocal conditions. Balachandran [1], Balachandran and Ilamaran [6], Balachandran and Chandrasekaran [3], Dauer and Balachandran [17], and Balachandran et al. [7] have studied the nonlocal Cauchy problem for various classes of integrodifferential equations. Physical motivation for this kind of problem is given in [18, 25]. It is well known [36] (when A= 0 and g = 0) that only the continuity of f is not sufficient to assure local existence of solutions, even when X is a Hilbert space. Therefore, one has to restrict either the function f or the semigroup operator. Usually restrictions on f are imposed, as either f should satisfy the local Lipschitz condition, or be monotonic, or be completely continuous. Here we assume that the nonlinear terms satisfy the boundedness condition.