Existence of Mild Solutions for a Semilinear Integrodifferential Equation with Nonlocal Initial Conditions

and Applied Analysis 3 concrete situations. Indeed, we consider an example with a particular choice of the function b and the operator A is defined by Aw t, ξ a1 ξ ∂2 ∂ξ2 w t, ξ b1 ξ ∂ ∂ξ w t, ξ c ξ w t, ξ , 1.3 where the given coefficients a1, b1, c satisfy the usual uniform ellipticity conditions. 2. Preliminaries Most of the notations used throughout this paper are standard. So, N, Z, R, and C denote the set of natural integers and real and complex numbers, respectively, N0 N ∪ {0}, R 0,∞ and R 0 0,∞ . In this work X and Y always are complex Banach spaces with norms ‖ · ‖X and ‖ · ‖Y ; the subscript will be dropped when there is no danger of confusion. We denote the space of all bounded linear operators from X to Y by L X,Y . In the case X Y , we will write briefly L X . Let A be an operator defined in X. We will denote its domain by D A , its domain endowed with the graph norm by D A , its resolvent set by ρ A , and its spectrum by σ A C \ ρ A . As we have already mentioned C I;X is the vector space of all continuous functions f : I → X. This space is a Banach space endowed with the norm ∥f ∥ ∞ sup t∈I ∥f t ∥ X. 2.1 In the same manner, for n ∈ Nwe write C I;X for denoting the space of all functions from I to X which are n-times differentiable. Further, C∞ I;X represents the space of all infinitely differentiable functions from I to X. We denote by L1 I;X the space of all equivalent classes of Bochner-measurable functions f : I → X such that ‖f t ‖X is integrable for t ∈ I. It is well known that this space is a Banach space with the norm