LINEAR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS ON A BANACH SPACE

Abstract

This chapter focuses on linear neutral functional differential equations on a Banach space. It presents a linear autonomous neutral differential equation defined on a real Banach space X by the following relations: d/dt[x(t) − ∑Bjx(t − hj)] = A × (t) + ∑Ajx (t − hj) if t ≥ 0 and by x(t) = φ(t) if t ∈ [−h, 0]. Here 0 < h1 < h2 < … hm = h are constants, A: X → X is a closed unbounded operator with dense domain, D(A), on X that generates a semi-group of class C0, Bj, 1 < j < m are bounded endomorphisms on X such that range (Bj) ⊂ D(A) for each j, {Aj}, 1 < j < m are bounded endomorphisms on X and φ: [−h,0] → X is a continuous mapping. The chapter also discusses the Banach space of continuous mappings.