In this paper, we investigate the existence and uniqueness of a piecewise asymptotically almost periodic mild solution to nonautonomous neutral Volterra integro-differential equations with impulsive effects in Banach space. The working tools are based on the Krasnoselskii’s fixed point theorem and semigroup theory. | Turk J Math (2017) 41: 1656 – 1672 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Piecewise asymptotically almost periodic solution of neutral Volterra integro-differential equations with impulsive effects Zhinan XIA∗ Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, Zhejiang, . China Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we investigate the existence and uniqueness of a piecewise asymptotically almost periodic mild solution to nonautonomous neutral Volterra integro-differential equations with impulsive effects in Banach space. The working tools are based on the Krasnoselskii’s fixed point theorem and semigroup theory. In order to illustrate our main results, we study the piecewise asymptotically almost periodic solution of the impulsive partial differential equations with Dirichlet conditions. Key words: Neutral Volterra integro-differential equations, impulsive effects, asymptotically almost periodicity, Krasnoselskii’s fixed point theorem 1. Introduction In this paper, we investigate the existence and uniqueness of a piecewise asymptotically almost periodic mild solution of neutral Volterra integro-differential equations with impulsive effects: ∫ t d D(t, u(t)) = A(t)D(t, u(t)) + k(t − s)g(s, u(s))ds + h(t, u(t)), t ∈ R, t ̸= ti , i ∈ Z, dt −∞ − ∆u(ti ) = u(t+ i ) − u(ti ) = γi u(ti ) + δi , () where A(t) : D ⊂ X → X are a family of closed linear operators on Banach space X , D(t, u(t)) = u(t) + f (t, u(t)), f, g, h : R × X → X are piecewise asymptotically almost periodic functions in t ∈ R uniformly − in the second variable, γi , δi are asymptotically almost periodic sequences, and u(t+ i ), u(ti ) represent the right-hand side and the left-hand side limits of u(·) at ti , respectively. There are many physical phenomena that are described by means of .
