Malaya Journal of Matematik, Vol. 7, No. 1, 27-33, 2019 https://doi.org/10.26637/MJM0701/0006 Existence of mild solutions to partial neutral differential equations with non-instantaneous impulses R. Poongodi 1 , V. T. Suveetha 2 and S. Dhanalakshmi 3 Abstract In this article, we study the existence of PC -mild solutions for the initial value problems for a class of semilinear neutral equations. These equations have non-instantaneous impulses in Banach space and the corresponding solution semigroup is noncompact. We assume that the nonlinear terms satisfies certain local growth condition and a noncompactness measure condition. Also we assume the non-instantaneous impulsive functions satisfy some Lipschitz conditions. Keywords Mild solutions, Non-instantaneous impulse, Noncompactness, Neutral systems. AMS Subject Classification 54B05. 1 Research Scholar, Department of Mathematics, Kongunadu Arts and Science College, Coimbatore, Tamil Nadu, India. 2, 3 Department of Mathematics, Kongunadu Arts and Science College, Coimbatore, Tamil Nadu, India. *Corresponding author: 1 pookasc@gmail.com 2 suveetha94@gmail.com Article History: Received 21 July 2018; Accepted 26 December 2018 c 2019 MJM. Contents 1 Introduction ........................................ 27 2 Preliminaries ....................................... 28 3 Main Results ........................................ 29 References ......................................... 32 1. Introduction In this article, we study the existence of PC -mild so- lutions for a class of partial neutral functional differential equations with non-instantaneous impulses described by the form d dt  x(t ) − g(t , x(t ))  = −Ax(t )+ f (t , x(t )), t ∈∪ n k=0 (s k , t k+1 ], x(t )= γ k (t , x(t )), t ∈∪ n k=1 (t k , s k ], x (0)= x 0 , (1.1) where A : D(A) ⊂ E → E is a closed linear operator, −A is the infinitesimal generator of a strong continuous semigroup ( C 0 -semigroup) U (t )(t ≥ 0) on a Banach space E , 0 < t 1 < t 2 < ··· < t n < t n+1 := a, a > 0 is a constant, s 0 := 0 and s k ∈ (t k , t k+1 ) for each k = 1, 2, ··· , n, f , g : [0, a] × E → E are ap- propriate functions, γ k : (t k , s k ] × E → E is non-instantaneous impulsive neutral function for all k = 1, 2, ··· , n, x 0 ∈ E . In mathematical models for both physical sciences and social sciences impulsive differential equations have became more important in recent years. There are several process and phenomena in the real world, which are subjected during their development to the short-term external influences. At definite points in time, many dynamic phenomena experience unfore- seen instantaneous, quick healing exhibited by a jump in their states. Their duration is negligible compared with the total duration of the studies phenomena and process. Therefore, it can be assumed that these external effects are “instantaneous”. For more facts on the results and applications of impulsive differential systems, one can refer to the books of Laksmikan- tham [15] the papers [1, 2, 4, 5-7, 10, 12, 17, 18, 22, 24, 25] and the references cited therein. Neutral Differential Equations arise in many areas of sci- ence and engineering have received much attention in the last ten years. These models turned out to be very serviceable in the situation where the system depends not only on the present states but also on the past states. See the monograph of [11,