Almost Automorphic Type Solutions For Two Classes Of Neutral Differential Equations With Delay

Neutral differential equations with delay have been widely used in physics,ecology,automatic control and other fields of science and engineering,so the study of their almost automorphic type solutions has aroused the interest of many scholars.Therefore,the existence and uniqueness of almost automorphic type solutions of two classes of neutral differential equations with delay are discussed in this paper.Pseudo almost automorphic solutions of differential equations are more general than almost automorphic solutions,so pseudo almost automorphic solutions for a class of neutral differential equations with piecewise constant arguments and delay are studied in the first part of this paper.Firstly,by decreasing the order of the second-order equations and according to the continuity of solutions of the differential equations with piecewise constant arguments at the integer points,a difference equation is constructed.Then the related lemmas are obtained by studying the existence and uniqueness of the solutions of the equivalent system and using the related properties of pseudo almost automorphic functions,Cauchy-Schwarz inequality and Lebesgue control convergence theorem.Finally,the existence and uniqueness of pseudo almost automorphic solutions of the considered differential equation with delay are discussed by using the related lemmas and pseudo almost automorphic sequence solutions of the difference equation.The existence and uniqueness of asymptotically almost automorphic mild solutions for another kind of neutral differential equations with delay are studied in the second part of this paper.Firstly,under appropriate hypothesis,the related propositions including the composite functions of binary asymptotically almost automorphic functions and asymptotically almost automorphic functions are still asymptotically almost automorphic,the integrals of bounded linear operators with certain characteristics acting on asymptotically almost automorphic functions are asymptotically almost automorphic are proved by using the definition and some basic properties of asymptotically almost automorphic functions,Cauchy convergence criteria of infinite integrals and Lebesgue control convergence theorem.Combined with the related propositions,Banach fixed point theorem and the definition of mild solution,the existence and uniqueness of asymptotically almost automorphic mild solutions to this kind of differential equation with delay are discussed.