The Existence And Uniqueness Of Almost Periodic Type Solutions For Several Classes Of Differential Equations

As well known, when the almost periodic type solutions of differential equations with piecewise constant argument are discussed, the almost periodic type sequence solutions of relevant difference equations are usually used. Especially in recent years, more and more mathematical workers began to study the existence and uniqueness of almost periodic type sequence solutions for difference equations, and the existence and uniqueness of almost periodic type solutions for some differential equations by joining difference equations and differential equations together. Moreover, the existence of various solutions for certain equations is discussed by using fixed point theory, and great progress is made.The existence and uniqueness of asympototically almost periodic solutions for two types differential equations are mainly discussed in this paper.Specific contents are as follows:A difference equation is constructed through the continuity of solutions of differential equations in the first part after this, the existence of asympototically almost periodic solutions for a class of differential equation with piecewise constant is discussed using asympototically almost periodic sequence solutions of relevant difference equations combinating exponential dichotomy.The exponential dichotomy theory and fixed point theory are used in the second part, and established the full conditions that assured a class of differential equations with limited delay exist the asympototically almost periodic solutionsSome of the results in the paper are promotion of the well-known results, thus the relevant conclusions will be more extensively applied. The others are obtained by changing the conditions of previous results, these methods can provide some reference for other problems.