| This paper mainly consists of two parts. First, the author introduces thecomposition theory of pseudo almost automorphic functions and weightedpseudo almost automorphic functions. Thus, if function h:→Xis a kind ofalmost automorphic kind functions, function f:×X→Yis also in this kind, sowhether the composition function f (, h())is in this kind? In this part, firstlywe introduce three kinds of composition theorems of pseudo almostautomorphic functions, and make a comparison of them according to theirconditions. Then we give two kinds of composition theorems of weightedpseudo almost automorphic functions, and we make the introduction besystematic through the comparison of them.In the second part, on base of the first part, the author gives two new kindsof composition theorems of weighted pseudo almost automorphic functions. Theidea of these two new theorems is benefited respectively from the study of thecomposition theorems of pseudo almost automorphic functions by Philippe.Cand the study of the composition theorems of weighted pseudo almostautomorphic functions by Ding.H.S. The former weakens the condition fromboundedness to compactness through some knowledge of functional analysis,which helps me give the theorem. The latter weakens the conditions of functionitself instead of strengthen of the weighted function, which helps me give thetheorem that gives a simple represent by assuming the weighted function in asmaller space. Then, under the study of the second part, we make a study of thecontinuity of the composite operator. We respectively give necessary conditionsof the continuity in one point and in the hole space of this operator, and that isinspired by the study of the operator of pseudo almost automorphic functions byPhilippe.C. Finally, we discuss the existence of weighted pseudo almostautomorphic solutions of the equationx ’(t)=Ax (t)+f (t, x (t))t∈and give the conditions under the composite theory in this paper. |
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