Pseudo Almost Periodic Type And Pseudo Almost Automorphic Type Functions And Their Applications

The notion of pseudo almost periodic functions was introduced by Chuanyi Zhang in his doctoral dissertation in1992. These functions are a natural generalization of Bohr al-most periodic functions. A pseudo almost periodic function can be decomposed uniquely as the sum of an almost periodic function and an ergodic perturbation function. Com-pared with Bohr almost periodic functions, these functions can represent more abundant dynamical behavior, so they have been widely used in the qualitative theory of various equations derived from neuron networks, biomathematics, heat conduction model and so on.In recent years, pseudo almost periodic functions were generalized from different perspectives and the generalized ones mainly include weighted pseudo almost periodic functions, Stepanov-like pseudo almost automorphic functions, weighted Stepanov-like pseudo almost automorphic functions etc. So far these pseudo almost periodic type and pseudo almost automorphic type functions have been applied in some abstract differential equations including nonlinear heat equation and semilinear evolution equation. This the-sis mainly focus on some basic properties of the pseudo almost periodic type functions mentioned above and their applications as well. Moreover, we introduce and study a new class of pseudo almost periodic functions of several variables. The specific content of this thesis is as follows:Firstly, we study the weighted pseudo almost periodic functions. A sufficient con-dition for translation invariance of weighted ergodic perturbation functions is given and some applications of the translation invariance in the theory of weighted pseudo almost periodic function space are presented. Under some suitable weights, the existence, trans-lation invariance and ergodicity of the mean of an almost periodic function are generalized to the weighted mean of the function. We also introduce the notion of weighted ergodic zero set and use it to generalize the definition of equivalence of different weighted pseudo almost periodic function spaces.Secondly, we study the weighted Stepanov-like pseudo almost automorphic func-tions. We found that when the weighted ergodic perturbation function space is translation invariant, the closure of the range of each weighted Stepanov pseudo almost automorphic function almost contains the range of its almost automorphic component. Then we can show that if a weighted Stepanov-like pseudo almost automorphic function is uniformly continuous with respect to the parameter, then so is its almost automorphic componen-t. Utilizing this result, we improve the composition theorem of weighted Stepanov-like pseudo almost automorphic functions by reducing some redundancy condition and certain compactness assumption on the range of a Stepanov-like almost automorphic function. To illustrate these results, the existence and uniqueness of weighted Stepanov-like pseudo almost automorphic solutions to two classes of Volterra integral equations are studied.Then the Stepanov-like pseudo almost automorphism of the solution of the monoton-ic evolution equation u’(t)+A(t)u(t)=f(t) is studied, where A(t) is a monotonic nonlinear operator and is uniformly continuous and pseudo almost automorphic in t, f is Stepanov pseudo almost automorphic. Replacing A(·) and f(·) with their almost automorphic com-ponents respectively, then the new equation is called the almost automorphic component of the original equation. To show Stepanov-like pseudo almost automorphism, we use the following new method:under suitable conditions, we firstly obtain the existence of a unique global solution of the original equation, and prove its almost automorphic com-ponent equation has a unique almost automorphic type solution at the same time; then utilizing the monotonicity of A(t) to show that the difference of the two solutions above is a Stepanov-like ergodic perturbation function.Finally, analogous to multivariate almost periodic functions, we introduce multivari-ate pseudo almost periodic functions and study their basic properties. These properties include that the closure of the range of a multivariate pseudo almost periodic function contains the range of its almost periodic component and that the space of multivariate pseudo almost periodic functions is completeness and so on. Additionally, the uniformly multivariate pseudo almost periodic functions are also defined. As an application, we s-tudied the pseudo almost periodicity of the weak solution of a semilinear elliptic equation-△u+∑jn=1cj(?)ju+f(x, u)=h(x) on Rn...