Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let $\mathrm{m}od\left(f\right)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x\left(t\right)$ of the system $$\dot{x}=f(t,x),t\in ℝ,x\in D\subset {ℝ}^n$$ is irregular with respect to $L_2$ (or partially irregular) if $(\mathrm{m}od\left(x\right)+L_1)\cap L_2=\left\{0\right\}$. Suppose that $f(t,x)=A\left(t\right)x+X(t,x),$ where $A\left(t\right)$ is an almost periodic $(n\times n)$-matrix and $\mathrm{m}od\left(A\right)\cap \mathrm{m}od\left(X\right)=\left\{0\right\}.$ We consider the existence problem for almost periodic irregular with respect to $\mathrm{m}od\left(A\right)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions...

In this work, we present a new concept of Stepanov weighted pseudo almost periodic and automorphic functions which is more generale than the classical one, and we obtain a new existence result of μ-pseudo almost periodic and μ-pseudo almost automorphic mild solutions for some nonautonomous evolution equations with Stepanov μ-pseudo almost periodic terms. An example is shown to illustrate our results.

In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic $PC$-mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic $PC$-mild solution of a two-dimensional impulsive fractional...