Admissible, bounded and periodic solutions of semilinear evolution equations on the line

Abstract

We investigate semilinear evolution equations of the form $u(t)=U(t,s)u(s)+{\int }_s^{t}U(t,\xi )f(\xi ,u(\xi ))d\xi$ for $t\ge s$ and $s\in ℝ$ in Banach space $X$. Under the assumptions that the evolution family ${(U(t,s))}_{t\ge s}$ has the exponential dichotomy and the function $f:ℝ\times X\to X$ has the Carathéodory property, we show that the semilinear evolution equations on the line has a unique admissible solution, bounded solution, periodic solution when the function $f$ satisfies the condition $\varphi$-Lipschitz and there exists a periodic solution when the function $f$ satisfies the condition $\parallel f(t,x)\parallel \le \phi (t)(1+\parallel x\parallel )$ for all $x\in X$ and almost everywhere $t\in ℝ$.