The existence of bounded solutions for abstract ODE's \[ x'(t)=A(t)x(t)+f(t),\quad tı\mathbf R \] \[ x''(t)+bx'(t)=A(t)x(t)+f(t),\quad tı\mathbf R \] and for abstract boundary problems for PDE \[ \begin{cases} u'_t(t,x)-u''_{xx}(t)=A(t)u(t,x)+g(t,x),\quad tı\mathbf R, xı[0,i] u(t,0)=u(t,x)=0,\quad tı\mathbf R; \end{cases} \] \[ \begin{cases} u''_(tt)(t,x)+bu'_t(t,x)-u''_{xx}(t,x)=A(t)u(t,x)+g(t,x),\quad tı\mathbf R, xı[0,i] u(t,0)=u(t,i)=0,\quad tı\mathbf R \end{cases} \] is considered. Here $A$ is a periodic operator valued function, $f$ is a bounded on $\mathbf R$ Banach valued function and $b\in\mathbf R$.