Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations - Slyusarchuk, V. Yu. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
Let E be a finite-dimensional Banach space, let C^0(R; E) be a Banach space of functions continuous and bounded on R and taking values in E; let K: C^0(R ,E) → C^0(R, E) be a c-continuous bounded mapping, let A: E → E be a linear continuous mapping, and let h ∈ C^0(R, E). We establish conditions for the existence of bounded solutions of the nonlinear equation dx(t)dt + (Kx)(t)Ax(t) = h(t), t ∈ R.