On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A 1(t)u,A 2(t)u,⋯,A ≉(t)u), (A i (t),i = 1, 2, ⋯,≉), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ≉ intoE and −A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A i(t),i = 1, 2, ⋯,≉), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.