On Certain Singular Ordinary Differential Equations of the First Order in Banach Spaces

Abstract

We consider in the interval $(0,T)$ certain linear and nonlinear singular ordinary differential equations of the first order, where the unknown function takes values in a Banach space and 0 is a singular point. Under suitable assumptions we prove that for each of these equations there exists a unique solution of the class $C^1$ in $(0,T)$ which is continuous at 0 or bounded in a right hand neighbourhood of 0. Moreover, in the linear case there is introduced another version of assumptions which guarantees that every solution of the considered linear equations 1105 the properties stated above.