Functional differential equations with maxima, via step by step contraction principle

T. A. Burton presented in some examples of integral equations a notion of progressive contractions on C([a,\infty \lbrack). In 2019, I. A. Rus formalized this notion (I. A. Rus, Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle, Advances in the Theory of Nonlinear Analysis and its Applications, 3 (2019) No. 3, 111–120), put “step by step” instead of “progressive” in this notion, and give some variant of step by step contraction principle in the case of operators with Volterra property on C([a,b],\mathbb{B)} and C([a,\infty \lbrack,\mathbb{B}) where \mathbb{B} is a Banach space. In this paper we use the abstract result given by I. A. Rus, to study some classes of functional differential equations with maxima.