On the Zeros of Monotone Operators of Retarded Type in a Banach Space
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Recent interest in the Cauchy problem for differential equations in a Banach space  has stimulated a similar interest for differential equations of a retarded type in a Banach space. The difficulty in imposing assumptions due to a different range and domain space has been overcome in  where existence of solutions is established using a monotoni- city type condition in terms of norm and weaker forms of differential inequalities. The theory of existence of solutions of differential equations has been used in [1,2,5,6,7] to obtain existence of zeros and fixed points for nonlinear operators from E into E. In this paper we consider the existence of zeros of operators which map into E but whose domain is the space of continuous functions on some real interval into E. Since such operators generate differential equations of retarded type, we call such operators retarded type. We employ generalized inner product [1,6] and adapt the methods analogous to [1,2,7]. Our results generalize those of . Furthermore, we obtain for the first time, existence of zeros of operators of retarded type.