On the Existence of Solutions of Differential Equations and Zeros of Operators in K-Banach Spaces

Abstract

**Please note that the full text is embargoed** ABSTRACT: The theory of existence of solutions of differential equations in a Banach space employing norm as a measure is sufficiently well known [5, 6, 8, 9]. Also utilizing this theory one can prove the existence of zeros of operators [2, 7, 8, 9, 11]. The advantage of using a generalized norm as a candidate in discussing the qualitative theory of differential equations is also known [1]. These thoughts naturally lead to the use of cone-valued norms as a measure since this approach unifies the existing theories as well as offers a more flexible mechanism for applications. In this paper, we wish to work in such a general setting and consequently we develop the appropriate theory of Banach spaces whose norm is cone-valued. Using this as a vehicle we then prove an existence theorem for differential equations in K-Banach spaces which is then utilized to prove the existence of zeros of nonlinear operators.