## On the Existence of Zeros of Lyapunov-Monotone Operators

##### Abstract

Consider a nonlinear operator T from a Banach space into itself.
The study of the existence of zeros of T plays an important role in yielding fixed points of nonlinear operators. The operator T has a zero if and only if the initial value problem [see pdf for notation],has a constant solution. If T is a monotone operator then
(1.1) has a unique solution [see
pdf for notation] defined on [see pdf for notation] and the solution operator [see pdf for notation] is nonexpansive for all [see pdf for notation]. Imposing further assumptions one can show that U(t) must have a common fixed point and that fixed point is the desired zero of T. Thus one can use the theory of differential equations and some known fixed point theorems on the solution operator to obtain the existence of zeros of T. See for example [1,3,4, 5,9,11]. In this paper, we introduce the notion of Lyapunov-monotone operators in terms of several Lyapunov-like functions and utilizing certain results in abstract cones that are recently proved [6], we establish the existence of zeros of such operators. This leads us to work with generalized Banach spaces which offer a flexible technic. The results obtained are so general
that when we employ as a candidate
a generalized norm, we still
get results more general than the
known ones [8,9].