## Abstract

In the investigation of accretive operators in Banach spaces X, the existence of zeros plays an important role, since it yields surjectivity results as well as fixed point theorems for operators S such that I-S is accretive. Let D⊂X and T: D→X an operator such that the initial value problems

(1) u′(t)=-Tu(t), u(0)=x εD are solvable. Then T has a zero iff (1) has a constant solution for some xεD. Under certain assumptions on D and T it is possible to show that (1) has a unique solution u(t,x) on [0,∞), for every xεD. In this case, define U(t): D→D by U(t)x=u(t,x). If T is accretive it turns out that U(t) is nonexpansive for every t≥0. This fact constitutes the basis for several authors concerned with this subject. They proceed with assumptions on D and X ensuring either that the U(t) must have a common fixed point x_{o} or that U(_{p}) has a fixed point x_{p} for every p≥0. In the first case, U(t)x_{o} is a constant solution of (1), whence Tx_{o}=0. In the second case, U(t)x_{p} is a p-periodic solution of (1). Hence, one has to impose additional conditions on T which imply that a p-periodic solution must be constant, for some p>0.

The main purpose of the present paper is to show that, in certain situations, either the operators U(t) are actually strict contractions or T may be approximated by operators T_{n} such that the corresponding U_{n}(t) are strict contractions. Thus, we obtain several results in general Banach spaces and a unification of some results in special spaces.

## Keywords

Banach Space Unique Solution Number Theory Point Theorem Additional Condition## Preview

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