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 xo or that U(p) has a fixed point xp for every p≥0. In the first case, U(t)xo is a constant solution of (1), whence Txo=0. In the second case, U(t)xp 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 Tn such that the corresponding Un(t) are strict contractions. Thus, we obtain several results in general Banach spaces and a unification of some results in special spaces.
KeywordsBanach Space Unique Solution Number Theory Point Theorem Additional Condition
Unable to display preview. Download preview PDF.
- BROWDER, F.: Nonlinear accretive operators in Banach spaces. Bull. Amer. Math. Soc.73, 470–776 (1967)Google Scholar
- —: Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces. Bull. Amer. Math. Soc.73, 867–874 (1967)Google Scholar
- —: Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull. Amer. Math. Soc.73, 875–882 (1967)Google Scholar
- GATICA, J.; KIRK, W.: Fixed point theorems for Lipschitzian pseudo-contractive mappings. Proc. Amer. Math. Soc.36, 111–115 (1972)Google Scholar
- LASOTA, A.; YORKE, J.A.: Bounds for periodic solutions of differential equations in Banach spaces. J. Diff. Eq.10, 83–91 (1971)Google Scholar
- MARTIN, R.H.: Differential equations on closed subsets of a Banach space. Trans. Amer. Math. Soc.179, 399–414 (1973)Google Scholar
- PETRYSHYN, W.V.: Projection methods in nonlinear numerical functional analysis. J. Math. Mech.17, 353–372 (1967)Google Scholar
- REICH, S.: Remarks on fixed points. Atti Accad. Lincei52, 689–697 (1972)Google Scholar
- VIDOSSICH, G.: How to get zeros of monotone and accretive operators using the theory of ordinary differential equations. Actas Sem. Anal. Func. Sao Paulo (to appear)Google Scholar
- -:Non-existence of periodic solutions and applications to zeros of nonlinear operators (preprint)Google Scholar