VECTOR SPACES OVER ANALYTIC FUNCTION FIELDS BEING ASSOCIATED TO ORDINARY DIFFERENTIAL EQUATIONS J. E. PALOMAR TARANC ´ ON Dep. Math. Inst. Jaume I C/. Europa, 3-2E, 12530-Burriana-(Castell´ on), Spain Abstract. It is a well-known fact, that the general solution of a linear ordinary differential equation belongs to a function vector space over the real (complex) number field generated by a finite exponential function set. This circumstance allows us to handle the large-time behavior and stability problems easily. However there is a wide class of non-linear differential equations whose so- lutions belong to some function space being also generated by an arbitrary exponential function set; so that the stability problems can be handled as in the linear case. To discern whether a given autonomous system belongs to such an equation class, sufficient conditions have been stated in a previous article. Now, we gen- eralize those conditions to a wider class of non-linear autonomous systems. These generalized conditions imply the existence of an ana- lytic function field K(X) for each smooth vector X, and a vector space V over the function field K(X) for the corresponding equa- tion dx(t))/dt=X(x(t)), such that each solution of this equation belongs to the vector space V. Thus, a vector space V over an analytic function field K(X) is associated to every member of a wide class of autonomous systems of ordinary differential equa- tions. The algebraic and topological structures for V and K(X) are also investigated. Finally, by means of a useful integral transform we build the general solution for those autonomous systems satisfying our gen- eralized conditions. 1. Introduction The large time behavior and stability problems in ordinary differential equations are not difficult to handle in linear autonomous systems, because the solution of a linear equation belongs to a vector space over the real (complex) number field generated by an exponential function set, say the function set  e λ j | j ∈ A  , where A stands for any finite index set. However, 1991 Mathematics Subject Classification. Primary 46A19, secondary 46N20, 34A25. Key words and phrases. Autonomous system, analytic function field, topological vector space, integral transform, Θ-linear equation. 1