On the structure of the set of solutions of nonlinear equations

Abstract

Let T be a mapping from a subset of a Banach space X into a Banach space Y. The present paper investigates the nature of the set of solutions of the equation T(x) = y for a given y E Y, i.e. when T-l(y) # 0 ? What are the topological properties of T-l(y)? A prototype for an answer to these questions is given by Peano existence theorem on the connectedness of the set of solutions of an ordinary differential equation in the real case. In its general setting, this problem was first attacked by Aronszajn [l] and Stampacchia [l 11; recently, by Browder-Gupta [5], Vidossich [12] and, above all, Browder [3, Sec. 51 who gives several interesting results in an excellent treatment. Customary, the structure of T-l(y) is studied under the assumption that T is a proper map (T is a proper map iff T-l(K) is compact whenever K is), since differential and integral equations in the finite dimensional case lead in a natural way to proper maps. But in the infinite dimensional case this correspondence holds rarely, so that one must find more suitable maps. Since the problem of what are the “good” mappings for the infinite dimensional case is still open, the results of the present paper may be of interest at least as first step toward the general solution. For, the investigation of T-l(y) is made here under the assumption that T is a closed map. Section 1 gives a general existence theorem which is employed to deduce a proof of Peano existence theorem for ordinary differential equations so elementary that it may be included in any first course on differential equations. Section 2 gives some theorems on the connectedness of T-l(y), generalizing in various ways theorems proved in [ 1, 3, 5 and 111, and solving positively in a special case the problem of Stampacchia treated in [12]. The elimination of the hypothesis “T is proper” in the present theorems is obtained by using in a non-trivial manner a theorem of Michael [lo] recently generalized by Dykes [7], according to which a continuous closed onto map with suitable domain is compact-covering (T : X -4 Y is a compact-coveting map iff for each compact KC Y there is a compact C C X such that T(C) = K. Note