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This paper considers Frechet differentiability almost everywhere in the sense of category of regular, locally Lipschitzian real-valued functions defined on open subsets of a Banach space. It is first shown that, for separable Banach spaces, Clarke's generalized gradient of such a function is a minimal, convex- and compact-valued, upper semicontinuous multifunction. Using a theorem of Christensen and Kenderov it is then shown that, for separable Asplund spaces, such a function is Frechet differentiable on a dense Gδ subset of its domain.

Examples have been given of Lipschitzian functions that are Gâteaux-differentiable everywhere, but nowhere Fréchet-differentiable. One such example has been reported, mistakenly, in several papers as having domain in L 2 ( [ 0 , π ] ) {L^2}([0,\pi ]) , when it should have been L 1 ( [ 0 , π ] ) {L^1}([0,\pi ]) . We discuss this example.

We construct measurable selections for closed set-valued maps into arbitrary complete metric spaces. We do not need to make any separability assumptions. We view the set-valued maps as point-valued maps into the hyperspace and our measurability assumptions arethe usual kinds of measurability of point-valued maps in this setting. We also discuss relationship of these measurability conditions to the ones usually considered in the theory of measurable selections.

For a Banach space X X and a subset A A of X X , c A {c_A} denotes the Čebyšev center of A A and P A x {P_A}x denotes the nearest point in A A to the point x x in X X . The space of all subsets of X X is furnished with the Hausdorff metric. The modulus of continuity of the function A → c A A \to {c_A} is computed in the case when X X is a Hilbert space and the sets A A are compact; the same is done for the function A → P A x A \to {P_A}x , for fixed x x , in the case when X X is uniformly convex and the sets A A are convex and closed.

Let X X be a metrizable space. A Vietoris-type topology, called the locally finite topology, is defined on the hyperspace 2 X {2^X} of all closed, nonempty subsets of X X . We show that the locally finite topology coincides with the supremum of all Hausdorff metric topologies corresponding to equivalent metrics on X X . We also investigate when the locally finite topology coincides with the more usual topologies on 2 X {2^X} and when the locally finite topology is metrizable.

This note points out that the « only if » part of Theorem 1′ (i) of the above paper [Annali di Matematica pura ed applicata, (IV)101(1974), pp. 229–236] is false as is part (ii) of Theorem 2. Counterexamples are given and a weak form of the « only if » part of Theorem 1′ (i) is established.

Let F: X → Y be a multifunction from X to Y. Then, given measure-theoretic or topological structures on X and Y, it is possible in various ways to define the measurability of F. The selection problem is to determine which structures on X and Y and which definitions of measurability of F ensure that F will have a measurable selector. This problem has been studied recently in papers by Castaing (2) and Kuratowski and Ryll-Nardzewski (6). In the latter paper, the problem is studied for its own interest. The former uses solutions of the problem to obtain general Filippov-type theorems. (See, for example, the corollaries following Theorems 2 and 3 of Castaing's paper.) For other material on Filippov's results see, among others, (3; 4; 5; 7; 9).

This paper is concerned with the existence of solutions to the initial value problem for generalized differential equations (orientor fields) when the right-hand side, F ( t , x ), may be unbounded. Two global and one local existence theorems are established when F satisfies Caratheodory type conditions involving weak integral boundedness conditions. The multiple-valued function F ( t , ·) is assumed to have closed graph and to be lower semicontinuous at each point x where F ( t , x ) is not convex.

Under certain restrictions we show that an invariant (n - 1)-manifold in the x-space bifurcates (branches off) from the fixed periodic solutions for values of iu near ,to. This process will be known as bifurcation from a periodic solution. Poincare [10] has shown that in the plane such behavior may occur. Various other authors have studied manifolds defined by solutions of ordinary differential equations. However, most of this work is concerned with the perturbation problem. For an extensive bibliography on this topic see W. T. Kyner [6]. The basic theorem of N. Levinson [9] established the existence of an invariant manifold based on x = -(t) in the extended phase space. This manifold was topologically a torus, that is, Hln S I = S I x S' x ... x S I (Sn is the unit n-sphere) in the (x,t)-space. It was assumed that the real parts of n-1 of the characteristic exponents based on +(t) were negative for all values of the parameter iu. This theorem has been generalized by various authors including Hufford [4], Kyner [5; 7], Diliberto and Hufford [3], and Diliberto [1; 2]. The assumption of stability (the real parts of n - 1 of the characteristic exponents are negative) has been replaced by the assumption of conditional stability (the real parts of n - 1 of the characteristic exponents are nonzero). These results differ from ours in three respects. The first is that our invariant manifold lies in the x-space