We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existenceâ€¦ (More)

We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous Îµ :M â†’ (0,+âˆž), and for every positive number r > 0,â€¦ (More)

Heterochrony, evolutionary modifications in the rates and/or the timing of development, is widely recognized as an important agent of evolutionary change. In this paper, we are concerned with theâ€¦ (More)

We prove the following new characterization of C (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a C smooth (Lipschitz) bump function if and only if it has anotherâ€¦ (More)

Weshow that ifX is a Banach space having an unconditional basis and a C p-smooth Lipschitz bump function, then for every C1-smooth functionf from X into a Banach space Y, and for every continuousâ€¦ (More)

We prove that every continuous mapping from a separable infinitedimensional Hilbert space X into R can be uniformly approximated by Câˆž smooth mappings with no critical points. This kind of result canâ€¦ (More)

In this note we give a subdifferential mean value inequality for every continuous GÃ¢teaux subdifferentiable function f in a Banach space which only requires a bound for one but not necessarily all ofâ€¦ (More)

We study the size of the sets of gradients of bump functions on the Hilbert space `2, and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in `2â€¦ (More)

We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have Cp-smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for everyâ€¦ (More)