Set-Valued Analysis

This paper is concerned with the dynamics of nonautonomous general dynamical systems (NAGDSs in short) and applications to differential inclusions on ℝ m . First, we show that if a NAGDS has a compact uniformly attracting set, then it has a pullback attractor $\mathcal{A}$ with the parametrically inflated pullback attractor $\mathcal{A}(\varepsilon_0)$ being uniformly forward attracting. Then, we establish some stability results for pullback attractors. Finally, we apply the abstract theory to nonautonomous differential inclusions on ℝ m to obtain some interesting results. In particular, the effects of small time delays to asymptotic stability is addressed.

It is shown that if k(x) is an upper semicontinuous and quasi lower semicontinuous function on a Banach space X, then k(x)B X* is the Clarke subdifferential of some locally Lipschitz function on X. Related results for approximate subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke subdifferential, one can obtain a maximal Clarke subdifferential map via its ‘local Lipschitz-constant’ function. Finally, some results concerning the characterization and calculus of local Lipschitz-constant functions are developed.

We consider the question of integration of a multivalued operator T, that is the question of finding a function f such that T⊑∂f. If ∂ is the Fenchel–Moreau subdifferential, the above problem has been completely solved by Rockafellar, who introduced cyclic monotonicity as a necessary and sufficient condition. In this article we consider the case where f is quasiconvex and ∂ is the lower subdifferential ∂<. This leads to the introduction of a property that is reminiscent to cyclic monotonicity. We also consider the question of the density of the domains of subdifferential operators.

We study nonexpansive set-valued maps in Banach and metric spaces. We are concerned, in particular, with the generic existence and approximation of fixed points, as well as with the structure of fixed point sets.

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ N evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.

This paper is primarily concerned with the problem of maximality for the sum A + B and composition L* ML in non-reflexive Banach space settings under qualifications constraints involving the domains of A, B, M. Here X, Y are Banach spaces with duals X*, Y*, A, B: X ⇉ X*, M: Y ⇉ Y* are multi-valued maximal monotone operators, and L: X → Y is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as simpler proofs, improvements of previously known results, and several new results on the topic are presented.

In this paper, we propose a Hausdorff metric to measure the “distance” between two linear inequality systems on a real normed space X. For this topology, which comes through a pseudo-metric in the set Σ of linear inequality systems, the closedness of the feasible set mapping is studied, and at the same time a characterization of the stability of the subset Σ c of consistent sytems is given.

We discuss several notions of first and second-order differentiability for integral functionals on a Hilbert space.

We modify the definitions of continuity and lower semicontinuity for single-valued mappings and upper and lower semicontinuity for set-valued mappings. For single-valued mappings we have a generalisation of Osgood's theorem and for set-valued mappings we have an extension of Fort's theorem and a generalisation of Michael's selection theorem producing a densely defined selection with a natural continuity property relative to the domain.