Convergence of set-valued mappings:
Equi-outer semicontinuity
- with Adib Bagh, Mathematics, University of California, Davis.
- Set-Valued Analysis, 4 (1996), 333-360.
Content:
Let's consider the following system:
where A is a mapping from X whose values are subsets of Y,
f in Y and D is a subset of X with X,Y metric spaces.
This mapping A could be a (partial) differential operator, or
represents the amalgamated version of a linear or nonlinear
system of equations; such systems might represent the
Karush-Kuhn-Tucker (KKT) conditions of an optimization problem
involving constraints or the Euler equation of a variational
problem. We allow for multi-valuedness, to deal with
KKT-conditions, differential systems with turbulence,
economic and biological models involving preference relations.
There are three basic questions that must be answered about such
systems (with equalities or inclusions): existence of a solution,
uniqueness of the solution, and consistency of the approximations.
Although the ensuing development can shed some light on existence
and uniqueness, we are concerned here with questions
related to approximations. This will be dealt with in the
following framework:
- let S(x) = A(x) - f if x in D, otherwise S(x) is empty.
The mapping S is then a set-valued mapping from (all of) X to Y.
The preceding system can now be formulated:
- find x in X such that 0 in S(x).
An approximating system would then take the form:
- find x in X such that 0 in Sn(x).
where Sn approximates S in some sense. We are going to
be interested in approximating systems where Sn is close to
S in terms of their graphs. The graph of a set-valued mapping
S is the subset of XxY:
- gph S := {(x, y) | y in S(x)}.
Convergence of Sn to S would then be defined in terms of the
convergence of their graphs. The basic reason for the interest in
graph convergence is that it's the ``weakest'' convergence notion
that will ``guarantee'' the convergence of solutions.
However graph convergence of mappings, even when they are
single-valued, isn't always easy to verify. This paper explores
the relationship between graph convergence and some other
convergence notions, in particular pointwise convergence.
After some preliminaries about set convergence and the
continuity of set-valued mappings, the notion of equi-semicontinuity
for collections of set-valued mappings is introduced to
answer the question raised about the relationship between
graph and pointwise convergence. Equi-outer semicontinuity with respect to
the Choquet-Wijsman convergence of sets is explored next.
The study of other convergence notions for set-valued mappings
culminates with a compactness result akin to the
Arzela-Ascoli theorem. The relationship between the notions of
equi-outer semicontinuity for mappings and equi-lower semi-continuity
of extended real-valued functions is analyzed, as well as
the relationship between Mosco-pointwise and
Mosco-graph-convergence of mappings.
The paper concludes with applications to the convergence of
- the subgradients of convex functions,
- maximal monotone operators,
- differential inclusions.
August 15, 1995