Summary: FIXED POINT THEOREM OF CONE EXPANSION AND
COMPRESSION OF FUNCTIONAL TYPE
RICHARD I. AVERY AND DOUGLAS R. ANDERSON1
Abstract. The fixed point theorem of cone expansion and compression of norm type is
generalized by replacing the norms with two functionals satisfying certain conditions to
produce a fixed point theorem of cone expansion and compression of functional type. We
conclude with an application verifying the existence of a positive solution to a discrete
second-order conjugate boundary value problem.
Dedicated to Allan Peterson on the occasion of his 60th birthday.
There are many fixed point theorems. See  for an introduction to the study and
applications of fixed point theorems. In this paper we will generalize the fixed point theorem
of cone expansion and compression of norm type. The generalization allows the user to
choose two functionals that satisfy certain conditions which are used in place of the norm.
In applications to boundary value problems the functionals will typically be the minimum
or maximum of the function over a specific interval. Hence in boundary value problem
applications the functionals usually do not satisfy the triangle inequality property of a
norm. The flexibility of using functionals instead of norms allows the theorem to be used
in a wider variety of situations. In particular, in applications to boundary value problems
it allows for improved sufficiency conditions for the existence of a positive solution.