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This paper presents three new existence results for semipositone
Fredholm integral equations of the form
where
is a constant. Existence in both
and
will be discussed.
Throughout this paper
is nonnegative
but our nonlinearity
may take negative values.
Problems of this type are referred to as semipositone problems in the
literature and they arise naturally in chemical reactor theory [4]. The
constant
is called the Thiele modulus and of physical interest is
the existence of positive solutions to
when
is small. The literature on positive solutions to Fredholm integral equations
(see [3]-[8]] and the references therein) is almost totally devoted to
when
takes nonnegative values (i.e. positone problems). Only a few
results (see [1, Chapter 4]) are available for the semipositone problem.
Existence in this paper will be established using
Krasnoselskii's fixed point theorem in a
cone, which we state here for the convenience of the reader.
Theorem 1.1. Let
be a Banach space and let
be a cone in
. Assume
and
are open subsets of
with
and
and let
be
continuous and completely continuous. In addition suppose either
or
hold. Then
has a fixed point in
.
Next: Bibliography
Up: EXISTENCE OF POSITIVE SOLUTIONS
Previous: EXISTENCE OF POSITIVE SOLUTIONS
Nobuki Takayama
2002-09-18