The existence of positive solutions for nonlinear secondorder and higherorder multipoint boundary value problems has been studied by several authors, for example, see [1–12] and the references therein. However, there are a few papers dealing with the existence of positive solutions for the
thorder multipoint boundary value problems with infinitely many singularities. Hao et al. [13] discussed the existence and multiplicity of positive solutions for the following
thorder nonlinear singular boundary value problems:
where
,
,
may be singular at
and/or
. Hao et al. established the existence of at least two positive solution for the boundary value problems if
is either superlinear or sublinear by applying the Krasnosel'skiiGuo theorem on cone expansion and compression.
In [14], Kaufmann and Kosmatov showed that there exist countably many positive solutions for the twopoint boundary value problems with infinitely many singularities of following form:
where
for some
and has countably many singularities in
.
In [15], Ji and Guo proved the existence of countably many positive solutions for the
thorder ordinary differential equation
with one of the following
point boundary conditions:
where
,
(
),
,
(
,
),
,
for some
and has countably many singularities in
.
Motivated by the result of [13–15], in this paper we are interested in the existence of countably many positive solutions for nonlinear
thorder threepoint boundary value problem
where
,
,
,
,
,
(
,
),
,
for some
and has countably many singularities in
. We show that the problem (1.5) has countably many solutions if
and
satisfy some suitable conditions. Our approach is based on the Krasnosel'skii fixed point theorem and LeggettWilliams fixed point theorem in cones.
Suppose that the following conditions are satisfied.
There exists a sequence
such that
,
,
, and
for all
There exists
such that
for all
.
Assuming that
satisfies the conditions

(we cite [15, Example
] to verify existence of
) and imposing growth conditions on the nonlinearity
, it will be shown that problem (1.5) has infinitely many solutions.
The paper is organized as follows. In Section 2, we provide some necessary background material such as the Krasnosel'skii fixedpoint theorem and LeggettWilliams fixed point theorem in cones. In Section 3, the associated Green's function for the
thorder threepoint boundary value problem is first given and we also look at some properties of the Green's function associated with problem (1.5). In Section 4, we prove the existence of countably many positive solutions for problem (1.5) under suitable conditions on
and
. In Section 5, we give two simple examples to illustrate the applications of obtained results.