Recently, in [

1–

4], Erbe, Kong, Jiang, Wang, and Weng considered the following singular functional differential equations:

where
and the existence of positive solutions to (1.1) is obtained. When
in (1.1), Agarwal and O'Regan in [5], Lin and Xu in [6] discussed the existence of positive solutions to (1.1) also. We notice that the nonlinearities
in all the above-mentioned references depend on
.

The more difficult case is that the term
depends on
for second-order functional differential equations with delay. When
has no singularity at
and
, there are many results on the following (1.2) (see [7–9] and references therein). Up to now, to our knowledge, there are fewer results on (1.2) when the term
is allowed to possess singularity for the term
at
and
, which is of more actual significance.

In this paper, motivated by above results, we consider the second-order initial-boundary value problems:

where
. By Leray-Schauder fixed-point theorem, the existence of positive solutions to (1.2) is obtained when
is singular at
and
.

For
and
, let
and
. Then,
and
are Banach spaces. Let
and
. Obviously,
and
are cones in
and
respectively. Now, we give a new definition.

Deffinition.

is said to be singular at
for
when
satisfies
for
and
is said to be singular at
for
when
satisfies
for
.

And one defines some functions which one has to use in this paper.

where
is a Green's function. It is clear that
for
and
on

We now introduce the definition of a solution to IBVP(1.2).

Deffinition.

A function
is said to be a solution to IBVP(1.2) if it satisfies the following conditions:

(1)
is continuous and nonnegative on
;

(2)
;

(3)
and
exist on
;

(4)
is Lebesgue integrable on
;

(5)
for
.

Furthermore, a solution
is said to be positive if
on
.

Let

be a solution to IBVP(1.2). Then, it can be represented as

for all solutions,
, to IBVP(1.2), where
. For
, let
on
throughout this paper. Obviously,
and
for all
.

Throughout this paper, we assume the following hypotheses hold.

(H_{1})
is continuous on
.

(H

_{2}) There exists

, such that

Lemma 1.3.

*Assume that (H*
_{
1
}
*)-(H*
_{
2
}
*) hold, then there exists a*
*, such that*
for all solutions,
, to (1.2).

Proof.

Suppose that the claim is false. (1.5) guarantees that there exists a sequence

of solutions to IBVP(1.2) such that

Without loss of generality, we may assume that

From

and (1.5), it follows that

which contradicts the assumption that
and hence the claim is true provided
is suitably small.

Remark 1.4.

holds provided that
is sufficiently small, where
is in Lemma 1.3.

There exist a nonnegative continuous function

defined on (0,1) and two nonnegative continuous functions

defined on, respectively,

, such that

where

and

) satisfy

Furthermore,

is nonincreasing and

is nondecreasing, that is,

Lemma 1.5 (see [7]).

Let
be the Banach space and let *X* be any nonempty, convex, closed, and bounded subset of
. If
is a continuous mapping of
into itself and
is relatively compact, then the mapping
has at least one fixed point (i.e., there exists an
with
) .

Using Lemma 1.5, we present the existence of at least one positive solution to (1.2) when
is singular at
and
(notice the new Definition 1.1). To some extent, our paper complements and generalizes these in [1–6, 8–10].