The periodic boundary value problems

where *f* is a continuous or *L*^{1}-Caratheodory type function have been extensively studied. A very popular technique to obtain the existence and multiplicity of positive solutions to the problem is Krasnosel'skii's fixed point theorem of cone expansion/compression type, see for example [1–4], and the references contained therein. In those papers, the following condition is an essential assumptions:

(*A*) The Green function *G*(*t*, *s*) associated with problem (1) is positive for all (*t*, *s*) ∈ [0, *T*] × [0, *T*].

Under condition (*A*), Torres get in [4] some existence results for (1) with jumping nonlinearities as well as (1) with a repulsive or attractive singularity, and the authors in [3] obtained the multiplicity results to (1) when *f*(*t*, *u*) has a repulsive singularity near *x* = 0 and *f*(*t*, *u*) is super-linear near *x* = +∞. In [2], a special case, *a*(*t*) ≡ *m*^{2} and
, was considered, the multiplicity results to (1) are obtained when the nonlinear term *f*(*t*, *u*) is singular at *u* = 0 and is super-linear at *u* = ∞.

Recently, in [5], the hypothesis (*A*) is weakened as

(*B*) The Green function *G*(*t*, *s*) associated with problem (1) is nonnegative for all (*t*, *s*) ∈ [0, *T*] × [0, *T*] but vanish at some interior points.

By defining a new cone, in order to apply Krasnosel'skii's fixed point theorem, the authors get an existence result when
and
is sub-linear at *u* = 0 and *u* = ∞ or
is super-linear at *u* = 0 and *u* = ∞ with
is convex and nondecreasing.

In [6], the author improve the result of [5] and prove the existence results of at least two positive solutions under conditions weaker than sub- and super-linearity.

In [7], the author study (1) with *f*(*t*, *u*) = *λb*(*t*)*f*(*u*) under the following condition:

(*C*) The Green function *G*(*t*, *s*) associated with problem (1) changes sign and
where *G* ^{-} is the negative part of *G*.

Inspired by those papers, here we study the problem:

where
is a constant and the associated Green's function may changes sign. The aim is to prove the existence of positive solutions to the problem.