#### Existence of positive periodic solutions of second-order differential equations with weak singularities

Cite this article as: Ma: Existence of positive periodic solutions of second-order differential equations with weak
singularities. Boundary Value Problems
Existence of positive periodic solutions of second-order differential equations with weak singularities
You Ma 0
0 School of Mathematical Sciences, Beijing Normal University , Beijing, 100875 , P.R. China
We establish the existence of positive periodic solutions of the second-order differential equation x + a(t)x = f (t, x) + c(t) via Schauder's fixed point theorem, where a ∈ L1(R/T Z; R+), c ∈ L1(R/T Z; R), f is a Caratheodory function and it is singular at x = 0. Our main results generalize some recent results by Torres (J. Differ. Equ. 232:277-284, 2007). MSC: 34B10; 34B18 © 2014 Ma; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
periodic solution; weak singularity; Schauder's fixed point theorem; indefinite weight
1 Introduction
In this paper, we are concerned with the existence of positive periodic solutions of the
second-order differential equation
x + a(t)x = f (t, x) + c(t)
c(t) dt < .
with a ∈ L(R/T Z; R+), c ∈ L(R/T Z; R), f ∈ Car(R/T Z × (, ∞); R) is a L Caratheodory
function, and f is singular at x = .
The interest on this type of equations began with the paper of Lazer and Solimini [].
They dealt with the case that a ≡ and f (x) = xλ , (.) reduces to the special equation
which was initially studied by Lazer and Solimini []. They proved that for λ ≥ (called a
strong force condition in the terminology first introduced by Gordon [, ]), a necessary
and sufficient condition for the existence of a positive periodic solution of (.) is that the
mean value of c is negative,
Moreover, if < λ < (weak force condition) they found examples of functions c with
negative mean values and such that periodic solutions do not exist.
If compared with the literature available for strong singularities, see [–] and the
references therein, the study of the existence of periodic solutions in the presence of a weak
singularity is much more recent and the number of references is considerably smaller. The
likely reason may be that with a weak singularity, the energy near the origin becomes
finite, and this fact is not helpful for obtaining the a priori bound needed for a classical
application of the degree theory, and also is not helpful for the fast rotation needed in
recent versions of the Poincaré-Birkhoff theorem. The first existence result with a weak
force condition appears in Rachunková et al. []. Since then, Eq. (.) with f having weak
singularities has been studied by several authors; see Torres [, ], Franco and Webb
[], Chu and Li [] and Li and Zhang [].
Recently, Torres [] showed how a weak singularity can play an important role if
Schauder’s fixed point theorem is chosen in the proof of the existence of positive periodic
solutions for (.). From now on, for a given function ξ ∈ L∞[, ∞], we denote the
essential supremum and infimum of ξ by ξ ∗ and ξ∗, respectively. We write ξ if ξ ≥ for a.e.
t ∈ [, T ] and it is positive in a set of positive measure. Under the following assumption:
(H) the linear equation u + a(t)u = is nonresonant and the corresponding Green’s
function obeys
G(t, s) ≥ , (t, s) ∈ [, T ] × [, T ],
Torres showed the following.
Theorem A [, Theorem ] Let (H) hold and define
G(t, s)c(s) ds.
Assume that
(H) there exist b ∈ L(, T ) with b
for all u > , a.e. t ∈ [, T ].
Theorem B [, Theorem ] Let (H) hold. Assume that
(H) there exist two functions b, bˆ ∈ L(, T ) with b, bˆ
that
bˆ (t) b(t)
≤ uλ ≤ f (t, u) ≤ uλ ,
u ∈ (, ∞), a.e. t ∈ [, T ].
Theorem C [, Theorem ] Let (H) and (H) hold. Let
G(t, s)bˆ (s) ds ,
G(t, s)b(s) ds .
then (.) has a positive T -periodic solution.
Obviously, (H) and (H) are too restrictive so that the above mentioned results are only
applicable to (.) with nonlinearity which is bounded at origin and infinity by a function
of the form uλ . Very recently, Ma et al. [] generalized Theorems A-C under some
conditions which allow the nonlinearity f to be bounded by two different functions uα and uβ .
Notice that b in (H), and b, bˆ in (H). Of course the natural question is: what
would happen if we allow that the functions b and bˆ may change sign?
It is worth remarking that if b ∈ L(R/T Z; R) changes its sign, then the existence of
T -periodic solutions for the equation
x = bx(t)
G(t, s)c(s) ds,
y+ = max{y, },
y– = max{–y, }.
Theorem . Let γ ∗ > . Assume (H) and
(A) there exists b ∈ C[, T ] with
meas I+ > ,
meas I– > ,
is still open; see Bravo and Torres [] and Hakl and Torres []. Notice that (.) plays
an important role in the study of stabilization of matter-wave breathers in Bose-Einstein
condensates [], the propagation of guided waves in optical fibers [], and in the
electromagnetic trapping of a neutral atom near a charged wire [].
In the sequel, we denote the set of continuous T -periodic functions by CT . Define
It is the purpose of this paper to general Theorems A-C under some assumptions which
allow the nonlinearity b and bˆ to change sign. The main tool is Schauder’s fixed point
theorem.
2 Existence of periodic solutions
I+ = t ∈ [, T ] | b(t) > ,
I– = t ∈ [, T ] | b(t) < ,
∀x ∈ (, ∞) and t ∈ I+
∀x ∈ (, ∞) and t ∈ I–.
then there exists a positive T -periodic solution of (.).
F [u](t) :=
K = x ∈ CT : r ≤ x(t) ≤ R for all t
G(t, s) f s, x(s) + c(s) ds =
By a direct application of Schauder’s fixed point theorem, the proof is finished if we prove
that F maps the closed convex set defined as
G(t, s)b+(s) ds,
G(t, s)b–(s) ds,
Then for x ∈ K , it follows from (.) that
F [u](t) ≤
Remark . If b in [, T ], then β– ≡ , and accordingly (β–)∗ = . In this case, (.)
is satisfied for all λ > and γ with γ∗ > . So, Theorem . generalizes Theorem A.
For u ∈ CT , let us define
I+u = t ∈ [, T ] | u(t) > ,
I–u = t ∈ [, T ] | u(t) < .
Theorem . Let γ∗ = . Assume (H) and
(A) there exist b, h ∈ C[, T ] and < λ < such that
h(t) b(t)
≤ xλ ≤ f (t, x) ≤ xλ ,
h(t) b(t)
≥ xλ ≥ f (t, x) ≥ xλ ,
h+ ∗s–λ – b– ∗sλ+ ≥ ,
(A) h and b satisfy
∀x ∈ (, ∞) and a.e. t ∈ I+b,
∀x ∈ (, ∞) and a.e. t ∈ I–b,
Then there exists a positive T -periodic solution of (.).
Proof We follow the same strategy and notation as in the proof of Theorem .. Again, we
need to fix r < R such that F(K ) ⊆ K . We define the functions
G(t, s)b+(s) ds,
G(t, s)h+(s) ds,
G(t, s)b–(s) ds,
G(t, s)h–(s) ds.
F [u](t) :=
G(t, s) f s, x(s) + c(s) ds =
Some easy computations prove that it is sufficient to find r < R such that
F [u](t) ≤
F [u](t) ≥
≥ r.
b+ ∗Rλ – (h–λ)∗ + γ ∗ ≤ R,
R
h+ ∗R–λ – b– ∗Rλ+ ≥ .
Theorem . Let γ ∗ ≤ . Assume (H) and (A) and
(A) there exists
Then there exists a positive T -periodic solution of (.).
As in the proof of [, Theorem ], we may take r = [ bh∗∗λ λ] –λ , then (.) can be reduced
by the condition (.).
Remark . As an application of Theorem ., let us consider the problem
u(t) =
t ∈ (, ),
u() = u(T ),
u () = u (T ),
c(t) ≡ ,
b(t) = t –
t ∈ [, ].
By [, Lemma .], the Green function of the linear problem
u(t) = ,
t ∈ (, ),
u() = u(T ),
u () = u (T ),
can be explicitly given by
Equation (.) yields
G(t, s)c(s) ds =
G(t, s)b–(s) ds
G(t, s) =
(sin t – sin t– )(sin s – sin s– )
≤ s ≤ t ≤ ,
≤ t ≤ s ≤ .
(sin t – sin t– )( – cos + cos – cos )
and consequently
has at least one positive -periodic solution.
Competing interests
The author declares that she has no competing interests.
Author’s contributions
YM completed the study, carried out the results of this article and drafted the manuscript, and checked the proofs and
verified the calculation. The author read and approved the final manuscript.
Acknowledgements
This work was supported by the NSFC (No. 11361054).
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