Periodic solutions for second order di erential equations with inde nite singularities

where g, h, r are T−periodic functions with g, h, r ∈ L([0, T],R). Chu and et. al in [16] studied the problem of twist periodic solutions to (1.1) for the case of r(t) ≡ 0. We notice that in [15] and [16], the functions of g(t) and h(t) are required to be g(t) ≥ 0 and h(t) ≥ 0 a.e. t ∈ [0, T]. Recently, the periodic problem for second order di erential equations with inde nite singularities has attract much attention from researchers (see [12] and [17]-[20]). For example, in [18], the authors considered the existence of positive periodic solutions to the equation like x′′(t) = α(t) xμ , (1.2)


Introduction
In the past years, the problem of existence of periodic solutions to second order di erential equations with de nite singularities, either attractive type or repulsive type, was extensively studied by many researchers [1]- [14]. In [15], Hakl and Torres investigated the problem of periodic solutions to the equation where g, h, r are T−periodic functions with g, h, r ∈ L([ , T], R). Chu and et. al in [16] studied the problem of twist periodic solutions to (1.1) for the case of r(t) ≡ . We notice that in [15] and [16], the functions of g (t) and h(t) are required to be g(t) ≥ and h(t) ≥ a.e. t ∈ [ , T]. Recently, the periodic problem for second order di erential equations with inde nite singularities has attract much attention from researchers (see [12] and [17]- [20]). For example, in [18], the authors considered the existence of positive periodic solutions to the equation like where the sign of weight function α(t) can change on [ , T], and µ ≥ . The equations like (1.2) with inde nite singularities can be used to model some important problems appearing in many physical contexts (see [21] and the references therein).
In this paper, we consider the problem of periodic solutions to the equation where f ∈ C(( , +∞), R), φ, α and β are T−periodic and in L([ , T], R), m, µ and γ are constants with m ≥ and µ ≥ γ > . By using a continuation theorem of Manásevich and Mawhin, some new results on the existence of periodic solutions to (1.3) are obtained. In (1.3), the signs of α(t), β(t) and φ(t) are all allowed to change. This means that the singularity associated to restoring force − α(t) x µ + β(t) x γ at x = is inde nite type(see [18,19]). The periodic problem for equation (1.3) has been investigated in recent paper [20]. However, in [20], β(t) ≡ , and the functions of α(t) and φ(t) are required to be either α(t) ≥ a.e. t ∈ [ , T],ᾱ > andφ > , or φ(t) ≥ a.e. t ∈ [ , T],ᾱ > andφ > . The signi cance of present paper relies in the following aspects. Firstly, the coe cient function f (x) associated to friction term f (x)x may have a singularity at x = . Secondly, compared with the case where the signs of functions φ(t), α(t) and β(t) are in de nite, the work of obtaining the estimates of periodic solutions to (1.3) is more di cult. In order to overcome this di culty, we propose a function F(x) = x f (s)ds. By analyzing some properties of F(x) at x = and x = +∞, we investigate the mechanism under which how the singularity associated to f (x) at x = in uences the priori estimates of periodic solutions. Moreover, the constant µ in (1.3) is allowed to be in ( , ). For this case, even if α(t) and β(t) are constant functions, the singular restoring force − α(t) x µ + β(t) x γ has a weak singularity at x = . Finally, by using a theorem in present paper, a new result on the existence of periodic solutions is obtained for Rayleigh-Plesset equation in the case of k ∈ ( , +∞). Equation

satis es the inequalities M < x(t) < M and |x (t)| < M , for all t ∈ [ , T]. 2.Each possible solution c to the equationᾱ
Remark 2.1 Ifφ > andᾱ > , then we have from the condition of m ≥ and µ > γ > that there are two constants D and D with < D < D such that

Lemma 2.2 ([5]). Let u ∈ [ , ω] → R be an arbitrary absolutely continuous function with u( ) = u(ω).Then the inequality
Now, we embed equation (1.3) into the following equations family with a parameter λ ∈ ( , ] and Proof. Let u ∈ D, then Integrating the above equality over the interval [ , T], we obtain i.e., If (2.12) does not hold, then and (2.14) From (2.11), (2.13) and µ > γ, and by using mean value theorem of integrals, we have that there exists a point η ∈ [ , T] such that which contradicts to (2.14). This contradiction veri es (2.12), and so (2.5) holds. Meanwhile, we can assert that (2.6) is true. In fact, multiplying two sides of (2.8) with u µ (t) and integrating it over the interval [ , T], we obtain that where A is de ned by (2.7). This implies that (2.6) holds.
Proof. Firstly, we will prove that there exist two constants γ > and γ > with γ > γ , such that i.e., which together with (3.5) gives It follows from (3.2) that there is a constant γ > such that In fact, if u ∈ D, then Let u attain its maximum over [ , T] at t ∈ [ , T], then u (t ) = and we deduce from (3.8) that This implies that condition 1 of Lemma 2.1 is satis ed. Also, we can deduce from Remark 2.1 that Proof. From the proof of Theorem 3.1, we see that it su ces for us to verify (3.4) and (3.3). In order to do it, let u ∈ D, u satis es (2.18). Let t and t be de ned as same as the ones in the proof of Theorem 3.1, that is It is easy to see that (2.25) can be deduced from (3.10), and (3.9) veri es (2.26). By using Lemma 2.5 and (2.5) in Lemma 2.4, we know that (3.11) i.e., (3.12) which together with (3.11) and gives It follows from (3.10) and the conditionφ > that there is a constant ρ > such that If