Existence and Multiplicity of Solutions for a Periodic Hill's Equation with Parametric Dependence and Singularities

Abstract

We deal with the existence and multiplicity of solutions for the periodic boundary value problem 𝑥^′′(𝑡)+𝑎(𝑡)𝑥(𝑡)=𝜆𝑔(𝑡)𝑓(𝑥)+𝑐(𝑡),𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇), where 𝜆 is a positive parameter. The function 𝑓∶(0,∞)→(0,∞) is allowed to be singular, and the related Green's function is nonnegative and can vanish at some points.

1. Introduction and Preliminaries

In the recent paper [1], the authors obtain existence, multiplicity, and nonexistence results for the periodic problem𝑥^′′(𝑡)−𝑘²𝑥(𝑡)=𝜆𝑔(𝑡)𝑓(𝑥),𝑥(0)=𝑥(2𝜋),𝑥^′(0)=𝑥^′(2𝜋), (1.1) depending on the parameter 𝜆>0. Although not explicitly mentioned in [1], we point out the important fact that the related Green’s function of (1.1) is strictly negative for all 𝑘>0.

The aim of this paper is to give complementary results to those of [1] for the case of a nonnegative related Green's function. In particular, we will deal with problem𝑥^′′(𝑡)+𝑎(𝑡)𝑥(𝑡)=𝜆𝑔(𝑡)𝑓(𝑥)+𝑐(𝑡),𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇), (1.2) assuming that its Green’s function is nonnegative (for instance, if 𝑎(𝑡)=𝑘², this means 0<𝑘≤𝜋/𝑇). Moreover, in order to give wider applicable results, we will also allow 𝑓(𝑥) to be singular at 𝑥=0 (the reader may have in mind the model 𝑓(𝑥)=1/𝑥^𝛼, for some 𝛼>0).

We note that analogous arguments have been developed in [2] for the fourth-order discrete equation 𝑢(𝑘+4)+𝑀𝑢(𝑘)=𝜆𝑔(𝑘)𝑓(𝑢(𝑘))+𝑐(𝑘),𝑢(𝑖)=𝑢(𝑇+𝑖),𝑖=0,…,3. (1.3)

The main tool used in this paper is Krasnoselskii's fixed point theorem in a cone, which is a classical tool extensively used in the related literature (see, for instance, [1, 3–5] and references therein). We will use cones of the form 𝒦=𝑥∈𝒞([0,𝑇],[0,∞))∶𝜑(𝑥)≥𝜎‖𝑥‖, (1.4) where 0<𝜎≤1 is a fixed constant and 𝜑∶𝒞([0,𝑇],[0,∞))→[0,∞) is a functional satisfying (i)𝜑(𝑥+𝑦)≥𝜑(𝑥)+𝜑(𝑦) for all 𝑥,𝑦∈𝒞([0,𝑇],[0,∞)), (ii)𝜑(𝜆𝑥)=𝜆𝜑(𝑥) for all 𝜆>0 and 𝑥∈𝒞([0,𝑇],[0,∞)).

In particular, in Section 2, we use the standard choice 𝜑(𝑥)=min_{𝑡∈[0,𝑇]}𝑥(𝑡), and in Section 3, we use 𝜑(𝑥)=∫^𝑇₀𝑥(𝑠)𝑑𝑠, which has been recently introduced in [3].

We say that the linear problem𝑥^′′+𝑎(𝑡)𝑥=0,𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇) (1.5) is nonresonant when its unique solution is the trivial one. It is well known that if (1.5) is nonresonant, then the nonhomogeneous problem 𝑥^′′+𝑎(𝑡)𝑥=ℎ(𝑡),a.e.𝑡∈[0,𝑇];𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇), (1.6) always has a unique solution which, moreover, can be written as 𝑥(𝑡)=^𝑇₀𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠, (1.7) where 𝐺(𝑡,𝑠) is Green's function related to (1.5). Thus, defining for each 𝜆>0 the operator 𝒯_𝜆∶𝒟𝒯_𝜆≡{𝑥∈𝒞([0,𝑇])∶𝑥(𝑡)>0∀𝑡∈[0,𝑇]}⟶𝒞([0,𝑇]), (1.8) given by𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+^𝑇₀𝐺(𝑡,𝑠)𝑐(𝑠)𝑑𝑠,𝑡∈[0,𝑇], (1.9) we have that 𝑥>0 is a solution of problem (1.2) if and only if 𝑥=𝒯_𝜆𝑥.

Throughout the paper, we will use the following notation: 𝛾(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑐(𝑠)𝑑𝑠,𝑚=min_{[]𝑡,𝑠∈0,𝑇}𝐺(𝑡,𝑠),𝑀=max_{[]𝑡,𝑠∈0,𝑇}𝐺(),𝑓𝑡,𝑠₀=lim_𝑥→0+𝑓(𝑥)𝑥,𝑓_∞=lim_𝑥→∞𝑓(𝑥)𝑥. (1.10)

For 𝑎∈𝐿¹(0,𝑇), we denote by 𝑎₊=max{𝑎,0} its positive part, and for 1≤𝑝≤∞, we denote by 𝑝 its conjugate, that is 1/𝑝+1/𝑝=1. Moreover, for an essentially bounded function ℎ∶[0,𝑇]→ℝ, we define ℎ_∗=infess_{[]𝑡∈0,𝑇}ℎ(𝑡),ℎ^∗=supess_{[]𝑡∈0,𝑇}ℎ(𝑡), (1.11) and for 𝑥∈𝒞([0,𝑇]), we will define ‖𝑥‖=sup_{[]𝑡∈0,𝑇}|𝑥(𝑡)|. (1.12)

The following section is devoted to prove the existence, multiplicity, and nonexistence of solutions of problem (1.2) by assuming that the related Green’s function is strictly positive, whereas in Section 3, we turn out to the case, where the related Green's function is non negative. We point out that in the recent paper [6], the existence of solution for problem (1.2) with a sign-changing Green's function is studied, but only considering a regular 𝑓 and 𝑐(𝑡)≡0.

2. Positive Green's Function

In this section, we assume the following hypotheses: (𝐻0)𝛾_∗>0 or 𝑐(𝑡)≡0,(𝐻1) problem (1.5) is nonresonant and the corresponding Green's function 𝐺(𝑡,𝑠) is strictly positive on [0,𝑇]×[0,𝑇],(𝐻2)𝑔∈𝐿¹([0,𝑇]), 𝑔(𝑡)≥0 for a.e. 𝑡∈[0,𝑇], and ∫^𝑇₀𝑔(𝑠)𝑑𝑠>0,(𝐻3)𝑓∶(0,∞)→(0,∞) is continuous,(𝐻4)𝑐∈𝐿¹([0,𝑇]).

Notice that condition (𝐻3) allows 𝑓 to be singular at 𝑥=0. In particular, (𝐻3) is satisfied when 𝑓(𝑥)=1/𝑥^𝛼, 𝛼>0 (the case 0<𝛼<1 is called a weak singularity, while 𝛼≥1 is called an strong singularity).

On the other hand, it is well known that for constant 𝑎(𝑡)≡𝑘², condition (𝐻1) is equivalent to 0<𝑘<𝜋/𝑇. For a time-dependent and nonnegative potential 𝑎(𝑡), Torres gave a sharp 𝐿^𝑝-criterium [5] based on an antimaximum principle obtained in a previous work by Torres and Zhang [7]. That criterium has been extended in [8] for sign-changing potentials with strictly positive average. The obtained result is the following.

Proposition 2.1 (see [8, Theorem  3.2]). Define 𝐾(𝛼,𝑇)=2𝜋𝛼𝑇^1+2/𝛼22+𝛼^1−2/𝛼Γ(1/𝛼)Γ(1/2+1/𝛼)²4,if1≤𝛼<∞,𝑇,if𝛼=∞, (2.1) where Γ is the usual Gamma function.
Assume that 𝑎∈𝐿^𝑝(0,𝑇) for some 1≤𝑝≤∞, ∫^𝑇₀𝑎(𝑡)𝑑𝑡>0, and moreover, 𝑎_+𝑝<𝐾2𝑝,𝑇. (2.2) Then, 𝐺(𝑡,𝑠)>0 for all (𝑡,𝑠)∈[0,𝑇]×[0,𝑇].

In [9], by studying antimaximum principles for the semilinear equation𝑢^′𝑝−2𝑢^′′+𝑎(𝑡)|𝑢|^𝑝−2𝑢=ℎ(𝑡),𝑢(0)=𝑢(𝑇),𝑢^′(0)=𝑢^′(𝑇), (2.3) the previous result has been extended to the potentials with nonnegative average as follows.

Lemma 2.2 (see [9, Theorem  3.4 and Remark  3.7]). Assume that 𝑎∈𝐿^𝑝(𝐼) for some 𝑝≥1, ∫^𝑇₀𝑎(𝑡)𝑑𝑡≥0, and moreover, 𝑎_+𝑝≤𝐾2𝑝,𝑇. (2.4) Then, 𝐺(𝑡,𝑠)>0 for a. e. (𝑡,𝑠)∈𝐼×𝐼.

Zhang constructs in [10] some examples of potentials a(𝑡) for which the related Green's function is strictly positive, but (2.2) does not hold. In consequence, the best Sobolev constant 𝐾(2𝑝,𝑇) is not an optimal estimate to ensure the positiveness of Green's function. For optimal conditions in order to get maximum or antimaximum principles, expressed using eigenvalues, Green's functions, or rotation numbers, the reader is referred to the recent work of Zhang [11].

Example 2.3. By Proposition 2.1, Hill's equation 𝑥^′′+𝑎(1+𝑏cos(𝑡))𝑥=0, (2.5) with the periodic boundary conditions 𝑥(0)=𝑥(2𝜋),𝑥^′(0)=𝑥^′(2𝜋) (2.6) satisfies (𝐻1), provided that 𝑎>0, and moreover, 𝑎(1+𝑏cos(𝑡))_+𝑝<𝐾2𝑝,2𝜋,forsome𝑝∈[1,∞], (2.7) where 𝐾 is given by (2.1). So, for each 𝑏∈ℝ, the condition (𝐻1) is fulfilled if 0<𝑎<𝑀(𝑏)∶=sup_{[]𝑝∈1,∞}𝐾2𝑝,2𝜋(1+𝑏cos(𝑡))_+𝑝. (2.8)
In particular, it is known that 𝑀(0)=1/4 and 𝑀(1)=0.16448 (since the maximum of 𝐾(2𝑝,2𝜋)/‖1+cos(𝑡)‖_𝑝 is attained at 𝑝≈2.1941, see [10, Example  4.4]). The graphic of 𝑀(𝑏) is showed in Figure 1.

Figure 1 

Graphic of 𝑀(𝑏).

From (𝐻1), it follows that 𝑚>0, and we define the cone𝒦∶=𝑥∈𝒞([0,𝑇],[0,∞))∶min_{[]𝑡∈0,𝑇}𝑥(𝑡)≥𝜎‖𝑥‖, (2.9) where𝑚𝜎=min𝑀,𝛾_∗𝛾^∗,if𝛾_∗𝑚>0or𝜎=𝑀,if𝑐(𝑡)≡0. (2.10) In both cases, 0<𝜎<1, and for 0<𝑟<𝑅, we define𝒦_𝑟,𝑅∶={𝑥∈𝒦∶𝑟≤‖𝑥‖≤𝑅}. (2.11)

Next, we give sufficient conditions for the solvability of problem (1.2).

Theorem 2.4. Assume that conditions (𝐻0), (𝐻1), (𝐻2), (𝐻3), and (𝐻4) are fulfilled. Then, for each 𝜆>0 and 0<𝑟<𝑅, the operator 𝒯_𝜆∶𝒦_𝑟,𝑅→𝒦 given by (1.9) is well defined and completely continuous.
Moreover, if either(i)‖𝒯_𝜆𝑥‖≤‖𝑥‖ for any 𝑥∈𝒦 with ‖𝑥‖=𝑟 and ‖𝒯_𝜆𝑥‖≥‖𝑥‖ for any 𝑥∈𝒦 with ‖𝑥‖=𝑅, or(ii)‖𝒯_𝜆𝑥‖≥‖𝑥‖ for any 𝑥∈𝒦 with ‖𝑥‖=𝑟 and ‖𝒯_𝜆𝑥‖≤‖𝑥‖ for any 𝑥∈𝒦 with ‖𝑥‖=𝑅, then 𝒯_𝜆 has a fixed point in 𝒦_𝑟,𝑅, which is a positive solution of problem (1.2).

Proof. Note that if 𝑥∈𝒦_𝑟,𝑅, then 0<𝜎𝑟≤𝑥(𝑡)≤𝑅 for all 𝑡∈[0,𝑇], so 𝒦_𝑟,𝑅⊂𝒟(𝒯_𝜆), and then 𝒯_𝜆∶𝒦_𝑟,𝑅→𝒞([0,𝑇]) is well defined. Standard arguments show that 𝒯_𝜆(𝒟(𝒯_𝜆))⊂𝒦 and that 𝒯_𝜆 is completely continuous. Then, from Krasnoselskii's fixed point theorem (see [12, p. 148]), it follows the existence of a fixed point for 𝒯_𝜆 in 𝒦_𝑟,𝑅 which is, by the definition of 𝒯_𝜆, a positive solution of problem (1.2).

Before proving the existence and multiplicity results for problem (1.2), we need some technical lemmas proved in the next subsection.

2.1. Auxiliary Results

Lemma 2.5. Assume that conditions (𝐻0), (𝐻1), (𝐻2), (𝐻3), and (𝐻4) are satisfied. Then, for each 𝑅>𝛾^∗, there exists 𝜆₀(𝑅)>0 such that for every 0<𝜆≤𝜆₀(𝑅), we have 𝒯_𝜆𝑥≤‖𝑥‖for𝑥∈𝒦with‖𝑥‖=R. (2.12)

Proof. Fix 𝑅>𝛾^∗, and let 𝑥∈𝒦 with ‖𝑥‖=𝑅. If 0<𝜆≤𝜆₀(𝑅)∶=𝑅−𝛾^∗𝑀max_{[]𝑢∈𝜎𝑅,𝑅}𝑓(𝑢)∫^𝑇₀𝑔(𝑠),𝑑𝑠 (2.13) then, for all 𝑡∈[0,𝑇] the following inequalities hold: 𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+𝛾(𝑡)≤𝜆𝑀max_{[]𝑢∈𝜎𝑅,𝑅}𝑓(𝑢)^𝑇₀𝑔(𝑠)𝑑𝑠+𝛾^∗≤𝑅=‖𝑥‖, (2.14) and thus ‖𝒯_𝜆𝑥‖≤‖𝑥‖.

Lemma 2.6. Assume that conditions (𝐻0), (𝐻1), (𝐻2), (𝐻3), and (𝐻4) are fullfiled. Then, for each 𝑟>0, there exists 𝜆₀(𝑟)>0 such that for every 𝜆≥𝜆₀(𝑟), we have 𝒯_𝜆𝑥≥‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=r. (2.15)

Proof. Fix 𝑟>0, and let 𝑥∈𝒦 with ‖𝑥‖=𝑟. If 𝜆≥𝜆₀(𝑟)𝑟∶=𝑚min_{[]𝑢∈𝜎𝑟,𝑟}𝑓(𝑢)∫^𝑇₀𝑔(𝑠),𝑑𝑠 (2.16) then 𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+𝛾(𝑡)≥𝜆𝑚min_{[]𝑢∈𝜎𝑅,𝑅}𝑓(𝑢)^𝑇₀𝑔(𝑠)𝑑𝑠+𝛾_∗≥𝑟=‖𝑥‖, (2.17) and thus ‖𝒯_𝜆𝑥‖≥‖𝑥‖.

Lemma 2.7. Suppose that conditions (𝐻1), (𝐻2), (𝐻3), and (𝐻4) are satisfied and 𝑐(𝑡)≡0. Then, if 𝑓₀=0, there exists 𝑟₀(𝜆)>0 such that for every 0<𝑟≤𝑟₀(𝜆), we have 𝒯_𝜆𝑥≤‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=r. (2.18)

Proof. Since 𝑓₀=0 for 𝜀=𝜀(𝜆)=1/𝜆𝑀∫^𝑇₀𝑔(𝑠)𝑑𝑠, there exists 𝑟₀(𝜆)>0 such that 𝑓(𝑢)≤𝜀𝑢 for each 0<𝑢≤𝑟₀(𝜆).
Fix 0<𝑟≤𝑟₀(𝜆), and let 𝑥∈𝒦 with ‖𝑥‖=𝑟. Then,𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠≤𝜆𝑀^𝑇₀𝑔(𝑠)(𝑠)‖𝑥‖𝜀𝑥𝑑𝑠≤𝜆𝑀𝜀^𝑇₀𝑔(𝑠)=‖𝑥‖,𝑑𝑠 (2.19) and thus ‖𝒯_𝜆𝑥‖≤‖𝑥‖.

Lemma 2.8. Assume that hypothesis (𝐻0), (𝐻1), (𝐻2), (𝐻3), and (𝐻4) hold. Then, if 𝑓₀=∞, there exists 𝑟₀(𝜆)>0 such that for every 0<𝑟≤𝑟₀(𝜆), we have 𝒯_𝜆𝑥≥‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=r. (2.20)

Proof. Since 𝑓₀=∞ for 𝐿=𝐿(𝜆)=1/𝜆𝑚𝜎∫^𝑇₀𝑔(𝑠)𝑑𝑠, there exists 𝑟₀(𝜆)>0 such that 𝑓(𝑢)≥𝐿𝑢 for each 0<𝑢≤𝑟₀(𝜆).
Fix 0<𝑟≤𝑟₀(𝜆), and let 𝑥∈𝒦 with ‖𝑥‖=𝑟. Then,𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+𝛾(𝑡)≥𝜆𝑚^𝑇₀𝑔(𝑠)𝐿𝑥(𝑠)𝑑𝑠+𝛾_∗≥𝜆𝑚𝐿𝜎‖𝑥‖^𝑇₀𝑔(𝑠)=‖𝑥‖,𝑑𝑠 (2.21) and thus ‖𝒯_𝜆𝑥‖≥‖𝑥‖.

Lemma 2.9. Suppose that conditions (𝐻0), (𝐻1), (𝐻2), (𝐻3), and (𝐻4) are satisfied. Then, if 𝑓_∞=0 then, there exists 𝑅₀(𝜆)>0 such that for every 𝑅≥𝑅₀(𝜆), we have 𝒯_𝜆𝑥≤‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=R. (2.22)

Proof. Since 𝑓_∞=0 for 𝜀(𝜆)=1/2𝜆𝑀∫^𝑇₀𝑔(𝑠)𝑑𝑠, there exists 𝑅₁(𝜆)>0 such that 𝑓(𝑢)≤𝜀𝑢 for each 𝑢≥𝑅₁(𝜆). We define 𝑅₀(𝜆)∶=max{𝑅₁(𝜆)/𝜎,2𝛾^∗}.
Fix 𝑅≥𝑅₀(𝜆), and let 𝑥∈𝒦 with ‖𝑥‖=𝑅. Then, 𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+𝛾(𝑡)≤𝜆𝑀^𝑇₀𝑔(𝑠)𝜀𝑥(𝑠)+𝛾^∗≤𝜆𝑀𝜀‖𝑥‖^𝑇₀𝑔(𝑠)𝑑𝑠+𝛾^∗=𝑅2+𝛾^∗≤𝑅2+𝑅2=𝑅=‖𝑥‖, (2.23) and thus ‖𝒯_𝜆𝑥‖≤‖𝑥‖.

Lemma 2.10. Assume that (𝐻0), (𝐻1), (𝐻2), (𝐻3), and (𝐻4) are fullfiled. Then, if 𝑓_∞=∞, there exists 𝑅₀(𝜆)>0 such that for every 𝑅≥𝑅₀(𝜆), we have 𝒯_𝜆𝑥≥‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=R. (2.24)

Proof. Since 𝑓_∞=∞ for 𝐿=𝐿(𝜆)=1/𝜆𝑚𝜎∫^𝑇₀𝑔(𝑠)𝑑𝑠, there exists 𝑅₁(𝜆)>0 such that 𝑓(𝑢)≥𝐿𝑢 for each 𝑢≥𝑅₁(𝜆). We define 𝑅₀(𝜆)∶=𝑅₁(𝜆)/𝜎.
Fix 𝑅≥𝑅₀(𝜆), and let 𝑥∈𝒦 with ‖𝑥‖=𝑅. Then, 𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+𝛾(𝑡)≥𝜆𝑚^𝑇₀𝑔(𝑠)𝐿𝑥(𝑠)𝑑𝑠+𝛾_∗≥𝜆𝑚𝐿𝜎‖𝑥‖^𝑇₀𝑔(𝑠)=‖𝑥‖,𝑑𝑠 (2.25) and thus ‖𝒯_𝜆𝑥‖≥‖𝑥‖.

In the sequel, we study separately the two different cases considered in condition (𝐻0); that is, 𝛾_∗>0 or 𝑐(𝑡)≡0.

2.2. The Case 𝛾_∗>0

Theorem 2.11. Assume that conditions (𝐻1), (𝐻2), (𝐻3), and (𝐻4) are fulfilled. If, moreover, 𝛾_∗>0, the following results hold:(1) there exists 𝜆₀>0 such that problem (1.2) has a positive solution if 0<𝜆<𝜆₀,(2) if 𝑓_∞=0, then problem (1.2) has a positive solution for every 𝜆>0,(3) if 𝑓_∞=∞, then there exists 𝜆₀>0 such that problem (1.2) has two positive solutions if 0<𝜆<𝜆₀,(4) if 𝑓₀>0 and 𝑓_∞>0, then there exists 𝜆₀>0 such that problem (1.2) has no positive solutions if 𝜆>𝜆₀.

Proof. Fix 0<𝑟<𝛾_∗. Then, for each 𝜆>0 and 𝑥∈𝒦 with ‖𝑥‖=𝑟, we have 𝒯_𝜆𝑥≥𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+𝛾(𝑡)≥𝛾_∗>𝑟=‖𝑥‖. (2.26)Part 1. Fix 𝑅>𝛾^∗(≥𝛾_∗>𝑟), and take 𝜆₀=𝜆₀(𝑅) given by Lemma 2.5. Then, from Theorem 2.4 (ii), it follows the existence of a positive solution for problem (1.2) if 0<𝜆<𝜆₀. Part 2. Fix 𝜆>0, and take 𝑅>max{𝑟,𝑅₀(𝜆)}, where 𝑅₀(𝜆) is given by Lemma 2.9. Then, from Theorem 2.4 (ii), it follows the existence of a positive solution for problem (1.2). Part 3.Fix 𝑅₂>𝑅₁>𝛾^∗(≥𝛾_∗>𝑟), and take 𝜆₀=min{𝜆₀(𝑅₁),𝜆₀(𝑅₂)}, where 𝜆₀(𝑅₁) and 𝜆₀(𝑅₂) are the given by Lemma 2.5.
Now, fix 0<𝜆<𝜆₀, and take 𝑅>max{𝑅₂,𝑅₀(𝜆)}, where 𝑅₀(𝜆) is given by Lemma 2.10. Therefore, from Theorem 2.4, it follows the existence of two positive solutions 𝑥₁ and 𝑥₂ for problem (1.2) such that 𝑟≤‖𝑥₁‖≤𝑅₁<𝑅₂≤‖𝑥₂‖≤𝑅. (2.27)Part 4. Since 𝑓₀>0 and 𝑓_∞>0, there exists 𝐿>0 such that 𝑓(𝑢)≥𝐿𝑢 for all 𝑢>0. Define 𝜆₀1∶=𝑚𝜎𝐿∫^𝑇₀𝑔(𝑠).𝑑𝑠 (2.28)
If for 𝜆>𝜆₀, there exists a positive solution 𝑥 of problem (1.2), we know that 𝑥∈𝒟(𝒯_𝜆) and, as consequence, 𝑥=𝒯_𝜆(𝑥)∈𝒦. Therefore, we deduce the following inequalities: ‖𝑥‖=𝒯_𝜆𝑥≥𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+𝛾(𝑡)≥𝜆𝑚^𝑇₀𝑔(𝑠)𝐿𝑥(𝑠)𝑑𝑠+𝛾_∗≥𝜆𝑚𝐿𝜎‖𝑥‖^𝑇₀𝑔(𝑠)>‖𝑥‖,𝑑𝑠 (2.29) and we attain a contradiction.

Example 2.12. Let us consider the forced Mathieu-Duffing-type equation 𝑥^′′+𝑎(1+𝑏cos(𝑡))𝑥−𝜆𝑥³=𝑐(𝑡), (2.30) which fits into expression (1.2) by defining 𝑎(𝑡)=𝑎(1+𝑏cos(𝑡)),𝑔(𝑡)=1 and 𝑓(𝑥)=𝑥³.
Equation (2.30), with 𝑐(𝑡)≡0, was studied in [13], where a sufficient condition for the existence of a 2𝜋-periodic solution is given. However, since the proof relies in the application of Schauder’s fixed point theorem in a ball centered at the origin, the trivial solution 𝑥(𝑡)≡0 is not excluded. The existence of a nontrivial solution was later obtained by Torres in [5, Corollary 4.2]. More precisely, Torres proves that if function 𝑎(𝑡)>0 for a.e. 𝑡∈[0,2𝜋] and ‖𝑎‖_𝑝<𝐾(2𝑝,2𝜋), then the homogeneous problem (𝑐(𝑡)≡0) (2.30) has at least two nontrivial one-signed 2𝜋-periodic solutions.
In this paper, as a consequence of Example 2.3 and Theorem 2.11, Part 3, we arrive at the following multiplicity result for the inhomogeneous (𝑐(𝑡)≢0) equation (2.30) with a not necessarily constant sign function 𝑎(𝑡).

Corollary 2.13. If condition (2.8) is satisfied and 𝛾_∗>0, then there exists 𝜆₀>0 such that (2.30) has at least two positive 2𝜋-periodic solutions, provided that 0<𝜆<𝜆₀.

2.3. The Case 𝑐(𝑡)≡0

Theorem 2.14. Assume that conditions (𝐻1), (𝐻2), (𝐻3), and (𝐻4) hold. If moreover 𝑐(𝑡)≡0 the following results hold: (1) if 𝑓₀=∞ or 𝑓_∞=∞, then there exists 𝜆₀>0 such that problem (1.2) has a positive solution if 0<𝜆<𝜆₀,(2) if 𝑓_∞=0 then there exists 𝜆₀>0 such that problem (1.2) has a positive solution for every 𝜆>𝜆₀,(3) if 𝑓₀=∞ and 𝑓_∞=0 then problem (1.2) has a positive solution for every 𝜆>0,(4) if 𝑓₀=∞ and 𝑓_∞=∞ then there exists 𝜆₀>0 such that problem (1.2) has two positive solutions if 0<𝜆<𝜆₀,(5) if 𝑓₀=0 and 𝑓_∞=∞, then problem (1.2) has a positive solution for every 𝜆>0,(6) if 𝑓₀=0 and 𝑓_∞=0 then there exists 𝜆₀>0 such that problem (1.2) has two positive solutions if 𝜆>𝜆₀,(7) if 𝑓₀>0 and 𝑓_∞>0, then there exists 𝜆₀>0 such that problem (1.2) has no positive solutions if 𝜆>𝜆₀.

Proof of Part 1. Fix 𝑅>𝛾^∗=0 and take 𝜆₀=𝜆₀(𝑅)>0 given by Lemma 2.5. In consequence, for all 0<𝜆≤𝜆₀(𝑅) we have 𝒯_𝜆𝑥≤‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=𝑅. (2.31)
Now, let 0<𝜆<𝜆₀ be fixed, and choose 0<𝑟<min{𝑅,𝑟₀(𝜆)}, where 𝑟₀(𝜆) is given by Lemma 2.8 when 𝑓₀=∞. In case of 𝑓_∞=∞, we get 𝑟>max{𝑅,𝑅₀(𝜆)}>𝑅, with 𝑅₀(𝜆) given by Lemma 2.10. In both situations, we arrive at 𝒯_𝜆𝑥≥‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=𝑟. (2.32)
Thus, Theorem 2.4 implies the existence of a positive solution for problem (1.2). Part 2.Fix 𝑟>0, and take 𝜆₀=𝜆₀(𝑟)>0 given by Lemma 2.6. Now, for each 𝜆>𝜆₀, take 𝑅>max{𝑟,𝑅₀(𝜆)}, with 𝑅₀(𝜆) given by Lemma 2.9, and apply Theorem 2.4.Part 3. For each 𝜆>0, take 𝑟₀(𝜆)<𝑅₀(𝜆) given by Lemmas 2.8 and 2.9, respectively, and apply Theorem 2.4.Part 4.Fix 𝑅₂>𝑅₁>𝛾^∗=0, and take 𝜆₀=min{𝜆₀(𝑅₁),𝜆₀(𝑅₂)} given by Lemma 2.5. Now, for each 0<𝜆<𝜆₀, take 𝑟<min{𝑅₁,𝑟₀(𝜆)} given by Lemma 2.8 and 𝑅>max{𝑅₂,𝑅₀(𝜆)} given by Lemma 2.10. Then, Theorem 2.4 implies the existence of two positive solutions 𝑥₁ and 𝑥₂ for problem (1.2) such that 𝑟≤𝑥₁≤𝑅₁<𝑅₂≤𝑥₂≤𝑅. (2.33)Part 5. Use Lemmas 2.7 and 2.10 and Theorem 2.4.Part 6. Use Lemmas 2.6, 2.7, and 2.9 and Theorem 2.4 twice.Part 7. The proof follows the same steps as Part 4 in Theorem 2.11.

Remark 2.15. Theorem 2.14 complements [1, Theorem  2.1], since it provides similar results for the problem 𝑥^′′+𝑘²𝑥=𝜆𝑔(𝑡)𝑓(𝑥), with 0<𝑘<𝜋/𝑇.

Example 2.16. Consider as a model the problem 𝑥^′′((𝑡1+𝑎1+𝑏cos))𝑥=𝜆𝑥^𝛼+𝜇𝑥^𝛽,𝑥(0)=𝑥(2𝜋),𝑥^′(0)=𝑥^′(2𝜋), (2.34) where 𝑎>0, 𝑏∈ℝ and 𝛼,𝛽,𝜇≥0. When 𝑏=1 and 𝜇=0, (2.34) is the Brillouin-beam focusing equation which has been widely studied in the literature (see [5, 10, 14] and references therein). Now, we have the following:

Corollary 2.17. Assume condition (2.8). Then, the following results are satisfied:(i) if 0≤𝛽<1, then problem (2.34) has a positive solution for every 𝜆>0.(ii) if 𝛽=1 and 𝜇>0, then there exists 𝜆₀>0 such that problem (2.34) has a positive solution for every 0<𝜆<𝜆₀, and there exists 𝜆₁>0 such that the problem has no positive solution for 𝜆>𝜆₁.(iii) if 𝛽>1 and 𝜇>0, then there exists 𝜆₀>0 such that problem (2.34) has two positive solutions for every 0<𝜆<𝜆₀.

Proof. Condition (2.8) implies that condition (𝐻1) is satisfied. Now, to prove (i), (ii) or (iii) it is enough to apply Theorem 2.14 Part 3, Part 1 and Part 7 or Part 4, respectively.

3. Nonnegative Green's Function

In this section instead of conditions (𝐻1) and (𝐻3), we assume (𝐻1) Problem (1.5) is nonresonant, the corresponding Green's function 𝐺(𝑡,𝑠) is nonnegative on [0,𝑇]×[0,𝑇], and 𝛽=min_{𝑡∈[0,𝑇]}∫^𝑇₀𝐺(𝑡,𝑠)𝑑𝑠>0,(𝐻3)𝑓∶[0,∞)→[0,∞) is continuous, and 𝑓(𝑢)>0 for all 𝑢>0.

Notice that 𝑥(𝑡)=∫^𝑇₀𝐺(𝑡,𝑠)𝑑𝑠 is the unique solution of the problem 𝑥^′′(𝑡)+𝑎(𝑡)𝑥(𝑡)=1,𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇), (3.1) and then (𝐻1) asks for this solution to be strictly positive. On the other hand, assumption (𝐻3) allows us to consider only regular problems. We will discuss to singular problems in Section 3.1 by means of a truncation technique.

For constant 𝑎(𝑡)≡𝑘², condition (𝐻1) is equivalent to 0<𝑘≤𝜋/𝑇. For nonconstant 𝑎(𝑡), condition (𝐻1) is satisfied provided that Lemma 2.2 holds.

On the other hand, under condition (𝐻1), it is allowed that𝑚=min_{[]𝑡,𝑠∈0,𝑇}𝐺(𝑡,𝑠)=0, (3.2) so 𝜎=𝑚/𝑀 can be equal to 0, and thus, the arguments used in the previous section do not work. So, by assuming that 𝛾_∗≥0, let us define 𝒦∶=𝑥∈𝒞([0,𝑇],[0,∞))∶^𝑇₀𝑥(𝑠)𝑑𝑠≥𝜎‖𝑥‖, (3.3) where 𝜎=min{𝛽/𝑇𝑀,∫^𝑇₀𝛾(𝑠)𝑑𝑠/𝑇‖𝛾‖} if ‖𝛾‖>0 or 𝜎=𝛽/𝑇𝑀 if ‖𝛾‖=0. As far as we know, the cone 𝒦 was introduced in [3].

Clearly, 0<𝜎≤1, and for 0<𝑟<𝑅, we define 𝒦_𝑟,𝑅∶=𝑥∈𝒦∶𝑟≤‖𝑥‖≤𝑅. (3.4)

Next, we prove the following result similar to Theorem 2.4.

Theorem 3.1. Assume that (𝐻0),(𝐻1),(𝐻2),(𝐻3), and (𝐻4) hold. Then, for each 𝜆>0 and 0<𝑟<𝑅, the operator 𝒯_𝜆∶𝒦_𝑟,𝑅→𝒦 given by (1.9) is well defined and completely continuous.
Moreover, if either (i)‖𝒯_𝜆𝑥‖≤‖𝑥‖ for any 𝑥∈𝒦 with ‖𝑥‖=𝑟 and ‖𝒯_𝜆𝑥‖≥‖𝑥‖ for any 𝑥∈𝒦 with ‖𝑥‖=𝑅, or (ii)‖𝒯_𝜆𝑥‖≥‖𝑥‖ for any 𝑥∈𝒦 with ‖𝑥‖=𝑟 and ‖𝒯_𝜆𝑥‖≤‖𝑥‖ for any 𝑥∈𝒦 with ‖𝑥‖=𝑅, then 𝒯_𝜆 has a fixed point in 𝒦_𝑟,𝑅, which is a nonnegative solution of problem (1.2).

Proof. If 𝑥∈𝒞([0,𝑇],[0,∞)) and assuming ‖𝛾‖>0 (the case ‖𝛾‖=0 being analogous), we obtain ^𝑇₀𝒯_𝜆𝑥(𝑡)𝑑𝑡=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠𝑑𝑡+^𝑇₀𝛾(𝑡)𝑑𝑡=𝜆^𝑇₀𝑔(𝑠)𝑓(𝑥(𝑠))^𝑇₀𝐺(𝑡,𝑠)𝑑𝑡𝑑𝑠+^𝑇₀𝛾(𝑠)𝑑𝑠≥𝜆𝛽^𝑇₀𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+^𝑇₀𝛾(𝑠)𝛽𝑑𝑠=𝜆𝑇𝑀𝑇𝑀^𝑇₀𝑔(𝑠)𝑓(𝑥(𝑠∫))𝑑𝑠+^𝑇0𝛾(𝑠)𝑑𝑠𝑇𝛾𝑇𝛾≥𝜎𝜆𝑇𝑀^𝑇₀𝑔(𝑠)𝑓(𝑥(𝑠𝛾≥𝜎𝒯))𝑑𝑠+𝑇_𝜆𝑥. (3.5) Thus, 𝒯_𝜆(𝒞([0,𝑇],[0,∞)))⊂𝒦, and it is standard to show that 𝒯_𝜆 is completely continuous. In consequence, from Krasnoselskii's fixed point theorem (see [12, p.148]), it follows the existence of a fixed point for 𝒯_𝜆 in 𝒦_𝑟,𝑅 which it is, by the definition of 𝒯_𝜆, a non negative solution of problem (1.2).

Now, we are going to give sufficient conditions to obtain ‖𝒯_𝜆𝑥‖≤‖𝑥‖ or ‖𝒯_𝜆𝑥‖≥‖𝑥‖. The combination of the next lemmas with Theorem 3.1 will allow us to prove existence and multiplicity results for problem (1.2).

Lemma 3.2. Suppose that the conditions (𝐻0),(𝐻1),(𝐻2),(𝐻3), and (𝐻4) are satisfied. Then, for each 𝑅>𝛾^∗, there exists 𝜆₀(𝑅)>0 such that for every 0<𝜆≤𝜆₀(𝑅), we have 𝒯_𝜆𝑥≤‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=R. (3.6)

Proof. Fix 𝑅>𝛾^∗, and let 𝑥∈𝒦 with ‖𝑥‖=𝑅. If 0<𝜆≤𝜆₀(𝑅)∶=𝑅−𝛾^∗𝑀max_{[]𝑢∈0,𝑅}𝑓(𝑢)∫^𝑇₀𝑔(𝑠),𝑑𝑠 (3.7) then 𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+𝛾(𝑡)≤𝜆𝑀max_{[]𝑢∈0,𝑅}𝑓(𝑢)^𝑇₀𝑔(𝑠)𝑑𝑠+𝛾^∗≤𝑅=‖𝑥‖, (3.8) and thus ‖𝒯_𝜆𝑥‖≤‖𝑥‖.

Lemma 3.3. Assume that (𝐻1),(𝐻2),(𝐻3), (𝐻4), and ∫^𝑇₀𝛾(𝑠)𝑑𝑠>0 are satisfied. Then, there exists 𝑟₀>0 such that for each 0<𝑟<𝑟₀, we have 𝒯_𝜆𝑥≥‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=r. (3.9)

Proof. Fix 0<𝑟<𝑟₀∶=1/𝑇∫^𝑇₀𝛾(𝑠)𝑑𝑠, and let 𝑥∈𝒦 with ‖𝑥‖=𝑟. Then, 𝒯_𝜆𝑥≥1𝑇^𝑇₀𝒯_𝜆𝑥(𝑡)𝜆𝑑𝑡=𝑇^𝑇₀𝐺()𝑔(𝑠)𝑓(𝑥(𝑠1𝑡,𝑠))𝑑𝑠𝑑𝑡+𝑇^𝑇₀𝛾(𝑡)𝑑𝑡≥𝑟₀>𝑟=‖𝑥‖, (3.10) and thus ‖𝒯_𝜆𝑥‖≥‖𝑥‖.

Lemma 3.4. Let (𝐻0),(𝐻1),(𝐻2),(𝐻3), and (𝐻4) be fulfilled. Then, if 𝑓_∞=0, there exists 𝑅₀(𝜆)>0 such that for every 𝑅≥𝑅₀(𝜆), we have 𝒯_𝜆𝑥≤‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=R. (3.11)

Proof. Define 𝑓(𝑢)=max_{0≤𝑧≤𝑢}𝑓(𝑧). Clearly, 𝑓 is a nondecreasing function on [0,∞); moreover, since 𝑓_∞=0, it is obvious that lim_𝑢→∞𝑓(𝑢)𝑢=0. (3.12)
Therefore, we have that for 𝜀(𝜆)=1/2𝜆𝑀∫^𝑇₀𝑔(𝑠)𝑑𝑠, there exists 𝑅₁(𝜆)>0 such that 𝑓(𝑢)≤𝜀𝑢 for each 𝑢≥𝑅₁(𝜆).
Define 𝑅₀(𝜆)∶=max{𝑅₁(𝜆),2𝛾^∗}, fix 𝑅≥𝑅₀(𝜆), and let 𝑥∈𝒦 with ‖𝑥‖=𝑅. Then 𝒯_𝜆𝑥(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+𝛾(𝑡)≤𝜆^𝑇₀𝐺()𝑔(𝑠)𝑓𝑥(𝑡)‖𝑥‖𝑡,𝑠(‖‖)𝑑𝑠+𝛾≤𝜆𝑀𝜀^𝑇₀𝑔(𝑠)𝑑𝑠+𝛾^∗=𝑅2+𝛾^∗≤𝑅2+𝑅2=𝑅=‖𝑥‖, (3.13) and thus ‖𝒯_𝜆𝑥‖≤‖𝑥‖.

Theorem 3.5. Assume (𝐻0),(𝐻1),(𝐻2),(𝐻3), and (𝐻4). The following results hold:(1) if ∫^𝑇₀𝛾(𝑠)𝑑𝑠>0. then there exists 𝜆₀>0 such that problem (1.2) has a nonnegative solution if 0<𝜆<𝜆₀, (2) if ∫^𝑇₀𝛾(𝑠)𝑑𝑠>0 and 𝑓_∞=0, then problem (1.2) has a nonnegative solution for every 𝜆>0.

Proof. The first assertion is a direct consequence of Lemmas 3.2 and 3.3. The second part follows from Lemmas 3.3 and 3.4

Now, we will impose a strong condition on function 𝑔 by assuming that 𝑔 is strictly positive on the whole interval. (𝐻2)𝑔∈𝐿¹([0,𝑇]), 𝑔(𝑡)≥𝑔_∗>0 for a.e. 𝑡∈[0,𝑇].

Lemma 3.6. Assume that conditions (𝐻0),(𝐻1),(𝐻2),(𝐻3), and (𝐻4) are satisfied. Then, if 𝑓₀=∞, there exists 𝑟₀(𝜆)>0 such that for every 0<𝑟≤𝑟₀(𝜆), we have 𝒯_𝜆𝑥≥‖𝑥‖,for𝑥∈𝒦with‖𝑥‖=r. (3.14)

Proof. Since 𝑓₀=∞ for 𝐿=𝐿(𝜆)=𝑇/𝜆𝛽𝜎𝑔_∗, there exists 𝑟₀(𝜆)>0 such that 𝑓(𝑢)≥𝐿𝑢 for each 0≤𝑢≤𝑟₀(𝜆).
Fix 0<𝑟≤𝑟₀(𝜆), and let 𝑥∈𝒦 with ‖𝑥‖=𝑟. Then, 𝒯_𝜆𝑥≥1𝑇^𝑇₀𝒯_𝜆𝑥(𝑡)𝜆𝑑𝑡=𝑇^𝑇₀𝐺()𝑔(𝑠)𝑓(𝑥(𝑠1𝑡,𝑠))𝑑𝑠𝑑𝑡+𝑇^𝑇₀𝛾(𝑡)≥𝜆𝑑𝑡𝑇^𝑇₀𝐺()𝑔(𝑠)𝑓(𝑥(𝑠≥𝜆𝑡,𝑠))𝑑𝑡𝑑𝑠𝑇𝛽𝑔_∗𝐿^𝑇₀𝑥(𝑠)≥𝜆𝑑𝑠𝑇𝛽𝑔_∗𝐿𝜎‖𝑥‖=‖𝑥‖, (3.15) and thus ‖𝒯_𝜆𝑥‖≥‖𝑥‖.

Now, we are in a position to present the main result of this section.

Theorem 3.7. Suppose that conditions (𝐻0),(𝐻1),(𝐻2),(𝐻3), and (𝐻4) are fulfilled. The following assertions are satisfied:(1) if 𝑓₀=∞, then there exists 𝜆₀>0 such that problem (1.2) has a nonnegative solution if 0<𝜆<𝜆₀, (2) if 𝑓₀=∞ and 𝑓_∞=0, then problem (1.2) has a nonnegative solution for every 𝜆>0, (3) if 𝑓₀>0 and 𝑓_∞>0, then there exists 𝜆₀>0 such that problem (1.2) has no nonnegative solutions if 𝜆>𝜆₀.

Proof. The first assertion is a direct consequence of Lemmas 3.2 and 3.6. The second part follows from Lemmas 3.4 and 3.6.
To prove Part 3, by using that 𝑓₀>0 and 𝑓_∞>0, we know that there exists 𝐿>0 such that 𝑓(𝑢)≥𝐿𝑢 for all 𝑢≥0.
By defining 𝜆₀𝑇∶=𝛽𝜎𝐿𝑔_∗, (3.16) we have that if there is any 𝜆>𝜆₀ for which there exists a nonnegative solution 𝑥 of problem (1.2), then 𝑥=𝒯_𝜆𝑥∈𝒦. So, we arrive at the following contradiction: 𝒯_𝜆𝑥≥1𝑇^𝑇₀𝒯_𝜆𝑥(𝑡)𝜆𝑑𝑡=𝑇^𝑇₀𝐺()𝑔(𝑠)𝑓(𝑥(𝑠1𝑡,𝑠))𝑑𝑠𝑑𝑡+𝑇^𝑇₀𝛾(𝑡)≥𝜆𝑑𝑡𝑇^𝑇₀𝐺()𝑔(𝑠)𝑓(𝑥(𝑠≥𝜆𝑡,𝑠))𝑑𝑡𝑑𝑠𝑇𝛽𝑔_∗𝐿^𝑇₀𝑥(𝑠)≥𝜆𝑑𝑠𝑇𝛽𝑔_∗𝐿𝜎‖𝑥‖>‖𝑥‖. (3.17)

3.1. Applications to Singular Equations

Despite the fact that in the previous results we deal with regular functions, it is possible to apply some of them to the singular equation𝑥^′′(𝑡)+𝑎(𝑡)𝑥(𝑡)=𝜆𝑔(𝑡)𝑓(𝑥(𝑡))+𝑐(𝑡),𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇), (3.18) by means of a truncation technique.

To this end, we will consider a function 𝑓 that satisfies (𝐻5)𝑓∶(0,∞)→(0,∞) is a continuous function such that 𝑓_∞=0.

Theorem 3.8. Assume that 𝛾_∗>0 and conditions (𝐻1), (𝐻2), (𝐻4), and (𝐻5) hold.
Then, problem (3.18) has a positive solution for every 𝜆>0.

Proof. Let 𝑟=𝛾_∗>0, and define the function 𝑓_𝑟(𝑢)=𝑓(𝑟)𝑓(𝑢),if0≤𝑢<𝑟,,if𝑢≥𝑟. (3.19)
From (𝐻5), it follows that 𝑓_𝑟 satisfies condition (𝐻3), and (𝑓_𝑟)_∞=0. Moreover, 𝛾_∗>0 implies that ∫^𝑇₀𝛾(𝑠)𝑑𝑠>0. As consequence, Theorem 3.5, Part 2, implies that the modified problem 𝑥^′′(𝑡)+𝑎(𝑡)𝑥(𝑡)=𝜆𝑔(𝑡)𝑓_𝑟(𝑥(𝑡))+𝑐(𝑡),𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇) (3.20) has a nonnegative solution 𝑥_𝑟 for all 𝜆>0. Such function is given by the expression 𝑥_𝑟(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑓_𝑟𝑥_𝑟(𝑠)𝑑𝑠+𝛾(𝑡). (3.21) The nonnegativeness of functions 𝐺, 𝑔 and 𝑓_𝑟 implies that the solution 𝑥_𝑟(𝑡)≥𝛾_∗=𝑟 for all 𝑡∈[0,𝑇]. Therefore, 𝑥_𝑟 is a positive solution of problem (3.18).

Remark 3.9. Theorem 3.8 is an alternative result to those obtained in [15, 16] by means of Schauder's fixed point theorem.

Example 3.10. Let us consider the repulsive singular differential equation 𝑥^′′+𝑎(𝑡)𝑥=𝜆𝑔(𝑡)𝑥^1/2−ln(𝑥)+𝑐(𝑡). (3.22) Since 𝑓(𝑥)=𝑥^1/2−ln(𝑥) satisfies (𝐻5), we can apply Theorem 3.8 to obtain the following.

Corollary 3.11. Assume that (𝐻1), (𝐻2), and (𝐻4) hold. If 𝛾_∗>0 then, (3.22) has a positive 𝑇-periodic solution for every 𝜆>0.

Acknowledgment

This work was partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, project no. MTM2010-15314.