Study on a kind of ϕ-Laplacian Liénard equation with attractive and repulsive singularities

In this paper, by application of the Manasevich-Mawhin continuation theorem, we investigate the existence of a positive periodic solution for a kind of ϕ-Laplacian singular Liénard equation with attractive and repulsive singularities.


Introduction
Liénard equation [] x + f (x)x + g(x) =  (.) appears as a simplified model in many domains in science and engineering. It was intensively studied during the first half of the th century as it can be used to model oscillating circuits or simple pendulums. For example, Van der Pol oscillator where g was on the singular case, i.e., when g(t, x) → +∞, as x →  + . By application of coincidence degree theory, the author obtained that (.) had at least one periodic solution. Afterwards, Jebelean and Mawhin investigated the following quasilinear equation of p-Laplacian type: where g satisfied slightly strong singularity, i.e., The authors proved that the above problem had at least one positive periodic solution through a basic application of the Manasevich-Mawhin continuation theorem. Recently, Xin and Cheng [] studied the following p-Laplacian Liénard equation with singularity and deviating argument: By applications of coincidence degree theory and some analysis skills, they obtained that (.) had at least one positive periodic solution.
All the aforementioned results concern singular Liénard equation and singular p-Laplacian Liénard equation. There are few results on the φ-Laplacian Liénard equation with singularity. Motivated by [, , ], in this paper, we further consider the following φ-Laplacian Liénard equation: where f : R → R is an L  -Carathéodory function, which means, it is measurable in the first variable and continuous in the second variable, and for every  < r < s, there exists h r,s ∈ L  [, T] such that |g(t, x(t))| ≤ h r,s for all x ∈ [r, s] and a.e. t ∈ [, T]; τ is a positive constant; e ∈ L  (R) is a T-periodic function; g : (, +∞) → R is the L  -function, the nonlinear term g of (.) can be with a singularity at origin, i.e., It is said that (.) is of attractive type (resp. repulsive type) if g(x) → +∞ (resp. g(x) → -∞) as x →  + . Moreover, φ : R → R is a continuous function and φ() = , which satisfies It is easy to see that φ represents a large class of nonlinear operators, including |u| p- u : R → R which is a p-Laplacian operator.
The remaining part of the paper is organized as follows. In Section , we give some preliminary lemmas. In Section , by employing the Manasevich-Mawhin continuation theorem, we state and prove the existence of a positive periodic solution for (.) with attractive singularity. In Section , we investigate the existence result for (.) with repulsive singularity. In Section , two numerical examples demonstrate the validity of the method. Our results improve and extend the results in [, , -].

Preliminary lemmas
For the T-periodic boundary value problem

Lemma . (Manasevich-Mawhin []) Let be an open bounded set in C
then the periodic boundary value problem (.) has at least one periodic solution on .
Next, we embed equation (.) into the following equation family with a parameter λ ∈ (, ]: By applications of Lemma ., we obtain the following result. Lemma . Suppose that (A  ) and (A  ) hold. Assume that there exist positive constants E  , E  , E  and E  < E  such that the following conditions hold: Then (.) has at least one T-periodic solution.

Main results (I): periodic solution of (1.4) with attractive singularity
In this section, we investigate the existence of a positive periodic solution for (.) with attractive singularity.
Theorem . Assume that conditions (A  ) and (A  ) hold. Suppose that the following conditions hold: There exist positive constants a, b and m such that Proof Firstly, we claim that there exists a point t  ∈ [, T] such that Let t, t be, respectively, the global minimum point and the global maximum point x(t) on [, T]; then x (t) =  and x (t) = , and we claim that Similarly, we can get From (H  ), (.) and (.), we have In view of x being a continuous function, we can get (.).
Multiplying both sides of (.) by x (t) and integrating over the interval [, T], we have Therefore, we can get It is easy to see that there exists a positive constant M  (independent of λ) such that From (.) and (.), we have On the other hand, integrating both sides of (.) over [, T], we have Therefore, from (.), (.) and (H  ), we have such that x (t  ) = , while φ() = , from (.), (.) and (.), we have Then we can get which is a contradiction. So, (.) holds. From (.), we have Multiplying both sides of (.) by x (t) and integrating on [ By (.) and (.), we can get Moreover, from (.), we have From (.), we have From (H  ), we know that there exists a constant M  >  such that Then condition () of Lemma . is satisfied. For a possible solution C to equation it satisfies E  < C < E  . Therefore, condition () of Lemma . holds. Finally, we consider condition () of Lemma . is also satisfied. In fact, from (H  ), we have So condition () is also satisfied. By application of Lemma ., we get that (.) has at least one positive periodic solution.

Main results (II): periodic solution of (1.4) with repulsive singularity
In this section, we consider (.) in the case that f (t, x) ≡ f (x). Then (.) can be written as We will discuss the existence of a positive periodic solution for (.) with repulsive singularity.
Theorem . Assume that conditions (A  ) and (A  ) hold. Suppose that the following conditions hold: There exists a constant γ * >  such that inf x∈R |f (x)| ≥ γ * > . Then (.) has at least one positive T-periodic solution.
Proof Firstly, we embed equation (.) into the following equation family with a parameter λ ∈ (, ]: (.) Integrating both sides of (.) over [, T], from (H *  ), we have From the continuity of g, we know there exists t *  ∈ [, T] such that We follow the same strategy and notation as in the proof of Theorem .. We know that there exists M *  >  such that Next, we prove that there exists a positive constant M *  such that x ≤ M *  . In fact, we get from (.) that Thus, from (.) we know that there exists some positive constant M *  such that The proof left is the same as that of Theorem ..