Existence of Homoclinic Orbits for a Singular Differential Equation Involving p-Laplacian

In the last years, homoclinic solutions for Hamiltonian systems and differential and difference systems have been studied by several authors. Based on variational methods and critical points theory, Rabinowitz [1] has given fundamental contributions to homoclinic solutions for Hamiltonian systems. Carriãro and Miyagaki [2] obtained the existence of homoclinic orbits for second-order timedependent Hamiltonian systems. Izydorek and Janczewska [3] obtained that a homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second-order differential equations. By means of an extension of Mawhin’s continuation theorem, Lu et al. [4] obtained the existence of homoclinic solutions for a class of second-order neutral functional differential systems. Ding and Guo [5] showed that there exists at least one homoclinic solution for the anomalous diffusion system. For more results about homoclinic solutions, see, e.g., [6–10] and relevant references. In recent years, homoclinic solution problems of secondorder singular differential equation have raised concerns. Bonheure and Torres [11] studied the existence of homoclinic solutions for the model scalar second-order boundary value problem


Introduction
In the last years, homoclinic solutions for Hamiltonian systems and differential and difference systems have been studied by several authors. Based on variational methods and critical points theory, Rabinowitz [1] has given fundamental contributions to homoclinic solutions for Hamiltonian systems. Carriãro and Miyagaki [2] obtained the existence of homoclinic orbits for second-order timedependent Hamiltonian systems. Izydorek and Janczewska [3] obtained that a homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second-order differential equations. By means of an extension of Mawhin's continuation theorem, Lu et al. [4] obtained the existence of homoclinic solutions for a class of second-order neutral functional differential systems. Ding and Guo [5] showed that there exists at least one homoclinic solution for the anomalous diffusion system. For more results about homoclinic solutions, see, e.g., [6][7][8][9][10] and relevant references.
In recent years, homoclinic solution problems of secondorder singular differential equation have raised concerns. Bonheure and Torres [11] studied the existence of homocli-nic solutions for the model scalar second-order boundary value problem where a, b, c ∈ Cðℝ, ℝÞ, p > 0: When c = 0, Equation (1) has a good variational structure and can be studied by variational method for Equation (1), see [1,12,13]. When c ≠ 0, variational method cannot be used to study Equation (1) because of the no good variational structure. Hence, based on the method of the upper and lower solutions and fixed point theorem on cones, the authors obtained the existence of homoclinic solutions for Equation (1) which is different from the variational methods used in [14][15][16]. Motivated by the above work, this paper is devoted to the study of the existence of homoclinic solutions to secondorder singular differential equation with p-Laplacian: where ϕ p ðsÞ = jsj p−2 s, p > 1, f , g, h, e ∈ Cðℝ, ℝÞ with hðt + TÞ = hðtÞ > 0. As in the literature, a solution uðtÞ of Equation (2) is called a homoclinic solution if uðtÞ ⟶ 0 as |t | ⟶ + ∞: When such a solution satisfies in addition to u ′ ðtÞ ⟶ 0 as |t | ⟶ + ∞, it is usually called a homoclinic solution or a pulse, although here, 0 is not a stationary solution of Equation (2). Since Equation.
(2) is a strongly nonlinear equation, the traditional methods (including fixed point theorem and lower and upper solutions) are no longer applicable to study homoclinic solutions to Equation (2), so a new continuation theorem due to Mana'sevich and Mawhin will be developed for studying Equation (2). The distinctive contributions of this paper are outlined as follows: (1) The problem (2) is a more general form compared with existing problems (see [1,[11][12][13] The following sections are organized as follows: In Section 2, some useful lemmas and notations are given. In Section 3, sufficient conditions are established for the existence of homoclinic solutions of (2). In Section 4, two examples are given to show the feasibility of our results. Finally, Section 5 concludes the paper.

Preliminary and Some Lemmas
In this section, we give some notations and lemmas which will be used in this paper. The set of all positive integers is denoted by N. Let with the norm jφj 0 = max (2) is a nonlinear operator, the famous Mawlin's continuation theorem [18] cannot be directly applied to (2). In order to generalize Mawlin's continuation theorem, Mana'sevich and Mawhin [17] obtained the following continuation theorem for nonlinear systems with p-Laplacian-like operators: (1) For each λ ∈ ð0, 1Þ, the problem has no solution on ∂Ω Theorem 1. Assume that Ω is an open bounded set in C T such that the above conditions (4) -(6) hold.
Then, problem has a solution in Ω.
Lemma 2 (see [19]). If u ∈ Cðℝ, ℝÞ, a > 0, μ 1 , μ 2 > 1 are constants, then for every t ∈ ℝ , the following inequality holds: Lemma 3 (see [4]). Let fu k g ∈ C 1 2kT be a sequence of 2kTperiodic functions, such that for each k ∈ N, u k satisfies where A 0 , A 1 are constants independent of k ∈ N. Then, there exist u 0 ∈ Cðℝ, ℝÞ and a subsequence fu k j g of fu k g such that for each j ∈ N For investigating the existence of homoclinic solutions to (2), for each k ∈ N, we firstly consider the existence of 2kT-periodic solutions u k ðtÞ for the following equation: Journal of Function Spaces where e k : ℝ ⟶ ℝ is a 2kT-periodic extension such that here ε 0 ∈ ð0, T/2Þ is a constant.
In the present paper, we list the following assumptions: (H 1 ). f : ℝ ⟶ ℝ is a continuous bounded nonnegative function (H 2 ). g : ℝ ⟶ ℝ is strictly monotone increasing and there are positive constants σ and n such that

Main Results
Let yðtÞ = 1 − xðtÞ, then (11) is changed into the following form: Obviously, the existence of 2kT-periodic solutions to (2) is a transfer to the existence of 2kT-periodic solutions to (15). For (15), consider the corresponding parameter equation: Here, we give the main results of the present paper in the following theorem.