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Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian via a geometric approach
- Xuelei Wang^{1, 2},
- Qihuai Liu^{3}Email author and
- Dingbian Qian^{1}
- Received: 28 August 2015
- Accepted: 4 February 2016
- Published: 15 February 2016
Abstract
New results on the existence and multiplicity of the solutions for some nonlinear boundary value problems with singular ϕ-Laplacian are obtained via a bend-twist fixed point theorem. These results improve related theorems in the previous literature. Moreover, the geometric approach in this paper provides a new method to investigate the existence and multiplicity of periodic motions of charged particles in a three-dimensional electromagnetic field.
Keywords
- periodic solution
- geometric approach
- relativistic oscillator
- singular ϕ-Laplacian
1 Introduction
Equation (1.1) includes many interesting physical and mechanical models. A significant example is that equation (1.1) describes the dynamics of a charged particle in electric and magnetic fields when the particle velocities are relativistic [1].
When the magnetic potential A is independent of time t, equation (1.3) corresponds to an autonomous Hamiltonian system of second-order differential equations, which is discussed by Moser and Zehnder [2], pp.16-17. These problems were also studied by many mathematicians and physicists (see [3–7]).
Recently, Bereanu and Mawhin obtained several interesting results on the existence and multiplicity of the solutions for various boundary value problems of equation (1.4) by using the Leray-Schauder degree, the lower and upper solutions, and the method of variational calculation; see [8–15] and other related works; we can refer to [16–19]. In particular, a universal existence theorem for Dirichlet problem and an existence theorem for periodic and Neumann problem were established when f satisfies some sign conditions [8]. Moreover, the problem for the existence of periodic solutions under the Hartman-type condition or the modified Hartman-type condition was considered in [20].
The existence of periodic solutions for nonlinear problems with singular ϕ-Laplacian was extended to the n-dimensional case by Brezis and Mawhin [26], where the approach is mostly variational, but requires the use of results on an auxiliary system based upon fixed point theory and Leray-Schauder degree. The multiplicity and existence of periodic solutions in the n-dimensional case was proved by Mawhin in [27] by using a Lusternik-Schnirelman-type multiplicity result for some indefinite functionals. We refer to [28, 29] for the related developments.
The purpose of this paper is to find some method to deal with the problem with higher-dimensional system (1.1). Our discussion is based on a geometric idea by phase-space analysis. We show that when f satisfies some general condition, then the solutions of (1.1) have some ‘bend-twist’ property in the generalized phase-space. This observation inspires us constructing a bend-twist fixed point theorem to prove the existence and multiplicity of periodic solutions for three-dimensional system (1.1).
More specifically, we extend the universal existence theorem for Dirichlet problem and the existence theorem for periodic (or Neumann) problem in [8] to the existence and multiplicity theorem for three-dimensional system, respectively. Even for the scalar equation (1.6), an interesting feature of our result is that the existence of periodic solutions is independent of the condition of \(g'(x)\) and the damping coefficient f.
The paper is organized as follows. Section 2 is devoted to introducing some preliminary results with respect to equation (1.1). A geometric fixed theorem (Theorem 2.1) is proved by simple topological degree argument. This theorem is the basic tool used in this paper. In Section 3.1, we obtain a universal existence theorem for equation (1.1) with the Dirichlet condition, which generalizes the result in [8]. This is not the case for other boundary condition; an existence and multiplicity result (see Theorem 3.2) is proved in Section 3.2 when f satisfies some local sign condition. Periodic motions of relativistic oscillators of charged particles in a three-dimensional electromagnetic field are investigated in Section 3.3. In Section 4, a new result on the existence and multiplicity of periodic solutions of generalized pendulum-type equations that does not need any information upon the differentiability of g is obtained by the geometric approach.
2 Bend-twist theorem and some preliminary results
In this section, we first introduce a geometric fixed point theorem, which will be used frequently in the subsequent sections. The geometric fixed point theorem is a small variation of the Poincaré-Miranda theorem (see [30] for instance), which goes back to Poincaré (1883) and has been used many times in the study of boundary value problems and periodic solutions. For example, see a recent paper [31] and the references therein.
Theorem 2.1
(Bend-twist theorem)
Since the only difference between Theorem 2.1 and Poincaré-Miranda theorem is the permutation of the indexes in \(\mathcal{F}\), that is, we have to consider \(F_{j_{i}}\) instead of \(F_{i}\), we omit the proof of Theorem 2.1. We can refer to [31] for an elementary proof based upon basic exterior calculus.
Next, we perform some preliminary results on the existence and uniqueness of a solution of initial value problem for equation (1.1) based on phase-plane analysis, which imply that the Poincaré mapping of (1.1) can be well defined.
Let \(C_{T}\) denote the Banach space of continuous functions on \([0,T]\) with uniform norm \(\|\cdot\|_{\infty}\), and \(\widetilde{C}_{T}\) denote the Banach space of continuous and T-periodic functions with zero mean value. We also denote the inverse function of ϕ by \(\phi^{-1}:\mathbb{R}^{3}\rightarrow B_{a}(0)\) (\(0< a<+\infty\)). Since \(\phi_{i}\) (\(i=1,2,3\)) is a monotonous homeomorphism with respect to the ith component of variable, the inverse function \(\phi_{i}^{-1}\) is also a monotonous homeomorphism with respect to the ith component of variable.
Lemma 2.1
Every solution of equation (1.1) can be uniquely defined on the interval \((-\infty,+\infty)\).
Proof
Since \(f:[0,T]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}\) is locally Lipschitz continuous, every solution of equation (2.1) with the initial value \((x_{0},y_{0})\) exists uniquely.
3 Three-dimensional systems with singular ϕ-Laplacian
3.1 The case of Dirichlet boundary condition
Theorem 3.1
In fact, the existence of Theorem 3.1 for Dirichlet boundary conditions is well known for more general systems from the work of Bereanu and Mawhin [12]. Therefore, we leave the proof for the reader while using directly Theorem 2.1 or topology degree theory, and we can also refer to the proof of Theorem 3.2.
3.2 Periodic or Neumann problems with nonlinearities
Lemma 3.1
If there exists a component function \(f_{i}:[0,T]\times\mathbb{R}^{3}\times\mathbb{R}^{3}\rightarrow \mathbb{R}\) such that \(f_{i}\) is always positive or negative on its domain, then equation (1.1) with periodic boundary condition or Neumann boundary condition has no solution.
Proof
In the following, we will show that a general sign condition upon f can guarantee the existence and multiplicity of solutions for equation (1.1) with periodic or Neumann boundary condition.
Theorem 3.2
Proof
Consequently, \(\mathcal{F}\) satisfies the bend-twist condition on parallelotope domain \(\mathcal{D}_{k}\). By Theorem 2.1, \(\mathcal{F}\) has at least a zero point in domain \(\mathcal{D}_{k}\). Hence, the existence of solutions is proved.
Finally, we show that the continuity condition on f can take place of the Lipschitz condition in our results by the method developed by Ding et al. [32].
We remark that conditions (3.3) and (3.4) for Neumann or periodic problems in Theorem 3.2 were first introduced by Bereanu and Mawhin, where they have established the existence result for one-dimensional nonlinear problems with singular ϕ-Laplacian [8], Theorem 2. Moreover, the proof of Theorem 3.2 involves an idea of shooting approach to nonlinear system with singular ϕ, and we can refer to [33].
Corollary 3.1
Proof
Corollary 3.2
Corollary 3.2 is easily deduced from Corollary 3.1.
Example 3.1
These results also hold in the case of one-dimensional plane problems with singular ϕ-Laplacian. An immediate consequence discussed by Bereanu and Mawhin ([8], Theorem 2) follows from Theorem 3.2.
Remark 3.1
It is not difficult to see that the results also hold for Neumann boundary conditions.
3.3 Relativistic equations of charged particles in an electromagnetic field
Theorem 3.3
Assume that (H_{1}) holds and the electric potential V is a continuous differentiable function. Then for any continuous T-periodic vector magnetic potential \(\mathrm{A}(t)\) with zero mean value of every component variable, system (1.3) admits at least one T-periodic motion.
Proof
Example 3.2
4 Generalized Liénard differential equations with periodic boundary value condition
In this section, we investigate the generalized Liénard differential equations. The existence, multiplicity, and dependence on a parameter are considered. The results are illustrated with some examples.
4.1 Existence of periodic solutions under a sign condition
In what follows, we will apply Theorem 2.1 to discuss the existence of T-periodic solutions for equation (4.1).
Theorem 4.1
If g satisfies the sign condition (H_{2}) and the function F is continuous differentiable, then for any given \(T>0\), equation (4.1) has at least one T-periodic solution.
Proof
Let \((\mathrm{x}(t;\mathrm{x}^{(0)},\mathrm{y}^{(0)}), \mathrm {y}(t;\mathrm{x}^{(0)},\mathrm{y}^{(0)}) )\) be a solution of equation (4.2). In the following, we will prove the existence of zero points of the mapping \(\mathcal{F}=\mathcal{P}-\mathrm{id}\), which corresponds to a T-periodic solution of (4.2).
Consequently, \(\mathcal{F}\) satisfies the bend-twist condition on the parallelotope domain \(\mathcal{D}\). By Theorem 2.1, \(\mathcal{F}\) has at least a zero point in \(\mathcal{D}\). Hence, the existence of solutions is proved. □
It is interesting to note that no assumption on the friction term \(\mathrm{F}(\mathrm{x})\) is required. Some direct consequences follow from Theorem 2.1.
Corollary 4.1
Proof
The limits \(\lim_{x_{i}\to\pm\infty}g_{i}(t,\mathrm{x})=\pm\infty\) (or \(\lim_{x_{i}\to\pm\infty}g_{i}(t,\mathrm{x})=\mp\infty\)) imply that there exist \(d_{i}>0\) (\(i=1,2,3\)) such that \(x_{i}g_{i}(t,\mathrm{x})>0\) (or \(x_{i}g_{i}(t,\mathrm{x})<0\)) for all \(t\in\mathbb{R}\) and \(|x_{i}|>d_{i}\). Therefore, sign condition (H_{2}) holds. By applying Theorem 4.1 we end the proof. □
Example 4.1
Notice that when \(i=1\), the equation drops into a one-dimensional system. The results discussed before also hold. A direct consequence discussed in [22] is the following example.
Example 4.2
4.2 Existence and multiplicity of periodic solutions without sign condition
In most cases, the function g does not satisfy sign condition (H_{1}). It is obvious that Theorem 4.1 cannot be applied to the relativistic pendulum-type equations since the potentials of pendulum-type equations are fluctuating and oscillating. But in this circumstance, the pendulum-type equations satisfy some sign property locally.
Theorem 4.2
The proof of Theorem 4.2 is a simple adaptation of that one of Theorem 3.2, and we do not repeat it here.
Theorem 4.3
Moreover, if \(\xi_{k+1}-\xi_{k}\geq(b_{k+1}+c_{k})/2+2aT\), \(k=1,2,\ldots ,n-1\), then equation (4.12) has at least n geometric distinct T-periodic solutions \(x_{k}(t)\) such that \(x_{k+1}(t)-x_{k}(t)>0\), \(k=1,2,\ldots,n-1\).
Proof
Corollary 4.2
For any values a, k and for any \(p(t)\in\widetilde{C}_{T}\), equation (4.15) has at least one T-periodic solution, provided that \(2cT<\pi\).
Proof
Obviously, we have generalized the result [23], Theorem 3. It is worth mentioning that for the particular sinx in equation (4.15), Torres has improved his method and obtained a better result [24], Corollary 3. However, for general g, his method strictly depends on the differentiability of the function g; see [24], Theorem 4. Note that our results do not need any information on the differentiability of g.
Remark 4.1
We can see that if \(x(t)\) is a T-periodic solution of equation (4.15), then \(x(t)+2k\pi\), \(k\in\mathbb{Z}\), also is a T-periodic solution. In fact, there exists a positive increasing sequence \(\{\xi_{k}=k\pi\}_{k=1}^{+\infty}\) that satisfies condition (H_{3}) of Theorem 4.3.
Furthermore, it is interesting to note that no assumption on the friction coefficient \(f(x)\) is required. So we obtain the following simple result.
Corollary 4.3
Example 4.3
In case \(g(x)\) does not satisfy (H_{3}), also T-periodic solutions of equation (4.12) can exist. In the following, we will consider the existence of T-periodic solutions with some other conditions on g.
Theorem 4.4
Proof
Now we have verified that \(\mathcal{F}\) satisfies the bend-twist condition on \(\mathcal{D}\). By Theorem 2.1 there exists at least one point \((x_{1},y_{1})\in\mathcal{D}\) such that \(\mathcal {F}(x_{1},y_{1})=0\), which corresponds to a fixed point of the Poincaré mapping.
Similarly, we get the following theorem.
Theorem 4.5
Proof
Now we have verified that \(\mathcal{F}\) satisfies the bend-twist condition on \(\mathcal{D}\). By Theorem 2.1 there exists at least one point \((x_{1},y_{1})\in\mathcal{D}\) such that \(\mathcal {F}(x_{1},y_{1})=0\), which is corresponding to a fixed point of the Poincaré mapping.
Theorem 4.6
Proof
Using Theorem 4.6, we have the following example.
Example 4.4
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation (Nos. 11271277, 11301106) and Guangxi Natural Science Foundation (Nos. 2014GXNSFBA118017, 2014GXNSFAA118004).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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