Abstract
An experimental conjecture on the existence of positive periodic solutions for the Brillouin electron beam focusing system \(x''+a(1+\cos 2t)x=\frac{1}{x}\) for \(a\in(0,\frac{1}{2})\) is proved by applications of the Manasevich-Mawhin theorem.
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1 Introduction
In this paper, we consider the 2π-periodic boundary value problem for the equation
where \(a>0\) is constant.
The equations arise in the study of electronics and govern the motion of a magnetically focused axially symmetric electron beam under the influence of a Brillouin flow [1]. When the negative pole in a traveling-wave tube is shielded completely by a magnetic field screen, the electron beam focusing system can be described by (1.1). Besides, from a mathematical point of view, equation (1.1) is a singular perturbation of the Mathieu equation.
Motivated by the results of laboratory experiments experts realized in [1], it was conjectured that (1.1) should have a positive periodic solutions if \(a\in(0,\frac{1}{4})\) [2]. In the last fifty years, many mathematicians have given birth to extensive literature about this topic (see [3–7]). Although numerical studies back up the experimental conjecture, an analytical proof of the existence of periodic solutions of (1.1) for \(a\in(0,\frac{1}{4})\) is still lacking.
The first analytic work on periodic solution of (1.1) was obtained by Ding [3]. Ding proved that (1.1) had at least one positive periodic solution if \(a\in(0,\frac{1}{16})\). Afterwards, Ye and Wang [4] obtained that (1.1) had at least one positive periodic solution if \(a\in(0,0.1442)\). In [5], Zhang investigated a kind of singular Liénard equation, and by applications of his theory, they extended the existence result of (1.1) to \(a\in(0,0.1532)\).
However, in the above works, authors were not able to prove or disprove the result which was conjectured in [1]. In this paper, we will show that (1.1) has at least one positive 2π-periodic solution when the parameter \(a\in(0,\frac{1}{2})\) other than \((0,\frac{1}{4})\).
2 Brillouin electron beam focusing system
Lemma 2.1
Manasevich-Mawhin [8]
Let Ω be an open bounded set in \(C^{1}_{T}:=\{x\in C^{1}(\mathbb{R},\mathbb{R}): x(t+T)-x(t)\equiv 0\}\). If
-
(i)
The problem
$$ { } \bigl(\phi\bigl(x'\bigr)\bigr)'=\lambda \tilde{f}\bigl(t,x,x'\bigr),\quad x\in C^{1}_{T}, $$(2.1)where \(\tilde{f}:[0,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) is assumed to be Carathéodory. For each \(\lambda\in(0,1)\), problem (2.1) has no solution on ∂Ω.
-
(ii)
The equation
$$F(a):=\frac{1}{T} \int^{T}_{0}\tilde{f}\bigl(t,x,x'\bigr) \,dt=0, $$has no solution on \(\partial\Omega\cap\mathbb{R}\).
-
(iii)
The Brouwer degree of F
$$\deg\{F,\Omega\cap\mathbb{R},0\}\neq0. $$
Then problem (2.1) has at least one periodic solution on Ω̄.
Lemma 2.2
[9]
Suppose that \(u\in C^{1}_{T}\) and there exists \(t_{0}\in[0,T]\) such that \(\vert u(t_{0}) \vert < d\). Then
Next, we prove that Brillouin electron beam focusing system (1.1) has at least one positive 2π-periodic solution if \(a\in (0,\frac{1}{2} )\). Firstly, we consider the following singular equation:
where \(a(t)\in C(\mathbb{R},[0,+\infty))\) and \(a(t+T)=a(t)\), \(\forall t\in \mathbb{R}\).
Theorem 2.1
Assume that \(\vert a \vert _{\infty}:=\max_{t\in[0,T]} \vert a(t) \vert <\frac{4\pi^{2}}{T^{2}}\) holds. Then (2.2) has at least one positive T-periodic solution.
Proof
Firstly, we consider the following (homotopy) family of (2.2):
Let \(x(t)\in C^{1}_{T}\) be an arbitrary solution of (2.3). Integrating (2.3) from 0 to T, we get
So, we know that there exist positive constants \(D_{1}< D_{2}\) and \(t_{0}\in(0,T)\) such that
Therefore, we have
Let us write \(x(t)=\bar{x}+\tilde{x}(t)\), here \(\tilde{x}(t):=x(t)-\bar{x}\), and \(\bar{x}:=\frac{1}{T}\int^{T}_{0}x(t)\,dt\). Obviously, \(\int^{T}_{0}\tilde{x}(t)\,dt=0\). Now (2.3) for \(\tilde{x}(t)\) is
since \(\bar{x}''=0\). Multiplying (2.7) by \(\bar{x}-\tilde{x}(t)\), we have
Integrating this equation over one period and making use of the T-periodicity of \(\tilde{x}(t)\), we get
So, we have
Since \(a(t)\geq0\), then \(-\bar{x}^{2}\int^{T}_{0}a(t)\,dt\leq0\). So, we have
For any \(\varepsilon>0\), there is \(g_{\varepsilon}^{+}\in L^{2}(0,T)\) and \(g_{\varepsilon}^{+}>0\)
for all \(x(t)>0\) and a.e. \(t\in[0,T]\). So, we have
where \(\vert a \vert _{\infty}=\max_{t\in[0,T]} \vert a(t) \vert \). Since \(D_{1}\leq x(t_{0})\leq D_{2}\), by Lemma 2.2, we have
By applications of Wirtinger’s inequality (in [10] Lemma 2.4) and (2.9), we have
where \(\Vert g_{\varepsilon}^{+} \Vert _{2}= (\int^{T}_{0} \vert g_{\varepsilon}^{+}(t) \vert ^{2}\,dt )^{\frac{1}{2}}\). Since \(\tilde{x}'(t)=x'(t)\), then we have
From \(\vert a \vert _{\infty}<\frac{4\pi^{2}}{T^{2}}\) for \(\varepsilon>0\) sufficiently small, there exists a positive constant \(M_{1}'\) such that
From (2.6) and by applying Hölder’s inequality, we have
On the other hand, from \(x(0)=x(T)\), we know that there is a point \(t_{1}\in[0,T]\) such that \(x'(t_{1})=0\), and then \(\vert x'(t) \vert = \vert x'(t_{1})+\int^{t}_{t_{1}} x''(s)\,ds \vert \leq \int^{T}_{0} \vert x''(s) \vert \,ds\). From (2.3) and (2.8), we have
i.e.,
Multiplying (2.3) by \(x'(t)\), we get
Let \(\tau\in[0,T]\) be as in (2.5). For any \(\tau\leq t\leq T\), we integrate (2.12) on \([\tau,t]\) and get
By (2.11) we have
With these inequalities we can derive from (2.13) that
So, we know that there exists \(M_{3}>0\) such that
since \(\lim_{x\to 0^{+}}\int^{x}_{1}\frac{1}{u}\,du=+\infty\). The case \(t\in[0,\tau]\) can be treated similarly.
Having in mind (2.5), (2.10), (2.11) and (2.15), we define
where \(0< E_{1}<\min\{M_{3},D_{1}\}\), \(E_{2}>\max\{M_{1},D_{2}\} \) and \(E_{3}>M_{2}\). Then condition (i) of Lemma 2.1 is satisfied. For a constant \(x \in\ker L\), \(x>0\), we have
Obviously, \(\bar{g}(x)<0\) for all \(x\in(0,E_{1})\), \(\bar{g}(x)>0\) for all \(x>E_{2}\), so condition (ii) of Lemma 2.1 holds. Set
we have \(xH(x,\mu)>0\). Thus \(H(x,\mu)\) is a homotopic transformation and
Thus assumption (iii) of Lemma 2.1 is also verified. Therefore (2.2) has at least one positive T-periodic solution. □
Next, we apply Theorem 2.1 to Brillouin electron beam focusing system (1.1). Equation (1.1) is of the form (2.2) with \(a(t)=a(1+\cos2t)\).
Theorem 2.2
If \(a\in (0,\frac{1}{2} )\), then (1.1) has at least one positive 2π-periodic solution.
Proof
If \(a<\frac{1}{2}\), then
i.e., \(\vert a \vert _{\infty}<\frac{4\pi^{2}}{T^{2}}\) holds. Theorem 2.1 implies that (1.1) has at least one 2π-periodic positive solution. □
Finally, we present an example to illustrate our result.
Example 2.1
Consider the second order differential equation with singularity:
It is clear that \(T=\pi\), \(a(t)=1+\cos t\). Obviously,
Therefore, (2.17) has at least one π-periodic solution by application of Theorem 2.1.
References
Bevc, V, Palmer, J, Süsskind, C: On the design of the transition region of axisymmetric, magnetically focused beam valves. J. Br. Inst. Radio Eng. 18, 696-708 (1958)
Ding, T: Applications of Qualitative Methods of Ordinary Differential Equations. Higher Education Press, BeiJing (2004)
Ding, T: A boundary value problem for the periodic Brillouin focusing system. Acta Sci. Natur. Univ. Pekinensis 11, 31-38 (1965)
Ye, Y, Wang, X: Nonlinear differential equations in electron beam focusing theory. Acta Math. Appl. Sin. 1, 13-41 (1978)
Zhang, M: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203, 254-269 (1996)
Torres, P: Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system. Math. Methods Appl. Sci. 23, 1139-1143 (2000)
Garrione, M, Zamora, M: Periodic solutions of the Brillouin electron beam focusing equation. Commun. Pure Appl. Anal. 13, 961-975 (2014)
Manásevich, R, Mawhin, J: Periodic solutions for nonlinear systems with p-Laplacian-like operator. J. Differ. Equ. 145, 367-393 (1998)
Xin, Y, Cheng, Z: Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument. Adv. Differ. Equ. 2016, 41 (2016)
Torres, P, Cheng, Z, Ren, J: Non-degeneracy and uniqueness of periodic solutions for 2n-order differential equations. Discrete Contin. Dyn. Syst., Ser. A 33, 2155-2168 (2013)
Acknowledgements
ZBC and SWY would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by the National Natural Science Foundation of China (No. 11501170), China Postdoctoral Science Foundation funded project (No. 2016M590886), Fundamental Research Funds for the Universities of Henan Province (NSFRF140142), Henan Polytechnic University Outstanding Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055).
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ZBC and SWY worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
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Cheng, Z., Yao, S. New results for Brillouin electron beam focusing system. Bound Value Probl 2017, 70 (2017). https://doi.org/10.1186/s13661-017-0800-2
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DOI: https://doi.org/10.1186/s13661-017-0800-2