Positive doubly periodic solutions of telegraph equations with delays
- Yongxiang Li^{1}Email author and
- Huanhuan Zhang
Received: 20 November 2014
Accepted: 29 May 2015
Published: 13 June 2015
Abstract
This paper deals with the existence of positive doubly periodic solutions for the nonlinear telegraph equation with delays \(\mathcal{L}u=f( t, x, u(t-\tau_{1}, x),\ldots,u(t-\tau_{n}, x) )\), \((t, x)\in\mathbb{R}^{2}\), where \(\mathcal{L}u:=u_{tt}-u_{xx}+c u_{t}+a(t, x) u \) is a linear telegraph operator acting on function \(u: \mathbb{R}^{2}\to \mathbb{R}\), \(c>0\) is a constant, \(a\in C(\mathbb{R}^{2}, (0, \infty))\) is 2π-periodic in t and x, \(f\in C(\mathbb{R}^{2}\times[0, \infty)^{n}, [0, \infty))\) is 2π-periodic in t and x, and \(\tau_{1}, \ldots, \tau_{n}\in[0, \infty)\) are constants. Some existence results of positive doubly 2π-periodic weak solutions are obtained under that \(f(t, x, \eta_{1}, \ldots, \eta_{n})\) satisfies some superlinear or sublinear growth conditions on \(\eta_{1}, \ldots, \eta_{n}\). The discussion is based on the fixed point index theory in cones.
Keywords
telegraph equation with delays doubly periodic solution cone fixed point index of cone mappingMSC
35B15 47H101 Introduction and main results
As is well known, the telegraph equation describes a great deal of physical systems. For instance, the propagation of electromagnetic waves in an electrically conducting medium, the motion of a string or membrane with external damping, the motion of a viscoelastic fluid under the Maxwell body theory, the damped wave equation in a thermally conducting medium, etc. (see [1, 2]). In these models, the existence of time periodic solutions is an important problem which has attracted many authors’ attention and concern, see [3–16] and the references therein. All of these works are on the telegraph equations without time delays. It has been widely argued and accepted [17, 18] that for various reasons, time delay should be taken into consideration in modeling. Obviously, the telegraph equation with time delay has more actual significance. For instance, in the control propagation of electromagnetic wave signals, the signal intensity \(u(t, x)\) is subjected to a telegraph equation with time delay, in which the time delay expresses that the control act has delays. The purpose of this paper is to discuss existence of positive doubly periodic solutions for the nonlinear telegraph equation (1.1) with time delays.
- (H1)
\(a\in C(\mathbb{T}^{2})\) and \(0< a(t, x)\le\frac{c^{2}}{4}\) for \((t, x)\in\mathbb{R}^{2}\);
- (H2)
\(f\in C(\mathbb{T}^{2}\times[0, \infty)^{n}, [0, \infty))\),
Theorem 1.1
- (F1)there exist positive constants \(c_{1}, \ldots, c_{n}\) satisfying \(c_{1}+\cdots+c_{n}<\underline{a}\) and \(\delta>0\) such thatfor \((t, x)\in\mathbb{R}^{2}\) and \(\eta_{1},\ldots,\eta_{n}\in[0, \delta]\);$$f(t, x, \eta_{1}, \ldots,\eta_{n})\le c_{1} \eta_{1}+\cdots+c_{n} \eta_{n} $$
- (F2)there exist positive constants \(d_{1}, \ldots, d_{n}\) satisfying \(d_{1}+\cdots+d_{n}>\overline{a}\) and \(H>0\) such thatfor \((t, x)\in\mathbb{R}^{2}\) and \(\eta_{1},\ldots,\eta_{n}\ge H\),$$f(t, x, \eta_{1}, \ldots,\eta_{n})\ge d_{1} \eta_{1}+\cdots+d_{n} \eta_{n} $$
Theorem 1.2
- (F3)there exist positive constants \(d_{1}, \ldots, d_{n}\) satisfying \(d_{1}+\cdots+d_{n}>\overline{a}\) and \(\delta>0\) such thatfor \((t, x)\in\mathbb{R}^{2}\) and \(\eta_{1},\ldots,\eta_{n}\in[0, \delta]\);$$f(t, x, \eta_{1}, \ldots,\eta_{n})\ge d_{1} \eta_{1}+\cdots+d_{n} \eta_{n} $$
- (F4)there exist positive constants \(c_{1}, \ldots, c_{n}\) satisfying \(c_{1}+\cdots+c_{n}<\underline{a}\) and \(H>0\) such thatfor \((t, x)\in\mathbb{R}^{2}\) and \(\eta_{1},\ldots,\eta_{n}\ge H\),$$f(t, x, \eta_{1}, \ldots,\eta_{n})\le c_{1} \eta_{1}+\cdots+c_{n} \eta_{n} $$
The proofs of Theorems 1.1 and 1.2 are based on the fixed point index theory in cones, which will be given in Section 3. Some preliminaries to discuss Equation (1.1) are presented in Section 2.
2 Preliminaries
For Equation (2.1), in [13] the present author using the above result and a perturbation method of positive operator has built the following existence-uniqueness and positive estimate result.
Lemma 2.1
(Lemma 2 in [13])
- (1)
The restriction of P on \(C(\mathbb{T}^{2})\), \(P: C(\mathbb {T}^{2})\to C(\mathbb{T}^{2})\) is a completely continuous operator.
- (2)If \(h(t, x)\ge0\), a.e. \((t, x)\in\mathbb{T}^{2}\), Ph has the positivity estimate$$ \underline{G} \|h\|_{1}\le(Ph) (t, x)\le\frac{\overline{G}}{ \underline{G} \|a\|_{1}} \|h \|_{1},\quad \forall(t, x)\in\mathbb{T}^{2}. $$(2.7)
Lemma 2.2
\(A(K_{0})\subset K \), and \(A: K\to K\) is completely continuous.
Proof
By Lemma 2.2, the positive doubly periodic solution of Equation (1.1) is equivalent to the nontrivial fixed point of A. We will find the nontrivial fixed point of A by using the fixed point index theory in cone K.
We recall some concepts and conclusions on the fixed point index in [19, 20]. Let E be a Banach space and \(K\subset E\) be a closed convex cone in E. Assume Ω is a bounded open subset of E with boundary ∂Ω, and \(K\cap\Omega \neq\emptyset\). Let \(A:K\cap\overline{\Omega}\to K\) be a completely continuous mapping. If \(Au\neq u\) for any \(u\in K\cap\partial\Omega\), then the fixed point index \(i (A, K\cap\Omega, K)\) has definition. One important fact is that if \(i (A, K\cap\Omega, K)\neq 0\), then A has a fixed point in \(K\cap\Omega\), see Theorem 2.3.2 in [20]. The following two lemmas in [20] are needed in our argument.
Lemma 2.3
(Lemma 2.3.1 in [20])
Let Ω be a bounded open subset of E with \(\theta\in\Omega \), and \(A: K\cap\overline{\Omega}\to K\) be a completely continuous mapping. If \(\lambda Au\neq u\) for every \(u\in K\cap\partial\Omega\) and \(0<\lambda\le1\), then \(i (A, K\cap\Omega, K)=1\).
Lemma 2.4
(Corollary 2.3.1 in [20])
Let Ω be a bounded open subset of E and \(A:K\cap\overline {\Omega}\to K\) be a completely continuous mapping. If there exists \(e\in K\setminus \{\theta\}\) such that \(u-Au\ne\mu e\) for every \(u\in K\cap\partial \Omega\) and \(\mu\ge0\), then \(i (A, K\cap\Omega, K)=0\).
In the next section, we will use Lemma 2.3 and Lemma 2.4 to prove Theorem 1.1 and Theorem 1.2.
3 Proofs of main results
Proof of Theorem 1.1
Proof of Theorem 1.2
Let \(\Omega_{1}, \Omega_{2}\subset C(\mathbb {T}^{2})\) be defined by (3.1). We prove that the operator A defined by (2.9) has a fixed point in \(K\cap(\Omega_{2}\setminus \overline{\Omega}_{1})\) when r is small enough and R large enough.
Example 3.1
Example 3.2
Declarations
Acknowledgements
This research was supported by NNSF of China (11261053) and NSF of Gansu Province (1208RJZA129).
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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