New results of positive doubly periodic solutions to telegraph equations

1.   Introduction

Consider the following telegraph equation

with doubly periodic boundary conditions

where $c > 0$ is a constant, $a, \ \omega\in C(\mathbb{R}^2, \mathbb{R}^+)$ is $2\pi$-periodic in $t$ and $x$ and $\gamma$ satisfies $0 < |\gamma| < 1$.

Telegraph equation is a typical partial differential equation, and there are important applications in the propagation of electromagnetic waves in an electrically conducting medium, the motion of a viscoelastic fluid under the Maxwell body theory, the motion of a string or membrane with external damping and the damped wave equation in a thermally conducting medium. For details and explanations, we refer the readers to Barbu ^[1] and Roussy-Pearcy ^[2].

Many authors have demonstrated increasing interest in the subject of telegraph equations with various boundary conditions by different methods: maximum principles and the method of upper and lower solutions, see Li ^[3], Mawhin-Ortega-Robles-Pérez ^[4,5,6] and Ortega-Robles-Pérez ^[7]; an integral equation approach, see Gilding-Kersner ^[8], and the fixed point theorem in a cone, see Li ^[9], and Wang and An ^[10,11].

In this paper, we choose different strategy of proof which relies essentially on the theory of Hilbert's metric, the eigenvalue theory, the fixed point index in a cone and the theory of $\alpha$-concave operator.

On the one hand, we notice that Hilbert introduced the Hilbert's metric in 1895 in an early paper ^[12] which is on the foundations of geometry. Birkhoff ^[13] made clear the usefulness of Hilbert's metric in algebra and analysis in 1957. In suitable metric spaces, Birkhoff proved that the Jentzch's theorem for integral operators with positive kernel and Perron-Frobenius theorem for non-negative matrices could both be verified by an application of the Banach contraction mapping theorem. Applications of Hilbert's metric to positive integral operators and ordinary differential equations are given by Bushell ^[14].

But many important kernels are not positive kernel, which plays an important role in Birkhoff ^[13]. For example, Green's functions of some boundary value problems are nonnegative, not positive; for instance, when $n = 3$, the Green's function $G_1(x, y)$ for the elliptic boundary value problem

satisfies that

where

For the case of one-dimension, it is well known that the Green's function $G_2(t, s)$ for the following boundary value problem

is that

It so follows from (1.3) that

Therefore, the theory of Hilbert's metric has been somewhat neglected, perhaps because the Green's function is lack of the strictly positivity for many boundary value problems. In this paper, we give a new attempt to consider the existence and uniqueness of positive solution for problem (1.1) with (1.2) by using the theory of Hilbert's metric.

Let $\mathbb{Z}, \mathbb{R}$, and $\mathbb{R}^+$ denote the set of all integers, real numbers and nonnegative real numbers, respectively and let

denote the the torus.

Throughout this paper, a doubly 2$\pi$-periodic function will be identified to be a function defined on $\mathbb{T}^2$. We also let

respectively denote the spaces of doubly periodic functions with the indicated degree of regularity. And $\mathcal{D}'(\mathbb{T}^2)$ is the space of distributions on $\mathbb{T}^2$.

We call a function $u\in L^1(\mathbb{T}^2)$ being a doubly periodic solution to (1.1) with (1.2), if $u$ satisfies (1.1) and (1.2) in the distribution sense, that is

For convenience, we assume that $a$ and $\omega$ satisfy

${\bf (H_{1})}$ $a\in C(\mathbb{T}^2), \ 0\leq a(t, x)\leq\frac{c^2}{4}$ for $(t, x)\in \mathbb{R}^2$, and $\int_{\mathbb{T}^2}a(t, x)dtdx > 0;$

${\bf (H_{2})}$ $\omega\in C(\mathbb{T}^2), \ \omega(t, x)\geq0$ for $(t, x)\in \mathbb{R}^2$, and $\int_{\mathbb{T}^2}\omega(t, x)dtdx > 0.$

Theorem 1.1. If ${\bf (H_{1})}$ and ${\bf (H_{2})}$ hold and $0 < |\gamma| < 1$, then problem (1.1) with (1.2) admits a unique positive doubly periodic solution.

It is not difficult to see that if $\gamma = -1$, then the approach to prove Theorem 1.1 is invalid. We hence need introduce some different techniques, such as contraction ratio $R(A)$ and projective diameter $D(A)$ for some positive operator $A$, to prove the existence and uniqueness of positive doubly periodic solution for the following problem

Theorem 1.2. If ${\bf (H_{1})}$ and ${\bf (H_{2})}$ hold, then problem (1.4) with (1.2) admits a unique positive doubly periodic solution.

Remark 1.1. The method used in the proof of Theorem 1.1 and Theorem 1.2 is invalid when we consider the case $\gamma > 1$. Therefore we need to introduce a different technique to approach this case.

On the other hand, we notice that the study of asymptotic behavior of solutions for various partial differential equations is also a hot topic, see Aviles ^[15], Feng-Zhang ^[16], Gidas-Spruck ^[17], Zhang-Feng ^[18,19] and the references therein. Recently, Feng ^[20] employs the fixed point theorem of cone expansion and compression of norm type to analyze the existence and asymptotic behavior of nontrivial radial convex solutions for a Monge-Ampère system. Comparing with the fixed point theorem used in ^[20], the eigenvalue theory is often a very effective approach to analyze the existence and asymptotic behavior of positive solutions for various boundary value problems.

Next, we employ the eigenvalue theory to analyze the existence and asymptotic behavior of positive doubly periodic solutions for

with (1.2), where $\lambda > 0$ is a parameter.

Theorem 1.3. Suppose that ${\bf (H_{1})}$ and ${\bf (H_{2})}$ hold. Then, for $\gamma > 0$, we have the following conclusions:

(i) If $\gamma > 1$, then for any $\lambda > 0$, problem (1.5) with (1.2) admits a positive doubly periodic solution $u_{\lambda}\in C(\mathbb{T}^2)$ with $u_{\lambda}\not\equiv 0$, and

(ii) If $0 < \gamma < 1$, then for any $\lambda > 0$, problem (1.5) with (1.2) admits a positive doubly periodic solution $u_{\lambda}\in C(\mathbb{T}^2)$ with $u_{\lambda}\not\equiv 0$, and

(iii) If $\gamma = 1$, then there exits $\lambda_* > 0$ such that problem (1.5) with (1.2) admits no positive doubly periodic solution for $0 < \lambda < \lambda_*$.

We also consider the more general equation

where $\lambda > 0$ is a parameter.

We first give new results of existence, nonexistence and asymptotic behavior of positive doubly periodic solutions for problem (1.8) with (1.2) in Theorems 3.1–3.3. We will use the eigenvalue theory again to prove these assertions.

Then we consider the multiplicity of positive doubly periodic solutions for problem (1.8) with (1.2) by employing the fixed point index in a cone, which is used in Hu and Wang ^[21] and Zhang ^[22].

We obtain the following theorem:

Theorem 1.4. Suppose that ${\bf (H_{1})}$ and ${\bf (H_{2})}$ hold. In addition, if $f:\ [0, +\infty)\rightarrow [0, +\infty)$ is continuous and satisfies the following conditions:

${\bf (H_{3})}\ \ \lim\limits_{u\rightarrow 0^+}\frac{f(u)}{u} = 0$;

${\bf (H_{4})}\ \ \lim\limits_{u\rightarrow +\infty}\frac{f(u)}{u} = 0$;

${\bf (H_{5})}$ $0 < \varliminf\limits_{u\rightarrow +\infty}f(u)\leq+\infty.$

Then, for any given $\tau > 0$, there exists $\delta > 0$ such that, for $\lambda > \delta$, problem (1.8) with (1.2) admits at least two positive doubly periodic solutions $u_{\lambda}^{(1)}(t, x), \ \ u_{\lambda}^{(2)}(t, x)\in C(\mathbb{T}^2)$ and $\max\limits_{(t, x)\in \mathbb{T}^2}u_{\lambda}^{(1)}(t, x) > \tau$.

Next, we apply the theory of $\alpha$-concave operator to analyze the uniqueness and continuity of positive doubly periodic solution on the parameter $\lambda$ for problem (1.8) with (1.2).

More precisely, we have:

Theorem 1.5. Suppose that ${\bf (H_{1})}, \ {\bf (H_{2})}$ and $f(u):\ [0, +\infty)\rightarrow [0, +\infty)$ is a nondecreasing function with $f(u) > 0$ for $u > 0$, and satisfies $f(\rho u)\geq\rho^{\alpha}f(u)$, for any $0 < \rho < 1,$ where $0\leq\alpha < 1.$ Then, for any $\lambda\in (0, \infty)$, problem (1.8) with (1.2) admits a unique positive doubly periodic solution $u_{\lambda}(t).$ Furthermore, such a solution $u_{\lambda}(t)$ satisfies the following properties:

(i) $u_{\lambda}(t)$ is strong increasing in $\lambda$. That is, $\lambda_{1} > \lambda_{2} > 0$ implies $u_{\lambda_{1}}(t)\gg u_{\lambda_{2}}(t)$ for $t\in J.$

(ii) $\lim\limits_{\lambda\rightarrow0^{+}}\|u_{\lambda}\| = 0, \ \ \lim\limits_{\lambda\rightarrow +\infty}\|u_{\lambda}\| = +\infty$.

(iii) $u_{\lambda}(t)$ is continuous with respect to $\lambda$. That is, $\lambda\rightarrow \lambda_{0} > 0$ implies $\|u_{\lambda}-u_{\lambda_{0}}\|\rightarrow0$.

Remark 1.2. We only need $f$ to be monotonic, but not continuous in Theorem 1.5.

The rest of the paper is organized as follows. In Section 2, we shall recall some necessary definitions, lemmas and theorems about Hilbert's metric, which will be used to prove Theorem 1.1 and 1.2. Section 3 is devoted to proving the existence, nonexistence and asymptotic behavior of positive doubly periodic solutions by using the eigenvalue theory and the inequality technique. The multiplicity of positive doubly periodic solutions for (1.8) with (1.2) is discussed in Section 4 and we will prove Theorem 1.4 there. In Section 5, we will give the proof of Theorem 1.5.

2.   Hilbert's metric and positive doubly periodic solution

In this section, we will give a new attempt to solve problem $(1.1)$ with (1.2) by using Hilbert's metric. As Bushell ^[14] pointed out that Hilbert's original definition of the projective metric is the logarithm of the cross-ratio for certain points in the interior of a convex cone in $\mathbb{R}^n$. Let us begin with the definition of the Hilbert metric in a general setting.

Let $E$ be a real Banach space with a closed solid cone $K^{0}$ ($K^{0}$ denotes the interior of $K$). Then for $u, v\in K^{0}$ we define

It is shown in Bushell ^[14] that

Definition 2.1. (Definition 2.2, Bushell ^[14]) If $A: K\rightarrow K$ we say that $A$ is non-negative, and if $A:K^{0}\rightarrow K^{0}$ we say that $A$ is positive.

Definition 2.2. (Definition 2.2, Bushell ^[14]) Hilbert's projective metric $d (\cdot, \cdot)$ is defined in $K^{0}$ by

Lemma 2.1. (Theorem 2.1, Bushell ^[14]) $\{K^{0}, d\}$ is a pseudo-metric space and $X = \{K^{0}\cap B, d\}$ is a metric space, where $B$ is the unit sphere in $E$.

Lemma 2.2. (Lemma 2.2, Bushell ^[14]) If $u, v\in K^{0}$, then $d(\xi u, \zeta v) = d(u, v)$ for all $\xi, \ \zeta > 0$.

Moreover, Bushell ^[24] proved that if the norm is monotone with respect to $P$ (i.e. $0\leq u\leq v\Rightarrow \|u\|\leq\|v\|$), then $X$ is complete.

Definition 2.3. (Definition 1, Botter ^[23]) If $A(\xi u) = \xi^\gamma Au$ for all $u\in K^{0}, \ \xi > 0$, we say that $A$ is positive homogeneous of degree $\gamma$ in $K^{0}$.

Definition 2.4. (Definition 3, Botter ^[23]) If $A: E\rightarrow E$, then $A$ is said to be monotone decreasing (increasing) if $u\leq v$ implies $Au\leq Av (Au\geq Av)$.

We first consider the linear equation

where $\lambda\in \mathbb{R}$, and $h(t, x)\in L^1(\mathbb{T}^2)$.

Let $\mathfrak{L}_\lambda$ denote the differential operator

acting on functions on $\mathbb{T}^2$. It follows from Lemma 1 in ^[9] that if $\lambda < 0$, then $\mathfrak{L}_\lambda$ admits the resolvent $\mathfrak{R}_\lambda$ defined by

where $u$ is the unique solution of (2.1). Moreover, if we restrict $\mathfrak{R}_\lambda$ on $L^p(\mathbb{T}^2)$ or $C(\mathbb{T}^2)$, then $\mathfrak{R}_\lambda$ is compact. Specifically, $\mathfrak{R}_\lambda: C(\mathbb{T}^2)\rightarrow C(\mathbb{T}^2)$ is completely continuous.

Letting $\mu = -\frac{c^2}{4}$, then the Green's function $G(t, x)$ of $\mathfrak{L}_\mu$ can be explicitly expressed, which was obtained in Lemma 5.2 of ^[7]. Thus, it follows from Lemma 5.1 in ^[7] that the unique solution of (2.1) can be defined by convolution product

By the definition of $G(t, x)$, one can get

where

Letting $h\in L^1(\mathbb{T}^2)$ with $h(t, x)\geq 0$ for $a.e.\ (t, x)\in \mathbb{T}^2$, then it follows from (2.2) that

Now take the Banach space to be $C(\mathbb{T}^2): = E$ with supremum norm.

Let $K\subset E$ by

Then

Thus, $E$ is an ordered Banach space with $K$.

For convenience, we denote the norm in Banach $E$ by $\|\cdot\|$, and in $L^p(\mathbb{T}^2)$ by $\|\cdot\|_p$ hereafter.

Next, we consider (2.1) with (1.2) when $-\lambda$ is replaced by $a(t, x)$. In ^[9], the author obtained the following conclusions.

Lemma 2.3. (Lemma 2, Li ^[9]) Suppose that $h(t, x)\in L^2(\mathbb{T}^2)$. Then (2.1) with (1.2) admits a unique solution$u: = Ph$, and $P: L^1(\mathbb{T}^2)\rightarrow C(\mathbb{T}^2)$ is a linear bounded operator with the following properties.

(i) $P: C(\mathbb{T}^2)\rightarrow C(\mathbb{T}^2)$ is a completely continuous operator.

(ii) If $h(t, x) > 0$ for $a.e.\ (t, x)\in \mathbb{T}^2$, then

Let $T: K\rightarrow E$ be defined by

It follows from Lemma 2.3 that $T: K\rightarrow E$ is completely continuous, and the doubly periodic solution of (1.1) with (1.2) is equivalent to the fixed point of $T$.

Proof of Theorem 1.1. Let $u\in K$ and

for $(t, x)\in \mathbb{R}^2$. Then $h\in E$ and $Tu = Ph$, and it follows from the proof of Lemma 2 in ^[9] that

Set $u\in K^{0}$, and define

Then it follows from ${\bf (H_2)}$, (2.2), (2.7) and (2.9) that

which shows that $Tu\in K^{0}$. Further $T: K^{0}\rightarrow K^{0}$.

Plainly $T$ is monotone increasing (decreasing) and positive homogeneous of degree $\gamma$ when $0 < \gamma < 1$ ($-1 < \gamma < 0$). Next we analyze the existence and uniqueness of positive fixed point of $T$ in $K^{0}$.

Case 1) $0 < \gamma < 1$

Since $0 < \gamma < 1$, one can prove

In fact, it follows from

and $T$ is monotone increasing and positive homogeneous of degree $\gamma$ that

which shows that

Thus

Consider the mapping $\hat{T}: E\rightarrow E$ defined by

We hence get from Lemma 2.2 that

It is not difficult to see that the norm is monotone with respect to $K$, so $E$ is complete. Thus $\hat{T}$ is a contraction in Hilbert's metric and so admits a unique fixed point in $E$; call it $v^*$.

Let

Next we prove that $T$ possesses a unique fixed point in $K^{0}$. In fact, obviously $u^*\in K^{0}$, and

Moreover, suppose there exists another point $u_1\in K^{0}$ such that $Tu_1 = x_1$. Then it follows from (2.10) that

This shows that $d(u^*, u_1) = 0$. So we get $u^* = \varsigma u_1$ for $\varsigma > 0$, and hence

which shows $\varsigma = 1, \ \ u^* = u_1$.

Case 2) Let $-1 < \gamma < 0$. Then for $u, v\in K^{0}$, it follows from

and $T$ is monotone decreasing and positive homogeneous of degree $\gamma$ that

which shows that

Thus

Let

Then one can prove $\hat{T}: E\rightarrow E$, and

So $\hat{T}$ admits a unique fixed point $v^*$ in $E$ by using Banach's contraction mapping theorem. Let

One can prove that $u^*$ is the unique fixed point of $T$ in $K^{0}$.

It is not difficult to see that if $\gamma = -1$, then the Banach's contraction mapping theorem is invalid in the proof of Theorem 1.1. We hence need introduce some different techniques to prove the existence and uniqueness of positive doubly periodic solution for problem (1.1) with (1.2).

Definition 2.5. (Definition 2, Botter ^[23]) The contraction ratio, $R(A)$, of $A$ is defined by

Definition 2.6. (Definition 3.3, Bushell ^[14]) If $A$ is positive, the projective diameter $D(A)$ of $A$ is defined by

Lemma 2.4. (Corollary, Bushell ^[14]) Let $A$ be a positive linear mapping. Then

Lemma 2.5. (Theorem 3.2, Bushell ^[14]) Let $A$ be a positive linear mapping in $X$. Then

Lemma 2.6. (Theorem 1, Botter ^[23]) Let $A$ be a monotone decreasing mapping which is positive homogeneous of degree $-\alpha (\alpha > 0)$. Then the contraction ratio, $R(A)$, does not exceed $\alpha$

Proof of Theorem 1.2. Define $\mathbb{B}:K^{0}\rightarrow K^{0}$ by

Then one can verify that $B$ is a monotone decreasing and positive homogeneous of degree $-1$. Lemma 2.6 tells us that

Let $T^*: K^{0}\rightarrow K^{0}$ be defined by

Plainly $T^*$ is a positive linear mapping of $E$ into $E$. And then it follows from Lemma 2.5 that

But $T^*$ admits finite projective diameter, so we get $R(T^*) < 1$.

Define $T:K^{0}\rightarrow K^{0}$ by

As in the proof of Theorem 1.1 one can define $\hat{T}: E\rightarrow E$ defined by

We hence get from Lemma 2.2 that

But $R(T^*)R(\mathbb{B}) < 1$ and hence $\hat T$ is a contraction. Therefore it follows from $E$ is complete that $\hat T$ possesses a unique point in $E$. From this we can construct (as in Theorem 1.1) an element $u$ of $K^{0}$ such that

This finishes the proof of Theorem 1.2

Remark 2.1. One of the contributions of this section is to give the application of Hilbert's projective metric to boundary value problems.

3.   Existence and asymptotic behavior of positive doubly periodic solution

In this section, we apply the eigenvalue theory to study the existence and asymptotic behavior of positive continuous solutions to problem $(1.5)$ with (1.2). So we first collect some known results of the eigenvalue theory, which will be used in the subsequent proofs.

Lemma 3.1. (Corollary of Theorem 1, Guo ^[26]) Let $A: E\rightarrow E$ be completely continuous. Suppose that $A\theta = \theta$,

and

Then the following two conclusions hold:

i) Every $\mu\neq 0$ is an eigenvalue of $A$, i.e., there exists $x_{\mu}\in E, \ x_{\mu}\neq 0$ such that $Ax_{\mu} = \mu x_{\mu}$;

ii) $\lim\limits_{\mu\rightarrow \infty} \|x_{\mu}\| = +\infty$.

Lemma 3.2. (Corollary of Theorem 2, Guo ^[26]) Let $A: E\rightarrow E$ be completely continuous. Suppose that $A\theta = \theta$,

and

Then the following two conclusions hold:

i) Every $\mu\neq 0$ is an eigenvalue of $A$, i.e., there exists $x_{\mu}\in E, \ x_{\mu}\neq 0$ such that $Ax_{\mu} = \mu x_{\mu}$;

ii) $\lim\limits_{\mu\rightarrow \infty} x_{\mu} = 0$.

Proof of Theorem 1.3. Here we only prove the conclusion (i) holds since the proof is similar when we verify (ii). Let

where

Then, it is easy to verify that $T$ is completely continuous from $K^*$ to $K^*$.

Moreover, it follows from ${\bf (H_1)}$ and the the definitions of $\underline{G}$ and $\overline{G}$ that $0 < \sigma < 1$.

Next, we prove that all the conditions of the Lemma 3.1 and Lemma 3.2 are satisfied.

On the one hand, for $u\in K^*$, it follows from (2.7) and (2.8) that

It hence follows from (3.2) that: for $\gamma > 1$, we get

for $0 < \gamma < 1$, we have

On the other hand, for any $u\in K^*$, we deduce from (2.7), (2.8) and (2.9) that

Thus, for $\gamma > 1$, we get

For $0 < \gamma < 1$, we get

Then from (3.3) and (3.6) or (3.4) and (3.7), together with Lemma 3.1 and Lemma 3.2, we respectively get that: for any $\lambda^* > 0$, there exists $u_{\lambda^*}\in E$ with $u_{\lambda^*}\neq \theta$ such that $Tu_{\lambda^*} = \lambda^*u_{\lambda^*}$; and

Moreover, we obtain from (3.5) that

which shows that

But

It so follows from (3.9) that

Thus, when $\gamma > 1$, we have

Similarly, when $0 < \gamma < 1$, we get from (3.10) that

Let $\lambda = \frac{1}{\lambda^*}$. Thus, we finish the proof of (i) and (ii) in Theorem 1.3.

Next, we give the proof of (iii) in Theorem 1.3. Assume $u$ is a positive solution for problem (1.1) with (1.2). We will prove that this leads to a contradiction for $0 < \lambda < \lambda_*$, where

Since $(Tu)(t, x) = \frac{1}{\lambda}u(t, x)$ for $x\in E$, it follows from (3.2) that

This is a contradiction, and our proof is finished.

Remark 3.1. The approach to prove Theorem 1.3 can be applied to the more general problem (1.8) with (1.2).

Remark 3.2. Although we also use the eigenvalue theory to study problem (1.8) with (1.2), since the nonlinear term is in a general form, some new techniques are needed. For detail to see the proof of Theorems 3.1–3.3.

To consider the existence and asymptotic behavior of positive continuous solutions for eigenvalue problem $(1.8)$ with (1.2), we need to introduce the following notations:

where $\gamma > 0$.

Theorem 3.3. Suppose that ${\bf (H_{1})}$ and ${\bf (H_{2})}$ hold. If $f: \mathbb{R}^+\rightarrow \mathbb{R}^+$ is continuous and

then for $\gamma > 0$ and $\gamma\neq 1$, we get the following conclusions:

(i) If $\gamma > 1$, then for any $\lambda > 0$, problem (1.8) with (1.2) admits a positive doubly periodic solution $u_{\lambda}$ with $u_{\lambda}\not\equiv 0$, and

(ii) If $0 < \gamma < 1$, then for any $\lambda > 0$, problem (1.8) with (1.2) admits a positive doubly periodic solution $u_{\lambda}$ with $u_{\lambda}\not\equiv 0$, and

Proof. Let $T^*: K\rightarrow E$ be defined by

It follows from Lemma 2.3 that $T^*: K\rightarrow E$ is completely continuous, and the doubly periodic solution of (1.8) with (1.2) is equivalent to the fixed point of $\lambda T^*$.

Similarly, one can verify that $T^*$ is completely continuous from $K^*$ to $K^*$, where $K^*$ is defined in (3.1).

Let $u\in K$ and $h^*(t, x) = \omega(t, x)f(u(t, x))$ for $(t, x)\in \mathbb{R}^2$. Then $h^*\in E$ and $T^*u = Ph^*$, and it follows from the proof of Lemma 2 in ^[9] that

Considering $f_{\infty} = \infty$, there exists $r_{1} > 0$ such that $f(u)\geq \varepsilon_{1}u^\gamma$, for $u\geq r_{1}$, where $\varepsilon_{1} > 0$ satisfies

Let $\partial K^*_{{r_{1}}} = \{u\in K^*:\|u\| = r_1\}$. Then, for $x\in K^*_{{r_{1}}}$, we get from (3.14) and (3.15) that

We so have the following two conclusions:

1) $\gamma > 1$

Since $\gamma > 1$, we get

2) $0 < \gamma < 1$

Since $0 < \gamma < 1$, we obtain

Next, turning to $f^{0} = 0$, there exists $r_{2}:0 < r_{2} < r_{1}$ so that $f(u)\leq \varepsilon_{2}u^\gamma$, for $0\leq u\leq r_{2}$, where $\varepsilon_{2} > 0$ satisfies

Thus, for $u\in \partial K^*_{r_{2}}$, we have from (3.14)

We hence obtain from (3.19) the following two conclusions:

1) $\gamma > 1$

Since $\gamma > 1$, we get

2) $0 < \gamma < 1$

Since $0 < \gamma < 1$, we get

Thus, observing $\gamma > 1$, (3.17), (3.20) and Lemma 3.1, we find that: for any $\bar\lambda > 0$, there exists $u_{\bar\lambda}\in E$ with $u_{\bar\lambda}\neq \theta$ such that $T^*u_{\bar\lambda} = \bar\lambda u_{\bar\lambda}$; and

Moreover, it follows from (3.16) that

which shows that

This proves that

when $\gamma > 1$.

When considering $0 < \gamma < 1$, then from (3.18) and (3.21), together with Lemma 3.2, we obtain that: for any $\bar\lambda > 0$, there exists $u_{\bar\lambda}\in E$ with $u_{\bar\lambda}\neq \theta$ such that $T^*u_{\bar\lambda} = \bar\lambda u_{\bar\lambda}$; and $\lim\limits_{\bar\lambda\rightarrow \infty}u_{\bar\lambda} = 0.$

Let $\lambda = \frac{1}{\bar\lambda}$. Then the proof of Theorem 3.3 is completed.

In Theorem 3.3, we consider the existence of positive solution for (1.8) with (1.2) in the case $\gamma > 1$ and $0 < \gamma < 1$. Next we discuss what happen in the case $\gamma = 1$?

In fact, we will obtain two nonexistence results when $\gamma = 1$.

Theorem 3.4. Suppose that ${\bf (H_{1})}, \ {\bf (H_{2})}$, $\gamma = 1$ and $f_0 = 0$. Then problem (1.8) with (1.2) possesses no positive doubly periodic solution for $0 < \lambda < 1$.

Proof. Let $u$ be a positive solutions to problem (1.8) with (1.2). We next show that this leads to a contradiction for $0 < \lambda < 1$. Since $(T^*u)(t, x) = \frac{1}{\lambda}u(t, x)$ for $(t, x)\in \mathbb{T}^2$, it follows from (3.19) that

which is a contradiction, and our proof is completed.

Theorem 3.5. Suppose that ${\bf (H_{1})}, \ {\bf (H_{2})}$, $\gamma = 1$ and $f_\infty = 0$. Then problem (1.8) with (1.2) possesses no positive doubly periodic solution for $\lambda > 1$.

Proof. Let $u$ be a positive solutions to problem (1.8) with (1.2). We next show that this leads to a contradiction for $\lambda > 1$. Since $(T^*u)(t, x) = \frac{1}{\lambda}u(t, x)$ for $(t, x)\in \mathbb{T}^2$, it follows from (3.16) that

This is a contradiction, and our proof is finished.

4.   Proof of Theorem 1.4

In this section, we consider the multiplicity of positive solutions for problem (1.8) with (1.2) by using a completely different method from that of Ortega-Robles-Pérez ^[7], Mawhin-Ortega-Robles-Pérez ^[4,5,6], Li ^[3], Gilding-Kersner ^[8], Li ^[9], and Wang and An ^[10,11], namely the following lemma about fixed point index in a cone, which is used in Hu and Wang ^[21] and Zhang ^[22].

Lemma 4.1. (^[25]) Let $E$ be a real Banach space and $K$ be acone in $E$. For $r > 0$, define $K_{r} = \{x\in K: \|x\| < r \}$. Assumethat $T: \bar K_{r}\rightarrow K$ is completely continuous such that$Tx\neq x$ for $x\in \partial K_{r} = \{x\in K:\|x\| = r\}$.

$(i)$ If $\|Tx\|\geq \|x\|$ for $x\in \partial K_{r}$, then $i(T, K_{r}, K) = 0$.

$(ii)$ If $\|Tx\|\leq \|x\|$ for $x\in \partial K_{r}$, then $i(T, K_{r}, K) = 1$.

Let $T_\lambda: K\rightarrow E$ be defined by

where $K$ is defined in (2.5) and $E: = C(\mathbb{T}^2)$.

It follows from Lemma 2.3 that $T_\lambda: K\rightarrow E$ is completely continuous, and the doubly periodic solution of (1.8) with (1.2) is equivalent to the fixed point of $T_\lambda$.

Proof of Theorem 1.4. Let $u\in K$ and

for $(t, x)\in \mathbb{R}^2$. Then $h_\lambda\in E$ and $T_\lambda u = Ph_\lambda$, and it follows from the proof of Lemma 2 in ^[9] that

For any given $\tau > 0$, it follows from ${\bf (H_{5})}$ that there exist $\eta > 0$ and $d > \tau$ such that

Letting $\delta = \frac{d}{\underline{G}\eta\|\omega\|_1}$, then for $\lambda > \delta$, (4.1) and Lemma 2.3 imply that $T_{\lambda}: K\rightarrow K$ is completely continuous.

Considering ${\bf (H_{3})}$, there exists $0 < r < d$ such that $f(u)\leq \frac{\Lambda}{2} u$ for $0\leq u\leq r$, where

So, for $u\in \partial K_{r} = \{u\in K:\|u\| = r\}$, we have from (4.1) and (4.2) that

Consequently, for $u \in\partial K_{r}$, we have $\|T_{\lambda}u\| < \|u\|$. Thus, by Lemma 4.1, we get

Now turning to ${\bf (H_{4})}$, there exists $\sigma > 0$, for $u > \sigma$, such that $f(u)\leq \frac{\Lambda}{2}u$. Letting $\eta = \max\limits_{ 0\leq u \leq \sigma}f(u)$, then

Choosing

So for $x \in\partial K_{R}$, from (4.1) and (4.2) we have

Thus, it follows from Lemma 4.1 that

On the other hand, for $u \in\bar K_{d}^{R} = \{u\in K: \|u\|\leq R, \min\limits_{(t, x)\in \mathbb{T}^2}u(t, x)\geq d\}$, (4.1), (4.2), (4.7) and (4.8) yield that

Furthermore, for $u \in\bar K_{d}^{R}$, from (4.1), (4.2), (4.3), and ${\bf (H_{5})}$, we obtain

Letting $u_{0}\equiv \frac{d+R}{2}$ and $H(x, u) = (1-x)T_{\lambda}u+xu_{0}$, then $H:[0, 1]\times\bar K_{d}^{R}\rightarrow K$ is completely continuous, and from the analysis above, we obtain for $(x, u)\in [0, 1]\times \bar K_{d}^{R}$

Therefore, for $x\in [0, 1], u\in \partial K_{d}^{R}$, we have $H(x, u)\neq u.$ Hence, by the normality property and the homotopy invariance property of the fixed point index, we obtain

Consequently, by the solution property of the fixed point index, $T_{\lambda}$ admits a fixed point $u_{\lambda}^{(1)}$ with $u_{\lambda}^{(1)} \in K_{d}^{R}$, and

On the other hand, it follows from (4.6), (4.10) and (4.12) together with the additivity of the fixed point index that

By the solution property of the fixed point index, $T_{\lambda}$ so admits a fixed point $u_{\lambda}^{(2)}$ and $u_{\lambda}^{(2)} \in K_{R}\backslash (\bar K_{r}\cup\bar K_{d}^{R})$. It is not difficult to see that $u_{\lambda}^{(1)}\neq u_{\lambda}^{(2)}$. This gives the proof of Theorem 1.4.

5.   Proof of Theorem 1.5

In this section, we intend to analyze the uniqueness and continuity of positive solution on the parameter $\lambda$ to problem (1.8) with (1.2). In order to prove Theorem 1.5, we need the following results and some definitions, which can be found in Guo-Lakshmikantham ^[27].

Let $E$ be a real Banach space, $K$ is a cone of $E$. Every cone $K \subset E$ induces an semi-order in $E$ given by $''\leq''$. That is, $x\leq y$ if and only if $y-x \in K$. If cone $K$ is solid and $y-x\in K^{\circ}$, we write $x\ll y$.

Definition 5.1. Let $K$ be a cone of a real Banach space $E$. $K$ is a solid cone, if $K^{\circ}$ is not empty, where $K^{\circ}$ is the interior of $K$.

Definition 5.2. Let $K$ be a solid cone of a real Banach space $E$. $A: K^{\circ}\rightarrow K^{\circ}$ is an operator. $A$ is called an $\alpha$-concave operator ($-\alpha$-convex operator), if

where $0\leq\alpha < 1$. The operator $A$ is increasing (decreasing), if $x_{1}, x_{2}\in K^{\circ}$ and $x_{1}\leq x_{2}$ imply $Ax_{1}\leq Ax_{2} \ (Ax_{1}\geq Ax_{2})$, and further, the operator $A$ is strong increasing (decreasing), if $x_{1}, x_{2}\in K^{\circ}$ and $x_{1} < x_{2}$ imply $Ax_{2}- Ax_{1}\in K^{\circ} (Ax_{1}- Ax_{2}\in K^{\circ})$. Let $x_{\lambda}$ be a proper element of the enginvalve $\lambda$ of $A$, that is $Ax_{\lambda} = \lambda x_{\lambda}$. $x_{\lambda}$ is called strong increasing (decreasing), if $\lambda_{1} > \lambda_{2}$ implies that $x_{\lambda_{1}}-x_{\lambda_{2}}\in K^{\circ}(x_{\lambda_{2}}-x_{\lambda_{1}}\in K^{\circ})$, which is denoted by $x_{\lambda_{1}}\gg x_{\lambda_{2}}(x_{\lambda_{2}}\gg x_{\lambda_{1}})$.

Lemma 5.1. Suppose that $P$ is a normal cone of a real Banach space, $A: P^{\circ}\rightarrow P^{\circ}$ is an $\alpha-$concave increasing (or $-\alpha-$convex decreasing) operator. Then $A$ has exactly one fixed point in $P^{\circ}$.

Proof of Theorem 1.5. Let $\Psi = \lambda A$, where $A: K\rightarrow E$ be defined by

$K$ is defined in (2.5) and $E: = C(\mathbb{T}^2)$. In fact, one can prove that the operator $\Psi$ maps $K$ into $K$. In view of $G(t, s) > 0, \ {\bf (H_2)}$ and $f(u) > 0$ for $u > 0,$ it is easy to see that $\Psi:\ K^{0}\rightarrow K^{0}$. We prove that $\Psi:\ K^{0}\rightarrow K^{0}$ is an $\alpha$-concave increasing operator.

In fact, it follows from (5.1), Lemma 2.3 and $f(\rho u)\geq\rho^{\alpha}f(u)$ that

where $0\leq\alpha < 1.$ Since $f(u)$ is nondecreasing, then

Lemma 5.1 yields that $\Psi$ admits a unique fixed point $u_{\lambda}\in K^{0}$. This shows that problem (1.8) with (1.2) admits a unique positive solution $u_{\lambda}$.

Next, we prove that (i)-(iii) hold. Let $\gamma = \frac{1}{\lambda}$, and denote $\lambda Au_{\lambda} = u_{\lambda}$ by $Au_{\gamma} = \gamma u_{\gamma}$. Assume $0 < \gamma_{1} < \gamma_{2}$. Then $u_{\gamma_{1}}\geq u_{\gamma_{2}}.$ In fact, set

We prove $\bar{\eta}\geq1.$ If it is not, then $0 < \bar{\eta} < 1,$ and further

This shows

which is a contradiction to (5.2).

It follows from the discussion above that

This proves that $u_{\gamma}(t)$ is strong decreasing in $\gamma$. Namely $u_{\lambda}(t)$ is strong increasing in $\lambda$. This gives the proof of Theorem 1.5 (i).

Let $\gamma_{2} = \gamma$ and fix $\gamma_{1}$ in (5.3), we get $u_{\gamma_{1}}\geq\frac{\gamma}{\gamma_{1}}u_{\gamma}$, for $\gamma > \gamma_{1}$. Further

where $N_{1} > 0$ denotes a normal constant. Noting that $\gamma = \frac{1}{\lambda}$, we get $\lim\limits_{\lambda\rightarrow 0^{+}}\|u_{\lambda}(t)\| = 0.$

Similarly, set $\gamma_{1} = \gamma$ and fix $\gamma_{2}$, again by (5.3) and the normality of $K$, we obtain $\lim\limits_{\lambda\rightarrow +\infty}\|u_{\lambda}(t)\| = +\infty.$

This finishes the proof of (ii).

Finally, we prove the continuity of $u_{\gamma}(t)$. For given $\gamma_{0} > 0.$ From (i),

Set

It is clear to see that $0 < l_{\gamma} < 1$ and $u_{\gamma}\geq l_{\gamma}u_{\gamma_{0}}.$ We so have

and further

It follows the definition of $l_{\gamma}$ that

Again from the definition of $l_{\gamma}$, we obtain

Noticing that $K$ is a normal cone, it follows from (5.5) and (5.6) that

Similarly, we have

where $N_{2} > 0$ denotes a normal constant. This shows that Theorem 1.5 (iii) holds. The proof of Theorem 1.5 is complete.

Remark 5.1. The idea of the proof for Theorem 1.5 comes from Theorem 2.2.7 in Guo-Lakshmikantham ^[27], but there is almost no paper studying the uniqueness of positive doubly periodic solution for telegraph equations.

Remark 5.2. In Theorem 1.5, even though we do not suppose that $A$ is continuous even completely continuous, we can prove that $u_{\lambda}$ depends continuously on $\lambda$.

Acknowledgments

This work is sponsored by the Beijing Natural Science Foundation of China (1212003) and the promoting the classified development of colleges and universities-application and cultivation of scientific research awards of BISTU (2021JLPY408).

Conflict of interest

The authors declare there is no conflicts of interest.