Existence of two homoclinic solutions for a nonperiodic difference equation with a perturbation

1.   Introduction

It is well-known that homoclinic solutions play an important role in analyzing the chaos of dynamical systems. Hence, it is significative to deal with homoclinic solutions of dynamical systems. In the present paper, let ${\bf Z}$ and ${\bf R}$ be the set of integers and real numbers, respectively. For parameter $\lambda > 0$, we investigate the existence of two nontrivial homoclinic solutions for the following nonperiodic fourth-order difference equation

Here $\omega$ is a given constant, $1\leq p < 2$. $f(t, u): {\bf Z}\times {\bf R}\rightarrow{\bf R}$ is continuous in $u$ and $\frac{\partial F(t, u)}{\partial u} = f(t, u)$. $a(t):{\bf Z}\rightarrow{\bf R}$ and $h(t): {\bf Z}\rightarrow{\bf R}^+$. Let $\Delta u(t) = u(t+1)-u(t)$ be the forward difference operator and define $\Delta^0 u(t) = u(t)$, $\Delta^i u(t) = \Delta(\Delta^{i-1} u(t))$ for $i\geq 1$. As usual, we say that a solution $u = \{u(t)\}$ of (1.1) is homoclinic (to 0) if $\lim\limits_{|t|\rightarrow +\infty}u(t) = 0$. In addition, if $u(t)\neq 0$, then $u$ is called a nontrivial homoclinic solution.

Difference equations possess theoretical background and practical significance. For example, ^[1] proposes a Dirichlet boundary value problem of difference equation to represent the amplitude of the motion of every particle in the string, ^[2] uses difference equations to study the impact of dispersal of a two-patch SIR disease model and ^[3] studies Wolbachia infection in mosquito population based on discrete models. Consequently, difference equations have attracted many researchers' attentions and rich results are obtained. To mention a few, ^[4,5,6] establish criteria for the existence and multiplicity of solutions, ^{[7,8,9,10,11]} focus on sign-changing solutions and ^{[12,13,14,15,16,17,18]} deal with homoclinic solutions or heteroclinic solutions.

Consider (1.1), it has been put forward as a discrete mathematical model for the study of pattern formation in physics and mechanics and deeply studied. For example, ^[19,20] investigate sign-changing of a special case of (1.1) by the invariant sets of descending flow. As mention to homoclinic solutions, which play an important role in analyzing the chaos of dynamical systems, there are many publications such as ^[21] studies (1.1) in a special form with periodic assumption and ^[22,23] prove that some special kind of (1.1) admits one nontrivial homoclinic solution. To the best of our knowledge, as considering homoclinic solutions for fourth order difference equations similar to (1.1), in the most known results of the existence of one non-zero homoclinic solution usually depend on periodic conditions or on the multiplicity with assumption of odevity on nonlinear terms. Meanwhile, using a compactness lemma and variational techniques, we achieve two nontrivial homoclinic solutions for (1.1) not only with neither periodic conditions nor odd-even requirements on nonlinear terms, but also with a perturbation. In some sense that our result improves and extends some known results.

Let constants $c_0$ be given in Lemma 2.2 and $c_s$ be the best constant for the embedding of a Hilbert space $X$, which is defined in Section 2, in $L^s$, $2\leq s < +\infty$ and $q(t): {\bf Z}\rightarrow {\bf R}^+$ with $\max\limits_{t\in{\bf Z}}\{q(t)\} = q > c_0c^2_2$. Write

Remark 1.1. $\mu^*$ defined as (1.2) is reasonable. We state the proof of it in Lemma 3.2.

With the above notations, now we establish our main result:

Theorem 1.1. Let $h(t): {\bf Z}\rightarrow {\bf R}^+$ with $\max\limits_{t\in{\bf Z}}\{h(t)\} = h > 0$. Assume $a(t): {\bf Z}\rightarrow {\bf R}$ and there exists a constant $a_1$ such that $\omega\leq 2\sqrt{a_1}$ and

Further, for $i\geq 1$, suppose $f(t, u): {\bf Z}\times {\bf R}\rightarrow {\bf R}$ is continuous in $u$ and the following assumptions hold:

$(F_1) \; f(t, u)\equiv 0$ for all $u < 0$, $t\in {\bf Z}$ and there exists $b(t): {\bf Z}\rightarrow {\bf R}^+$ with $\max\limits_{t\in {\bf Z}}\{b(t)\} = b < \frac{c_0c_2^2}{2}$ such that

and

$(F_2)$

$(F_3)$ there exist two constants $\theta > 2$ and $\frac{c_0c_2^2(\theta-2)}{4\theta} < d_0 < \frac{c_0c_2^2(\theta-2)}{2\theta}$, such that

Then either $(i) \; i = 1$ and $\mu^* < 1$ or $(ii) \; i > 1$, there exists $\Lambda_0 > 0$ such that (1.1) admits at least two distinct homoclinic solutions for every $\lambda\in (0, \Lambda_0)$.

Remark 1.2. The assumptions in Theorem 1.1 are feasible. For example, take $b(t) = \frac{c_0c_2^2}{4}$ for all $t\in {\bf Z}$. It follows that $0 < \max\limits_{t\in {\bf Z}}\{b(t)\} = \frac{c_0c_2^2}{4} < \frac{c_0c_2^2}{2}$. Let

then

Consequently,

and

which means the assumptions $(F_1)$ and $(F_2)$ are satisfied.

Moreover, from the expression of $F(t, u)$, for all $t\in {\bf Z}$ and $u\geq 0$, we have

Obviously, $\frac{x^{i+1}}{1+{(x+\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}+\sqrt{2}})}^2} > 0$ for $x > 0$, which follows that $\int_0^u\frac{x^{i+1}}{1+{(x+\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}+\sqrt{2}})}^2}dx\geq0$. Then one can verify that condition $(F_3)$ is satisfied.

Remark 1.3. Our result improves and generalizes some known results. For example, similarly, ^[23] gives that there exists one homoclinic solution for the following difference equation

(1.1) can be regarded as a special case of (1.6) when $p = 2$. Our result Theorem 1.1 points out that (1.1) has at least two nontrivial homoclinic solutions. Meanwhile, according to ^[23], (1.1) possesses one homoclinic solution. Therefore, our result improves the result in ^[23] in some sense.

The organization of the paper is as follows: After this introduction, we present some basic lemmas and establish the corresponding variational functional to (1.1) in Section 2. Section 3 provides the detailed proof of our main result.

2.   Variational structure and basic lemmas

In this section, we give some notations and basic lemmas to prepare for the proof of our main result Theorem 1.1.

Denote $u = {\{u(t)\}}_{t\in {\bf Z}} = (\cdots, u(-t), \cdots, u(-1), u(0), u(1), \cdots, u(t), \cdots)$. Let the set of all two-sided sequences $S = \left\{u = \{u(t)\}: u(t)\in {\bf R}, t\in {\bf Z}\right\}$, then $S$ is a vector space with $au+bv = \{au(t)+bv(t)\}$ for $u, v\in S$, $a, b\in {\bf R}$. Define a subspace $E$ of $S$ as

For any $u, v \in E$, define

For later use, we define another Hilbert space $(X, { < u, v > }_X)$, where

and, for any $u, v\in X$, the inner product is given by

Then the corresponding norm is

In what follows, let

denote the space of functions whose s-th powers are summable on Z and

Thus the following embedding between $L^s$ spaces holds,

Now we start to consider the variational functional of (1.1). Define a functional $J: E\rightarrow {\bf R}$ as

Then the continuity of $f$ indicates that $J\in C^1(E, {\bf R})$ and, for any $u, v\in E$, its derivative is expressed as

which means that $u\in E$ is a critical point of $J$ if and only if $u$ is a homoclinic solution of (1.1).

Recall the definition of Cerami sequence and the variant version of the mountain pass theorem from critical point theory, which are helpful for us to seek critical points of (2.1).

Definition 2.1. Let $J \in C^1(E, {\bf R})$. A sequence $\{u_n\}\in E$ is called a Cerami sequence ($(C)_c$ sequence for short) for $J$ if $J(u_n) \rightarrow c$ for some $c \in {\bf R}$ and $(1 +\|u_n\|)J'(u_n)\rightarrow 0$ as $n \rightarrow \infty$. If any ${(C)}_c$ sequence for $J$ possesses a convergent subsequence, then $J$ satisfies the ${(C)}_c$ condition.

Lemma 2.1. (Mountain pass theorem) (^[24]) Let $E$ be a real Banach space with its dual space $E^*$, and suppose that $I\in C^1(E, {\bf R})$ satisfies

for some $\mu < \eta$, $\rho > 0$ and $e\in E$ with $\|e\| > \rho$. Let $\hat{c}\geq \eta$ be characterized by

where $\Gamma = \{\gamma\in C([0, 1], E): \gamma(0) = 0, \gamma(1) = e\}$ is the set of continuous paths joining 0 and $e$, then there exists a sequence $\{u_n\}\subset E$ such that

Remark 2.1. Similar to ^[23], Lemma 2.1 allows us to find a ${(C)}_c$ sequence for $J$.

In the following, we establish some compactness conditions.

Lemma 2.2. Suppose $a(t)\geq a_1 > 0$ and $\omega < 2\sqrt{a_1}$, then for any $u\in E$

is true for some constant $c_0 > 0$.

Proof. Since $\omega < 2\sqrt{a_1}$, we divide it into two cases.

Case 1. $\omega < 0$. Take $c_0 = \min\{-\omega, a_1, 1\}$, obviously,

which implies that (2.2) is true.

Case 2. $0\leq \omega < 2\sqrt{a_1}$. In this case, it is easy to get $\omega^2 < 4a_1$, then there exists $k\in (0, 3)$ satisfying

Hence,

Consider

Denote

then (2.3) ensures

which indicates that

Therefore, for $k\in (0, 3)$, we have

Analogous to ^[25], for any $u(t)\in X$, we take

then

Denote $\xi = 1-e^{i\zeta}\triangleq \varphi(\zeta)$, then $\zeta = \varphi^{-1}(\xi)$. Now we have

and

Thanks to (2.5), there has

Choose $c_0 = \sqrt{1-\frac{k}{3}} > 0$, then (2.6) leads to ${\|u\|}^2_E\geq c_0{\|u\|}^2_X$, that is, (2.2) holds for $0\leq \omega < 2\sqrt{a_1}$. Therefore, Lemma 2.2 is true and the proof is completed.

With the help of Lemma 2.2, we obtain that ${ < u, u > }_E$ is positive for all nonzero $u\in E$ and $E$ is a Hilbert space. Here and hereafter, we write

Now we state the main compactness lemma and present its proof in detail.

Lemma 2.3. Let (1.3) hold and $\omega < 2\sqrt{a_1}$. Then, for $2\leq s\leq +\infty$, $E$ is compactly embedded in $L^s$.

Proof. First, we prove Lemma 2.3 holds for $s = 2$.

Define

From (1.3), $\alpha(A)$ increases and $\alpha(A)\rightarrow +\infty$ as $|t|\rightarrow +\infty$.

Let $K$ be a bounded subset of $E$. It follows that, if $u\in K$, there exists a constant $M > 0$ such that ${\|u\|}_E\leq M$. Thanks to (2.2), we have

Hence, we have

Write $\hat{b}\triangleq M^2+|\omega|\cdot \frac{4M^2}{c_0}$. For any $\epsilon > 0$, take $A_0$ large enough such that

Since $K\subset E$ is bounded by $M$, there are $u_1, u_2, \cdots, u_m\in K$ such that for any $u\in K$, there exists some $u_l$ ($1\leq l\leq m$) satisfying

Combining (2.7) with (2.8), it yields

which implies ${\|u-u_l\|}_{L^2}\rightarrow 0$.

Next we verify that our claim is true for $s = +\infty$. Notice that for any $n\in {\bf N}$, $T\in {\bf Z}$, $2\leq k\in {\bf N}$ and $u\in E$, thanks to the Newton-Lebnitz formula of indefinite summation and step by step summation ^[26], we have

and

Hence, for all $T\leq t\leq T+k-1$, we have

On the other hand, owing to $a^\theta +b^\theta \leq {(a+b)}^\theta$ holds for $a, b \geq 0$ and $\theta \geq 1$, it follows

In view of $\frac{\sqrt{a}+\sqrt{b}}{\sqrt{2}}\leq \sqrt{a+b}$ ($a, b \geq 0$), similar to (2.10), we have

Therefore, for all $T\leq t\leq T+k-1$ and $2\leq k\in {\bf N}$, with the aid of (2.10) and (2.11), we have

which implies that

holds for any $u, v\in K$, $A > 0$ and all $|t| > A$. For any $\epsilon > 0$, take first $n$ large enough such that

Notice that $2\leq k\in {\bf N}$, then, for any $\epsilon > 0$, choose $A_0$ large enough such that

Therefore, we can draw a conclusion that

By the same method of (2.8), it follows that

Combing (2.13) with (2.14), we obtain

Finally, we accomplish the proof of Lemma 2.3 by verifying it is correct for $2 < s < +\infty$. Given an arbitrary $u\in E$, there has

which implies that $K$ is precompact in $L^s$. Combing (2.9), (2.15) and (2.16), we complete the proof of Lemma 2.3 immediately.

3.   Proof of the main results

In this section, we intend to prove the main result at length. Now we are in the position to state the following several lemmas which guarantee that the functional $J$, defined by (2.1), has the mountain pass geometry at first.

Lemma 3.1. Let $a(t)$ and $\omega$ satisfy the assumptions in Theorem 1.1 and the conditions $(F_1)$ and $(F_2)$ hold. Then there exist $\Lambda_0 > 0$ and constants $\rho, \eta > 0$ such that

for every $\lambda\in (0, \Lambda_0)$.

Proof. For every $\epsilon > 0$, notice that the condition $(F_1)$ implies that there exists a constant $\delta > 0$ such that $f(t, s)\leq (b+\epsilon)s^i\leq (b+\epsilon)s$ holds for $0 < s < \delta$. From $(F_2)$, $\lim\limits_{s\rightarrow\infty}\frac{f(t, s)}{s^i} = q(t)$ leads to $\lim\limits_{s\rightarrow\infty}\frac{f(t, s)}{s^{i+1}} = 0$, which implies that there exists a constant $M > 0$ big sufficiently such that $\frac{f(t, s)}{s^{i+1}}\leq \epsilon$, that is, $f(t, s)\leq \epsilon s^{i+1}$ with $s > M$. Further, since $f(t, s)$ is continuous, it is not difficult to choose a constant $C$ such that $\frac{f(t, s)}{s^{i+1}}\leq C$ for $\delta\leq s\leq M$. Therefore,

which indicates that there exist $C_\epsilon > 0$ and $r\geq i+2$ such that

For $2\leq r < +\infty$, let $c_r$ be the best constants for the embedded of $X$ in $L^r$. With the aid of Lemma 2.2, we get

that is,

Together with (3.1), for all $u\in E$, one can obtain

Therefore,

By the last equation in (3.3), select $\epsilon = \frac{c_0c_2^2}{2}-b > 0$ and denote $t = {\|u\|}_E\geq 0$, we define

Since $r > 2$ and $1\leq p < 2$, it is easy to find that $g(t)$ will get its maximum value at $t = {\left(\frac{rc_0^{\frac{r}{2}}c_r^{r}(2-p)}{4C_\epsilon(r-p)}\right)}^{\frac{1}{r-2}}\triangleq \rho > 0$. Hence

Combing with (3.3), it yields that there exists $\Lambda_0 = \frac{Mc_0^{\frac{p}{2}}c_p^{p}}{h} > 0$ such that we can find a constant $\eta > 0$ which satisfies $J(u)|_{{\|u\|}_E = \rho}\geq \eta$ for every $\lambda\in (0, \Lambda_0)$.

Lemma 3.2. Let $\rho$ and $\Lambda_0$ be defined in Lemma 3.1. Suppose that the conditions $(F_1)$ and $(F_2)$ hold, then for every $\lambda\in (0, \Lambda_0)$ there exists $e\in E$ with ${\|e\|}_E > \rho$ such that $J(e) < 0$ holds for either $i = 1$ and $\mu^* < 1$ or $i > 1$.

Proof. We give the proof in two cases.

Case Ⅰ. If $i = 1$ and $\mu^* < 1$, we first declare $\mu^*$, defined as (1.2), is reasonable. Let $u\in E$ satisfy $\sum\limits_{t = -\infty}^{+\infty}q(t)u^2(t) = 1$. Then

which means that ${\|u\|}_{L^2}^2\geq \frac{1}{q}$. In view of Lemma 2.2, we get

Thus $\mu^*\geq \frac{c_0c_2^2}{q} > 0$. Aim to get $\mu^*$ is attainable, let $\{u_n\}\in E$ be a minimizing sequence of (1.2), then $\{u_n\}$ is bounded and satisfies $\sum\limits_{t = -\infty}^{+\infty}q(t)u_n^2(t) = 1$. Choose a subsequence of $\{u_n\}$, without loss of generality, still denoted by $\{u_n\}$. In view of Lemma 2.3, there exists $\phi_1 \in E$ such that $u_n\rightharpoonup \phi_1$ weakly in $E$ and $u_n\rightarrow \phi_1$ strongly in $L^2$. Hence

Therefore,

which indicates that $\mu^* = \sum\limits_{t = -\infty}^{+\infty}\left[{\left(\Delta^2\phi_1(t-1)\right)}^2-\omega {\left(\Delta\phi_1(t-1)\right)}^2 +a(t)\phi_1^2(t)\right] = {\|\phi_1\|}^2_E$.

Since $\mu^* < 1$, it is not difficult to choose $0\leq \varphi \in E$ with $\sum\limits_{t = -\infty}^{+\infty}q(t)\varphi^2(t) = 1$ such that ${\|\varphi\|_E} < 1$. Using the given condition $(F_2)$, we have

which tells us that $J(s\varphi)\rightarrow -\infty$ as $s\rightarrow +\infty$. Then there exists $e\in E$ with ${\|e\|}_E > \rho$ such that $J(e) < 0$.

Case Ⅱ. If $i > 1$, in view of $q(t): {\bf Z}\rightarrow {\bf R}^+$, we find there exists $\psi\in E$ such that

In the same manner as Case Ⅰ, we have

Therefore, there exists $e\in E$ with ${\|e\|}_E > \rho$ such that $J(e) < 0$. The proof is completed.

Notice Lemma 3.1 and Lemma 3.2 show that $J$ meets all conditions in Lemma 2.1, hence $J$ possesses a $(C)_c$ sequence $\{u_n\}\subset E$ for the mountain pass level $\beta$ which is defined by

and $\Gamma = \{\gamma\in C([0, 1], E)|\gamma(0) = 0, \gamma(1) = e\}$.

In the following, we set out to look for homoclinic solutions for (1.1). Denote $B_\rho = \{u\in E: {\|u\|}_E < \rho\}$, where $\rho$ is given by Lemma 3.1. We first seek for a critical point of $J$ by showing $J$ attains a local minimum for small $\lambda$.

Lemma 3.3. Let $\rho$ and $\Lambda_0$ be defined in Lemma 3.1. Assume $a(t)$, $\omega$ and $h(t)$ satisfy Theorem 1.1 and $(F_1)$ hold. Then, for $\lambda\in (0, \Lambda_0)$, (1.1) possesses a homoclinic solution $u_0\in E$ such that

Proof. Since $h(t):{\bf Z}\rightarrow {\bf R}^+$, it is convenient to select $\zeta\in E$ such that $\sum\limits_{t = -\infty}^{+\infty}h(t){|\zeta(t)|}^p > 0$. For $\kappa > 0$ small enough, $(F_1)$ induces $F(t, \kappa\zeta(t)) > 0$ is correct for all $t\in {\bf Z}$. Then for $1\leq p < 2$, one has

Write $m\triangleq\inf\{J(u): u\in \bar{B_\rho}\}$, then $m < 0$. Thus there exists a minimizing sequence $\{u_n\}\subset E$ such that $J(u_n)\rightarrow m$ and $J'(u_n)\rightarrow 0$ as $n\rightarrow \infty$. Therefore, Lemma 2.3 ensures that $J$ admits a critical point $u_0\in E$ which satisfies $J'(u_0) = 0$ and $J(u_0) = m < 0$.

In view of Lemma 3.3, it is necessary for us to show that there exists another $\bar{u}\in E$ such that $J'(\bar{u}) = 0$ and $\bar{u}\neq u_0$ to accomplish the proof of Theorem 1.1.

Proof of Theorem 1.1. We complete the proof in two steps.

Step 1. The ${(C)}_c$ sequence $\{u_n\}\in E$ of $J$, defined by

is bounded. Let $n$ be large enough. By the condition $(F_3)$ and Lemma 3.1, it follows that

Obviously, for $\theta > 2$ and $p < 2$, (3.4) implies ${\|u_n\|}_E$ is bounded for all $\lambda\in (0, \Lambda_0)$.

Step 2. Now it is time for us to verify that $J$ has another critical point $\bar{u}$ which satisfies $J'(\bar{u}) = 0$ and $J(\bar{u}) = \beta > 0$. Since the ${(C)}_c$ sequence $\{u_n\}\subset E$ of $J$ is bounded, from Lemma 2.3, there exists $\bar{u}\in E$ satisfying, up to a subsequence,

Together with the Hölder inequality, it follows that

and

On the other hand, the definition of $J(u)$ indicates that

Hence $u_n\rightarrow \bar{u}$ strongly in $E$. Moreover, $J(\bar{u}) = \beta > 0$ and $\bar{u}$ is another homoclinic solution of (1.1). Consequently, $u_0$ and $\bar{u}$ are two distinct homoclinic solutions of (1.1). And the proof of Theorem 1.1 is finished.

Acknowledgements

The author sincerely thanks the handling editor and the anonymous referees for their valuable comments and suggestions.

Conflict of interest

All authors declare no conflicts of interest in this paper.