Wavelet estimations of a density function in two-class mixture model

1.   Introduction

Difference equations represent the discrete counterpart of ordinary differential equations, and they are applied in various research fields, such as computer science, economics and biology ^[1,2,3,4]. Many authors have obtained excellent results for difference equations ^[5,6,7,8]. In this paper, we focus on the discrete nonlinear Schrödinger (DNLS) equations which are widely utilized to describe a multitude of physical and biological phenomena, such as nonlinear optics ^[9,10], biomolecular chains ^[11], the lattice dynamics of solids and the localization of electromagnetic waves in photonic crystals with a nonlinear response ^[12]. For more reviews on this theme, we refer to ^{[13,14,15,16,17,18,19,20]}.

In this paper, we consider the existence of discrete gap solitons for the following periodic (DNLS) equation

where $L$ is a second order difference operator given by

$\{a_{n}\}$ and $\{b_{n}\}$ are real valued T-periodic sequences and $\omega\in \mathbb{R}$. $f_{n}(\cdot)\in C(\mathbb{R}, \mathbb{R})$, $f_{n}(0) = 0$ and $f_{n+T}(t) = f_{n}(t)$.

We are interested in the existence of gap solitons. This problem appears when we consider the standing wave solutions for the following DNLS equation

where $\Delta\psi_{n} = \psi_{n+1}+\psi_{n-1}-2\psi_{n}$ is the discrete Laplacian in one spatial dimension, $f_{n}(\cdot)$ is continuous from $\mathbb{C}$ into $\mathbb{C}$ and $f_{n}(0) = 0$ for every $n\in\mathbb{Z}$. The sequence $\{V_{n}\}$ of real numbers and the sequence $\{f_{n}(\cdot)\}$ of functions are assumed to be T-periodic, i.e., $V_{n+T} = V_{n}$, $f_{n+T}(\cdot) = f_{n}(\cdot)$. The typical saturable nonlinearities are as follows:

and

where $l_{n}$ is a real valued T-periodic sequence. The saturable nonlinearities can describe optical pulse propagation in various doped fibers ^[21,22]. In Eq (1.3), we suppose the nonlinearity $f_{n}$ is gauge invariant, i.e.,

Spatially localized standing waves are often called breathers or solitons. Since solitons are spatially localized time-periodic solutions and decay to zero at infinity, $\psi_{n}$ has the following form:

where $\{u_{n}\}$ is a real valued sequence and $\omega\in\mathbb{R}$ is the temporal frequency. Then, Eq (1.3) becomes

In Eq (1.1), the operator $L$ is a bounded and self-adjoint operator in the space $l^{2}$. The spectrum $\sigma(L)$ has a band structure, i.e., $\sigma(L)$ is a union of a finite number of closed intervals ^[23]. Thus the complement $\mathbb{R}\backslash \sigma(L)$ consists of a finite number of open intervals called spectral gaps, two of which are semi-infinite. Generally, a soliton for Eq (1.3) with the temporal frequency $\omega$ belonging to a spectral gap is called a gap soliton. In this work, we fix one spectral gap by $(-\infty, \beta)$ and consider exploring nontrivial solutions that are not equal to $0$ identically for the case $\omega\in(-\infty, \beta)$.

In 2006, Pankov ^[24] first studied the gap solitons for the periodic DNLS equation, Eq (1.3) with the function $f_{n}(t) = l_{n}|t^{2}|t$ given by the linking theorem in combination with periodic approximations. Since then, the DNLS equations with saturable nonlinearities have also been studied very intensively. These pioneering works have produced many novel and interesting results on the existence of solutions ^{[21,25,26,27,28]}. In 2008, Pankov and Rothos ^[25] considered Eq (1.1) with $a_{n} = -1$ and $b_{n} = 2$ when the nonlinearity $f_{n}(t) = f(t)$ is saturable at $\infty$. Pankov obtained the following result.

Theorem 1.1^[25]. Assume that the nonlinearity $f(t)$ satisfies the following assumptions:

$(h1) f(t) = o(t)$ as $t\rightarrow0$;

$(h2) \lim\limits_{|t|\rightarrow \infty}\frac{f(t)}{t} = l < \infty$;

$(h3) f(t)\in C(\mathbb{R})$ and $f(t)t < f'(t)t^{2}$ for $t\neq0$;

$(h4) g(t) = f(t)-lt$ is bounded.

Assume $\omega < 0$ and $l+\omega > 0$. Then there exists a non-trivial ground-state solution $u\in l^{2}$ of Eq (1.1).

In 2010, Pankov ^[26] considered the existence of gap solitons for Eq (1.1) with saturable nonlinearities when $\omega$ belongs to a spectral gap $(\alpha, \beta)$ of the linear part and $0\notin\sigma(L)$. The main method of the proof involves applying critical point theory in combination with periodic approximations of solutions.

Theorem 1.2^[26]. Assume the conditions $(i)-(iv)$ hold:

$(i)$ $\forall n\in\mathbb{Z}$, the function $f_{n}:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, $f_{n}(t) = f_{n+T}(t)$, $f_{n}(0) = 0$ and $f_{n}(t) = o(t)$ as $t\rightarrow0$;

$(ii) \lim\limits_{|t|\rightarrow \infty}\frac{f_{n}(t)}{t} = l_{n}$ exists and is finite, and the function $g_{n}(t) = f_{n}(t)-l_{n}t$ is bounded;

$(iii)$ For every $r_{0} > 0$, there exists $\delta_{0} = \delta_{0}(r_{0}) > 0$ such that

for $|t|\geq r_{0}$ and $F_{n}(t)\geq0$ for $t\in\mathbb{R}$.

$(iv) \lambda = \min\limits_{n\in\mathbb{Z}}l_{n} > \hat{\theta}$, where $\hat{\theta}$ is the bottom of the positive part of $\sigma(L)$.

Then there exists a nonzero solution $u\in l^{2}$ of Eq (1.1).

Under the condition that the nonlinearity is saturable, Zhou and Yu obtained a new sufficient condition for the existence of homoclinic solutions of the system by using the mountain pass lemma in combination with periodic approximations in 2010 ^[27]. They proved that it is also necessary in some special cases.

Theorem 1.3^[27]. Assume that $\omega\in(-\infty, \beta)$, $f_{n}$ is continuous in $t$, $f_{n}(t) = f_{n+T}(t)$ for any $n\in\mathbb{Z}$ and $t\in\mathbb{R}$, $f_{n}(t) = o(t)$ as $t\rightarrow0$. And the following conditions hold.

$(H1) \frac{f_{n}(t)}{t}$ is strictly increasing in $(0, +\infty)$ and strictly decreasing in $(-\infty, 0)$. Moreover, $\lim\limits_{|t|\rightarrow \infty}\frac{f_{n}(t)}{t} = d_{n} < \infty$;

$(H2) tf_{n}(t)-2F_{n}(t)\rightarrow \infty$ as $|t|\rightarrow \infty$, and $\limsup\limits_{t\rightarrow0} \frac{f_{n}^{2}(t)}{tf_{n}(t)-2F_{n}(t)} = p_{n} < \infty$.

If $d_{n} > \beta-\omega$ for $n\in\mathbb{Z}$, then Eq (1.1) has at least a nontrivial solution $u\in l^{2}$.

Other related gap solitons results can be found in the literature ^{[17,18,19,28,29,30,31]}. The existence of solutions for Eq (1.1) has been widely considered under the condition of various assumptions on the saturable nonlinearity $f_{n}(t)$. To prove the existence of solutions for Eq (1.1), the main required assumptions are $(h1)$, $(h3)$, $(iii)$, $(H1)$ and $(H2)$ which are given in most papers ^[19,30,32]. The conditions $(h1)$ and $(h3)$ imply the condition $(H1)$, that the function $t\rightarrow \frac{f_{n}(t)}{|t|}$ is strictly increasing in $(-\infty, 0)$ and $(0, +\infty)$. According to $(H1)$, it follows that $tf_{n}(t)\geq2F_{n}(t)\geq0$ for all $t\in\mathbb{R}$, that is, $tf_{n}(t)$ has an unchanging sign. We must point out that one essential assumption, $(H1)$ must be used to prove the critical functional satisfying the Palais-Smale (P.S.) condition in many studies. Moreover, the assumption $(H2)$, i.e., $tf_{n}(t)-2F_{n}(t)\rightarrow +\infty$ need also be used during the proof in some studies, such as in Theorem 1.3 from ^[27]. In these situations, most of the authors established the existence of solutions for Eq (1.1); see ^[27,31] for examples. Regarding the case that $tf_{n}(t)$ is allowed to be sign changing, there seems to be only a few papers that apply this for the DNLS equations. For the case in which $tf(t)$ is sign changing, we can find some results ^{[19,33,34,35]}. However, most of these papers deal with the nonlinearity in the sup-cubic case rather than the saturable nonlinearity at infinity.

Motivated by the above works, in this paper, we further consider the gap soliton problems of Eq (1.1) with saturable nonlinearity $f_{n}(t)$ at infinity. Let $\delta = \beta-\omega > 0$; we assume the following assumptions are satisfied:

$(F1)$ $\lim\limits_{t\rightarrow0}\frac{f_{n}(t)}{t} = q$ uniformly for all $n\in\mathbb{Z}$, where $|q| < \frac{\delta}{8}$;

$(F2)$There exists a constant $d > 0$ such that $\lim\limits_{|t|\rightarrow \infty}\frac{f_{n}(t)}{t} = d < +\infty$ uniformly for all $n\in\mathbb{Z}$ and the function $g_{n}(t) = f_{n}(t)-dt$ is bounded for each $n\in\mathbb{Z}$;

$(F3)$There exists a constant $\gamma\in(0, \delta)$ such that $tf_{n}(t)-4F_{n}(t)\geq-\gamma t^{2}$, $\forall n\in\mathbb{Z}$.

The term $t f_{n}(t)$ is allowed to be sign-changing based on the assumption $(F3)$ and the assumption $(F1)$ implies that the nonlinear term represents the mixed nonlinearities that can be superlinear or asymptotically linear at the origin. By using the mountain pass lemma in combination with periodic approximations, we establish the existence of gap solitons for Eq (1.1) in $l^{2}$.

Remark 1.1. Before proceeding further, we will first give a function that satisfies the conditions $(F1)-(F3)$, but not $(h1)$: $f_{n}(t) = o(t)$ as $t\rightarrow0$. For all $n\in\mathbb{Z}$, let $f_{n}(t) = \frac{t^{3}-2t}{1+t^{2}}$ for $t\in\mathbb{R}$. $(F1)-(F2)$ are obvious, but the aforementioned function does not satisfy $(h1)$. Since $F_{n}(t) = \frac{t^{2}}{2}-\frac{3}{2}\ln(1+t^{2})$, we can find a constant $\gamma = 5 > 0$ such that $tf_{n}(t)-4F_{n}(t)+5 t^{2} = \frac{4t^{4}+t^{2}+6(1+t^{2})\ln(1+t^{2})}{(1+t^{2})}\geq0$ and $t f_{n}(t) = \frac{t^{4}-2t^{2}}{1+t^{2}}$ is sign-changing.

2.   Preliminaries

In order to establish the variational framework associated with Eq (1.1) and apply the critical point theory, we will give some basic notations and lemmas that will be used to prove our main results. In the Hilbert space $E = l^{2}$, we consider the functional

where $(\cdot, \cdot)$ is the inner product in $l^{2}$, and $F_{n}(t) = \int_{0}^{t}f_{n}(s)ds$, $n\in\mathbb{Z}$. The corresponding norm in $E$ is denoted by $\|\cdot\|$. Then $J\in C^{1}(E, \mathbb{R})$ and we can compute the Frećhet derivative as

for all $u, \; v\in E$.

Equation (1.1) is the corresponding Euler-Lagrange equation for $J$. Thus, the nonzero critical points of $J$ are nontrivial solutions of Eq (1.1).

Let $S$ be the following set of all two-sided sequences:

Then, $S$ is a vector space with $au+bv = \{au_{n}+bv_{n}\}$ for $u, v\in S$ and $a, b\in\mathbb{R}$.

For any fixed positive integer $k$, we define the subspace $E_{k}$ of $S$ that consists of all $kT$-periodic sequences:

Obviously, $E_{k}$ is the $2kT$-dimensional Hilbert space. $E_{k}$ can be equipped with the inner product $(\cdot, \cdot)_{k}$ and norm $\|\cdot\|_{k}$ as follows:

and

We also define a norm $\|\cdot\|_{k\infty}$ in $E_{k}$ by

Consider the following functional $J_{k}$ in $E_{k}$:

then,

for all $u, \; v\in E_{k}$, where $L_{k}$ is the operator $L$ acting in $E_{k}$. This is a $C^{1}$-functional in a finite dimensional space and its critical points are exactly $kT$-periodic solutions of Eq (1.1).

We notice that $\sigma(L_{k})$ is finite, $\sigma(L_{k})\subset\sigma(L)$ and $\|L_{k}\|\leq\|L\|$ for all $k\in\mathbb{Z}$, as described in ^[23]. Furthermore, $\cup_{k\in\mathbb{Z}}\sigma(L_{k})$ is a dense subset of $\sigma(L)$.

Then, we have that

Let $X$ be a real Banach space and $J\in C^{1}(X, \mathbb{R})$. A sequence $\{u_{n}\}\subset X$ is called a P.S. sequence for $J$ if $\{J(u_{n})\}$ is bounded and $J'(u_{n})\rightarrow0$ as $n\rightarrow \infty$. We say that $J$ satisfies the P.S. condition if any P.S. sequence for $J$ possesses a convergent subsequence in $X$.

Let $B_{r}$ be the open ball centered at $0$ of radius $r$ in $H$, and $\partial B_{r}$ denotes its boundary. The following lemma will play an important role in the proof of our main results.

Lemma 2.1 (Mountain Pass Lemma ^[36]). Let $H$ be a real Hilbert space, and assume that $J\in C^{1}(H, \mathbb{R})$ satisfies the P.S. condition if $J(0) = 0$ and the following conditions hold.

$(G_{1})$ There exist constants $\rho > 0$ and $\alpha > 0$ such that $J(x)\geq \alpha$ for all $x\in\partial B_{\rho}$, where $B_{\rho} = \{x\in H: \|x\| < \rho\}$.

$(G_{2})$ There exists $e\in H\backslash \bar{B}_{\rho}$ such that $J(e) < 0$.

Then $c = \inf\limits_{h\in\Gamma}\sup\limits_{s\in[0, 1]}J(h(s))\geq \alpha$ is a positive critical value of $J$, where

3.   Main results

If some of these conditions, i.e., $(F1)-(F3)$ are violated or changed, Eq (1.1) has no nontrivial solutions in $l^{2}$. Thus, we have the following proposition.

Proposition 3.1. Assume that the conditions $(F1)$ and $(F2)$ hold and $\delta\rightarrow +\infty$. Then, Eq $(1.1)$ has no nontrivial solutions in $l^{2}$.

Proof. Arguing by contradiction, we assume that Eq $(1.1)$ has a nontrivial solutions for $u\in l^{2}$, then, $u$ is a critical point of $J$ that satisfies

Since $(F1)$ and $(F2)$ are satisfied, they imply that there exists a finite constant $a_{1} > 0$ such that $|tf_{n}(t)|\leq a_{1}|t|^{2}$ uniformly for all $n\in\mathbb{Z}$ and $t\in\mathbb{R}$. By Eq $(2.3)$, we have that

This is impossible from $\delta\rightarrow +\infty$.

Remark 3.1. If we consider that the conditions $(F1)$, $(F2)$ and $a_{1} < \beta-\omega$ are satisfied, Eq $(1.1)$ still has no nontrivial solutions in $l^{2}$(see^[27]).

We define the linear operator as

Let $G_{n}(t) = \int_{0}^{t}g_{n}(s)ds$ be the primitive functions of $g_{n}$ for all $n\in\mathbb{Z}$. Then we can represent the functional $J_{k}$ and its derivative in the following forms:

and

for all $u, \; v\in E_{k}$.

Lemma 3.1. Assume that the condition $(F2)$ holds and $\omega\notin\sigma(\tilde{L}_{k})$; then, the functional $J_{k}$ satisfies the P.S. condition.

Proof. Let $\{u^{(j)}\}\subset E_{k}$ be a sequence such that $J_{k}(u^{(j)})$ is bounded and $J'_{k}(u^{(j)})\rightarrow0$ as $j\rightarrow \infty$. Since $E_{k}$ is finite dimensional, it is enough to show that the sequence $\{u^{(j)}\}$ is bounded.

Let $E_{k}^{+}$ and $E_{k}^{-}$ be the positive spectral subspace and the negative spectral subspace of the operator $\tilde{L}_{k}-\omega$ in $E_{k}$, respectively; then, $E_{k} = E_{k}^{+}\bigoplus E_{k}^{-}$.

Hence, we can find a positive constant $\eta > 0$ such that

For each $j\in\mathbb{Z}$, we write $u^{(j)} = u^{(j)+}+u^{(j)-}$; then,

In view of $(F2)$, we can take $\varepsilon = \frac{\eta}{2} > 0$ and a sufficiently large $M > 0$ such that

We have that

Let $M_{0} = \max\limits_{n\in Q_{k}^{(j)}}\{|g_{n}(u_{n}^{(j)})|\}$. Then

The above inequality implies that

Combining Eq $(3.4)$, the Cauchy-Schwartz inequality and Eq $(3.6)$, we obtain the following inequality

Similarly, we get

Since $\|u^{(j)}\|_{k}^{2} = \|u^{(j)+}\|_{k}^{2}+\|u^{(j)-}\|_{k}^{2}$ and $\|u^{(j)+}\|_{k}+\|u^{(j)-}\|_{k}\leq\sqrt{2}\|u^{(j)}\|_{k}$, by Eqs $(3.7)$ and $(3.8)$, we have that

Hence, the sequence $\{u^{(j)}\}$ is bounded.

In the following theorem, we will use Lemma 2.1 and Lemma 3.1 to prove the existence of the nonzero critical point $u^{(k)}$ of $J_{k}$ in $E_{k}$.

Theorem 3.1. Assume that the conditions $(F1)$-$(F2)$ hold and $\omega\notin\sigma(\tilde{L}_{k})$. If $d > \beta-\omega$, then $J_{k}$ has at least one nonzero critical point $u^{(k)}$ in $E_{k}$. Moreover, $\|u^{(k)}\|_{k}$ is bounded and there exist positive constants $\xi$ and $\mu$ such that

Proof. The functional $J_{k}$ satisfies the P.S. condition according to Lemma 3.1. Next, we need only to verify that $J_{k}$ satisfies the conditions $G_{1}$ and $G_{2}$ in Lemma 2.1. In fact, $J_{k}(0) = 0$; let $\varepsilon = \frac{\delta}{4}-\frac{q}{2} > 0$; then, there exists some positive constant $\rho > 0$ such that

Since $\|u\|_{k\infty}\leq\|u\|_{k}$, we have that $\sum\limits_{n = -kT}^{kT-1}F_{n}(u_{n})\leq \frac{\delta}{4}\|u\|_{k}^{2}$ for all $u\in E_{k}$ with $\|u\|_{k}\leq\rho$; then,

Taking $\alpha = \frac{\delta}{4}\rho^{2}$, we get $J_{k}(u)\geq \alpha$ for all $u\in\partial B_{\rho}$; hence, $J_{k}$ satisfies the condition $G_{1}$ of Lemma 2.1.

Next, we will verify the condition $G_{2}$.

Let $\beta_{k}$ be the lowest point in $\sigma(L_{k})$. According to the spectral theory of periodic difference operators, endpoints of spectral gaps are either $T$-periodic or $T$-antiperiodic, eigenvalues of the difference operator $L$. Since $\beta$ is one of endpoints, we see that either $\beta_{k} = \beta$ for all integers $k\geq 1$, or that $\beta_{2k} = \beta$ for all integers $k \geq 1$. Since $\omega\notin\sigma(\tilde{L}_{k})$ and $d > \beta-\omega$, it is easy to verify that $E_{k}^{-}\neq\emptyset$ for $k\geq 1$.

We denote $\lambda$ by the lowest positive point in $\sigma(\tilde{L}_{k}-\omega)$ in $E_{k}$. Let $z^{k}\in E_{k}^{+}$ be a unit eigenvector of $\tilde{L}_{k}-\omega$ with the eigenvalue $\lambda$. We can find that $y\neq0$ in $E_{k}^{-}$. Let $u = z^{k}+\tau\frac{y}{\|y\|_{k}}$, where $\tau\in\mathbb{R}$. We have that

By $(F2)$, $|G_{n}(t)|$ has, at most, linear growth for all $n\in\mathbb{Z}$; we have that

Thus, we can choose $\tau_{0}\in\mathbb{R}$ such that $\|u_{0}\|_{k} = \sqrt{1+\tau_{0}^{2}} > \rho$ and $J_{k}(u_{0}) = J_{k}\left(z^{k}+\tau_{0}\frac{y}{\|y\|_{k}}\right) < 0$.

All of the assumptions of Lemma 2.1 have been verified; then, $J_{k}$ possesses a critical value $c_{k}\geq a$ with

where

Given that a critical point $u^{(k)}$ of $J_{k}$ corresponds to $c_{k}$ in $E_{k}$, $u^{(k)}$ is nonzero as $c_{k} > 0$. We denote $h\in\Gamma_{k}$ as $h(s) = s\left(z^{k}+\tau_{0}\frac{y}{\|y\|_{k}}\right)$ for $s\in[0, 1]$; then,

In fact,

Hence,

Let $0\leq\gamma < \delta$. We see from $(F3)$, Eq $(2.1)$ and $(2.2)$ that

So the sequence $\{u^{(k)}\}$ is bounded in $E_{k}$. In particular, Eq $(3.16)$ implies that $|u_{n}^{(k)}|\leq2\sqrt{\frac{M_{1}}{\delta-\gamma}}$ for $n\in\mathbb{Z}$, that is,

On the other hand, from Eq $(2.4)$, we have that

From $(F1)$, we have that $\varepsilon = \frac{\delta}{4}-|q| > 0$ and find a positive number $\xi$ such that

Consequently, from Eq $(3.18)$, we get $|u^{(k)}_{n}|\geq\xi$ for $n\in\mathbb{Z}$, so

Now, we can give the proof of our main result.

Theorem 3.2. Assume that the conditions $(F1)$-$(F3)$ hold. If $d > \beta-\omega$ and $\omega\notin\sigma(L-d)$, then Eq $(1.1)$ possesses at least one nontrivial solution $u$ in $l^{2}$.

Proof $u^{(k)} = \{u_{n}^{(k)}\}\in E_{k}$ is a critical point obtained via Theorem 3.1, and there exists $n_{k}\in\mathbb{Z}$ such that

Note that

By the periodicity of the coefficients in Eq (3.21), we see that $\{u_{n+T}^{(k)}\}$ is also a solution of Eq (3.21). Making some shifts if necessary, and without loss of generality, we can assume that $0\leq n_{k}\leq T-1$ in Eq (3.20). Moreover, passing to a subsequence of $\{u^{(k)}\}$ if necessary, we can also assume that $n_{k} = n^{\ast}$ for $k\geq1$ and some integer $n^{\ast}$ such that $0\leq n^{\ast}\leq T-1$. It follows from Eq (3.20) that we can choose a subsequence, still denoted by $\{u^{(k)}\}$, such that

Obviously, from Eq (3.21), given $k\rightarrow \infty$, we can obtain the following:

that is, $u$ is a nonzero solution of Eq (1.1), as Eq(3.20) implies $\xi\leq|u_{n^{\ast}}^{(k)}|$. Now, we will verify that $u = u_{n}\in l^{2}$. For each $s\in\mathbb{N}$, let $k > s$. Then, it follows from (3.17) that

Let $k\rightarrow \infty$. We obtain $\sum\limits_{n = -s}^{s}(u_{n})^{2}\leq\frac{4M_{1}}{\delta-\gamma}$. By the arbitrariness of $s$, we know that $u = {u_{n}}\in l^{2}$. The proof is completed.

4.   Conclusions

In this paper, through the use of variational methods, we consider the existence of gap solitons for the class of difference equations described by Eq (1.1) in one spatial dimension. Under general asymptotically linear conditions on nonlinearity $f_{n}(\cdot)$ at infinity, we find a gap soliton for Eq (1.1). Most of the pioneering work requires that the function $t\rightarrow \frac{f_{n}(t)}{|t|}$ is strictly increasing in $(-\infty, 0)$ and $(0, +\infty)$, and that $tf_{n}(t)\geq2F_{n}(t)\geq0$ for all $t\in\mathbb{R}$; see Theorems 1.1, 1.2 and 1.3. Obviously, the assumptions of these conditions are stronger than ours and $tf_{n}(t)$ has an unchanging sign, but our conditions require that $t f_{n}(t)$ is sign-changing according to $(F3)$. This is different from the previous assumptions of $(h3)$, $(iii)$ or $(H2)$. Moreover, for Theorems 1.1, 1.2 and 1.3, the corresponding authors have decided to assume $f(t) = o(t)$ at the origin; we give a general assumption $(F1)$ in this paper; it implies that the nonlinear term represents the mixed nonlinearities that can be superlinear or asymptotically linear at the origin. We must point out that our work is different from previous works, and that the results of this paper improve, extend and complement some related results in the literature ^[25,26,27]. Our solutions are obtained by applying the variational approach in combination with periodic approximations; additionally, the functional $J$ is required to satisfy the P.S. conditions; hence, some technical methods are used to deal with the process of proof.

The dynamic behavior of solitons has recently attracted much interest. In future work, we will analyze the existence problem for multiple solitons in Schrödinger lattice systems and the stability of gap solitons in lattices, as well as investigate the nonlinear evolution of unstable solitons under perturbations. Moreover, we plan to explore the stability of the solitons by investigating the linearization and to verify our results via direct simulations.

Acknowledgments

This research was funded by the Key Scientific Research Projects of Colleges and Universities in Henan Province (Grant No. 22B110006), the Scientific Research Projects of Lüliang City (Grant No. Rc2020213) and the Scientific and Technology Key Project of Henan Province of China (Grant No. 222102110028).

Conflict of interest

All authors declare that they have no conflict of interest.