DAU-Net: A medical image segmentation network combining the Hadamard product and dual scale attention gate

1.   Introduction and main result

In this article, we study the existence of infinitely many homoclinic solutions of the following fractional discrete $p$-Laplacian equations:

where $s\in(0, 1)$ and $p\in(1, \infty)$ are fixed constants, $V(a)\in\mathbb{R}^+$, $\lambda$ is a positive parameter, $f(a, \cdot)$ is a continuous function for all $a\in\mathbb{Z}$ and $(-\Delta_{1})^{s}_{p}$ is the fractional discrete $p$-Laplacian given by

where the discrete kernel $K_{s, p}$ has the following property: There exist two constants $0 < c_{s, p}\le C_{s, p}$, such that

The fractional operator has received more attention recent decades because of its many applications in the real world. Many scholars have paid attention to this kind of problem, and have produced a lot of classical works, see for example ^[1,2,3]. As a classical fractional operator, the fractional Laplacian has wide applications in various fields such as optimization, population dynamics and so on. The fractional Laplacian on $\mathbb{R}$ can be defined for $0 < s < 1$ and $\nu\in C_{0}^{\infty}(\mathbb{R}^{N})$ as

where $C_{N, s}$ is a positive constant and $P.V.$ denotes the Cauchy principle value. In various cases involving differential equations, the Laplace operator is replaced by either the fractional Laplace operator or other more general operators, and hence the existence results have been obtained by employing variational approaches, see for instance ^[4,5,6,7]. These alternative approaches have been found to offer improved descriptions of numerous phenomena observed in the natural world. Correspondingly, it is necessary to give some qualitative results by employing numerical analysis. The nonlocal feature of the fractional Laplacian is one of the important aspects to be considered in numerical methods, which makes it necessary to study the existence of solutions.

Let $\mathbb{Z}_{\mathcal{H}}$ denote a grid of fixed size $\mathcal{H} > 0$ on $\mathbb{R}$, i.e., $\mathbb{Z}_{\mathcal{H}} = \left\lbrace \mathcal{H}a | a\in \mathbb{Z}\right\rbrace$. In ^[8], the definition of the fractional discrete Laplacian on $\mathbb{Z}_{\mathcal{H}}$ is given by

where $s\in(0, 1)$, $\nu\in \ell_{s} = \left\lbrace \nu:\mathbb{Z}_{\mathcal{H}}\to \mathbb{R} \big| \sum_{\omega\in \mathbb{Z}}\frac{|\nu(\omega)|}{(1+|\omega|)^{1+2s}} < \infty \right\rbrace$ and

The above discrete kernel $K_{s}^{\mathcal{H}}$ has the following property: There exist two constants $0 < c_{s} \leq C_{s}$, such that for all $a\in \mathbb{Z} \backslash \{0\}$ there holds

In ^[8], Ciaurri et al. also proved that if $\nu$ is bounded then $\lim_{s \to 1^{-}}(-\Delta_{\mathcal{H}})^{s}\nu(a) = -\Delta_{\mathcal{H}}\nu(a)$, where $\Delta_{\mathcal{H}}$ is there discrete Laplacian on $\mathbb{Z}_{\mathcal{H}}$, i.e.

Moreover, under some suitable conditions and $\mathcal{H}\to0$, the fractional Laplacian can be approximated by the fractional discrete Laplacian.

Subsequently, let us give some existence results on the fractional difference equations. Xiang et al. ^[9] first investigated the fractional discrete Laplacian equations based on variational methods:

where $f(a, \cdot)$ is a continuous function for all $a\in\mathbb{Z}$, $\lambda > 0$, $V(a)\in\mathbb{R}^{+}$ and

Using the mountain pass theorem and Ekeland's variational principle under some suitable conditions, they obtained two homoclinic solutions for problem (1.3). It is evident that when $p = 2$, the discrete fractional $p$-Laplace operator corresponds to the discrete fractional Laplace operator. After that, Ju et al. ^[10] studied the following fractional discrete $p$-Laplacian equations

where $a\in\ell^{\frac{p}{p-q}}$, $b\in\ell^{\infty}$, $1 < q < p < r < \infty$, $\lambda > 0$, $V(a)\in\mathbb{R}^{+}$, $(-\Delta_{1})^{s}_{p}$ is the fractional discrete $p$-Laplacian. Under certain conditions, they employed the Nehari manifold method to achieve the existence of at least two homoclinic solutions for problem (1.4). In ^[11], Ju et al. investigated the existence of multiple solutions for the fractional discrete $p$-Laplacian equations with various nonlinear terms via different Clark's theorems. In a recent study conducted by Ju et al. in ^[12], it was demonstrated that using the fountain theorem and the dual fountain theorem under the same hypotheses, two separate sequences of homoclinic solutions were derived for the fractional discrete Kirchhoff-Schrödinger equations. Based on the findings from ^[8], it could be deduced that Eq.(1.3) can be reformulated as the renowned discrete version of the Schrödinger equation

It is worth mentioning that in ^[13], Agarwal et al. first employed the variational methods to analyze Eq.(1.5). Here, We give some literature on the study of difference equations using the critical point theory, see ^[14,15,16].

In particular, we observe that both the nonlinear terms in ^[11,12] have the following symmetry condition:

$(\mathcal{S})$ $f(a, \nu)$ is odd in $\nu$.

Therefore, in this paper, we consider the nonlinear term without condition $(\mathcal{S})$, and study the existence of multiple homoclinic solutions of problem (1.1). For this, let us first recall if the solution $\nu$ of Eq.(1.1) satisfies $\nu(d)\to 0$ as $|d|\to \infty$, then $\nu$ is called a homoclinic solution. Suppose that $V(a)$ and $f(a, u(a))$ in problem (1.1) satisfy the following assumptions:

$(\mathcal{V})$ $V\in\ell^{1}$ and there is a constant $V_{0}\in\left(0, \inf_{a\in\mathbb{Z}}V(a) \right]$; ($\ell^{1}$ is defined in next section)

$(\mathcal{F})$ $\left| f(a, u)\right| \le C\left(\left|u\right|^{p-1}+\left|u\right|^{t-1}\right)$ for any $a \in \mathbb{Z}$ and $u \in \mathbb{R}$, where $p < t < \infty$ and $C > 0$ is a constant.

Set

where $F(a, u) = \int_{0}^{u}f(a, \eta)d\eta$ and $C_{s}$, $C_{b}$ will appear in next section. Here we give the main conclusion of our paper as follows.

Theorem 1.1. Suppose that $(\mathcal{V})$ and $(\mathcal{F})$ are satisfied. Furthermore, the following inequality holds: $A < \theta B$. Then, for every $\lambda\in\left(\frac{C_{b}\left\|V\right\|_{1}}{pB}, \frac{C_{s}V_{0}}{pA}\right)$, problem (1.1) possesses infinitely many nontrivial homoclinic solutions. In addition, their critical values and their $\ell^{\infty}$-norms tend to zero.

The rest of this article is arranged as follows: In Section 2, we introduce some definitions and give some preliminary results. In Section 3, we give the proof of Theorem 1.1. In Section 4, we give an example to demonstrate the main result.

Here we illustrate some notations used in this paper:

● $C$, $C_{s, \omega}$, $C_{s}$, $C_{b}$ and $C_{\infty}$ are diverse positive constants.

● $\hookrightarrow$ denotes the embedding.

● $\to$ denotes the strong convergence.

2.   Preliminaries

First we give some basic definitions.

Let $1\le \omega\le\infty$, we give the definition of the space $\left(\ell^{\omega}, \left\| \cdot \right\|_{\omega} \right)$ as follows:

Through the corresponding conclusions in ^[17], we know that $\ell^{\omega}$ is a Banach space. Moreover, $\ell^{\omega_{1}} \hookrightarrow \ell^{\omega_{2}}$ and $\left\|\mu \right\|_{\omega_{2}}\le\left\|\mu \right\|_{\omega_{1}}$ if $1\le\omega_{1}\le\omega_{2}\le\infty$.

Next, we give the variational framework and some lemmas of this paper.

The space $\left(Q, \left\| \cdot \right\|_{Q} \right)$ is defined by

Lemma 2.1. (see [10, Lemma 2.1]) If $\xi\in\ell^{\omega}$, then $[\xi]_{s, \omega} \leq C_{s, \omega}\|\xi\|_{\omega} < \infty$.

Lemma 2.2. (see [11, Lemma 2.2]) Under the hypothesis $(\mathcal{V})$, $\left(Q, \left\| \cdot \right\|_{Q} \right)$ is a reflexive Banach space, and

is an equivalent norm of $Q$.

Through Lemma 2.2, we obtain that there exist $0\le C_{s}\le C_{b}$ such that

Lemma 2.3. Under the hypothesis $(\mathcal{V})$, $Q\hookrightarrow \ell^{r}$ is continuous for all $p\le r \le\infty$.

Proof. Using the above conclusions and $(\mathcal{V})$, we can deduce that

As desired.

Lemma 2.4. (see [10, Lemma 2.4]) If $\mathcal{W}\subset Q$ is a compact subset, then for $\forall\; \iota > 0$, $\exists\; a_{0}\in \mathbb{N}$ such that

For all $u\in Q$, we define

where

and

Clearly

Lemma 2.5. (see [10, Lemma 2.5]) Under the hypothesis $(\mathcal{V})$, then $D(\sigma)\in C^{1}(Q, \mathbb{R})$ with

for all $\sigma, \xi\in Q$.

Lemma 2.6. (see [12, Lemma 2.6]) Under the hypotheses $(\mathcal{V})$ and $(\mathcal{F})$, then $E(\sigma)\in C^{1}(Q, \mathbb{R})$ with

for all $\sigma, \xi\in Q$.

Combining Lemma 2.5 and Lemma 2.6, we know that $K(\sigma)\in C^{1}(Q, \mathbb{R})$.

Lemma 2.7. Under the hypotheses $(\mathcal{V})$ and $(\mathcal{F})$, then for $\forall\; \lambda > 0$, every critical point of $K$ is a homoclinic solution of problem (1.1).

Proof. Assume $\sigma$ be a critical point of $K$, we get for $\forall\; \xi\in Q$

For each $a\in\mathbb{Z}$, we define $e_{d}\in Q$ as follows:

Taking $\xi = e_{d}$ in (2.3), we have

So $\sigma$ is a solution of problem (1.1). Moreover, by Lemma 2.3 and $\sigma\in Q$, we know $\sigma(a)\to 0$ as $\left|a\right|\to \infty$. Thus, $\sigma$ is a homoclinic solution of problem (1.1).

3.   Proof of main result

In this section, we shall use the following Thoerem 3.1 to prove our main result. In fact, this theorem is a special version of Ricceri's variational principle [18, Lemma 2.5].

Theorem 3.1. (see [19, Lemma 2.1]) Let $Q$ be a reflexive Banach space, $K(\mu): = D(\mu)+\lambda E(\mu)$ for each $\mu\in Q$, where $D, E\in C^{1}(Q, \mathbb{R})$, $D$ is coercive, and $\lambda$ is a real positive parameter. For every $\gamma > \inf_{Q}D(\mu)$, let

and

If $\rho < +\infty$, then for every $\lambda\in(0, \frac{1}{\rho})$, the following conclusions holds only one:

$(a)$ there exists a global minimum of $D$ which is a local minimum of $K$.

$(b)$ there exists a sequence $\{\mu_{m}\}$ of pairwise distinct critical points (local minima) of $K$, with $\lim_{m\to\infty}D(\mu_{m}) = \inf_{Q}D(\mu)$, which converges to a global minimum of $D$.

Remark 3.1. Obviously, $\rho\ge0$. In addition, when $\rho = 0$, we think that $\frac{1}{\rho} = +\infty$.

Proof of Theorem 1.1. Let us recall

where $F(a, u) = \int_{0}^{u}f(a, \omega)d\omega$. Fix $\lambda\in\left(\frac{C_{b}\left\|V\right\|_{1}}{pB}, \frac{C_{s}V_{0}}{pA}\right)$ and set $K, D, E$ as in Section 2. By Lemma 2.2, Lemma 2.5 and Lemma 2.6, we know $Q$ be a reflexive Banach space and $D, E\in C^{1}(Q, \mathbb{R})$. Because of

as $\left\|\mu\right\|_{Q}\to+\infty$, i.e. $D$ is coercive. Now, we show that $\rho < +\infty$. For this purpose, let $\{\delta_{n}\}$ be a positive sequence such that $\lim_{n\to\infty}\delta_{n} = 0$ and

Put

for all $n\in\mathbb{N}$. Clearly, $\lim_{n\to\infty}\gamma_{n} = 0$. For $n > 0$ is big enough, by Lemma 2.2 and (2.1), we can derive that

Since $D(0) = E(0) = 0$, for each $n$ large enough, by (3.1), we get

Therefore, by (2.2), we acquire that

From (3.2), we get

Next, we verify that $0$ is not a local minimum of $K$. First, suppose that $B = +\infty$. Choosing $M$ such that $M > \frac{C_{b}\left\|V \right\|_{1}}{p\lambda}$ and let $\left\lbrace h_{n}\right\rbrace$ be a sequence of positive numbers, with $\lim_{n\to\infty}h_{n} = 0$, there exists $n_{1}\in\mathbb{N}$ such that for all $n\ge n_{1}$

Therefore, let $\left\lbrace l_{n}\right\rbrace$ be a sequence in $Q$ defined by

It is easy to infer that $\left\|l_{n}(a)\right\|_{Q}\to 0$ as $n\to\infty$. By $(\mathcal{V})$, (2.1) and (3.3), we obtain

So, $K(l_{n}) < 0 = K(0)$ for each $n\ge n_{1}$ big enough. Next, suppose that $B < +\infty$. Since $\lambda > \frac{C_{b}\left\|V\right\|_{1}}{pB}$, there exists $\varepsilon > 0$ such that $\varepsilon < B-\frac{C_{b}\left\|V\right\|_{1}}{p\lambda}$. Hence, also choosing $\left\lbrace h_{n}\right\rbrace$ be a sequence of positive numbers, with $\lim_{n\to\infty}h_{n} = 0$, there is $n_{2}\in\mathbb{N}$ such that for all $n\ge n_{2}$

Arguing as before and by choosing $\left\lbrace l_{n}\right\rbrace$ in $Q$ as above, we get

So, $K(l_{n}) < 0 = K(0)$ for each $n\ge n_{2}$ big enough. In general, $0$ is not a local minimum of $K$. By Theorem 3.1, $(a)$ is not valid, then we have a sequence $\left\lbrace \mu_{n}\right\rbrace\subset Q$ of critical points of $K$ such that

and

By Lemma 2.3, we gain

as $n\to\infty$. By Lemma 2.7, the problem (1.1) admits infinitely many nontrivial homoclinic solutions. In addition, their critical values and their $\ell^{\infty}$-norms tend to zero. This completes the proof.

4.   Example

Here, we give an example of a nonlinear term which can apply Theorem 1.1.

Example 4.1. We define

Obviously, we know that $\Phi(n+1) < \Psi(n) < \Phi(n)$ for all $n\in\mathbb{N}_{+}$ and $\lim_{n\to\infty}\Psi(n) = \lim_{n\to\infty}\Phi(n) = 0$. Set

And for each $a\in\mathbb{N}_{+}$, let $f(a, \cdot)$ is a nonnegative continuous function such that

There are many nonlinear terms that satisfy the above conditions. Here we give one of them as an example.

where $e_{M\times N}$ is the indicator function on $M\times N$. Then

and

Now it is easy to see that all the assumptions of Theorem 1.1 are satisfied, hence the corresponding conclusion can be delivered by Theorem 1.1.

Acknowledgements

B. Zhang was supported by the National Natural Science Foundation of China (No. 12171152), the Shandong Provincial Natural Science Foundation, PR China (No. ZR2023MA090) and the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.