A Schrödinger-type algorithm for solving the Schrödinger equations via Phragmén–Lindelöf inequalities

Abstract

In this article, we consider the numerical method for solving the Schrödinger equations via Phragmén–Lindelöf inequalities under the order induced by a symmetric cone with the function involved being monotone. Based on the Phragmén–Lindelöf inequalities, the underlying system of inequalities is reformulated as a system of smooth equations, and a Schrödinger-type method is proposed to solve it iteratively so that a solution of the system of the Schrödinger equations is found. By means of the Schrödinger type inequalities, the algorithm is proved to be well defined and to be globally convergent under weak assumptions and locally quadratically convergent under suitable assumptions. Preliminary numerical results indicate that the algorithm is effective.

1 Introduction

In this paper, we consider the following Schrödinger equation (see [18, 19]):

$$ - \Delta u +V(x)u= \biggl(\frac{1}{ \vert x \vert ^{\alpha }}\ast \vert u \vert ^{p} \biggr) \vert u \vert ^{p-2}u, \quad x \in \Re ^{n}, $$

(1.1)

where $n\geq 3$, $0<\alpha <n$, $2-\frac{\alpha }{n}< p\leq 2^{\ast } _{\alpha }=\frac{2n-\alpha }{n-2}$ and

(A1):

$V\in C(\mathbb{R}^{n},\mathbb{R^{+}})$ is coercive, that is,

$$ \lim_{|x|\to +\infty }V(x)=+\infty . $$

In 2016, Qiao et al. [19] first considered the bound and ground states for the nonlinear Schrödinger equations under the condition

(A2):

$\inf_{\mathbb{R}^{n}} V>0$, and there exists a positive constant r satisfying

$$ \operatorname{meas}\bigl\{ x\in \mathbb{R}^{n}, \vert x-y \vert \leq r, V(x)\leq M\bigr\} \to 0 $$

as $|y|\to \infty$, where $M>0$ and meas stands for the Lebesgue measure.

The nonlinear system (1.1) has been proved to possess wide application fields in many real world problems such as anomalous diffusion [2, 4, 15], disease models [6, 9, 21], ecological models [26], synchronization of chaotic systems [1, 27], etc.

Put

$$ \nu (x):=\frac{1}{2} \bigl\Vert u(x) \bigr\Vert ^{2} $$

is the Nevanlinna norm (see [8]), problem (1.1) is equivalent to the following Schrödinger problem defined by

$$ \min \nu (x), $$

(1.2)

where $x\in \Re ^{n}$.

The Phragmén–Lindelöf inequalities (see [23]) have the main objective to solve the so-called Schrödinger Phragmén–Lindelöf subproblem model to get the trial step $\tau _{k}$

$$\begin{aligned} \operatorname{Min}\quad & \mathcal{TW}_{k}(\tau ) = \frac{1}{2} \bigl\Vert u(x_{k})+\nabla S (x_{k}) \tau \bigr\Vert ^{2}, \\ &\Vert \tau \Vert \leq \triangle . \end{aligned}$$

In 2015, a modified Phragmén–Lindelöf inequality was proved by Wan [23]:

$$\begin{aligned} \operatorname{Min}\quad & \phi _{k}(\tau ) = \frac{1}{2} \bigl\Vert u(x_{k})+\nabla u(x_{k})\tau \bigr\Vert ^{2}, \\ &\Vert \tau \Vert \leq c^{p} \bigl\Vert u(x_{k}) \bigr\Vert ^{\gamma }, \end{aligned}$$

where p, c, and γ are three positive numbers.

Recently, another adaptive Schrödinger Phragmén–Lindelöf inequality has been defined by Qiao et al. [17]:

$$ \begin{aligned} \operatorname{Min}\quad & \mathcal{TM}_{k}(\tau ) = \frac{1}{2} \bigl\Vert u(x_{k})+\mathcal{B}_{k} \tau \bigr\Vert ^{2}, \\ &\Vert \tau \Vert \leq c^{p} \bigl\Vert u(x_{k}) \bigr\Vert , \end{aligned} $$

(1.3)

where $\mathcal{B}_{k}$ is defined by

$$ \mathcal{B}_{k+1}=\mathcal{B}_{k} - \frac{\mathcal{B}_{k} s_{k} s_{k} ^{\mathcal{T}} \mathcal{B}_{k}}{s_{k}^{\mathcal{T}} \mathcal{B}_{k} s _{k}} + \frac{y_{k} {y_{k}}^{\mathcal{T}}}{{y_{k}}^{\mathcal{T}}s_{k}}, $$

(1.4)

$y_{k}=u(x_{k+1})-u(x_{k})$ and $s_{k}=x_{k+1}-x_{k}$. This Schrödinger Phragmén–Lindelöf method can possess the global convergence without the nondegeneracy (see [1, 7, 11, 26] etc.), which shows that this paper made a further progress in theory. And there exist many applications about the Schrödinger Phragmén–Lindelöf inequalities (see [3, 25, 27, 28] etc.) for nonsmooth optimizations and other problems.

We further consider the Schrödinger Phragmén–Lindelöf model for the nonlinear system $u(x)$ at $x_{k}$ (see [17])

$$ \vartheta (x_{k}+\tau )=u(x_{k})+\nabla u(x_{k})^{\mathcal{T}}d+ \frac{1}{2}\mathcal{T}_{k}d^{2}, $$

(1.5)

where $\nabla u(x_{k})$ is the Jacobian matrix of $u(x)$ at $x_{k}$.

It is well known that the above model (1.5) can be written as the following extension (see [20, 23, 24]):

$$ \vartheta (x_{k}+\tau )=u(x_{k})+\nabla u(x_{k})^{\mathcal{T}}d+ \frac{3}{2}\bigl(s_{k-1}^{\mathcal{T}} \tau \bigr)^{2}s_{k-1}. $$

(1.6)

If we set the Schrödinger Phragmén–Lindelöf matrix $\nabla u(x_{k})$, then we can use the Schrödinger Phragmén–Lindelöf matrix $\mathcal{B}_{k}$ instead of it. Thus, our Schrödinger Phragmén–Lindelöf model can be defined as follows:

$$ \begin{aligned} \operatorname{Min}\quad& \mathcal{N}_{k}(\tau ) = \frac{1}{2} \biggl\Vert u(x_{k})+\mathcal{B}_{k}d+ \frac{3}{2}\bigl(s_{k-1}^{\mathcal{T}}\tau \bigr)^{2}s_{k-1} \biggr\Vert ^{2}, \\ &\Vert \tau \Vert \leq c^{p} \bigl\Vert u(x_{k}) \bigr\Vert ^{\gamma }, \end{aligned} $$

(1.7)

where $\mathcal{B}_{k}=\mathcal{H}_{k}^{-1}$ and $\mathcal{H}_{k}$ is generated by

$$\begin{aligned} \mathcal{H}_{k+1} =&\mathcal{V}_{k}^{\mathcal{T}} \mathcal{H}_{k} \mathcal{V}_{k}+\rho _{k}s_{k}s_{k}^{\mathcal{T}} \\ =& \mathcal{V}_{k}^{\mathcal{T}}\bigl[\mathcal{V}_{k-1}^{\mathcal{T}} \mathcal{H}_{k-1}\mathcal{V}_{k-1}+\rho _{k-1}s_{k-1}s_{k-1}^{ \mathcal{T}} \bigr]\mathcal{V}_{k}+\rho _{k}s_{k}s_{k}^{\mathcal{T}} \\ =& \cdots \\ =& \bigl[\mathcal{V}_{k}^{\mathcal{T}}\cdots \mathcal{V}_{k-{m}+1}^{ \mathcal{T}} \bigr]\mathcal{H}_{k-{m}+1}[\mathcal{V}_{k-{m}+1} \cdots \mathcal{V}_{k}] \\ &{}+\rho _{k-{m}+1}\bigl[\mathcal{V}_{k-1}^{\mathcal{T}}\cdots \mathcal{V} _{k-{m}+2}^{\mathcal{T}}\bigr]s_{k-{m}+1}s^{\mathcal{T}}_{k-{m}+1}[ \mathcal{V}_{k-{m}+2}\cdots \mathcal{V}_{k-1}] \\ &{}+ \cdots \\ &{}+\rho _{k}s_{k}s_{k}^{\mathcal{T}}, \end{aligned}$$

(1.8)

where $\rho _{k}=\frac{1}{s_{k}^{\mathcal{T}}y_{k}}$, $\mathcal{V} _{k}=I-\rho _{k}y_{k}s_{k}^{\mathcal{T}}$ (see [23] etc.).

Let $\tau _{k}^{p}$ be the solution of (1.7). Define

$$ A\tau _{k}\bigl(\tau _{k}^{p} \bigr)=\nu \bigl(x_{k}+\tau _{k}^{p}\bigr)-\nu (x_{k}), $$

(1.9)

and predict reduction by

$$ \mathcal{P}\tau _{k}\bigl(\tau _{k}^{p} \bigr)=\mathcal{N}_{k}\bigl(\tau _{k}^{p}\bigr)- \mathcal{N}_{k}(0). $$

(1.10)

Based on definitions of $A\tau _{k}(\tau _{k}^{p})$ and $P\tau _{k}(\tau _{k}^{p})$, we design their ratio by

$$ r_{k}^{p} = \frac{A\tau _{k}(\tau _{k}^{p})}{\mathcal{P}\tau _{k}(\tau _{k}^{p})}. $$

(1.11)

Therefore, the Schrödinger-type algorithm for solving (1.1) is stated as follows.

Algorithm

Initial::

Let $\mathfrak{B}_{0}=\mathfrak{H}_{0}^{-1}\in \Re ^{n}\times \Re ^{n}$ be a symmetric and positive definite matrix. $x_{0}\in \Re ^{n}$ and $\varrho =0$. ρ, c, and ϵ are three positive constants. Let $l:=0$;

Step 1::

Stop if $\|\chi (x_{l})\|<\epsilon$ holds;

Step 2::

Solve (1.7) with $\triangle =\triangle _{l}$ to obtain $\varsigma _{l}^{\varrho }$;

Step 3::

Compute $A\varsigma _{l}(\varsigma _{l}^{\varrho })$, $\mathcal{P}\varsigma _{l}(\varsigma _{l}^{\varrho })$, and the ratio $r_{l}^{\varrho }$. If $r_{l}^{\varrho }<\rho$, let $\varrho =\varrho +1$, go to Step 2. If $r_{l}^{\varrho }\geq \rho$, go to the next step;

Step 4::

Set $x_{l+1}=x_{l}+\varsigma _{l}^{\varrho }$, $y_{l}=\chi (x_{l+1})-\chi (x_{l})$, update $\mathfrak{B}_{l+1}= \mathfrak{H}_{l+1}^{-1}$ by (1.8) if $y_{l}^{\mathfrak{T}} \varsigma _{l}^{p}>0$, otherwise set $\mathfrak{B}_{l+1}=\mathfrak{B} _{l}$;

Step 5::

Let $l:=l+1$ and $\varrho =0$. Go to Step 1.

In this paper, we further focus on convergence results of the above algorithm under the following assumptions.

Assumptions

(A):

Define the set Ω by

$$ \varOmega =\bigl\{ x|\varphi (x)\leq \varphi (x_{0})\bigr\} . $$

(1.12)

It is easy to see that Ω is bounded.

(B):

The nonlinear system $\chi (x)$ is twice continuously differentiable in $\varOmega _{1}$, which is an open convex set containing Ω.

(C):

The following Phragmén–Lindelöf relation

$$ \bigl\Vert \bigl[\nabla \chi (x_{l})- \mathfrak{B}_{l}\bigr]\chi (x_{l}) \bigr\Vert =O\bigl( \bigl\Vert \varsigma _{l}^{p} \bigr\Vert \bigr) $$

(1.13)

holds.

(D):

The sequence matrices $\{\mathfrak{B}_{l}\}$ are uniformly bounded in $\varOmega _{1}$.

It follows from Assumption (B) that (see [10, 22])

$$ \bigl\Vert \nabla \chi (x_{l})^{\mathfrak{T}}\nabla \chi (x_{l}) \bigr\Vert \leq M_{L}, $$

(1.14)

where $M_{L}$ is a positive real number.

2 Convergence results

We first have the following new Phragmén–Lindelöf inequalities.

Lemma 2.1

Let $\tau _{k}^{p}$ be the solution of (1.1). Then

$$ \mathcal{P}\tau _{k}\bigl(\tau _{k}^{p} \bigr)\leq -\frac{1}{2} \bigl\Vert \mathcal{B}_{k}u(x _{k}) \bigr\Vert \min \biggl\{ \triangle _{k}, \frac{ \Vert \mathcal{B}_{k}u(x_{k}) \Vert }{M_{l} ^{2}}\biggr\} +O\bigl(\triangle _{k}^{2}\bigr) $$

(2.1)

holds.

Proof

Define

$$ \mathcal{J}(u)=\frac{1}{2} \int _{\Re ^{n}} \vert \nabla u \vert ^{2} + u^{2}\,dx- \frac{1}{2p} \int _{\Re ^{n}} \int _{\Re ^{n}} \frac{ \vert u(x) \vert ^{p} \vert u(y) \vert ^{p}}{ \vert x-y \vert ^{ \alpha }} \,dx \,dy. $$

It follows from (1.7) that

$$ j_{0}< j_{1}=\inf_{\mathcal{N}^{-}} \mathcal{J}(u)< j_{0}+\frac{p-1}{2p}S _{\alpha , p}^{\frac{p}{p-1}}. $$

Consider $V(x)$ is a minimizer for both $S_{\alpha , p}$. By the continuity of $\mathcal{J}$, we know that

$$ \mathcal{J}(u_{0}+tV)< j_{0}+\frac{p-1}{2p}S_{\alpha , p}^{ \frac{p}{p-1}}, $$

where $0\leq t<\gamma$.

So

$$\begin{aligned} \mathcal{J}(u_{0}+tV) &=\frac{1}{2} \Vert u_{0}+tV \Vert ^{2}-\frac{1}{2p} \tilde{B}(u_{0}+tV) - \int _{\Re ^{n}}h(u_{0}+tV)\,dx \\ &=\mathcal{J}(u_{0})+\frac{t^{2}}{2}\biggl[ \Vert V \Vert ^{2}-\frac{t^{p-2}}{p} \tilde{B}(V)\biggr] +\tilde{B}(u_{0})+ \tilde{B}(tV)-\tilde{B}(u_{0}+tV) \\ &< j_{0}+\frac{p-1}{2p}S_{\alpha , p}^{\frac{p}{p-1}}. \end{aligned}$$

It follows from $t\geq \gamma$ that

$$\begin{aligned} \mathcal{J}(u_{0}+tV) &=\frac{1}{2} \Vert u_{0}+tV \Vert ^{2}-\frac{1}{2p} \tilde{B}(u_{0}+tV) - \int _{\Re ^{n}}h(u_{0}+tV)\,dx \\ &=\frac{1}{2} \Vert u_{0} \Vert ^{2}+t \int _{\Re ^{n}}\nabla u_{0}\nabla V+u_{0}V \,dx +\frac{t^{2}}{2} \Vert V \Vert ^{2}-\frac{1}{2p} \tilde{B}(u_{0}) \\ &\quad{}+\frac{1}{2p}\bigl[\tilde{B}(u_{0})+\tilde{B}(tV)- \tilde{B}(u_{0}+tV)\bigr] - \frac{1}{2p}\tilde{B}(tV) \\ &\quad{}- \int _{\Re ^{n}}hu_{0}\,dx- \int _{\Re ^{n}}htV \,dx \\ &=\mathcal{J}(u_{0})+\frac{t^{2}}{2}\biggl[ \Vert V \Vert ^{2}-\frac{t^{2(p-1)}}{2p} \tilde{B}(V)\biggr] +\frac{1}{2p}\biggl[ \tilde{B}(u_{0})+\tilde{B}(tV) \\ &\quad{}-\tilde{B}(u_{0}+tV)+2p \int _{\Re ^{n}} \int _{\Re ^{n}}\frac{ \vert u _{0}(x) \vert ^{p} \vert u_{0}(y) \vert ^{p-2}u_{0}(y)}{ \vert x-y \vert ^{\alpha }} \,dx \,dy\biggr] \\ &< j_{0}+\frac{p-1}{2p}S_{\alpha , p}^{\frac{p}{p-1}}. \end{aligned}$$

Here, we use that $\langle J'(u_{0}), tV\rangle =0$ and $V(x)$ is a solution of (1.1). By the definition of $\tau _{k}^{p}$ [14, 16] and it being the solution of (1.7), we get

$$\begin{aligned} \mathcal{P}\tau _{k}\bigl(\tau _{k}^{p}\bigr) \leq & \mathcal{P}\tau _{k}\biggl(-\alpha \frac{\triangle _{k}}{ \Vert \mathcal{B}_{k}u(x_{k}) \Vert } \mathcal{B}_{k}u(x _{k})\biggr) \\ =&\frac{1}{2}\biggl[\alpha ^{2} \triangle _{k}^{2} \frac{ \Vert \mathcal{B}_{k} \mathcal{B}_{k}u(x_{k}) \Vert ^{2}}{ \Vert \mathcal{B}_{k}u(x_{k}) \Vert ^{2}}+\alpha ^{4} \triangle _{k}^{4} \frac{9}{4}\frac{(s_{k-1}^{\mathcal{T}} \mathcal{B}_{k}u(x_{k}))^{4}}{ \Vert \mathcal{B}_{k}u(x_{k}) \Vert ^{4}} \\ &{}+ 3\alpha ^{2} \triangle _{k}^{2} \frac{(s_{k-1}^{\mathcal{T}} \mathcal{B}_{k}u(x_{k}))^{2}}{ \Vert \mathcal{B}_{k}s_{k-1} \Vert ^{2}}u(x_{k})^{ \mathcal{T}}s_{k-1}-2\alpha \triangle _{k} \frac{(u(x_{k})^{ \mathcal{T}}\mathcal{B}_{k}\mathcal{B}_{k}u(x_{k}))}{ \Vert \mathcal{B} _{k}u(x_{k}) \Vert } \\ &{}- 3\alpha ^{3}\triangle _{k}^{3} \frac{(s_{k-1}^{\mathcal{T}} \mathcal{B}_{k}u(x_{k}))^{2}s_{k-1}^{\mathcal{T}}\mathcal{B}_{k} \mathcal{B}_{k}u(x_{k})}{ \Vert \mathcal{B}_{k}u(x_{k}) \Vert ^{3}}\biggr] \\ =& \frac{1}{2}\biggl[\alpha ^{2} \triangle _{k}^{2} \frac{ \Vert \mathcal{B}_{k} \mathcal{B}_{k}u(x_{k}) \Vert ^{2}}{ \Vert \mathcal{B}_{k}u(x_{k}) \Vert ^{2}} -2 \alpha \triangle _{k} \frac{(u(x_{k})^{\mathcal{T}}\mathcal{B}_{k} \mathcal{B}_{k}u(x_{k}))}{ \Vert \mathcal{B}_{k}u(x_{k}) \Vert }+O\bigl( \triangle _{k}^{2}\bigr)\biggr] \\ \leq & -\alpha \triangle _{k} \bigl\Vert \mathcal{B}_{k}u(x_{k}) \bigr\Vert +\frac{1}{2} \alpha ^{2} \triangle _{k}^{2} M_{l}^{2}+O\bigl(\triangle _{k}^{2}\bigr) \end{aligned}$$

for any $\alpha \in [0,1]$.

Therefore

$$\begin{aligned} \mathcal{P}\tau _{k}\bigl(\tau _{k}^{p}\bigr) \leq & \min_{0\leq \alpha \leq 1}\biggl[- \alpha \triangle _{k} \bigl\Vert \mathcal{B}_{k}u(x_{k}) \bigr\Vert + \frac{1}{2} \alpha ^{2} \triangle _{k}^{2} M_{l}^{2}\biggr]+O\bigl(\triangle _{k}^{2} \bigr) \\ \leq &-\frac{1}{2} \bigl\Vert \mathcal{B}_{k}u(x_{k}) \bigr\Vert \min \biggl\{ \triangle _{k},\frac{ \Vert \mathcal{B}_{k}u(x_{k}) \Vert }{M_{l}^{2}}\biggr\} +O \bigl(\triangle _{k}^{2}\bigr). \end{aligned}$$

 □

Lemma 2.2

Let Assumptions (A), (B), (C), and (D) hold. Then

$$ \bigl\vert A\tau _{k}\bigl(\tau _{k}^{p} \bigr)-\mathcal{P}\tau _{k}\bigl(\tau _{k}^{p} \bigr) \bigr\vert =O\bigl( \bigl\Vert \tau _{k}^{p} \bigr\Vert ^{2}\bigr), $$

where $\tau _{k}$ is the solution of (1.7).

Proof

It follows from (1.9) and (1.10) that

$$\begin{aligned} \bigl\vert A\tau _{k}\bigl(\tau _{k}^{p} \bigr)-\mathcal{P}\tau _{k}\bigl(\tau _{k}^{p} \bigr) \bigr\vert =& \bigl\vert \nu \bigl(x_{k}+\tau _{k}^{p}\bigr)-\mathcal{N}_{k}\bigl(\tau _{k}^{p}\bigr) \bigr\vert \\ =& \frac{1}{2} \biggl\vert \bigl\Vert u(x_{k})+\nabla u(x_{k}) \tau _{k}^{p} +O\bigl( \bigl\Vert \tau _{k}^{p} \bigr\Vert ^{2}\bigr) \bigr\Vert ^{2} \\ &{}- \biggl\Vert u(x_{k})+\mathcal{B}_{k}\tau _{k}^{p}+\frac{3}{2}\bigl(s_{k-1}^{ \mathcal{T}} \tau _{k}^{p}\bigr)^{2}s_{k-1} \biggr\Vert ^{2} \biggr\vert \\ =& \bigl\vert u(x_{k})^{\mathcal{T}}\nabla u(x_{k}) \tau _{k}^{p}-u(x_{k})^{ \mathcal{T}} \mathcal{B}_{k}\tau _{k}^{p}+O\bigl( \bigl\Vert \tau _{k}^{p} \bigr\Vert ^{2}\bigr) \\ &{}+O\bigl( \bigl\Vert \tau _{k}^{p} \bigr\Vert ^{3}\bigr)+O\bigl( \bigl\Vert \tau _{k}^{p} \bigr\Vert ^{4}\bigr) \bigr\vert \\ \leq & \bigl\Vert \bigl[\nabla u(x_{k})-\mathcal{B}_{k} \bigr]u(x_{k}) \bigr\Vert \bigl\Vert \tau _{k}^{p} \bigr\Vert +O\bigl( \bigl\Vert \tau _{k}^{p} \bigr\Vert ^{2}\bigr) \\ &{}+O\bigl( \bigl\Vert \tau _{k}^{p} \bigr\Vert ^{3}\bigr)+O\bigl( \bigl\Vert \tau _{k}^{p} \bigr\Vert ^{4}\bigr) \\ =&O\bigl( \bigl\Vert \tau _{k}^{p} \bigr\Vert ^{2}\bigr). \end{aligned}$$

 □

Theorem 2.1

Let Assumptions (A), (B), (C), and (D) hold. Then Algorithm either finitely stops or generates an infinite sequence $\{x_{k}\}$ satisfying

$$ \lim_{k\rightarrow \infty } \bigl\Vert u(x_{k}) \bigr\Vert =0, $$

(2.2)

where $\{x_{k}\}$ is defined as in Algorithm.

Proof

We know that $t^{-}(u)$ is a continuous function of u. Consequently, the manifold $\varLambda ^{-}$ disconnects $D^{1,2}( {\Re ^{n}})$ in exactly two connected components $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$, where

$$\begin{aligned}& \mathcal{U}_{1}= \biggl\{ u\in D^{1,2}\bigl({\Re ^{n}}\bigr): u=0 \text{ or } \Vert u \Vert _{D}< t^{-} \biggl(\frac{u}{ \Vert u \Vert _{D}} \biggr) \biggr\} , \\& \mathcal{U}_{2}= \biggl\{ u\in D^{1,2}\bigl({\Re ^{n}}\bigr): \Vert u \Vert _{D}>t^{-} \biggl( \frac{u}{ \Vert u \Vert _{D}} \biggr) \biggr\} . \end{aligned}$$

So $D^{1,2}({\Re ^{n}})=\varLambda ^{-}\cup \mathcal{U}_{1} \cup \mathcal{U}_{2}$. In particular, $u_{0}\in \varLambda ^{+} \subset \mathcal{U}_{1}$. Since

$$ t^{-} \biggl(\frac{u_{0}+tW}{ \Vert u_{0}+tW \Vert _{D}} \biggr)\frac{u_{0}+tW}{ \Vert u _{0}+tW \Vert _{D}}\in \varLambda , $$

we have

$$ 0< t^{-} \biggl(\frac{u_{0}+tW}{ \Vert u_{0}+tW \Vert _{D}} \biggr)< C_{0} $$

uniformly for $t\in \mathbb{R}$.

On the other hand,

$$ \Vert u_{0}+tW \Vert _{D}\geq t \Vert W \Vert _{D}- \Vert u_{0} \Vert _{D}\geq C_{0}, $$

where $t\geq \tilde{t}$.

So that we can fix a positive number $t_{0}$ such that

$$ \Vert u_{0}+t_{0}W \Vert _{D}> t^{-}\biggl(\frac{u_{0}+t_{0}W}{ \Vert u_{0}+t_{0}W \Vert _{D}}\biggr), $$

which yields that

$$ u_{0}+t_{0}W\in \mathcal{U}_{2}. $$

Combining this and the fact $u_{0}\in \mathcal{U}_{1}$, we know that

$$ u_{0}+t_{1}W\in \varLambda ^{-} $$

for some $0< t_{1}< t_{0}$.

So

$$ c_{1}=\inf_{\varLambda ^{-}}I(u)\leq \max_{0\leq t\leq t_{0}}I(u_{0}+tW) < c _{0}+\frac{N+2-\alpha }{4N-2\alpha }S^{\frac{2N-\alpha }{N+2-\alpha }} _{H,L}. $$

And there exists a minimizing sequence $\{u_{n}\}\subset \varLambda ^{-}$ satisfying

$$\begin{aligned}& I(u_{n})< c_{1}+\frac{1}{n}; \\& I(w)\geq I(u_{n})-\frac{1}{n} \Vert u-w \Vert _{D}, \end{aligned}$$

where $w\in \varLambda ^{-}$.

So that

$$\begin{aligned} c_{1}+1 &>I(u_{n})=\frac{1}{2} \Vert u_{n} \Vert ^{2}_{D}-\frac{1}{2\cdot 2^{ \ast }_{\alpha }}B(u_{n})- \int _{\Re ^{n}}h(x)u_{n} \,dx \\ &\geq \biggl(\frac{1}{2}-\frac{1}{2\cdot 2^{\ast }_{\alpha }} \biggr) \Vert u _{n} \Vert ^{2}_{D} - \biggl(1- \frac{1}{2\cdot 2^{\ast }_{\alpha }} \biggr) \Vert h \Vert _{H^{-1}} \Vert u_{n} \Vert _{D}, \end{aligned}$$

which implies $\|u_{n}\|$ has an upper bound.

It follows from $\{u_{n}\}\subset \varLambda ^{-}$ that

$$ \Vert u_{n} \Vert ^{2}_{D}\leq \bigl(2\cdot 2^{\ast }_{\alpha }-1\bigr) \frac{ \Vert u_{n} \Vert ^{2^{\ast }_{\alpha }}_{D}}{S^{2^{\ast }_{\alpha }}_{H,L}}. $$

Thus, $\|u_{n}\|_{D}$ has a uniform positive lower bound.

Similarly,

$$ I(u_{n})\to c_{1}, \qquad I'(u_{n}) \to 0\quad \text{in } H^{-1}. $$

By Lemma 2.2 and

$$ c_{1}< c_{0}+\frac{N+2-\alpha }{4N-2\alpha }S^{\frac{2N-\alpha }{N+2- \alpha }}_{H,L}, $$

we obtain that

$$ \int _{\Re ^{n}}h(x)u_{1}\,dx>0 \quad \text{and} \quad u_{1}\in \varLambda ^{+}, $$

which leads to a contradiction.

In the case $h>0$. Applying Lemma 2.1 to $u_{1}$ and $|u_{1}|$, we know that there exists $t^{-}(|u_{1}|)$ such that

$$ t^{-}\bigl( \vert u_{1} \vert \bigr) \vert u_{1} \vert \in \varLambda ^{-}. $$

Moreover,

$$ t^{-}\bigl( \vert u_{1} \vert \bigr)\geq t_{0}\bigl( \vert u_{1} \vert \bigr)=t_{0}(u_{1}) = \biggl[\frac{A(u_{1})}{(2^{ \ast }_{\alpha }-1)B(u_{1})} \biggr]^{\frac{1}{2^{\ast }_{\alpha }-2}}. $$

So

$$ \int _{\Re ^{n}}h(x)u_{1}\,dx= \int _{\Re ^{n}}h(x) \vert u_{1} \vert dx, $$

which implies that $u_{1}\geq 0$. According to the maximum principle, we get $u_{1}>0$.

It is easy to see that $\|u_{n}\|$ is bounded, which yields that

$$ \Vert u_{n} \Vert ^{2}= \Vert w_{n} \Vert ^{2}+ \Vert v \Vert ^{2}+o(1),\quad n\to \infty , $$

and

$$\begin{aligned} & \int _{\Re ^{n}} \frac{ \vert w_{n}(x) \vert ^{p} \vert w_{n}(y) \vert ^{p}}{ \vert x-y \vert ^{\alpha }} \,dw \\ &\quad = \int _{\Re ^{n}} \frac{ \vert w_{n}(x) \vert ^{p} \vert w_{n}(y) \vert ^{p}}{ \vert x-y \vert ^{\alpha }} \,dw + \int _{\Re ^{n}} \frac{ \vert w(x) \vert ^{p} \vert w(y) \vert ^{p}}{ \vert x-y \vert ^{\alpha }} \,dw+o_{n}(1) \end{aligned}$$

as $n\to \infty$.

So

$$\begin{aligned} c\leftarrow \mathcal{J}(w_{n}) &=\frac{1}{2} \Vert w_{n} \Vert ^{2}- \frac{1}{2p} \int _{\Re ^{n}} \frac{ \vert w_{n}(x) \vert ^{p} \vert w_{n}(y) \vert ^{p}}{ \vert x-y \vert ^{ \alpha }} \,dw - \int _{\Re ^{n}}h(x)w_{n} \,dx \\ &=\frac{1}{2} \Vert w_{n} \Vert ^{2}-\frac{1}{2p} \int _{\Re ^{n}} \frac{ \vert w_{n}(x) \vert ^{p} \vert w _{n}(y) \vert ^{p}}{ \vert x-y \vert ^{\alpha }} \,dw - \int _{\Re ^{n}}h(x)w_{n} \,dx \\ &\quad{}+\frac{1}{2} \Vert v \Vert ^{2}-\frac{1}{2p} \int _{\Re ^{n}} \frac{ \vert v(x) \vert ^{p} \vert v(y) \vert ^{p}}{ \vert x-y \vert ^{ \alpha }} \,dw - \int _{\Re ^{n}}h(x)v \,dx+o_{n}(1) \\ &=\mathcal{J}(v)+\frac{1}{2} \Vert w_{n} \Vert ^{2}-\frac{1}{2p}\tilde{B}(w _{n})+o_{n}(1) \end{aligned}$$

and

$$ \frac{1}{2} \Vert w_{n} \Vert ^{2}- \frac{1}{2p}\tilde{B}(w_{n})+o_{n}(1) < \frac{p-1}{2p}S_{\alpha , p}^{\frac{p}{p-1}}. $$

(2.3)

Notice that

$$ o(1)=\bigl\langle J'(u_{n}),u_{n}\bigr\rangle = \bigl\langle J'(v),v\bigr\rangle + \Vert w_{n} \Vert ^{2}-\tilde{B}(w_{n})+o(1), $$

which yields that

$$ \Vert w_{n} \Vert ^{2}- \tilde{B}(w_{n})=o(1). $$

(2.4)

It follows from (2.4) that

$$ \Vert w_{n} \Vert ^{2}=\tilde{B}(w_{n})\leq \frac{ \Vert w_{n} \Vert ^{2p}}{S_{\alpha , p} ^{p}} $$

and

$$\begin{aligned} \frac{1}{2}\frac{p-1}{p}S_{\alpha , p}^{\frac{p}{p-1}} &= \frac{1}{2}\biggl(1- \frac{1}{p}\biggr)S^{\frac{p}{p-1}} \\ &\leq \frac{1}{2}\biggl(1-\frac{1}{p}\biggr) \Vert w_{n} \Vert ^{2} \\ &=\frac{1}{2} \Vert w_{n} \Vert ^{2}- \frac{1}{2p}\tilde{B}(w_{n})+o_{n}(1) \\ &< \frac{p-1}{2p}S_{\alpha , p}^{\frac{p}{p-1}}, \end{aligned}$$

which also leads to a contradiction.

Suppose that

$$ \lim_{k\rightarrow \infty } \bigl\Vert \mathcal{B}_{k}u(x_{k}) \bigr\Vert =0 $$

(2.5)

holds. Using Assumption (C) we get (2.2). It follows from (2.5) that the subsequence $\{k_{j}\}$ satisfies

$$ \bigl\Vert \mathcal{B}_{k_{j}}u(x_{k_{j}}) \bigr\Vert \geq \varepsilon . $$

(2.6)

Set

$$ K=\bigl\{ k| \bigl\Vert \mathcal{B}_{k}u(x_{k}) \bigr\Vert \geq \varepsilon \bigr\} . $$

So we assume that

$$ \bigl\Vert u(x_{k}) \bigr\Vert \geq \varepsilon $$

holds, where $k\in K$.

It follows from the definition of Algorithm and Lemma 2.1 that

$$ \sum_{k\in K}\bigl[\nu (x_{k})-\nu (x_{k+1})\bigr]\geq -\sum_{k\in K}\rho \mathcal{P}\tau _{k}\bigl(\tau _{k}^{p_{k}}\bigr) \geq \sum_{k\in K}\rho \frac{1}{2}\min \biggl\{ c^{p_{k}}\varepsilon , \frac{\varepsilon }{M_{l}^{2}}\biggr\} \varepsilon . $$

Lemma 2.2 gives us that the sequence $\{\nu (x_{k})\}$ is convergent, which yields that

$$ \sum_{k\in K}\rho \frac{1}{2}\min \biggl\{ c^{p_{k}}\varepsilon ,\frac{ \varepsilon }{M_{l}^{2}}\biggr\} \varepsilon < +\infty . $$

Then $p_{k} \rightarrow +\infty$ when $k\rightarrow +\infty$ and $k\in K$. It follows that

$$ \begin{aligned} \min\quad & q_{k}(\tau ) =\frac{1}{2} \biggl\Vert u(x_{k})+\mathcal{B}_{k}\tau +\frac{3}{2}\bigl(s _{k-1}^{\mathcal{T}}\tau \bigr)^{2}s_{k-1} \biggr\Vert ^{2}, \\ &\mbox{s.t.} \quad \Vert \tau \Vert \leq c^{p_{k}-1} \bigl\Vert u(x_{k}) \bigr\Vert \end{aligned} $$

(2.7)

is unacceptable.

If we put $x_{k+1}'=x_{k}+\tau _{k}'$, then we have

$$ \frac{\nu (x_{k})-\nu (x_{k+1}')}{-\mathcal{P}\tau _{k}(\tau _{k}')}< \rho . $$

(2.8)

By applying Lemma 2.1 and the definition of $\triangle _{k}$, we know that

$$ -\mathcal{P}\tau _{k}\bigl(\tau _{k}'\bigr) \geq \frac{1}{2}\min \biggl\{ c^{p_{k}-1} \varepsilon , \frac{\varepsilon }{M_{l}^{2}}\biggr\} \varepsilon . $$

By applying Lemma 2.2, we know that

$$ \nu \bigl(x_{k+1}'\bigr)-\nu (x_{k})- \mathcal{P}\tau _{k}\bigl(\tau _{k}'\bigr)=O \bigl( \bigl\Vert \tau _{k}' \bigr\Vert ^{2} \bigr)=O\bigl(c^{2(p_{k}-1)}\bigr). $$

So

$$ \biggl\vert \frac{\nu (x_{k+1}')-\nu (x_{k})}{\mathcal{P}\tau _{k}(\tau _{k}')}-1 \biggr\vert \leq \frac{O(c^{2(p_{k}-1)})}{0.5\min \{c^{p_{k}-1}\varepsilon ,\frac{ \varepsilon }{M_{l}^{2}}\}\varepsilon +O(c^{2(p_{k}-1)}\varepsilon ^{2})}. $$

By applying $p_{k}\rightarrow +\infty$ as $k\rightarrow +\infty$, we know that

$$ \frac{\nu (x_{k})-\nu (x_{k+1}')}{-\mathcal{P}\tau _{k}(\tau _{k}')} \rightarrow 1, \quad k\in K, $$

which also gives a contradiction to (2.8). □

3 Numerical results

This section reports some numerical results of Algorithm.

3.1 Problems

Define

$$ u(x)=\bigl(\upsilon _{1}(x),\upsilon _{2}(x),\ldots , \upsilon _{n}(x)\bigr)^{ \mathcal{T}}. $$

Problem 1

The Schrödinger differential function (see [12])

$$ \upsilon _{l}(x)=2\Biggl(n+l(1-\cos x_{l})-\sin x_{l}-\sum_{j=1}^{n} \cos x _{j}\Biggr) (2\sin x_{l}-\cos x_{l}), $$

where $l=1,2,3,\ldots ,n$.

Initial guess:

$$ x_{0}=\biggl(\frac{101}{100n},\frac{101}{100n},\ldots , \frac{101}{100n}\biggr)^{ \mathcal{T}}. $$

Problem 2

Logarithmic function

$$ \upsilon _{l}(x)=\ln (x_{l}+1)-\frac{x_{l}}{n}, $$

where $l=1,2,3,\ldots ,n$.

Initial points:

$$ x_{0}=(1,1,\ldots ,1)^{\mathcal{T}}. $$

Problem 3

Schrödinger differential function (see [5, pp. 471–472])

$$\begin{aligned}& \upsilon _{1}(x) = (2-0.2x_{1})x_{1}-x_{2}+1, \\& \upsilon _{l}(x) = (2-0.2x_{l})x_{l}-x_{i-1}+x_{i+1}+1, \\& \upsilon _{n}(x) = (2-0.2x_{n})x_{n}-x_{n-1}+1, \end{aligned}$$

where $l=1,2,3,\ldots ,n$.

Initial points:

$$ x_{0}=(-1,-1,\ldots ,-1)^{\mathcal{T}}. $$

Problem 4

Trigexp function (see [5, p. 473])

$$\begin{aligned}& \upsilon _{1}(x) = 3x_{1}^{3}+x_{2}-4+2 \sin (x_{1}-x_{2})\sin (x_{1}+x _{2}), \\& \upsilon _{l}(x) = -2x_{i-1}e^{x_{i-1}-x_{l}}+3x_{l} \bigl(4+3x_{l}^{2}\bigr)+2x _{i+1} \\& -\sin (x_{l}-x_{i+1})\sin (x_{l}+x_{i+1})-2, \\& \upsilon _{n}(x) = -2x_{n-1}e^{x_{n-1}-x_{n}}+3x_{n}-2, \end{aligned}$$

where $l=1,2,3,\ldots ,n$.

Initial guess:

$$ x_{0}=(0,0,\ldots ,0)^{\mathcal{T}}. $$

Problem 5

Let $u(x)$ be the gradient of

$$ h(x)=\sum_{l=1}^{n}\bigl(e^{x_{l}}-x_{l} \bigr). $$

Then

$$ \upsilon _{l}(x)=e^{x_{l}}-1, $$

where $l=1,2,3,\ldots ,n$.

Initial points:

$$ x_{0}=\biggl(\frac{1}{n},\frac{2}{n},\ldots ,1 \biggr)^{\mathcal{T}}. $$

Parameters: $c=0.2$, $\epsilon =10^{-2}$, $\rho =0.03$, $p=3$, $m=6$ $\mathcal{H}_{0}$ is the unit matrix.

The method for (1.3) and (1.7): the Dogleg method [13, 25].

Code experiments: run on a PC with Intel Pentium(R) Xeon(R) E5507 CPU 2.27 GHz, 6.00 GB of RAM, and Windows 7 operating system.

Code software: MATLAB r2017a.

Stop rules: the program stops if $\|u(x)\|\leq 1e-4$ holds.

Other cases: we will stop the program if the iteration number is larger than ten hundred.

3.2 Results and discussion

The column meaning in the following tables:

Dim: the dimension. NI: the number of iterations.

NG: the norm function number. Time: the CPU-time in seconds.

Numerical results of Table 1 show the performance of these two algorithms about NI, NG, and Time. It is not difficult to see that both of these algorithm can successfully solve all these ten nonlinear problems.

Table 1 Experiment results

It is easy to see that the NI and the NG of Algorithm have won since their performance profile plot is on top right. And the Time of Algorithm YL has superiority over Algorithm. Both of these two algorithms have good robustness.

4 Conclusions

In this paper, we considered the numerical method for solving the Schrödinger equations via Phragmén–Lindelöf inequalities under the order induced by a symmetric cone with the function involved being monotone. Based on the Phragmén–Lindelöf inequalities, the underlying system of inequalities was reformulated as a system of smooth equations, and a Schrödinger-type method was proposed to solve it iteratively so that a solution of the system of the Schrödinger equations was found. By means of the Schrödinger type inequalities, the algorithm was proved to be well defined and to be globally convergent under weak assumptions and locally quadratically convergent under suitable assumptions. Preliminary numerical results indicate that the algorithm was effective.