1 Introduction

Motivated by uncertainty problems, risk measures and the superhedging in finance, Xue established the fundamental theory of Schrödingerean expectation theory (see [1]), where the minimally thin sets associated with a Schrödinger operator are introduced. In the Schrödingerean expectation framework, the notion of the corresponding Schrödingerean stochastic calculus of Itô type were also established (see [2]). As in [3], the set

$$ \Omega \times \mathbf{R}= \bigl\{ P=(X,y)\in {\mathbf{{R}}}^{n}; X\in \Omega ,y \in \bf {R} \bigr\} $$

in \(\mathbf{R}^{n}\) is simply denoted by \(\mathcal{C}_{n}(\Gamma )\). We call it a cylinder (see [3]). On that basis, the theory and applications of the Schrödingerean TOPSIS equation have been developed rapidly (see [2, 4–11] and the references therein).

In this paper, we consider the following Schrödingerean TOPSIS equation:

$$ T (-\triangle )^{s}u+a(P) u(P)=0 $$
(1.1)

in \(\mathcal{C}_{n}(\Gamma )\), where \(0< s\leq 1\) and the potential a satisfies the following condition:

$$ 0< \inf_{P=(r,\Theta )\in C_{n}(\Omega )}a(P)< \lim_{r\rightarrow \infty }a(P)=a_{\infty }< \infty . $$

Under the Lipschitz assumptions on the potential a, Yang (see [11]) has proved the wellposedness of such equations with the fixed-point iteration. Moreover, Liu (see [8]) has studied the Markov chains when coefficients are integral-Lipschitz, Zhang and Wu (see [9]) considered the modified Laplace equations with some good boundaries, Wang et al. (see [10]) studied stochastic functional differential equations with infinite delay. We can also refer the reader to Miyamoto (see [3]), Chen (see [5] and the references therein).

Let \(\alpha > 0\) and \(1\leq p < \infty \). Then the weighted weak space \(\aleph^{p} _{\alpha }(\Gamma ) \) on cylinders can be defined by

$$ \Vert u \Vert _{\aleph^{p}_{\alpha }(\Gamma )}:= \biggl( \int_{ \Gamma }\bigl\vert u(y) \bigr\vert ^{p}\,d\wp_{\alpha }(y) \biggr) ^{\frac{1}{q}}< \infty , $$

where u are weak solutions of (1.1) on cylinders, \(d\wp_{\alpha }(y)=\operatorname{dist}(y,\partial \Gamma )^{\alpha }\,dy\) and \(1/p+1/q=1\). Let dy denote the Lebesgue measure on \(\mathbf{R}^{n}\) and \(\operatorname{dist}(y,\partial \Gamma )\) denote the Euclidean distance from z to the boundary of Γ. We let \(\aleph^{p}_{\alpha }=\aleph^{p}_{\alpha }( \mathcal{C}_{n}(\Gamma ))\). Then we can check that \(dV_{\alpha }(y) = y^{\alpha }_{n}\,dy\) in \(\mathcal{C}_{n}(\Gamma )\).

Weak spaces are not studied as extensively as their holomorphic counterparts and many results on spaces has been done for bounded domains (see [12, 13]), for example, are good references for holomorphic Bergman spaces. \(\aleph^{p}_{0}(\Gamma )\) is studied in [5] and [3, 6] on the setting of upper half-space and bounded smooth domain in \(\mathbf{R}^{n}\), respectively. \(\aleph^{p} _{\alpha }(B)\), where B is the open unit ball and the upper half plane in \(\mathbf{R}^{n}\), are studied in [7] and [1], respectively.

For nonnegative functions \(g_{1}\) and \(g_{2}\), we often write \(g_{1} \le g_{2} \) or \(g_{2} \ge g_{1}\) if \(g_{1}\leq cg_{2}\), where c is an inessential positive constant. Also, we write \(g_{1}\approx g_{2}\) if \(g_{1}\le g_{2}\) and \(g_{2} \le g_{1}\). Throughout this paper, we shall use the same letter C to denote various constants which may be different from line to line.

2 Preliminary results

In this section, we first recall one definition and some previous results about the generalized Poisson kernel and Green function in the half space, which will be available later.

Let \(z\in \mathbf{R}^{n} \) and \(r > 0\). Let \(B(y,r)\) denote the open ball in \(\mathbf{R}^{n}\). Let \(V(B(0,1))\) be the volume of the unit ball in \(\mathbf{R}^{n}\), \(w \in \overline{\mathcal{C}_{n}( \Gamma )}\), \(\overline{w}=(w^{\prime }, -w_{n}) \) and \(z \in \mathcal{C}_{n}(\Gamma )\). Then the extended Poisson kernel \(P(y,w)\) in \(\mathcal{C}_{n}(\Gamma )\) can be defined by

$$\begin{aligned} P_{z}(w):=P(y,w)=\frac{1}{nV(B(0,1))} \frac{z_{n}+w_{n}}{\vert z-\overline{w} \vert ^{n}}. \end{aligned}$$
(2.1)

It is easy to see that (see [14] for details and related facts)

$$\begin{aligned} \int_{\partial \mathcal{C}_{n}(\Gamma )}P(y,w)\,dw^{\prime }=1, \end{aligned}$$
(2.2)

for each \(z\in \mathcal{C}_{n}(\Gamma )\) and for every \(w \in \overline{ \mathcal{C}_{n}(\Gamma )}\).

Let \(\vec{\beta }=(\beta_{1},\beta_{2},\ldots ,\beta_{n})\) be a multi-index with \(\beta_{j}\in \mathbf{N}\cup \{0\}\) for \(j=1,2, \ldots ,n\) and f be a homogeneous polynomial of degree \(\vert \vec{\beta } \vert +2\). Then we see from (2.1) that

$$\begin{aligned} D_{y}^{\vec{\beta }}P(y,w):=D_{z_{1}}^{\beta_{1}} \cdots D_{z_{1}}^{ \beta_{1}}P(y,w) =\frac{f(y-\overline{w})}{\vert z-\overline{w} \vert ^{n+2\vert \vec{\beta } \vert +1}}, \end{aligned}$$
(2.3)

where \(\vec{\beta }=\beta_{1}+\beta_{2}+\cdots +\beta_{n}\).

The following lemma collects so-called Poisson-Schrödinger type estimates (see [4]), which play important roles in our discussions.

Lemma 2.1

If β⃗ is a multi-index, u is the weak solution of (1.1) and bounded by M on \(B(y,r)\), then there exists a positive constant C depending on β⃗ such that

$$ \bigl\vert D^{\vec{\beta }}u(y) \bigr\vert \le \frac{CM}{r^{\vert \vec{\beta } \vert +1}} . $$

3 Main results

For the rest of this paper, we assume \(\alpha >0\), \(p, q\in (0,\infty )\) and u is the weak solution of (1.1).

First we prove that equation (1.1) has at least a weak solution.

Theorem 3.1

If a changes its sign, then (1.1) has at least a weak solution \(u_{\lambda }\).

Proof

For convenience, let

$$ d_{n}=\biggl(I-\frac{\mu_{n}}{\sigma_{n}}\mathcal{G}^{*}\mathcal{G} \biggr)v_{n}. $$

Using Lemma 2.1 it follows that \((I-\frac{\mu_{n}}{\sigma_{n}} \mathcal{G}^{*}\mathcal{G})\) is nonexpansive and averaged. Hence,

$$\begin{aligned} \Vert t_{n+1}-t_{n} \Vert &\le \frac{\sigma_{n+1}}{1+\sigma_{n+1}}\Vert d_{n+1}-d_{n} \Vert + \biggl\vert \frac{\sigma_{n+1}}{1+\sigma_{n+1}}-\frac{\sigma_{n}}{1+\sigma _{n}} \biggr\vert \Vert d_{n} \Vert \\ &\quad {}+ \frac{T}{1+\sigma_{n+1}} \bigl\{ (1-\sigma_{n+1})w_{n+1}+ \sigma_{n+1}d _{n+1}- \bigl[(1-\sigma_{n})w_{n}+ \sigma_{n}d_{n}\bigr] \bigr\} \\ &\quad {}+ \biggl\vert \frac{1}{1+\sigma_{n+1}}-\frac{1}{1+\sigma_{n}} \biggr\vert \bigl\Vert T\bigl[(1- \sigma_{n})w_{n}+\sigma_{n}d_{n} \bigr] \bigr\Vert \\ &\le \frac{\sigma_{n+1}}{1+\sigma_{n+1}}\Vert d_{n+1}-d_{n} \Vert + \biggl\vert \frac{\sigma_{n+1}}{1+\sigma_{n+1}}-\frac{\sigma_{n}}{1+\sigma _{n}} \biggr\vert \Vert d_{n} \Vert \\ &\quad {}+ \frac{1-\sigma_{n+1}}{1+\sigma_{n+1}}\Vert w_{n+1}-w_{n}\Vert + \frac{ \sigma_{n+1}}{1+\sigma_{n+1}}\Vert d_{n+1}-d_{n} \Vert + \frac{ \sigma_{n}-\sigma_{n+1}}{1+\sigma_{n+1}}\Vert w_{n} \Vert \\ &\quad {}+ \frac{\sigma_{n+1}-\sigma_{n}}{1+\sigma_{n+1}}\Vert d_{n} \Vert + \biggl\vert \frac{1}{1+\sigma_{n+1}}-\frac{1}{1+\sigma_{n}} \biggr\vert \bigl\Vert T\bigl[(1- \sigma_{n})w_{n}+\sigma_{n}d_{n}\bigr]\bigr\Vert . \end{aligned}$$
(3.1)

Moreover,

$$\begin{aligned} \Vert d_{n+1}-d_{n} \Vert &= \biggl\Vert \biggl(I- \frac{\mu_{n+1}}{\sigma_{n+1}}\mathcal{G}^{*} \mathcal{G}\biggr)v_{n+1}-\biggl(I-\frac{ \mu_{n}}{\sigma_{n}} \mathcal{G}^{*}\mathcal{G}\biggr)v_{n} \biggr\Vert \\ &\le \Vert v_{n+1}-v_{n} \Vert \\ &= \bigl\Vert P_{S_{i}}\bigl[(1-\alpha_{n+1})w_{n+1}- \gamma_{n}\mathcal{G}^{*} \mathcal{G}w_{n+1} \bigr] \\ &\quad {}-P_{S_{i}}\bigl[(1-\alpha_{n})w_{n}- \gamma_{n} \mathcal{G}^{*}\mathcal{G}w_{n}\bigr] \bigr\Vert \\ &\le \bigl\Vert \bigl(I-\gamma_{n+1}\mathcal{G}^{*} \mathcal{G}\bigr) w_{n+1}-\bigl(I- \gamma_{n+1} \mathcal{G}^{*}\mathcal{G}\bigr) w_{n}+(\gamma_{n}- \gamma_{n+1}) \mathcal{G}^{*}\mathcal{G} w_{n} \bigr\Vert \\ &\quad {}+ \alpha_{n+1}\Vert {-}w_{n+1} \Vert +\alpha_{n} \Vert w_{n}\Vert \\ &\le \Vert w_{n+1}-w_{n} \Vert +\vert \gamma_{n}-\gamma_{n+1}\vert \bigl\Vert \mathcal{G}^{*}\mathcal{G}w_{n} \bigr\Vert \\ &\quad {} +\alpha_{n+1}\Vert {-}w _{n+1} \Vert +\alpha_{n}\Vert w_{n}\Vert . \end{aligned}$$
(3.2)

Substituting (3.2) in (3.1), we infer that

$$\begin{aligned} \Vert t_{n+1}-t_{n} \Vert & \le \biggl\vert \frac{\sigma_{n+1}}{1+ \sigma_{n+1}}-\frac{\sigma_{n}}{1+\sigma_{n}} \biggr\vert \Vert d_{n} \Vert + \frac{\sigma_{n}-\sigma_{n+1}}{1+\sigma_{n+1}}\Vert w_{n} \Vert +\frac{\sigma_{n+1}-\sigma_{n}}{1+\sigma_{n+1}}\Vert d_{n} \Vert \\ &\quad {}+ \Vert w_{n+1}-w_{n} \Vert +\biggl\vert \frac{1}{1+\sigma_{n+1}}-\frac{1}{1+ \sigma_{n}} \biggr\vert \bigl\Vert T\bigl[(1- \sigma_{n})w_{n}+\sigma_{n}d_{n}\bigr] \bigr\Vert \\ &\quad {}+ \vert \gamma_{n}-\gamma_{n+1}\vert \Vert w_{n} \Vert +\alpha _{n+1}\Vert {-}w_{n+1} \Vert +\alpha_{n}\Vert w_{n}\Vert . \end{aligned}$$
(3.3)

By virtue of \(\lim_{n\rightarrow \infty }(\sigma_{n+1}-\sigma _{n})=0\), it follows that

$$ \lim_{n\rightarrow \infty }\biggl(\biggl\vert \frac{\sigma_{n+1}}{1+ \sigma_{n+1}}- \frac{\sigma_{n}}{1+\sigma_{n}} \biggr\vert \biggr)=0. $$

Moreover, \(\{ w_{n} \} \), and \(\{ v_{n} \} \) are bounded, and so is \(\{ d_{n} \} \). Therefore, (3.2) reduces to

$$ \lim_{n\rightarrow \infty }\sup \bigl(\Vert t_{n+1}-t_{n} \Vert - \Vert w_{n+1}-w_{n} \Vert \bigr)\le 0. $$
(3.4)

Applying (3.3) and Karamata regular variation theory, we get

$$ \lim_{n\rightarrow \infty }\Vert t_{n}-w_{n} \Vert =0. $$
(3.5)

Combining (3.4) with (3.2), we obtain

$$ \lim_{n\rightarrow \infty }\Vert x_{n+1}-x_{n} \Vert =0. $$

Using the convexity of the norm and (3.5), we deduce that

$$\begin{aligned} \Vert w_{n+1}-\hat{w} \Vert ^{2} &\le(1-\sigma_{n})\Vert w_{n}- \hat{w} \Vert ^{2}+\sigma_{n}\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\le\sigma_{n}\biggl\Vert {-}\alpha_{n}\hat{w}+(1-\alpha_{n})\biggl[w_{n}-\frac{ \gamma_{n}}{1-\alpha_{n}} \mathcal{G}^{*}\mathcal{G}w_{n} -\biggl(\hat{w}- \frac{ \gamma_{n}}{1-\alpha_{n}}\mathcal{G}^{*}\mathcal{G}\hat{w}\biggr)\biggr]\biggr\Vert ^{2} \\ &\le(1-\sigma_{n})\Vert w_{n}-\hat{w} \Vert ^{2}+\sigma_{n}\alpha _{n}\Vert {-}\hat{w} \Vert ^{2}+(1-\alpha_{n})\sigma_{n}\biggl[\Vert w_{n}-\hat{w} \Vert ^{2} \\ &\quad {}+ \frac{\gamma_{n}}{1-\alpha_{n}}\biggl(\frac{\gamma_{n}}{1-\alpha_{n}}-\frac{2}{ \rho (\mathcal{G}^{*}\mathcal{G})}\biggr)\bigl\Vert \mathcal{G}^{*}\mathcal{G}w _{n}-\mathcal{G}^{*}\mathcal{G}\hat{w}\bigr\Vert ^{2}\biggr] \\ &\le \Vert w_{n}-\hat{w} \Vert ^{2}+ \sigma_{n}\alpha_{n}\Vert {-}\hat{w} \Vert ^{2} \\ &\quad {}+ \sigma_{n}\gamma_{n}\biggl(\frac{\gamma_{n}}{1-\alpha_{n}}- \frac{2}{ \rho (\mathcal{G}^{*}\mathcal{G})}\biggr)\bigl\Vert \mathcal{G}^{*}\mathcal{G}w _{n}-\mathcal{G}^{*}\mathcal{G}\hat{w}\bigr\Vert ^{2}, \end{aligned}$$

which implies that

$$\begin{aligned} \bigl\Vert \mathcal{G}^{*} \mathcal{G}w_{n}-\mathcal{G}^{*}\mathcal{G} \hat{w}\bigr\Vert ^{2} &\le \Vert w_{n}-\hat{w} \Vert ^{2}-\Vert w _{n+1}-\hat{w} \Vert ^{2}+\sigma_{n}\alpha_{n}\Vert {-}\hat{w} \Vert ^{2} \\ &\le \Vert w_{n+1}-w_{n} \Vert \bigl(\Vert w_{n}-\hat{w} \Vert + \Vert w_{n+1}-\hat{w} \Vert \bigr)+\sigma_{n}\alpha_{n}\Vert {-}\hat{w} \Vert ^{2}. \end{aligned}$$

Since

$$\begin{aligned}& \lim_{n\rightarrow \infty }\inf \sigma_{n}\gamma_{n}\biggl( \frac{2}{ \rho (\mathcal{G}^{*}\mathcal{G})}-\frac{\gamma_{n}}{1-\alpha_{n}}\biggr)>0, \\& \lim_{n\rightarrow \infty }\alpha_{n}=0\quad \mbox{and}\quad \lim _{n\rightarrow \infty }\Vert w_{n+1}-w_{n} \Vert =0, \end{aligned}$$

we have the following result:

$$ \lim_{n\rightarrow \infty }\bigl\Vert \mathcal{G}^{*}\mathcal{G}w _{n}-\mathcal{G}^{*}\mathcal{G}\hat{w}\bigr\Vert =0. $$

Applying the property of the projection \(P_{S_{i}}\), one can easily show that

$$\begin{aligned} &\Vert u_{\lambda }-\hat{w\tau } \Vert ^{2} \\ &\quad =\bigl\Vert P_{S_{i}}\bigl[(1-\alpha_{n}) \tau_{n}-\gamma_{n}\mathcal{G}^{*} \mathcal{G} \tau_{n}\bigr]-P_{S_{i}}[\hat{w\tau }-tG*G\hat{w\tau }\bigr]\bigr\Vert ^{2} \\ &\quad \le \bigl\langle (1-\alpha_{n})w_{n}- \gamma_{n}\mathcal{G}^{*} \mathcal{G}\tau_{n}- \bigl(\hat{w\tau }-\gamma_{n}\mathcal{G}^{*} \mathcal{G} \hat{w\tau }\bigr),v_{n}-\hat{w\tau } \bigr\rangle \\ &\quad =\frac{1}{2}\bigl(\bigl\Vert \tau_{n}- \gamma_{n}\mathcal{G}^{*}\mathcal{G}\tau _{n}- \bigl(\hat{w\tau }-\gamma_{n}\mathcal{G}^{*}\mathcal{G}\hat{w \tau }\bigr)- \alpha_{n}\tau_{n}\bigr\Vert ^{2}+ \Vert v_{n}-\hat{w\tau }\Vert ^{2} \\ &\quad \quad {} -\bigl\Vert (1-\alpha_{n})w_{n}- \gamma_{n}\mathcal{G}^{*}\mathcal{G}\tau _{n}- \bigl(\hat{w\tau }-\gamma_{n}\mathcal{G}^{*}\mathcal{G}\hat{w \tau }\bigr)-v _{n}+\hat{w\tau }\bigr\Vert ^{2}\bigr) \\ &\quad \le \frac{1}{2}\bigl(\Vert \tau_{n}-\hat{w\tau } \Vert ^{2}+2\alpha_{n} \Vert {-}\tau_{n}\Vert \bigl\Vert \tau_{n}-\gamma_{n}\mathcal{G}^{*} \mathcal{G}\tau_{n}-\bigl(\hat{w\tau }-\gamma_{n} \mathcal{G}^{*} \mathcal{G}\hat{w\tau }\bigr) -\alpha_{n} \tau_{n}\bigr\Vert \\ & \quad \quad {}+\Vert v_{n}-\hat{w\tau }\Vert ^{2}-\bigl\Vert \tau_{n}-v_{n}-\gamma _{n} \mathcal{G}^{*}\mathcal{G}(\tau_{n}-\hat{w\tau })- \alpha_{n}\tau _{n}\bigr\Vert ^{2}\bigr) \\ &\quad \le \frac{1}{2}\bigl(\Vert \tau_{n}-\hat{w\tau } \Vert ^{2}+\alpha_{n}M+ \Vert v_{n}-\hat{w\tau }\Vert ^{2}-\Vert \tau_{n}-v_{n}\Vert ^{2} \\ &\quad \quad {} +2\gamma_{n} \bigl\langle \tau_{n}-v_{n}, \mathcal{G}^{*}\mathcal{G}( \tau_{n}-\hat{w\tau }) \bigr\rangle \\ &\quad \quad {} +2\alpha_{n} \langle \tau_{n}, \tau_{n}-v_{n} \rangle -\bigl\Vert \gamma _{n} \mathcal{G}^{*}\mathcal{G}(\tau_{n}-\hat{w\tau })+ \alpha_{n}\tau _{n}\bigr\Vert ^{2}\bigr) \\ &\quad \le \frac{1}{2}\bigl(\Vert \tau_{n}-\hat{w\tau } \Vert ^{2}+\alpha_{n}M+ \Vert v_{n}-\hat{w\tau }\Vert ^{2}-\Vert \tau_{n}-v_{n}\Vert ^{2} \\ &\quad \quad {} +2\gamma_{n}\Vert \tau_{n}-v_{n} \Vert \bigl\Vert \mathcal{G}^{*} \mathcal{G}(\tau_{n}- \hat{w\tau })\bigr\Vert +2\alpha_{n}\Vert \tau _{n}\Vert \Vert \tau_{n}-v_{n}\Vert \bigr) \\ &\quad \le \Vert \tau_{n}-\hat{w\tau }\Vert ^{2}+ \alpha_{n}M-\Vert \tau _{n}-v_{n}\Vert ^{2} \\ &\quad \quad {} -\Vert \tau_{n}-v_{n}\Vert ^{2}+4\gamma_{n}\Vert \tau_{n}-v _{n} \Vert \bigl\Vert \mathcal{G}^{*}\mathcal{G}(\tau_{n}- \hat{w\tau })\bigr\Vert \\ &\quad \quad {} +4\alpha_{n}\Vert \tau_{n}\Vert \Vert \tau_{n}-v_{n}\Vert , \end{aligned}$$

where \(M>0\) satisfying

$$ M\ge \sup_{k} \bigl\{ 2\Vert {-}\tau_{n}\Vert \bigl\Vert \tau_{n}-\gamma _{n}\mathcal{G}^{*} \mathcal{G}\tau_{n}-\bigl(\hat{w\tau }-\gamma_{n} \mathcal{G}^{*}\mathcal{G}\hat{w\tau }\bigr)-\alpha_{n} \tau_{n}\bigr\Vert \bigr\} . $$

So we complete the proof of Theorem 3.1. □

Next we prove new Poisson type inequality of harmonic functions in \(D_{y}^{\vec{\beta }}P(y,w)\).

Theorem 3.2

Let β⃗ be a multi-index such that

$$ \bigl(\vert \vec{\beta } \vert +n-2\bigr)p>\alpha +n+1 $$

and \(w\in \mathcal{C}_{n}(\Gamma )\). If

$$ u(y)=D_{y}^{\vec{\beta }}P(y,w) $$

in \(\mathcal{C}_{n}(\Gamma )\), then

$$ \Vert u \Vert _{\aleph_{\alpha }^{p}}\approx \tau_{n}^{\frac{n+\alpha +1}{p-n-\vert \vec{\beta } \vert +2}}. $$

Proof

First, we see from (2.3) that

$$ u(y)=\frac{f(y-\overline{w})}{\vert z-\overline{w} \vert ^{n+2\vert \vec{\beta } \vert +1}}, $$

where f is a homogeneous polynomial of degree \(\vert \vec{\beta } \vert +2\). Then we get

$$\begin{aligned} \Vert u \Vert _{\aleph_{\alpha }^{p}}^{p} =& \int_{\mathcal{C}_{n}(\Gamma )}\frac{\vert f(y-\overline{w})\vert ^{p+1}}{ \vert z-\overline{w}\vert ^{(n+2 \vert \vec{\beta } \vert )p} }z_{n}^{\alpha }\,dy \\ =& \int_{\mathcal{C}_{n}(\Gamma )}\frac{\vert f(y+(0,\tau_{n})) \vert ^{p+1}}{\vert z+(0,\tau_{n})\vert ^{(n+2 \vert \vec{\beta } \vert )p}}z_{n}^{\alpha }\,dy \\ =&\frac{\tau_{n}^{n+\alpha +(\vert \vec{\beta } \vert +1)p+1}}{\tau_{n}^{(n+2\vert \vec{\beta } \vert )p+1}} \int_{\mathcal{C}_{n}(\Gamma )}\frac{\vert f(y+(0,1)) \vert ^{p+1}}{\vert z+(0,1) \vert ^{(n+2\vert \vec{\beta } \vert )p+1}}z_{n}^{\alpha +1}\,dy \end{aligned}$$
(3.6)

from the change of variables \(z\mapsto (y^{\prime }+w^{\prime },z_{n})\) and then \(z\mapsto \tau_{n}z\), where we used the homogeneity of f.

Since f is a polynomial of degree \(1+\vert \vec{\beta } \vert \), we know that

$$\begin{aligned} 0 < &I \lesssim \int_{\mathcal{C}_{n}(\Gamma )}\frac{z_{n}^{\alpha +1}}{\vert z+(0,1) \vert ^{(n+\vert \vec{\beta } \vert -3)p}}\,dy \\ \lesssim & \int_{0}^{\infty }\frac{z_{n}^{\alpha +1}}{(y_{n}+1)^{(n+\vert \vec{\beta } \vert -1)p-n+2}} \int_{\partial \mathcal{C}_{n}(\Gamma )}\frac{z _{n}+1}{\vert z+(0,1) \vert ^{n}}\,dy^{\prime }\,dy_{n} \\ \lesssim & \int_{0}^{\infty }\frac{2}{(y_{n}+1)^{(n+\vert \vec{\beta } \vert -1)p-n- \alpha +2}}\,dy_{n} \\ < &\infty \end{aligned}$$

from (2.2), where I denotes the integral in (3.6) and we used the fact \((\vert \vec{\beta } \vert +n-1)p>\alpha +n\).

So

$$ \Vert u \Vert _{\aleph_{\alpha }^{p}}^{p}\approx \frac{1}{\tau_{n}^{(n+\vert \vec{\beta } \vert -1)p-(n+\alpha )+1}}, $$

which yields

$$ \Vert u \Vert _{\aleph_{\alpha }^{p}}\approx \tau_{n}^{(n+\alpha +1)/(p-n-\vert \vec{\beta } \vert +1)}. $$

Then we complete the proof. □

The following result implies that convergence in \(\aleph_{\alpha } ^{p}\)-norm implies the uniform convergence on each compact subset of \(\mathcal{C}_{n}(\Gamma )\) and point evaluation is a bounded linear functional on \(\aleph_{\alpha }^{p}\). Therefore we can see that \(\aleph_{\alpha }^{p}\) is a Banach space with \(\aleph_{\alpha }^{p}\)-norm.

Lemma 3.3

Let \(\alpha >0\), \(p>0\) and \(z\in \mathcal{C}_{n}(\Gamma )\). If \(u\in \aleph^{p}_{\alpha }\), then we have

$$ \bigl\vert u(y) \bigr\vert \le \frac{\Vert u \Vert ^{\alpha }_{\aleph^{p}_{\alpha }}}{y^{\frac{n+ \alpha +1}{p}}_{n}}. $$

Proof

Let \(r =\frac{z_{n}}{2}\). Note that \(\tau_{n} \approx z_{n}\), \(\tau_{n}\) ranges over all point in \(B(y,r)\).

Hence, we get

$$\begin{aligned} \Vert w_{n+1}-\hat{w\tau } \Vert ^{2}&\le (1-\sigma_{n})\Vert w_{n}-\hat{w\tau } \Vert ^{2}+\sigma_{n} \Vert v_{n}-\hat{w\tau } \Vert ^{2} \\ &\le \Vert \tau_{n}-\hat{w\tau }\Vert ^{2}+ \alpha_{n}M-\sigma_{n} \Vert \tau_{n}-v_{n} \Vert ^{2} \\ &\quad {}-\Vert \tau_{n}-v_{n}\Vert ^{2}+4 \gamma_{n}\Vert \tau_{n}-v_{n}\Vert \bigl\Vert \mathcal{G}^{*}\mathcal{G}(\tau_{n}-\hat{w\tau })\bigr\Vert \\ &\quad {}+4\alpha_{n}\Vert \tau_{n}\Vert \Vert \tau_{n}-v_{n}\Vert , \end{aligned}$$

which means that

$$\begin{aligned} \sigma_{n}\Vert \tau_{n}-v_{n} \Vert ^{2} &\le \Vert w_{n+1}-\tau _{n}\Vert \bigl(\Vert w_{n}-\hat{w\tau } \Vert +\Vert w_{n+1}- \hat{w\tau } \Vert \bigr) \\ &\quad {}+\alpha_{n}M-\sigma_{n}\Vert \tau_{n}-v_{n} \Vert ^{2} \\ &\quad {}-\Vert \tau_{n}-v_{n}\Vert ^{2}+4 \gamma_{n}\Vert \tau_{n}-v_{n}\Vert \bigl\Vert \mathcal{G}^{*}\mathcal{G}(\tau_{n}-\hat{w\tau })\bigr\Vert \\ &\quad {}+4\alpha_{n}\Vert \tau_{n}\Vert \Vert \tau_{n}-v_{n}\Vert . \end{aligned}$$

Since

$$\begin{aligned}& \lim_{n\rightarrow \infty }\alpha_{n}=0, \\& \lim_{n\rightarrow \infty }\Vert w_{n+1}-\tau_{n} \Vert =0, \end{aligned}$$

and

$$ \lim_{n\rightarrow \infty }\bigl\Vert \mathcal{G}^{*}\mathcal{G} \tau_{n}-\mathcal{G}^{*}\mathcal{G}\hat{w\tau }\bigr\Vert =0. $$

We infer that

$$ \lim_{n\rightarrow \infty }\Vert w_{n}-v_{n} \Vert =0. $$

Finally, we show that \(\tau_{n}\rightarrow \hat{w\tau }\). Using the property of the projection \(P_{S_{i}}\), we derive that

$$\begin{aligned} &\Vert u_{\lambda }-\hat{w\tau } \Vert ^{2} \\ &\quad =\biggl\Vert P_{S_{i}}\biggl[(1-\alpha_{n}) \biggl(\tau_{n}-\frac{\gamma_{n}}{1-\alpha _{n}}\mathcal{G}^{*}\mathcal{G} \tau_{n}\biggr)\biggr] \\ &\quad \quad {} -P_{S_{i}}\biggl[\alpha_{n} \hat{w\tau }+(1-\alpha_{n}) \biggl(\hat{w\tau }-\frac{\gamma_{n}}{1-\alpha _{n}} \mathcal{G}^{*}\mathcal{G}\tau_{n}\biggr)\biggr]\biggr\Vert ^{2} \\ &\quad \le \biggl\langle (1-\alpha ) \biggl(I-\frac{\gamma_{n}}{1-\alpha_{n}}(\tau_{n}-\hat{w\tau })\biggr)-\alpha_{n}\hat{w\tau},v_{n}-\hat{w\tau } \biggr\rangle \\ &\quad \le (1-\alpha_{n})\Vert \tau_{n}-\hat{w\tau }\Vert \Vert v_{n}- \hat{w\tau }\Vert +\alpha_{n} \langle \hat{w \tau },\hat{w\tau }-v _{n} \rangle \\ &\quad \le \frac{1-\alpha_{n}}{2}\bigl(\Vert \tau_{n}-\hat{w\tau }\Vert ^{2}+ \Vert v_{n}-\hat{w\tau }\Vert ^{2} \bigr)+\alpha_{n} \langle \hat{w\tau }, \hat{w\tau }-v_{n}\rangle , \end{aligned}$$

which is equal to

$$\begin{aligned} &\Vert u_{\lambda }-\hat{w\tau } \Vert ^{2}\le \frac{1-\alpha_{n}}{1+ \alpha_{n}}\Vert \tau_{n}-\hat{w\tau }\Vert ^{2}+ \frac{2\alpha_{n}}{1- \alpha_{n}} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle . \end{aligned}$$
(3.7)

It follows from (3.5) and (3.7) that

$$\begin{aligned} \Vert w_{n+1}-\hat{w\tau } \Vert &\le(1-\sigma_{n})\Vert w_{n}- \hat{w\tau } \Vert +\sigma_{n}\Vert v_{n}-\hat{w\tau }\Vert \\ &\le(1-\sigma_{n})\Vert w_{n}-\hat{w\tau } \Vert + \sigma_{n}\biggl(\frac{1- \alpha_{n}}{1+\alpha_{n}}\Vert \tau_{n}-\hat{w\tau }\Vert ^{2}+\frac{2 \alpha_{n}}{1-\alpha_{n}} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle \biggr) \\ &\le\biggl(1-\frac{2\alpha_{n}\gamma_{n}}{1+\alpha_{n}}\biggr)\Vert \tau_{n}- \hat{w\tau } \Vert ^{2}+\frac{2\alpha_{n}\gamma_{n}}{1-\alpha_{n}} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle . \end{aligned}$$
(3.8)

Since \(\frac{\gamma_{n}}{1-\alpha_{n}}\in (0,\frac{2}{\rho (G*G)})\), we observe that \(\alpha_{n}\in (0,\frac{\gamma_{n}\rho (G*G)}{2})\). Then

$$ \frac{2\alpha_{n}\gamma_{n}}{1-\alpha_{n}}\in \biggl(0,\frac{2\gamma_{n}(2- \gamma_{n}\rho (G*G))}{\gamma_{n}\rho (G*G)}\biggr), $$

that is to say

$$ \frac{2\alpha_{n}\gamma_{n}}{1-\alpha_{n}} \langle \hat{w\tau }, \hat{w\tau }-v_{n} \rangle \le \frac{2\gamma_{n}(2-\gamma_{n} \rho (G*G))}{\gamma_{n}\rho (G*G)} \langle \hat{w\tau }, \hat{w\tau }-v_{n} \rangle . $$

By virtue of \(\sum_{n=1}^{\infty } \frac{\sigma_{n}}{\gamma_{n}}<\infty \), \(\gamma_{n}\in (0,\frac{2}{ \rho (G*G)})\) and \(\langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle \) is bounded, we obtain that

$$ \sum_{n=1}^{\infty }\biggl(\frac{2\gamma_{n}(2-\gamma_{n}\rho (G*G))}{ \gamma_{n}\rho (G*G)} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle \biggr) \langle \hat{w \tau },\hat{w\tau }-v_{n} \rangle < \infty , $$

which implies that

$$ \sum_{n=1}^{\infty }\frac{2\alpha_{n}\gamma_{n}}{1-\alpha_{n}} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle \le \infty . $$

Moreover,

$$ \sum_{n=1}^{\infty } \frac{2\alpha_{n}\gamma_{n}}{1-\alpha_{n}} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle = \sum_{n=1}^{\infty }\frac{2\alpha_{n}\gamma_{n}}{1+\alpha_{n}} \frac{1+ \alpha_{n}}{1-\alpha_{n}} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle . $$
(3.9)

It follows that all the conditions are satisfied. Combining (3.8) and (3.9) and Lemma 2.1, we can show that \(\tau_{n}\rightarrow \hat{w\tau }\).

Now we repeat some calculations in (3.8) and (3.9) to have

$$\begin{aligned} \Vert z_{n}-\hat{u} \Vert \le \max \bigl\{ \Vert z_{n}-\hat{u} \Vert , \Vert {-}\hat{u} \Vert \bigr\} . \end{aligned}$$

Consequently, \({z_{n}}\) is bounded, and so is \({v_{n}}\). Let \(T=2P_{S_{i}}-I\). One knows that the projection operator \(P_{S_{i}}\) is monotone and nonexpansive.

Therefore,

$$\begin{aligned} z_{n+1} =&\frac{I+T}{2} \biggl[ (1-\sigma_{n})z_{n}+ \sigma_{n}\biggl(1-\frac{ \mu_{n}}{\sigma_{n}}\mathcal{G}^{*} \mathcal{G}\biggr)v_{n} \biggr] \\ =&\frac{I-\sigma_{n}}{2}z_{n}+\frac{\sigma_{n}}{2}\biggl(I- \frac{\mu_{n}}{ \sigma_{n}}\mathcal{G}^{*}\mathcal{G}\biggr)v_{n} + \frac{T}{2}\biggl[(1-\sigma _{n})z_{n}+ \sigma_{n}\biggl(I-\frac{\mu_{n}}{\sigma_{n}}\mathcal{G}^{*} \mathcal{G}\biggr)v_{n}\biggr], \end{aligned}$$

that is,

$$ z_{n+1}=\frac{1-\sigma_{n}}{2}z_{n}+\frac{1+\sigma_{n}}{2}t_{n}, $$

where

$$ t_{n}=\frac{\sigma_{n}(I-\frac{\mu_{n}}{\sigma_{n}}\mathcal{G}^{*} \mathcal{G})v_{n}+T[(1-\sigma_{n})z_{n} +\sigma_{n}(I-\frac{\mu_{n}}{ \sigma_{n}}\mathcal{G}^{*}\mathcal{G})v_{n}]}{1+\sigma_{n}}. $$

Indeed,

$$\begin{aligned} \Vert t_{n+1}-t_{n} \Vert &\le \frac{\sigma_{n+1}}{1+\sigma_{n+1}}\biggl\Vert \biggl(I-\frac{\mu_{n+1}}{\sigma _{n+1}}\mathcal{G}^{*} \mathcal{G}\biggr)v_{n+1}-\biggl(I-\frac{\mu_{n}}{\sigma _{n}} \mathcal{G}^{*}\mathcal{G}\biggr)v_{n} \biggr\Vert \\ &\quad {}+ \biggl\vert \frac{\sigma_{n+1}}{1+\sigma_{n+1}}-\frac{\sigma_{n}}{1+ \sigma_{n}} \biggr\vert \biggl\Vert \biggl(I-\frac{\mu_{n}}{\sigma_{n}}\mathcal{G} ^{*}\mathcal{G} \biggr)v_{n} \biggr\Vert \\ &\quad {}+ \frac{T}{1+\sigma_{n+1}} \biggl\{ (1-\sigma_{n+1})z_{n+1}+ \sigma_{n+1}\biggl(I-\frac{ \mu_{n+1}}{\sigma_{n+1}}\mathcal{G}^{*} \mathcal{G}\biggr)v_{n+1} \biggr\} \\ &\quad {}+ \biggl\vert \frac{1}{1+\sigma_{n+1}}-\frac{1}{1+\sigma_{n}} \biggr\vert \biggl\Vert T\biggl[(1- \sigma_{n})z_{n}+\sigma_{n} \biggl(I-\frac{\mu_{n}}{\sigma_{n}}\mathcal{G} ^{*}\mathcal{G} \biggr)u_{\lambda }\biggr] \biggr\Vert . \end{aligned}$$

After taking a weighted Ostrowski type inequality (see [15–17]), we have

$$\begin{aligned} \bigl\vert u(y) \bigr\vert ^{p} =& \biggl\vert \frac{1}{V(B(y,r))} \int_{B(y,r)}u(w)\,dw \biggr\vert ^{p+1} \\ \leq &\frac{1}{V(B(y,r))} \int_{B(y,r)}\bigl\vert u(w) \bigr\vert ^{p+1}\,dw \\ \approx &\frac{1}{y^{n}_{n}} \int_{B(y,r)}\bigl\vert u(w) \bigr\vert ^{p+1} \frac{w^{\alpha }_{n}}{y^{\alpha }_{n}}\,dw. \end{aligned}$$

So

$$ \bigl\vert u(y) \bigr\vert \le \frac{\Vert u \Vert ^{q}_{\aleph^{p}_{\alpha }}}{y^{\frac{n+\alpha +1}{p}}_{n}}. $$

The proof is complete. □

Unlike the cases of bounded domains, the next theorem shows that if \(p\ne q\), then there is no inclusion between \(\aleph^{p}_{\alpha }\) and \(\aleph^{q}_{\alpha }\).

Lemma 3.4

Let \(\alpha >0\) and \(p,q>0\). If \(p\ne q\), then \(\aleph^{p}_{\alpha }\) does not contain \(\aleph^{q}_{\alpha }\).

Proof

Suppose that \(\aleph^{p}_{\alpha }\subset \aleph^{q}_{\alpha }\). Then we see from Lemma 3.4 that convergence in any \(\aleph^{p} _{\alpha }\)-norm implies uniform convergence on compact subsets. Therefore we know from the closed graph theorem that the identity map from \(\aleph^{p}_{\alpha }\) to \(\aleph^{q}_{\alpha }\) is continuous. Hence we get

$$\begin{aligned} \Vert v \Vert ^{p}_{\aleph^{q}_{\alpha }}\lesssim \Vert v \Vert ^{q}_{\aleph^{p}_{ \alpha }} \end{aligned}$$
(3.10)

as v ranges over all functions in \(\aleph^{p}_{\alpha }\).

To show that (3.10) fails, there exists a nonnegative integer k large enough such

$$\begin{aligned} (n + k - 2)p > n +\alpha +1,\quad (n + k -2)q > n + \alpha . \end{aligned}$$
(3.11)

Set \(u(y)=D^{k}_{z_{n}}P(y,0)\) for \(z\in \mathcal{C}_{n}(\Gamma )\). It is obvious that u is also harmonic in \(\mathcal{C}_{n}(\Gamma )\), since u is a partial derivative of harmonic function. Therefore we see from (2.3) that

$$ u(y)=\frac{f(y)}{\vert z \vert ^{n+2k+1}} $$

for some homogeneous polynomial f of degree \(k+2\). Let \(u_{\delta }(y)=u(y+(0, \delta ))\), where \(\delta >0\). It is easy to see from Theorem 3.2 that for \(\delta >0\)

$$ \Vert u_{\delta } \Vert _{\aleph^{p}_{\alpha }}\lesssim \delta^{(n+\alpha )(p-n-k+1)} $$

and

$$ \Vert u_{\delta } \Vert _{\aleph^{q}_{\alpha }}\lesssim \delta^{(n+\alpha )(q-n-k+1)}, $$

because (3.11) holds.

Hence we get

$$\begin{aligned} \frac{\Vert u_{\delta } \Vert ^{p}_{\aleph^{q}_{\alpha }}}{\Vert u_{\delta } \Vert ^{q} _{\aleph^{p}_{\alpha }}}\approx \delta^{(n+\alpha )(1/q-1/p)} \end{aligned}$$
(3.12)

for \(\delta >0\). Since \(p \ne q\), the right side of (3.12) is not bounded as a function of δ. Thus (3.11) fails and the proof is complete. □

4 Conclusions

In this paper, we proved that there exists a weak solution for Schrödingerean technique for order performance by similarity equations. Meanwhile, the boundary behaviors of it were also obtained via the abstract theory of fuzzy multi-criterion decision making. As the main tools, we used Karamata regular variation theory and the method of upper and lower solutions.