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Existence and boundary behavior of weak solutions for Schrödingerean TOPSIS equations
Received: 16 November 2017
Accepted: 5 January 2018
Published: 19 January 2018
Abstract
In this paper, we prove that there exists a weak solution for Schrödingerean technique for order performance by similarity (TOPSIS) equations on cylinders. Meanwhile, the boundary behaviors of it are also obtained via the abstract theory of fuzzy multi-criterion decision making. As the main tools, we use Karamata regular variation theory and the method of upper and lower solutions.
Keywords
Schrödingerean TOPSIS equationsboundary behaviorcylinder
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
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).
$$ \Omega \times \mathbf{R}= \bigl\{ P=(X,y)\in {\mathbf{{R}}}^{n}; X\in \Omega ,y \in \bf {R} \bigr\} $$
In this paper, we consider the following Schrödingerean TOPSIS equation:
in \(\mathcal{C}_{n}(\Gamma )\), where \(0< s\leq 1\) and the potential a satisfies the following condition:
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).
$$ T (-\triangle )^{s}u+a(P) u(P)=0 $$
(1.1)
$$ 0< \inf_{P=(r,\Theta )\in C_{n}(\Omega )}a(P)< \lim_{r\rightarrow \infty }a(P)=a_{\infty }< \infty . $$
Let \(\alpha > 0\) and \(1\leq p < \infty \). Then the weighted weak space \(\aleph^{p} _{\alpha }(\Gamma ) \) on cylinders can be defined by
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 \(\mathcal{C}_{n}(\Gamma )\).
$$ \Vert u \Vert _{\aleph^{p}_{\alpha }(\Gamma )}:= \biggl( \int_{ \Gamma }\bigl\vert u(y) \bigr\vert ^{p}\,d\wp_{\alpha }(y) \biggr) ^{\frac{1}{q}}< \infty , $$
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
It is easy to see that (see [14] for details and related facts)
for each \(z\in \mathcal{C}_{n}(\Gamma )\) and for every \(w \in \overline{ \mathcal{C}_{n}(\Gamma )}\).
$$\begin{aligned} P_{z}(w):=P(y,w)=\frac{1}{nV(B(0,1))} \frac{z_{n}+w_{n}}{\vert z-\overline{w} \vert ^{n}}. \end{aligned}$$
(2.1)
$$\begin{aligned} \int_{\partial \mathcal{C}_{n}(\Gamma )}P(y,w)\,dw^{\prime }=1, \end{aligned}$$
(2.2)
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
where \(\vec{\beta }=\beta_{1}+\beta_{2}+\cdots +\beta_{n}\).
$$\begin{aligned} D_{y}^{\vec{\beta }}P(y,w):=D_{z_{1}}^{\beta_{1}} \cdots D_{z_{1}}^{ \beta_{1}}P(y,w) =\frac{f(y-\overline{w})}{\vert z-\overline{w} \vert ^{n+2\vert \vec{\beta } \vert +1}}, \end{aligned}$$
(2.3)
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
Using Lemma 2.1 it follows that \((I-\frac{\mu_{n}}{\sigma_{n}} \mathcal{G}^{*}\mathcal{G})\) is nonexpansive and averaged. Hence,
$$ d_{n}=\biggl(I-\frac{\mu_{n}}{\sigma_{n}}\mathcal{G}^{*}\mathcal{G} \biggr)v_{n}. $$
$$\begin{aligned} \Vert t_{n+1}-t_{n} \Vert &\le \frac{\sigma_{n+1}}{1+\sigma_{n+1}}\Vert d_{n+1}-d_{n} \Vert + \biggl\vert \frac{\sigma_{n+1}}{1+\sigma_{n+1}}-\frac{\sigma_{n}}{1+\sigma _{n}} \biggr\vert \Vert d_{n} \Vert \\ &\quad {}+ \frac{T}{1+\sigma_{n+1}} \bigl\{ (1-\sigma_{n+1})w_{n+1}+ \sigma_{n+1}d _{n+1}- \bigl[(1-\sigma_{n})w_{n}+ \sigma_{n}d_{n}\bigr] \bigr\} \\ &\quad {}+ \biggl\vert \frac{1}{1+\sigma_{n+1}}-\frac{1}{1+\sigma_{n}} \biggr\vert \bigl\Vert T\bigl[(1- \sigma_{n})w_{n}+\sigma_{n}d_{n} \bigr] \bigr\Vert \\ &\le \frac{\sigma_{n+1}}{1+\sigma_{n+1}}\Vert d_{n+1}-d_{n} \Vert + \biggl\vert \frac{\sigma_{n+1}}{1+\sigma_{n+1}}-\frac{\sigma_{n}}{1+\sigma _{n}} \biggr\vert \Vert d_{n} \Vert \\ &\quad {}+ \frac{1-\sigma_{n+1}}{1+\sigma_{n+1}}\Vert w_{n+1}-w_{n}\Vert + \frac{ \sigma_{n+1}}{1+\sigma_{n+1}}\Vert d_{n+1}-d_{n} \Vert + \frac{ \sigma_{n}-\sigma_{n+1}}{1+\sigma_{n+1}}\Vert w_{n} \Vert \\ &\quad {}+ \frac{\sigma_{n+1}-\sigma_{n}}{1+\sigma_{n+1}}\Vert d_{n} \Vert + \biggl\vert \frac{1}{1+\sigma_{n+1}}-\frac{1}{1+\sigma_{n}} \biggr\vert \bigl\Vert T\bigl[(1- \sigma_{n})w_{n}+\sigma_{n}d_{n}\bigr]\bigr\Vert . \end{aligned}$$
(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
which implies that
$$\begin{aligned} \Vert w_{n+1}-\hat{w} \Vert ^{2} &\le(1-\sigma_{n})\Vert w_{n}- \hat{w} \Vert ^{2}+\sigma_{n}\Vert v_{n}-\hat{w}\Vert ^{2} \\ &\le\sigma_{n}\biggl\Vert {-}\alpha_{n}\hat{w}+(1-\alpha_{n})\biggl[w_{n}-\frac{ \gamma_{n}}{1-\alpha_{n}} \mathcal{G}^{*}\mathcal{G}w_{n} -\biggl(\hat{w}- \frac{ \gamma_{n}}{1-\alpha_{n}}\mathcal{G}^{*}\mathcal{G}\hat{w}\biggr)\biggr]\biggr\Vert ^{2} \\ &\le(1-\sigma_{n})\Vert w_{n}-\hat{w} \Vert ^{2}+\sigma_{n}\alpha _{n}\Vert {-}\hat{w} \Vert ^{2}+(1-\alpha_{n})\sigma_{n}\biggl[\Vert w_{n}-\hat{w} \Vert ^{2} \\ &\quad {}+ \frac{\gamma_{n}}{1-\alpha_{n}}\biggl(\frac{\gamma_{n}}{1-\alpha_{n}}-\frac{2}{ \rho (\mathcal{G}^{*}\mathcal{G})}\biggr)\bigl\Vert \mathcal{G}^{*}\mathcal{G}w _{n}-\mathcal{G}^{*}\mathcal{G}\hat{w}\bigr\Vert ^{2}\biggr] \\ &\le \Vert w_{n}-\hat{w} \Vert ^{2}+ \sigma_{n}\alpha_{n}\Vert {-}\hat{w} \Vert ^{2} \\ &\quad {}+ \sigma_{n}\gamma_{n}\biggl(\frac{\gamma_{n}}{1-\alpha_{n}}- \frac{2}{ \rho (\mathcal{G}^{*}\mathcal{G})}\biggr)\bigl\Vert \mathcal{G}^{*}\mathcal{G}w _{n}-\mathcal{G}^{*}\mathcal{G}\hat{w}\bigr\Vert ^{2}, \end{aligned}$$
$$\begin{aligned} \bigl\Vert \mathcal{G}^{*} \mathcal{G}w_{n}-\mathcal{G}^{*}\mathcal{G} \hat{w}\bigr\Vert ^{2} &\le \Vert w_{n}-\hat{w} \Vert ^{2}-\Vert w _{n+1}-\hat{w} \Vert ^{2}+\sigma_{n}\alpha_{n}\Vert {-}\hat{w} \Vert ^{2} \\ &\le \Vert w_{n+1}-w_{n} \Vert \bigl(\Vert w_{n}-\hat{w} \Vert + \Vert w_{n+1}-\hat{w} \Vert \bigr)+\sigma_{n}\alpha_{n}\Vert {-}\hat{w} \Vert ^{2}. \end{aligned}$$
Since
we have the following result:
$$\begin{aligned}& \lim_{n\rightarrow \infty }\inf \sigma_{n}\gamma_{n}\biggl( \frac{2}{ \rho (\mathcal{G}^{*}\mathcal{G})}-\frac{\gamma_{n}}{1-\alpha_{n}}\biggr)>0, \\& \lim_{n\rightarrow \infty }\alpha_{n}=0\quad \mbox{and}\quad \lim _{n\rightarrow \infty }\Vert w_{n+1}-w_{n} \Vert =0, \end{aligned}$$
$$ \lim_{n\rightarrow \infty }\bigl\Vert \mathcal{G}^{*}\mathcal{G}w _{n}-\mathcal{G}^{*}\mathcal{G}\hat{w}\bigr\Vert =0. $$
Applying the property of the projection \(P_{S_{i}}\), one can easily show that
where \(M>0\) satisfying
$$\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}$$
$$ M\ge \sup_{k} \bigl\{ 2\Vert {-}\tau_{n}\Vert \bigl\Vert \tau_{n}-\gamma _{n}\mathcal{G}^{*} \mathcal{G}\tau_{n}-\bigl(\hat{w\tau }-\gamma_{n} \mathcal{G}^{*}\mathcal{G}\hat{w\tau }\bigr)-\alpha_{n} \tau_{n}\bigr\Vert \bigr\} . $$
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
and
\(w\in \mathcal{C}_{n}(\Gamma )\). If
in
\(\mathcal{C}_{n}(\Gamma )\), then
$$ \bigl(\vert \vec{\beta } \vert +n-2\bigr)p>\alpha +n+1 $$
$$ u(y)=D_{y}^{\vec{\beta }}P(y,w) $$
$$ \Vert u \Vert _{\aleph_{\alpha }^{p}}\approx \tau_{n}^{\frac{n+\alpha +1}{p-n-\vert \vec{\beta } \vert +2}}. $$
Proof
First, we see from (2.3) that
where f is a homogeneous polynomial of degree \(\vert \vec{\beta } \vert +2\). Then we get
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.
$$ u(y)=\frac{f(y-\overline{w})}{\vert z-\overline{w} \vert ^{n+2\vert \vec{\beta } \vert +1}}, $$
$$\begin{aligned} \Vert u \Vert _{\aleph_{\alpha }^{p}}^{p} =& \int_{\mathcal{C}_{n}(\Gamma )}\frac{\vert f(y-\overline{w})\vert ^{p+1}}{ \vert z-\overline{w}\vert ^{(n+2 \vert \vec{\beta } \vert )p} }z_{n}^{\alpha }\,dy \\ =& \int_{\mathcal{C}_{n}(\Gamma )}\frac{\vert f(y+(0,\tau_{n})) \vert ^{p+1}}{\vert z+(0,\tau_{n})\vert ^{(n+2 \vert \vec{\beta } \vert )p}}z_{n}^{\alpha }\,dy \\ =&\frac{\tau_{n}^{n+\alpha +(\vert \vec{\beta } \vert +1)p+1}}{\tau_{n}^{(n+2\vert \vec{\beta } \vert )p+1}} \int_{\mathcal{C}_{n}(\Gamma )}\frac{\vert f(y+(0,1)) \vert ^{p+1}}{\vert z+(0,1) \vert ^{(n+2\vert \vec{\beta } \vert )p+1}}z_{n}^{\alpha +1}\,dy \end{aligned}$$
(3.6)
Since f is a polynomial of degree \(1+\vert \vec{\beta } \vert \), we know that
from (2.2), where I denotes the integral in (3.6) and we used the fact \((\vert \vec{\beta } \vert +n-1)p>\alpha +n\).
$$\begin{aligned} 0 < &I \lesssim \int_{\mathcal{C}_{n}(\Gamma )}\frac{z_{n}^{\alpha +1}}{\vert z+(0,1) \vert ^{(n+\vert \vec{\beta } \vert -3)p}}\,dy \\ \lesssim & \int_{0}^{\infty }\frac{z_{n}^{\alpha +1}}{(y_{n}+1)^{(n+\vert \vec{\beta } \vert -1)p-n+2}} \int_{\partial \mathcal{C}_{n}(\Gamma )}\frac{z _{n}+1}{\vert z+(0,1) \vert ^{n}}\,dy^{\prime }\,dy_{n} \\ \lesssim & \int_{0}^{\infty }\frac{2}{(y_{n}+1)^{(n+\vert \vec{\beta } \vert -1)p-n- \alpha +2}}\,dy_{n} \\ < &\infty \end{aligned}$$
So
which yields
$$ \Vert u \Vert _{\aleph_{\alpha }^{p}}^{p}\approx \frac{1}{\tau_{n}^{(n+\vert \vec{\beta } \vert -1)p-(n+\alpha )+1}}, $$
$$ \Vert u \Vert _{\aleph_{\alpha }^{p}}\approx \tau_{n}^{(n+\alpha +1)/(p-n-\vert \vec{\beta } \vert +1)}. $$
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
which means that
$$\begin{aligned} \Vert w_{n+1}-\hat{w\tau } \Vert ^{2}&\le (1-\sigma_{n})\Vert w_{n}-\hat{w\tau } \Vert ^{2}+\sigma_{n} \Vert v_{n}-\hat{w\tau } \Vert ^{2} \\ &\le \Vert \tau_{n}-\hat{w\tau }\Vert ^{2}+ \alpha_{n}M-\sigma_{n} \Vert \tau_{n}-v_{n} \Vert ^{2} \\ &\quad {}-\Vert \tau_{n}-v_{n}\Vert ^{2}+4 \gamma_{n}\Vert \tau_{n}-v_{n}\Vert \bigl\Vert \mathcal{G}^{*}\mathcal{G}(\tau_{n}-\hat{w\tau })\bigr\Vert \\ &\quad {}+4\alpha_{n}\Vert \tau_{n}\Vert \Vert \tau_{n}-v_{n}\Vert , \end{aligned}$$
$$\begin{aligned} \sigma_{n}\Vert \tau_{n}-v_{n} \Vert ^{2} &\le \Vert w_{n+1}-\tau _{n}\Vert \bigl(\Vert w_{n}-\hat{w\tau } \Vert +\Vert w_{n+1}- \hat{w\tau } \Vert \bigr) \\ &\quad {}+\alpha_{n}M-\sigma_{n}\Vert \tau_{n}-v_{n} \Vert ^{2} \\ &\quad {}-\Vert \tau_{n}-v_{n}\Vert ^{2}+4 \gamma_{n}\Vert \tau_{n}-v_{n}\Vert \bigl\Vert \mathcal{G}^{*}\mathcal{G}(\tau_{n}-\hat{w\tau })\bigr\Vert \\ &\quad {}+4\alpha_{n}\Vert \tau_{n}\Vert \Vert \tau_{n}-v_{n}\Vert . \end{aligned}$$
Since
and
We infer that
$$\begin{aligned}& \lim_{n\rightarrow \infty }\alpha_{n}=0, \\& \lim_{n\rightarrow \infty }\Vert w_{n+1}-\tau_{n} \Vert =0, \end{aligned}$$
$$ \lim_{n\rightarrow \infty }\bigl\Vert \mathcal{G}^{*}\mathcal{G} \tau_{n}-\mathcal{G}^{*}\mathcal{G}\hat{w\tau }\bigr\Vert =0. $$
$$ \lim_{n\rightarrow \infty }\Vert w_{n}-v_{n} \Vert =0. $$
Finally, we show that \(\tau_{n}\rightarrow \hat{w\tau }\). Using the property of the projection \(P_{S_{i}}\), we derive that
which is equal to
$$\begin{aligned} &\Vert u_{\lambda }-\hat{w\tau } \Vert ^{2} \\ &\quad =\biggl\Vert P_{S_{i}}\biggl[(1-\alpha_{n}) \biggl(\tau_{n}-\frac{\gamma_{n}}{1-\alpha _{n}}\mathcal{G}^{*}\mathcal{G} \tau_{n}\biggr)\biggr] \\ &\quad \quad {} -P_{S_{i}}\biggl[\alpha_{n} \hat{w\tau }+(1-\alpha_{n}) \biggl(\hat{w\tau }-\frac{\gamma_{n}}{1-\alpha _{n}} \mathcal{G}^{*}\mathcal{G}\tau_{n}\biggr)\biggr]\biggr\Vert ^{2} \\ &\quad \le \biggl\langle (1-\alpha ) \biggl(I-\frac{\gamma_{n}}{1-\alpha_{n}}(\tau_{n}-\hat{w\tau })\biggr)-\alpha_{n}\hat{w\tau},v_{n}-\hat{w\tau } \biggr\rangle \\ &\quad \le (1-\alpha_{n})\Vert \tau_{n}-\hat{w\tau }\Vert \Vert v_{n}- \hat{w\tau }\Vert +\alpha_{n} \langle \hat{w \tau },\hat{w\tau }-v _{n} \rangle \\ &\quad \le \frac{1-\alpha_{n}}{2}\bigl(\Vert \tau_{n}-\hat{w\tau }\Vert ^{2}+ \Vert v_{n}-\hat{w\tau }\Vert ^{2} \bigr)+\alpha_{n} \langle \hat{w\tau }, \hat{w\tau }-v_{n}\rangle , \end{aligned}$$
$$\begin{aligned} &\Vert u_{\lambda }-\hat{w\tau } \Vert ^{2}\le \frac{1-\alpha_{n}}{1+ \alpha_{n}}\Vert \tau_{n}-\hat{w\tau }\Vert ^{2}+ \frac{2\alpha_{n}}{1- \alpha_{n}} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle . \end{aligned}$$
(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
that is to say
$$ \frac{2\alpha_{n}\gamma_{n}}{1-\alpha_{n}}\in \biggl(0,\frac{2\gamma_{n}(2- \gamma_{n}\rho (G*G))}{\gamma_{n}\rho (G*G)}\biggr), $$
$$ \frac{2\alpha_{n}\gamma_{n}}{1-\alpha_{n}} \langle \hat{w\tau }, \hat{w\tau }-v_{n} \rangle \le \frac{2\gamma_{n}(2-\gamma_{n} \rho (G*G))}{\gamma_{n}\rho (G*G)} \langle \hat{w\tau }, \hat{w\tau }-v_{n} \rangle . $$
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
which implies that
$$ \sum_{n=1}^{\infty }\biggl(\frac{2\gamma_{n}(2-\gamma_{n}\rho (G*G))}{ \gamma_{n}\rho (G*G)} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle \biggr) \langle \hat{w \tau },\hat{w\tau }-v_{n} \rangle < \infty , $$
$$ \sum_{n=1}^{\infty }\frac{2\alpha_{n}\gamma_{n}}{1-\alpha_{n}} \langle \hat{w\tau },\hat{w\tau }-v_{n} \rangle \le \infty . $$
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,
that is,
where
$$\begin{aligned} z_{n+1} =&\frac{I+T}{2} \biggl[ (1-\sigma_{n})z_{n}+ \sigma_{n}\biggl(1-\frac{ \mu_{n}}{\sigma_{n}}\mathcal{G}^{*} \mathcal{G}\biggr)v_{n} \biggr] \\ =&\frac{I-\sigma_{n}}{2}z_{n}+\frac{\sigma_{n}}{2}\biggl(I- \frac{\mu_{n}}{ \sigma_{n}}\mathcal{G}^{*}\mathcal{G}\biggr)v_{n} + \frac{T}{2}\biggl[(1-\sigma _{n})z_{n}+ \sigma_{n}\biggl(I-\frac{\mu_{n}}{\sigma_{n}}\mathcal{G}^{*} \mathcal{G}\biggr)v_{n}\biggr], \end{aligned}$$
$$ z_{n+1}=\frac{1-\sigma_{n}}{2}z_{n}+\frac{1+\sigma_{n}}{2}t_{n}, $$
$$ t_{n}=\frac{\sigma_{n}(I-\frac{\mu_{n}}{\sigma_{n}}\mathcal{G}^{*} \mathcal{G})v_{n}+T[(1-\sigma_{n})z_{n} +\sigma_{n}(I-\frac{\mu_{n}}{ \sigma_{n}}\mathcal{G}^{*}\mathcal{G})v_{n}]}{1+\sigma_{n}}. $$
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
as v ranges over all functions in \(\aleph^{p}_{\alpha }\).
$$\begin{aligned} \Vert v \Vert ^{p}_{\aleph^{q}_{\alpha }}\lesssim \Vert v \Vert ^{q}_{\aleph^{p}_{ \alpha }} \end{aligned}$$
(3.10)
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
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 \(\delta >0\)
and
because (3.11) holds.
$$ u(y)=\frac{f(y)}{\vert z \vert ^{n+2k+1}} $$
$$ \Vert u_{\delta } \Vert _{\aleph^{p}_{\alpha }}\lesssim \delta^{(n+\alpha )(p-n-k+1)} $$
$$ \Vert u_{\delta } \Vert _{\aleph^{q}_{\alpha }}\lesssim \delta^{(n+\alpha )(q-n-k+1)}, $$
Hence we get
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. □
$$\begin{aligned} \frac{\Vert u_{\delta } \Vert ^{p}_{\aleph^{q}_{\alpha }}}{\Vert u_{\delta } \Vert ^{q} _{\aleph^{p}_{\alpha }}}\approx \delta^{(n+\alpha )(1/q-1/p)} \end{aligned}$$
(3.12)
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.
Declarations
Acknowledgements
The authors are thankful to the honorable reviewers for their valuable suggestions and comments, which improved the paper. This work was supported by the Natural Science Foundation of Heilongjiang Province (No. A2016209040).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
Department of Computer Science and Technology, Harbin Engineering University, Harbin, China
(2)
College of Power and Energy Engineering, Harbin Engineering University, Harbin, China
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