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.
Similar content being viewed by others
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).
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).
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 \(\mathcal{C}_{n}(\Gamma )\).
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 \(w \in \overline{ \mathcal{C}_{n}(\Gamma )}\).
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 \(\vec{\beta }=\beta_{1}+\beta_{2}+\cdots +\beta_{n}\).
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
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,
Moreover,
Substituting (3.2) in (3.1), we infer that
By virtue of \(\lim_{n\rightarrow \infty }(\sigma_{n+1}-\sigma _{n})=0\), it follows that
Moreover, \(\{ w_{n} \} \), and \(\{ v_{n} \} \) are bounded, and so is \(\{ d_{n} \} \). Therefore, (3.2) reduces to
Applying (3.3) and Karamata regular variation theory, we get
Combining (3.4) with (3.2), we obtain
Using the convexity of the norm and (3.5), we deduce that
which implies that
Since
we have the following result:
Applying the property of the projection \(P_{S_{i}}\), one can easily show that
where \(M>0\) satisfying
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
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.
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 \((\vert \vec{\beta } \vert +n-1)p>\alpha +n\).
So
which yields
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
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
Since
and
We infer that
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
It follows from (3.5) and (3.7) that
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
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
Moreover,
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
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
Indeed,
After taking a weighted Ostrowski type inequality (see [15–17]), we have
So
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 \(\aleph^{p}_{\alpha }\).
To show that (3.10) fails, there exists a nonnegative integer k large enough such
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.
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. □
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.
References
Xue, G, Yuzbasi, E: Fixed point theorems for solutions of the stationary Schrödinger equation on cones. Fixed Point Theory Appl. 2015, 34 (2015)
Chu, T, Lin, Y: Improved extensions of the TOPSIS for group decision-making under fuzzy environment. J. Inf. Optim. Sci. 23(2), 273-286 (2002)
Miyamoto, I: Harmonic functions in a cylinder which vanish on the boundary. Jpn. J. Math. 22(2), 241-255 (1996)
Çelen, A: Comparative analysis of normalization procedures in TOPSIS method: with an application to Turkish deposit banking market. Informatica 25(2), 185-208 (2014)
Chen, T, Li, Y: The extended TOPSIS method with interval-valued fuzzy sets and experimental analysis on separation measures. Adv. Fuzzy Sets Syst. 4(3), 269-292 (2009)
Jiang, H, Wang, Z, Xie, M: TOPSIS method with objective weight based on attribute measure and its application. Math. Econ. 27(2), 1-7 (2010)
Khorshid, S: Utilizing the hierarchical fuzzy TOPSIS and entropy method in a SWOT analysis. Adv. Fuzzy Sets Syst. 10(1), 1-32 (2011)
Liu, Q: An extended TOPSIS method for multiple attribute decision making problems with unknown weight based on 2-dimension uncertain linguistic variables. J. Intell. Fuzzy Syst. 27(5), 2221-2230 (2014)
Zhang, J, Wu, Q: Technique for order preference by similarity to ideal solution (TOPSIS) applied in vague sets. Trans. Beijing Inst. Techol. 26(10), 937-940 (2006)
Wang, L, Dun, C, Yang, R: Multi-objective \((Q,r)\) model based on hybrid differential evolution algorithm and entropy-based TOPSIS. Control Decis. 26(12), 1913-1916, 1920 (2011)
Yang, W, Shi, J, Pang, Y: Fuzzy multi-attribute group decision making method based on TOPSIS with partial weight information. Mohu Xitong yu Shuxue 28(2), 144-151 (2014)
Coifman, RR, Rochberg, R: Representation theorems for holomorphic and harmonic functions in \(L^{p}\). Astérisque 77, 11-66 (1980)
Gasiorowicz, S: Elementary Particle Physics. Wiley, New York (1966)
Axler, S, Bourdon, P, Ramey, W: Harmonic Function Theory. Springer, New York (1992)
Chen, Y, Yu, Y: An inequality of Ostrowski type for twice differentiable functions. Math. Pract. Theory 35(12), 188-192 (2005)
Dragomir, S, Wang, S: An inequality of Ostrowski-Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules. Comput. Math. Appl. 33(11), 15-20 (1997)
Dragomir, S, Wang, S: A new inequality of Ostrowski’s type in \(L_{1}\) norm and applications to some special means and to some numerical quadrature rules. Tamkang J. Math. 28(3), 239-244 (1997)
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).
Author information
Authors and Affiliations
Contributions
All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Wang, Y., Sun, J., Kou, L. et al. Existence and boundary behavior of weak solutions for Schrödingerean TOPSIS equations. Bound Value Probl 2018, 12 (2018). https://doi.org/10.1186/s13661-018-0927-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-018-0927-9