Abstract
The main aim of this paper is to study and establish some new coincidence point and common fixed point theorems for solutions of the stationary Schrödinger equation on cones. An interesting application is to investigate the existence and uniqueness for solutions of the Dirichlet problem with respect to the Schrödinger operator on cones and the growth property of them.
Similar content being viewed by others
1 Introduction and main results
Let R and \(\mathbf{R}_{+}\) be the set of all real numbers and the set of all positive real numbers, respectively. We denote by \(\mathbf{R}^{n}\) (\(n\geq2\)) the n-dimensional Euclidean space. A point in \(\mathbf{R}^{n}\) is denoted by \(P=(X,x_{n})\), \(X=(x_{1},x_{2},\ldots,x_{n-1})\). The Euclidean distance of two points P and Q in \(\mathbf{R}^{n}\) is denoted by \(|P-Q|\). Also \(|P-O|\) with the origin O of \(\mathbf{R}^{n}\) is simply denoted by \(|P|\). The boundary, the closure, and the complement of a set S in \(\mathbf{R}^{n}\) are denoted by ∂S, \(\overline{\mathbf{S}}\), and \(\mathbf{S}^{c}\), respectively.
For \(P\in\mathbf{R}^{n}\) and \(r>0\), let \(B(P,r)\) denote the open ball with center at P and radius r in \(\mathbf{R}^{n}\).
We introduce a system of spherical coordinates \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \(\mathbf{R}^{n}\) which are related to cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).
The unit sphere and the upper half unit sphere in \(\mathbf{R}^{n}\) are denoted by \(\mathbf{S}^{n-1}\) and \(\mathbf{S}_{+}^{n-1}\), respectively. For simplicity, a point \((1,\Theta)\) on \(\mathbf{S}^{n-1}\) and the set \(\{\Theta; (1,\Theta)\in\Omega\}\) for a set Ω, \(\Omega\subset\mathbf{S}^{n-1}\), are often identified with Θ and Ω, respectively. By \(C_{n}(\Omega)\), we denote the set \(\mathbf{R}_{+}\times\Omega\) in \(\mathbf{R}^{n}\) with the domain Ω on \(\mathbf{S}^{n-1}\) (\(n\geq2\)). We call it a cone. Then \(T_{n}\) is a special cone obtained by putting \(\Omega=\mathbf{S}_{+}^{n-1}\). We denote the sets \(I\times\Omega\) and \(I\times\partial{\Omega}\) with an interval on R by \(C_{n}(\Omega;I)\) and \(S_{n}(\Omega;I)\). By \(S_{n}(\Omega; r)\) we denote \(C_{n}(\Omega)\cap S_{r}\). By \(S_{n}(\Omega)\) we denote \(S_{n}(\Omega; (0,+\infty))\), which is \(\partial{C_{n}(\Omega)}-\{O\}\).
We shall say that a set \(E\subset C_{n}(\Omega)\) has a covering \(\{r_{j}, R_{j}\}\) if there exists a sequence of balls \(\{B_{j}\}\) with centers in \(C_{n}(\Omega)\) such that \(E\subset\bigcup_{j=0}^{\infty} B_{j}\), where \(r_{j}\) is the radius of \(B_{j}\) and \(R_{j}\) is the distance from the origin to the center of \(B_{j}\).
Let \(C_{n}(\Omega)\) be an arbitrary domain in \(\mathbf{R}^{n}\) and denote the class of nonnegative radial potentials \(a(P)\), i.e. \(0\leq a(P)=a(r)\), \(P=(r,\Theta)\in C_{n}(\Omega)\), such that \(a\in L_{\mathrm{loc}}^{b}(C_{n}(\Omega))\) with some \(b> {n}/{2}\) if \(n\geq4\) and with \(b=2\) if \(n=2\) or \(n=3\).
This article is devoted to the stationary Schrödinger equation
where Δ is the Laplace operator and . The class of these solution is denoted by \(H(a,\Omega)\). Note that they are the (classical) harmonic functions on cones in the case \(a=0\). Under these assumptions the operator \(\operatorname{Sch}_{a}\) can be extended in the usual way from the space \(C_{0}^{\infty}(C_{n}(\Omega))\) to an essentially self-adjoint operator on \(L^{2}(C_{n}(\Omega))\) (see [1], Chapter 13). We will denote it \(\operatorname{Sch}_{a}\) as well. The latter has a Green-Sch function \(G_{\Omega}^{a}(P,Q)\). Here \(G_{\Omega}^{a}(P,Q)\) is positive on \(C_{n}(\Omega)\) and its inner normal derivative \(\partial G_{\Omega}^{a}(P,Q)/{\partial n_{Q}}\geq0\), where \({\partial}/{\partial n_{Q}}\) denotes the differentiation at Q along the inward normal into \(C_{n}(\Omega)\). We denote this derivative by \(PI_{\Omega}^{a}(P,Q)\), which is called the Poisson-Sch kernel with respect to \(C_{n}(\Omega)\).
For positive functions \(h_{1}\) and \(h_{2}\), we say that \(h_{1}\lesssim h_{2}\) if \(h_{1}\leq Mh_{2}\) for some constant \(M>0\). If \(h_{1}\lesssim h_{2}\) and \(h_{2}\lesssim h_{1}\), we say that \(h_{1}\approx h_{2}\).
Let Ω be a domain on \(\mathbf{S}^{n-1}\) with smooth boundary. Consider the Dirichlet problem
where \(\Lambda_{n}\) is the spherical part of the Laplace opera \(\Delta_{n}\)
We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by \(\varphi(\Theta)\), \(\int_{\Omega}\varphi^{2}(\Theta)\,dS_{1}=1\). In order to ensure the existence of λ and a smooth \(\varphi(\Theta)\). We put a rather strong assumption on Ω: if \(n\geq3\), then Ω is a \(C^{2,\alpha}\)-domain (\(0<\alpha<1\)) on \(\mathbf{S}^{n-1}\) surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [2], pp.88-89, for the definition of a \(C^{2,\alpha}\)-domain).
For any \((1,\Theta)\in\Omega\), we have (see [3], pp.7-8)
which yields
where \(P=(r,\Theta)\in C_{n}(\Omega)\) and \(\delta(P)=\operatorname{dist}(P,\partial{C_{n}(\Omega)})\).
We consider solutions of an ordinary differential equation
It is well known (see, for example, [4]) that if the potential , then (1.2) has a fundamental system of positive solutions \(\{V,W\}\) such that V is nondecreasing with (see [5–7])
and W is monotonically decreasing with
We will also consider the class , consisting of the potentials such that there the finite limit \(\lim_{r\rightarrow\infty}r^{2} a(r)=k\in[0,\infty)\) exists, and moreover, \(r^{-1}|r^{2} a(r)-k|\in L(1,\infty)\). If , then the (sub-) superfunctions are continuous (see [8]).
In the rest of paper, we assume that and we shall suppress denotation of this assumption for simplicity.
Denote
then the solutions to (1.2) have the asymptotic (see [9])
We denote the Green-Sch potential with a positive measure v on \(C_{n}(\Omega)\) by
The Poisson-Sch integral \(PI_{\Omega}^{a} \mu(P)\ (\mbox{resp. }PI_{\Omega}^{a}[g](P)) \not\equiv+\infty\) (\(P\in C_{n}(\Omega)\)) of μ (resp. g) on \(C_{n}(\Omega)\) is defined as follows:
where
μ is a positive measure on \(\partial{C_{n}(\Omega)}\) (resp. g is a continuous function on \(\partial{C_{n}(\Omega)}\) and \(d\sigma_{Q}\) is the surface area element on \(S_{n}(\Omega)\)) and \({\partial}/{\partial n_{Q}}\) denotes the differentiation at Q along the inward normal into \(C_{n}(\Omega)\).
We define the positive measure \(\mu'\) on \(\mathbf{R}^{n}\) by
Remark 1
If \(d\mu(Q)=|g(Q)|\,d\sigma_{Q}\) (\(Q=(t,\Phi)\in S_{n}(\Omega)\)), where \(g(Q)\) is a continuous function on \(\partial{C_{n}(\Omega)}\), then we have (see [10, 11])
Let \(\epsilon>0\), \(0\leq\alpha\leq n\), and λ be any positive measure on \(\mathbf{R}^{n}\) having finite total mass. For each \(P=(r,\Theta)\in\mathbf{R}^{n}-\{O\}\), the maximal function \(M(P;\lambda,\alpha)\) is defined by (see [12–15])
The set
is denoted by \(E(\epsilon; \lambda, \alpha)\).
As on cones, Qiao [16], Corollaries 2.1 and 2.2, have proved the following result. For similar results, we refer the reader to papers by Qiao and Deng (see [17, 18]).
Theorem A
Let g be a continuous function on \(\partial{C_{n}(\Omega)}\) satisfying
Then \(PI_{\Omega}^{0}[g](P)\in H(0,\Omega)\) and
Theorem B
Letgbe a continuous function on \(\partial{C_{n}(\Omega)}\)satisfying (1.4). Then the function \(PI_{\Omega}^{0}[g](P)\) (\(P=(r,\Theta)\)) satisfies
and (1.5) holds.
Now we state our first result.
Theorem 1
Letϵbe a sufficiently small positive number andμbe a positive measure on \(\partial{C_{n}(\Omega)}\)such that
Then there exists a covering \(\{r_{j},R_{j}\}\)of \(E(\epsilon; \mu',n-\alpha)\) (\(\subset C_{n}(\Omega)\)) satisfying
such that
Corollary 1
Letμbe a positive measure on \(S_{n}(\Omega)\)such that \(PI_{\Omega}^{a} \mu(P)\not\equiv+\infty \) (\(P\in C_{n}(\Omega)\)). Then for a sufficiently largeLand a sufficiently smallϵwe have
From (1.3) and Remark 1 we know that the following result generalizes Theorem A in the case \(d\mu(Q)=|g(Q)|\,d\sigma_{Q}\).
Corollary 2
Let g be a continuous function on \(\partial{C_{n}(\Omega)}\) satisfying
Then \(PI_{\Omega}^{a} \mu(P)\in H(a,\Omega)\) and
Our next aim is concerned with the solutions of the Dirichlet problem for the Schrödinger operator \(\operatorname{Sch}_{a}\) on \(C_{n}(\Omega)\) and the growth property of them.
Theorem 2
Letα, ϵbe defined as in Theorem 1andgbe a continuous function on \(\partial{C_{n}(\Omega)}\)satisfying
where \(d_{\sigma_{\Phi}}\)is the surface area element of∂Ω at \(\Phi\in\partial{\Omega}\). Then the function \(PI_{\Omega}^{a}[g](P)\) (\(P=(r,\Theta)\)) satisfies
and there exists a covering \(\{r_{j},R_{j}\}\)of \(E(\epsilon; \mu'',\alpha)\)satisfying (1.5) such that
Remark 2
In the case \(a=0\), (1.8) is equivalent to (1.4) from (1.3). In the case \(\alpha=n\), (1.6) is a finite sum, then the set \(E(\epsilon; \mu'',0)\) is a bounded set and (1.9) holds in \(C_{n}(\Omega)\), which generalizes Theorem B.
2 Some lemmas
Lemma 1
(see [1], p.354)
for any \(P=(r,\Theta)\in C_{n}(\Omega)\)and any \(Q=(t,\Phi)\in S_{n}(\Omega)\)satisfying \(0<\frac{t}{r}\leq\frac{4}{5}\) (resp. \(0<\frac{r}{t}\leq\frac{4}{5}\));
for any \(P=(r,\Theta)\in C_{n}(\Omega)\)and any \(Q=(t,\Phi)\in S_{n}(\Omega; (\frac{4}{5}r,\frac{5}{4}r))\).
Lemma 2
Letμbe a positive measure on \(S_{n}(\Omega)\)such that there is a sequence of points \(P_{i}=(r_{i},\Theta_{i})\in C_{n}(\Omega)\), \(r_{i}\rightarrow +\infty \) (\(i\rightarrow+\infty\)) satisfying \(PI_{\Omega}^{a}\mu(P_{i})<+\infty\) (\(i=1,2,\ldots\)). Then for a positive numberl,
and
Proof
Take a positive number l satisfying \(P_{1}=(r_{1},\Theta_{1})\in C_{n}(\Omega)\), \(r_{1}\leq\frac{4}{5}l\). Then from (2.2), we have
which gives (2.4). For any positive number ϵ, from (2.4), we can take a number \(R_{\epsilon}\) such that
If we take a point \(P_{i}=(r_{i},\Theta_{i})\in C_{n}(\Omega)\), \(r_{i}\geq \frac{5}{4}R_{\epsilon}\), then we have from (2.1)
If R (\(R>R_{\epsilon}\)) is sufficiently large, then
which gives (2.5). □
Lemma 3
Let \(\epsilon>0\), \(0\leq\alpha\leq n\)andλbe any positive measure on \(\mathbf{R}^{n}\)having a finite total mass. Then \(E(\epsilon; \lambda, \alpha)\)has a covering \(\{r_{j},R_{j}\}\) (\(j=1,2,\ldots\)) satisfying
Proof
Set
If \(P=(r,\Theta)\in E_{j}(\epsilon; \lambda, \beta)\), then there exists a positive number \(\rho(P)\) such that
Since \(E_{j}(\epsilon; \lambda, \beta)\) can be covered by the union of a family of balls \(\{B(P_{j,i},\rho_{j,i}):P_{j,i}\in E_{k}(\epsilon; \lambda, \beta)\}\) (\(\rho_{j,i}=\rho(P_{j,i})\)). By the Vitali lemma (see [19]), there exists \(\Lambda_{j} \subset E_{j}(\epsilon; \lambda, \beta)\), which is at most countable, such that \(\{B(P_{j,i},\rho_{j,i}):P_{j,i}\in\Lambda_{j} \}\) are disjoint and \(E_{j}(\epsilon; \lambda, \beta) \subset \bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i},5\rho_{j,i})\).
Therefore
On the other hand, note that \(\bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i},\rho_{j,i}) \subset\{P=(r,\Theta):2^{j-1}\leq r<2^{j+2}\} \), so that
Hence we obtain
Since \(E(\epsilon; \lambda, \beta)\cap\{P=(r,\Theta)\in\mathbf{R}^{n}; r\geq4\}=\bigcup_{j=2}^{\infty}E_{j}(\epsilon;\lambda, \beta)\). Then \(E(\epsilon; \lambda, \beta)\) is finally covered by a sequence of balls \(\{B(P_{j,i},\rho_{j,i}), B(P_{1},6)\}\) (\(j=2,3,\ldots\) ; \(i=1,2,\ldots\)) satisfying
where \(B(P_{1},6)\) (\(P_{1}=(1,0,\ldots,0)\in\mathbf{R}^{n}\)) is the ball which covers \(\{P=(r,\Theta)\in\mathbf{R}^{n}; r<4\}\). □
3 Proof of Theorem 1
Take any point \(P=(r,\Theta)\in C_{n}(\Omega; (R,+\infty))-E(\epsilon; \mu', \alpha)\), where R (\(\leq\frac{4}{5}r\)) is a sufficiently large number and ϵ is a sufficiently small positive number.
Write
where
and
The relation \(G_{\Omega}^{a}(P,Q)\leq G_{\Omega}^{0}(P,Q)\) implies this inequality (see [20])
By (2.1), (2.2), and Lemma 2, we have the following growth estimates:
where
and
We first have
from Lemma 2.
Next, we shall estimate \(PI_{\Omega}^{a}(22)(P)\). Take a sufficiently small positive number c such that \(S_{n}(\Omega;(\frac{4}{5}r,\frac{5}{4}r))\subset B(P,\frac{1}{2}r)\) for any \(P=(r,\Theta)\in\Lambda(c)\), where
and divide \(C_{n}(\Omega)\) into two sets \(\Lambda(c)\) and \(C_{n}(\Omega)-\Lambda(c)\).
If \(P=(r,\Theta)\in C_{n}(\Omega)-\Lambda(c)\), then there exists a positive \(c'\) such that \(|P-Q|\geq c'r\) for any \(Q\in S_{n}(\Omega)\), and hence
from Lemma 2.
We shall consider the case \(P\in\Lambda(c)\). Now put
Since \(S_{n}(\Omega)\cap\{Q\in\mathbf{R}^{n}: |P-Q|< \delta(P)\}=\varnothing\), we have
where \(i(P)\) is a positive integer satisfying \(2^{i(P)-1}\delta(P)\leq\frac{r}{2}<2^{i(P)}\delta(P)\).
By (1.1) we have \(r\varphi(\Theta)\lesssim\delta(P)\) (\(P=(r,\Theta)\in C_{n}(\Omega)\)), and hence
for \(i=0,1,2,\ldots,i(P)\).
Since \(P=(r,\Theta)\notin E(\epsilon; \mu', \alpha)\), we have from (1.3)
and
So
Combining (3.2)-(3.6), we finally find that if L is sufficiently large and ϵ is sufficiently small, then \(PI_{\Omega}^{a}\mu(P)=o(V(r)\varphi^{1-\alpha}(\Theta))\) as \(r\rightarrow\infty\), where \(P=(r,\Theta)\in C_{n}(\Omega; (R,+\infty))-E(\epsilon; \mu', \alpha)\). Finally, there exists an additional finite ball \(B_{0}\) covering \(C_{n}(\Omega; (0,R])\), which, together with Lemma 3, gives the conclusion of Theorem 1.
4 Proof of Theorem 2
For any fixed \(P=(r,\Theta)\in C_{n}(\Omega)\), take a number R satisfying \(R>\max(1,\frac{5}{4}r)\). By (1.7) and (2.2), we have
Thus \(PI_{\Omega}^{a}[g](P)\) is finite for any \(P\in C_{n}(\Omega)\). Since \(PI_{\Omega}^{a}(P,Q)\in H(a,\Omega)\in H(a,\Omega)\) for any \(Q\in S_{n}(\Omega)\), \(PI_{\Omega}^{a}[g](P)\in H(a,\Omega)\).
Now we study the boundary behavior of \(PI_{\Omega}^{a}[g](P)\). Let \(Q'=(t',\Phi')\in\partial{C_{n}(\Omega)}\) be any fixed point and L be any positive number such that \(L>\max\{t'+1,\frac{4}{5}R\}\).
Set \(\chi_{S(L)}\) is the characteristic function of \(S(L)=\{Q=(t,\Phi)\in\partial{C_{n}(\Omega)},t\leq L\}\) and write
where
and
Notice that \(PI_{\Omega}^{a}(1)[g](P)\) is the Poisson-Sch integral of \(g(Q)\chi_{S(\frac{5}{4}L)}\), we have
Since \(\lim_{\Theta\rightarrow\Phi'}\varphi(\Theta)=0\), \(PI_{\Omega}^{a}(2)[g](P)=O(V(r)\varphi(\Theta))\), and therefore tends to zero. So the function \(PI_{\Omega}^{a}[g](P)\) can be continuously extended to \(\overline{C_{n}(\Omega)}\) such that
for any \(Q'=(t',\Phi')\in \partial{C_{n}(\Omega)}\) from the arbitrariness of L. Further, (1.9) is the conclusion of Theorem 1. Thus we complete the proof of Theorem 2.
Change history
22 January 2020
The Editors-in-Chief have retracted this article [1] because the results presented are invalid. The article also shows significant overlap with a number of previously published articles [2–5] and evidence of both peer review and authorship manipulation.
The authors have not responded to any correspondence regarding this retraction.
References
Levin, B, Kheyfits, A: Asymptotic behavior of subfunctions of time-independent Schrödinger operator. In: Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis, Chapter 11, pp. 323-397. Science Press, Beijing (2008)
Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977)
Miranda, C: Partial Differential Equations of Elliptic Type. Springer, Berlin (1970)
Verzhbinskii, GM, Maz’ya, VG: Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary. I. Sib. Mat. Zh. 12(2), 874-899 (1971)
Xue, GX: A remark on the a-minimally thin sets associated with the Schrödinger operator. Bound. Value Probl. 2014, 133 (2014)
Xue, GX: Rarefied sets at infinity associated with the Schrödinger operator. J. Inequal. Appl. 2014, 247 (2014)
Yoshida, H: Nevanlinna norm of a subharmonic function on a cone or on a cylinder. Proc. Lond. Math. Soc. (3) 54(2), 267-299 (1987)
Simon, B: Schrödinger semigroups. Bull. Am. Math. Soc. 7(2), 447-526 (1982)
Hartman, P: Ordinary Differential Equations. Wiley, New York (1964)
Miyamoto, I, Yoshida, H: On a-minimally thin sets at infinity in a cone. Hiroshima Math. J. 37(1), 61-80 (2007)
Miyamoto, I, Yoshida, H: On harmonic majorization of the Martin function at infinity in a cone. Czechoslov. Math. J. 55(4), 1041-1054 (2005)
Qiao, L, Deng, GT: Integral representations of harmonic functions in a cone. Sci. Sin., Math. 41(6), 535-546 (2011) (in Chinese)
Qiao, L, Pan, GS: Generalization of the Phragmén-Lindelöf theorems for subfunctions. Int. J. Math. 24(8), 1350062 (2013)
Qiao, L, Ren, YD: Integral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone. Monatshefte Math. 173(4), 593-603 (2014)
Qiao, L, Zhao, T: Boundary limits for fractional Poisson a-extensions of \(L^{p}\) boundary function in a cone. Pac. J. Math. 272(1), 227-236 (2014)
Qiao, L: Integral representations for harmonic functions of infinite order in a cone. Results Math. 61(4), 63-74 (2012)
Qiao, L, Deng, GT: A lower bound of harmonic functions in a cone and its application. Sci. Sin., Math. 44(6), 671-684 (2014) (in Chinese)
Qiao, L, Deng, GT: The Riesz decomposition theorem for superharmonic functions in a cone and its application. Sci. Sin., Math. 44(12), 1247-1256 (2014) (in Chinese)
Stein, EM: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Ancona, A: First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains. J. Anal. Math. 72(3), 45-92 (1997)
Acknowledgements
The last half of this work was done while the second author stayed at Istanbul University during the program ‘Nonlinear Partial Differential Equations’. We are grateful to Istanbul University and the steering committee of the program. The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions, which helped to improve the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
The Editors-in-Chief have retracted this article [1] because the results presented are invalid. The article also shows significant overlap with a number of previously published articles [2-5] and evidence of both peer review and authorship manipulation. The authors have not responded to any correspondence with regards to this retraction.
[1] Xue, G. & Yuzbasi, E. Fixed Point Theory Appl (2015) 2015: 34. https://doi.org/10.1186/s13663-015-0275-8
[2] Lei Qiao, Guan-Tie Deng Taiwanese J. Math. Volume 15, Number 5 (2011), 2213-2233 https://projecteuclid.org/euclid.twjm/1500406431
[3] Qiao, L. and Deng, G. (2012), Integral representation for the solution of the stationary Schrödinger equation in a cone. Math. Nachr., 285: 2029-2038. doi:10.1002/mana.201100251
[4] Lei Qiao, Bai-Yun Su, Guan-Tie Deng, Taiwanese J. Math. Volume 16, Number 5 (2012), 1733-1748 https://projecteuclid.org/euclid.twjm/1500406793
[5] Li, Z. & Ychussie, B. Fixed Point Theory Appl (2015) 2015: 89. https://doi.org/10.1186/s13663-015-0342-1
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Xue, G., Yuzbasi, E. RETRACTED ARTICLE: Fixed point theorems for solutions of the stationary Schrödinger equation on cones. Fixed Point Theory Appl 2015, 34 (2015). https://doi.org/10.1186/s13663-015-0275-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13663-015-0275-8