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Journal articles on the topic 'Liouville theorems'

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Published: 1 April 2023

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1

Korniłowicz, Artur, Adam Naumowicz, and Adam Grabowski. "All Liouville Numbers are Transcendental." Formalized Mathematics 25, no. 1 (March 28, 2017): 49–54. http://dx.doi.org/10.1515/forma-2017-0004.

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Summary In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and π [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville’s theorem on Diophantine approximation.

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2

Glagoleva, R. Ya. "Phragmen-Liouville-type theorems and Liouville theorems for a linear parabolic equation." Mathematical Notes of the Academy of Sciences of the USSR 37, no. 1 (January 1985): 67–70. http://dx.doi.org/10.1007/bf01652519.

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3

Tuna, Hüseyin, and Aytekin Eryilmaz. "Livšic’s theorem for q-Sturm—Liouville operators." Studia Scientiarum Mathematicarum Hungarica 53, no. 4 (December 2016): 512–24. http://dx.doi.org/10.1556/012.2016.53.4.1348.

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In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livšic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.

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4

Araya, Ataklti, and Ahmed Mohammed. "On Cauchy–Liouville-type theorems." Advances in Nonlinear Analysis 8, no. 1 (August 24, 2017): 725–42. http://dx.doi.org/10.1515/anona-2017-0158.

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Abstract In this paper we explore Liouville-type theorems to solutions of PDEs involving the ϕ-Laplace operator in the setting of Orlicz–Sobolev spaces. Our results extend Liouville-type theorems that have been obtained recently.

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5

Bear, H. S. "Liouville theorems for heat functions." Communications in Partial Differential Equations 11, no. 14 (January 1986): 1605–25. http://dx.doi.org/10.1080/03605308608820476.

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6

Jin, Zhiren. "Liouville theorems for harmonic maps." Inventiones Mathematicae 108, no. 1 (December 1992): 1–10. http://dx.doi.org/10.1007/bf02100594.

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7

Awadalla, Muath, Muthaiah Subramanian, Kinda Abuasbeh, and Murugesan Manigandan. "On the Generalized Liouville–Caputo Type Fractional Differential Equations Supplemented with Katugampola Integral Boundary Conditions." Symmetry 14, no. 11 (October 29, 2022): 2273. http://dx.doi.org/10.3390/sym14112273.

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In this study, we examine the existence and Hyers–Ulam stability of a coupled system of generalized Liouville–Caputo fractional order differential equations with integral boundary conditions and a connection to Katugampola integrals. In the first and third theorems, the Leray–Schauder alternative and Krasnoselskii’s fixed point theorem are used to demonstrate the existence of a solution. The Banach fixed point theorem’s concept of contraction mapping is used in the second theorem to emphasise the analysis of uniqueness, and the results for Hyers–Ulam stability are established in the next theorem. We establish the stability of Ulam–Hyers using conventional functional analysis. Finally, examples are used to support the results. When a generalized Liouville–Caputo (ρ) parameter is modified, asymmetric results are obtained. This study presents novel results that significantly contribute to the literature on this topic.

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8

Chen, Wenxiong, and Leyun Wu. "Liouville Theorems for Fractional Parabolic Equations." Advanced Nonlinear Studies 21, no. 4 (October 14, 2021): 939–58. http://dx.doi.org/10.1515/ans-2021-2148.

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Abstract In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space R n - 1 × R \mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.

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9

Gol'dshtein, Vladimir, and Alexander Ukhlov. "On the functional properties of weak (p,q)-quasiconformal homeomorphisms." Ukrainian Mathematical Bulletin 16, no. 3 (October 21, 2019): 329–44. http://dx.doi.org/10.37069/1810-3200-2019-16-3-2.

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We study the functional properties of weak (p,q)-quasiconformal homeomorphisms such as Liouville-type theorems, the global integrability, and the Hölder continuity. The proof of Liouville-type theorems is based on the duality property of weak (p,q)-quasiconformal homeomorphisms.

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10

Zhu, Meijun. "Liouville theorems on some indefinite equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 3 (1999): 649–61. http://dx.doi.org/10.1017/s0308210500021569.

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11

Gao, Liu, Lingen Lu, and Guilin Yang. "Liouville theorems of subelliptic harmonic maps." Annals of Global Analysis and Geometry 61, no. 2 (November 22, 2021): 293–307. http://dx.doi.org/10.1007/s10455-021-09811-3.

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12

Kaneko, Hiroshi. "Liouville theorems based on symmetric diffusions." Bulletin de la Société mathématique de France 124, no. 4 (1996): 545–57. http://dx.doi.org/10.24033/bsmf.2292.

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13

Wang, Minqiu, and Songting Yin. "Some Liouville Theorems on Finsler Manifolds." Mathematics 7, no. 4 (April 15, 2019): 351. http://dx.doi.org/10.3390/math7040351.

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We give some Liouville type theorems of L p harmonic (resp. subharmonic, superharmonic) functions on a complete noncompact Finsler manifold. Using the geometric relationship between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility. These improve the recent literature (Zhang and Xia, 2014).

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14

Caristi, G., L. D’Ambrosio, and E. Mitidieri. "Liouville theorems for some nonlinear inequalities." Proceedings of the Steklov Institute of Mathematics 260, no. 1 (April 2008): 90–111. http://dx.doi.org/10.1134/s0081543808010070.

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15

Chen, Q., J. Jost, and G. Wang. "Liouville theorems for Dirac-harmonic maps." Journal of Mathematical Physics 48, no. 11 (November 2007): 113517. http://dx.doi.org/10.1063/1.2809266.

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16

D’Ambrosio, Lorenzo. "Liouville theorems for anisotropic quasilinear inequalities." Nonlinear Analysis: Theory, Methods & Applications 70, no. 8 (April 2009): 2855–69. http://dx.doi.org/10.1016/j.na.2008.12.028.

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17

Zhu, Xiangrong, and Meng Wang. "Liouville theorems for quasi-harmonic functions." Nonlinear Analysis: Theory, Methods & Applications 73, no. 9 (November 2010): 2890–96. http://dx.doi.org/10.1016/j.na.2010.06.045.

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18

Zhuo, Ran, and FengQuan Li. "Liouville type theorems for Schrödinger systems." Science China Mathematics 58, no. 1 (November 21, 2014): 179–96. http://dx.doi.org/10.1007/s11425-014-4925-9.

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19

Priola, Enrico, and Jerzy Zabczyk. "Liouville theorems for non-local operators." Journal of Functional Analysis 216, no. 2 (November 2004): 455–90. http://dx.doi.org/10.1016/j.jfa.2004.04.001.

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20

Mikeš, Josef, Vladimir Rovenski, and Sergey Stepanov. "A Contribution of Liouville-Type Theorems to the Geometry in the Large of Hadamard Manifolds." Mathematics 10, no. 16 (August 11, 2022): 2880. http://dx.doi.org/10.3390/math10162880.

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A complete, simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. In this article, we prove Liouville-type theorems for isometric and harmonic self-diffeomorphisms of Hadamard manifolds, as well as Liouville-type theorems for Killing–Yano, symmetric Killing and harmonic tensors on Hadamard manifolds.

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21

KOU, CHUNHAI, HUACHENG ZHOU, and CHANGPIN LI. "EXISTENCE AND CONTINUATION THEOREMS OF RIEMANN–LIOUVILLE TYPE FRACTIONAL DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 22, no. 04 (April 2012): 1250077. http://dx.doi.org/10.1142/s0218127412500770.

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In this paper we study the existence and continuation of solution to the general fractional differential equation (FDE) with Riemann–Liouville derivative. If no confusion appears, we call FDE for brevity. We firstly establish a new local existence theorem. Then, we derive the continuation theorems for the general FDE, which can be regarded as a generalization of the continuation theorems of the ordinary differential equation (ODE). Such continuation theorems for FDE which are first obtained are different from those for the classical ODE. With the help of continuation theorems derived in this paper, several global existence results for FDE are constructed. Some illustrative examples are also given to verify the theoretical results.

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22

Duong, Anh Tuan, and Quoc Hung Phan. "Optimal Liouville-type theorems for a system of parabolic inequalities." Communications in Contemporary Mathematics 22, no. 06 (May 27, 2019): 1950043. http://dx.doi.org/10.1142/s0219199719500433.

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We establish optimal Liouville-type theorems for the system of parabolic inequalities [Formula: see text] and for the scalar inequality [Formula: see text] in the whole space [Formula: see text] and in [Formula: see text]. Our optimal Liouville-type theorems are proved for two different classes of solutions: the nontrivial nonnegative and the positive.

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23

Cai, Guocai, Hongjing Pan, and Ruixiang Xing. "A Note on Parabolic Liouville Theorems and Blow-Up Rates for a Higher-Order Semilinear Parabolic System." International Journal of Differential Equations 2011 (2011): 1–9. http://dx.doi.org/10.1155/2011/896427.

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We improve some results of Pan and Xing (2008) and extend the exponent range in Liouville-type theorems for some parabolic systems of inequalities with the time variable onR. As an immediate application of the parabolic Liouville-type theorems, the range of the exponent in blow-up rates for the corresponding systems is also improved.

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24

Wang, Lei, and Meijun Zhu. "Liouville theorems on the upper half space." Discrete & Continuous Dynamical Systems - A 40, no. 9 (2020): 5373–81. http://dx.doi.org/10.3934/dcds.2020231.

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25

García, A. G., and M. A. Hernández-medina. "Sampling theorems and difference sturm—liouville problems." Journal of Difference Equations and Applications 6, no. 6 (January 2000): 695–717. http://dx.doi.org/10.1080/10236190008808253.

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26

Huynh, Nhat Vy, Phuong Le, and Dinh Phu Nguyen. "Liouville theorems for Kirchhoff equations in RN." Journal of Mathematical Physics 60, no. 6 (June 2019): 061506. http://dx.doi.org/10.1063/1.5096238.

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27

Chen, Wenxiong, Lorenzo D’Ambrosio, and Yan Li. "Some Liouville theorems for the fractional Laplacian." Nonlinear Analysis: Theory, Methods & Applications 121 (July 2015): 370–81. http://dx.doi.org/10.1016/j.na.2014.11.003.

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28

Wang, Weimin, and Li Hong. "Liouville-type theorems for semilinear elliptic systems." Nonlinear Analysis: Theory, Methods & Applications 75, no. 13 (September 2012): 5380–91. http://dx.doi.org/10.1016/j.na.2012.04.057.

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29

Bonfiglioli, Andrea, and Ermanno Lanconelli. "Liouville-type theorems for real sub-Laplacians." manuscripta mathematica 105, no. 1 (May 2001): 111–24. http://dx.doi.org/10.1007/pl00005872.

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30

Chen, Wenxiong, and Jiuyi Zhu. "Indefinite fractional elliptic problem and Liouville theorems." Journal of Differential Equations 260, no. 5 (March 2016): 4758–85. http://dx.doi.org/10.1016/j.jde.2015.11.029.

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31

Moon, Dong Joo, Huili Liu, and Seoung Dal Jung. "Liouville type theorems for p-harmonic maps." Journal of Mathematical Analysis and Applications 342, no. 1 (June 2008): 354–60. http://dx.doi.org/10.1016/j.jmaa.2007.12.018.

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32

Birindelli, Isabeau, and Enzo Mitidieri. "Liouville theorems for elliptic inequalities and applications." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 6 (1998): 1217–47. http://dx.doi.org/10.1017/s0308210500027293.

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In this paper we prove nonexistence of positive C2 solutions for systems of semilinear elliptic inequalities, for polyharmonic semilinear inequalities in cones and, under better conditions on the nonlinearity, for bounded positive solutions of elliptic semilinear equations in half spaces. Using a blow-up argument, these results allow us to prove a-priori bounds for a class of semilinear elliptic systems of equations in bounded domains.

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33

DʼAmbrosio, Lorenzo, and Enzo Mitidieri. "Liouville theorems for elliptic systems and applications." Journal of Mathematical Analysis and Applications 413, no. 1 (May 2014): 121–38. http://dx.doi.org/10.1016/j.jmaa.2013.11.052.

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34

Gu, Yi, and Lei Zhang. "Degree counting theorems for singular Liouville systems." ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA - CLASSE DI SCIENZE 21, no. 2 (December 2020): 1103–35. http://dx.doi.org/10.2422/2036-2145.201812_007.

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35

Zhang, Zhengce. "Liouville-Type Theorems for Some Integral Systems." Applied Mathematics 01, no. 02 (2010): 94–100. http://dx.doi.org/10.4236/am.2010.12012.

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36

Caristi, G., E. Mitidieri, and S. I. Pohozaev. "Some Liouville theorems for quasilinear elliptic inequalities." Doklady Mathematics 79, no. 1 (February 2009): 118–24. http://dx.doi.org/10.1134/s1064562409010360.

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37

Colding, Tobias H., and William P. Minicozzi. "Liouville theorems for harmonic sections and applications." Communications on Pure and Applied Mathematics 51, no. 2 (February 1998): 113–38. http://dx.doi.org/10.1002/(sici)1097-0312(199802)51:2<113::aid-cpa1>3.0.co;2-e.

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38

Barrett, Louis C., and Dennis N. Winslow. "Interlacing theorems for interface Sturm-Liouville systems." Journal of Mathematical Analysis and Applications 129, no. 2 (February 1988): 533–59. http://dx.doi.org/10.1016/0022-247x(88)90270-3.

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39

Harrabi, Abdellaziz. "High-order Bahri–Lions Liouville-type theorems." Annali di Matematica Pura ed Applicata (1923 -) 198, no. 5 (March 28, 2019): 1675–92. http://dx.doi.org/10.1007/s10231-019-00839-8.

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40

Atsuji, Atsushi. "The submartingale property and Liouville type theorems." manuscripta mathematica 154, no. 1-2 (December 19, 2016): 129–46. http://dx.doi.org/10.1007/s00229-016-0907-2.

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41

Klimek, Malgorzata. "Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions." Symmetry 13, no. 12 (November 28, 2021): 2265. http://dx.doi.org/10.3390/sym13122265.

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In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

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42

Zung, Pham Tien. "On Bellman-Golubov theorems for the Riemann-Liouville operators." Journal of Function Spaces and Applications 7, no. 3 (2009): 289–300. http://dx.doi.org/10.1155/2009/862572.

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43

IQBAL, SAJID, GHULAM FARID, JOSIP PEČARIĆ, and ARTION KASHURI. "Hardy-Type Inequalities for an Extension of the Riemann- Liouville Fractional Derivative Operators." Kragujevac Journal of Mathematics 45, no. 5 (2021): 797–813. http://dx.doi.org/10.46793/kgjmat2105.797i.

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In this paper we present variety of Hardy-type inequalities and their reﬁnements for an extension of Riemann-Liouville fractional derivative operators. Moreover, we use an extension of extended Riemann-Liouville fractional derivative and modiﬁed extension of Riemann-Liouville fractional derivative using convex and monotone convex functions. Furthermore, mean value theorems and n-exponential convexity of the related functionals is discussed.

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44

Adalar, İbrahi̇m, and Ahmet Sinan Ozkan. "Inverse problems for a conformable fractional Sturm–Liouville operator." Journal of Inverse and Ill-posed Problems 28, no. 6 (December 1, 2020): 775–82. http://dx.doi.org/10.1515/jiip-2019-0058.

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AbstractIn this paper, a Sturm–Liouville boundary value problem which includes conformable fractional derivatives of order α, {0<\alpha\leq 1} is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study the half-inverse problem and prove a Hochstadt–Lieberman-type theorem.

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45

Wei, Xian-Biao, Yan-Hsiou Cheng, and Yu-Ping Wang. "The Partial Inverse Spectral and Nodal Problems for Sturm–Liouville Operators on a Star-Shaped Graph." Mathematics 10, no. 21 (October 26, 2022): 3971. http://dx.doi.org/10.3390/math10213971.

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We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subset and prove uniqueness theorems from paired-dense nodal subsets in interior subintervals having a central vertex. In particular, we obtain some uniqueness theorems by replacing the information of nodal data on some fixed edge with part of the eigenvalues under some conditions.

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46

Liu, Dan, Xuejun Zhang, and Mingliang Song. "Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero." Journal of Mathematics 2021 (September 14, 2021): 1–10. http://dx.doi.org/10.1155/2021/4221459.

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We deal with the following Sturm–Liouville boundary value problem: − P t x ′ t ′ + B t x t = λ ∇ x V t , x , a.e. t ∈ 0,1 x 0 cos α − P 0 x ′ 0 sin α = 0 x 1 cos β − P 1 x ′ 1 sin β = 0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

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47

NIKOLOV, NIKOLAI, and SVILENA HRISTOVA. "Liouville type theorems on Z 2 and Z 1." Carpathian Journal of Mathematics 29, no. 2 (2013): 217–22. http://dx.doi.org/10.37193/cjm.2013.02.07.

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48

Lupaş, Alina Alb, and Georgia Irina Oros. "Differential sandwich theorems involving Riemann-Liouville fractional integral of $ q $-hypergeometric function." AIMS Mathematics 8, no. 2 (2022): 4930–43. http://dx.doi.org/10.3934/math.2023246.

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<abstract><p>The development of certain aspects of geometric function theory after incorporating fractional calculus and $ q $-calculus aspects is obvious and indisputable. The study presented in this paper follows this line of research. New results are obtained by applying means of differential subordination and superordination theories involving an operator previously defined as the Riemann-Liouville fractional integral of the $ q $-hypergeometric function. Numerous theorems are stated and proved involving the fractional $ q $-operator and differential subordinations for which the best dominants are found. Associated corollaries are given as applications of those results using particular functions as best dominants. Dual results regarding the fractional $ q $-operator and differential superordinations are also considered and theorems are proved where the best subordinants are given. Using certain functions known for their remarkable geometric properties applied in the results as best subordinant, interesting corollaries emerge. As a conclusion of the investigations done by applying the means of the two dual theories considering the fractional $ q $-operator, several sandwich-type theorems combine the subordination and superordiantion established results.</p></abstract>

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49

Alzabut, Jehad, James Viji, Velu Muthulakshmi, and Weerawat Sudsutad. "Oscillatory Behavior of a Type of Generalized Proportional Fractional Differential Equations with Forcing and Damping Terms." Mathematics 8, no. 6 (June 25, 2020): 1037. http://dx.doi.org/10.3390/math8061037.

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In this paper, we study the oscillatory behavior of solutions for a type of generalized proportional fractional differential equations with forcing and damping terms. Several oscillation criteria are established for the proposed equations in terms of Riemann-Liouville and Caputo settings. The results of this paper generalize some existing theorems in the literature. Indeed, it is shown that for particular choices of parameters, the obtained conditions in this paper reduce our theorems to some known results. Numerical examples are constructed to demonstrate the effectiveness of the our main theorems. Furthermore, we present and illustrate an example which does not satisfy the assumptions of our theorem and whose solution demonstrates nonoscillatory behavior.

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50

RIGOLI, MARCO, and ALBERTO G. SETTI. "ENERGY ESTIMATES AND LIOUVILLE THEOREMS FOR HARMONIC MAPS." International Journal of Mathematics 11, no. 03 (May 2000): 413–48. http://dx.doi.org/10.1142/s0129167x00000211.

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We obtain lower and upper energy estimates for harmonic maps between Riemannian manifolds under natural curvature conditions leading to various Liouville-type theorems. Some of the methods described may also be applied to vanishing-type problems for vector bundle-valued harmonic forms.

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