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Journal articles on the topic 'Nonlinear Liouville Theorems'

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Published: 14 March 2023

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1

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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2

Branding, Volker. "Nonlinear Dirac Equations, Monotonicity Formulas and Liouville Theorems." Communications in Mathematical Physics 372, no. 3 (November 13, 2019): 733–67. http://dx.doi.org/10.1007/s00220-019-03608-z.

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Abstract We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.

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3

Berestycki, Henri, I. Capuzzo Dolcetta, and Louis Nirenberg. "Superlinear indefinite elliptic problems and nonlinear Liouville theorems." Topological Methods in Nonlinear Analysis 4, no. 1 (September 1, 1994): 59. http://dx.doi.org/10.12775/tmna.1994.023.

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4

D'Ambrosio, Lorenzo, and Sandra Lucente. "Nonlinear Liouville theorems for Grushin and Tricomi operators." Journal of Differential Equations 193, no. 2 (September 2003): 511–41. http://dx.doi.org/10.1016/s0022-0396(03)00138-4.

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5

Phan, Quoc Hung. "Liouville-type theorems for nonlinear degenerate parabolic equation." Journal of Evolution Equations 16, no. 3 (January 7, 2016): 519–37. http://dx.doi.org/10.1007/s00028-015-0311-5.

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6

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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7

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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8

García-Melián, Jorge, Alexander Quaas, and Boyan Sirakov. "Liouville theorems for nonlinear elliptic equations in half-spaces." Journal d'Analyse Mathématique 139, no. 2 (October 2019): 559–83. http://dx.doi.org/10.1007/s11854-019-0066-y.

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9

Burgos-Pérez, M. Á., and J. García-Melián. "Liouville theorems for elliptic systems with nonlinear gradient terms." Journal of Differential Equations 265, no. 12 (December 2018): 6316–51. http://dx.doi.org/10.1016/j.jde.2018.07.034.

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10

Dung, Nguyen Thac, Pham Duc Thoan, and Nguyen Dang Tuyen. "Liouville theorems for nonlinear elliptic equations on Riemannian manifolds." Journal of Mathematical Analysis and Applications 496, no. 1 (April 2021): 124803. http://dx.doi.org/10.1016/j.jmaa.2020.124803.

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11

Luca, Rodica. "Existence and multiplicity of positive solutions for a singular Riemann-Liouville fractional differential problem." Filomat 34, no. 12 (2020): 3931–42. http://dx.doi.org/10.2298/fil2012931l.

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We investigate the existence and multiplicity of positive solutions for a nonlinear Riemann-Liouville fractional differential equation with a nonnegative singular nonlinearity, subject to Riemann-Stieltjes boundary conditions which contain fractional derivatives. In the proofs of our main results, we use an application of the Krein-Rutman theorem and some theorems from the fixed point index theory.

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12

Agarwal, Ravi P., and Rodica Luca. "Positive Solutions for a Semipositone Singular Riemann–Liouville Fractional Differential Problem." International Journal of Nonlinear Sciences and Numerical Simulation 20, no. 7-8 (November 18, 2019): 823–31. http://dx.doi.org/10.1515/ijnsns-2018-0376.

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AbstractWe study the existence of multiple positive solutions for a nonlinear singular Riemann–Liouville fractional differential equation with sign-changing nonlinearity, subject to Riemann–Stieltjes boundary conditions which contain fractional derivatives. In the proof of our main theorem, we use various height functions of the nonlinearity of equation defined on special bounded sets, and two theorems from the fixed point index theory.

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13

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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14

Damag, Faten H., Adem Kılıçman, and Awsan T. Al-Arioi. "On Hybrid Type Nonlinear Fractional Integrodifferential Equations." Mathematics 8, no. 6 (June 16, 2020): 984. http://dx.doi.org/10.3390/math8060984.

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In this paper, we introduce and investigate a hybrid type of nonlinear Riemann Liouville fractional integro-differential equations. We develop and extend previous work on such non-fractional equations, using operator theoretical techniques, and find the approximate solutions. We prove the existence as well as the uniqueness of the corresponding approximate solutions by using hybrid fixed point theorems and provide upper and lower bounds to these solutions. Furthermore, some examples are provided, in which the general claims in the main theorems are demonstrated.

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15

Hu, Liang-Gen. "Liouville-type theorems for the fourth order nonlinear elliptic equation." Journal of Differential Equations 256, no. 5 (March 2014): 1817–46. http://dx.doi.org/10.1016/j.jde.2013.12.001.

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16

Wang, Lin Feng. "Liouville theorems and gradient estimates for a nonlinear elliptic equation." Journal of Differential Equations 260, no. 1 (January 2016): 567–85. http://dx.doi.org/10.1016/j.jde.2015.09.003.

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17

Felmer, Patricio, and Alexander Quaas. "Fundamental solutions and Liouville type theorems for nonlinear integral operators." Advances in Mathematics 226, no. 3 (February 2011): 2712–38. http://dx.doi.org/10.1016/j.aim.2010.09.023.

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18

Wu, Jia-Yong. "Gradient estimates for a nonlinear parabolic equation and Liouville theorems." manuscripta mathematica 159, no. 3-4 (October 11, 2018): 511–47. http://dx.doi.org/10.1007/s00229-018-1073-5.

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19

Abdo, Mohammed S. "Qualitative Analyses of ψ-Caputo Type Fractional Integrodifferential Equations in Banach Spaces." Journal of Advances in Applied & Computational Mathematics 9 (April 28, 2022): 1–10. http://dx.doi.org/10.15377/2409-5761.2022.09.1.

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In this research paper, we develop and extend some qualitative analyses of a class of a nonlinear fractional integro-differential equation involving ψ-Caputo fractional derivative (ψ-CFD) and ψ-Riemann-Liouville fractional integral (ψ-RLFI). The existence and uniqueness theorems are obtained in Banach spaces via an equivalent fractional integral equation with the help of Banach’s fixed point theorem (B’sFPT) and Schaefer’s fixed point theorem (S’sFPT). An example explaining the main results is also constructed.

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20

Liu, Kui, Michal Fečkan, and Jinrong Wang. "Hyers–Ulam Stability and Existence of Solutions to the Generalized Liouville–Caputo Fractional Differential Equations." Symmetry 12, no. 6 (June 4, 2020): 955. http://dx.doi.org/10.3390/sym12060955.

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The aim of this paper is to study the stability of generalized Liouville–Caputo fractional differential equations in Hyers–Ulam sense. We show that three types of the generalized linear Liouville–Caputo fractional differential equations are Hyers–Ulam stable by a ρ -Laplace transform method. We establish existence and uniqueness of solutions to the Cauchy problem for the corresponding nonlinear equations with the help of fixed point theorems.

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21

Chaona, Zhu. "Gradient Estimates and Liouville-Type Theorems for a Nonlinear Elliptic Equation." Journal of Partial Differential Equations 31, no. 3 (June 2018): 237–51. http://dx.doi.org/10.4208/jpde.v31.n3.4.

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22

Quaas, A., J. García-Melián, and M. Á. Burgos-Pérez. "Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems." Discrete and Continuous Dynamical Systems 36, no. 9 (May 2016): 4703–21. http://dx.doi.org/10.3934/dcds.2016004.

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23

Birindelli, Isabeau, Giulio Galise, and Fabiana Leoni. "Liouville theorems for a family of very degenerate elliptic nonlinear operators." Nonlinear Analysis 161 (September 2017): 198–211. http://dx.doi.org/10.1016/j.na.2017.06.002.

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24

Bonfiglioli, A., and F. Uguzzoni. "Nonlinear Liouville theorems for some critical problems on H-type groups." Journal of Functional Analysis 207, no. 1 (February 2004): 161–215. http://dx.doi.org/10.1016/s0022-1236(03)00138-1.

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25

Huang, Guangyue, and Bingqing Ma. "Gradient estimates and Liouville type theorems for a nonlinear elliptic equation." Archiv der Mathematik 105, no. 5 (September 28, 2015): 491–99. http://dx.doi.org/10.1007/s00013-015-0820-z.

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26

Tan, Jinggang, and Xiaohui Yu. "Liouville type theorems for nonlinear elliptic equations on extended Grushin manifolds." Journal of Differential Equations 269, no. 1 (June 2020): 523–41. http://dx.doi.org/10.1016/j.jde.2019.12.014.

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27

Xie, Wenzhe, Jing Xiao, and Zhiguo Luo. "Existence of Solutions for Riemann-Liouville Fractional Boundary Value Problem." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/540351.

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By using the method of upper and lower solutions and fixed point theorems, the existence of solutions for a Riemann-Liouville fractional boundary value problem with the nonlinear term depending on fractional derivative of lower order is obtained under the classical Nagumo conditions. Also, some results concerning Riemann-Liouville fractional derivative at extreme points are established with weaker hypotheses, which improve some works in Al-Refai (2012). As applications, an example is presented to illustrate our main results.

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28

Li, Chenkuan, and Joshua Beaudin. "On the Nonlinear Integro-Differential Equations." Fractal and Fractional 5, no. 3 (July 30, 2021): 82. http://dx.doi.org/10.3390/fractalfract5030082.

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The goal of this paper is to study the uniqueness of solutions of several nonlinear Liouville–Caputo integro-differential equations with variable coefficients and initial conditions, as well as an associated coupled system in Banach spaces. The results derived are new and based on Banach’s contractive principle, the multivariate Mittag–Leffler function and Babenko’s approach. We also provide a few examples to demonstrate the use of our main theorems by convolutions and the gamma function.

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29

Yu, Changlong, and Jufang Wang. "Positive Solutions of Nonlocal Boundary Value Problem for High-Order Nonlinear Fractional -Difference Equations." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/928147.

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We study the nonlinear -difference equations of fractional order , , , , , where is the fractional -derivative of the Riemann-Liouville type of order , , , , and . We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems. Finally, we give examples to illustrate the results.

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30

Usami, Hiroyuki. "Applications of Riccati-type inequalities to asymptotic theory of elliptic problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 5 (September 21, 2009): 1071–89. http://dx.doi.org/10.1017/s0308210507000959.

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We show how one-dimensional generalized Riccati-type inequalities can be employed to analyse asymptotic behaviour of solutions of elliptic problems. We give Liouville-type theorems as well as necessary conditions for the existence of solutions of specified asymptotic behaviour of nonlinear elliptic problems.

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31

Fazly, Mostafa. "Liouville Type Theorems for Stable Solutions of Certain Elliptic Systems." Advanced Nonlinear Studies 12, no. 1 (January 1, 2012): 1–17. http://dx.doi.org/10.1515/ans-2012-0101.

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AbstractWe establish Liouville type theorems for elliptic systems with various classes of nonlinearities on ℝis necessarily constant, whenever the dimension N < 8 + 3α +We also consider the case of bounded domains Ω ⊂ ℝ

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32

Alsaedi, Ahmed, Amjad F. Albideewi, Sotiris K. Ntouyas, and Bashir Ahmad. "On Caputo–Riemann–Liouville Type Fractional Integro-Differential Equations with Multi-Point Sub-Strip Boundary Conditions." Mathematics 8, no. 11 (October 31, 2020): 1899. http://dx.doi.org/10.3390/math8111899.

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In this paper, we derive existence and uniqueness results for a nonlinear Caputo–Riemann–Liouville type fractional integro-differential boundary value problem with multi-point sub-strip boundary conditions, via Banach and Krasnosel’skii⏝’s fixed point theorems. Examples are included for the illustration of the obtained results.

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33

Song, Mingliang, and Shuyuan Mei. "Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations." Journal of Function Spaces 2021 (January 28, 2021): 1–17. http://dx.doi.org/10.1155/2021/6668037.

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The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

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34

Tanigawa, Tomoyuki. "Oscillation Theorems for Differential Equations Involving Even Order Nonlinear Sturm–Liouville Operator." gmj 14, no. 4 (December 2007): 737–68. http://dx.doi.org/10.1515/gmj.2007.737.

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Abstract We are concerned with the oscillatory and nonoscillatory behavior of solutions of differential equations involving an even order nonlinear Sturm–Liouville operator of the form where α and β are distinct positive constants. We first give the criteria for the existence of nonoscillatory solutions with specific asymptotic behavior on infinite intervals, and then derive necessary and sufficient conditions for all solutions of (∗) to be oscillatory by eliminating all nonoscillatory solutions of (∗).

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35

Hsini, M. "Liouville type theorems for nonlinear elliptic equations involving operator in divergence form." Journal of Mathematical Physics 53, no. 10 (October 2012): 103706. http://dx.doi.org/10.1063/1.4753979.

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36

Chen, Huyuan, and Patricio Felmer. "On Liouville type theorems for fully nonlinear elliptic equations with gradient term." Journal of Differential Equations 255, no. 8 (October 2013): 2167–95. http://dx.doi.org/10.1016/j.jde.2013.06.009.

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37

Punzo, Fabio. "Liouville theorems for fully nonlinear elliptic equations on spherically symmetric Riemannian manifolds." Nonlinear Differential Equations and Applications NoDEA 20, no. 3 (November 22, 2012): 1295–315. http://dx.doi.org/10.1007/s00030-012-0209-7.

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38

Abolarinwa, Abimbola. "Gradient estimates for a weighted nonlinear elliptic equation and Liouville type theorems." Journal of Geometry and Physics 155 (September 2020): 103737. http://dx.doi.org/10.1016/j.geomphys.2020.103737.

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39

Abolarinwa, Abimbola. "Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation." Journal of Mathematical Analysis and Applications 473, no. 1 (May 2019): 297–312. http://dx.doi.org/10.1016/j.jmaa.2018.12.049.

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40

Huang, Guangyue, and Zhi Li. "Liouville type theorems of a nonlinear elliptic equation for the V-Laplacian." Analysis and Mathematical Physics 8, no. 1 (March 1, 2017): 123–34. http://dx.doi.org/10.1007/s13324-017-0168-6.

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41

Ahmad, Bashir, Sotiris K. Ntouyas, Jessada Tariboon, and Ahmed Alsaedi. "A STUDY OF NONLINEAR FRACTIONAL-ORDER BOUNDARY VALUE PROBLEM WITH NONLOCAL ERDELYI-KOBER AND GENERALIZED RIEMANN-LIOUVILLE TYPE INTEGRAL BOUNDARY CONDITIONS." Mathematical Modelling and Analysis 22, no. 2 (March 18, 2017): 121–39. http://dx.doi.org/10.3846/13926292.2017.1274920.

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We investigate a new kind of nonlocal boundary value problems of nonlinear Caputo fractional differential equations supplemented with integral boundary conditions involving Erdelyi-Kober and generalized Riemann-Liouville fractional integrals. Existence and uniqueness results for the given problem are obtained by means of standard fixed point theorems. Examples illustrating the main results are also discussed.

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42

Samadi, Ayub, Sotiris K. Ntouyas, Bashir Ahmad, and Jessada Tariboon. "On a Coupled Differential System Involving (k,ψ)-Hilfer Derivative and (k,ψ)-Riemann–Liouville Integral Operators." Axioms 12, no. 3 (February 22, 2023): 229. http://dx.doi.org/10.3390/axioms12030229.

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We investigate a nonlinear, nonlocal, and fully coupled boundary value problem containing mixed (k,ψ^)-Hilfer fractional derivative and (k,ψ^)-Riemann–Liouville fractional integral operators. Existence and uniqueness results for the given problem are proved with the aid of standard fixed point theorems. Examples illustrating the main results are presented. The paper concludes with some interesting findings.

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43

Manigandan, M., Subramanian Muthaiah, T. Nandhagopal, R. Vadivel, B. Unyong, and N. Gunasekaran. "Existence results for coupled system of nonlinear differential equations and inclusions involving sequential derivatives of fractional order." AIMS Mathematics 7, no. 1 (2021): 723–55. http://dx.doi.org/10.3934/math.2022045.

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<abstract><p>In this article, we investigate new results of existence and uniqueness for systems of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order and along with new kinds of coupled discrete (multi-points) and fractional integral (Riemann-Liouville) boundary conditions. Our investigation is mainly based on the theorems of Schaefer, Banach, Covitz-Nadler, and nonlinear alternatives for Kakutani. The validity of the obtained results is demonstrated by numerical examples.</p></abstract>

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44

Aghajani, Asadollah, Craig Cowan, and Vicenţiu D. Rădulescu. "Positive supersolutions of fourth-order nonlinear elliptic equations: explicit estimates and Liouville theorems." Journal of Differential Equations 298 (October 2021): 323–45. http://dx.doi.org/10.1016/j.jde.2021.07.005.

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45

Zhu, Xiaobao. "Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds." Nonlinear Analysis: Theory, Methods & Applications 74, no. 15 (October 2011): 5141–46. http://dx.doi.org/10.1016/j.na.2011.05.008.

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46

Ma, Bingqing, and Fanqi Zeng. "Hamilton–Souplet–Zhang's gradient estimates and Liouville theorems for a nonlinear parabolic equation." Comptes Rendus Mathematique 356, no. 5 (May 2018): 550–57. http://dx.doi.org/10.1016/j.crma.2018.04.003.

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47

Liu, Pan. "Liouville-type theorems for the stationary compressible barotropic and incompressible inhomogeneous Navier–Stokes equations." Journal of Mathematical Physics 63, no. 12 (December 1, 2022): 123101. http://dx.doi.org/10.1063/5.0085031.

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The present paper is dedicated to the Liouville-type problem for the three-dimensional stationary compressible barotropic Navier–Stokes equations and incompressible inhomogeneous Navier–Stokes equations. We show that velocity u is trivial under some suitable assumptions stated in terms of BMO−1 spaces.

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48

Borisut, Piyachat, Poom Kumam, Idris Ahmed, and Kanokwan Sitthithakerngkiet. "Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems." Symmetry 11, no. 6 (June 22, 2019): 829. http://dx.doi.org/10.3390/sym11060829.

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In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): c D 0 + q u ( t ) = f ( t , u ( t ) ) , t ∈ [ 0 , T ] , u ( k ) ( 0 ) = ξ k , u ( T ) = ∑ i = 1 m β i R L I 0 + p i u ( η i ) , where n - 1 < q < n , n ≥ 2 , m , n ∈ N , ξ k , β i ∈ R , k = 0 , 1 , ⋯ , n - 2 , i = 1 , 2 , ⋯ , m , and c D 0 + q is the Caputo fractional derivatives, f : [ 0 , T ] × C ( [ 0 , T ] , E ) → E , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be R (with the absolute value) or C ( [ 0 , T ] , R ) with the supremum-norm. R L I 0 + p i is the Riemann–Liouville fractional integral of order p i > 0 , η i ∈ ( 0 , T ) , and ∑ i = 1 m β i η i p i + n - 1 Γ ( n ) Γ ( n + p i ) ≠ T n - 1 . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results.

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49

De Nitti, Nicola, Francis Hounkpe, and Simon Schulz. "On Liouville-type theorems for the 2D stationary MHD equations." Nonlinearity 35, no. 2 (December 21, 2021): 870–88. http://dx.doi.org/10.1088/1361-6544/ac3f8b.

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Abstract We establish new Liouville-type theorems for the two-dimensional stationary magneto-hydrodynamic incompressible system assuming that the velocity and magnetic field have bounded Dirichlet integral. The key tool in our proof is observing that the stream function associated to the magnetic field satisfies a simple drift–diffusion equation for which a maximum principle is available.

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50

Harrabi, Abdellaziz, and Belgacem Rahal. "Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions." Advances in Nonlinear Analysis 8, no. 1 (December 20, 2016): 193–202. http://dx.doi.org/10.1515/anona-2016-0168.

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Abstract In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.

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