-
Previous Article
On carrying-capacity construction, metapopulations and density-dependent mortality
- DCDS-B Home
- This Issue
-
Next Article
Advection control in parabolic PDE systems for competitive populations
Eigenvectors of homogeneous order-bounded order-preserving maps
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA |
The existence of eigenvectors associated with the cone spectral radius is shown for homogenous, order-preserving, continuous maps that have compact and order-bounded powers (iterates). The order-boundedness makes it possible to show the existence of eigenvectors for perturbations of the maps using Hilbert's projective metric, while the power compactness or similar compactness properties together with a uniform continuity condition let the eigenvectors of the perturbations converge to an eigenvector of the original map.
References:
[1] |
M. Akian, S. Gaubert and R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, arXiv: 1112.5968 [math. FA], 2012. |
[2] |
H. Bauer, Probability Theory and Elements of Measure Theory, sec. ed. , Academic Press, London, 1981. |
[3] |
G. Birkhoff,
Uniformly semi-primitive multiplicative processes, Trans. Amer. Math. Soc., 104 (1962), 37-51.
doi: 10.1090/S0002-9947-1962-0146100-6. |
[4] |
E. Bohl,
Eigenwertaufgaben bei monotonen Operatoren und Fehlerabschätzungen für Operatorgleichungen, Arch. Rat. Mech. Anal., 22 (1966), 313-332.
|
[5] |
E. Bohl, Monotonie: Lösbarkeit und Numerik bei Operatorgleichungen, Springer, Berlin Heidelberg, 1974. |
[6] |
F. F. Bonsall,
Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.
|
[7] |
R. S. Cantrell and C. Cosner,
Effects of domain size on the persistence of populations in a diffusive food-chain model with Beddington-DeAngelis functional response, Natur. Resource Modeling, 14 (2001), 335-367.
doi: 10.1111/j.1939-7445.2001.tb00062.x. |
[8] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons, Ltd. , Chichester, 2003. |
[9] |
R. S. Cantrell, C. Cosner and V. Hutson,
Permanence in ecological systems with spatial heterogeneity, Proc. Royal Soc. Edinburgh Sect. A, 123 (1993), 533-559.
doi: 10.1017/S0308210500025877. |
[10] |
R. S. Cantrell, C. Cosner and V. Hutson,
Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.
doi: 10.1216/rmjm/1181072101. |
[11] |
L. Collatz,
Einschließungssatz für die Eigenwerte von Integralgleichungen, Math. Z., 47 (1941), 395-398.
|
[12] |
L. Collatz,
Einschließungssatz für die charakteristischen Zahlen von Matrizen, Math. Z., 48 (1942), 221-226.
doi: 10.1007/BF01180013. |
[13] |
K. D. Deimling, Nonlinear Functional Analysis, Springer, Berlin Heidelberg, 1985. |
[14] |
R. M. Dudley, Real Analysis and Probability, sec. ed. , Cambridge University Press, Cambridge, 2002. |
[15] |
J. H. M. Evers, S. C. Hille and A. Muntean,
Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Differential Equations, 259 (2015), 1068-1097.
doi: 10.1016/j.jde.2015.02.037. |
[16] |
K.-H. Förster and B. Nagy,
On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator, Linear Algebra and its Applications, 120 (1980), 193-205.
doi: 10.1016/0024-3795(89)90378-9. |
[17] |
P. Gwiazda and A. Marciniak-Czochra,
Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 (2010), 733-773.
doi: 10.1142/S021989161000227X. |
[18] |
P. Gwiazda, A. Marciniak-Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space (tentative title), in preparation. |
[19] |
K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros,
Models for pair formation in bisexual populations, J. Math. Biol., 26 (1988), 635-649.
doi: 10.1007/BF00276145. |
[20] |
K. P. Hadeler,
Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102.
doi: 10.1007/BF00046676. |
[21] |
S. C. Hille and D. T. H. Worm,
Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures, Integral Equations Operator Theory, 63 (2009), 351-371.
doi: 10.1007/s00020-008-1652-z. |
[22] |
W. Jin and H. R. Thieme,
An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius, Disc. Cont. Dyn. Systems B, 21 (2016), 447-470.
doi: 10.3934/dcdsb.2016.21.447. |
[23] |
M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. |
[24] |
M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989. |
[25] |
M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin Heidelberg, 1984. |
[26] |
U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications, De Gruyter, Berlin, 2015. |
[27] |
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space (Russian), Uspehi Mat. Nauk (N. S. ), 3 (1948), 3-95, English Translation, AMS Translation, 1950 (1950), 128pp. |
[28] |
B. Lemmens, B. Lins, R. D. Nussbaum and M. Wortel, Denjoy-Wolff theorems for Hilbert's and Thompson's metric spaces, J. Analyse Math. , (to appear), arXiv: 1410.1056v2[math. DS] |
[29] |
B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge, Cambridge University Press, 2012.
doi: 10.1017/CBO9781139026079.![]() ![]() ![]() |
[30] |
B. Lemmens and R. D. Nussbaum,
Continuity of the cone spectral radius, Proc. Amer. Math. Soc., 141 (2013), 2741-2754.
doi: 10.1090/S0002-9939-2013-11520-0. |
[31] |
J. Mallet-Paret and R. D. Nussbaum},
Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discr. Cont. Dyn. Sys. (DCDS-A), 8 (2002), 519-562.
doi: 10.3934/dcds.2002.8.519. |
[32] |
J. Mallet-Paret and R. D. Nussbaum,
Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory and Appl., 7 (2010), 103-143.
doi: 10.1007/s11784-010-0010-3. |
[33] |
J. Mallet-Paret and R. D. Nussbaum,
Asymptotic fixed point theory and the beer barrel theorem, J. Fixed Point Theory Appl., 4 (2008), 203-245.
doi: 10.1007/s11784-008-0095-0. |
[34] |
H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equation, (Montecatini, 1981), Lecture Notes in Math. , Springer, Berlin, 1048 (1984), 60-110. |
[35] |
R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Fixed Point Theory, (E. Fadell and G. Fournier, eds. ), Springer, Berlin New York, 886 (1981), 309-330. |
[36] |
R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. , Providence, 75 (1988), ⅳ+137 pp. |
[37] |
R. D. Nussbaum,
Eigenvectors of order-preserving linear operators, J. London Math. Soc., 58 (1998), 480-496.
doi: 10.1112/S0024610798006425. |
[38] |
H. H. Schaefer,
Positive Transformationen in lokalkonvexen halbgeordneten Vektorräumen, Math. Ann., 129 (1955), 323-329.
doi: 10.1007/BF01362375. |
[39] |
H. H Schaefer,
Halbgeordnete lokalkonvexe Vektorräume. Ⅱ, Math. Ann., 138 (1959), 259-286.
doi: 10.1007/BF01342907. |
[40] |
H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. |
[41] |
H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps, arXiv: 1302.3905v1[math. FA], 2013. |
[42] |
H. R. Thieme, Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces, arXiv: 1406.6657v2[math. FA], 2014 (revised 2016). |
[43] |
H. R. Thieme, Spectral radii and Collatz-Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm, Ordered Structures and Applications, (tentative title), Positivity Ⅶ (Zaanen Centennial Conference) (M. de Jeu, B. de Pagter, O. van Gaans, M. Veraar, eds. ), Birkhäuser, (2016), 415-467. |
[44] |
H. R. Thieme,
Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations, J. Dynamics and Differential Equations, 28 (2016), 1115-1144.
doi: 10.1007/s10884-015-9463-9. |
[45] |
A. C. Thompson,
On the eigenvectors of some not-necessarily-linear transformations, Proc. London Math. Soc., 15 (1965), 577-598.
|
[46] |
A. J. Tromba, The beer barrel theorem, a new proof of the asymptotic conjecture in fixed point theory, Functional Differential Equations and Approximations of Fixed Points, (H. -O. Peitgen, H. -O. Walther, eds. ), 484–488, Lecture Notes in Math. 730, Springer, Berlin Heidelberg 1979. |
[47] |
H. Wielandt,
Unzerlegbare, nicht negative Matrizen, Math. Z., 52 (1950), 642-648.
doi: 10.1007/BF02230720. |
[48] |
K. Yosida, Functional Analysis, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123 Springer-Verlag New York Inc. , New York, 1968. |
[49] |
P. P. Zabreiko, M. A. Krasnosel'skii and Yu. V. Pokornyi,
On a class of linear positive operators, Functional Analysis and Its Applications, 5 (1971), 272-279.
|
show all references
References:
[1] |
M. Akian, S. Gaubert and R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, arXiv: 1112.5968 [math. FA], 2012. |
[2] |
H. Bauer, Probability Theory and Elements of Measure Theory, sec. ed. , Academic Press, London, 1981. |
[3] |
G. Birkhoff,
Uniformly semi-primitive multiplicative processes, Trans. Amer. Math. Soc., 104 (1962), 37-51.
doi: 10.1090/S0002-9947-1962-0146100-6. |
[4] |
E. Bohl,
Eigenwertaufgaben bei monotonen Operatoren und Fehlerabschätzungen für Operatorgleichungen, Arch. Rat. Mech. Anal., 22 (1966), 313-332.
|
[5] |
E. Bohl, Monotonie: Lösbarkeit und Numerik bei Operatorgleichungen, Springer, Berlin Heidelberg, 1974. |
[6] |
F. F. Bonsall,
Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.
|
[7] |
R. S. Cantrell and C. Cosner,
Effects of domain size on the persistence of populations in a diffusive food-chain model with Beddington-DeAngelis functional response, Natur. Resource Modeling, 14 (2001), 335-367.
doi: 10.1111/j.1939-7445.2001.tb00062.x. |
[8] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, John Wiley & Sons, Ltd. , Chichester, 2003. |
[9] |
R. S. Cantrell, C. Cosner and V. Hutson,
Permanence in ecological systems with spatial heterogeneity, Proc. Royal Soc. Edinburgh Sect. A, 123 (1993), 533-559.
doi: 10.1017/S0308210500025877. |
[10] |
R. S. Cantrell, C. Cosner and V. Hutson,
Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.
doi: 10.1216/rmjm/1181072101. |
[11] |
L. Collatz,
Einschließungssatz für die Eigenwerte von Integralgleichungen, Math. Z., 47 (1941), 395-398.
|
[12] |
L. Collatz,
Einschließungssatz für die charakteristischen Zahlen von Matrizen, Math. Z., 48 (1942), 221-226.
doi: 10.1007/BF01180013. |
[13] |
K. D. Deimling, Nonlinear Functional Analysis, Springer, Berlin Heidelberg, 1985. |
[14] |
R. M. Dudley, Real Analysis and Probability, sec. ed. , Cambridge University Press, Cambridge, 2002. |
[15] |
J. H. M. Evers, S. C. Hille and A. Muntean,
Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Differential Equations, 259 (2015), 1068-1097.
doi: 10.1016/j.jde.2015.02.037. |
[16] |
K.-H. Förster and B. Nagy,
On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator, Linear Algebra and its Applications, 120 (1980), 193-205.
doi: 10.1016/0024-3795(89)90378-9. |
[17] |
P. Gwiazda and A. Marciniak-Czochra,
Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 (2010), 733-773.
doi: 10.1142/S021989161000227X. |
[18] |
P. Gwiazda, A. Marciniak-Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space (tentative title), in preparation. |
[19] |
K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros,
Models for pair formation in bisexual populations, J. Math. Biol., 26 (1988), 635-649.
doi: 10.1007/BF00276145. |
[20] |
K. P. Hadeler,
Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102.
doi: 10.1007/BF00046676. |
[21] |
S. C. Hille and D. T. H. Worm,
Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures, Integral Equations Operator Theory, 63 (2009), 351-371.
doi: 10.1007/s00020-008-1652-z. |
[22] |
W. Jin and H. R. Thieme,
An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius, Disc. Cont. Dyn. Systems B, 21 (2016), 447-470.
doi: 10.3934/dcdsb.2016.21.447. |
[23] |
M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. |
[24] |
M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989. |
[25] |
M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin Heidelberg, 1984. |
[26] |
U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications, De Gruyter, Berlin, 2015. |
[27] |
M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space (Russian), Uspehi Mat. Nauk (N. S. ), 3 (1948), 3-95, English Translation, AMS Translation, 1950 (1950), 128pp. |
[28] |
B. Lemmens, B. Lins, R. D. Nussbaum and M. Wortel, Denjoy-Wolff theorems for Hilbert's and Thompson's metric spaces, J. Analyse Math. , (to appear), arXiv: 1410.1056v2[math. DS] |
[29] |
B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge, Cambridge University Press, 2012.
doi: 10.1017/CBO9781139026079.![]() ![]() ![]() |
[30] |
B. Lemmens and R. D. Nussbaum,
Continuity of the cone spectral radius, Proc. Amer. Math. Soc., 141 (2013), 2741-2754.
doi: 10.1090/S0002-9939-2013-11520-0. |
[31] |
J. Mallet-Paret and R. D. Nussbaum},
Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discr. Cont. Dyn. Sys. (DCDS-A), 8 (2002), 519-562.
doi: 10.3934/dcds.2002.8.519. |
[32] |
J. Mallet-Paret and R. D. Nussbaum,
Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory and Appl., 7 (2010), 103-143.
doi: 10.1007/s11784-010-0010-3. |
[33] |
J. Mallet-Paret and R. D. Nussbaum,
Asymptotic fixed point theory and the beer barrel theorem, J. Fixed Point Theory Appl., 4 (2008), 203-245.
doi: 10.1007/s11784-008-0095-0. |
[34] |
H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equation, (Montecatini, 1981), Lecture Notes in Math. , Springer, Berlin, 1048 (1984), 60-110. |
[35] |
R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Fixed Point Theory, (E. Fadell and G. Fournier, eds. ), Springer, Berlin New York, 886 (1981), 309-330. |
[36] |
R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. , Providence, 75 (1988), ⅳ+137 pp. |
[37] |
R. D. Nussbaum,
Eigenvectors of order-preserving linear operators, J. London Math. Soc., 58 (1998), 480-496.
doi: 10.1112/S0024610798006425. |
[38] |
H. H. Schaefer,
Positive Transformationen in lokalkonvexen halbgeordneten Vektorräumen, Math. Ann., 129 (1955), 323-329.
doi: 10.1007/BF01362375. |
[39] |
H. H Schaefer,
Halbgeordnete lokalkonvexe Vektorräume. Ⅱ, Math. Ann., 138 (1959), 259-286.
doi: 10.1007/BF01342907. |
[40] |
H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. |
[41] |
H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps, arXiv: 1302.3905v1[math. FA], 2013. |
[42] |
H. R. Thieme, Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces, arXiv: 1406.6657v2[math. FA], 2014 (revised 2016). |
[43] |
H. R. Thieme, Spectral radii and Collatz-Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm, Ordered Structures and Applications, (tentative title), Positivity Ⅶ (Zaanen Centennial Conference) (M. de Jeu, B. de Pagter, O. van Gaans, M. Veraar, eds. ), Birkhäuser, (2016), 415-467. |
[44] |
H. R. Thieme,
Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations, J. Dynamics and Differential Equations, 28 (2016), 1115-1144.
doi: 10.1007/s10884-015-9463-9. |
[45] |
A. C. Thompson,
On the eigenvectors of some not-necessarily-linear transformations, Proc. London Math. Soc., 15 (1965), 577-598.
|
[46] |
A. J. Tromba, The beer barrel theorem, a new proof of the asymptotic conjecture in fixed point theory, Functional Differential Equations and Approximations of Fixed Points, (H. -O. Peitgen, H. -O. Walther, eds. ), 484–488, Lecture Notes in Math. 730, Springer, Berlin Heidelberg 1979. |
[47] |
H. Wielandt,
Unzerlegbare, nicht negative Matrizen, Math. Z., 52 (1950), 642-648.
doi: 10.1007/BF02230720. |
[48] |
K. Yosida, Functional Analysis, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123 Springer-Verlag New York Inc. , New York, 1968. |
[49] |
P. P. Zabreiko, M. A. Krasnosel'skii and Yu. V. Pokornyi,
On a class of linear positive operators, Functional Analysis and Its Applications, 5 (1971), 272-279.
|
[1] |
Vladimir Müller, Aljoša Peperko. On the Bonsall cone spectral radius and the approximate point spectrum. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5337-5354. doi: 10.3934/dcds.2017232 |
[2] |
Mickaël D. Chekroun, Jean Roux. Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3957-3980. doi: 10.3934/dcds.2013.33.3957 |
[3] |
Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447 |
[4] |
Vladimir Müller, Aljoša Peperko. Lower spectral radius and spectral mapping theorem for suprema preserving mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4117-4132. doi: 10.3934/dcds.2018179 |
[5] |
Shige Peng. Law of large numbers and central limit theorem under nonlinear expectations. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 4-. doi: 10.1186/s41546-019-0038-2 |
[6] |
Raffaele Chiappinelli. Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkoff-Kellogg theorem. Conference Publications, 2007, 2007 (Special) : 260-268. doi: 10.3934/proc.2007.2007.260 |
[7] |
P. Magal, H. R. Thieme. Eventual compactness for semiflows generated by nonlinear age-structured models. Communications on Pure and Applied Analysis, 2004, 3 (4) : 695-727. doi: 10.3934/cpaa.2004.3.695 |
[8] |
Ryszard Rudnicki, Radoslaw Wieczorek. Does assortative mating lead to a polymorphic population? A toy model justification. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 459-472. doi: 10.3934/dcdsb.2018031 |
[9] |
Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial and Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975 |
[10] |
Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22 |
[11] |
Rui Zou, Yongluo Cao, Gang Liao. Continuity of spectral radius over hyperbolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3977-3991. doi: 10.3934/dcds.2018173 |
[12] |
Marianne Akian, Stéphane Gaubert, Antoine Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009 |
[13] |
Daniele Mundici. The Haar theorem for lattice-ordered abelian groups with order-unit. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 537-549. doi: 10.3934/dcds.2008.21.537 |
[14] |
Salahuddin. System of generalized mixed nonlinear ordered variational inclusions. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 445-460. doi: 10.3934/naco.2019026 |
[15] |
Mustapha Mokhtar-Kharroubi, Quentin Richard. Spectral theory and time asymptotics of size-structured two-phase population models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2969-3004. doi: 10.3934/dcdsb.2020048 |
[16] |
Jacek Banasiak, Amartya Goswami. Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 617-635. doi: 10.3934/dcds.2015.35.617 |
[17] |
Johan Rosenkilde. Power decoding Reed-Solomon codes up to the Johnson radius. Advances in Mathematics of Communications, 2018, 12 (1) : 81-106. doi: 10.3934/amc.2018005 |
[18] |
Chen Ling, Liqun Qi. Some results on $l^k$-eigenvalues of tensor and related spectral radius. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 381-388. doi: 10.3934/naco.2011.1.381 |
[19] |
Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143 |
[20] |
Zaihong Jiang, Li Li, Wenbo Lu. Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4231-4258. doi: 10.3934/dcdss.2021126 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]