American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities
  • CPAA Home
  • This Issue
  • Next Article
    Well-posedness of abstract distributed-order fractional diffusion equations
March  2014, 13(2): 623-634. doi: 10.3934/cpaa.2014.13.623

Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials

Ziheng Zhang 1, and  Rong Yuan 2,

1. 

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received  January 2013 Revised  April 2013 Published  October 2013

In this paper we investigate the existence of infinitely many homoclinic solutions for the following damped vibration problems \begin{eqnarray} \ddot q+A \dot q-L(t)q+W_q(t,q)=0, \end{eqnarray} where $A$ is an antisymmetric constant matrix, $L\in C(R,R^{n^2})$ is a symmetric and positive definite matrix for all $t\in R$, $W\in C^1(R\times R^n,R)$. The novelty of this paper is that, for the case that $W$ is subquadratic at infinity, we establish two new criteria to guarantee the existence of infinitely many homoclinic solutions for (DS) via the genus properties in critical point theory. Recent results in the literature are generalized and significantly improved.
Keywords: critical point, Homoclinic solutions, genus., variational methods.
Mathematics Subject Classification: Primary: 34C37, 35A15; Secondary: 35B3.
Citation: Ziheng Zhang, Rong Yuan. Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 623-634. doi: 10.3934/cpaa.2014.13.623
References:
[1]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, paper title is capitalized., Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16 (2003), 639-642. doi: 10.1016/S0893-9659(03)00059-4.  Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. Appl. Nonlinear Anal., 1 (1994), 97-129.  Google Scholar

[4]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.1090/S0894-0347-1991-1119200-3.  Google Scholar

[5]

A. Daouas, Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption, Nonlinear Anal., 74 (2011), 3407-3418. doi: 10.1016/j.na.2011.03.001.  Google Scholar

[6]

Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113. doi: 10.1016/0362-546X(94)00229-B.  Google Scholar

[7]

Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and supequadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413. doi: 10.1016/j.na.2008.10.116.  Google Scholar

[8]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[9]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, (1994), 1-10. doi: 10.1.1.27.6093.  Google Scholar

[10]

X. Lv and J. Jiang, Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials, Nonlinear Analysis: Real World Applications, 13 (2012), 1152-1158. doi: 10.1016/j.nonrwa.2011.09.008.  Google Scholar

[11]

X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 72 (2010), 390-398. doi: 10.1016/j.na.2009.06.073.  Google Scholar

[12]

Y. Lv and C. Tang, Existence of even homoclinic orbits for a class of Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198. doi: 10.1016/j.na.2006.08.043.  Google Scholar

[13]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.  Google Scholar

[14]

H. Poincaré, Les méthodes nouvelles de la mécanique céleste,, Gauthier-Villars, (): 1897.   Google Scholar

[15]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Reg. Conf. Ser. in. Math., vol. 65, American Mathematical Society, Provodence, RI, 1986.  Google Scholar

[16]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A., 114 (1990), 33-38. doi: 10.1017/S0308210500024240.  Google Scholar

[17]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: 10.1007/BF02571356.  Google Scholar

[18]

A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Proceedings of the Second World Congress of Nonlinear Analysis, Part 8 (Athens, 1996), Nonlinear Anal., 30 (1997), 4849-4857. doi: SO362-546X(97)00142-9.  Google Scholar

[19]

J. Sun, H. Chen and J. J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 20-29. doi: 10.1016/j.jmaa.2010.06.038.  Google Scholar

[20]

J. Sun, J. J. Nieto and M. Otero-Novoa, On homoclinic orbits for a class of damped vibration systems,, Advances in Difference Equations, 2012 ().  doi: 10.1186/1687-1847-2012-102.  Google Scholar

[21]

X. Tang and X. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite subquadratic potentials, Nonlinear Anal., 74 (2011), 6314-6325. doi: 10.1016/j.na.2011.06.010.  Google Scholar

[22]

X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 1103-1119. doi: 10.1017/S0308210509001346.  Google Scholar

[23]

L. Wan and C. Tang, Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Discrete. Cont. Dyn. Syst. Ser. B, 15 (2011), 255-271. doi: 10.3934/dcdsb.2011.15.255.  Google Scholar

[24]

J. Wang, J. Xu and F. Zhang, Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials, Comm. Pure. Appl. Anal., 10 (2011), 269-286. doi: 10.3934/cpaa.2011.10.269.  Google Scholar

[25]

J. Wang, F. Zhang and J. Xu, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 366 (2010), 569-581. doi: 10.1016/j.jmaa.2010.01.060.  Google Scholar

[26]

J. Wei and J, Wang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials, J. Math. Anal. Anal., 366 (2010), 694-699. doi: 10.1016/j.jmaa.2009.12.024.  Google Scholar

[27]

X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398. doi: 10.1016/j.na.2011.03.059.  Google Scholar

[28]

M. Yang and Z. Han, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials, Nonlinear Analysis: Real World Applications, 12 (2011), 2742-2751. doi: 10.1016/j.nonrwa.2011.03.019.  Google Scholar

[29]

M. Yang and Z. Han, Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities, Nonlinear Anal., 74 (2011), 2635-2646. doi: 10.1016/j.na.2010.12.019.  Google Scholar

[30]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order non-autonomous systems, Electron. J. of Differential Equations, 128 (2009), 1-9.  Google Scholar

[31]

Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal., 72 (2010), 894-903. doi: 10.1016/j.na.2009.07.021.  Google Scholar

[32]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems, Nonlinear Anal., 71 (2009), 4125-4130. doi: 10.1016/j.na.2009.02.071.  Google Scholar

[33]

Z. Zhang and R. Yuan, Homoclinic solutions of some second order non-autonomous systems, Nonlinear Anal., 71 (2009), 5790-5798. doi: 10.1016/j.na.2009.05.003.  Google Scholar

[34]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems, Nonlinear Analysis: Real World Applications, 11 (2010), 4185-4193. doi: 10.1016/j.nonrwa.2010.05.005.  Google Scholar

[35]

W. Zhu, Existence of homoclinic solutions for a class of second order systems, Nonlinear Anal., 75 (2012), 2455-2463. doi: 10.1016/j.na.2011.10.043.  Google Scholar

[36]

W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287. doi: 10.1016/S0893-9659(03)90130-3.  Google Scholar

show all references

References:
[1]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, paper title is capitalized., Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16 (2003), 639-642. doi: 10.1016/S0893-9659(03)00059-4.  Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. Appl. Nonlinear Anal., 1 (1994), 97-129.  Google Scholar

[4]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.1090/S0894-0347-1991-1119200-3.  Google Scholar

[5]

A. Daouas, Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption, Nonlinear Anal., 74 (2011), 3407-3418. doi: 10.1016/j.na.2011.03.001.  Google Scholar

[6]

Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113. doi: 10.1016/0362-546X(94)00229-B.  Google Scholar

[7]

Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and supequadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413. doi: 10.1016/j.na.2008.10.116.  Google Scholar

[8]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[9]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, (1994), 1-10. doi: 10.1.1.27.6093.  Google Scholar

[10]

X. Lv and J. Jiang, Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials, Nonlinear Analysis: Real World Applications, 13 (2012), 1152-1158. doi: 10.1016/j.nonrwa.2011.09.008.  Google Scholar

[11]

X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 72 (2010), 390-398. doi: 10.1016/j.na.2009.06.073.  Google Scholar

[12]

Y. Lv and C. Tang, Existence of even homoclinic orbits for a class of Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198. doi: 10.1016/j.na.2006.08.043.  Google Scholar

[13]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.  Google Scholar

[14]

H. Poincaré, Les méthodes nouvelles de la mécanique céleste,, Gauthier-Villars, (): 1897.   Google Scholar

[15]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Reg. Conf. Ser. in. Math., vol. 65, American Mathematical Society, Provodence, RI, 1986.  Google Scholar

[16]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A., 114 (1990), 33-38. doi: 10.1017/S0308210500024240.  Google Scholar

[17]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: 10.1007/BF02571356.  Google Scholar

[18]

A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Proceedings of the Second World Congress of Nonlinear Analysis, Part 8 (Athens, 1996), Nonlinear Anal., 30 (1997), 4849-4857. doi: SO362-546X(97)00142-9.  Google Scholar

[19]

J. Sun, H. Chen and J. J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 20-29. doi: 10.1016/j.jmaa.2010.06.038.  Google Scholar

[20]

J. Sun, J. J. Nieto and M. Otero-Novoa, On homoclinic orbits for a class of damped vibration systems,, Advances in Difference Equations, 2012 ().  doi: 10.1186/1687-1847-2012-102.  Google Scholar

[21]

X. Tang and X. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite subquadratic potentials, Nonlinear Anal., 74 (2011), 6314-6325. doi: 10.1016/j.na.2011.06.010.  Google Scholar

[22]

X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 1103-1119. doi: 10.1017/S0308210509001346.  Google Scholar

[23]

L. Wan and C. Tang, Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Discrete. Cont. Dyn. Syst. Ser. B, 15 (2011), 255-271. doi: 10.3934/dcdsb.2011.15.255.  Google Scholar

[24]

J. Wang, J. Xu and F. Zhang, Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials, Comm. Pure. Appl. Anal., 10 (2011), 269-286. doi: 10.3934/cpaa.2011.10.269.  Google Scholar

[25]

J. Wang, F. Zhang and J. Xu, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 366 (2010), 569-581. doi: 10.1016/j.jmaa.2010.01.060.  Google Scholar

[26]

J. Wei and J, Wang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials, J. Math. Anal. Anal., 366 (2010), 694-699. doi: 10.1016/j.jmaa.2009.12.024.  Google Scholar

[27]

X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398. doi: 10.1016/j.na.2011.03.059.  Google Scholar

[28]

M. Yang and Z. Han, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials, Nonlinear Analysis: Real World Applications, 12 (2011), 2742-2751. doi: 10.1016/j.nonrwa.2011.03.019.  Google Scholar

[29]

M. Yang and Z. Han, Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities, Nonlinear Anal., 74 (2011), 2635-2646. doi: 10.1016/j.na.2010.12.019.  Google Scholar

[30]

R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order non-autonomous systems, Electron. J. of Differential Equations, 128 (2009), 1-9.  Google Scholar

[31]

Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal., 72 (2010), 894-903. doi: 10.1016/j.na.2009.07.021.  Google Scholar

[32]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems, Nonlinear Anal., 71 (2009), 4125-4130. doi: 10.1016/j.na.2009.02.071.  Google Scholar

[33]

Z. Zhang and R. Yuan, Homoclinic solutions of some second order non-autonomous systems, Nonlinear Anal., 71 (2009), 5790-5798. doi: 10.1016/j.na.2009.05.003.  Google Scholar

[34]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems, Nonlinear Analysis: Real World Applications, 11 (2010), 4185-4193. doi: 10.1016/j.nonrwa.2010.05.005.  Google Scholar

[35]

W. Zhu, Existence of homoclinic solutions for a class of second order systems, Nonlinear Anal., 75 (2012), 2455-2463. doi: 10.1016/j.na.2011.10.043.  Google Scholar

[36]

W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287. doi: 10.1016/S0893-9659(03)90130-3.  Google Scholar

[1]

D. Motreanu, Donal O'Regan, Nikolaos S. Papageorgiou. A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1791-1816. doi: 10.3934/cpaa.2011.10.1791
[2]

Salvatore A. Marano, Sunra J. N. Mosconi. Multiple solutions to elliptic inclusions via critical point theory on closed convex sets. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3087-3102. doi: 10.3934/dcds.2015.35.3087
[3]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[4]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483
[5]

Antonio Ambrosetti, Massimiliano Berti. Applications of critical point theory to homoclinics and complex dynamics. Conference Publications, 1998, 1998 (Special) : 72-78. doi: 10.3934/proc.1998.1998.72
[6]

Anna Maria Candela, Giuliana Palmieri. Some abstract critical point theorems and applications. Conference Publications, 2009, 2009 (Special) : 133-142. doi: 10.3934/proc.2009.2009.133
[7]

Domingo González, Gamaliel Blé. Core entropy of polynomials with a critical point of maximal order. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 115-130. doi: 10.3934/dcds.2019005
[8]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[9]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[10]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036
[11]

Flaviano Battelli, Claudio Lazzari. On the bifurcation from critical homoclinic orbits in n-dimensional maps. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 289-303. doi: 10.3934/dcds.1997.3.289
[12]

Samir Adly, Daniel Goeleven, Dumitru Motreanu. Periodic and homoclinic solutions for a class of unilateral problems. Discrete & Continuous Dynamical Systems, 1997, 3 (4) : 579-590. doi: 10.3934/dcds.1997.3.579
[13]

S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493
[14]

Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021037
[15]

Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153
[16]

Urszula Foryś, Yuri Kheifetz, Yuri Kogan. Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 511-525. doi: 10.3934/mbe.2005.2.511
[17]

Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
[18]

Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075
[19]

O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185
[20]

Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences