February  2022, 15(2): 387-407. doi: 10.3934/dcdss.2021068

Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition

School of Basic Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) - 175005, India

* Corresponding author: Muslim Malik

Received  August 2020 Revised  March 2021 Published  February 2022 Early access  June 2021

In this manuscript, we investigate the existence, uniqueness and controllability results of a Sobolev type fuzzy differential equation with non-instantaneous impulsive conditions. Non-linear functional analysis, Banach fixed point theorem and fuzzy theory are the main techniques used to establish these results. In support, an example is given to validate the obtained analytical findings.

Citation: Muslim Malik, Anjali Rose, Anil Kumar. Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 387-407. doi: 10.3934/dcdss.2021068
References:
[1]

S. Agarwal and D. Bahuguna, Existence of solutions to Sobolev-type partial neutral differential equations, International Journal of Stochastic Analysis, 163 (2006), 1-10.  doi: 10.1155/JAMSA/2006/16308.

[2]

R. P. AgarwalD. BaleanuJ. J. NietoD. F. Torres and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, Journal of Computational and Applied Mathematics, 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.

[3]

N. U. AhmedK. L. Teo and S. H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis, 54 (2003), 907-925.  doi: 10.1016/S0362-546X(03)00117-2.

[4]

S. Arora, M. T. Mohan and J. Dabas, Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces, Mathematical Control and Related Fields, (2020). doi: 10.3934/mcrf. 2020049.

[5]

G. Arthi and K. Balachandran, Controllability of second order impulsive evolution systems with infinite delay, Nonlinear Analysis: Hybrid Systems, 11 (2014), 139-153.  doi: 10.1016/j.nahs.2013.08.001.

[6]

K. BalachandranS. Kiruthika and J. J. Trujillo, On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Computers and Mathematics with Applications, 62 (2011), 1157-1165.  doi: 10.1016/j.camwa.2011.03.031.

[7]

K. Balachandran and J. Y. Park, Sobolev type integrodifferential equation with nonlocal condition in Banach spaces, Taiwanese Journal of Mathematics, 7 (2003), 155-163.  doi: 10.11650/twjm/1500407525.

[8]

K. Balachandran and J. P. Dauer, Controllability of functional differential systems of Sobolev type in Banach spaces, Kybernetika, 34 (1998), 349-357. 

[9]

P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of a fuzzy solution for nonlinear neutral functional differential equiations, Computers and Mathematics with Applications, 42 (2001), 961-967.  doi: 10.1016/S0898-1221(01)00212-7.

[10]

P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions, Computers and Mathematics with Applications, 47 (2004), 1115-1122.  doi: 10.1016/S0898-1221(04)90091-0.

[11]

B. Bede and S. G. Gal, Generalizations of the differentiability of fuzzy number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and System, 151 (2005), 581-599.  doi: 10.1016/j.fss.2004.08.001.

[12]

A. Bencsik, B. Bede, J. Tar and J. Fodor, Fuzzy differential equations in modeling hydraulic differential servo cylinders, in Third Romanian-Hungarian Joint Symposium on Applied Computational Intelligence (SACI), Timisoara, Romania, (2006).

[13]

A. Boudaoui and A. Slama, Existence and controllability results for Sobolev-type fractional impulsive stochastic differential equations with infinite delay, Journal of Mathematics and Applications, 40 (2017), 37-58.  doi: 10.7862/rf.2017.3.

[14]

J. Casasnovas and F. Rossell, Averaging fuzzy biopolymers, Fuzzy Sets and Systems, 152 (2005), 139-158.  doi: 10.1016/j.fss.2004.10.019.

[15]

P. Diamond and P. E. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scientific, (1994). doi: 10.1142/2326.

[16]

D. Dubois and H. Prade, Towards fuzzy differential calculus part 1: Integration of fuzzy mappings, Fuzzy Sets and System, 8 (1982), 1-7.  doi: 10.1016/0165-0114(82)90025-2.

[17]

D. Dubois and H. Prade, Towards fuzzy differential calculus part 2: Integration on fuzzy intervals, Fuzzy Sets and System, 8 (1982), 105-116.  doi: 10.1016/0165-0114(82)90001-X.

[18]

M. GuoX. Xue and R. Li, Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets and Systems, 138 (2003), 601-615.  doi: 10.1016/S0165-0114(02)00522-5.

[19]

J. H. JeongJ. S. KimH. E. Youm and J. H. Park, Exact controllability for fuzzy differential equations using extremal solutions, Journal of Computational Analysis and Applications, 23 (2017), 1056-1069. 

[20]

A. Kandel and W. J. Byatt, Fuzzy differential equations, Proceedings of the International Conference on Cybernetics and Society, Tokyo, Japan, 1 (1978), 1213-1216. 

[21]

M. Kumar and S. Kumar, Controllability of impulsive second order semilinear fuzzy integrodifferential control systems with nonlocal initial conditions, Applied Soft Computing, 39 (2016), 251-265.  doi: 10.1016/j.asoc.2015.10.006.

[22]

S. Kumar and R. Sakthivel, Constrained controllability of second order retarded nonlinear systems with nonlocal condition, IMA Journal of Mathematical Control and Information, 37 (2020), 441-454.  doi: 10.1093/imamci/dnz007.

[23]

Y. KwunJ. KimM. Park and J. Park, Nonlocal controllability for the semilinear fuzzy integrodifferential equations in n-dimensional fuzzy vector space, Advances in Difference Equations, 2009 (2009), 1-6.  doi: 10.1155/2009/734090.

[24]

Y. KwunJ. KimM. Park and J. Park, Controllability for the impulsive semilinear nonlocal fuzzy integrodifferential equations in n-dimensional fuzzy vector spacey, Advances in Difference Equations, 2010 (2010), 1-22.  doi: 10.1186/1687-1847-2010-983483.

[25]

B. Liu, A survey of credibility theory, Fuzzy Optimization and Decision Making, 5 (2006), 387-408.  doi: 10.1007/s10700-006-0016-x.

[26]

M. MalikR. DhayalS. Abbas and A. Kumar, Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A, Matemáticas, 113 (2019), 103-118.  doi: 10.1007/s13398-017-0454-z.

[27]

A. Meraj and D. N. Pandey, Approximate controllability of nonlocal non-autonomous Sobolev type evolution equations, An International Journal of Optimization and Control: Theories and Applications (IJOCTA), 9 (2019), 86-94.  doi: 10.11121/ijocta.01.2019.00644.

[28]

A. Meraj and D. N. Pandey, Approximate controllability of non-autonomous Sobolev type integro-differential equations having nonlocal and non-instantaneous impulsive conditions, Indian Journal of Pure and Applied Mathematics, 51 (2020), 501-518.  doi: 10.1007/s13226-020-0413-9.

[29]

M. Mizumoto and K. Tanaka, Some Properties of Fuzzy Numbers, North-Holland, 1979.

[30]

M. MuslimA. Kumar and R. Sakthivel, Exact and trajectory controllability of second order evolution systems with impulses and deviated arguments, Mathematical Methods in the Applied Sciences, 41 (2018), 4259-4272.  doi: 10.1002/mma.4888.

[31]

M. Muslim and R. P. Agarwal, Exact controllability of an integro-differential equation with deviated argument, Functional Differential Equations, 21 (2014), 31-45. 

[32]

J. H. ParkJ. S. ParkY. C. Ahn and Y. C. Kwun, Controllability for the impulsive semilinear fuzzy integrodifferential equations, Springer, 40 (2007), 704-713.  doi: 10.1007/978-3-540-71441-5_76.

[33]

J. H. Park, J. S. Park and Y. C. Kwun, Controllability for the semilinear fuzzy integrodifferential equations with nonlocal conditions, in International Conference on Fuzzy Systems and Knowledge Discovery, Springer, Berlin, Heidelberg, (2006), 221-230. doi: 10.1007/11881599_25.

[34]

G. ShenR. SakthivelY. Ren and M. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collectanea Mathematica, 71 (2020), 63-82.  doi: 10.1007/s13348-019-00248-3.

[35]

R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM Journal on Mathematical Analysis, 3 (1972), 527-543.  doi: 10.1137/0503051.

[36]

J. WangM. Feckan and A. Debbouche, Time optimal control of a system governed by non-instantaneous impulsive differential equations, Journal of Optimization Theory and Applications, 182 (2019), 573-587.  doi: 10.1007/s10957-018-1313-6.

[37]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

[38]

H. J. Zimmermann, Fuzzy set theory and its applications, Springer Science and Business Media, (2011). doi: 10.1007/978-94-010-0646-0.

show all references

References:
[1]

S. Agarwal and D. Bahuguna, Existence of solutions to Sobolev-type partial neutral differential equations, International Journal of Stochastic Analysis, 163 (2006), 1-10.  doi: 10.1155/JAMSA/2006/16308.

[2]

R. P. AgarwalD. BaleanuJ. J. NietoD. F. Torres and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, Journal of Computational and Applied Mathematics, 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.

[3]

N. U. AhmedK. L. Teo and S. H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis, 54 (2003), 907-925.  doi: 10.1016/S0362-546X(03)00117-2.

[4]

S. Arora, M. T. Mohan and J. Dabas, Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces, Mathematical Control and Related Fields, (2020). doi: 10.3934/mcrf. 2020049.

[5]

G. Arthi and K. Balachandran, Controllability of second order impulsive evolution systems with infinite delay, Nonlinear Analysis: Hybrid Systems, 11 (2014), 139-153.  doi: 10.1016/j.nahs.2013.08.001.

[6]

K. BalachandranS. Kiruthika and J. J. Trujillo, On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Computers and Mathematics with Applications, 62 (2011), 1157-1165.  doi: 10.1016/j.camwa.2011.03.031.

[7]

K. Balachandran and J. Y. Park, Sobolev type integrodifferential equation with nonlocal condition in Banach spaces, Taiwanese Journal of Mathematics, 7 (2003), 155-163.  doi: 10.11650/twjm/1500407525.

[8]

K. Balachandran and J. P. Dauer, Controllability of functional differential systems of Sobolev type in Banach spaces, Kybernetika, 34 (1998), 349-357. 

[9]

P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of a fuzzy solution for nonlinear neutral functional differential equiations, Computers and Mathematics with Applications, 42 (2001), 961-967.  doi: 10.1016/S0898-1221(01)00212-7.

[10]

P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions, Computers and Mathematics with Applications, 47 (2004), 1115-1122.  doi: 10.1016/S0898-1221(04)90091-0.

[11]

B. Bede and S. G. Gal, Generalizations of the differentiability of fuzzy number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and System, 151 (2005), 581-599.  doi: 10.1016/j.fss.2004.08.001.

[12]

A. Bencsik, B. Bede, J. Tar and J. Fodor, Fuzzy differential equations in modeling hydraulic differential servo cylinders, in Third Romanian-Hungarian Joint Symposium on Applied Computational Intelligence (SACI), Timisoara, Romania, (2006).

[13]

A. Boudaoui and A. Slama, Existence and controllability results for Sobolev-type fractional impulsive stochastic differential equations with infinite delay, Journal of Mathematics and Applications, 40 (2017), 37-58.  doi: 10.7862/rf.2017.3.

[14]

J. Casasnovas and F. Rossell, Averaging fuzzy biopolymers, Fuzzy Sets and Systems, 152 (2005), 139-158.  doi: 10.1016/j.fss.2004.10.019.

[15]

P. Diamond and P. E. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scientific, (1994). doi: 10.1142/2326.

[16]

D. Dubois and H. Prade, Towards fuzzy differential calculus part 1: Integration of fuzzy mappings, Fuzzy Sets and System, 8 (1982), 1-7.  doi: 10.1016/0165-0114(82)90025-2.

[17]

D. Dubois and H. Prade, Towards fuzzy differential calculus part 2: Integration on fuzzy intervals, Fuzzy Sets and System, 8 (1982), 105-116.  doi: 10.1016/0165-0114(82)90001-X.

[18]

M. GuoX. Xue and R. Li, Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets and Systems, 138 (2003), 601-615.  doi: 10.1016/S0165-0114(02)00522-5.

[19]

J. H. JeongJ. S. KimH. E. Youm and J. H. Park, Exact controllability for fuzzy differential equations using extremal solutions, Journal of Computational Analysis and Applications, 23 (2017), 1056-1069. 

[20]

A. Kandel and W. J. Byatt, Fuzzy differential equations, Proceedings of the International Conference on Cybernetics and Society, Tokyo, Japan, 1 (1978), 1213-1216. 

[21]

M. Kumar and S. Kumar, Controllability of impulsive second order semilinear fuzzy integrodifferential control systems with nonlocal initial conditions, Applied Soft Computing, 39 (2016), 251-265.  doi: 10.1016/j.asoc.2015.10.006.

[22]

S. Kumar and R. Sakthivel, Constrained controllability of second order retarded nonlinear systems with nonlocal condition, IMA Journal of Mathematical Control and Information, 37 (2020), 441-454.  doi: 10.1093/imamci/dnz007.

[23]

Y. KwunJ. KimM. Park and J. Park, Nonlocal controllability for the semilinear fuzzy integrodifferential equations in n-dimensional fuzzy vector space, Advances in Difference Equations, 2009 (2009), 1-6.  doi: 10.1155/2009/734090.

[24]

Y. KwunJ. KimM. Park and J. Park, Controllability for the impulsive semilinear nonlocal fuzzy integrodifferential equations in n-dimensional fuzzy vector spacey, Advances in Difference Equations, 2010 (2010), 1-22.  doi: 10.1186/1687-1847-2010-983483.

[25]

B. Liu, A survey of credibility theory, Fuzzy Optimization and Decision Making, 5 (2006), 387-408.  doi: 10.1007/s10700-006-0016-x.

[26]

M. MalikR. DhayalS. Abbas and A. Kumar, Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A, Matemáticas, 113 (2019), 103-118.  doi: 10.1007/s13398-017-0454-z.

[27]

A. Meraj and D. N. Pandey, Approximate controllability of nonlocal non-autonomous Sobolev type evolution equations, An International Journal of Optimization and Control: Theories and Applications (IJOCTA), 9 (2019), 86-94.  doi: 10.11121/ijocta.01.2019.00644.

[28]

A. Meraj and D. N. Pandey, Approximate controllability of non-autonomous Sobolev type integro-differential equations having nonlocal and non-instantaneous impulsive conditions, Indian Journal of Pure and Applied Mathematics, 51 (2020), 501-518.  doi: 10.1007/s13226-020-0413-9.

[29]

M. Mizumoto and K. Tanaka, Some Properties of Fuzzy Numbers, North-Holland, 1979.

[30]

M. MuslimA. Kumar and R. Sakthivel, Exact and trajectory controllability of second order evolution systems with impulses and deviated arguments, Mathematical Methods in the Applied Sciences, 41 (2018), 4259-4272.  doi: 10.1002/mma.4888.

[31]

M. Muslim and R. P. Agarwal, Exact controllability of an integro-differential equation with deviated argument, Functional Differential Equations, 21 (2014), 31-45. 

[32]

J. H. ParkJ. S. ParkY. C. Ahn and Y. C. Kwun, Controllability for the impulsive semilinear fuzzy integrodifferential equations, Springer, 40 (2007), 704-713.  doi: 10.1007/978-3-540-71441-5_76.

[33]

J. H. Park, J. S. Park and Y. C. Kwun, Controllability for the semilinear fuzzy integrodifferential equations with nonlocal conditions, in International Conference on Fuzzy Systems and Knowledge Discovery, Springer, Berlin, Heidelberg, (2006), 221-230. doi: 10.1007/11881599_25.

[34]

G. ShenR. SakthivelY. Ren and M. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collectanea Mathematica, 71 (2020), 63-82.  doi: 10.1007/s13348-019-00248-3.

[35]

R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM Journal on Mathematical Analysis, 3 (1972), 527-543.  doi: 10.1137/0503051.

[36]

J. WangM. Feckan and A. Debbouche, Time optimal control of a system governed by non-instantaneous impulsive differential equations, Journal of Optimization Theory and Applications, 182 (2019), 573-587.  doi: 10.1007/s10957-018-1313-6.

[37]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

[38]

H. J. Zimmermann, Fuzzy set theory and its applications, Springer Science and Business Media, (2011). doi: 10.1007/978-94-010-0646-0.

[1]

Avadhesh Kumar, Ankit Kumar, Ramesh Kumar Vats, Parveen Kumar. Approximate controllability of neutral delay integro-differential inclusion of order $ \alpha\in (1, 2) $ with non-instantaneous impulses. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021058

[2]

Ichrak Bouacida, Mourad Kerboua, Sami Segni. Controllability results for Sobolev type $ \psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022028

[3]

Liang Bai, Juan J. Nieto, José M. Uzal. On a delayed epidemic model with non-instantaneous impulses. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1915-1930. doi: 10.3934/cpaa.2020084

[4]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084

[5]

Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1329-1343. doi: 10.3934/dcdss.2020368

[6]

Ramalingam Sakthivel, Palanisamy Selvaraj, Yeong-Jae Kim, Dong-Hoon Lee, Oh-Min Kwon, Rathinasamy Sakthivel. Robust $ H_\infty $ resilient event-triggered control design for T-S fuzzy systems. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022028

[7]

Zhaoxia Duan, Jinling Liang, Zhengrong Xiang. $ H_{\infty} $ control for continuous-discrete systems in T-S fuzzy model with finite frequency specifications. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022064

[8]

Baskar Sundaravadivoo. Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2561-2573. doi: 10.3934/dcdss.2020138

[9]

Luisa Malaguti, Stefania Perrotta, Valentina Taddei. $ L^p $-exact controllability of partial differential equations with nonlocal terms. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021053

[10]

Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317

[11]

Thomas French. Follower, predecessor, and extender set sequences of $ \beta $-shifts. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4331-4344. doi: 10.3934/dcds.2019175

[12]

Mohd Raiz, Amit Kumar, Vishnu Narayan Mishra, Nadeem Rao. Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022007

[13]

Victor Vargas. On involution kernels and large deviations principles on $ \beta $-shifts. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2699-2718. doi: 10.3934/dcds.2021208

[14]

Pablo Amster, Mariel Paula Kuna, Dionicio Santos. Stability, existence and non-existence of $ T $-periodic solutions of nonlinear delayed differential equations with $ \varphi $-Laplacian. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022070

[15]

Melvin Faierman. Fredholm theory for an elliptic differential operator defined on $ \mathbb{R}^n $ and acting on generalized Sobolev spaces. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1463-1483. doi: 10.3934/cpaa.2020074

[16]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070

[17]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations and Control Theory, 2022, 11 (3) : 621-633. doi: 10.3934/eect.2021017

[18]

Xiaohui Zhang, Xuping Zhang. Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022081

[19]

Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations. Mathematical Control and Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015

[20]

Yinuo Wang, Chuandong Li, Hongjuan Wu, Hao Deng. Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1767-1776. doi: 10.3934/dcdss.2022005

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (318)
  • HTML views (447)
  • Cited by (0)

Other articles
by authors

[Back to Top]