July  2017, 22(5): 1999-2010. doi: 10.3934/dcdsb.2017118

Averaging of fuzzy integral equations

Department of optimal control and economic cybernetics, Odessa National University, Dvoryanskaya 2, Odessa, 65082, Ukraine

Received  January 2016 Revised  February 2016 Published  March 2017

The integral equations are encountered in various fields of science and in numerous applications, including elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine. In this paper the fuzzy integral equation is considered and the existence and uniqueness theorem, the theorem of continuous dependence on the right-hand side and initial fuzzy set are proved. Also the possibility of using the scheme of full averaging for fuzzy integral equation with a small parameter is considered.

Citation: Natalia Skripnik. Averaging of fuzzy integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118
References:
[1]

T. AllahviranlooA. AmirteimooriM. Khezerloo and S. Khezerloo, A new method for solving fuzzy Volterra integro-differential equations, Australian J.Basic Appl. Sci., 5 (2011), 154-164. Google Scholar

[2]

J.-P. Aubin, Fuzzy differential inclusions, Probl. Control Inf. Theory, 19 (1990), 55-67. Google Scholar

[3]

R. J. Aumann, Integrals of set -valued functions, J. Math. Anal. Appl., 12 (1965), 1-12. doi: 10.1016/0022-247X(65)90049-1. Google Scholar

[4]

V. A. Baidosov, Differential inclusions with fuzzy right-hand side, Sov. Math. Dokl., 40 (1990), 567-569. Google Scholar

[5]

V. A. Baidosov, Fuzzy differential inclusions, J. Appl. Math. Mech., 54 (1990), 8-13. doi: 10.1016/0021-8928(90)90080-T. Google Scholar

[6]

K. Balachandran and K. Kanagarajan, Existence of solutions of fuzzy delay integro-differential equations with nonlocal condition, J. Korea Society Industrial Appl. Math., 9 (2005), 65-74. Google Scholar

[7]

P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for the nonlinear fuzzy integro-differential equations, Appl. Math. Lett., 14 (2001), 455-462. doi: 10.1016/S0893-9659(00)00177-4. Google Scholar

[8]

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

[9]

N. N. Bogolyubov and Yu. A. Mitropol'skij, Asymptotic Methods in the Theory of Nonlinear Oscillations International Monographs on Advanced Mathematics and Physics Hindustan Publishing Corp. , Delhi, Gordon and Breach Science Publishers, New York, 1961.Google Scholar

[10]

A. N. Filatov, Averaging in Systems of Differential and Integrodifferential Equations (in Russian) Fan, Tashkent, 1971.Google Scholar

[11]

A. N. Filatov, Asymptotic Methods in the Theory of Differential and Integrodifferential Equations (in Russian) Fan, Tashkent, 1974.Google Scholar

[12]

M. FriedmanM. Ma and A. Kandel, Numerical solutions of fuzzy differential equations and integral equations, Fuzzy Sets Syst., 106 (1999), 35-48. doi: 10.1016/S0165-0114(98)00355-8. Google Scholar

[13]

M. GhanbariR. Toushmalni and E. Kamrani, Numerical solution of linear Fredholm fuzzy integral equation of the second kind by block-pulse functions, Australian J. Basic Appl. Sci., 3 (2009), 2637-2642. Google Scholar

[14]

E. Hullermeier, An approach to modeling and simulation of uncertain dynamical system, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 5 (1997), 117-137. doi: 10.1142/S0218488597000117. Google Scholar

[15]

M. JahantighT. Allahviranloo and M. Otadi, Numerical solution of fuzzy integral equations, Applied Math. Sci., 2 (2008), 33-46. Google Scholar

[16]

O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301-317. doi: 10.1016/0165-0114(87)90029-7. Google Scholar

[17]

S. V. Kats and N. V. Skripnik, Averaging of set-valued integral equations with constant delay, Perspective directions of world science: The 33-th International conference "Innovative Potential of World Science the XXI Century", 2 (2015), 83-84. Google Scholar

[18]

M. Kisielewicz, Method of averaging for differential equations with compact convex valued solutions, Rend. Math., 9 (1976), 397-408. Google Scholar

[19]

M. A. Krasnosel'skii and s. G. Krein, On the principle of averaging in nonlinear mechanics (in Russian), Usp. Mat. Nauk, 10 (1955), 147-152. Google Scholar

[20]

N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1947.Google Scholar

[21]

Y. C. KwunM. J. KimB. Y. Lee and J. H. Park, Existence of solutions for the semilinear fuzzy integrodifferential equations using by successive iteration, J. Korean Institute of Intelligent Systems, 18 (2008), 543-548. doi: 10.5391/JKIIS.2008.18.4.543. Google Scholar

[22]

Y. C. Kwun, J. S. Kim, M. J. Park and J. H. Park, Nonlocal controllability for the semilinear fuzzy integrodifferential equations in n-dimensional fuzzy vector space Adv. Difference Equ. 2009 (2009), Article ID 734090, 16 pages.Google Scholar

[23]

Y. C. Kwun, J. S. Kim, M. J. Park and J. H. Park, Controllability for the impulsive semilinear nonlocal fuzzy integrodifferential equations in n-dimensional fuzzy vector space, Adv. Difference Equ. 2010 (2010), Article ID 983483, 22 pages.Google Scholar

[24]

Y. C. Kwun and D. G. Park, Optimal control problem for fuzzy differential equations, Proceedings of the Korea-Vietnam Joint Seminar, (1998), 103-114. Google Scholar

[25]

V. Lakshmikantham, T. Gnana Bhaskar and D. J. Vasundhara, Theory of Set Differential Equations in Metric Spaces Cambridge Scient ific Publishers, Cambridge, 2006.Google Scholar

[26]

V. Lakshmikantham and R. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions Taylor -Francis, 2003.Google Scholar

[27]

Yu. A. Mitropol'skij, Method of averaging in nonlinear mechanics (in Russian) Naukova dumka, Kiev, 1971.Google Scholar

[28]

Yu. A. Mitropol'skij and G. N. Khoma, Mathematical Justification of Asymptotic Methods of Nonlinear Mechanics (in Russian) Naukova dumka, Kiev, 1983.Google Scholar

[29]

I. V. Molchanyuk and A. V. Plotnikov, Linear control systems with a fuzzy parameter, Nonlinear Oscil., 9 (2006), 59-64. doi: 10.1007/s11072-006-0025-2. Google Scholar

[30]

I. V. Molchanyuk and A. V. Plotnikov, Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters, Ukr. Math. J., 61 (2009), 457-466. doi: 10.1007/s11253-009-0214-0. Google Scholar

[31]

J. Mordeson and W. Newman, Fuzzy integral equations, Inform. Sci., 87 (1995), 215-229. doi: 10.1016/0020-0255(95)00126-3. Google Scholar

[32]

N. Parandin and M. A. Fariborzi Araghi, The approximate solution of linear fuzzy Fredholm integral equations of the second kind by using iterative interpolation, World Academy Sci., Engineering Technology, 25 (2009), 978-984. Google Scholar

[33]

J. H. ParkJ. S. Park and Y. C. Kwun, Controllability for the semilinear fuzzy integrodifferential equations with nonlocal conditions, Fuzzy Systems and Knowledge Discovery, Lecture Notes in Computer Science, 4223 (2006), 221-230. doi: 10.1007/11881599_25. Google Scholar

[34]

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

[35]

J. Y. Park and H. K. Han, Existence and uniqueness theorem for a solution of fuzzy differential equations, Int. J. Math. Math. Sci., 22 (1999), 271-279. doi: 10.1155/S0161171299222715. Google Scholar

[36]

J. Y. Park and H. K. Han, Fuzzy differential equations, Fuzzy Sets Syst., 110 (2000), 69-77. doi: 10.1016/S0165-0114(98)00150-X. Google Scholar

[37]

J. Y. ParkY. C. Kwun and J. U. Jeong, Existence of solutions of fuzzy integral equations in Banach spaces, Fuzzy Sets Syst., 72 (1995), 373-378. doi: 10.1016/0165-0114(94)00296-J. Google Scholar

[38]

N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko and N. V. Skripnik, Differential Equations with Impulse Effects: Multivalued Right-Hand Sides with Discontinuities Walter De Gruyter GmbHCo. , Berlin/Boston, 2011.Google Scholar

[39]

N. D. Phu and T. T. Tung, Some results on sheaf-solutions of sheaf set control problems, Nonlinear Anal., 67 (2007), 1309-1315. doi: 10.1016/j.na.2006.07.018. Google Scholar

[40]

A. V. Plotnikov and T. A. Komleva, Linear problems of optimal control of fuzzy maps, Intelligent Information Management, 1 (2009), 139-144. doi: 10.4236/iim.2009.13020. Google Scholar

[41]

A. V. PlotnikovT. A. Komleva and A. V. Arsiry, Necessary and sufficient optimality conditions for a control fuzzy linear problem, Int. J. Industrial Math., 1 (2009), 197-207. Google Scholar

[42]

A. V. PlotnikovT. A. Komleva and I. V. Molchanyuk, Linear control problems of the fuzzy maps, J. Software Engineering Applications, 3 (2010), 191-197. doi: 10.4236/jsea.2010.33024. Google Scholar

[43]

A. V. Plotnikov and N. V. Skripnik, The generalized solutions of the fuzzy differential inclusions, Int. J. Pure Appl. Math, 56 (2009), 165-172. Google Scholar

[44]

A. V. Plotnikov and N. V. Skripnik, Differential Equations with "Clear" and Fuzzy Multivalued Right-Hand Sides. Asymptotics Methods (in Russian) AstroPrint, Odessa, 2009.Google Scholar

[45]

V. A. Plotnikov, A. V. Plotnikov and A. N. Vityuk, Differential Equations with a Multivalued Right-Hand Side: Asymptotic Methods Astroprint, Odessa, 1999.Google Scholar

[46]

Sh. Sadigh Behzadi, Solving fuzzy nonlinear Volterra-Fredholm integral equations by using homotopy analysis and Adomian decomposition methods, J. Fuzzy Set Valued Anal. 2011 (2011), Art. ID 00067, 13 pages.Google Scholar

[47]

M. M. ShamivandA. Shahsavaran and S M. Tari, Solution to Fredholm fuzzy integral equations with degenerate kernel, Int. J. Contemp. Math. Sci., 6 (2011), 535-543. Google Scholar

[48]

N. V. Skripnik, The full averaging of fuzzy differential inclusions, Iranian J. Optimization, 1 (2009), 302-317. Google Scholar

[49]

N. V. Skripnik, The partial averaging of fuzzy differential inclusions, J. Adv. Res. Differ. Equ., 3 (2011), 52-66. Google Scholar

[50]

N. V. Skripnik, The partial averaging of fuzzy impulsive differential inclusions, Diff. Int. Equ., 24 (2011), 743-758. Google Scholar

[51]

N. V. Skripnik, Averaging of multivalued integral equations, J. Math.Sci., 201 (2014), 384-390. Google Scholar

[52]

I. Tise, Set integral equations in metric spaces, Math. Morav., 13 (2009), 95-102. Google Scholar

[53]

V. S. Vasil'kovskaya and A. V. Plotnikov, Integrodifferential systems with fuzzy noise, Ukr. Math. J., 59 (2007), 1482-1492. doi: 10.1007/s11253-008-0005-z. Google Scholar

[54]

C. Wu and M. Ma, On the integrals, series and integral equations of fuzzy set-valued functions, J. Harbin Inst. Technol., 21 (1990), 11-19. Google Scholar

[55]

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

show all references

References:
[1]

T. AllahviranlooA. AmirteimooriM. Khezerloo and S. Khezerloo, A new method for solving fuzzy Volterra integro-differential equations, Australian J.Basic Appl. Sci., 5 (2011), 154-164. Google Scholar

[2]

J.-P. Aubin, Fuzzy differential inclusions, Probl. Control Inf. Theory, 19 (1990), 55-67. Google Scholar

[3]

R. J. Aumann, Integrals of set -valued functions, J. Math. Anal. Appl., 12 (1965), 1-12. doi: 10.1016/0022-247X(65)90049-1. Google Scholar

[4]

V. A. Baidosov, Differential inclusions with fuzzy right-hand side, Sov. Math. Dokl., 40 (1990), 567-569. Google Scholar

[5]

V. A. Baidosov, Fuzzy differential inclusions, J. Appl. Math. Mech., 54 (1990), 8-13. doi: 10.1016/0021-8928(90)90080-T. Google Scholar

[6]

K. Balachandran and K. Kanagarajan, Existence of solutions of fuzzy delay integro-differential equations with nonlocal condition, J. Korea Society Industrial Appl. Math., 9 (2005), 65-74. Google Scholar

[7]

P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for the nonlinear fuzzy integro-differential equations, Appl. Math. Lett., 14 (2001), 455-462. doi: 10.1016/S0893-9659(00)00177-4. Google Scholar

[8]

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

[9]

N. N. Bogolyubov and Yu. A. Mitropol'skij, Asymptotic Methods in the Theory of Nonlinear Oscillations International Monographs on Advanced Mathematics and Physics Hindustan Publishing Corp. , Delhi, Gordon and Breach Science Publishers, New York, 1961.Google Scholar

[10]

A. N. Filatov, Averaging in Systems of Differential and Integrodifferential Equations (in Russian) Fan, Tashkent, 1971.Google Scholar

[11]

A. N. Filatov, Asymptotic Methods in the Theory of Differential and Integrodifferential Equations (in Russian) Fan, Tashkent, 1974.Google Scholar

[12]

M. FriedmanM. Ma and A. Kandel, Numerical solutions of fuzzy differential equations and integral equations, Fuzzy Sets Syst., 106 (1999), 35-48. doi: 10.1016/S0165-0114(98)00355-8. Google Scholar

[13]

M. GhanbariR. Toushmalni and E. Kamrani, Numerical solution of linear Fredholm fuzzy integral equation of the second kind by block-pulse functions, Australian J. Basic Appl. Sci., 3 (2009), 2637-2642. Google Scholar

[14]

E. Hullermeier, An approach to modeling and simulation of uncertain dynamical system, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 5 (1997), 117-137. doi: 10.1142/S0218488597000117. Google Scholar

[15]

M. JahantighT. Allahviranloo and M. Otadi, Numerical solution of fuzzy integral equations, Applied Math. Sci., 2 (2008), 33-46. Google Scholar

[16]

O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301-317. doi: 10.1016/0165-0114(87)90029-7. Google Scholar

[17]

S. V. Kats and N. V. Skripnik, Averaging of set-valued integral equations with constant delay, Perspective directions of world science: The 33-th International conference "Innovative Potential of World Science the XXI Century", 2 (2015), 83-84. Google Scholar

[18]

M. Kisielewicz, Method of averaging for differential equations with compact convex valued solutions, Rend. Math., 9 (1976), 397-408. Google Scholar

[19]

M. A. Krasnosel'skii and s. G. Krein, On the principle of averaging in nonlinear mechanics (in Russian), Usp. Mat. Nauk, 10 (1955), 147-152. Google Scholar

[20]

N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1947.Google Scholar

[21]

Y. C. KwunM. J. KimB. Y. Lee and J. H. Park, Existence of solutions for the semilinear fuzzy integrodifferential equations using by successive iteration, J. Korean Institute of Intelligent Systems, 18 (2008), 543-548. doi: 10.5391/JKIIS.2008.18.4.543. Google Scholar

[22]

Y. C. Kwun, J. S. Kim, M. J. Park and J. H. Park, Nonlocal controllability for the semilinear fuzzy integrodifferential equations in n-dimensional fuzzy vector space Adv. Difference Equ. 2009 (2009), Article ID 734090, 16 pages.Google Scholar

[23]

Y. C. Kwun, J. S. Kim, M. J. Park and J. H. Park, Controllability for the impulsive semilinear nonlocal fuzzy integrodifferential equations in n-dimensional fuzzy vector space, Adv. Difference Equ. 2010 (2010), Article ID 983483, 22 pages.Google Scholar

[24]

Y. C. Kwun and D. G. Park, Optimal control problem for fuzzy differential equations, Proceedings of the Korea-Vietnam Joint Seminar, (1998), 103-114. Google Scholar

[25]

V. Lakshmikantham, T. Gnana Bhaskar and D. J. Vasundhara, Theory of Set Differential Equations in Metric Spaces Cambridge Scient ific Publishers, Cambridge, 2006.Google Scholar

[26]

V. Lakshmikantham and R. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions Taylor -Francis, 2003.Google Scholar

[27]

Yu. A. Mitropol'skij, Method of averaging in nonlinear mechanics (in Russian) Naukova dumka, Kiev, 1971.Google Scholar

[28]

Yu. A. Mitropol'skij and G. N. Khoma, Mathematical Justification of Asymptotic Methods of Nonlinear Mechanics (in Russian) Naukova dumka, Kiev, 1983.Google Scholar

[29]

I. V. Molchanyuk and A. V. Plotnikov, Linear control systems with a fuzzy parameter, Nonlinear Oscil., 9 (2006), 59-64. doi: 10.1007/s11072-006-0025-2. Google Scholar

[30]

I. V. Molchanyuk and A. V. Plotnikov, Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters, Ukr. Math. J., 61 (2009), 457-466. doi: 10.1007/s11253-009-0214-0. Google Scholar

[31]

J. Mordeson and W. Newman, Fuzzy integral equations, Inform. Sci., 87 (1995), 215-229. doi: 10.1016/0020-0255(95)00126-3. Google Scholar

[32]

N. Parandin and M. A. Fariborzi Araghi, The approximate solution of linear fuzzy Fredholm integral equations of the second kind by using iterative interpolation, World Academy Sci., Engineering Technology, 25 (2009), 978-984. Google Scholar

[33]

J. H. ParkJ. S. Park and Y. C. Kwun, Controllability for the semilinear fuzzy integrodifferential equations with nonlocal conditions, Fuzzy Systems and Knowledge Discovery, Lecture Notes in Computer Science, 4223 (2006), 221-230. doi: 10.1007/11881599_25. Google Scholar

[34]

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

[35]

J. Y. Park and H. K. Han, Existence and uniqueness theorem for a solution of fuzzy differential equations, Int. J. Math. Math. Sci., 22 (1999), 271-279. doi: 10.1155/S0161171299222715. Google Scholar

[36]

J. Y. Park and H. K. Han, Fuzzy differential equations, Fuzzy Sets Syst., 110 (2000), 69-77. doi: 10.1016/S0165-0114(98)00150-X. Google Scholar

[37]

J. Y. ParkY. C. Kwun and J. U. Jeong, Existence of solutions of fuzzy integral equations in Banach spaces, Fuzzy Sets Syst., 72 (1995), 373-378. doi: 10.1016/0165-0114(94)00296-J. Google Scholar

[38]

N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko and N. V. Skripnik, Differential Equations with Impulse Effects: Multivalued Right-Hand Sides with Discontinuities Walter De Gruyter GmbHCo. , Berlin/Boston, 2011.Google Scholar

[39]

N. D. Phu and T. T. Tung, Some results on sheaf-solutions of sheaf set control problems, Nonlinear Anal., 67 (2007), 1309-1315. doi: 10.1016/j.na.2006.07.018. Google Scholar

[40]

A. V. Plotnikov and T. A. Komleva, Linear problems of optimal control of fuzzy maps, Intelligent Information Management, 1 (2009), 139-144. doi: 10.4236/iim.2009.13020. Google Scholar

[41]

A. V. PlotnikovT. A. Komleva and A. V. Arsiry, Necessary and sufficient optimality conditions for a control fuzzy linear problem, Int. J. Industrial Math., 1 (2009), 197-207. Google Scholar

[42]

A. V. PlotnikovT. A. Komleva and I. V. Molchanyuk, Linear control problems of the fuzzy maps, J. Software Engineering Applications, 3 (2010), 191-197. doi: 10.4236/jsea.2010.33024. Google Scholar

[43]

A. V. Plotnikov and N. V. Skripnik, The generalized solutions of the fuzzy differential inclusions, Int. J. Pure Appl. Math, 56 (2009), 165-172. Google Scholar

[44]

A. V. Plotnikov and N. V. Skripnik, Differential Equations with "Clear" and Fuzzy Multivalued Right-Hand Sides. Asymptotics Methods (in Russian) AstroPrint, Odessa, 2009.Google Scholar

[45]

V. A. Plotnikov, A. V. Plotnikov and A. N. Vityuk, Differential Equations with a Multivalued Right-Hand Side: Asymptotic Methods Astroprint, Odessa, 1999.Google Scholar

[46]

Sh. Sadigh Behzadi, Solving fuzzy nonlinear Volterra-Fredholm integral equations by using homotopy analysis and Adomian decomposition methods, J. Fuzzy Set Valued Anal. 2011 (2011), Art. ID 00067, 13 pages.Google Scholar

[47]

M. M. ShamivandA. Shahsavaran and S M. Tari, Solution to Fredholm fuzzy integral equations with degenerate kernel, Int. J. Contemp. Math. Sci., 6 (2011), 535-543. Google Scholar

[48]

N. V. Skripnik, The full averaging of fuzzy differential inclusions, Iranian J. Optimization, 1 (2009), 302-317. Google Scholar

[49]

N. V. Skripnik, The partial averaging of fuzzy differential inclusions, J. Adv. Res. Differ. Equ., 3 (2011), 52-66. Google Scholar

[50]

N. V. Skripnik, The partial averaging of fuzzy impulsive differential inclusions, Diff. Int. Equ., 24 (2011), 743-758. Google Scholar

[51]

N. V. Skripnik, Averaging of multivalued integral equations, J. Math.Sci., 201 (2014), 384-390. Google Scholar

[52]

I. Tise, Set integral equations in metric spaces, Math. Morav., 13 (2009), 95-102. Google Scholar

[53]

V. S. Vasil'kovskaya and A. V. Plotnikov, Integrodifferential systems with fuzzy noise, Ukr. Math. J., 59 (2007), 1482-1492. doi: 10.1007/s11253-008-0005-z. Google Scholar

[54]

C. Wu and M. Ma, On the integrals, series and integral equations of fuzzy set-valued functions, J. Harbin Inst. Technol., 21 (1990), 11-19. Google Scholar

[55]

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

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