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 and 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. 

[2]

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

[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.

[4]

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

[5]

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

[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. 

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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. 

[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.

[15]

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

[16]

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

[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. 

[18]

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

[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. 

[20]

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

[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.

[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.

[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.

[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. 

[25]

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

[26]

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

[27]

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

[28]

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

[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.

[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.

[31]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[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.

[45]

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

[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.

[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. 

[48]

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

[49]

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

[50]

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

[51]

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

[52]

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

[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.

[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. 

[55]

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

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. 

[2]

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

[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.

[4]

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

[5]

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

[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. 

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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. 

[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.

[15]

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

[16]

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

[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. 

[18]

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

[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. 

[20]

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

[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.

[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.

[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.

[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. 

[25]

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

[26]

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

[27]

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

[28]

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

[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.

[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.

[31]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[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.

[45]

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

[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.

[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. 

[48]

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

[49]

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

[50]

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

[51]

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

[52]

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

[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.

[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. 

[55]

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

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Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349

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