Journal of Mathematical Sciences and Applications
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Journal of Mathematical Sciences and Applications. 2013, 1(1), 1-5
DOI: 10.12691/jmsa-1-1-1
Open AccessArticle

Existence and Uniqueness Theorem for Fuzzy Integral Equation

Andrej V. Plotnikov1, 2, and Natalia V. Skripnik2

1Department of Applied Mathematics, Odessa State Academy Civil Engineering and Architecture, Odessa, Ukraine

2Department of Optimal Control and Economic Cybernetics, Odessa National University named after I.I. Mechnikov, Odessa, Ukraine

Pub. Date: March 02, 2013

Cite this paper:
Andrej V. Plotnikov and Natalia V. Skripnik. Existence and Uniqueness Theorem for Fuzzy Integral Equation. Journal of Mathematical Sciences and Applications. 2013; 1(1):1-5. doi: 10.12691/jmsa-1-1-1

Abstract

In this article we consider fuzzy integral equations and prove the existence and uniqueness theorem.

Keywords:
fuzzy integral equation existence uniqueness fuzzy differential equation

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  Zadeh, , “Fuzzy sets,” Inf. Control, (8), 338-353, 1965.
 
[2]  Kaleva, O., “Fuzzy differential equations,” Fuzzy Sets Syst., 24 (3), 301-317, 1987.
 
[3]  Lakshmikantham, V., Gnana Bhaskar, T. and Vasundhara, Devi J. Theory of set differential equations in metric spaces, Cambridge Scientific Publishers, Cambridge, 2006.
 
[4]  Lakshmikantham, V. and Mohapatra, R. Theory of fuzzy differential equations and inclusions, Taylor - Francis, 2003.
 
[5]  Park, J.Y., and Han, H.K., “Existence and uniqueness theorem for a solution of fuzzy differential equations,” Int. J. Math. Math. Sci., 22 (2), 271-279, 1999.
 
[6]  Park, J.Y., and Han, H.K., “Fuzzy differential equations,” Fuzzy Sets Syst., 110 (1), 69-77, 2000.
 
[7]  Plotnikov, A.V. and Skripnik, N.V., Differential equations with ''clear'' and fuzzy multivalued right-hand sides. Asymptotics Methods (in Russian), AstroPrint, Odessa, 2009.
 
[8]  Sadigh Behzadi, Sh., “Solving Fuzzy Nonlinear Volterra-Fredholm Integral Equations by Using Homotopy Analysis and Adomian Decomposition Methods,” Journal of Fuzzy Set Valued Analysis, Volume 2011 Article ID jfsva-00067, 13 Pages, 2011.
 
[9]  Jahantigh, M., Allahviranloo, T. and Otadi, M., “Numerical Solution of Fuzzy Integral Equations,” Applied Mathematical Sciences, 2 (1), 33-46, 2008.
 
[10]  Friedman, M., Ma, M. and Kandel, A., “Numerical Solutions of fuzzy differential equations and integral equations,” Fuzzy Sets and Systems, 106, 35-48, 1999.
 
[11]  Ghanbari, M., Toushmalni, R. and Kamrani, E., “Numerical Solution of Linear Fredholm Fuzzy Integral Equation of the Second Kind by Block-pulse Functions,” Australian Journal of Basic and Applied Sciences, 3 (3), 2637-2642, 2009.
 
[12]  Mordeson J. and Newman, W., “Fuzzy Integral Equations,” Information Sciences, 87 (4), 215-229, 1995.
 
[13]  Park, J.Y., Kwun, Y.C. and Jeong, J.U., “Existence of solutions of fuzzy integral equations in Banach spaces,” Fuzzy Sets and Systems, 72, 373-378, 1995.
 
[14]  Parandin, N. and Fariborzi Araghi, M. A., “The Approximate Solution of Linear Fuzzy Fredholm Integral Equations of the Second Kind by Using Iterative Interpolation,” World Academy of Science, Engineering and Technology, 25, 978-984, 2009.
 
[15]  Shamivand, M.M., Shahsavaran, A. and Tari, S.M., “Solution to Fredholm Fuzzy Integral Equations with Degenerate Kernel,” Int. J. Contemp. Math. Sciences, 6 (11), 535-543, 2011.
 
[16]  Wu, C. and Ma, M., “On the integrals,series and integral equations of fuzzy set-valued functions,” J. Harbin Inst. Technol., 21, 11-19, 1990.
 
[17]  Allahviranloo, T., Amirteimoori, A., Khezerloo, M., and Khezerloo, S., “A new method for solving fuzzy volterra integro-differential equations,” Australian Journal of Basic and Applied Sciences, 5 (4), 154-164, 2011.
 
[18]  Balachandran, K., and Kanagarajan, K., “Existence of solutions of fuzzy delay integrodifferential equations with nonlocal condition,” Journal of Korea Society for Industrial and Applied Mathematics, 9 (2), 65-74, 2005.
 
[19]  Balasubramaniam, P., and Muralisankar, S., “Existence and uniqueness of fuzzy solution for the nonlinear fuzzy integrodifferential equations,” Appl. Math. Lett., 14 (4), 455-462, 2001.
 
[20]  Balasubramaniam, P., and Muralisankar, S., “Existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions,” Comput. Math. Appl., 47, 1115-1122, 2004.
 
[21]  Aubin, J.-P., “Fuzzy differential inclusions,” Probl. Control Inf. Theory, 19 (1), 55-67, 1990.
 
[22]  Baidosov, V. A., “Differential inclusions with fuzzy right-hand side,” Sov. Math., 40 (3), 567-569, 1990.
 
[23]  Baidosov, V. A., “Fuzzy differential inclusions,” J. Appl. Math. Mech., 54 (1), 8-13, 1990.
 
[24]  Hullermeier, E., “An approach to modeling and simulation of uncertain dynamical system,” Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 7, 117-137, 1997.
 
[25]  Plotnikov, A. V., and Skripnik, N. V., 2009, The generalized solutions of the fuzzy differential inclusions, Int. J. Pure Appl. Math., 56(2), 165-172.
 
[26]  Skripnik, N.V., “The full averaging of fuzzy differential inclusions,” Iranian Journal of Optimization, 1, 302-317, 2009.
 
[27]  Skripnik, N.V., “The partial averaging of fuzzy differential inclusions,” J. Adv. Res. Differ. Equ., 3 (1), 52-66, 2011.
 
[28]  Skripnik, N.V., “The partial averaging of fuzzy impulsive differential inclusions,” Differential and Integral Equations, 24 (7-8), 743-758, 2011.
 
[29]  Kwun, Y.C., and Park, D.G., “Optimal control problem for fuzzy differential equations,” Proceedings of the Korea-Vietnam Joint Seminar, 103-114, 1998.
 
[30]  Phu, N.D., and Tung, T.T., “Some results on sheaf-solutions of sheaf set control problems,” Nonlinear Anal., 67(5), 1309-1315, 2007.
 
[31]  Plotnikov, A.V., Komleva, T.A., and Arsiry, A.V., “Necessary and sufficient optimality conditions for a control fuzzy linear problem,” Int. J. Industrial Mathematics, 1 (3), 197-207, 2009.
 
[32]  Kwun, Y.C., Kim, M.J., Lee, B.Y., and Park, J.H., “Existence of solutions for the semilinear fuzzy integrodifferential equations using by successive iteration,” Journal of Korean Institute of Intelligent Systems, 18, 543-548, 2008.
 
[33]  Kwun, Y.C., Kim, J.S., Park, M.J., and Park, J.H., “Nonlocal controllability for the semilinear fuzzy integrodifferential equations in n-dimensional fuzzy vector space,” Adv. Difference Equ., vol. 2009, Article ID 734090, 16 pages, 2009.
 
[34]  Kwun, Y.C., Kim, J.S., Park, M.J., and Park, J.H., “Controllability for the impulsive semilinear nonlocal fuzzy integrodifferential equations in n-dimensional fuzzy vector space,” Adv. Difference Equ., vol. 2010, Article ID 983483, 22 pages, 2010.
 
[35]  Park, J.H., Park, J.S., and Kwun, Y.C., “Controllability for the semilinear fuzzy integrodifferential equations with nonlocal conditions,” Fuzzy Systems and Knowledge Discovery, Lecture Notes in Computer Science, vol. 4223/2006, 221-230, 2006.
 
[36]  Park, J.H., Park, J.S., Ahn, Y.C., and Kwun, Y.C., “Controllability for the impulsive semilinear fuzzy integrodifferential equations,” Adv. Soft Comput., 40, 704-713, 2007.
 
[37]  Molchanyuk, I.V., and Plotnikov, A.V., “Linear control systems with a fuzzy parameter,” Nonlinear Oscil., 9 (1), 59-64, 2006.
 
[38]  Molchanyuk, I.V., and Plotnikov, A.V., “Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters,” Ukr. Math. J., 61 (3), 457-463, 2009.
 
[39]  Plotnikov, A.V., and Komleva, T.A., “Linear problems of optimal control of fuzzy maps,” Intelligent Information Management, 1 (3), 139-144, 2009.
 
[40]  Plotnikov, A.V., Komleva, T.A., and Molchanyuk, I.V., “Linear control problems of the fuzzy maps,” J. Software Engineering & Applications, 3 (3), 191-197, 2010.
 
[41]  Vasil'kovskaya, V.S., and Plotnikov, A.V., “Integrodifferential systems with fuzzy noise,” Ukr. Math. J., 59(10), 1482-1492, 2007.
 
[42]  Puri, M.L., and Ralescu, D.A., “Fuzzy random variables,” J. Math. Anal. Appl., 114(2), 409-422, 1986.
 
[43]  Dubois, D. and Prade, H., “Towards fuzzy differential calculus. I. Integration of fuzzy mappings,” Fuzzy Sets and Systems, 8 (1), 1-17, 1982.
 
[44]  Dubois, D. and Prade, H., “Towards fuzzy differential calculus. II. Integration on fuzzy intervals,” Fuzzy Sets and Systems, 8 (2), 105-116, 1982.