AIMS Mathematics, 2019, 4(5): 1430-1449. doi: 10.3934/math.2019.5.1430.

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Existence and uniqueness solutions of fuzzy integration-differential mathematical problem by using the concept of generalized differentiability

1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
2 Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran
3 Faculty of Engineering and Natural Sciences, Bahcesehir University Istanbul, Turkey

In this study, we demonstrate studies on two types of solutions linear fuzzy functional integration and differential equation under two kinds Hukuhara derivative by using the concept of generalized differentiability. Various types of solutions to are generated by applying of two separate concepts of fuzzy derivative in formulation of differential problem. Some patterns are presented to describe these results.
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Keywords existence and uniqueness solutions; integration-differential mathematical problem; generalized differentiability

Citation: M. R. Nourizadeh, N. Mikaeilvand, T. Allahviranloo. Existence and uniqueness solutions of fuzzy integration-differential mathematical problem by using the concept of generalized differentiability. AIMS Mathematics, 2019, 4(5): 1430-1449. doi: 10.3934/math.2019.5.1430

References

  • 1. B. Ahmad and S. Sivasundaram, Dynamics and stability of impulsive hybrid setvalued integration and differential equations with delay, Nonlinear Analysis: Theory, Methods & Applications, 65 (2006), 2082-2093.
  • 2. A. Ahmadian, M. Suleiman, S. Salahshour, et al. A Jacobi operational matrix for solving a fuzzy linear differential equation, Adv. Differ. Equ-NY, 2013 (2013), 104.
  • 3. R. P. Agarwal, V. Lakshmikantham and J. J. Nieto, On the concept of solution for differential equations with uncertainty, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 2859-2862.
  • 4. R. P. Agarwal, D. O'Regan and V. Lakshmikantham, Viability theory and fuzzy differential equations, Fuzzy Set. Syst., 151 (2005), 536-580.
  • 5. T. Allahviranloo, S. Abbasbandy, O. Sedaghatfar, et al. A New Method for Solving Fuzzy Integration and differential Equation Under Generalized Differentiability, Neural Computing and Applications, 21 (2012), 191-196.    
  • 6. T. Allahviranloo, S. Hajighasemi, M. Khezerloo, et al. Existence and Uniqueness of Solutions of Fuzzy Volterra Integration and differential Equations, International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, (2010), 491-500.
  • 7. T. Allahviranloo, A. Amirteimoori, M. Khezerloo, et al. A New Method for Solving Fuzzy Volterra Integration and differential Equations, Australian Journal of Basic and Applied Sciences, 5 (2011), 154-164.
  • 8. T. Allahviranloo, S. Salahshour and S. Abbasbandy, Solving fuzzy differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci., 17 (2012), 1372-1381.    
  • 9. T. Allahviranloo, S. Abbasbandy, O. Sedaghatfar, et al. A New Method for Solving Fuzzy Integration and differential Equation Under Generalized Differentiability, Neural Computing and Applications, 21 (2012), 191-196.    
  • 10. T. Allahviranloo, M. Ghanbari, E. Haghi, et al. A note on "Fuzzy linear systems", Fuzzy Set. Syst., 177 (2011), 87-92.    
  • 11. T. Allahviranloo, S. Abbasbandy, N. Ahmady, et al. Improved predictor corrector method for solving fuzzy initial value problems, Inform. Sciences, 179 (2009), 945-955.    
  • 12. T. Allahviranloo, N. A. Kiani and N. Motamedi, Solving fuzzy differential equations by differential transformation method, Inform. Sciences, 179 (2009), 956-966.    
  • 13. T. Allahviranloo, S. Abbasbandy, S. Salahshour, et al. A new method for solving fuzzy linear differential equations, Computing, 92 (2011), 181-197.    
  • 14. T. Allahviranloo and S. Salahshour, A new approach for solving first order fuzzy differential equation, International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, 81 (2010), 522-531.
  • 15. T. Allahviranloo and S. Salahshour, Euler method for solving hybrid fuzzy differential equation, Soft Comput., 15 (2011), 1247-1253.    
  • 16. L. C. Barros, R. C. Bassanezi and P. A. Tonelli, Fuzzy modeling in population dynamics, Ecological Modeling, 128 (2000), 27-33.    
  • 17. B. Bede and S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Set. Syst., 151 (2005), 581-599.    
  • 18. B. Bede, I. J. Rudas and A. L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Inform. Sciences, 177 (2007), 1648-1662.    
  • 19. B. Bede and L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Set. Syst., 230 (2013), 119-141.    
  • 20. B. Bede, A note on "two-point boundary value problems associated with non-linear fuzzy differential equations", Fuzzy Set. Syst., 157 (2006), 986-989.    
  • 21. J. J. Buckley and T. Feuring, Fuzzy differential equations, Fuzzy Set. Syst., 110 (2000), 43-54.    
  • 22. S. S. L. Chang and L. Zadeh, On fuzzy mapping and control, IEEE T. Syst. Man Cy., 2 (1972), 30-34.
  • 23. Y. Chalco-Cano and H. Roman-Flores, On new solutions of fuzzy differential equations, Chaos Soliton. Fract., 38 (2008),112-119.    
  • 24. Y. Chalco-Cano, A. Rufian-Lizana, H. Román-Flores, et al. Calculus for interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Set. Syst., 219 (2013), 49-67.    
  • 25. M. Chen and C. Han, Some topological properties of solutions to fuzzy differential systems, Inform. Sciences, 197 (2012), 207-214.    
  • 26. D. Dubois and H. Prade, Towards fuzzy differential calculus part 3: Differentiation, Fuzzy Set. Syst., 8 (1982), 225-233.    
  • 27. L. J. Jowers, J. J. Buckley and K. D. Reilly, Simulating continuous fuzzy systems, Inform. Sciences, 177 (2007), 436-448.    
  • 28. O. Kaleva, Fuzzy differential equations, Fuzzy Set. Syst., 24 (1987), 301-317.    
  • 29. A. Khastan, J. J. Nieto and R. Rodríguez-López, Variation of constant formula for first order fuzzy differential equations, Fuzzy Set. Syst., 177 (2011), 20-33.    
  • 30. A. Khastan, J. J. Nieto, R. Rodríguez-López, Fuzzy delay differential equations under generalized differentiability, Inform. Sciences, 275 (2014), 145-167    
  • 31. S. Khezerloo, T. Allahviranloo, S. H. Ghasemi, et al. Expansion Method for Solving Fuzzy Fredholm-Volterra Integral Equations, International Conference on Information Processing and Management of Uncertainty in KnowledgeBased Systems, 81 (2010), 501-511.
  • 32. V. B. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992.
  • 33. V. Lakshmikantham and R. N. Mohapatra, Theory of fuzzy differential equations and inclusions, Taylor Francis, London, 2003.
  • 34. V. Lakshmikantham and S. Leela, Fuzzy differential systems and the new concept of stability, Nonlinear Dynamics and Systems Theory, 1 (2001), 111-119.
  • 35. V. Lupulescu, On a class of fuzzy functional differential equations, Fuzzy Set. Syst., 160 (2009), 1547-1562.    
  • 36. V. Lupulescu, Initial value problem for fuzzy differential equations under dissipative conditions, Inform. Sciences, 178 (2008), 4523-4533.    
  • 37. J. K. Hale, Theory of functional differential equations, Springer, New York, 1997.
  • 38. M. T. Malinowski, Interval differential equations with a second type Hukuhara derivative, Applied Mathematics Letters, 24 (2011), 2118-2123.    
  • 39. M. T. Malinowski, Random fuzzy differential equations under generalized Lipschitz condition, Nonlinear Analysis: Real World Applications, 13 (2012), 860-881.    
  • 40. M. T. Malinowski, Existence theorems for solutions to random fuzzy differential equations, Nonlinear Analysis: Theory, Methods & Applications, 73 (2010), 1515-1532.
  • 41. M. T. Malinowski, On random fuzzy differential equations, Fuzzy Set. Syst., 160 (2009), 3152-3165.    
  • 42. M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, et al. Fuzzy differential equations and the extension principle, Inform. Sciences, 177 (2007), 3627-3635.    
  • 43. J. J. Nieto, A. Khastan and K. Ivaz, Numerical solution of fuzzy differential equations under generalized differentiability, Nonlinear Analysis: Hybrid Systems, 3 (2009), 700-707.    
  • 44. J. J. Nieto, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Set. Syst., 102 (1999), 259-262.    
  • 45. J. J. Nieto and R. Rodríguez-López, Bounded solutions for fuzzy differential and integral equations, Chaos Soliton. Fract., 27 (2006), 1376-1386.    
  • 46. J. Y. Park and J. U. Jeong, On existence and uniqueness of solutions of fuzzy integration and differential equations, Indian J. Pure Appl. Math., 34 (2003), 1503-1512.
  • 47. P. Prakash, J. J. Nieto, J. H. Kim, et al. Existence of solutions of fuzzy neutral differential equations in Banach spaces, Dynamical Systems and Applications, 14 (2005), 407-418.
  • 48. M. L. Puri and D. Ralescu, Differential for fuzzy functions, J. Math. Anal. Appl., 91 (1983), 552-558.    
  • 49. S. Salahshour and T. Allahviranloo, Applications of fuzzy Laplace transforms, Soft Computing, 17 (2013), 145-158.    
  • 50. S. Salahshour and T. Allahviranloo, Application of fuzzy differential transform method for solving fuzzy Volterra integral equations, Appl. Math. Model., 37 (2013), 1016-1027.    
  • 51. S. Salahshour and M. Khan, Exact solutions of nonlinear interval Volterra integral equations, International Journal of Industrial Mathematics, 4 (2012), 375-388.
  • 52. S. Salahshour, M. Khezerloo, S. Hajighasemi, et al. Solving Fuzzy Integral Equations of the Second Kind by Fuzzy Laplace Transform Method, International Journal of Industrial Mathematics, 4 (2012), 21-29.
  • 53. L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 1311-1328.
  • 54. S. Song and C. Wu, Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations, Fuzzy Set. Syst., 110 (2000), 55-67.    
  • 55. M. R. Nourizadeh, T. Allahviranloo, N. Mikaeilvand, Positive solutions of fuzzy fractional Volterra integro-differential equations with the Fuzzy Caputo Fractional Derivative using the Jacobi polynomials operational matrix, International Journal of Computer Science and Network Security, 18 (2018), 241-252.
  • 56. X. P. Xue and Y. Q. Fu, On the structure of solutions for fuzzy initial value problem, Fuzzy Set. Syst., 157 (2006), 212-229.    
  • 57. L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.    

 

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