The concept of Hukuhara Derivative and Aumann Integrals for Intuitionistic fuzzy number valued functions

Abstract

In this paper we have frstly defned a metric in intuitionistic fuzzy environment and studied its properties. Then we have proved that the metric space of fuzzy number valued functions is complete under this metric. We have studied the concept of Aumann integration for intuitionistic fuzzy number valued functions in terms of α and β cuts. We have given the relation between Hukuhara derivative and Aumann integral for intuitionistic fuzzy valued functions by using the fundamental theorem of calculus.  

Keywords

• Intuitionistic fuzzy sets,Intuitionistic fuzzy valued functions,Hukuhara differentiability,Aumann integral,Intuitionistic Hausdorff metric

References

1. [1] Lotf A. Z., ”Fuzzy Sets,” Information and Control, 8, (1965), pp. 338-353.
2. [2] Krassimir T. A., ”Intuitionistic Fuzzy Sets,” Fuzzy Sets and Systems, 20, 1, (1986), pp. 87–96.
3. [3] Joseph A. G. ”L-Fuzzy Sets,” Journal of Mathematical Analysis and Applications, 18, 1,(1967), 145-174.
4. [4] Jerry M. M. ”Advances in Type-2 Fuzzy Sets and Systems,” Information Sciences, 177, 1, (2007), pp. 84-110.
5. [5] Krassimir T. A., Intuitionistic Fuzzy Sets: Theory and Applications, Physica, Heidelberg, 1999.
6. [6] Laecio C. B., Rodney C. B, Pedro A. T., ”Fuzzy Modelling in Population Dynamics,” Ecol. Model, 128, (2000), pp. 27-33.
7. [7] Deng-Feng L., ”Multiattribute Decision Making Models and Methods Using Intuitionistic Fuzzy Sets,” Journal of Computer and System Sciences, 70, 1 (2005), pp. 73-85.
8. [8] Jun Y., ”Multicriteria Fuzzy Decision-Making Method Based on a Novel Accuracy Function under Interval-Valued Intuitionistic Fuzzy Environment,” Expert Systems with Applications, 36, 3, (2009), pp. 6899-6902.

9. [9] Li D., Cheng C., ”New Similarity Measures of Intuitionistic Fuzzy Sets and Application to Pattern Recognitions,” Pattern Recognition Letters, 23, 1-3, (2002), pp. 221-225.
10. [10] Supriya K. D., Ranjit B., Akhil R. R., ”An Application of Intuitionistic Fuzzy Sets in Medical Diagnosis,” Fuzzy Sets and Systems, 117, 2, (2001), pp. 209-213.
11. [11] Athar K., ”Homeopathic Drug Selection Using Intuitionistic Fuzzy Sets,” Homeopathy, 98,1, (2009), pp. 35-39.
12. [12] Ming-Hung S., Ching-Hsue C., Jing-Rong C., ”Using Intuitionistic Fuzzy Sets for Faulttree Analysis on Printed Circuit Board Assembly,” Microelectronics Reliability, 46, 12, (2006), pp. 2139-2148.
13. [13] Laecio C. B., Rodney C. B, Pedro A. T., ”Fuzzy Modelling in Population Dynamics,” Ecol. Model, 128, (2000), pp. 27-33.
14. [14] Oktay D., ”Statistical Fuzzy Approximation to Fuzzy Differentiable Functions by Fuzzy Linear Operators,” Hacettepe Journal of Mathematics and Statistics, 39, 4, (2010), pp. 497-514.
15. [15] Omer A., ¨ Omer O., ”A Prey Predator Model with Fuzzy Initial Values,” Hacettepe Journal of Mathematics and Statistics, 41, 3, (2012), pp. 387-395.
16. [16] Senol D., Lawrence. M. B., ”Intuitionistic Textures Revisited,” Hacettepe Journal of Mathematics and Statistics, 34, (2005), 115-130.
17. [17] Jin Han P., ”Intuitionistic Fuzzy Metric Spaces”, Chaos, Solitons Fractals 22, 5, (2004), pp. 1039-1046.
18. [18] Qian L., Zeshui X., ”Fundamental Properties of Intuitionistic Fuzzy Calculus,” Knowledge-Based Systems, 76, (2015), pp. 1-16.
19. [19] M. Oberguggenberger, S. Pittschmann, ”Differential Equations with Fuzzy Parameters,” Math. Mod. Syst,, 5, (1999), 181-202.
20. [20] Omer A., Tahir K., ¨ Omer O., Burhan T., ”An Algorithm for the Solution of Second Order ¨ Fuzzy Initial Value Problems,” Expert Systems with Applications, 40, (2013), 953-957.
21. [21] Omer A., Tahir K., Selami B., Burhan T., ”Solving a Second Order Fuzzy Initial Value ¨ Problem Using the Heaviside Function,” Turk. J. Math. Comput. Sci., 4, (2016), 16-25.
22. [22] Phil D., Peter E. K., Metric Spaces of Fuzzy Sets: Theory and Applications. World scientifc, 1994.
23. [23] Masuo H., ”Integration des Applications Mesurables dont la Valeur est un Compact Convexe,” Funkcialaj Ekvacioj, 10, 3 (1967), pp. 205-223.
24. [24] Madan L. P., Dan A. R., ”Differentials of Fuzzy Functions,” Journal of Mathematical Analysis and Applications, 91, 2, (1983), 552-558.
25. [25] Osmo K., ”Fuzzy Differential Equations,” Fuzzy Sets and Systems, 24, 3, (1987), pp. 301-317.
26. [26] Eyke H., ”An Approach to Modelling and Simulation of Uncertain Dynamical Systems,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5, 02, (1997), pp. 117-137.
27. [27] Michio S., ”Theory of Fuzzy Integrals and its Applications,” Ph.D. Dissertation, Tokyo Institute of Technology, 1974.
28. [28] Dan R., Gregory A., ”The Fuzzy Integral,” Journal of Mathematical Analysis and Applications, 75, 2, (1980), pp. 562-570.
29. [29] Didier D., Henri P., ”Towards Fuzzy Differential Calculus Part 1: Integration of Fuzzy Mappings.” Fuzzy Sets and Systems, 8, 1, (1982), 1-17.
30. [30] Robert J. A., ”Integrals of Set-Valued Functions,” Journal of Mathematical Analysis and Applications, 12, 1, (1965), pp. 1-12.
31. [31] Jan V. T., ”Convex Analysis: An Introductory Text,” John WileySons, Chichester, UK, 1948.
32. [32] Carl P. S., Lawrence B., Mathematics for Economists, vol. 7, Norton, New York, 1994.
33. [33] Omer A. and Selami B., ”Initial Value Problems in Intuitionistic Fuzzy Environment,” Proceedings of the the 5th International Fuzzy Systems Symposium, Ankara, Turkey, October 14-15, 2017.
34. [34] Omer A., Selami B., ”Intuitionistic Fuzzy Initial Value Problems - An Application”, Hacettepe Journal of Mathematics and Statistics, (2017) Doi: 10.15672/HJMS.2018.598.