FIXED POINTS OF SOFT SET-VALUED MAPS WITH APPLICATIONS TO DIFFERENTIAL INCLUSIONS

Abstract

In this paper, a notion of soft set-valued maps in Hausdorff fuzzy metric space is introduced. To this end, we establish fixed point theorems of set-valued mappings whose range set lies in a family of soft sets. Consequently, a few significant fixed point results of fuzzy, multivalued and single-valued mappings are pointed out and discussed. Some illustrative nontrivial examples which dwell upon the generality of our results are also provided. As an application, sufficient conditions for solvability of multi-valued boundary value problems involving both Riemann-Liouville and Caputo fractional derivatives with non-local fractional integro-differential boundary conditions are investigated to indicate a usability of the ideas presented herein.

Keywords

• Fuzzy set
• fuzzy mapping
• fuzzy metric space
• Hausdorff fuzzy metric
• soft set
• soft set-valued map
• e-soft fixed point
• fractional differential inclusion

References

1. [1] Abbas, M., Altun, I., and Gopal, D. (2009). Common fixed point theorems for non compatible mappings in fuzzy metric spaces. Bulletin of Mathematical Analysis and Applications, 1(2), 47-56.
2. [2] Abdeljawad, T., Agarwal, R. P., Karapinar, E., and Kumari, P. S. (2019). Solutions of the nonlinear integral equation and fractional differential equation using thetechnique of a fixed point with a numerical experiment in extended b-metric space. Symmetry, 11(5), 686.
3. [3] Atanassov, K. T. (1999). Intuitionistic fuzzy sets. In Intuitionistic fuzzy sets (pp. 1-137). Physica, Heidelberg.
4. [4] Atangana, A., and Owolabi, K. M. (2018). New numerical approach for fractional differential equations. Mathematical Modelling of Natural Phenomena, 13(1), 3.
5. [5] Belarbi, S., and Dahmani, Z. (2013). Some applications of Banach fixed point and Leray Schauder theorems for fractional boundary value problems. Journal of Dynamical Systems and Geometric Theories, 11(1-2), 59-73.
6. [6] Benchohra, M., Hamani, S., and Ntouyas, S. K. (2009). Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Analysis: Theory, Methods and Applications, 71(7-8), 2391-2396.
7. [7] Cagman, N., Karataş, S., and Enginoglu, S. (2011). Soft topology. Computers and Mathematics with Applications, 62(1), 351-358.
8. [8] Castaing, C., and Valadier, M. (2006). Convex analysis and measurable multifunctions (Vol. 580). Springer.

9. [9] Covitz, H., and Nadler, S. B. (1970). Multi-valued contraction mappings in generalized metric spaces. Israel Journal of Mathematics, 8(1), 5-11.
10. [10] Fatimah, F., Rosadi, D., Hakim, R. F., and Alcantud, J. C. R. (2018). N-soft sets and their decision making algorithms. Soft Computing, 22(12), 3829-3842.
11. [11] Frigon, M., and O'Regan, D. (2002). Fuzzy contractive maps and fuzzy fixed points. Fuzzy Sets and Systems, 129(1), 39-45.
12. [12] Goguen, J. A. (1967). L-fuzzy sets. Journal of mathematical analysis and applications, 18(1), 145-174.
13. [13] Gregori, V., Morillas, S., and Sapena, A. (2011). Examples of fuzzy metrics and applications. Fuzzy Sets and Systems, 170(1), 95-111.
14. [14] Gregori, V., and Mi~nana, J. J. (2016). On fuzzy -contractive sequences and fixed point theorems. Fuzzy Sets and Systems, 300, 93-101.
15. [15] Gregori, V., and Sapena, A. (2002). On fixed-point theorems in fuzzy metric spaces. Fuzzy sets and systems, 125(2), 245-252.
16. [16] George, A., and Veeramani, P. (1997). On some results of analysis for fuzzy metric spaces. Fuzzy sets and systems, 90(3), 365-368.
17. [17] Grabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy sets and systems, 27(3), 385-389.
18. [18] Heilpern, S. (1981). Fuzzy mappings and fixed point theorem. Journal of Mathematical Analysis and Applications, 83(2), 566-569.
19. [19] Houas, M., and Dahmani, Z. (2016). On existence of solutions for fractional differential equations with nonlocal multi-point boundary conditions. Lobachevskii Journal of Mathematics, 37(2), 120-127.
20. [20] Hussain, N., Kutbi, M. A., and Salimi, P. (2020). Global optimal solutions for proximal fuzzy contractions. Physica A: Statistical Mechanics and its Applications, 123925.
21. [21] Kiany, F., and Amini-Harandi, A. (2011). Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces. Fixed Point Theory and Applications, 2011(1), 94.
22. [22] Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J. (2006). Theory and applications of fractional differential equations (Vol. 204). Elsevier Science Limited.
23. [23] Kramosil, I., and Michalek, J. (1975). Fuzzy metrics and statistical metric spaces. Kybernetika, 11(5), 336-344.
24. [24] Mehmood, N., and Ahmad, N. (2019). Existence results for fractional order boundary value problem with nonlocal non-separated type multi-point integral boundary conditions. AIMS Mathematics, 5(1): 385{398.
25. [25] Mehmood, F., Ali, R., and Hussain, N. (2019). Contractions in fuzzy rectangular b-metric spaces with application. Journal of Intelligent and Fuzzy Systems, (Preprint), 1-11.
26. [26] Michael, E. (1966). A selection theorem. Proceedings of the American Mathematical Society, 17(6), 1404-1406.
27. [27] Mihet, D. (2008). Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 159(6), 739-744.
28. [28] Mihet, D. (2010). Fixed point theorems in fuzzy metric spaces using property EA. Nonlinear Analysis: Theory, Methods and Applications, 73(7), 2184-2188.
29. [29] Mishra, S. N., Sharma, N., and Singh, S. L. (1994). Common fixed points of maps on fuzzy metric spaces. International Journal of Mathematics and Mathematical Sciences, 17(2), 253-258.
30. [30] Mohammed, S. S., and Azam, A. (2019). Fixed points of soft-set valued and fuzzy set-valued maps with applications. Journal of Intelligent and Fuzzy Systems, vol. 37, no. 3, 3865-3877.
31. [31] Mohammed, S. S., and Azam, A. (2019). Integral type contractions of soft set valued maps with application to neutral differential equation. AIMS Mathematics, 5(1), 342-358.
32. [32] Mohammed, S. S. and Azam, A. (2020). An algorithm for fuzzy soft set based decision making approach. Yugoslav Journal of Operations Research, 30(1), 59-70.
33. [33] Mohammed, S. S. (2020). On Bilateral fuzzy contractions. Functional Analysis, Approximation and Computation, 12 (1), 1-13.
34. [34] Molodtsov, D. (1999). Soft set theoryfirst results. Computers and Mathematics with Applications, 37(4-5), 19-31.
35. [35] Nadler, S. B. (1969). Multi-valued contraction mappings. Pacific Journal of Mathematics, 30(2), 475-488.
36. [36] Ntouyas, S. K. (2013). Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions. Opuscula Mathematica, 33(1), 117-138.
37. [37] Ntouyas, S. K., Alsaedi, A., and Ahmad, B. (2019). Existence Theorems for Mixed RiemannLiouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions. Fractal and Fractional, 3(2), 21.
38. [38] Phiangsungnoen, S., Sintunavarat, W., and Kumam, P. (2014). Fuzzy fixed point theorems in Hausdor_ fuzzy metric spaces. Journal of Inequalities and Applications, 2014(1), 201.
39. [39] Raki_c, D., Mukheimer, A., Do_senovi_c, T., Mitrovi_c, Z. D., and Radenovic, S. (2020). On some new fixed point results in fuzzy b-metric spaces. Journal of Inequalities and Applications, 2020(1), 1-14.
40. [40] Rakic, D., Dosenovic, T., Mitrovic, Z. D., de la Sen, M., and Radenovic, S. (2020). Some fixed point theorems of Ciric type in fuzzy metric spaces. Mathematics, 8(2), 297.
41. [41] Riaz, M., Cagman, N., Zareef, I., and Aslam, M. (2019). N-soft topology and its applications to multi-criteria group decision making. Journal of Intelligent and Fuzzy Systems, 36(6), 6521-6536.
42. [42] Rodriguez-Lopez, J., and Romaguera, S. (2004). The Hausdor_ fuzzy metric on compact sets. Fuzzy sets and systems, 147(2), 273-283.
43. [43] Saini, R. K., and Singh, S. B. (2012). Fuzzy version of some fixed point theorems on expansion type maps in fuzzy metric spaces. Thai Journal of Mathematics, 5(2), 245-252.
44. [44] Schweizer, B., and Sklar, A. (1960). Statistical metric spaces. Pacific J. Math, 10(1), 313-334.
45. [45] Sedghi, S., Shobkolaei, N., Do_senovic, T., and Radenovic, S. (2018). Suzuki-type of common fixed point theorems in fuzzy metric spaces. Math. Slovaca, 68(2), 451-462.
46. [46] Shoaib, A., Azam, A., and Shahzad, A. (2018). Common Fixed Point Results for the Family of Multivalued Mappings Satisfying Contractions on a Sequence in Hausdorff Fuzzy Metric Space. Journal of Computational Analysis and Applications, 24(4).
47. [47] Toufik, M., and Atangana, A. (2017). New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models. The European Physical Journal Plus, 132(10), 444.
48. [48] Wardowski, D. (2013). Fuzzy contractive mappings and fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 222, 108-114.
49. [49] Yan, R., Sun, S., Sun, Y., and Han, Z. (2013). Boundary value problems for fractional differential equations with nonlocal boundary conditions. Advances in Difference Equations, 2013(1), 176.
50. [50] Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338-353.