Sushma BASİL, Santhi ANTONY

Solution of Fuzzy Volterra Integral and Fractional Differential Equations via Fixed Point Theorems

In this paper we present fuzzy coupled fixed point results in the turf of complete b-metric spaces via nonlinear F-contraction; in follow we derive some interesting results as byproducts. Eventually, we apply our results in solving fuzzy Volterra integral equations and Caputo-Hadamard type of fractional differential equations.

Keywords:

Fuzzy mapping, Fuzzy coupled fixed point, b-metric space Fuzzy Volterra integral equations, Caputo-Hadamard fractional derivative,

PDF

___

1] H.M. Abu-Donia, Common fixed point theorems for fuzzy mappings in metric space under φ-contraction condition, Chaos Solitons Fractals. 34(2) (2007) 538-543.
[2] J. Ahmad, A. Al-Rawashdeh, A. Azam, New fixed point theorems for generalized F-contraction in complete metric spaces, Fixed Point Theory Appl. 80 (2015) 1-18.
[3] T. Allahviranloo, P. Salehi, M. Nejatiyan, Existence and uniqueness of the solution of nonlinear fuzzy Volterra integral equations, Iranian Journal of Fuzzy Systems. 12(2) (2015) 75-86.
[4] B. Alqahtani, A. Fulga, F. Jarad and E. Karapınar, Nonlinear F-contractions on b-metric spaces and differential equations in the frame of fractional derivatives with Mittag-Leffler Kernel, Chaos, Solitons and Fractals. 128(C) (2019) 349-354.
[5] A. Azam, M. Arshad, P. Vetro, On a pair of fuzzy contractive mappings, Math. Comput. Model. 52 (2010) 207-214.
[6] H. Aydi, M.F. Bota, E. Karapınar, S. Moradi, A common fixed point for weak-ϕ-contractions on b-metric spaces, Fixed Point Theory. 13(2) (2012) 337-346.
[7] I.A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal. 30, (1989) 26-37.
[8] T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65(7) (2006) 1379-1393.
[9] A. Boutiara, K. Guerbati, M. Benbachir, Caputo-Hadamard fractional differential equation with three-point boundary conditions in Ba- nach spaces, AIMS Mathematics. 5(1) (2019) 259-272.
[10] L.B. Ciric, V. Lakshmikantham, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Analysis TMA, 70(12) (2009) 4341-4349.
[11] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1(1) (1993) 5-11.
[12] M.A. Darwish, On the existence of a fuzzy integral equation of Urysohn-Volterra Type, Discussiones Mathematicae, Differential Inclu- sions, Control and Optimization. 28(1) (2008), 75-82.
[13] S. Heilpern, Fuzzy mappings and fixed point theorem, Journal of Mathematical Analysis and Applications, 83 (1981) 566-569.
[14] F. Jarad, D. Baleanu and A. Abdeljawad, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ. 142 (2012) 1-8.
[15] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst. 24(3) (1987) 301-317.
[16] A.A. Kilbas, H.M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier. 204 (2006).
[17] M.A. Kutbi, E. Karapınar, J. Ahmad and A. Azam, Some fixed point results for multivalued mappings in b-metric spaces. 126 (2014) 1-11.
[18] B.S. Lee, S.J. Cho, A fixed point theorem for contractive type fuzzy mappings, Fuzzy sets Syst. 61(3) (1994), 309-312.
[19] L.V. Nguyena, N.H. Hoc, On nonlinear F-contractive fuzzy mappings, Journal of Intelligent and Fuzzy Systems. 36(6) (2019) 6481-6491.
[20] H. Piri, P. Kumam, Wardowski Type Fixed Point Theorems in Complete Metric Spaces, Fixed Point Theory Appl. 45 (2016) 1-12.
[21] H. Qawaqneh, M.S. Noorani, W. Shatanawi, H. Aydi, H. Alsamir, Fixed Point Results for Multi-Valued Contractions in b-Metric Spaces and an Application, Mathematics, 7(132) (2019) 1-13.
[22] W. Sintunavarat, P. Kumam, Y. J. Cho, Coupled fixed point theorems for nonlinear contractions without mixed monotone property, Fixed Point Theory and Applications. 170 (2012) 1-16.
[23] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 94 (2012) 1-6.
[24] D. Wardowski, N. Dung, Fixed points of F-weak contraction on complete metric spaces, Demonstr. Math. XLVII. (2014) 146-155.
[25] D. Wardowski, Solving existence problems via F-contractions, Proc. Am. Math. Soc 146(4), (2018) 1585-1598.
[26] A. Yacine, B. Nouredine, Boundary value problem for Caputo-Hadamard fractional differential equations, Surveys in Mathematics and its Applications, 12 (2017) 103-115.
[27] L.A. Zadeh, Fuzzy sets, Inf. Control. 8 (1965) 338-353.
[28] L. Zhu, C.X. Zhu and X.J. Huang, Coupled coincidence and common fixed point theorems for single-valued and fuzzy mappings, Iranian Journal of Fuzzy Systems. 12(1) (2015) 75-87.