SOLUTION TO A SYSTEM OF NON-LINEAR FUZZY DIFFERENTIAL EQUATION WITH GENERALIZED HUKUHARA DERIVATIVE VIA FIXED POINT THEOREM
Keywords:
Simulation function, Fuzzy mappings, Fuzzy coupled fixed point, Fuzzy differential equationAbstract
In this manuscript, we define a new class of control functions classified as ascendant functions. Consequently, we furnish
a fuzzy coupled fixed point result, that is different from one available in the literature, using the notion of simulation function; in follow, we validate the result through a non-trivial example. As an inference, we use the result to analyze the existence of a solution for a non-linear system of fuzzy initial value problem involving generalized Hukuhara derivative.
References
H. M. Abu-Donia, Common fixed point theorems for fuzzy mappings in metric space under φ-contraction condition, Chaos Solit. 34(2) (2007) 538-543.
H. Afshari, E. Karapinar, A solution of the fractional differential equations in the setting of b-metric space, Carpathian Math. Publ. 13 (3) (2021) 764-774.
H. Afshari, E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces, Adv. Differ. Equ. 616 (2020).
J. Ahmad, G. Marino, S. A. Al-Mezel, Common α-fuzzy fixed point results for F-contractions with Applications, Mathematics. 9(3) (2021) 1-14.
H. Argoubi, B. Samet, C. Vetro, Nonlinear contractions involving simulation functions in a metric space with a partial order, J. Nonlinear Sci. Appl. 8(6) (2015) 1082-1094.
A. Azam, I. Beg, Common fixed points of fuzzy maps, Math. Comput. Model. 49 (2009) 1331-1336.
A. Azam, M. Arshad, P. Vetro, On a pair of fuzzy φ-contractive mappings, Math. Comput. Model. 52 (2010) 207-214.
S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fundam. Math. 3 (1922) 133-181.
T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65(7) (2006) 1379-1393.
V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. Theory Methods Appl. 70(12) (2009) 4341-4349.
S. Heilpern, Fuzzy mappings and fixed point theorem, J. Math. Anal. Appl. 83 (1981) 566-569.
O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst. 24(3) (1987) 301-317.
T. Kamaran, Common fixed points theorems for fuzzy mappings, Chaos Solit. 38(5) (2008) 1378-1382.
E. Karapinar, J. Martinez-Moreno, N. Shahzad, A. F. Rolda´n Lo´pez de Hierro, Extended Proinov X-contractions in metric spaces and fuzzy metric spaces satisfying the property N C by avoiding the monotone condition, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116:140 (2022). [15] F. Khojasteh, S. Shukla, S. Radenovic, A new approach to the study of fixed points theory via simulation functions, Filomat. 29(6) (2015) 1189-1194.
B.S. Lee, S. J. Cho, A fixed point theorem for contractive type fuzzy mappings, Fuzzy sets Syst. 61(3) (1994), 309-312.
Y. Lin, J.H. Liu, Semilinear integro-differential equations with nonlocal Cauchy problem, Nonlinear Anal. Theory Methods Appl. 26(5) (1996) 1023-1033.
L. Liu, A. Mao, Y. Shi, New fixed point theorems and application of mixed monotone mappings in partially ordered metric spaces, J. Funct. Spaces, 2018(2) (2018) 1-11.
S. B. Nadler Jr, Multivalued contraction mappings, Pac. J. Appl. Math. 30(2) (1969) 475-488.
J.J. Nieto, A. Ouahab, R. Rodrguez-Lpez, Random fixed point theorems in partially ordered metric spaces, J. Fixed Point Theory Appl. 2016(98) (2016) 1-19.
S.K. Ntouyas, P.Ch. Tsamatos, Global existence for semilinear evolution integro differential equations with delay and nonlocal conditions, Appl. Anal. 64 (1997) 99-105.
W. Sintunavarat, P. Kumam, Y. J. Cho, Coupled fixed point theorems for nonlinear contractions without mixed monotone property, J. Fixed Point Theory Appl. 170 (2012).
L. A. Zadeh, Fuzzy sets, Inf. Control. 8 (1965) 338-353.
