Generalized contraction theorems approach to fuzzy differential equations in fuzzy metric spaces

Iqra Shamas; Saif Ur Rehman; Thabet Abdeljawad; Mariyam Sattar; Sami Ullah Khan; Nabil Mlaiki; Iqra Shamas; Saif Ur Rehman; Thabet Abdeljawad; Mariyam Sattar; Sami Ullah Khan; Nabil Mlaiki

doi:10.3934/math.2022628

AIMS Mathematics

2022, Volume 7, Issue 6: 11243-11275. doi: 10.3934/math.2022628

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Research article

Generalized contraction theorems approach to fuzzy differential equations in fuzzy metric spaces

1.
Institute of Numerical Sciences, Department of Mathematics, Gomal University, Dera Ismail Khan 29050, Pakistan
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
4.
Department of Mechanical Engineering, Institute of Space Technology, P. O. Box 2750, Islamabad, Pakistan

Received: 12 January 2022 Revised: 23 March 2022 Accepted: 28 March 2022 Published: 11 April 2022
MSC : 47H10, 54H25

Fixed point theory is one of the most interesting areas of research in mathematics. In this direction, we study some unique common fixed point results for a pair of self-mappings without continuity on fuzzy metric spaces under the generalized contraction conditions by using "the triangular property of fuzzy metric". Moreover, we present weak-contraction and generalized Ćirić-contraction theorems. The results are supported by suitable examples. Further, we establish a supportable application of the fuzzy differential equations to ensure the existence of a unique common solution to validate our main work.
- fixed point,
- common fixed point,
- fuzzy differential equations,
- contraction conditions,
- fuzzy metric space
Citation: Iqra Shamas, Saif Ur Rehman, Thabet Abdeljawad, Mariyam Sattar, Sami Ullah Khan, Nabil Mlaiki. Generalized contraction theorems approach to fuzzy differential equations in fuzzy metric spaces[J]. AIMS Mathematics, 2022, 7(6): 11243-11275. doi: 10.3934/math.2022628

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Abstract

Fixed point theory is one of the most interesting areas of research in mathematics. In this direction, we study some unique common fixed point results for a pair of self-mappings without continuity on fuzzy metric spaces under the generalized contraction conditions by using "the triangular property of fuzzy metric". Moreover, we present weak-contraction and generalized Ćirić-contraction theorems. The results are supported by suitable examples. Further, we establish a supportable application of the fuzzy differential equations to ensure the existence of a unique common solution to validate our main work.

References

[1]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
[2]	I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975), 336–344.
[3]	A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Set. Syst., 64 (1994), 395–399. https://doi.org/10.1016/0165-0114(94)90162-7 doi: 10.1016/0165-0114(94)90162-7
[4]	M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Set. Syst., 27 (1988), 385–389. https://doi.org/10.1016/0165-0114(88)90064-4 doi: 10.1016/0165-0114(88)90064-4
[5]	F. Kiany, A. Amini-Haradi, Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces, Fixed Point Theory Appl., 2011 (2011), 94. https://doi.org/10.1186/1687-1812-2011-94 doi: 10.1186/1687-1812-2011-94
[6]	J. P. Aubin, J. Siegel, Fixed points and stationary points of dissipative multivalued maps, Proc. Amer. Math. Soc., 78 (1980), 391–398. https://doi.org/10.1090/S0002-9939-1980-0553382-1 doi: 10.1090/S0002-9939-1980-0553382-1
[7]	M. Fakhar, Endpoints of set-valued asymptotic contractions in metric spaces, Appl. Math. Lett., 24 (2010), 428–431. https://doi.org/10.1016/j.aml.2010.10.028 doi: 10.1016/j.aml.2010.10.028
[8]	V. Gregori, A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Set. Syst., 125 (2002), 245–252. https://doi.org/10.1016/S0165-0114(00)00088-9 doi: 10.1016/S0165-0114(00)00088-9
[9]	A. A. Harandi, Endpoints of set-valued contractions in metric spaces, Nonlinear Anal. Theor., 72 (2010), 132–134. https://doi.org/10.1016/j.na.2009.06.074 doi: 10.1016/j.na.2009.06.074
[10]	N. Hussain, A. A. Harandi, Y. J. Cho, Approximate endpoints for set-valued contractions in metric spaces, Fixed Point Theory Appl., 2010 (2010), 614867. https://doi.org/10.1155/2010/614867 doi: 10.1155/2010/614867
[11]	N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., 141 (1989), 177–188. https://doi.org/10.1016/0022-247X(89)90214-X doi: 10.1016/0022-247X(89)90214-X
[12]	S. U. Rehman, H. Aydi, G.-X. Chen, S. Jabeen, S. U. Khan, Some set-valued and multi-valued contraction results in fuzzy cone metric spaces, J. Inequal. Appl., 2021 (2021), 110. https://doi.org/10.1186/s13660-021-02646-3 doi: 10.1186/s13660-021-02646-3
[13]	K. Wlodarczyk, D. Klim, R. Plebaniak, Existence and uniqueness of endpoints of closed set-valued asymptotic contractions in metric spaces, J. Math. Anal. Appl., 328 (2007), 46–57. https://doi.org/10.1016/j.jmaa.2006.05.029 doi: 10.1016/j.jmaa.2006.05.029
[14]	C. D. Bari, C. Vetro, A fixed point theorem for a family of mappings in fuzzy metric space, Rend. Circ. Mat. Palermo, 52 (2003), 315–321. https://doi.org/10.1007/bf02872238 doi: 10.1007/bf02872238
[15]	I. Beg, M. Abbas, Invariant approximation for fuzzy nonexpansive mappings, Math. Bohem., 136 (2011), 51–59. https://doi.org/10.21136/MB.2011.141449 doi: 10.21136/MB.2011.141449
[16]	I. Beg, S. Sedghi, N. Shobe, Fixed point theorems in fuzzy metric spaces, Int. J. Anal., 2013 (2013), 934145. https://doi.org/10.1155/2013/934145 doi: 10.1155/2013/934145
[17]	C. D. Bari, C. Vetro, Fixed points, attractors and weak fuzzy contractive mappings in a fuzzy metric space, J. Fuzzy Math., 1 (2005), 973–982.
[18]	M. Imdad, J. Ali, Some common fixed point theorems in fuzzy metric spaces, Math. Commun., 11 (2006), 153–163.
[19]	A. F. R. L-de Hierro, A. F. Fulga, A. Karapinar, N. Shahzad, Proinov-type fixed-point results in non-archimedean fuzzy metric spaces, Mathematics, 9 (2021), 1594. https://doi.org/10.3390/math9141594 doi: 10.3390/math9141594
[20]	M. Jleli, E. Karapinar, B. Samet, On cyclic $(\psi, \phi)-$contractions in Kaleva-Seikkala's type fuzzy metric spaces, J. Intell. Fuzzy Syst., 27 (2014), 2045–2053. https://doi.org/10.3233/IFS-141170 doi: 10.3233/IFS-141170
[21]	X. Li, S. U. Rehman, S. U. Khan, H. Aydi, J. Ahmad, N. Hussain, Strong coupled fixed point results and applications to Urysohn integral equations, Dynam. Syst. Appl., 30 (2021), 197–218. https://doi.org/10.46719/dsa20213023 doi: 10.46719/dsa20213023
[22]	B. D. Pant, S. Chauhan, Common fixed point theorems for two pairs of weakly compatible mappings in menger spaces and fuzzy metric spaces, Sci. Stud. Res., 21 (2011), 81–96.
[23]	J. Rodriguez-Lopez, S. Romaguera, The Haudorff fuzzy metric on compact sets, Fuzzy Set. Syst., 147 (2008), 273–283. https://doi.org/10.1016/j.fss.2003.09.007 doi: 10.1016/j.fss.2003.09.007
[24]	S. U. Rehman, I. Shamas, N. Jan, A. Gumaei, M. Al-Rakhami, Some coincidence and common fixed point results in fuzzy metric space with an application to differential equations, J. Funct. Space., 2021 (2021), 9411993. https://doi.org/10.1155/2021/9411993 doi: 10.1155/2021/9411993
[25]	S. Chauhan, M. A. Khan, S. Kumar, Unified fixed point theorems in fuzzy metric spaces via common limit range property, J. Inequal. Appl., 2013 (2013), 182. https://doi.org/10.1186/1029-242X-2013-182 doi: 10.1186/1029-242X-2013-182
[26]	A.-F. Roldán-López-de-Hierro, E. Karapinar, S. Manro, Some new fixed point theorems in fuzzy metric spaces, J. Intell. Fuzzy Syst., 27 (2014), 2257–2264. https://doi.org/10.3233/IFS-141189 doi: 10.3233/IFS-141189
[27]	Z. Sadeghi, S. M. Vaezpour, C. Park, R. Saadati, C. Vetro, Set-valued mappings in partially ordered fuzzy metric spaces, J. Inequal. Appl., 2014 (2014), 157. https://doi.org/10.1186/1029-242X-2014-157 doi: 10.1186/1029-242X-2014-157
[28]	I. Shamas, S. U. Rehman, H. Aydi, T. Mahmood, E. Ameer, Unique fixed-point results in fuzzy metric spaces with an application to Fredholm integral equations, J. Funct. Space., 2021 (2021), 4429173. https://doi.org/10.1155/2021/4429173 doi: 10.1155/2021/4429173
[29]	I. Shamas, S. U. Rehman, N. Jan, A. Gumaei, M. Al-Rakhami, A new approach to fuzzy differential equations using weakly-compatible self-mappings in fuzzy metric spaces, J. Funct. Space., 2021 (2021), 6123154. https://doi.org/10.1155/2021/6123154 doi: 10.1155/2021/6123154
[30]	T. Som, Some results on common fixed point in fuzzy metric spaces, Soochow Journal of Mathematics, 33 (2007), 553–561.
[31]	S. U. Rehman, R. Chinram, C. Boonpok, Rational type fuzzy-contraction results in fuzzy metric spaces with an application, J. Math., 2021 (2021), 6644491. https://doi.org/10.1155/2021/6644491 doi: 10.1155/2021/6644491
[32]	B. Schweizer, A. Sklar, Statical metric spaces, Pac. J. Math., 10 (1960), 313–334. https://doi.org/10.2140/pjm.1960.10.313 doi: 10.2140/pjm.1960.10.313
[33]	A. R. Alharbi, M. B. Almatrafi, A. R. Seadawy, Construction of the numerical and analytical wave solutions of the Joseph-Egri dynamical equation for the long waves in nonlinear dispersive systems, Int. J. Mod. Phys. B, 34 (2020), 2050289. https://doi.org/10.1142/S0217979220502896 doi: 10.1142/S0217979220502896
[34]	J. Chesters, R. Mottonen, K. E. Watkins, Transcranial direct current stimulation over left inferior frontal cortex improves speech fluency in adults who stutter, Brain, 141 (2018), 1161–1171. https://doi.org/10.1093/brain/awy011 doi: 10.1093/brain/awy011
[35]	Y. S. Ozkan, A. R. Seadawy, E. Yasar, Multi-wave, breather and interaction solutions to $(3+1)$ dimensional Vakhnenko-Parkes equation arising at propagation of high-frequency waves in a relaxing medium, J. Taibah Univ. Sci., 15 (2021), 666–678. https://doi.org/10.1080/16583655.2021.1999053 doi: 10.1080/16583655.2021.1999053
[36]	A. R. Seadawy, K. K. Ali, R. Nuruddeen, A variety of soliton solutions for the fractional Wazwaz-Benjamin-Bona-Mahony equations, Results Phys., 12 (2019), 2234–2241. https://doi.org/10.1016/j.rinp.2019.02.064 doi: 10.1016/j.rinp.2019.02.064
[37]	U. Younas, A. R. Seadawy, M. Younis, S. T. R. Rizvi, Dispersive of propagation wave structures to the Dullin-Gottwald-Holm dynamical equation in a shallow water waves, Chinese J. Phys., 68 (2020), 348–364. https://doi.org/10.1016/j.cjph.2020.09.021 doi: 10.1016/j.cjph.2020.09.021
[38]	V. Lakshmikantham, R. Mohapatra, Theory of fuzzy differential equations and inclusion, London: CRC Press, 2003. https://doi.org/10.1201/9780203011386

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