Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria.

2 School of Arts and Sciences, American University of Nigeria, Yola, Adamawa State, Nigeria.

Abstract

In this paper, the notion of $p$-hybrid $L$-fuzzy contractions in the framework of $b$-metric space is introduced. Sufficient conditions for existence of common $L$-fuzzy fixed points under such contractions are also investigated. The established ideas are generalizations of many concepts in fuzzy mathematics. In the case where our postulates are reduced to their classical variants, the concept presented herein merges and extends several significant and well-known fixed point theorems in the setting of both single-valued and multi-valued mappings in the corresponding literature of discrete and computational mathematics.  A few of these special cases are pointed out and discussed. In support of our main hypotheses, a nontrivial example is provided.

Keywords

[1] M. Alansari, S.S. Mohammed, A. Azam and N. Hussain, On Multivalued Hybrid Contractions with Applications, J. Funct. Spaces, Vol. 2020, Article ID 8401403.

[2] A.E. Al-Mazrooei and J. Ahmad, Fixed point theorems for fuzzy mappings with applications, J. Intell. Fuzzy Syst., 36(4), (2019), pp. 3903-3909.

[3] A. Azam, M. Arshad and P. Vetro, On a pair of fuzzy $varphi$-contractive mappings, Math. Comput. Modelling, 52(2) (2010), pp. 207-214.

[4] A. Azam and I. Beg, Common fixed points of fuzzy maps, Math. Comput. Modelling, 49(7),(2009), pp. 1331-1336.

[5] I.A. Bakhtin, The contraction mapping principle in almost metric spaces, Functional Analysis, 30,(1989), pp. 26-37.

[6] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3(1),(1922), pp. 133-181.

[7] V. Berinde, Generalized contractions in quasimetric spaces, In Seminar on Fixed Point Theory; Babes-Bolyai University: Cluj-Napoca, Romania, 1993, pp. 3–9.

[8] M. Boriceanu, Fixed point theory for multivalued generalized contraction on a set with two $b$-metrics, Stud. Univ. Babeş-Bolyai, Math., 4(3), (2009), pp. 126-132.

[9] N. Bourbaki, Topologie Generale, Herman, Paris, France, 4(1),(1974), pp. 8-15.

[10] S. Czerwik, Contraction mappings in $ b $-metric spaces, Acta Math. Inform. Univ. Ostrav., 1(1), (1993), pp. 5-11.

[11] M.S. El Naschie, Wild topology, hyperbolic geometry and fusion algebra of high energy particle physics, Chaos Solitons Fractals, 13(9), (2002), pp. 1935-1945.

[12] M.S. El Naschie, On the unification of the fundamental forces and complex time in the $E^infty$ space, Chaos Solitons Fractals, 11(7), (2000), pp. 1149-1162.

[13] J.A. Goguen, $L$-fuzzy sets, J. Math. Anal. Appl., 18(1), (1967), pp. 145-174.

[14] S. Heilpern, Fuzzy mappings and fixed point theorem, J. Math. Anal. Appl., 83(2),(1981), pp. 566-569.

[15] N. Hussain, D. Doric, Z. Kadelburg and S. Radenovic, Suzuki-type fixed point results in metric type spaces, Fixed Point Theory Appl., 2012(1),(2012), pp. 126-134.

[16] N. Hussain and Z.D. Mitrović, On multi-valued weak quasi-contractions in $b$-metric spaces, Journal of Nonlinear Sciences and Applications, 10(7),(2017), pp. 3815-3823.

[17] T. Kamran, M. Samreen and Q. UL Ain, A generalization of $b$-metric space and some fixed point theorems, Mathematics, 5(2), (2017), pp. 19-26.

[18] H. Kaneko and S. Sessa, Fixed point theorems for compatible multi-valued and single-valued mappings, Int. J. Math. Comput. Sci., 12(2),(1989), pp. 257-262.

[19] E. Karapinar and A. Fulga, New Hybrid Contractions on $b$-Metric Spaces, Mathematics, 7(7), (2019), pp. 578-586.

[20] E. Karapinar, A short survey on the recent fixed point results on $b$-metric spaces, Constr. Math. Anal., 1(1), (2018), pp. 15-44.

[21] B.S. Lee, G.M. Sung, S.J. Cho and D.S. Kim, A common fixed point theorem for a pair of fuzzy mappings, Fuzzy Sets Syst., 98, (1998), pp.133-136.

[22] S.N. Mishra, S.L. Singh and R. Talwar, Nonlinear hybrid contractions on Menger and uniform spaces, Indian Journal of Pure and applied mathematics, 25,(1994), pp. 1039-1052.

[23] S.S. Mohammed and A. Azam, Fixed points of soft-set valued and fuzzy set-valued maps with applications, J. Intell. Fuzzy Syst., 37(3), (2019), pp. 3865-3877.

[24] S.B. Nadler, Multi-valued contraction mappings, Pac. J. Math., 30(2), (1969), pp. 475-488.

[25] S.A. Naimpally, S.L. Singh and J.H.M. Whitfield, Coincidence theorems for hybrid contractions, Math. Nachr., 127(1), (1986), pp.177-180.

[26] J.Y. Park and J.U. Jeong, (1997), pp. Fixed point theorems for fuzzy mappings, Fuzzy Sets Syst., 87(1), 111-116.

[27] H.K. Pathak, S.M. Kang and Y.J. Cho, Coincidence and fixed point theorems for nonlinear hybrid generalized contractions, Czech. Math. J., 48(2), (1998), pp. 341-357.

[28] V. Popa, Coincidence and fixed points theorems for noncontinuous hybrid contractions, In Nonlinear Analysis Forum, 7, (2002), pp. 153-158.

[29] D. Rakić, T. Došenović, Z.D. Mitrović, M. de la Sen and S. Radenović, Some fixed point theorems of Ćirić type in fuzzy metric spaces, Mathematics, 8(2), (2020), pp. 297-302.

[30] D. Rakić, A. Mukheimer, T. Došenović, Z.D. Mitrović and S. Radenović, On some new fixed point results in fuzzy $b$-metric spaces, J. Inequal. Appl., 2020(1), (2020),pp. 1-14.

[31] K.P. Rao, P.R. Swamy and J. R. Prasad, A common fixed point theorem in complex valued $b$-metric spaces, Bulletin of Mathematics and Statistics research, 1(1), (2013), pp. 1-8.

[32] M. Rashid, A. Azam and N. Mehmood, $L$-Fuzzy fixed points theorems for $L$-fuzzy mappings via $beta_{FL}$-admissible pair, The Scientific World Journal, Vol. 2014, Article ID 853032,(2014), 8 pages.

[33] M. Rashid, M.A. Kutbi and A. Azam, Coincidence theorems via alpha cuts of $L$-fuzzy sets with applications, Fixed Point Theory Appl., 2014(1), (2014), pp. 212-230.

[34] I.A. Rus, Generalized contractions and applications, Cluj University Press, 2(6),(2001), pp. 60-71.

[35] M.S. Shagari and A. Azam, Fixed point theorems of fuzzy set-valued maps with applications, Problemy Analiza, 9(27),(2020), pp. 2-17.

[36] C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008), pp. 266-272.

[37] S.L. Singh and B. Prasad, Some coincidence theorems and stability of iterative procedures, Comput. Math. Appl., 55(11), (2008), pp. 2512-2520.

[38] L.A. Zadeh, Fuzzy sets, Inf. Control, 8(3), (1965), pp. 338-353.