Some fuzzy common fixed point theorems using common limit in the range property with an application. (English) Zbl 1474.54126
Summary: In the present paper, we prove some common fixed point theorems for mappings satisfying common limit in the range property in \(M\)-fuzzy metric space. Further, we prove fixed point theorem for \(\phi \)-contractive conditions in aforesaid spaces with the illustration of an example. As an application of our result, we study the existence and uniqueness of the solution of integral equation (Volterra integral equations of the second kind) with instances.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 47H10 | Fixed-point theorems |
| 54A40 | Fuzzy topology |
Keywords:
\(M\)-fuzzy metric spaces; common limit in the range property ((CLR) property); weakly compatible mappingsReferences:
| [1] | M. Aamri and D. El Moutawakil,Some new common fixed point theorems under strict contractive conditions, Journal of Mathematical Analysis and Applications, 270 (1) (2002), 181-188. · Zbl 1008.54030 |
| [2] | J. Chu and P. J. Torres,Application of Schauder’s fixed point theorem to singular differential equations, Bulletin of the London Mathematical Society, 39 (2007), 653660. · Zbl 1128.34027 |
| [3] | B. Deshpande and A. Handa,Application of coupled fixed point technique in solving integral equations on modified intuitionistic fuzzy metric spaces, Advances in Fuzzy Systems, (2014), Article ID: 348069, 11 pages. · Zbl 1323.45001 |
| [4] | B. Deshpande, S. Sharma and A. Handa,Common coupled fixed point theorems for nonlinear contractive condition on intuitionistic fuzzy metric spaces with application to integral equations, Journal of the Korean Society of Mathematical Education. Series B. The Pure and Applied Mathematics, 20 (3) (2013), 159-180. · Zbl 1304.54075 |
| [5] | B. C. Dhage,Generalized metric space and mapping with fixed point, Bulletin of the Calcutta Mathematical Society, 84 (4) (1992), 329-336. · Zbl 0782.54037 |
| [6] | S. Gahler,2-metrische Raume und ihre topologische Struktur, Mathematische Nachrichten, 26 (1963), 115-148. · Zbl 0117.16003 |
| [7] | M. Gangopadhyay, P.Dan and M. Saha,An application of random fixed point theorem in integral equation, IOSR Journal of Mathematics, 9 (4)(2014), 57-61. |
| [8] | A. George and P. Veeramani,On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (3) (1994), 395-399. · Zbl 0843.54014 |
| [9] | M. Gugnani, M. Aggarwal and R. Chugh,Common fixed point results in G-metric spaces and applications, International Journal of Computer Applications, 43 (11) (2012), 38-42. |
| [10] | G. Jungck,Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far Eastern Journal of Mathematics and Science, 4 (1996), 199-215. · Zbl 0928.54043 |
| [11] | N. Mani,GeneralizedCψβ-rational contraction and fixed point theorem with application to second order differential equation, Mathematica Moravica, 22 (1) (2018), 43-54. · Zbl 1488.54147 |
| [12] | U. Mishra, C. Vetro and P. Kumam,On modifiedα-φ-fuzzy contractive mappings and an application to integral equations, Journal of Inequalities and Applications, 2016 (2016), Article ID: 67, 15 pages. · Zbl 1337.54044 |
| [13] | Z. Mustafa and B. Sims,Some Remarks ConcerningD-Metric Spaces, Proceedings of the International Conferences on Fixed Point Theory and Applications, Valencia (Spain), July 2003, 189-198. · Zbl 1079.54017 |
| [14] | R. Muthuraj and R. Pandiselvi,Common fixed point theorems for weakly compatible of four mappings in generalized fuzzy metric spaces, International Journal of Mathematics and its Applications, 3 (1) (2015), 39-47. |
| [15] | S. V. R. Naidu, K. P. R. Rao, and N. Srinivasa Rao,On the topology ofD-metric spaces and generation ofD-metric spaces from metric spaces, International Journal of Mathematics and Mathematical Sciences, 2004 (51) (2004), 2719-2740. · Zbl 1080.54023 |
| [16] | V. Popa,A general fixed point theorem for two hybrid pairs of mappings satisfying a mixed implicit relation and applications, Mathematica Moravica, 21 (2) (2017), 103-114. · Zbl 1474.54219 |
| [17] | K.P.R. Rao, G.N.V. Kishore, K.V. Siva Parvathi,A quadruple fixed point theorem for contractive type condition by using ICS mapping and application to integral equation, Mathematica Moravica, 18 (2) (2014), 21-34. · Zbl 1349.54134 |
| [18] | R. Saadati,Existence and uniqueness of solutions for a class of integral equations by common fixed point theorems in IFMT-spaces, Journal of Inequalities and Applications, 2016 (2016), Article ID: 205, 8 pages. · Zbl 1346.54028 |
| [19] | B. Schweizer and A. Sklar,Statistical metric spaces, Pacific Journal of Mathematics, 10 (1) (1960), 313-334. · Zbl 0091.29801 |
| [20] | S.Sedghi, N. Shobe and A. Aliouche,A common fixed point theorem for weakly compatible mappings in fuzzy metric space, General Mathematics, 18 (3) (2010), 3-12. · Zbl 1289.54159 |
| [21] | S. Sedghi, N. Shobe and H. Zhou,A Common fixed point theorem inD∗-metric spaces, Fixed Point Theory and Applications, 2007 (2007), Article ID: 27906, 1-13. · Zbl 1147.54024 |
| [22] | S. Sedghi and N. Shobe,Fixed point theorems in M-fuzzy metric spaces with property (E), Advances in Fuzzy Mathematics, 1(1) (2006), 55-65. |
| [23] | D. Singh, M. Sharma and R. Sharma,Fixed point theorems in M-fuzzy metric spaces, Annals of Fuzzy Mathematics and Informatics, 5 (1) (2013), 147-155. · Zbl 1302.54047 |
| [24] | W. Sintunavarat and P. Kumam,Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, Journal of Applied Mathematics, 2011 (2011), Article ID: 637958, 14 pages. · Zbl 1226.54061 |
| [25] | S. Sharma and B. Deshpandey,Fixed point theorem and its application to best approximation theory, Bulletin of the Calcutta Mathematical Society, 93 (2) (2001), 155-166 · Zbl 1002.47029 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
