Document Type : Research Paper
Authors
1 Faculty of Mathematics, Yazd University, Yazd, Iran and Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.
2 Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.
Abstract
The main purpose of this paper is to study the approximate best proximity pair of cyclic maps and their diameter in fuzzy normed spaces defined by Bag and Samanta. First, approximate best point proximity points on fuzzy normed linear spaces are defined and four general lemmas are given regarding approximate fixed point and approximate best proximity pair of cyclic maps on fuzzy normed spaces. Using these results, we prove theorems for various types of well-known generalized contractions in fuzzy normed spaces. Also, we apply our results to get an application of approximate fixed point and approximate best proximity pair theorem of their diameter.
Keywords
[2] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems, 151 (2005), pp. 513-547.
[3] T. Bag and S.K. Samanta, Some fixed point theorems in fuzzy normed linear spaces, Inf. Sci., 177 (2007), pp. 3271-3289.
[4] M. Berinde, Approximate fixed point theorems, Mathematica, LI (1) (2006).
[5] V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9 (2004).
[6] V. Berinde, On the approximation of fixed points of weak contractive mappings, Math., 19 (2003), pp. 7-22.
[7] S.K. Chatterjea, Fixed-point theorems, C.R. Acad. Bulgare Sci. 25 (1972), pp. 727-730.
[8] S.C. Cheng and J.N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc., 86 (1994), pp. 429-436.
[9] A. Chitra and P.Y. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc., 74 (1969), pp. 660-665.
[10] Lj. B. Ciric, A generalization of Banach’s contraction principle, Proc. Am. Math. Soc., 45 (1974), pp. 267-273.
[11] C. Felbin, Finite dimensional fuzzy normeded linear spaces, Fuzzy Sets and Systems, 48 (1992), pp. 239-248.
[12] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984), pp. 215-229.
[13] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 10 (1968), pp. 71-76.
[14] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12 (1984), pp. 143-154.
[15] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), pp. 326-334.
[16] S.A.M. Mohsenalhosseini, Approximate fixed point theorems in fuzzy norm spaces for an operator, Advances in Fuzzy Systems, 2013, Article ID 613604, 8 pages.
[17] S.A.M. Mohsenalhosseini, Approximate best proximity pairs in metric space for Contraction Maps, Advances in Fixed Point Theory, 4 (2014), pp. 310-324.
[18] S.A.M. Mohsenalhosseini, H. Mazaheri, and M.A. Dehghan, Approximate best proximity pairs in metric space, Abstract and Applied Analysis, (2011), Article ID 596971, 9 pages.
[19] S.A.M. Mohsenalhosseini, H. Mazaheri, and M.A. Dehghan, Approximate fixed point in fuzzy normed spaces for nonlinear maps, Iranin Journal of Fuzzy Systems, 10 (2013), pp. 135-142.
)
