Best proximity points for weak $\mathcal{MT}$-cyclic Kannan contractions

Year 2018, , 43 - 48, 30.06.2018

Hossein Lakzian , Ing-Jer Lin

https://doi.org/10.33401/fujma.405536

Cited By: 3

Abstract

In this paper, we introduce a notion of weak $% \mathcal{MT}$-cyclic Kannan contractions with respect to a $\mathcal{MT}$% -function $\varphi$ and then we shall prove some new convergent and existence theorems of best proximity point theorems for these contractions in uniformly Banach spaces.

Keywords

Cyclic map, Best proximity point, $\mathcal{MT}$-cyclic contraction, Weak $\mathcal{MT}$-cyclic Kannan contraction

References

• [1] W.A. Kirk, P.S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (2003) 79-89.
• [2] A. Anthony Eldred, P. Veeramani, Existence and convergence of best proximity points , J. Math. Anal. Appl. 323 (2006) 1001-1006.
• [3] A. Anthony Eldred, J. Anuradha, P. Veeramani, On the equivalence of the Mizoguchi-Takahashi fixed point theorem to Nadler’s theorem, Appl. Math. Letters 22 (2009) 1539-1542.
• [4] M.A. Al-Thagafi, Naseer Shahzad, Convergence and existence results for best proximity points, Nonlinear Analysis 70 (2009) 3665-3671.
• [5] T. Suzuki, M. Kikkawa, C. Vetro, The existence of the best proximity points in metric spaces with the property UC, Nonlinear Anal. 71 (2009) 2918-2926.
• [6] W.-S. Du, Some new results and generalizations in metric fixed point theory, Nonlinear Anal. 73 (2010) 1439-1446.
• [7] W.-S. Du, Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi’s condition in quasi ordered metric spaces, Fixed Point Theory and Applications (2010), Article ID 876372, doi: 10.1155/2010/876372.
• [8] W.-S. Du, H. Lakzian, Nonlinear conditions and new inequalities for best proximity points, Journal of Inequalities and Applications, 2012 (2012), 206. https://doi.org/10.1186/1029-242x-2012-206.
• [9] S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales, Fund. Math. 3 (1922) 133-181.
• [10] S. Karpagam, Sushama Agrawal, Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps, Nonlinear Anal. Vol. 74, (4), pp. 1040-1046, 2011.
• [11] S. Karpagam, S. Agarwal, Best proximity point theorems for p-cyclic Meir-Keeler contractions, Fixed Point Theory and Applications, 2009, 2009:197308.
• [12] S. Kargapam, B. Zlatanov, Best proximity points of p-cyclic orbital Meir-Keeler contraction maps, Nonlinear Analysis: Modeling and Control, 21(6), 2016, 790-806.
• [13] C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. 69 (2008) 3790-3794.
• [14] M. A. Petric, Best proximity point theorems for weak cyclic Kannan contractions, Filomat, Vol. 25, pp. 145-154, 2011.
• [15] R. Espinola, A new approach to relatively nonexpansive mappings, Proc. Amer. Math. Soc. 136 (6) (2008) 1987-1995.
• [16] R. Espinola, A. Fernandez-Leon, On best proximity points in metric and Banach spaces, Canadian Journal of Mathematics (in press).
• [17] A. Amini-Harandi, A. P. Farajzadeh, D. O’Regan, and R. P. Agarwal, Coincidence point, best approximation and best proximity theorems for condensing set-valued maps in hyperconvex metric spaces, Fixed Point Theory and Applications, 2008, Article ID:543154, 8 pages.
• [18] A. Amini-Harandi, A. P. Farajzadeh, D. O’Regan, and R. P. Agarwal, Best proximity pairs for upper semicontinuous set-valued maps in hyperconvex metric spaces, Fixed Point Theory and Applications, vol. 2008, Article ID 648985, 5 pages, 2008.
• [19] R. Kannan, Some results on fixed points-II, Amer. Math. Monthly, 76 (1969), 405-408.
• [20] Z. He, W.-S. Du, I.-J. Lin, The existence of fixed points for new nonlinear multivalued maps and their applications, Fixed Point Theory and Applications 2011, 2011:84, doi:10.1186/1687-1812-2011-84.
• [21] S. Sadiq Basha, N. Shahzad, R. Jeyaraj, Optimal approximate solutions of fixed point equations, Abstract and Applied Analysis,Volume 2011, Article ID 174560, 9 pages, doi:10.1155/2011/174560.
• [22] I.-J. Lin, H. Lakzian and Y. Chou, On Convergence Theorems for Nonlinear Mappings SatisfyingMT 􀀀C Conditions, Appl. Math. Sciences, Vol. 6, 2012, no. 67, 3329 - 3337.
• [23] I.-J. Lin, H. Lakzian and Y. Chou, On Best Proximity Point Theorems for New Cyclic Maps, Int. Math. Forum, Vol. 7, 2012, no. 37, 1839 - 1849.
• [24] C. Vetro, Best proximity points: convergence and existence theorems for p-cyclic mappings, Non-linear Analysis: Theory, Methods and Applications, 73(7), 2010, 2283-2291.
• [25] N. V. Dung and S. Radenovi´c, Remarks on theorems for cyclic quasi-contractions in uniformly convex Banach spaces, Krag. J. Math. Volume 40 (2) (2016), Pages 272-279.
• [26] A. Kostic, V. Rakocevic, S. Radenovic, Best proximity points involving simulation functions with w0-distance, RACSAM /doi.org/10.1007/s13398- 018-0512-1. (2018).