# Simultaneous Determination of Distance Between Sets by Multivalued Kannan Type Coupling

## Main Article Content

### Abstract

In this paper we define a multivalued Kannan type coupling between two subsets of a metric space and use it to obtain the distance between the two subsets through the determination of two pairs of points simultaneously. The problem is a multivalued coupled proximity point problem which falls under the general category of global optimization and is approached from the standpoint of fixed point theory. We use UC-property which is a geometric property that holds automatically for appropriate pairs of subsets of uniformly convex Banach spaces and is adapted to metric spaces by certain postulations. The main results are illustrated with examples. Corresponding results are obtained in Banach spaces. The work is in the domain of setvalued analysis.

## Article Details

### References

- A. Abkar, M. Gabeleh, Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theory Appl. 151 (2011), 418-424.
- A. Abkar, M. Gabeleh, The existence of best proximity points for multivalued non-self-mappings, Rev. R. Acad. Cienc. Exactas Fs. Nat., Ser. A Mat., RACSAM, 107 (2013), 319-325.
- R. P. Agarwal, D. O”˜Regan, D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Springer, New York, 2009.
- I. Altun, D. Turkoglu, Some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation, Filomat 22 (2008), 13-21.
- M. A. Ahmed, Common fixed point theorems for weakly compatible mappings, Rocky Mountain J. Math. 33 (2003), 1189-1203.
- C. D. Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic Meir - Keeler contractions, Nonlinear Anal. 69 (2008), 3790-3794.
- A. Bejenaru, A. Pitea, Fixed point and best proximity point theorems in partial metric spaces, J. Math. Anal. 7(4) (2016), 25-44.
- B. S. Choudhury, P. Maity, P. Konar, A global optimality result using nonself mappings, Opsearch 51(2) (2014), 312-320.
- B. S. Choudhury, N. Metiya, P. Maity, Coincidence point results of multivalued weak C-contractions on metric spaces with a partial order, J. Nonlinear Sci. Appl. 6 (2013), 7-17.
- B. S. Choudhury, P. Maity, Cyclic coupled fixed point result using Kannan type contractions, J. Operators 2014 (2014), Art. ID 876749, 5 pages.
- B. S. Choudhury, P. Maity, Best proximity point results in generalized metric spaces, Vietnam J. Math. 44 (2016), 339-349.
- B. S. Choudhury, P. Maity, N. Metiya, Best proximity point results in setvalued analysis, Nonlinear Anal. Modelling Control. 21 (2016), 293-305.
- B. S. Choudhury, P. Maity, K. Sadarangani, A best proximity point theorem using discontinuous functions, J. convex anal. 24 (2017), 41-53.
- B. Damjanovic, D. Dragan, Multivalued generalizations of the Kannan fixed point theorem, Filomat 25(1) (2011), 125-131.
- A. A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), 1001- 1006.
- E. Karapinar, I. M. Erhan, Best proximity point on different type contractions, Appl. Math. Info. Sci. 5(3) (2011), 558-569.
- B. Fisher, Common fixed points of mappings and setvalued mappings, Rostock Math. Colloq. 18 (1981), 69-77.
- M. Gabeleh, Best proximity points: global minimization of multivalued non-self mappings, Optim. Lett. 8 (2014), 1101- 1112.
- J. Gornicki, Fixed point theorems for Kannan type mappings, J. Fixed Point Theory Appl. 19 (3) (2017), 2145-2152.
- A. Ilchev, B. Zlatanov, Error estimates for approximation of coupled best proximity points for cyclic contractive maps, Appl. Math. Comput. 290 (2016), 412-425.
- R. Kannan, Some result on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76.
- R. Kannan, Some result on fixed points-II, Amer. Math. Monthly 76 (1969), 405-408.
- M. Petric, Best proximity point theorems for weak cyclic Kannan contraction, Filomat 25(1) (2011), 145-154.
- V. Pragadeeswarar, M. Marudai, P. Kumam, Best proximity point theorems for multivalued mappings on partially ordered metric spaces, Nonlinear Sci. Appl. 9 (2016), 1911-1921.
- W. Shatanawi, A. Pitea, Best proximity point and best proximity coupled point in a complete metric space with (P)- property, Filomat 29(1) (2015), 63-74.
- T. Suzuki, M. Kikkawa, C. Vetro, The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal. 71 (2009), 2918-2926.