Best proximity points of discontinuous operator in partially ordered metric spaces
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Authors
B. S. Choudhury
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, West Bengal, India.
M. Jleli
- Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia.
P. Maity
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, West Bengal, India.
Abstract
In this paper we establish best proximity point results for monotone multivalued mappings in partially ordered metric
spaces. We consider three notions of monotonicity of multivalued mappings. The main theorem is obtained by utilizing
property UC and MT-functions. There is no requirement of continuity on the multivalued function which is illustrated with two
supporting examples of the results established in this paper. There are two corollaries. Some existing results are extended to the
domain of partially ordered metric spaces through one of the corollaries.
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ISRP Style
B. S. Choudhury, M. Jleli, P. Maity, Best proximity points of discontinuous operator in partially ordered metric spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 1, 308--315
AMA Style
Choudhury B. S., Jleli M., Maity P., Best proximity points of discontinuous operator in partially ordered metric spaces. J. Nonlinear Sci. Appl. (2017); 10(1):308--315
Chicago/Turabian Style
Choudhury, B. S., Jleli, M., Maity, P.. "Best proximity points of discontinuous operator in partially ordered metric spaces." Journal of Nonlinear Sciences and Applications, 10, no. 1 (2017): 308--315
Keywords
- Best proximity point
- multivalued cyclic mapping
- multivalued approximately monotone increasing mapping
- multivalued partly monotone increasing mapping.
MSC
References
-
[1]
A. Abkar, M. Gabeleh, Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theory Appl., 150 (2011), 188–193.
-
[2]
A. Abkar, M. Gabeleh, Generalized cyclic contractions in partially ordered metric spaces, Optim. Lett., 6 (2012), 1819– 1830.
-
[3]
R. P. Agarwal, M. A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., 87 (2008), 109–116.
-
[4]
M. A. Al-Thagafi, N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal., 70 (2009), 3665–3671.
-
[5]
B. S. Choudhury, A. Kundu, (\(\psi,\alpha,\beta\))-weak contractions in partially ordered metric spaces, Appl. Math. Lett., 25 (2012), 6–10.
-
[6]
B. S. Choudhury, P. Maity, Coupled fixed point results in generalized metric spaces, Math. Comput. Modelling, 54 (2011), 73–79.
-
[7]
B. S. Choudhury, P. Maity, P. Konar, A global optimality result using nonself mappings, Opsearch, 51 (2014), 312–320.
-
[8]
B. S. Choudhury, P. Maity, N. Metiya, Best proximity point theorems with cyclic mappings in setvalued analysis, Indian J. Math., 57 (2015), 79–102.
-
[9]
B. S. Choudhury, P. Maity, N. Metiya, Best proximity point results in set-valued analysis, Nonlinear Anal. Model. Control, 21 (2016), 293–305.
-
[10]
L. Ćirić, Fixed point theorems for multi-valued contractions in complete metric spaces, J. Math. Anal. Appl., 348 (2008), 499–507 .
-
[11]
L. Ćirić, N. Cakić, M. Rajović, J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl., 2008 (2008 ), 11 pages.
-
[12]
M. De la Sen, S. L. Singh, M. E. Gordji, A. Ibeas, R. P. Agarwal, Best proximity and fixed point results for cyclic multivalued mappings under a generalized contractive condition, Fixed Point Theory Appl., 2013 (2013 ), 21 pages.
-
[13]
C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal., 69 (2008), 3790–3794.
-
[14]
W.-S. Du, On coincidence point and fixed point theorems for nonlinear multivalued maps, Topology Appl., 159 (2012), 49–56.
-
[15]
W.-S. Du, H. Lakzian, Nonlinear conditions for the existence of best proximity points, J. Inequa. Appl., 2012 (2012 ), 7 pages.
-
[16]
A. A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001–1006.
-
[17]
K. Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z., 122 (1969), 234–240.
-
[18]
M. Jleli, B. Samet , Best proximity point results for MK-proximal contractions on ordered sets, J. Fixed Point Theory Appl., 17 (2015), 439–452.
-
[19]
E. Karapınar, Best proximity points of cyclic mappings, Appl. Math. Lett., 25 (2012), 1761–1766.
-
[20]
E. Karapınar, G. Petruşel, K. Tas, Best proximity point theorems for KT-types cyclic orbital contraction mappings, Fixed Point Theory, 13 (2012), 537–546.
-
[21]
W. A. Kirk, S. Reich, P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24 (2003), 851–862.
-
[22]
N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., 141 (1989), 177–188.
-
[23]
S. B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475–488.
-
[24]
K. Neammanee, A. Kaewkhao, Fixed points and best proximity points for multi-valued mapping satisfying cyclical condition, Int. J. Math. Sci. Appl., 1 (2011), 9 pages.
-
[25]
V. Pragadeeswarar, M. Marudai, Best proximity points: approximation and optimization in partially ordered metric spaces, Optim. Lett., 7 (2013), 1883–1892.
-
[26]
W. Sintunavarat, P. Kumam, Coupled best proximity point theorem in metric spaces, Fixed Point Theory Appl., 2012 (2012 ), 16 pages.
-
[27]
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.
-
[28]
M. Turinici , Abstract comparison principles and multivariable Gronwall-Bellman inequalities, J. Math. Anal. Appl., 117 (1986), 100–127.