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March  2020, 28(1): 549-557. doi: 10.3934/era.2020028

Existence of best proximity points satisfying two constraint inequalities

Duraisamy Balraj 1, , Muthaiah Marudai 1, , Zoran D. Mitrovic 2, , Ozgur Ege 3,, and  Veeraraghavan Piramanantham 1,

1. 

Department of Mathematics, Bharathidasan University, Trichirapalli, Tamilnadu, India
2. 

University of Banja Luka, Faculty of Electrical Engineering, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina
3. 

Department of Mathematics, Ege University, Bornova, 35100, Izmir, Turkey

* Corresponding author: ozgur.ege@ege.edu.tr

Received  December 2019 Published  March 2020

In this paper, we prove the existence of best proximity point and coupled best proximity point on metric spaces with partial order for weak proximal contraction mappings such that these critical points satisfy some constraint inequalities.

Keywords: Best proximity, constraint inequalities, coupled best proximity point, weak proximal contraction, Kannan mapping.
Mathematics Subject Classification: Primary: 47H09; Secondary: 47H10.
Citation: Duraisamy Balraj, Muthaiah Marudai, Zoran D. Mitrovic, Ozgur Ege, Veeraraghavan Piramanantham. Existence of best proximity points satisfying two constraint inequalities. Electronic Research Archive, 2020, 28 (1) : 549-557. doi: 10.3934/era.2020028
References:
[1]

M. A. Alghamdi, N. Shahzad and F. Vetro, Best proximity points for some classes of proximal contractions, Abstr. Appl. Anal., 2013 (2013), 713252, 10PP. doi: 10.1155/2013/713252.  Google Scholar

[2]

D. Balraj and V. Piramanantham, Best proximity points for generalized proximal cyclic coupled mappings, Int. J. Res. Anal. Rev., 6 (2019), 869-880.   Google Scholar

[3]

S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727-730.   Google Scholar

[4]

B. S. Choudhury and P. Maity, Cyclic coupled fixed point result using Kannan type contractions, J. Operators, 2014 (2014), 876749, 1–5. doi: 10.1155/2014/876749.  Google Scholar

[5]

L. B. Ciric, Generalized contractions and fixed point theorem, Publ. Inst. Math., 12 (1971), 19-26.   Google Scholar

[6]

A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001-1006.  doi: 10.1016/j.jmaa.2005.10.081.  Google Scholar

[7]

M. Jleli and B. Samet, A fixed point problem under two constraint inequalities, Fixed Point Theory Appl., 2016 (2016), 1-14.  doi: 10.1186/s13663-016-0504-9.  Google Scholar

[8]

J. G. Kadwin and M. Marudai, Fixed point and best proximity point results for generalised cyclic coupled mappings, Thai J. Math., 14 (2016), 431-441.   Google Scholar

[9]

R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76.   Google Scholar

[10]

P. Kumam, V. Pragadeeswarar, M. Marudai and K. Sitthithakerngkiet, Coupled best proximity points in ordered metric spaces, Fixed Point Theory Appl., 2014 (2014), 1-13.  doi: 10.1186/1687-1812-2014-107.  Google Scholar

[11]

V. Lakshmikantham and L. B. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341-4349.  doi: 10.1016/j.na.2008.09.020.  Google Scholar

show all references

References:
[1]

M. A. Alghamdi, N. Shahzad and F. Vetro, Best proximity points for some classes of proximal contractions, Abstr. Appl. Anal., 2013 (2013), 713252, 10PP. doi: 10.1155/2013/713252.  Google Scholar

[2]

D. Balraj and V. Piramanantham, Best proximity points for generalized proximal cyclic coupled mappings, Int. J. Res. Anal. Rev., 6 (2019), 869-880.   Google Scholar

[3]

S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727-730.   Google Scholar

[4]

B. S. Choudhury and P. Maity, Cyclic coupled fixed point result using Kannan type contractions, J. Operators, 2014 (2014), 876749, 1–5. doi: 10.1155/2014/876749.  Google Scholar

[5]

L. B. Ciric, Generalized contractions and fixed point theorem, Publ. Inst. Math., 12 (1971), 19-26.   Google Scholar

[6]

A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001-1006.  doi: 10.1016/j.jmaa.2005.10.081.  Google Scholar

[7]

M. Jleli and B. Samet, A fixed point problem under two constraint inequalities, Fixed Point Theory Appl., 2016 (2016), 1-14.  doi: 10.1186/s13663-016-0504-9.  Google Scholar

[8]

J. G. Kadwin and M. Marudai, Fixed point and best proximity point results for generalised cyclic coupled mappings, Thai J. Math., 14 (2016), 431-441.   Google Scholar

[9]

R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76.   Google Scholar

[10]

P. Kumam, V. Pragadeeswarar, M. Marudai and K. Sitthithakerngkiet, Coupled best proximity points in ordered metric spaces, Fixed Point Theory Appl., 2014 (2014), 1-13.  doi: 10.1186/1687-1812-2014-107.  Google Scholar

[11]

V. Lakshmikantham and L. B. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341-4349.  doi: 10.1016/j.na.2008.09.020.  Google Scholar

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