# Existence and Convergence Theorems For Best Proximity Points of Proximal Multi-Valued Nonexpansive Mappings

## Authors

• Panitarn Sarnmeta Ph.D. Degree Program in Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200,
• Suthep Suantai Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200

## Keywords:

Proximal multi-valued nonexpansive, Best proximity point

## Abstract

The concepts of proximal contraction and proximal nonexpansive mapping have been investigated and extended in many direction. However, most of these works concern only single-valued mappings. So, in this paper, we introduce a concept of proximal nonexpansive for non-self set-valued mappings and prove the existence of best proximity point for such mappings under appropriate conditions. We also provide an algorithm to approximate a best proximity point of such mappings, and prove its convergence theorem. Moreover, a numerical example supporting our main results is also given.

## References

S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math. 3 (1922), 133 – 181, http://eudml.org/doc/213289.

S. S. Basha, Best proximity points: optimal solutions, J. Optim. Theory Appl. 151 (2011), 210 – 216, DOI: 10.1007/s10957-011-9869-4.

S. S. Basha and N. Shahzad, Best proximity point theorems for generalized proximal contractions, Fixed Point Theory Appl. 42 (2012), DOI: 10.1186/1687-1812-2012-42.

P. Z. Daffer and H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995), 655 – 666, DOI: 10.1006/jmaa.1995.1194.

T. D. Benavides and P. L. Ramí­rez, Fixed point theorems for multivalued nonexpansive mappings without uniform convexity, Abstr. Appl. Anal. 2003 (2003), 375 – 386, DOI: 10.1155/S1085337503203080.

E. L. Dozo, Multivalued nonexpansive mappings and Opial's condition, Proc. Amer. Math. Soc. 38 (1973), 286 – 292, DOI: 10.1090/S0002-9939-1973-0310718-0.

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.

K. Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 122 (1969), 234 – 240, DOI: 10.1007/BF01110225.

A. Fernández-León, Best proximity points for proximal contractions, J. Nonlinear Convex Anal. 15 (2014), 313 – 324, http://hdl.handle.net/11441/43082.

M. Gabeleh, Best proximity points for weak proximal contractions, Bull. Malays. Math. Sci. Soc. 38(1) (2015), 143 – 154, DOI: 10.1007/s40840-014-0009-9.

M. Gabeleh, Best proximity point theorems via proximal non-self mappings, J. Optim. Theory Appl. 164 (2015), 565 – 576, DOI: 10.1007/s10957-014-0585-8.

K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, Contemp. Math. 21 (1983), 115 – 123, DOI: 10.2307/2047831.

E. Karapinar, Best proximity points of cyclic mappings, Appl. Math. Lett. 25 (2012), 1761 – 1766, DOI: 10.1016/j.aml.2012.02.008.

W. K. Kim and K. H. Lee, Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl. 316 (2006), 433 – 446, DOI: 10.1016/j.jmaa.2005.04.053.

N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989), 177 – 188, DOI: 10.1016/0022-247X(89)90214-X.

S. B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475 – 487, DOI: 10.1007/BF02771543.

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591 – 597, DOI: 10.1090/S0002-9904-1967-11761-0.

S. Pirvbavafa and S. M. Vaezpour, Equilibria of free abstract economies via best proximity point theorems, Boletí­n de la Sociedad Matemática Mexicana 24(2) (2018), 471 – 481, DOI: 10.1007/s40590-017-0175-5.

Sh. Rezapour, M. Derafshpour and N. Shahzad, Best proximity points of cyclic (phi)-contractions on reflexive Banach spaces, Fixed Point Theory Appl. 2010 (2010), Article ID 946178, DOI: 10.1155/2010/946178.

C.-K. Zhong, J. Zhu and P.-H. Zhao, An extension of multi-valued contraction mappings and fixed points, Proc. Amer. Math. Soc. 128 (2000), 2439 – 2444, DOI: 10.1090/S0002-9939-99-05318-6.

30-09-2019
CITATION

## How to Cite

Sarnmeta, P., & Suantai, S. (2019). Existence and Convergence Theorems For Best Proximity Points of Proximal Multi-Valued Nonexpansive Mappings. Communications in Mathematics and Applications, 10(3), 369–377. https://doi.org/10.26713/cma.v10i3.1199

Research Article