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

Panitarn Sarnmeta, Suthep Suantai

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.


Keywords


Proximal multi-valued nonexpansive; Best proximity point

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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.




DOI: http://dx.doi.org/10.26713%2Fcma.v10i3.1199

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