Cyclic uniform Lipschitzian mappings and proximal uniform normal structure
Tóm tắt
For
$$x\in A\cup B,$$
define
$$d_n=d\left( T^nx,T^{n-1}x\right) , n\ge 1$$
where A and B are subsets of a metric space and T is a cyclic map on
$$A\cup B$$
. In this paper, we introduce a new class of mappings called cyclic uniform Lipschitzian mappings for which
$$\{d_n\}$$
is not necessarily a non-increasing sequence and therein prove the existence of a best proximity pair. We also introduce a notion called proximal uniform normal structure and using the same we prove the existence of a best proximity pair for such mappings. Some open problems in this direction are also discussed.
Tài liệu tham khảo
Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 7(10), 3665–3671 (2009)
Casini, E., Maluta, E.: Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure. Nonlinear Anal. 9(1), 103–108 (1985)
Downing, D.J., Turett, B.: Some properties of the characteristic of convexity relating to fixed point theory. Pac. J. Math 104(2), 343–350 (1983)
Edelstein, M.: A theorem on fixed points under isometries. Am. Math. Monthly 70, 298–300 (1963)
Eldred A, A., Kirk, W.A., Veeramani, P.: proximal normal structure and relatively nonexpansive mappings. Stud. Math. 1713, 283–293 (2005)
Eldred A, A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323(2), 1001–1006 (2006)
Espínola, R.: A new approach to relatively non-expansive mappings. Proc. Am. Math. Soc. 136(6), 1987–1995 (2008)
Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171–174 (1972)
Goebel, K., Kirk, W.A.: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant. Stud. Math. 47, 135–140 (1973)
Goebel, K., Kirk, W.A., Thele, R.L.: Uniformly Lipschitzian families of transformations in Banach spaces. Can. J. Math. 26, 1245–1256 (1974)
Górnicki, K.M.: Fixed points of uniformly Lipschitzian mappings. Bull. Polish Acad. Sci. Math. 36(1–2), 57–63 (1988)
Kosuru G Sankara, R., Veeramani, P.: On existence of best proximity pair theorems for relatively nonexpansive mappings. J. Nonlinear Convex Anal. 11(1), 71–77 (2010)
Kosuru, G. Sankara, R., Veeramani, P.: A note on existence and convergence of best proximity points for pointwise cyclic contractions. Numer. Funct. Anal. Optim. 32(7), 821–830 (2011)
Kosuru, G. Sankara, R.: Extensions of Edelstein’s theorem on contractive mappings. Numer. Funct. Anal. Optim. 36(7), 887–900 (2015)
Kirk, W.A.: Sims, Brailey. Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht (2001)
Megginson, R.E.: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183. Springer, New York (1998)
Suzuki, T., Kikkawa, M., Vetro, C.: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 71(7–8), 2918–2926 (2009)