Error estimates for approximating fixed points and best proximity points for noncyclic and cyclic contraction mappings

Document Type : Research Article

Author

A. Safari-Hafshejani

Department of Pure Mathematics, Payame Noor University (PNU), P. O. Box: 19395-3697, Tehran, Iran.

10.22067/ijnao.2023.79068.1188

Abstract

In this article, we find a priori and a posteriori error estimates of the fixed point for the Picard iteration associated with a noncyclic contraction map, which is defined on a uniformly convex Banach space with a modulus of convexity of power type. As a result, we obtain priori and posteriori error estimates of Zlatanov for approximating the best proximity points of
cyclic contraction maps on this type of space.

Keywords

Fixed point

Noncyclic contraction map

Uniformly convex Ba-nach space

Modulus of convexity

Priori and posteriori errors estimates

20.1001.1.24236977.2023.13.3.2.7

Subjects

• Numerical Analysis

---

References

[1] Beauzamy, B. Introduction to Banach spaces and their geometry, North–Holland Publishing Company, Amsterdam, 1979.
[2] Berinde, V. Iterative approximation of fixed points, Springer, Berlin, 2007.
[3] Deville, R., Godefroy, G. and Zizler, V. Smoothness and renormings in Banach spaces, Longman Scientific Technical, Harlow; copublished in the United States with John Wiley Sons, Inc., New York, 1993.
[4] Eldred, A. and Veeramani, P. Existence and convergence of best prox-imity points, J. Math. Anal. Appl. 323(2) (2006), 1001–1006.
[5] Fabian, M., Habala, P., Hajek, P., Montesinos, V., Pelant, J. and Zizler, V. Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8. Springer-Verlag, New York, 2001.
[6] Fallahi, K., Ghahramani, H. and Soleimani Rad, G. Integral type con-tractions in partially ordered metric spaces and best proximity point, Iran. J. Sci. Technol., Trans. A, Sci. 44 (2020), 177–183.
[7] Fallahi, K. and Soleimani Rad, G. Best proximity points theorem in b-metric spaces endowed with a graph, Fixed Point Theory, 21(2) (2020), 465–474.
[8] Gabeleh, M. and Shahzad, N. Best proximity pair and fixed point results for noncyclic mappings in convex metric spaces, Filomat, 30(12) (2016), 3149–3158.
[9] Goebel, K. and Kirk, W.A. Topics in metric fixed point theory, Cam-bridge University Press, 1990.
[10] Hristov, M., Ilchev, A. and Zlatanov, B. On the best proximity points for p-cyclic summing contractions, Mathematics, 8(7) (2020), 1–11.
[11] Ilchev, A. Error estimates for approximating best proximity points for Kannan cyclic contractive maps, AIP Conference Proceedings, 2048(1) (2018), 050002.

12] Kirk, W., Srinivasan, P. and Veeramani, P. Fixed points for map-pings satisfying cyclical contractive conditions, Fixed Point Theory, 4(1) (2003), 79–89.
[13] Meir, A. On the Uniform Convexity of Lp Spaces 1 < p < 2, Illinois J. Math. 28(3) (1984), 420–424.
[14] Safari-Hafshejani, A. The existence of best proximity points for gener-alized cyclic quasi-contractions in metric spaces with the UC and ultra-metric properties, Fixed Point Theory, 23(2) (2022), 507–518.
[15] Safari-Hafshejani, A., Amini-Harandi, A. and Fakhar, M. Best proximity points and fixed points results for noncyclic and cyclic Fisher quasi-contractions, Numer. Funct. Anal. Optim. 40(5) (2019), 603–619.
[16] Scheu, G. On the optimization of the error estimate of a fixed point theorem, J. Math. Anal. Appl. 57 (1977), 298–322.
[17] Zlatanov, B. Error estimates for approximating best proximity points for cyclic contractive maps, Carpathian J. Math. 32(2) (2016), 265–270.