Stability and Iterative Procedures for Quadrupled Fixed Point in Partially Ordered Metric Spaces

Subject Areas : Operation Research

Samuel Aniki ¹

1 - Department of Mathematics, Faculty of Science, Confluence University of Science and Technology, Osara, Kogi State, Nigeria

Received: 2021-12-13 Accepted : 2022-04-16 Published : 2021-12-01

Keywords: Stability, Quadrupled fixed point, Metric space, mixed monotone, iteration procedure,

Abstract :

The concept of fixed point is of great interest in mathematics as well as in many areas of applied sciences. Previously, the question of fixed points in partially ordered metric spaces had been studied, and quadrupled fixed point theorem is a continuation of the tripled fixed point theorem. This work presents the stability for quadrupled fixed point iterative procedure with the aid of some mathematical analysis properties, and establishes results for mixed monotone mappings which satisfy its contractive-type conditions. The results of this work extend some of the results in literature.

References:

Amini-Harandi, A. (2012). Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem. Mathematical and Computer Modeling, 57, 2343-2348.

Aniki, S.A. & Rauf, K. (2019). Stability results on coupled fixed point iterative procedures in complete metric spaces. Islamic University Multidisciplinary Journal, 6(3), 175-186. ISSN 2617-6513.

Aniki, S.A. & Rauf, K. (2020). Stability theorem and results for quadrupled fixed point of contractive type single valued operators. Iranian Journal of Optimization, 12(2).

Abbas, M., Aydi H. & Karapinar E. (2011). Tripled fixed point of multivalued nonlinear contraction mappings in partially ordered metric spaces. Abstract and Applied Analysis, (2), 1-12.

Bhaskar, T.G. & Lakshmikantham, V. (2006). Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis, 65(7), 1379-1393.

Berinde V. (2003). Summable almost stability of fixed point iteration procedures. Carpathian Journal of Mathematics, 19(2), 81–88.

Berinde, V. & Borcut, M. (2011). Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Analysis, 74(15), 4889-4897.

Ciric L.B. & Lakshmikantham V. (2009). Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stochastic Analysis and Applications, 27(6), 1246–1259.

Choudhury, B.S. & Kundu, A.A. (2010) Coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Analysis, 73(8), 2524-2531.

Imoru, C.O. & Olatinwo M.O. (2003). On the stability of Picard andMann iteration processes. Carpathian Journal of Mathematics, 19(2), 155–160.

Jachymski J.R. (1997). An extension of A. Ostrowski’s theorem on the round-off stability of iterations. Aequationes Mathematicae, 53(3), 242–253.

Karapinar, E. (2010). Coupled foxed point theorems for nonlinear contractions in cone metric spaces. Computer & Mathematics with Applications, 59(12), 3656-3668.

Lakshmikantham, V. & Ciric, L.B. (2009). Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Analysis, 70(12), 4341-4349.

Olatinwo, M.O., Owojori, O.O. & Imoru, C.O. (2006). Some stability results for fixed point iteration processes. Australian Journal of Mathematical Analysis and Applications, 2(1), 1–7.

Osilike, M.O. (1996). A stable iteration procedure for quasi-contractive maps. Indian Journal of Pure and Applied Mathematics, 27(1), 25–34.

Osilike, M.O. (1995). Stability results for fixed point iteration procedure. Journal of the Nigerian Mathematical  Society, 14, 17–29.

Ostrowski, A.M. (1967). The round-off stability of iterations. Journal of Applied Mathematics and Mechanics, 47(1), 77–81.

Rauf, K. & Aniki, S.A. (2021). Quadrupled fixed point theorems for contractive type mappings in partially ordered cauchy spaces. Confluence Journal of Pure and Applied Sciences, 4(1), 18-30.

Rao, K.P.R. & Kishore, G.N.V. (2011). A Unique Common tripled fixed point theorem in partially ordered cone metric spaces. Bulletin of Mathematical Analysis and Applications, 3(4), 213-222.

Rhoades, B.E. (1990). Fixed point theorems and stability results for fixed point iteration procedures. Indian Journal of Pure and Applied Mathematics, 21(1), 1–9.

Rhoades, B.E. (1993) Fixed point theorems and stability results for fixed point iteration procedures II. Indian Journal of Pure and Applied Mathematics, 24(11), 691–703.

Sabetghadam, F., Masiha, H.P. & Sanatpour, A.H. (2009). Some coupled fixed point theorems in cone metric spaces. Fixed Point Theory and Applications, Article ID 125426. doi:10.1155/2009/125426

Timis, I. (2014). Stability of tripled fixed point iteration procedures for mixed monotone mappings. Carpathian Mathematical Publications, 6(2), 377-388.