Existence and Convergence of Fixed points of Generalized alpha non-expensive mappings in metric spaces:

Ali  Asghar; Mohammad Showkat Rahim Chowdhury; Noor  Muhammad; Muhammad Suhail Aslam

doi:10.32350/sir.63.01

Ali Asghar Department of Mathematics, Institute of Southern Punjab, Multan, Pakistan
Mohammad Showkat Rahim Chowdhury Department of Mathematics and Statistics, Faculty of Science, The University of Lahore, Lahore, Pakistan
Noor Muhammad Department of Mathematics, Institute of Southern Punjab, Multan, Pakistan
Muhammad Suhail Aslam Department of Mathematics and Statistics, Faculty of Science, The University of Lahore, Lahore, Pakistan

DOI: https://doi.org/10.32350/sir.63.01

Keywords: convergence, condition (C), fixed point, non-expansive mapping, opial property, α-non-expansive

Abstract

Abstract Views: 245

We give certain conditions about different kinds of mappings. These conditions will be median of non-expansive mappings and quasi-non-expansive mappings. We will establish some fixed point results of some generalized non-expansive mappings in metric spaces. Moreover, we will also establish few existence and convergence results about generalization of non-expansive mappings. Finally, we present some useful lemmas and propositions.

Downloads

Download data is not yet available.

References

Connell EH. Properties of fixed point spaces. Proc Amer Math Soc.1959;10(6):974-979. https://doi.org/10.2307/2033633

Subrahmanyam PV. Completeness and fixed-points. Monatsh Math. 1975;80:325-330. https://doi.org/10.1007/BF01472580

Ishtiaq U, Saleem N, Uddin F, Sessa S, Ahmad K, di Martino F. Graphical views of intuitionistic fuzzy double-controlled metric-like spaces and certain fixed-point results with application. Symmetry. 2022;14(11):e2364. http://doi.org/10.3390/sym14112364

Browder FE. Fixed-point theorems for noncompact mapping in Hilbert space. Proc. Natl. Acad. Sci. USA. 1965;53(6):1272-1276. https://doi.org/10.1073/pnas.53.6.1272

Gohde D. Zum Prinzip def kontraktiven Abbildung. Math Nachr.1965;30(3-4):251-258. https://doi.org/10.1007/978-3-322-94888-5_2

Goebel K, Kirk WA. Iteration processes for non-expansive mappings. Contempt. Math. 1983;21:115-123.

Kirk WA. A fixed point theorem for mappings which do not increase distances. Amer Math Monthly. 1965;72:1004-1006. https://doi.org/10.2307/2313345

Browder FE. Non-expansive nonlinear operators in a Banach space. Proc Natl Acad Sci. 1965;54(6):1041-1044. https://doi.org/10.1073/pnas.54.4.1041

Saleem, N, Agwu IK, Ishtiaq U. Radenovi´c, S. Strong convergence theorems for finite family of enriched strictly pseudocontractive mappings and φT-Enriched lipschitizian mappings using a new modified mixed- type ishikawa iteration scheme with error. Symmetry. 2022;14(5):e1032. https://doi.org/10.3390/sym14051032

Suzuki T. Krasnoselskii and Mann’s type sequence and Ishikawa’s strong convergence theorem. Paper presented at W. Taka- hashi, T. Tanaka, eds., Proceedings of the third international conference on Nonlinear Analysis and convex Analysis; 2004, Yokohama.

Suzuki T. Strong convergence of Krasnoselskii and Mann,s type sequence for one parameter non-expansive semigroup without Bochner integrals. J Math Anal Appl. 2005;305(1):227-239. https://doi.org/10.1016/j.jmaa.2004.11.017

Dulst DV. Equivalent norms and the fixed point property for non-expansive mappings. J London Math Soc. 1982;25:139-144. https://doi.org/10.1112/jlms/s2-25.1.139

Suzuki T. Strong convergence theorems for infinite family of non-expansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005;2005:e685918. https://doi.org/10.1155/FPTA.2005.103

Suzuki T. Fixed point theorems and convergence theorems for some generalized non expansive mappings. J Math Anal Appl. 2008;340(2):1088-1095. https://doi.org/10.1016/j.jmaa.2007.09.023

Shukla R, Pant R, De la Sen M. Generalized α non-expansive mappings in Banach Spaces. Fixed Point Theory Appl. 2017:4. https://doi.org/10.1186/s13663-017-0597-9

Edelstein M, Brien RCO. Non-expansive mappings, asymptotic regularity and successive approximations. J London Math Soc. 1978;s-217(3):547-554. https://doi.org/10.1112/jlms/s2-17.3.547

Goebel K, Kirk WA. Topics in metric fixed point theory. Cambridge University Press; 1990.

Prus S. Geometrical background of metric fixed point theory. In: Kirk WA, Sims B, eds. Handbook of metric fixed point theory. Kluwer Academic Publishers; 2001:93-132.

Bailon JB. Quelques aspects de la theory des points fixes dans les spaces de Banach-II. Paper presented at: Seminar d, Analysis fonctionnelle. Ecole Polytech, Palaiseau; 1979. http://www.numdam.org/item/SAF_1978-1979____A7_0.pdf

Diaz JB, Metcalf FT. On the structure of the set of sub sequential limit points of successive approximations. Bull Amer Math Soc. 1967;73(4):516-519.

Muhammad N, Asghar A, Irum S, Akgul A, Khalil EM, Mustaffa. Approximation of fixed point of generalized non-expansive mapping via faster iterative scheme in metric domain [J]. Aims Math. 2023;8(2):2856-2870. https://doi.org/10.3934/math.2023149

Ishikawa S. Fixed points and iteration of non-expansive mappings in a Banach space. Proc Amer Math Soc. 1976;59:65-71.

Asghar A, Qayyum A, Muhammad N. Different types of topological structures by graphs. Eur J Math Anal. 2023;3:e3. https://doi.org/10.28924/ada/ma.3.3

Opial Z. Weak convergence of the sequence of successive approximations for non-expansive mappings. Bull Amer Math Soc. 1967;73:591-597.

Kirk WA. Caristi’s fixed point theorem and metric convexity. Colloq Math. 1976; 36:81-86.

Usman A, Haifa AA, Umar I, Khalil A, Shajib A. Solving nonlinear fractional differential equations for contractive and weakly compatible mappings in neutrosophic metric spaces. J Func Spac. 2022;2022:e1491683. https://doi.org/10.1155/2022/1491683

Existence and Convergence of Fixed points of Generalized alpha non-expensive mappings in metric spaces