Fixed point results for generalized convex orbital Lipschitz operators

• Autorzy: Zhou Mi , Li Guohui , Saleem Naeem , Popescu Ovidiu , Secelean Nicolae Adrian

Identyfikatory

DOI

10.1515/dema-2024-0082

• Języki publikacji: EN

Abstrakty

EN

Krasnoselskii’s iteration is a classical and important method for approximating the fixed point of an operator that satisfies certain conditions. Many authors have used this approach to obtain several famous fixed point theorems for different types of operators. It is well known that Kirk’s iteration can be seen as a generalization of Krasnoselskii’s iteration, in which the iterates are generated by a certain generalized averaged mapping. This approximation method is of great practical significance because the iterative formula contains more information related to the operator in question. The purpose of this study is to define weak (𝛼𝑛,𝛽𝑖) -convex orbital Lipschitz operators. These concepts not only extend the previously introduced Popescu-type convex orbital (𝜆,𝛽)-Lipschitz operators in Fixed-point results for convex orbital operators, (Demonstr. Math. 56 (2023), 20220184), but also encompass many classical contractive operators. Popescu also proved a fixed point result for his proposed operator using the graphic contraction principle and obtained an approximation of the fixed point with Krasnoselskii’s iterates. To extend Popescu’s main results from Krasnoselskii’s iterative scheme to Kirk’s iterative scheme, several fixed point theorems are established, in which an appropriate Kirk’s iterative algorithm can be used to approximate the fixed point of a k-fold averaged mapping associated with our presented convex orbital Lipschitz operators. These results not only generalize, but also complement the existing results documented in the previous literature.

Słowa kluczowe

EN

fixed point; convex orbital Lipschitz operator; weak (α n , β i )-convex orbital Lipschitz operator; weakly Picard operator; Ulam-Hyers stability; well-posedness

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20240082
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Zhou Mi

• Center for Mathematical Research, University of Sanya, Sanya 572022, China

autor

Li Guohui

• School of New Energy and Intelligent Networked Automobile, University of Sanya, Sanya 572022, China

autor

Saleem Naeem

• Department of Mathematics, University of Management and Technology, Lahore 54700, Pakistan
• Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa-0204, South Africa

autor

Popescu Ovidiu

• Department of Mathematics and Computer Sciences, Transilvania University of Braşov, Braşov, Romania

autor

Secelean Nicolae Adrian

• Department of Mathematics and Computer Science, Faculty of Sciences, Lucian Blaga University of Sibiu, Sibiu 550012, Romania

Bibliografia

• [1] V. Berinde, A. Petruşel, I. A. Rus, and M. A. Şerban, The retraction-displacement condition in the theory of fixed-point equation with a convergent iterative algorithm, in: T. Rassias, V. Gupta (Eds.), Mathematical Analysis, Approximation Theory and Their Applications, Springer, Cham, 2016.
• [2] S. Reich and A. J. Zaslavski, Generic well-Posedness of fixed point problems, Vietnam J. Math. 46 (2018), 5–13, DOI: https://doi.org/10.1007/s10013-017-0251-1.
• [3] S. Reich and A. J. Zaslavski, Genericity in Nonlinear Analysis, Springer, New York, 2014.
• [4] A. M. Ostrowski, The round off stability of iterations, Z. Angew. Math. Mech. 47 (1967), no. 2, 77–81, DOI: https://doi.org/10.1002/zamm.19670470202.
• [5] I. A. Rus and M. A. Şerban, Basic problems of the metric fixed-point theory and the relevance of a metric fixed-point theorem, Carpathian J. Math. 29 (2013), 239–258, DOI: http://dx.doi.org/10.37193/CJM.2013.02.04.
• [6] I. A. Rus, Results and problems in Ulam stability of operatorial equations and inclusions, in: T. Rassias (Ed.) Handbook of Functional Equations, Springer, New York, 2014.
• [7] I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory 10 (2009), no. 2, 305–320.
• [8] O. Popescu, Fixed-point results for convex orbital operators, Demonstr. Math. 56 (2023), 20220184, DOI: https://doi.org/10.1515/dema-2022-0184.
• [9] A. Petruşel and G. Petruşel, Fixed point results for decreasing convex orbital operators, J. Fixed Point Theory Appl. 23 (2021), 35, DOI: https://doi.org/10.1007/s11784-021-00873-1.
• [10] A. Petruşel and I. A. Rus, Graphic contraction principle and applications, in: T. M. Rassias and P. M. Pardalos (Eds.) Mathematical Analysis and Applications, Springer, Cham, 2019.
• [11] M. A. Krasnoselskii, Two remarks about the method of successive approximations, Uspehi Mat. Nauk (NS) 10 (1955), no. 1(2), 123–127.
• [12] W. Nithiarayaphaks and W. Sintunavarat, On approximating fixed points of weak enriched contraction mappings via Kirkas iterative algorithm in Banach spaces, Carpathian J. Math., 39 (2023), no. 2, 423–432, DOI: https://doi.org/10.37193/CJM.2023.02.07.
• [13] M. Zhou, N. Saleem, and M. Abbas, Approximating fixed points of weak enriched contractions using Kirk’s iteration scheme of high order, J. Inequal. Appl. 2024 (2024), 23, DOI: https://doi.org/10.1186/s13660-024-03097-2.
• [14] W. A. Kirk, On successive approximations for nonexpansive mappings in Banach spaces, Glasg. Math. J. 12 (1971), no. 1, 6–9, DOI: https://doi.org/10.1017/S0017089500001063.
• [15] K. Goebel and W. A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973), 135–140, DOI: https://doi.org/10.4064/sm-47-2-134-140.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2026).