Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems

• Autorzy: Chairatsiripong Chonjaroen , Yambangwai Damrongsak , Thianwan Tanakit

Identyfikatory

DOI

10.1515/dema-2022-0234

• Języki publikacji: EN

Abstrakty

EN

In this article, weak and strong convergence theorems of the M-iteration method for 𝒢-nonexpansive mapping in a uniformly convex Banach space with a directed graph were established. Moreover, weak convergence theorem without making use of Opial’s condition is proved. The rate of convergence between the M-iteration and some other iteration processes in the literature was also compared. Specifically, our main result shows that the M-iteration converges faster than the Noor and SP iterations. Finally, the numerical examples to compare convergence behavior of the M-iteration with the three-step Noor iteration and the SP-iteration were given. As application, some numerical experiments in real-world problems were provided, focused on image deblurring and signal recovering problems.

Słowa kluczowe

EN

G-nonexpansive mapping; M-iteration method; signal recovering problem; image deblurring problem; directed graph

PL

metoda iteracji; odzyskiwanie sygnału; rozmycie obrazu; graf skierowany

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2023
• Tom: Vol. 56, nr 1
• Strony: art. no. 20220234
• Opis fizyczny: Bibliogr. 33 poz., rys. wykr.

Twórcy

autor

Chairatsiripong Chonjaroen

• Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand

autor

Yambangwai Damrongsak

• Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand

autor

Thianwan Tanakit

• Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand

Bibliografia

• [1] S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math. 3 (1922), 133–181.
• [2] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1359–1373.
• [3] R. P. Kelisky and T. J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967), 511–520.
• [4] S. M. A. Aleomraninejad, S. Rezapour, and N. Shahzad, Some fixed-point result on a metric space with a graph, Topol. Appl. 159 (2012), 659–663.
• [5] M. R. Alfuraidan and M. A. Khamsi, Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph, Fixed Point Theory Appl. 2015 (2015), 44.
• [6] M. R. Alfuraidan, Fixed points of monotone nonexpansive mappings with a graph, Fixed Point Theory Appl. 2015 (2015), 49.
• [7] J. Tiammee, A. Kaewkhao, and S. Suantai, On Browder’s convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs, Fixed Point Theory Appl. 2015 (2015), 187.
• [8] O. Tripak, Common fixed-points of G-nonexpansive mappings on Banach spaces with a graph, Fixed Point Theory Appl. 2016 (2016), 87.
• [9] S. Khatoon, and I. Uddin, Convergence analysis of modified Abbas iteration process for two G-nonexpansive mappings, Rend. Circ. Mat. Palermo II. Ser. 70 (2021), 31–44.
• [10] S. Khatoon, I. Uddin, J. Ali, and R. George, Common fixed-points of two G-nonexpansive mappings via a faster iteration procedure, J. Funct. Spaces 2021 (2021), Article ID 9913540, 8 pages, DOI: https://doi.org/10.1155/2021/9913540.
• [11] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), no. 1, 217–229.
• [12] R. Glowinski and P. L. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanic, SIAM, Philadelphia, 1989.
• [13] S. Haubruge, V. H. Nguyen, and J. J. Strodiot, Convergence analysis and applications of the Glowinski Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl. 97 (1998), 645–673.
• [14] W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235 (2011), 3006–3014.
• [15] K. Ullah and M. Arshad, Numerical reckoning fixed-points for Suzuki’s generalized nonexpansive mappings via iteration process, Filomat 32 (2018), no. 1, 187–196.
• [16] F. Gursoy and V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, arXiv (2014), no. 1403.2546v2, 1–16.
• [17] R. P. Agarwal, D. O’Regan, and D. R. Sahu, Iterative construction of fixed-points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (2007), 61–79.
• [18] C. Garodia and I. Uddin, A new iterative method for solving split feasibility problem, J. Appl. Anal. Comput. 10 (2020), no. 3, 986–1004.
• [19] C. Garodia and I. Uddin, A new fixed-point algorithm for finding the solution of a delay differential equation, AIMS Mathematics 5 (2020), no. 4, 3182–3200.
• [20] C. Garodia, I. Uddin, and S. H. Khan, Approximating common fixed-points by a new faster iteration process, Filomat 34 (2020), no. 6, 2047–2060.
• [21] R. Johnsonbaugh, Discrete Mathematics, 7th. Prentice Hall, New Jersey, 1997.
• [22] Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597.
• [23] H. F. Senter and W. G. Dotson, Approximating fixed-points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375–380.
• [24] S. Shahzad and R. Al-Dubiban, Approximating common fixed-points of nonexpansive mappings in Banach spaces, Georgian Math. J. 13 (2006), no. 3, 529–537.
• [25] P. Sridarat, R. Suparaturatorn, S. Suantai, and Y. J. Cho, Convergence analysis of SP-iteration for G-nonexpansive mappings with directed graphs, Bull. Malay. Math. Sci. Soc. 42 (2019), 2361–2380.
• [26] J. Schu, Weak and strong convergence to fixed-points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc. 43 (1991), no. 1, 153–159.
• [27] S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 331 (2005), 506–517.
• [28] D. Yambangwai, S. Aunruean, and T. Thianwan, A new modified three-step iteration method for G-nonexpansive mappings in Banach spaces with a graph, Numer. Algor. 84 (2020), 537–565.
• [29] M. G. Sangago, Convergence of Iterative Schemes for Nonexpansive Mappings, Asian-European J. Math. 4 (2011), no. 4, 671–682.
• [30] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasicontractive operators, Fixed Point Theory Appl. 2 (2004), 97–105.
• [31] R. L. Burden and J. D. Faires, Numerical Analysis, 9th edn. Brooks/Cole Cengage Learning, Boston, 2010.
• [32] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, Berlin, 2017.
• [33] J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bulletin de la Société Mathématique de France 93 (1965), 273–299

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).