Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)

• Autorzy: Paimsang Papinwich , Thianwan Tanakit

Identyfikatory

DOI

10.1515/dema-2024-0070

• Języki publikacji: EN

Abstrakty

EN

This article aims to provide a modified Noor iterative scheme to approximate the fixed points of generalized nonexpansive mappings with property (E) called MN-iteration. We establish the strong and weak convergence results in a uniformly convex Banach space. Additionally, numerical experiments of the iterative technique are demonstrated using a signal recovery application in a compressed sensing situation. Ultimately, an illustrative analysis regarding Noor, SP-, and MN-iteration procedures is obtained via polysomnographic techniques. The images obtained are called polynomiographs. Polynomiographs have importance for both the art and science aspects. The obtained graphs describe the pattern of complex polynomials and also the convergence properties of the iterative method. They can also be used to increase the functionality of the existing polynomiography software.

Słowa kluczowe

EN

signal recovery; polynomiography; MN-iteration; García-Falset mapping; weak and strong convergence

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20240070
• Opis fizyczny: Bibliogr. 50 poz., rys., tab.

Twórcy

autor

Paimsang Papinwich

• 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] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal Appl. 340 (2008), no. 2, 1088–1095.
• [2] J. García-Falset, E. Llorens-Fuster, and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375 (2011), no. 1, 185–195.
• [3] H. F. Senter and W. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), no. 2, 375–380.
• [4] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul. 4 (2005). no. 4, 1168–1200.
• [5] A. Dixit, D. R. Sahu, A. K. Singh, and T. Som, Application of a new accelerated algorithm to regression problems, Soft Comput. 24 (2020), 1539–1552.
• [6] O. Ege and I. Karaca, Banach fixed point theorem for digital images, J. Nonlinear Sci. Appl. 8 (2015), 237–245.
• [7] J. Franklin, Methods of Mathematical Economics, Springer Verlag, New York, 1980.
• [8] W. Inthakon, S. Suantai, P. Sarnmeta, and D. Chumpungam, A new machine learning algorithm based on optimization method for regression and classification problems, Mathematics 8 (2020), no. 6, 1007.
• [9] S. Khatoon, I. Uddin, and D. Baleanu, Approximation of fixed point and its application to fractional differential equation, J. Appl. Math. Comput. 66 (2021), no. 1–2, 507–525.
• [10] J.L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–519.
• [11] D. R. Sahu, Applications of accelerated computational methods for quasi-nonexpansive operators to optimization problems, Soft Comput. 24 (2020), no. 23, 17887–17911.
• [12] D. R. Sahu, J. C. Yao, M. Verma, and K. K. Shukla, Convergence rate analysis of proximal gradient methods with applications to composite minimization problems, Optimization 70 (2021), no. 1, 75–100.
• [13] S. Suantai, K. Kankam, and P. Cholamjiak, A novel forward-backward algorithm for solving convex minimization problem in Hilbert spaces, Mathematics 8 (2020), no. 1, 42.
• [14] R. Suparatulatorn, P. Charoensawan, and K. Poochinapan, Inertial self-adaptive algorithm for solving split feasible problems with applications to image restoration, Math. Methods Appl. Sci. 42 (2019), no. 18, 7268–7284.
• [15] R. Suparatulatorn and A. Khemphet, Tseng type methods for inclusion and fixed point problems with applications, Mathematics 7 (2019), no. 12, 1175.
• [16] R. Suparatulatorn, A. Khemphet, P. Charoensawan, S. Suantai, and N. Phudolsitthiphat, Generalized self-adaptive algorithm for solving split common fixed point problem and its application to image restoration problem, Int. J. Comput. Math. 97 (2020), no. 7, 1431–1443.
• [17] L.U. Uko, Remarks on the generalized Newton method, Math. Program. 59 (1993), 404–412.
• [18] L.U. Uko, Generalized equations and the generalized Newton method, Math. Program. 73 (1996), 251–268.
• [19] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. 54 (1965), no. 4, 1041–1044.
• [20] D. Göhde, Zum prinzip der kontraktiven abbildung, Math. Nachr. 30 (1965), no. 3–4, 251–258.
• [21] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), no. 9, 1004–1006.
• [22] B. Rhoades, Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl. 183 (1994), no. 1, 118–120.
• [23] B. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001), no. 4, 2683–2693.
• [24] 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.
• [25] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 4 (1974), 147–150.
• [26] W. R. Mann, Mean value methods in iteration, Amer. Math. Soc. 4 (1953), 506–510.
• [27] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), no. 1, 217–229.
• [28] 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), no. 9, 3006–3014.
• [29] D. Thakur, B. S. Thakur, and M. Postolache, New iteration scheme for numerical reckoning fixed points of nonexpansive mappings, J. Inequal. Appl. 2014 (2014), 1–15.
• [30] V. Berinde and F. Takens, Iterative Approximation of Fixed Points, vol. 1912, Springer, Berlin, 2007.
• [31] J. Ali, F. Ali, and P. Kumar, Approximation of fixed points for Suzukias generalized nonexpansive mappings, Mathematics 7 (2019), no. 6, 522.
• [32] B. S. Thakur, D. Thakur, and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, Appl. Math. Comput. 275 (2016), 147–155.
• [33] G. I. Usurelu and M. Postolache, Convergence analysis for a three-step Thakur iteration for Suzuki-type nonexpansive mappings with visualization, Symmetry 11 (2019), no. 12, 1441.
• [34] G. I. Usurelu, A. Bejenaru, and M. Postolache, Operators with property (E) as concerns numerical analysis and visualization, Numer. Funct. Anal. Optim. 41 (2020), no. 11 1398–1419.
• [35] B. S. Thakur, D. Thakur, and M. Postolache, A new iteration scheme for approximating fixed points of nonexpansive mappings, Filomat 30 (2016), no. 10, 2711–2720.
• [36] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. 91 (2008), 166–177.
• [37] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010), no. 1, 79–88.
• [38] P. Kocourek, W. Takahashi, and J-C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math. 14 (2010), no. 6, 2497–2511.
• [39] G. I. Usurelu and M. Postolache, Algorithm for generalized hybrid operators with numerical analysis and applications, J. Nonlinear Var. Anal. 6 (2022), no. 3, 255–277.
• [40] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc. 43 (1991), no. 1, 153–159.
• [41] S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311 (2005), no. 2, 506–517.
• [42] X.D. Liu, J.H. Zhang, Z.J. Li, and J.X. Zhang, Generalized secant methods and their fractal patterns, Fractals 17 (2009), no. 2, 211–215.
• [43] M. Szyszkowicz, A survey of several root-finding methods in the complex plane, Comput. Graph. Forum 10 (1991), no. 2, 141–144.
• [44] R. Ye, Another choice for orbit traps to generate artistic fractal images, Comput. Graph. 26 (2002), no. 4, 629–633.
• [45] B. Kalantari, Polynomiography and applications in art, education and science, Comput. Graph. 28 (2004), no. 3, 417–430.
• [46] A. Husain, M. N. Nanda, M. S. Chowdary, and M. Sajid, Fractals: an eclectic survey, part-I, Fractal Fract. 6 (2022), no. 2, 89.
• [47] A. Husain, M. N. Nanda, M. S. Chowdary, and M. Sajid, Fractals: An eclectic survey, part II, Fractal Fract. 6 (2022), no. 7, 379.
• [48] A. Tomar, V. Kumar, U. S. Rana, and M. Sajid, Fractals as Julia and Mandelbrot Sets of complex cosine functions via fixed point iterations, Symmetry 15 (2023), no. 2, 478.
• [49] B. Kalantari, Polynomial Root-Finding and Polynomiography, World Sci. Publishing Co., Hackensack, 2009.
• [50] A. S. Householder, The Numerical Treatment of a Single Nonlinear Equation, McGraw-Hill, New York, 1970.

Uwagi

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