Strong and Weak Convergence of Cesàro Mean for Some Iteration Methods

Year 2023, , 236 - 253, 31.05.2023

Lale Cona , Çiğdem Kaygusuz

https://doi.org/10.35193/bseufbd.1190411

Abstract

In this article, firstly, summability techniques and iteration methods are investigated. In addition, studies on ergodic theory which are related to the Cesàro mean summability technique are examined for some iteration methods. Finally, a study on the strong convergence the Cesàro mean with the Halpren iteration for asymptotic non-expanding transformations is discussed.

Keywords

Cesàro Summability, Ergodic Theory, Iteration Methods, Strong Convergence, Fixed Point Approaches

Bazı İterasyon Yöntemleri için Cesàro Ortalamasının Kuvvetli ve Zayıf Yakınsaklığı

Abstract

Bu makalede, ilk olarak toplanabilme teknikleri ve iterasyon yöntemleri araştırılmıştır. Ayrıca, bazı iterasyon yöntemleri için Cesàro anlamında toplanabilme tekniğine bağlı olarak ergodik teori üzerine yapılan çalışmalar incelenmiştir. Son olarak asimptotik genişlemeyen dönüşümler için Halpren iterasyonu ile Cesàro ortalamasının güçlü yakınsaklığı üzerine yapılan bir çalışma irdelenmiştir.

Keywords

Cesàro Toplanabilme, Ergodik Teori, İterasyon Yöntemleri, Kuvvetli Yakınsaklık, Sabit Nokta Yaklaşımları

References

• Grandi, G. (1703). Quadraturo circuli et hyperbolaeper infinitas hyperbolas geometrice exhibita, Pisa.
• Cauchy, A. L. (1821). Analyse algebrique. Chez Debure Freres, Paris, 576 pp.
• Cesaro, E. (1890). Sur la multiplication des series. Bulletin des Sciences Mathematiques,14, 114–120.
• Flett, T. M. (1957). On an extension of absolute summability and some theorems of Littlewood and Paley. Proceedings of the London Mathematical Society, (3) 7, 113–141.
• Das, G. (1969). Tauberian theorems for absolute Nörlund summability. Proceedngs of the London Mathematical Society, 9 (3), 357-394.
• Kishore, N.  Hotta, G. C. (1970). On |(N,) ̅P_n | summabilitiy factors. Acta Scientarium Mathematicarum (SZEGED), 31, 9–12.
• Tanovic-Miller, N. (1979). On strong summability. Glasnik Mathematicki, 34 (14), 87–97.
• Nörlund, N. E. (1919). Sur une application des fonctiony permutables. Lunds Universitets Arsskrift, 2, 16, 1–10.
• Boos, J. (2000). Classical and Modern Methods in Summability. Oxford university Press, New York.
• Ceng, L. C., Ansari, Q.  Yao, J. C. (2011). Some iterative methods for finding fixed points and for solving constrained convex minimization problems, Nonlinear Analysis. Theory, Methods Applications, 74, 5286–5302.
• Argyros, I. K. ve Hilout, S. (2013). Computational methods in nonlinear analysis: Efficient algorithms, fixed point theory and applications. World Scientific Publishing Company Incorporated, Hackensack, NJ.
• Argyros, I. (2007). Computational theory of iterative methods. Elsevier Science.
• Border, K. C. (1989). Fixed point theorems with applications to economics and game theory, Cambridge University press, Cambridgei UK.
• Borwein, J. ve Sims, B. (2011). The douglas-rachford algorithm in the absence of convexity. H. H. Bauschke Burachik, R. S. Combettes, P. L. Elser, V. Luke, D. R. ve Wolkowicz, H., ed. Fixed-point algorithms for ınverse problems in science and engineering. Springer New York, 93–109. Shehu, Y. (2011). Iterative methods for family of strictly pseudocontractive mappings and system of generalized mixed equilibrium problems and variational inequality problems. Fixed Point Theory and Applications, 852789.
• Liouville, J. (1837). Second memoire: Sur le developpement des fonctions ou parties de fonctions en series dont les divers termes sont assujest à satisfare à une meme équ ation différentielles dusecond membre contenant un parametre variable. Jurnal de Mathématiques pures et appliques, 2, 16–35.
• Mann, W. R. (1953). Mean value methods in iteration. Proceedings of the American Mathematical Society, 4, 506–510.
• Baillon, J. B. (1975). Un the´ore`me de type ergodique pour les contractions non line´airs dans un espaces de Hilbert. Comptes Rendus Academie Sciences Paris A B, 280, 1511–41.
• Zhu, Z.  Chen, R. (2014). Strong Convergence on Iterative Methods of Cesàro Meansfor Nonexpansive Mapping in Banach Space. Hindawi Publishing Corporation Abstract and Applied Analysis, Article ID 205875.
• Song, Y. (2011). Halpern iteration of Cesàro means for asymptotically nonexpansive mappings. ScienceAsia, 37, 145–151.
• Song, Y. (2010). Mann iteration of Cesa`ro means for asymptotically nonexpansive mappings. Nonlinear Analysis, 72, 176–82.
• Madox, I. J. (1970). Elements of Functional Analysis. Cambridge Univercity Press, Cambridge.
• Bayraktar, M. (2006). Fonksiyonel Analiz, Atatürk Üniversitesi Yayınları, Erzurum.
• Musayev, B. ve Alp, M. (2000). Fonksiyonel analiz. Kütahya.
• Takahashi, W. (2000). Nonlinear functional analysis – Fixed point theory and its applications. Yokohama Publishers, Yokohama.
• Kreyszig, E. (1978). Introductory functional analysis with applications, John Wiley-Sons, Newyork.
• Petersen, G. M. (1966). Regular matrix transformations. Mc Graw Publishing Company Limited, London-New York-Toronto.
• Fekete, M. (1911). Zur theorie der divergenten reihen. Mathematical es Termezs Ertesitö (Budapest), 29, 719–726.
• Mazhar, S. M. (1966). On the Summability factors of infinite series. Publicationes Mathematicae Debrecen, 13, 229–236.
• Picard, E. (1890). Memoire sur la théoire dés équations aux dérivées partielles et la méthode des approximations successives. Jurnal deMathématiques pures et appliquées, 6, 145–210.
• Berinde, V. (2007). Iterative approximation of fixed points. Springer, Berlin.
• Krasnoselskii, M. A. (1955). Two remarks on the method of succesive approximations, Uspekhi Matematicheskikh Nauk, 10, 123–127.
• Schaefer, H. (1957). Über die Methode Sukzessiver Approximationen. jahresbericht der Deutschen Mathematiker-Vereinigung, 59, 131–140.
• Halpern, B. (1967). Fixed points of nonexpansive maps. Bulletin of the American Mathematical Society, 73, 957–61.
• Ishikawa, S. (1974). Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44, 147–150.
• Noor, M. A. (2000). New approximation schemes for general variational inequlities, Journal of Mathematics and Applications, 251, 217–229.
• Agarwal, R. P., O’Regan, D.  Sahu, D. R. (2007). Iterative construction of fixed points of nearly asymtotically nonexpansive mapping, Journal of Nonlinear and Convex Analysis, 8 (1), 61–79.
• Thianwan, S. (2009). Common fixed points of new iterations for two asymptotically nonexpansive nonself –mappings in a Banach Space. Journal of Compatational and Applied Mathematics, 224, 688–695.
• Karahan, I.  Özdemir, M. (2013). A General iterative method for approximation of fixed points and their applications. Advances in Fixed Point Theory, 3 (3), 510–526. ouzara, N. E. H. (2017). Some Fixed-Point Results for a New Three Steps Iteration Process in Banach Spaces. Fixed Point Theory, 18 (2), 625–640.
• Banach, S. (1922). Sur les operations dans les ensembles abstraits et leur applications aux equations integrals. Fundamenta Mathematicae, 3 (1), 133–181.
• Kannan, R. (1968). Some results on fixed points. Bulletin of Calcutta Mathematical Society, 60, 71–76.
• Chatterjea, S. K. (1972). Fixed Point Theorems, Comptes rendus de I’Academie. Bulgare des Sciences, 25, 727–730.
• Zamfirescu, T. (1972). Fix point theorems in metric spaces. Archivum Mathematicum, 23 (1972), 292–298.
• Berinde, V. (2004). Approximating fixedpoints of weak contractions using the picard iteration. Nonlinear Analysis Forum, 9, 43–53.
• Zhang, J.  Cui, Y. (2014). Iterative algorithms based on hybrid method and Cesàro mean of asymptotically nonexpansive mappings for equilibrium problems. Fixed Point Theory and Applications, 16 (2014), 1–16.
• Combettes, P.L.  Hirstoaga, S. A. (2005). Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis, 6, 117–136.
• Goebel, K.  Kirk, W. A. (1972). A fixed-point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society, 35, 171–4.
• Kirk, W. A., Yanez, C. M. ve Shin, S. S. (1998). Asymptotically nonexpansive mappings. Nonlinear Analysis, 33, 1–12.
• Bruck, R. E. (1979). A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel Journal of Mathematics, 32 (2-3), 107–116.
• Hirano, N. ve Takahashi, W. (1979). Nonlinear ergodic theorems for nonexpansive mappings in Hilbert spaces. Kodai Mathematical Journal, 2, 11–25.
• Shimizu, T.  Takahashi, W. (1996). Strong convergence theorem for asymptotically non-expansive mappings. Nonlinear Analysis, 26, 265–72.
• Shioji, N. ve Takahashi, W. (1999a). A strong convergence theorem for asymptotically non-expansive mappings in Banach spaces. Archivum Mathematicum, 72, 354–9.
• Moore, C. ve Nnoli, B. V. C. (2001). Strong convergence of averaged approximants for Lipschitz pseudocontractive maps. Journal of Mathematical Analysis and Applications, 260, 269–78.
• Wittmann, R. (1992). Approximation of fixed points of nonexpansive mappings. Archivum Mathematicum, 59, 486–91.
• Reich, S. (1974). Some fixed-point problems. Atti della Accademia Nazionale dei Lincei, 57, 194–8.
• Reich, S. (1983). Some problems and results in fixed point theory. Contemporary mathematics, 21, 179–87.
• Reich, S. (1994). Approximating fixed points of nonexpansive mappings. Panamerican Mathematical Journal, 4, 23–8.
• Song, Y. (2007) Iterative approximation to common fixed points of a countable family of nonexpansive mappings. Applied analysis, 86, 1329–37.
• Song, Y. (2008a). A new sufficient condition for the strong convergence of Halpern type iterations. Applied Mathematics and Computation, 198, 721–8.
• Song, Y. ve Xu, Y. (2008). Strong convergence theorems for nonexpansive semigroup in Banach spaces. Journal of Mathematical Analysis and Applications, 338, 152–61.
• Song, Y. ve Chen, R. (2006a). Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings. Applied Mathematics and Computation, 180, 275–87.
• Song, Y. ve Chen, R. (2008). Strong convergence of an iterative method for non-expansive mappings. Mathematische Nachrichten, 281, 1196–204.
• Xu, H. K. (2002). Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society, 66, 240–56.
• Xu, H. K. (2004). Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications, 298 (1), 279–291.
• Song, Y.  Chen, R. (2007). Viscosity approximate methods to Cesaro means for nonexpansive mappngs. Applied Mathematics and Computation, 186 (2), 1120–1128.
• Yao, Y., Liou, Y. C.  Zhou, H. (2009). Strong convergence of an iterative method for nonexpansive mappings with new control conditions, Nonlinear Analysis. Theory, Methods and Applications, 70 (6), 2332–2336.
• Song, G. ve Zhang, H. (2011). Reproducing kernel Banach spaces with the ℓ1 norm II: Error analysis for regularized least square regression. Neural computation, 23 (10), 2713–2729.
• Liu, L. S. (1995). Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 194, 114–25.
• Xu, H. K. (2003). An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications, 116, 659–78.
• Matsushita, S. ve Kuroiwa, D. (2004). Strong convergence of averaging iterations of non-expansive nonselfmappings. Journal of Mathematical Analysis and Applications, 294, 206–14.
• Lions, P. L. (1977). Approximation de points fixes de contraction. Comptes Rendus Academie Sciences Paris A B, 284, 1357–9.