The equivalence of convergence results between modified Ishikawa and modified Mann iterations

• Autorzy: Mogbademu A. A. , Kim J. K.
• Języki publikacji: EN

Abstrakty

EN

In [11], the author discussed a new class of nearly weak uniformly L-Lipschitzian mappings and prove some strong convergence results of the modified Ishikawa iteration with errors in real Banach spaces. And the author has given the open problem as follows: Are there any difference on convergence between the Mann iteration and Ishikawa iteration? Can we prove the equivalence on convergence between these two iterations? In this paper, we given an affirmative answer to the open problem.

Słowa kluczowe

EN

modified Mann iteration process with errors; modified Ishikawa iteration process with error; Banach space; fixed point; nearly weak uniformly L-Lipschitzian

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2019
• Tom: nr 62
• Strony: 93--102
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Mogbademu A. A.

• Department of Mathematics University of Lagos Lagos, Nigeria

autor

Kim J. K.

• Department of Mathematics Educations Kyungam University Masan, Kyungnam, Republic of Korea

Bibliografia

• [1] Banerjee S., Choudhury B.C., The equivalence between the convergences of Ishikawa and Mann iterations for an ф- strongly pseudocontractive operators, Nonlinear Funct. Analy. and Appl., 12(1)(2007), 61-74.
• [2] Chang S., Some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 129(2000), 845-853.
• [3] Chang S.S., Cho Y.J., Kim J.K., Some results for uniformly L-Lipschitzian mappings in Banach spaces, Appl. Math. Lett., 22(2009), 121-125.
• [4] Harder A.M., Hicks T.L., Stability results for fixed point iteration procedures, Math. Japon., 33(1988), 693-706.
• [5] Ishikawa S., Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1974), 147-150.
• [6] Kim J.K., Sahu D.R., Nam Y.M., Convergence theorem for fixed points of nearly uniformly L-Lipschitzian asymptotically generalized Ф-hemicon- tractive mappings, Nonl. Anal., 71(2009), e2833- e2838.
• [7] Mann W.R., Mean value methods in iteration, Proc. Amer. Math. Soc., 4(1953), 506-610.
• [8] Mogbademu A.A., A convergence theorem for Multistep iterative scheme for nonlinear maps, Publ. Inst. Math. (Beograd) (N.S.), 98(112)(2015), 281-285.
• [9] Mogbademu A.A., Strong convergence results for nonlinear mappings in Banach spaces, Creat. Math. Inform., 25(1)(2016), 79-85.
• [10] Mogbademu A.A., Fixed points of nearly weak uniformly L-Lipschitzian mappings in real Banach spaces, Creat. Math. Inform., 27(1)(2018), 63-70.
• [11] Mogbademu A.A., Ishikawa iteration with errors for nearly weak uniformly L-Lipschitzian mappings, Transyvania J. Math. Mech., 10(1)(2018), 23-30.
• [12] Moore C., Nnoli B.C., Iterative solution of nonlinear equations involving set-valued uniformly accretive operators, Computers and Mathematics with Applications, 42(1-2)(2001), 131-140.
• [13] Ofoedu E.U., Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space, J. Math. Anal. Appl., 321(2006), 722-728.
• [14] Rhoades B.E., Soltuz S.M., The equivalence between Mann-Ishikawa iterations and multistep iteration, Nonl. Anal.: Theory, Methods and Applications, 58(2004), 218-228.
• [15] Sahu D.R., Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces, Comment. Math. Univ. Carolinae, 46(4)(2005), 653-666.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).