CONVERGENCE OF MULTI-STEP ITERATIVE SEQUENCE FOR NONLINEAR UNIFORMLY L-LIPSCHITZIAN MAPPINGS

Year 2015, Volume: 3 Issue: 2, 89 - 99, 01.10.2015

Adesanmi Alao Mogbademu

Abstract

In this paper, by using the proof method of Xue, Ra q and Zhou[19] some strong convergence results of multi-step iterative sequence are proved for nearly uniformly L- Lipschitzian mappings in real Banach spaces. Our results generalise and improve some recent known results.

Keywords

Multistep iteration; asymptotically pseudo-contractive; uniformly L􀀀 Lipschitzian; Banach spaces

References

• [1] Chang, S. S. Some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 129 , 99 (2000), 845-853.
• [2] Chang, S. S., Cho, Y. J., Lee, B. S. and Kang, S. H. Iterative approximation of xed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces , J. Math. Anal. Appl., 224 (1998), 165-194.
• [3] Chang, S. S., Cho, Y. J., and Kim, J. K. Some results for uniformly L-Lipschitzian mappings in Banach spaces , Appl. Math. Lett., 22, (2009), 121-125.
• [4] Goebel, K. and Kirk, W. A. A xed point theorem for asymptotically nonexpansive mappings , Proc. Amer. Math. Soc., Vol. 35 , (1972), 171-174.
• [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. and Nam, Y. M.Convergence theorem for xed points of nearly uniformly L􀀀 Lipschitzian asymptotically generalized 􀀀 hemicontractive mappings, Nonl. Anal., 71, 99(2009), e2833- e2838.
• [7] Mann, W.R. Mean value methods in iteration, Proc. Amer. Math. Soc., 4, 99 (1953), 506-610 .
• [8] Moore, C. and Nnoli, B. V. C. Iterative solution of nonlinear equations involving set-valued uniformly accretive operators , Comput. Math. Anal. Appl., 42, (2001), 131-140 .
• [9] Mogbademu, A.A. and Xue, Z. Some convergence results for nonlinear maps in Banach spaces, Int. J. Open Problems Compt. Math., Vol. 6, (2013), 1- 10.
• [10] Mogbademu, A.A. Convergence theorem of modied Noor iteration for nonlinear maps in Banach spaces, J. Adv. Math. Stud., Vol. 7 (2014), nos. 1,56-64.
• [11] Noor, M.A. Three-step iterative algorithms for multi-valued quasi variational inclusions J. Math. Anal. Appl., 225 (2001), 589-604.
• [12] Noor, M.A., Kassias, T. M. and Huang, Z. Three-step iterations for nonlinear accretive operator equations , J. Math. Anal. Appl., 274 (2002), 59-68.
• [13] Ofoedu, E.U. Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseu- docontractive mapping in real Banach space , J. Math. Anal. Appl., 321 (2006), 722-728.
• [14] Olaleru, J.O. and Mogbademu, A.A. Modied Noor iterative procedure for uniformly con- tinuous mappings in Banach spaces, Boletin de la Asociacion Matematica Venezolana, Vol. XVIII, No. 2 (2011), 127- 135.
• [15] Raq, A., Acu, A. M. and Sofonea, F. An iterative algorithm for two asymptotically pseu- docontractive mappings , Int. J. Open Problems Compt. Math., Vol. 2 (2009), 371- 382.
• [16] Rhoades, B.E. and Soltuz, S.M. The equivalence between Mann-Ishikawa iterations and multistep iteration , Nonl. Anal.: Theory, Methods and Applications, Vol. 58 (2004), 218- 228.
• [17] Schu, J. Iterative construction of xed points of asymptotically nonexpansive mappings , J. Math. Anal. Appl., 158 (1999), 407-413.
• [18] Sahu, D. R. Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces , Comment. Math. Univ. Carolin 46 (4) (2005), 653-666.
• [19] Xue, Z., Raq, A. and Zhou, H. On the convergence of multi-step iteration for uniformly continuous 􀀀 Hemicontractive mappings, Abstract and Applied Analysis, Vol. 2012, Article ID 386983, (2012), 1-9.