Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NECESSARY AND SUFFICIENT CONDITION FOR THE CONVERGENCE OF THE MANN SEQUENCE FOR A CLASS OF NONLINEAR OPERATORS

  • Chidume, C.E. (The Avdus Salam International Centre for Theoretical Physics) ;
  • Nnoli, B.V.C. (Department of Mathematics, University of Jos)
  • Published : 2002.05.01
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Abstract

Let E be a real Banach space. Let T : E longrightarrow E be a map with F(T) : = { x $\in$ E : Tx = x} $\neq$ 0 and satisfying the accretive-type condition $\lambda\$\mid$x-Tx\$\mid$^2$, for all $x\inE,\;x^*\inf(T)\;and\;\lambda >0$. We prove some necessary and sufficient conditions for the convergence of the Mann iterative sequence to a fixed point of T.

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References

  1. F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math Anal. Appl. 20 (1967), 297-228. https://doi.org/10.1016/0022-247X(67)90085-6
  2. C. E. Chidume, The solution by iteration of nonlinear equations in certain Banach spaces, J. Nig. Math. Soc. 3 (1984), 57-62.
  3. M. K. Ghosh and L. Debnath, Convergence of Ishikawa iterates of quasinonexpansive mappings, J. Math Anal. Appl. 207 (1997), 96-103. https://doi.org/10.1006/jmaa.1997.5268
  4. J. P. Gossez and E. Lami Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific. J. Math. 40 (1972), 565-573. https://doi.org/10.2140/pjm.1972.40.565
  5. T. L. Hicks and J. R. Kubicek, On the Mann iteration process in Hilbert space, J. Math Anal. Appl. 59 (1977), 498-504. https://doi.org/10.1016/0022-247X(77)90076-2
  6. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  7. S. Maruster, The solution by iteration of nonlinear equations, Proc. Amer. Math. Soc. 66 (1977), 69-73. https://doi.org/10.1090/S0002-9939-1977-0473935-9
  8. Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. https://doi.org/10.1090/S0002-9904-1967-11761-0
  9. M. O. Osilike and A. Udomene, Demiclosedness principle and convergence theorems for stictly pseudocontractive mappings of the Browder-Petryshyn type, J. Math Anal. Appl. 256 (2001), no. 2, 431-445. https://doi.org/10.1006/jmaa.2000.7257
  10. Tan and Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math Anal. Appl. 178 (1993), 301-308. https://doi.org/10.1006/jmaa.1993.1309

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