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CONVERGENCE OF APPROXIMATING PATHS TO SOLUTIONS OF VARIATIONAL INEQUALITIES INVOLVING NON-LIPSCHITZIAN MAPPINGS

  • Jung, Jong-Soo ;
  • Sahu, Daya Ram
  • Published : 2008.03.31

Abstract

Let X be a real reflexive Banach space with a uniformly $G\hat{a}teaux$ differentiable norm, C a nonempty closed convex subset of X, T : C $\rightarrow$ X a continuous pseudocontractive mapping, and A : C $\rightarrow$ C a continuous strongly pseudocontractive mapping. We show the existence of a path ${x_t}$ satisfying $x_t=tAx_t+(1- t)Tx_t$, t $\in$ (0,1) and prove that ${x_t}$ converges strongly to a fixed point of T, which solves the variational inequality involving the mapping A. As an application, we give strong convergence of the path ${x_t}$ defined by $x_t=tAx_t+(1-t)(2I-T)x_t$ to a fixed point of firmly pseudocontractive mapping T.

Keywords

pseudocontractive mapping;strongly pseudocontractive mapping;firmly pseudocontractive mapping;nonexpansive mapping;fixed points;uniformly $G\hat{a}teaux$ differentiable norm;variational inequality

References

  1. F. E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875-882 https://doi.org/10.1090/S0002-9904-1967-11823-8
  2. L. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Aacademic Publishers, 1990
  3. K. Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 365-374 https://doi.org/10.1007/BF01171148
  4. K. Deimling, Nonlinear Functional Analysis, Spring-Verlag, Berlin, 1985
  5. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, Inc., 1984
  6. K. S. Ha and J. S. Jung, Strong convergence theorems for accretive operators in Banach space, J. Math. Anal. Appl. 147 (1990), no. 2, 330-339 https://doi.org/10.1016/0022-247X(90)90351-F
  7. J. S. Jung and S. S. Kim, Strong convergence theorems for nonexpansive nonselfmappings in Banach space, Nonlinear Anal. 33 (1998), no. 3, 321-329 https://doi.org/10.1016/S0362-546X(97)00526-9
  8. T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508-520 https://doi.org/10.2969/jmsj/01940508
  9. R. H. Martin, Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973), 399-414 https://doi.org/10.2307/1996511
  10. C. H. Morales, On the fixed point theory for local k-pseudocontractions, Proc. Amer. Math. Soc. 81 (1981), no. 1, 71-74 https://doi.org/10.2307/2043988
  11. C. H. Morales, Sreong convergence theorems for pseudo-contractive mapping in Banach spaces, Houston J. Math. 16 (1990), no. 4, 549-557
  12. C. H. Morales and J. S. Jung, Convergence of paths for pseudo-contractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000), 3411-3419 https://doi.org/10.1090/S0002-9939-00-05573-8
  13. A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl. 241 (2000), no. 1, 46-55 https://doi.org/10.1006/jmaa.1999.6615
  14. J. G. O'Hara, P. Pillay and H. K. Xu, Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear Anal. 64 (2006), 2022-2042 https://doi.org/10.1016/j.na.2005.07.036
  15. Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bul. Amer. Math. Soc. 73 (1967), 591-597 https://doi.org/10.1090/S0002-9904-1967-11761-0
  16. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287-292 https://doi.org/10.1016/0022-247X(80)90323-6
  17. J. Schu, Approximating fixed points of Lipschitzian pseudocontractive mappings, Houston J. Math. 19 (1993), no. 1, 107-115
  18. B. K. Sharma and D. R. Sahu, Firmly pseudo-contractive mappings and fixed points, Comment. Math. Univ. Carolinae 38 (1997), no. 1, 101-108
  19. H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), no. 1, 240-256
  20. V. Barbu and Th. Precupanu, Convexity and Optimization in Banach spaces, Editura Academiei R. S. R., Bucharest, 1978
  21. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990
  22. H. K. Xu, Approximating curves of nonexpansive nonself-mappings in Banach spaces, C. R. Acad. Sci. Paris Ser. I. Math. 325 (1997), 151-156 https://doi.org/10.1016/S0764-4442(97)84590-9

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