Articles

APPROXIMATION OF FIXED POINTS AND VARIATIONAL SOLUTIONS FOR PSEUDO-CONTRACTIVE MAPPINGS IN#br# BANACH SPACES

  • Yekini SHEHU
Expand
  • Department of Mathematics, University of Nigeria, Nsukka, Nigeria

Received date: 2012-10-25

  Revised date: 2013-08-25

  Online published: 2014-03-20

Abstract

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gˆateaux differentiable norm. Assume that every nonempty closed con-vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map-pings which is also a unique solution to variational inequality problem involving-strongly
pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con-vex optimization problems, and split feasibility problems. Our result extends many recent important results.

Cite this article

Yekini SHEHU . APPROXIMATION OF FIXED POINTS AND VARIATIONAL SOLUTIONS FOR PSEUDO-CONTRACTIVE MAPPINGS IN#br# BANACH SPACES[J]. Acta mathematica scientia, Series B, 2014 , 34(2) : 409 -423 . DOI: 10.1016/S0252-9602(14)60015-1

References

[1] Xu H K. Inequality in Banach spaces with Applications, Nonlinear Anal, 1991, 16: 1127–1138

[2] Xu Z B, Roach G F. Characteristic inequalities of uniformly smooth Banach spaces. J Math Anal Appl, 1991, 157: 189–210

[3] Mann W R. Mean value methods in iterations. Bull Amer Math Soc, 1953, 4: 506–510. MR 14: 988f

[4] Browder F E, Petryshyn W E. Construction of fixed points of nonlinear mappings in Hilbert spaces. J Math Anal Appl, 1967, 20: 197–228

[5] Kato T. Nonlinear semigroups and evolution equations. Japan: J Math Soc, 1967, 19: 508–520

[6] Browder F E. Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull Amer Math Soc, 1967, 73: 875–882

[7] Browder F E. Nonlinear operators and nonlinear equations of evolution in Banach spaces//Proc of Symposia in Pure Math, 1976, 18 (part 2)

[8] Ray W O. An elementary proof of surjective for a class of accretive operators. Proc Amer Math Soc, 1979, 75: 255–258

[9] Caristi J. The fixed point theory for mappings satisfying inwardness conditions. PhD Thesis. Iowa City: The University of Iowa, 1975

[10] Martin R H, Jr. A global existence theorem for authonomous differential equations in Banach spaces. Proc Amer Math Soc, 1970, 26: 307–314

[11] Martin R H, Jr. Nonlinear Operators and Differential Equations in Banach Spaces. New York: Interscience, 1976

[12] Browder F E. The solvability of nonlinear functional equations. Duke Math J, 1963, 30: 557–566

[13] Browder F E. Nonlinear elliptic boundary value problems. Bull Amer Math Soc, 1963, 69: 862–874

[14] Browder F E. Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces. Bull Amer Math Soc, 1967, 73: 470–475

[15] Browder F E. Nonlinear monotone and accretive operators in Banach space. Proc Natl Acad Sci USA, 1968, 61 388–393

[16] Deimling K. Nonlinear Functional Analysis. Springer-Verlag, 1985

[17] Chidume C E. Iterative approximation of fixed points of Lipschitz pseudo-contractive maps. Proc Amer Math Soc, 129 (2001), 2245–2251. MR 2002e:47071

[18] Chidume C E, Moore C. Fixed point iteration for pseudo-contractive maps. Proc Amer Math Soc, 1999, 127(4): 1163–1170. MR 99f: 47068

[19] Chidume C E, Mutangadura S. An example on the Mann iteration method for Lipschitz pseudocontractions. Proc Amer Math Soc, 2001, 129(8): 2359–2363. MR 20022f: 47104

[20] Chidume C E, Zegeye H. Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps. Proc Amer Math Soc, 2004, 132: 831–840

[21] Morales C H. Zeros of accretive operators satisfying certain boundary conditions. J Math Anal Appl, 1985, 105: 167–175

[22] Morales C H. Variational inequalities for -pseudo-contractive mappings. Nonlinear Anal TMA, 2012, 75: 477–484

[23] Reich S. Strong convergence theorems for resolvents of accretive mappings in Banach spaces. J Math Anal Appl, 1980, 75: 287–292

[24] Song Y, Chen R. Convergence theorems of iterative algorithms for continuous pseudocontractive mappings.
Nonlinear Anal, 2007, 67: 3058–3063

[25] Turret B. A dual view of a theorem of Baillon//Singh S P, Burry J H. Nonlinear Analysis and Applications. New York: Marcel Dekker, 1982: 279–286

[26] Yu Y. An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings. Journal of Applied Mathematics Volume 2012, Article ID 341953, 11 pages, doi:10.1155/2012/341953

[27] Schu J. Approximating fixed points of Lipschitzian pseudo-contractive mappings. Houston J Math, 1993, 19: 107–115. MR 94f: 47066

[28] Yao Y, Zhou H, Liou Y C. Modified Mann’s algorithm based on the CQ method for pseudo-contractive mappings. J Appl Math Inform, 2010, 28(5/6): 1499–1506

[29] Zegeye H, Shahzad N. Strong convergence of an iterative method for pseudo-contractive and monotone mappings. J Global Optim, 2012, 54(1): 173–184

[30] Zegeye H, Shahzad N, Alghamdi M A. Convergence of Ishikawa´s iteration method for pseudocontractive mappings. Nonlinear Anal, 2011, 74(18): 7304–7311

[31] Ofoedu E U, Zegeye H. Further investigation on iteration processes for pseudocontractive mappings with application. Nonlinear Anal, 2012, 75(10: 153–162

[32] Osilike M O. Iterative solution of nonlinear equations of the -strongly accretive type. J Math Anal Appl, 1996, 200: 259–271

[33] Ishikawa S. Fixed points by a new iteration method. Proc Amer Math Soc, 1974, 44(1): 147–150

[34] Qihou L. On Naimpally and Singh’s open questions. J Math Anal Appl, 1987, 124: 157–164. MR 88j: 47078

[35] Qihou L. The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings. J Math Anal Appl, 1990, 148: 55–62. MR 92b: 47094

[36] Chidume C E, Abbas M, Ali B. Convergence of the Mann iteration algorithm for a class of pseudocontractive
mappings. Appl Math Comput, 2007, 194: 1–6

[37] Bruck R E. A strong convergent iterative method for the solution of 0 2 Ax for a maximal monotone operator A in a Hilbert space. J Math Anal Appl, 1974, 48: 114–126

[38] K¨ornlein D, Kohlenbach U. Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces. Nonlinear Anal, 2011, 74: 5253–5267

[39] Li X, Huang N, O’Regan D. Viscosity approximation methods for pseudo-contractive semigroups in Banach
spaces. Nonlinear Anal, 2012, 75: 3776–3786

[40] Moore C, Nnoli B V C. Iterative solution of nonlin-ear equations involving set-valued uniformly accretive operators. Comput Math Appl, 2001 42: 131–140

[41] Zegeye H, Shahzad N, Mekonen T. Viscosity approximation methods for pseudo-contractive mappings in Banach spaces. Appl Math Comp, 2007, 185: 538–546

[42] Zegeye H, Shahzad N. Strong convergence theorems for a common zero of a countably infinite family of -inverse strongly accretive mappings. Nonlinear Anal TMA, 2009, 71: 531–538

Outlines

/