Approximative Method of Fixed Point for Φ-Pseudocontractive Operator and an Application to Equation with Accretive Operator ()
Abstract
In this paper, Φ-pseudo-contractive
operators and Φ-accretive operators, more general than the strongly pseudo-contractive
operators and strongly accretive operators, are introduced. By setting up a new
inequality, authors proved that if is a uniformly continuous Φ-pseudo-contractive operator then T has
unique fixed point q and the Mann iterative sequence with random errors
approximates to q. As an application, the iterative
solution of nonlinear equation with Φ-accretive operator is obtained. The
results presented in this paper improve and generalize some corresponding
results in recent literature.
Share and Cite:
Wen, Y. , Feng, A. and Xu, Y. (2014) Approximative Method of Fixed Point for Φ-Pseudocontractive Operator and an Application to Equation with Accretive Operator.
Journal of Applied Mathematics and Physics,
2, 21-25. doi:
10.4236/jamp.2014.21004.
Conflicts of Interest
The authors declare no conflicts of interest.
References
|
[1]
|
C. E. Chidume, “Approximation of Fixed Points of Strongly Pseudo-Contractive Mappings,” Proceedings of the American Mathematical Society, Vol. 120, 1994, pp. 545-551. http://dx.doi.org/10.1090/S0002-9939-1994-1165050-6
|
|
[2]
|
S. S. Chang, Y. J. Cho and B. S. Lee, “Iterative Approximations of Fixed Points and Solutions for Strongly Accretive and Strongly Pseudo-Contractive Mappings in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 224, 1998, pp. 149-165. http://dx.doi.org/10.1006/jmaa.1998.5993
|
|
[3]
|
Z. Q. Liu, M. Bounias and S. M. Kang, “Iterative Approximations of Solutions to Nonlinear Equations of -Strongly Accretive in Banach Spaces,” Rocky Mountain Journal of Mathematics, Vol. 32, 2002, pp. 981-997.
|
|
[4]
|
E. Asplund, “Positivity of Duality Mappings,” American Mathematical Society, Vol. 73, 1967, pp. 200-203.
http://dx.doi.org/10.1090/S0002-9904-1967-11678-1
|
|
[5]
|
Y. G. Xu, “Ishikawa and Mann Iterative Processes with Errors for Nonlinear Strongly Accretive Operator Equations,” Journal of Mathematical Analysis and Applications, Vol. 224, 1998, pp. 91-101. http://dx.doi.org/10.1006/jmaa.1998.5987
|
|
[6]
|
W. R. Mann, “Mean Value Methods in Iteration,” Proceedings of the American Mathematical Society, Vol. 4, 1953, pp. 506- 510. http://dx.doi.org/10.1090/S0002-9939-1953-0054846-3
|
|
[7]
|
K. Deimling, “Nonlinear Functional Analysis,” Springer-Verlag, Berlin, 1985. http://dx.doi.org/10.1007/978-3-662-00547-7
|