Iterative process with errors for fixed points of multivalued \(\Phi\)-hemicontractive operators in uniformly smooth Banach spaces. (English) Zbl 0956.47025
The author considers a uniformly smooth Banach space and a (not necessarily continuous) multivalued \(\Phi\)-hemicontractive mapping \(T:E\to 2^E\). He proves that, under suitable conditions, the multivalued Ishikawa iterative sequence with errors strongly converges to the unique fixed point of \(T\). A related result deals with the strong convergence of the Ishikawa iterative sequence with errors to the solution of the equation \(f\in Tx\) when \(T: E\to 2^E\) is multivalued \(\Phi\)-strongly accretive.
Reviewer: Ulrich Kosel (Freiberg)
MSC:
| 47H10 | Fixed-point theorems |
| 47H04 | Set-valued operators |
| 47J25 | Iterative procedures involving nonlinear operators |
Keywords:
uniformly smooth Banach space; multivalued \(\Phi\)-hemicontractive mapping; multivalued Ishikawa iterative sequence with errorsReferences:
| [1] | Huang, Z., Approximating fixed points of ø-hemicontractive mappings by the Ishikawa iteration process with errors in uniformly smooth Banach spaces, Computers Math. Applic., 36, 2, 13-21 (1998) · Zbl 0938.47042 |
| [2] | Ding, X. P., Iterative process with errors to locally strictly pseudocontractive maps in Banach spaces, Computers Math. Applic., 32, 10, 91-97 (1996) · Zbl 0930.47037 |
| [3] | Ding, X. P., Iterative process with errors for fixed points of multivalued operator of monotone type, J. Sichuan Normal University (Natural Science), 20, 2, 54-59 (1997) |
| [4] | Osilike, M. O., Iterative solution of nonlinear equations of the ∅-strongly accretive type, J. Math. Anal. Appl., 200, 259-271 (1996) · Zbl 0860.65039 |
| [5] | Osilike, M. O., Iterative construction of fixed points of multi-valued operators of the accretive type II, Soochow J. Math., 24, 141-146 (1998) · Zbl 0911.47058 |
| [6] | Zhou, H., Iterative solution of nonlinear equations involving strongly accretive operators without the Lipschitz assumption, J. Math. Anal. Appl., 213, 296-307 (1997) · Zbl 0896.47048 |
| [7] | Zhou, H.; Jia, Y., Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption, (Proc. Amer. Math. Soc., 125 (6) (1997)), 1705-1709 · Zbl 0871.47045 |
| [8] | Chidume, C. E., Iterative solution of nonlinear equations with strongly accretive operators, J. Math. Anal. Appl., 192, 502-518 (1995) · Zbl 0868.47040 |
| [9] | Chidume, C. E., Global iteration schemes for strongly pseudo-contractive maps, (Proc. Amer. Math. Soc., 126 (9) (1998)), 2641-2649 · Zbl 0901.47046 |
| [10] | Chang, S. S., On Chidume’s open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces, J. Math. Anal. Appl., 216, 94-111 (1997) · Zbl 0909.47049 |
| [11] | Chidume, C. E.; Moore, C., The solution by iteration of nonlinear equations in uniformly smooth Banach spaces, J. Math. Anal. Appl., 215, 132-146 (1997) · Zbl 0906.47050 |
| [12] | Reich, S., An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal., 2, 85-92 (1978) · Zbl 0375.47032 |
| [13] | Chang, S. S.; Cho, Y. J.; Lee, B. S.; Jung, J. S.; Kang, S. M., Iterative approximations of fixed points and solutions for strongly accretive and strong pseudo-contractive mappings, J. Math. Anal. Appl., 224, 149-165 (1998) · Zbl 0933.47040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
