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New fixed point theorems for 1-set-contractive operators in Banach spaces
- Source :
-
Nonlinear Analysis . Aug2007, Vol. 67 Issue 3, p938-944. 7p. - Publication Year :
- 2007
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Abstract
- Abstract: W.V. Petryshyn [Remark on condensing and -set-contractive mappings, J. Math. Anal. Appl. 39 (1972) 717–741] and R.D. Nussbaum [Degree theory for local condensing maps, J. Math. Anal. Appl. 37 (1972) 741–766] defined the topological degree of 1-set-contractive fields and studied fixed point theorems of 1-set-contractive operators by virtue of the potential tool. Following them, G.Z. Li [The fixed point index and the fixed point theorems of 1-set-contraction mappings, Proc. Amer. Math. Soc. 104 (1988) 1163–1170] introduced the concept of semi-closed 1-set-contractive operators and studied the fixed point theorems for such a class of operators. In this paper, we continue to study semi-closed 1-set-contractive operators and investigate the boundary conditions under which the topological degrees of 1-set-contractive fields, , are equal to 1. Correspondingly, we can obtain some new fixed point theorems for 1-set-contractive operators which improve and extend many famous theorems such as the Leray–Schauder theorem, Rothe’s theorem, Altman’s theorem, Petryshyn’s theorem, etc. On the other hand, this class of 1-set-contractive operators includes strict set-contractive operators, condensing operators, semi-contractive operators and others (see [G.Z. Li, The fixed point index and the fixed point theorems of 1-set-contraction mappings, Proc. Amer. Math. Soc. 104 (1988) 1163–1170; L.S. Liu, Approximation theorems and fixed point theorems for various class of 1-set-contractive mappings in Banach spaces, Acta Math. Sinica 17 (2001) 103–112]). So the results in this paper remain valid for the above-mentioned operators. In addition, our conclusions and methods are different from the ones in many recent works (see [L.S. Liu, Approximation theorems and fixed point theorems for various class of 1-set-contractive mappings in Banach spaces, Acta Math. Sinica 17 (2001) 103–112; N. Shahzad, S. Latif, Random fixed points for several classes of 1-ball-contractive and 1-set-contractive random maps, J. Math. Anal. Appl. 237 (1999) 83–92; T.C. Lin, Random approximations and random fixed point theorems for continuous 1-set-contractive random maps, Proc. Amer. Math. Soc. 123 (1995) 1167–1176]). [Copyright &y& Elsevier]
- Subjects :
- *FIXED point theory
*COMPLEX variables
*BANACH spaces
*MATHEMATICS
Subjects
Details
- Language :
- English
- ISSN :
- 0362546X
- Volume :
- 67
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Nonlinear Analysis
- Publication Type :
- Academic Journal
- Accession number :
- 24709125
- Full Text :
- https://doi.org/10.1016/j.na.2006.06.051