New fixed point theorems for 1-set-contractive operators in Banach spaces

Abstract

W.V. Petryshyn [Remark on condensing and k -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 A and investigate the boundary conditions under which the topological degrees of 1-set-contractive fields, deg ( I − A , Ω , p ) , 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]).