Taoudi, Mohamed Aziz Krasnosel’skii type fixed point theorems under weak topology features. (English) Zbl 1225.47071 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 1, 478-482 (2010). Summary: We prove the following Krasnosel’skii type fixed point theorem. Let \(M\) be a nonempty bounded closed convex subset of a Banach space \(X\). Suppose that \(A:M\rightarrow X\) and \(B:X\rightarrow X\) are two weakly sequentially continuous mappings satisfying 8.5mm (i) \(AM\) is relatively weakly compact;(ii) \(B\) is a strict contraction;(iii) \((x=Bx+Ay\), \(y\in M)\Rightarrow x\in M\). Then \(A+B\) has at least one fixed point in \(M\).This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature. Cited in 26 Documents MSC: 47H10 Fixed-point theorems 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. Keywords:fixed point theorem; measure of weak noncompactness; strict contractions; nonexpansive mappings; Krasnosel’skij type fixed point theorem PDFBibTeX XMLCite \textit{M. A. Taoudi}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 1, 478--482 (2010; Zbl 1225.47071) Full Text: DOI References: [1] Krasnosel’skii, M. A., Some problems of nonlinear analysis, Amer. Math. Soc. Trans. Ser. 2, 10, 2, 345-409 (1958) · Zbl 0080.10403 [2] Smart, D. R., Fixed Point Theorems (1980), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0427.47036 [3] Barroso, C. S., Krasnoselskii’s fixed point theorem for weakly continuous maps, Nonlinear Anal., 55, 25-31 (2003) · Zbl 1042.47035 [4] Barroso, C. S.; Teixeira, E. V., A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Anal., 60, 4, 625-650 (2005) · Zbl 1078.47014 [5] Arino, O.; Gautier, S.; Penot, J. P., A fixed point theorem for sequentially continuous mappings with applications to ordinary differential equations, Funkc. Ekvac., 27, 273-279 (1984) · Zbl 0599.34008 [6] Cain, G. L.; Nashed, M. Z., Fixed points and stability for a sum of two operators in locally convex spaces, Pacific J. Math., 39, 3, 581-592 (1971) · Zbl 0229.47044 [7] Burton, T. A., A fixed point theorem of Krasnosel’skii, Appl. Math. Lett., 11, 85-88 (1998) · Zbl 1127.47318 [8] Banas, J.; Rivero, J., On measures of weak noncompactness, Ann. Mat. Pura Appl., 151, 213-224 (1988) · Zbl 0653.47035 [9] De Blasi, F. S., On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. Roum., 21, 259-262 (1977) · Zbl 0365.46015 [10] Barroso, C. S., The approximate fixed point property in Hausdorff topological vector spaces and applications, Discrete Contin. Dyn. Syst., 25, 2, 467-479 (2009) · Zbl 1178.47034 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.