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Krasnosel’skii type fixed point theorems under weak topology features. (English) Zbl 1225.47071

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.

MSC:

47H10 Fixed-point theorems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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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
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