An extension of the notion of zero-epi maps to the context of topological spaces. (English) Zbl 1009.47055
The authors introduce the class of hyper-solvable equations whose concept may be regarded as an extension to the context of topological spaces of the known notion of 0-epi maps. After collecting some notation, definitions and preliminary results they give a homotopy principle for hyper-solvable equations. Moreover, they provide examples showing how these equations arise in the framework of Leray-Schauder degree, Lefschetz number theory and essential compact vector fields in the sense of A. Granas.
Reviewer: Jürgen Appell (Würzburg)
MSC:
| 47H10 | Fixed-point theorems |
| 54C55 | Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 47H11 | Degree theory for nonlinear operators |
| 55M20 | Fixed points and coincidences in algebraic topology |
Keywords:
fixed points; absolute neighborhood retracts; continuation principle; homotopy invariance; hyper-solvable equations; 0-epi maps; homotopy principle; Leray-Schauder degree; Lefschetz number theory; essential compact vector fieldsReferences:
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| [2] | Brown, R. F.: The Lefschetz Fixed Point Theorem. London: Scott, Foresman and Comp. 1971. · Zbl 0216.19601 |
| [3] | Granas, A.: The Theory of Compact Vector Fields and some Applications to the Topology of Functional Spaces (Rozprawy Matematyczne: Vol. 30). Warszawa: Polish Sci. Publ. (PWN) 1962. · Zbl 0111.11001 |
| [4] | Furi, M., Martelli, M. and A. Vignoli: On the solvability of nonlinear operator equations in normed spaces. Ann. Mat. Pura Appl. (4) 124 (1980), 321 - 343. · Zbl 0456.47051 · doi:10.1007/BF01795399 |
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